Knihovna PřF MU 1987 3145016110 • ii Jvtia f^lKálnlsh kateder Invenl.írnf Číslo............................................ X^tL utouatura................................. /č ý in«n«mi t,.. ks...................................JL:J.J£/Ha_ 3145016110 PLASMA DIAGNOSTICS WITH MICROWAVES i WILEY SERIES IN PLASMA PHYSICS SANBORN C. BROWN, advisory editor research laboratory of electronics massachusetts institute of technology PLASMA DIAGNOSTICS WITH MICROWAVES heald and wharton ■ plasma diagnostics with microwaves mcdaniel ■ collision phenomena in ionized gases M. A. Heald I >cpartment of Physics Swarthmore College Swarthmore Pennsylvania ('. B. Wharton (loiieral Atomic Division I Icneral Dynamics Corporation Siin Diego, California Universita J. E. Pur kyne přtrodovédwcká fakulta Knihovní siřodisko /yn Hlav. Inv, L C j Vy Depo v knih. ...'ffižíŽQh. Zjjr/. ■i ;U'i. Inv. t. 7" ______.2.__ Íigii9tui»a_______ _ _ John Wiley <& Sons Inc., New York - London ■ Sydney Copyright © 1965 by John Wiley & Sons, Tnc. All Rights Reserved. This book or any plrt thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 64-23839 PRINTED IN THE UNITED STATES OK AMERICA To Jane and Gloria, who have played well I he roles of both wife and midwife during this long labor. Preface The subject of plasma diagnostics is concerned with making significant, nonperturbing measurements and expressing them in numbers. The term "diagnostics," of course, comes from the medical profession. The word was first borrowed by scientists engaged in testing nuclear explosions about fifteen years ago to describe measurements in which they deduced the progress of various physical processes from the observable external symptoms. The word crept into the jargon of the then-classified Sherwood Program, the AEC program of controlled nuclear fusion research. Now the term "plasma diagnostics" is applied to a wide variety of plasma measurements. The term implies that the diagnostic measurement does not itself change the state of the plasma, that two or more diagnostic measurements may thus be made simultaneously and, in most cases, that timc-rcsolved data can be obtained from a single transient event. In the case of microwave diagnostics, the interpretation often is difficult and requires not only an understanding of the formal theory of electromagnetic interactions with plasmas, but also development of an intuitive skill in selecting meaningful simplifications. Many of the cases of wave propagation are much too complicated to permit exact formulation and solution; a feeling for how things scale from similar, more tractable cases is often essential. Consequently, we shall find it necessary to develop a fairly complete basic theory of many facets of wave interactions on which to base our scaling and approximating to specific cases. Historically, the subject or microwave diagnostics is not new. The laboratory experiments of Balthazar van der Pol (1920) to demonstrate that charged particles have a large influence on electromagnetic wave propagation did much to settle an important controversy of the day, vt Preface vii whether or not an ionosphere was responsible for distant radio propagation. His calculations of plasma conductivity and refractive index yielded the same equations to be found in our Chapter 1 for the Lorentz plasma, including a density-dependent term that was later named the plasma frequency by Tonks and Langmuir (1929). Subsequently, workers in the field of ionospheric propagation, with such landmarks as Appleton (1932) and Mitra (1952), have perfected to a high degree the use of radio-wave probes for sounding the ionosphere. The improved microwave technology following World War II opened new expanses of the frequency spectrum. The pioneering theoretical work of Margenau (1946) and experimental work of Biondi and Brown (1949) and others in the M.l.T. group, in developing resonant cavity lechniques, rekindled interest in plasma measurements with electromagnetic waves. Faraday-rotation measurements with waves beamed through controlled fusion plasmas, performed by R. F. Post and others in Berkeley in 1952, stimulated the development of microwave diagnostics as a standard measuring technique in Project Sherwood research (Wharton B1 1955; Heald, 1956). Laboratory experimental techniques have come a long way since the days of van der Pol, who measured the shift in standing waves on a Lecher wire terminated by small capacitive discs immersed in the plasma. His microwave source was a Blondlot arc, running under kerosene, producing n lew milliwatts of damped 200 Mc wave trains. Nevertheless, embellishments of those early day techniques are still used for diagnosing low-density plasmas, and several of the experiments described in Chapters 5 and 6 luive recognizable similarities. Our aim in writing this book has been to bring together, on the one kmd. a summary of the basic theory of the interaction of electromagnetic waves with plasmas and, on the other, a description of the practical bxperimental techniques that exploit this interaction. The book is written mainly in the context of the plasmas of controlled fusion research, which niv characteristically hot (implying a high degree of ionization and low intcrparticle collision rates) and large (relative to the wavelength of an I |c tromagnetic wave at the plasma frequency a>„), However, most of I In material is relevant also to the plasmas found in the fields of M.H.D. power generation, space vehicle propulsion and communication, iono-Iphcric radio propagation, microwave devices, classical gas discharges, iind radio astronomy. We have limited the detailed discussions to "high-Im■quciicy" techniques that use waves at frequencies of the order of the cirri ron plasma frequency. Also, we have given somewhat more attention In free-Space beam techniques than to those employing resonant cavities llnd waveguides. via Preface Preface ix The formal theory ofelectromagnetic interactions in plasma often makes use of mathematical techniques that are beyond the experimentalist's training. We make no apologies about writing the book from an experimentalist's viewpoint, so that many of the theoretical discussions are somewhat cavalier and inelegant. On the other hand, we have endeavored to go beyond the mere displaying of useful theoretical formulas. The theory presented is developed sufficiently to indicate the assumptions and ranges of validity of the more sophisticated theoretical presentations in the literature as well as to serve as a primer for these treatments. Thus for instance, we present a hydromagnetie treatment of wave propagation in Section 3,3 although, in general, more useful "practical" results are obtained by the corresponding kinetic treatment of Section 3.4. The hope, then, is that this volume may be of some value to those seeking an introduction to plasma physics beyond the narrow topic of microwave diagnostics, without offending the specialist seeking solutions to specific problems. The similarities and differences of electromagnetic and spacecharge wave propagation in bounded plasmas and in infinite, homogeneous plasmas are carefully delineated, including the dependence of wave properties on temperature and magnetic field strength. Because the effects of finite temperature on spacecharge wave propagation have been included, we are able to give an introduction to wave growth and electrostatic instabilities and to collisionless (Landau) damping. Some of the effects of instabilities and turbulence on wave scattering and electromagnetic radiation are also discussed. There is considerable experimental material included. We have attempted to present the broad picture of wave propagation and radiation experiments, still retaining enough detail to permit an experimentalist new in the field to proceed with a diagnostic experiment. The descriptions of the techniques, therefore, are fairly complete. Numerous illustrative examples have been chosen from the literature and from our own work. Some of the latter are new and are published here for the first time. Our subject is specialized; this volume was not visualized as a text in formal courses, although portions of it may well fit into certain courses. No formal problems have been included although problem material is present. In the plasma physics field, workers will come from a number of diverse backgrounds. An argument or notation that is familiar and elementary to a microwave tube engineer, for example, may not be familiar to a specialist in gaseous discharges. And the terminology of fluid mechanics or ionosphere research may be unfamiliar to workers in the former fields, and vice versa. The literature abounds with different Rotational conventions, often not explicitly staled. Consequently, we have attempted to maximize this book's use for reference and individual study by defining many terms, by cross-referencing, and by including or giving references to elementary and applied material from diverse fields that might be passed over in a textbook. The first six chapters are devoted to the propagation of externally generated waves in a plasma, the first five being primarily theoretical. Chapter 1 presents the well-known Appleton-Hartrec theory of wave propagation, including a brief discussion of the effect of heavy-ion motion. The next two chapters examine the role of collision processes, especially electron-ion Coulomb collisions, and summarize the modifications that occur when electron thermal speeds are comparable with the wave phase velocity. Chapters 4 and 5 consider the boundaries and spatial non-uniformity of real plasmas. Chapter 4 deals with the propagation of free-space beams, and includes a discussion of the choice of antenna systems with which to probe a plasma sample of given size. Chapter 5 meanwhile deals with a plasma confined in a resonant cavity or waveguide, or acting as a waveguiding structure itself, for either electromagnetic or spacecharge waves. Finally Chapter 6 gives an extensive discussion of the practical applications of these active microwave probing techniques. Chapters 7 and 8 present, respectively, the theory of microwave radiation generated by a plasma, by both thermal and nonthermal processes, and the practical application of passive radiometric techniques. Chapter 9 is devoted to extensive descriptions and photographs of much Of the hardware and special circuits useful for microwave diagnostics, but not readily available commercially. With some regret we have omitted a listing of commercial suppliers of generators, detectors, and components, although references to listings in the literature have been cited. While this information is of great use to an experimentalist entering the field, we feared that a list would soon be obsolete and subject to unintended favoritism. A brief survey of a number of other plasma diagnostic techniques besides microwaves is given in Chapter 10. The main emphasis has been placed on techniques that yield information similar to that obtainable with microwaves, that is, plasma electron properties, although other techniques were included for completeness. Correlative measurements are indicated where possible. A summary of wave propagation in general dielectric materials is given in Appendix A. This can serve as a starting point for those readers unfamiliar with microwaves or needing a review. A brief summary of some of I he properties of tensors and matrices is given in Appendix B, since these operations are used repeatedly throughout the text. An unusually large number of references is cited, many from the very recent research literature. To a large extent, this book is a review of research in progress rather than a text in a well established field. Therefore we have felt it necessary, especially in the theoretical chapters, to provide the means of following up our introductory discussions with more detailed research papers. It is inevitable that many of these references will be superseded by new work in the near future. A bibliography of important general references is given at the end of the book. Literature references in the text are identified by author and year. Bibliographic details are then given in listings following the general bibliography. The references for Chapters 1 to 8 and the appendices are compiled into a single alphabetical author list. However, since the material covered in Chapters 9 and 10 is somewhat foreign to that oT the other chapters, separate reference lists are given for these two chapters. The reference lists may be used as an author index. We have attempted to follow the notation and terminology of plasma waves admirably systematized by the M.T.T, group (Allis et al., 1963), except in a few cases where strong tradition decrees otherwise. Familiarity with our notation for general wave propagation may be obtained by a quick glance through Appendix A, We have tried to avoid obscure normalized parameters, so often found in theoretical journal papers, in an attempt to preserve some physical insight into the equations. Because we make much use of complex notation for familiar coefficients (such as conductivity, dielectric constant, and refractive index) we have used the special symbol t, the steady-state solution is t = Z7-z E. The resulting current density is «/ = —nei— —j-E, which is of the Ohm's-law form J=aE, having a conductivity ,ne* men (1.3.2) (1.3.3) (1.3.4) I or wave propagation in a linear medium, as shown in Appendix A, a complex conductivity may be replaced by a complex dielectric constant which, in this case, is I 1-- (1.3.5) where cap is the plasma frequency defined by (1.2.7), and the symbol U is used to denote explicitly a complex quantity. This analysis may be applied to a plane electromagnetic wave traveling in the z direction and varying as exp(yW—yz), where y = a+jfi is the complex propagation coefficient, and a and /S are the attenuation and phase coefficients.2 The dispersion relation for such a wave is (1.3.6) For low frequencies ujp, the w;ive is propagated without attenuation: « = 0 \ to j C o;2»tu,,2 c 1 In* phase and group velocities, in this case, are I 2\ Vl (1.3.8) (1.3.9) (1.3.10) Which are respectively greater and less than the vacuum velocity of light c. The refractive index p, for high frequencies, ^£j£w(i_4f, (1.3.11) Vj, to \ / In U'ns Hum unity, in contrast to the index of ordinary dielectrics. Appendix A lor n review ol Ihe basic electromagnetic wave theory and a ■.....,'......ol' our notation. 6 Electromagnetic wave propagation in a cold plasma Chap. 1 1,3 Electromagnetic wave propagation (no magnetic field) 7 1.3.2 Conductivity with collisional damping: Lorentz conductivity. The influence of discrete positive ions and neutral molecules in a plasma can be represented to good approximation by including a viscous damping term, proportional to velocity, in the electron equation of motion. With this addition to (1.3.1), the equation of motion becomes ml=-eE-vmi. (1.3.12) The form of the damping term vm£ anticipates the fact, discussed at length in Chapter 2, that direct physical meaning can be given to the parameter v as a collision frequency (strictly, collision frequency for momentum transfer).. In brief, we argue that on the average an electron loses its directed momentum m£ at each collision. Thus, if the electron averages v collisions per second, — vm£ represents the time rate of change of momentum and, hence, the statistical average force exerted on the electron by the massive ion-neutral component of the plasma. More careful examination of the averaging process (Section 2.3) shows that this form of damping term is strictly correct only when the collision frequency is independent of electron velocity, a rather special case. For the present, we need only regard v as a phenomcnological damping constant, having the dimensions of radian frequency. The steady-state solution of (1.3.12) for oscillatory fields, obtained by the substitution 8jdt —> jw, is eE moi(ot —Jv) The current density /= — ne£ yields a complex conductivity «e2 v —jw ne* m(v+ju>) m vs + u>2 (1.3.13) (1.3.14) which is known as the Lorentz conductivity. Lorentz dielectric constant is ■KT—JK{=l~j-=1 The equivalent complex oj(oj — jv) 2 vjo) 2 + W 2 (1.3.20) I -im<**)=(4(i ^)4[(i-^)%(^-2-)TT ^ l\ ar+tt-f 2[\ a> + v / V): the three frequency regions. We consider ii'. it I unction of frequency the electrical properties of a plasma charac-Iri ./.-.' by the parameters plasma frequency w? (proportional to square imil "I electron density m'4) and effective collision frequency v (in general, ■ inn, lion of electron density and temperature; see Sections 2.4.3 and i i .aiming the Lorentz conductivity (1.3.14) and no magnetic field. 1 ' io ciisc of a highly ionized, liigh-temperature plasma for which v«wv, I unit distinguish three frequency regions. 1 Mm i if,, where I is i he complex annular wave number used by many authors. ■• V|i|h'iulix A. 8 Electromagnetic wave propagation in a cold plasma Chap. 1 LOW frequencies w < v. In this region the conductivity is largely real and, to a first approximation, is e2 ;—, (1.3.22) "' mv a familiar relation from the elementary kinetic theory of conductors. Expanding in the limit m«v, vz«a,)1'4ocw"'^ as in the familiar metallic skin-effect problem. We note, incidentally, that for a highly ionized gas the temperature variation of the electron-ion collision frequency is such that the low-frequency conductivity increases with the three-halves power of the temperature (see Section 2.5.3). A one-kilovolt plasma is approximately equivalent to room-temperature copper. intermediate frequencies Koxoi,. In this region the plasma will not propagate an electromagnetic wave, after the manner of a waveguide beyond cutoff. Expanding, now, in the limit y2«tt)2«(i>p2, and using (A.52) and (A.53), we obtain: X~u>\ 8^2 2to/J (1.3.26) (1.3.27) (1.3.28) Thus, the penetration depth of this evanescent wave is practically constant in the interior of this frequency region, and is comparable to the wavelength of a free-space wave at the plasma frequency—a few millimeters in typical laboratory plasmas. 1.3 Electromagnetic wave propagation (no magnetic field) 9 high frequencies w>a>p. Here, the plasma becomes a relatively low-loss dielectric. In the limit v2«w2 — / and v2«o>2(a)2 — a*/)2/to/, using (A.50) and (A.51), we obtain: <**} [H 2o.>2-V)2J~l1 «>*) x 2m* \ ox2} (1.3.29) (1.3.30) (1.3.31) Note that the refractive index is quite insensitive to collisional damping and that the attenuation is very small for the assumed conditions. Numerical values of a, and y are shown as functions of frequency in Tig. 1.3 for a typical laboratory plasma. Fig. 1.4 shows the attenuation length S as a function of frequency for two different collision frequencies lm temperatures). The magnitudes of the quantities plotted change jtmoothly and slowly at the boundaries of the three frequency regions with I In- exception of the attenuation parameters, a, x, and S, across the plasma resonance, u> = p. Expansion of the complete expression (1.3.21) for x in llie neighborhood of = 10"'1. In many laboratory devices the thickness of the plasma is of the order of 100 cjuip. ~2 ■1 0 1 1.3 Electromagnetic wave propagation (no magnetic field) 11 by ihe large factor (2cu„/i')''i in the small frequency change (2vco?)-'. Thus, ti fractional change of a few per cent in frequency or, alternatively, hi electron density (-3 diagrams, normalized to various quantities, is made in connection with spacecharge waves in Chapter 5. s 1.3.5 The critical electron density. For rf measurements in a time-varying laboratory plasma, the physical situation is better described with the following emphasis. Consider a fixed test frequency «j, to which there corresponds a critical density nc, defined by the plasma frequency relation urn (1.3.32) For densities below this critical value, the medium is a nearly transparent dielectric; ahove, the medium is opaque and highly reflecting. The indices p and x are shown as functions of electron density in Fig. 1.6. 1,4 Wave propagation with magnetic field We now consider the uniform Lorentz plasma to he immersed in a static magnetic field B0. The situation is considerably more complicated because of the vector relations involved in the magnetic force on a moving charge ?vxB, The equation of motion, corresponding to (1.3.12), becomes the Langevin equation m\=— eE(0 - ev x B0 - vmv (1.4.1) where v is the vector velocity of the electron replacing the one-dimensional Note that here, as before, we neglect the time-dependent (wave) magnetic field associated with the time-dependent electric field. Since the magnetic force compares to the electric force as vjc, this neglect is usually well justified, unless the plasma approaches relativistic temperatures in which case other corrections are necessary as well (Chapter 3). 1.4.1 Wave propagation along the magnetic field: circularly polarized waves. Before proceeding with a general solution, it is instructive to examine the special case of propagation along the field, which we take to be the z direction. Thus the wave electric field is expected to be in the x-y plane. The vector equation oT motion (1.4.1) represents three scalar equations. For an oscillatory electric field varying as exp jtot, as usual, these equations are explicitly (> + v>v+ (||k= -4 & (1.4.2a) (-— \vx + (jca + v)»v— —— Ev \ ml m (>+v)t,=--E„, tn (1.4.2b) (1.4.2c) 1.4 Wave propagation with magnetic field 13 The third equation is not coupled to the other two. Physically, it would represent a possible electromagnetic wave propagating perpendicular to I lie magnetic field or, alternatively, a spacecharge oscillation of the sort discussed in Section 1.2. in either case, the transverse motion is not affected and thus (1.4.2c) may be ignored. The first two equations are i oupled together by the vxB term in a way that obscures the dependence of VOu E, needed to evaluate the conductivity and propagation constants.4 The x*y symmetry of (1.4.2a) and (1.4.2b) suggests consideration of minting or circularly polarized fields. The notation used to express such vectors can be a matter of some subtlety. Consider a vector rotating with lime in a right-handed sense about the z axis. The y component is equal in magnitude to the component and lags it by 90°. In terms of the usual convention that the actual physical quantity is given by the real part of a complex quantity (phasor), we may write, for instance, Ex = Eq exp jcut Ey=~jE0 eg&jmi (1.4.3) (1.4.4) wlicrc Ea is a time-independent amplitude. Similarly, left- and right-hand toiating unit vectors would be written ai - a* +ya3, ar — ax —]'&y, (1.4.5) where a*, a„ are the unit vectors in the x and y directions. It is understood Lhul any quantity with which these unit vectors are used contains the time I'uclor cxp(+jW). We can now express an arbitrary field (of unrestricted polarization) in ■tonus of circularly polarized, rather than cartesian, components. Such an in Inirury field may be expanded formally in either system, with the identity3 E = *XEX + nvEv + sl3Es (1.4.6) Hv equating coefficients of the cartesian unit vectors, we find E: + Er Ej — £r (1.4.7) In Sivliim 1.4.7, a general method is developed for representing the conductivity as I leimor. 1 Wr elioosu I he numerical factor of \ 2 in the circularly polarized terms largely for IMtoliN or rotational convenience, (n effect, we thus regard Ex, E:, etc., as rms .......ilutlon, Other conventions will he found in the literature. 14 Electromagnetic wave propagation in a cold plasma Chap. 1 and, conversely,6 A/2 V2 (1.4.8) Note that a wave traveling in the +z direction with right-hand circular polarization in time has a left-handed space dependence. Tn the steady state, the electron velocities resulting from circularly polarized electric fields will also be circularly polarized, (1.4.9) (1.4.10) For these circularly polarized velocities the v X B0 term may be evaluated as v * Vi y) * a"=Vf ~ a" (1.4.11) that is, the term appears formally as being "parallel" to v so that (1.4.2a) and (1.4,2b) become independent of each other, X J m J 1 m I. , eB0\ e _ ( Jai + v-J-\vr=--Er. V m J m (1.4.12a) (1.4.12b) Using — nes- J = cE, we then obtain two conductivities for the circularly polarized fields ne 1 m v+j(cu + wb) ne2 1 in which we have introduced \e\Bn (1.4.13) (1.4.14) (1.4.15) 6 It is worth noting that this formalism is equivalent to using complex (phasor) notation in the real x-y plane, as well as in the usual complex time domain. Note in (1.4.3) and (1.4.4) Ex = Rc[E„ cxpjiot] and E„ = Rc[-jEa cxpyeu/] = Im[£0 cxp Jmt], 1.4 Wave propagation with magnetic field 15 Right circularly polarized wave (- sign) Electron gyration Static Bo Left circularly polarized wave (+ sign) I' l< p. 1.7 Geometry of circularly polarized waves propagating parallel or antiparallel in magnetic field. The handedness is defined with respect to the magnetic field. fol the electron cyclotron frequency or gyrofrequency, the angular frequency willi which an electron gyrates in a magnetic field.7 The equivalent complex dielectric constants are readily obtained by the usual relation (A.22) Z,,t=1-J = I 1 aiEf, (uj + tt>4) —JV cu„2(m ± w6) _ (1.4.16) a>[{m±a,bf + v2)j J \cU[(cU±a.&)3 + "2]J ( itcularly polarized electromagnetic waves, traveling in the ±z direction, 11 up circularly polarized electric fields, to which the conductivities or .electric constants just calculated apply. The geometry of these waves N nIiowii in Fig. 1.7. Therefore, the refractive and attenuation indices of hi i - vclotron macs may be calculated from the dielectric constant using ,46) and (A.47): k-Re(«,,rK) -V(m ± <*>b) 2 a>[(u±u>by + v2]j I / f . ">P2(<" ± <"„) (1.4.17) ft- + iiiaiility is not to be confused with the Larmor frequency, which occurs in other ill h \l problems involving precession of electrons in a magnetic field, and is defined M'i i Hi '/ri tl hat is, one-half I he cyclotron frequency). 16 Electromagnetic wave propagation in a cold plasma Chap. 1 Xi.r= -Im(*,/2) oJp2(oj + tu„) 1.4 Wave propagation with magnetic field 17 T£ w[(a>±cob)2 + v2] 1/f *>„"(<» ±°>>) V ■ 2\\ ^[(coi^ + v2]/ + (1.4.18) o^t^,,)2-}-^] IT Numerical illustrations of (1.4.17) and (1.4.18) are given by Figs. 1.8 and 1.9. Further examples are given in Sections 1.4.9 to 1.4.12. The evaluation of the effective collision frequency v is discussed in Sections 2.4.3 and 2.5.3. Normalized magnetic field FIG. 1.8 Refractive and attenuation indices as functions of magnetic field for circularly polarized waves propagated along the field, for various plasma densities. Right-hand polarization (sense of electron gyration), solid curves; left-hand, dashed. No collisions (r/w = 0). 1 1 o)p2/o)2 = 1.07 \ - / 0.5/ l\ / f\\\ 0 0.5 1.0 1.5 (b) FIG. 1.9 As Fig. 1.8 but nonzero collision rate, x/cu=0.01. If the collision frequency is sufficiently small, asm Fig. 1.8, simplified ex-1111 unit ins may be obtained. For propagating frequencies {kt > 0) in the limit [\w(w±a,b)-wpa\, (1.4.19) 18 Electromagnetic wave propagation in a cold plasma Chap. 1 1.4 Wave propagation with magnetic field 19 approximations (A.50) and (A.51) yield IM, 1- 0)(oi + CUi,) X! The refractive index (1.4.20) has a zero (cutoff condition) for and a pole (resonance condition) at the cyclotron resonance (1.4.20) (1.4.21) (1.4.22) (1.4.23) for the right-hand wave, which is the sense of electron gyration. Fig. 1.10 sketches the zeros and poles in the mv2—ci>6 plane; the cut-off regions are indicated by cross-hatching. The refractive index of the left-hand circularly polarized wave (+ sign in propagation formulas), which rotates in an opposite sense to electron gyration, shows no cyclotron resonance. The refractive index is always less than one. At low fields and high densities the wave is cut off. The refractive index of the right-hand wave (— sign), as a function of magnetic field, is seen in Fig. 1.8 to start at n 1) a right-hand wave always sees p> 1, and near the resonance region, oibjw>\, sees a very high refractive index. The wavelength is then very small and the wave velocity slow. This condition of right-hand polarization and a>„ > m is often called the "whistler mode" of propagation (Helliwell and Morgan, 1959). Such waves tend to be ducted along the magnetic field lines, the propagation vector being confined to small angles in respect to the field lines (see Section 1.4.11). Most laboratory plasmas, however, have appreciable losses at the conditions appropriate for whistler propagation, and the wave attenuation is very large. Further mention of the whistler mode is made in Chapter 6. 1.4.2 Faraday rotation of angle of polarization. The significance of the independence of (1.4.12a) and (1.4.12b) for circularly polarized waves is that a general wave, with arbitrary state of polarization, propagates along I he magnetic field in a plasma in a manner which may be analyzed by ()2 = njnc iniday rotation index áfi~fi, — nr, neglecting collisions, for Ja>=0.S 20 Electromagnetic wave propagation in a cold plasma Chap. 1 and (c) superposing the two components to obtain the resultant wave at any desired point. Specifically, if a linearly polarized, plane wave is incident on a magnetized plasma along the field, its direction of linear polarization will be rotated. The wave may be thought of as consisting of two countsrrotating waves or equal amplitude. As these waves pass through the plasma, they travel at different velocities, as determined by the upper and lower signs of (1.4.20). After the wave emerges from the plasma, its plane of linear polarization has been rotated, since one of the circularly polarized components has traveled electrically farther than the other. The rotation angle is 'p-=i(ft-^)rf=(^™1ar)W/A = (Zlfi)7r^A (1.4.24) where d is the distance traveled in the plasma. An example is given in Fig. 1.11. Note that this rotation is with respect to the magnetic field and independent of the sense of propagation along the field. Thus, the propagation is nonreciprocal (Goldstein, 1958). The attenuation of the two counterrotating components will also be different in general. When the difference is significant, one of the components in the emerging wave is smaller than the other, so that the wave is elliptically polarized. If one of the waves is completely cut off, as shown at densities greater than 0.5 and 2.5 in Fig. 1.11, the emerging wave will be purely circularly polarized, and no Faraday rotation effects will be measurable. 1.4.3 Arbitrary direction of propagation: coordinate systems. We are concerned with waves propagating in the direction specified by the vector (a) (b) FIG. 1.12 Alternative coordinate systems for propagation in an arbitrary direction to the magnetic field. 1.4 Wave propagation with magnetic field 21 propagation constant y.9 The vector nature of the equation of motion (1.4.1) forces us to define a coordinate system with care. Two such coordinate systems are common. We shall have occasion to use both. In one case, the z axis is aligned with the wave propagation direction y, and the static magnetic field B0 is taken in the y-z plane (Fig. 1.12a). In the other, the z axis is aligned with B0, and y is taken in the x-z plane (Fig. 1,12b). In both cases, 6 is, in magnitude, the angie between y and B(). Thus, 0 = 0 corresponds to propagation along the field, 9 = 90" to propagation across it. 1.4.4 Magnetic field at angle B with respect to propagation: Appleton's equation. We now seek a general solution for propagation at an arbitrary angle with respect to the magnetic field, using the coordinate system of Fig. 1.11a in which the propagation is in the z direction with phase factor expfjW —yz), and the static magnetic field lies in the y-z plane [Bn = B0 (0, sinfl, cos —.— Es m (1.4.25a) (1.4.25b) (1.4.25c) Identifying the cyclotron frequency (1.4.15) and replacing velocity by current density (J= -«ev), we have (jca + v)Jx + ojb cosr? Jv — wb sinfl Jz~~ Ex ■ oib COSfl Jx + (jco + l')/v Mi sint? Jx in written in matrix form /co + v (11%, COS0 — cob COS# + V w6 sinfl 0 ne rn ne2 m ne2 + (>»-; v)/, = —I, m ob sinfl 0 yu> + v ^X Ex Jy = Ev Jz. (1.4.26a) (1.4.26b) (1.4.26c) (1.4.27) This is seen to be an expression of Ohm's law, with a tensor reciprocal-.....iluctivity (resistivity),11 o"I-J = E. (1.4.28) " I lie direction of y is the wai>e-nnrmal direction, perpendicular to planes of constant I'liiiII Is not necessarily the direction of energy flow. Sec Section 1.4.11. " An elementary discussion of tensor algebra is given in Appendix 13. 22 Electromagnetic wave propagation in a cold plasma Chap. 1 In principle we could invert the matrix by standard methods to obtain the conductivity tensor a, find the equivalent dielectric constant tensor k = 1— _/a/W0, and then substitute in the anisotropic dispersion relation (A.74). This approach is followed in Section 1,4.7, Here, however, we use a different but equivalent approach, which is somewhat less complicated algebraically, ft is possible to obtain a formal expression for the conductivity tensor in terms of the propagation constant from Maxwell's equations. We then demand that the two expressions for the conductivity tensor be self-consistent. We assume plane waves having the phase factor exp(/W — yz), where y=«+/3 = jfunic is the complex propagation constant, in a medium carrying explicit current density J (dielectric constant and relative permeability unity). In component form, Maxwell's curl equations become [see (A.70) and (A.71)]: ~yEx= -jai^Hy 0= —jut fi^Hs yHv=Jx+jwz0Ex -yH^Jy+jUXisEy Q=Jz+jo>£aEs (1.4.29a) (1.4.29b) (1.4.29c) (1.4.30a) (1.4.30b) (1.4.30c) We note that the wave can have no longitudinal component of H, regardless of the direction of the static magnetic field; however, a longitudinal component of E is not excluded so that the Poynting vector need not be in the direction of y (see Section 1.4.11). Eliminating Hx, Hy, we have, in matrix notation, £2-l 0 0 0 /l2-l 0 0 0-1 ~EX- w E„ = which is an expression for Ohm's law in the form a-E = J. Now for (1.4.28) and (1.4.32) to be self-consistent E=a-1.J=d1a.K (1.4.31) (1.4.32) or (a"1-d-1).E = 0 (1.4.33) 1.4 Wave propagation with magnetic field 23 where 1 is the unit or identity tensor. Equation (1.4.33) represents three simultaneous homogeneous equations; the determinant of the coefficients must vanish to yield a solution. Substituting from (1.4.27) and (1.4.31) and carrying out the matrix multiplication (see Appendix B), we have (jb(£i~- l)cos0 + 1) + /«jp2/co 0 «j„(/iz-l)sin0 0 -(Jo)+v)+jp2lu>2 Y=(t>bl T = 1 -jvjw Y,. = Y cosd (1.4.35) M2 = f2- 1 YT= TsinS The determinantal equation (1.4.34) becomes T+ XjMs -jYL -jYr/M2 jYL Y+XjM2 0 =0. (1.4.36) -jYT 0 -(T-X)IM' M6 ' X II is most convenient to solve for XjM2. Rewriting and factoring out W'< t - X)jX, we have the quadratic the solution of which is iiikI finally /i2=l + M2=l L 2tJ-X)\-[4(V- X'f^ L I bus the result of solving the determinantal equation (1.4.34) is (1.4.39) I . v (tob2/wz) sin20 tu 2(1 -w/l^-jvluil (cV/to4) sin*0 ojb 4(1 —cu^/cu2—/v/w)2 Oi J+-7t cos20 (1.4.40) 24 Electromagnetic wave propagation in a cold plasma Chap, 1 This equation is Appleton's equation, and was first derived to describe propagation of radio waves in the ionosphere (Applcton, 1932; Ratcliffe, 1959). It gives the propagation indices for plane waves in a magnetized cold plasma that is uniform over dimensions that are large, compared to a wavelength. The sense of the + sign has been chosen to agree with that of (1.4.16) for the special case of propagation along the field (0 = 0). Equation (1.4.40) is often called the Appleton-Hartree equation, a somewhat ambiguous identification that recalls Hartree's incorrect version, based on an analysis including the so-called Lorentz polarization correction (see Appendix, Section A.2.2).10 Numerical examples of Appleton's equation are given in Sections 1.4.10 to 1.4.12. The effective collision frequency v is to be evaluated by the methods discussed in Sections 2.4.3 and 2.5.3. 1.4.5 Wave polarization. A characteristic wave is defined as one which does not change its state of polarization as it propagates through the medium. Ratios of field components are then independent of position. A more general wave may be represented as a sum of characteristic waves. For instance, we saw in Sections 1.4.1 and 1.4.2 that circularly polarized waves propagate along the magnetic field without alteration, while a linearly polarized wave experiences Faraday rotation. The concept of characteristic waves is closely related to the concept of normal modes or ei gen functions arising in the formal mathematical theory of oscillatory systems. Characteristic electromagnetic waves in a magnetized plasma, in general, are elliptically polarized. By assuming plane waves with all field components varying with the same phase factor exp(_///"' Ey cob((l2— 1) cosQ (1.4.43) Using the solution (1.4.40), we obtain (a»i./cu)cOS0 U(l [- K*/a.')sin L4(1-VK- An interesting property is C0S2f r (1.4.44) (1.4.45) For the special case studied in Section 1.4.1 of propagation along the field, R - +./ as required. The coefficient R specifies the wave polarization except for the longitudinal component Es. From the third equation of (1.4.33) [using (1.4.34)], we may evaluate a longitudinal polarization coefficient ^ Es co(,(ji3 — l)sin# j(w/:coi')(wsja>)sm0 . V (a>//cu2)sin^ w 2(1-/«.)J 4(1 >2+—T cos20 (1.4.46) 'I lie wave polarizations vary in a complicated manner with the many i1 it'iititctcrs. Curves for R have been published in connection with t.......mlicric investigations (Snyder and Helliwell, 1952; Consoli et al., 961). I I <> Propagation across the magnetic field. The important special cases id propagation along (t9 = 0) and across (0 = 90") the magnetic field are 26 Electromagnetic wave propagation in a cold plasma Chap. 1 1.4 Wave propagation with magnetic field 27 easily obtained from the general Appleton equation (1.4.40). The 6 = 0 case has already been discussed at length in Sections 1.4.1 and 1.4.2. For 0 = 90°, again two characteristic waves are obtained: ordinary extraordinary Parti = 1 - 'SI* 1 -jvlw (1.4.47) = l 1-7-- Wj,S[(cu Jp>2 jvjco 2____2. co3(u>2 ~ w2 - oj2 - ff H- v2(2co2 - w, -J ■ ■ y'K* + - 2a>„2 + 2(a>2 - a>„2 - w,3 -i^f + v2{2io2 X (1.4.48) The labels ordinary and extraordinary are conventional in ionospheric terminology, but are in conflict with the terminology of crystal optics and some magnetohydrodynamic literature (Allis, Buchsbaum, and Bers, 1963). Examination of (1.4.40) and (1.4.44) shows that the ordinary wave is linearly polarized with E parallel to the magnetic field. The ordinary wave is so named because it has the same dispersion relation as if no magnetic field were present [compare (1.3.15) and (1,4,47)].11 The field-free case was discussed at some length in Sections 1.3.3 and 1.3.4. The extraordinary wave is polarized with E perpendicular to the magnetic field and is linearly polarized in the sense that it is excited by a linearly polarized wave outside the plasma. However, from (1.4.46), we find that there is a component of E in the direction of propagation; thus, actually, E is elliptically polarized in the plane perpendicular to the magnetic field and including the direction of propagation. To obtain explicitly the indices jw and x, ft frz—L—jM, then by (A.46) and (A.47) ±L + (L2 + M2)'= (1.4.49) Tn the limit where collisional damping may be neglected, (1.4.48) reduces to •V) 2(tu2 — telp2 — < Wp2jw2— Oia"\(Ji 2jo-2 (1.4.50) 11 This independence no longer holds when finite electron thermal velocities are considered; see Chapter 3. 3 1 Vet > 1 *»ex< 1 /fa iBP 11 Electron density (co;)/o>) = n/nc I'lG. J.13 Propagating regions for extraordinary wave across magnetic field (0 = 90"), neglecting collisions. The wave is cut off in the shaded regions. O.h 1.0 1.5 Magnetic field 0)4/(1) 2.0 ii. 1.14 Kefraclive and allennation indices as fiuictions oT magnetic field, at ........■, plasma densities, for extraordinary wave propagating across the field, H'lllei ling collisions. 28 Electromagnetic wave propagation in a cold plasma Chap. 1 Cutoff of the extraordinary wave occurs at the two conditions Ct>j,2/fc>2= 1 -\-o)bju) COj,2/^ = 1 — OJij/oj. The cyclotron resonance is displaced to the new condition co3 = ai(,2 + cup2. (1.4.51) (1.4.52) (1.4.53) The poles and zeros of jl are sketched in Fig. 1.13. Figure 1.14 shows the refractive index plotted vs. cob, assuming no collisions. For ui.}?l,w2> 1, the index is real (waves propagate) if the magnetic field is sufficiently large. For ix)p2jcu'z< 1, there are cutoffs and resonances, depending on the value of ojb. At aipsjut'2—l the index remains at unity for all nonzero values of magnetic field. Figure 1.15 shows the index of refraction plotted vs. en,,2, assuming no collisions. For tuj,>tu, note that the index remains 0.5 1.0 1.5 2.0 Electron density (u>pjw)" = n/nc FiG. 1.15 Refractive and attenuation indices as functions of electron density, at various magnetic field strengths, for extraordinary wave propagating across the field, neglecting collisions. 1.4 Wave propagation with magnetic field 29 real (waves propagate) even at densities greater than twice the normal cutoff density. A linearly polarized wave incident on a magnetized plasma and propagating across the field is converted, in general, into an elliptically polarized wave. The situation is analyzed in the same way as the Faraday rotation of Section 1.4.2 by resolving the incident wave into component characteristic waves which propagate with different velocities. The effect is analogous to the Cotton-Mouton effect in classical optics. 1.4.7 The conductivity tensor. We seek a general conductivity relation between current density J and electric field E, for a plasma in a magnetic lield. Because of the vector nature of the magnetic force q\xB, this conductivity is anisotropic with respect to the magnetic field and is therefore a tensor. Although the conductivity tensor is a general relation, valid, within the assumptions of this chapter, for arbitrary electric fields of any origin,12 it offers in particular an alternative procedure to that of Section 1.4.4 for calculating the propagation constants of characteristic electromagnetic waves in a plasma. To derive the conductivity tensor, it is more convenient to use the second coordinate system of Fig. 1.12, in which the z axis is aligned with the magnetic field. The expansion of the general equation of motion U.4.1) is thus identical to (1.4.25) with the simplification 0—*O. The reciprocal conductivity is obtained immediately from (1.4.27) as tn ne2 jui + V Oib 0 — cuh jco + v 0 0 0 joi + v (1.4.54) where the coefficient is understood to multiply each term of the matrix. A matrix may be inverted formally by transposing the co-factors, and dividing by the determinant (see Appendix B). Carrying out this operation, we obtain tie2! in (ju> + v) [Ua> + v)2+ U)b2] 0 + v)B 0Jb(j) 0 O + v)2 0 0 0 + v)2 + a)„2J (1.4.55) ' Refinements are considered in Chapter 3 which introduce spatial dispersion, whereby the conductivity is itself a function of wave direction and propagation LiMisltinls. 30 Electromagnetic wave propagation in a cold plasma Chap. 1 and thus the general tensor conductivity is where 0" -fa* ox 0 0 0 °ll - . ne2 w - -jv . ne m ((x>—jv)2 — ojb2 . lit m (oo-jv)2-Ml? an = ~ I The corresponding tensor dielectric constant is (1.4.56) (1.4.57) (1.4.58) (1.4.59) 1 K± 0 " k = W 6 = 0 0 0 where k„ = 1 - l ->/« (1.4.60) (1.4.61) (1.4.62) (1.4.63) Having now obtained general expressions for the conductivity and dielectric constant, we may find wave propagation constants by using the dispersion relation for plane waves in an anisotropic medium (A.74). Without loss of generality we take the propagation direction to lie in the x-z plane with direction cosines (sinfl, 0, cost?); then (A.74) becomes kx—jj.2 cas2 6 +JKX jkx fx2 sinfl cose? 0 jl2 sind cos0 0 This equation is quadratic in p.2 H2 sin26 =0. (1.4.64) (1.4.65) 1.4 Wave propagation with magnetic field 31 with coefficients A = kx sin20 + k|; cos20 B= -(/cx2-«rx2)sin2e-/f±/f|((l+cos26i) C=kn(k±z-kx2). Alternatively the equation may be solved for tan20, yielding "\[fi2 -(fl + k>)][A2 - («L - * X )] (1.4.66) tan20 = — (/l2-^)^2-^2-^2)] (1.4.67) This latter form is particularly convenient, as it permits one to extract easily the propagation formulas for the special cases of 6 = 0 and 90°. Also one finds cutoff and resonance conditions as a function of angle by setting il2 — 0 or oo, respectively. Both (1.4.65) and (1.4.67) are alternative forms of Appleton's equation (1.4.40). Further discussion and numerical examples are given in Sections 1.4.9 to 1.4.12. 1.4.8 Conductivity in rotating coordinates. Some algebraic simplification of the conductivity and dielectric constant tensors is obtained by expressing them in the rotating coordinates introduced in Section 1.4.1 (Astrom, 1950; Turner, 1954). By (1.4.6) an arbitrary electric field is resolved into the independent component vectors E, = (a,+yay)£,/v'2 Er = (ax-jny) ETlV2 E2=a,£, (1.4.68) rather than the usual cartesian components (Ex, Ey, E2). Continuing in the coordinate system with z axis aligned with the magnetic field, and using (1.4.12), we find for the equations of motion t(> + v) +j<»b]vi = -—£■( [(> + v)-jmb]vr = - - Er (joi + v)v,,=--E,, m or in matrix form with J= — ties Fcd + wb —jv 0 0 0 0) — OJh —jv 0 0 0 oi-jv. . in Jn~e~2 'EC j, - j,. .Em. (1.4.69) (1.4.70) I 32 Electromagnetic wave propagation in a cold plasma Chap. I Inverting the matrix yields the conductivity tensor in rotating coordinates with a = °t= —j 0 0" 0 0 .0 0 a\ _ ne3 1 °r = —J m (a)-jv) + U}„ 2 1 . ne m (cü—jv) — üib .ne2 \ <*i = —j ■ m oi— jv Comparison with (1.4.57) to (1.4.59) shows at + ar (1.4.71) (1.4.72) (1.4.73) (1.4.74) 2 -(//<, «■ + «,. «1 =-»- k, =ki + 'cx :«a — «» (1.4.76) (1.4.77) (1.4.78) (1.4.79) (1.4.80) In this notation, Appleton's equation in the form (1.4.67) becomes ^0=-«^~K^-K<\. (,.4.81) 1.4 Wave propagation with magnetic field 33 The conversion of the conductivity or dielectric constant tensors from fixed to rotating coordinates may be accomplished directly by a unitary matrix transformation (Turner, 1954). Consider a vector which is given in terms of the usual cartesian com ponents. There exists a transformation matrix U which operates on the vector to express it in components in a second coordinate system without changing its physical meaning. For example, the equivalence stated in (1.4.6) is given in matrix notation by 1 V2 1 -j 0 i J o 0 0 V2 'Ei 'Er Ey — Er E, which is of the form U-E = E' The reverse transformation is "1 1 1 V2 0 j -J o _0 0 VI. ~EX- Er - E„ E, A of the form U-*-E'=E. U and U~3 are reciprocal matrices such that u-1.u=u.u-,=i. (1.4.82) (1.4.83) (1.4.84) (1.4.85) (1.4.86) Consider the relation J = 5-E; then, with the primes denoting the rotating coordinate system, CT'.E' = J,= U.J=U.a-E=U.o.U1E' and we obtain the transformation relation for the conductivity tensor ct^U-o-U-1. (1.4.87) If in cartesian coordinates in general "xx °"*2 (1.4.88) 34 Electromagnetic wave propagation in a cold plasma Chap. 1 then in the rotating coordinates using (1.4.82) to (1.4.85) ~(?xx + ay?) +j(ax» ~ oux) (oxx - oyv) ~J( rex —-- (1.4.91) (1.4.92) (1.4.93) (1.4.94) cutoffs For simplicity, we here neglect collisions (v/o 0). We introduce special frequencies: fejs _K/2) + [K/2)2 + wps]« (1.4.95) ]wa= +K/2) + [K/2)2 + cop2]^ (1.4.96) resonance wuh,s(to,*+«V)M (1.4.97) where co6 and o}„ are, of course, the electron cyclotron and plasma frequencies of (1.4.15) and (1.2.7). The resonance at wKh is known as the 1.4 Wave propagation with magnetic field 35 upper hybrid (cf. Section 1.5.2). In terms of these special frequencies the principal propagation formulas become: 0 = 0: (at + ai1)(ai ± co2)l - 0 = 90° fii,r = fiord " l**ex (a — u>2 u>2(u? — 0) and resonance (jt,-^ oo) for the respective waves. Figure 1.16 sketches qualitatively the behavior of the propagation indices as functions of frequency. The plasma medium may be thought of as a sort of filter with pass- arid stopbands (Mower, 1956). A useful point of view in laboratory plasma physics is to regard electron density, rather than wave frequency, as the independent variable. All quantities may then be normalized to the (fixed) wave frequency. In particular, the density may be normalized to the critical density nc, defined in Section 1.3.5 (njnc = w/jw2). Accordingly, the special frequencies of (1.4.95) to (1.4.97) correspond to the special densities: cutoffs nJnr = l+atJc - wbjo) resonance nhfne = \ ■ The propagation indices of the principal waves become: (1.4.101) (1.4.102) (1.4.103) 0=0: fii (1.4.104) (1.4.105) 8 = 90°: m aj^iiS; (1.4.106) !(„„)(„ „yp (, 4 ,07) r L nc(nh-n) J figure 1.17 shows qualitatively the behavior of the propagation indices as functions of density. Note, in particular, that for strong magnetic fields SUCh that oi„> co, fl2 and nn become negative, so that the respective cutoffs fiord It 1 0 Wave frequency os Uuh ^>2 FIG. 1.16 Qualitative variations of refractive index with frequency for principal waves, showing slopbands (shaded) and passbands. The locations of cutoffs and resonances arc unchanged as <»,, is greater or less than uip. 36 »2 nil Electron density n FIG. 1.17 Qualitative variation of refractive index with electron density for principal waves. Note disappearance of slopbands for R and X waves when u>bjco > I. 37 38 Electromagnetic wave propagation in a cold plasma Chap. 1 and resonances at rt = «2 and nh no longer occur in (1.4.105) and (1.4.107). For low fields where wb/cu)2 co2 2fl| ' *l(ft2-«lXft3-*r) 0*2 — «!lXKlAZ — KlKr) (1.4.108) (1.4,109) Cutoffs (p. -4- 0) and resonances (/^ -> co) for the various waves are sharply defined only as vjtu —> 0. They may be found from either (1.4.108) or (1.4.109): cutoffs co2 (1.4.110) resonances 1 •(« J)cos20 (1.4.111) The cutoffs are thus independent of propagation direction, but the resonances are not. The poles and zeros of the refractive index, bounding the pass- and stopbands, are shown in Fig. 1.18. The most notable new feature is the second family of resonances in the upper right portion of the figure. Appleton's equation in the form (1.4.108) is algebraically awkward because of the square root. It simplifies in either of the two limits in which the radical can be expanded binomially. These are known as the quasi-longitudinal (QL) and quasi-transverse (QT) approximations depending upon whether the term involving cos0 or sinfl, respectively, is dominant (Booker, 1935; Whitehead, 1952). Putting _ (co,,/co)(sin20/cosf>) _kikx — k,ke slnz0 2(1 — W^joj2 — /V/co) 2cosfl (1.4.112) 1.4 Wave propagation with magnetic field 39 .c 3 £ i 0 1 2 Electron density (c.i)p/o>)2 — n/nr_ FIG. 1.18 Cutoffs and resonances for oblique propagation, neglecting collisions. L and J? refer to the QL approximation, O and X to the QT. As an example, the nonpropagating regions for 0 = 30° are shaded (positive slope for O, negative slope for X). and expanding binomially to first order, we have: QL(|p2|«l): l-;-±-cos0(l+p+_Lp2+...) CO CO QT (|J32|»1>: S2 ~ 1 {'■ord ~ j- 1 —jvjui ± (cod/co)C0St? cosfJ(p 1+...) co Zoi 1- co„2/co2 1 — jvja> + (l— cop2/cu2 —jV/fti)COt20 (1.4.113) (1.4.114) r,2 ~ 1 Hi' 'CO 1 — co^/or — Jt>jo> vl K5>a)sin!!g (1.4.115) to 1 — to„2/coz—yV/co 40 Electromagnetic wave propagation in a cold plasma Chap. 1 Comparison with (1.4.16). (1.3.15), and (1.4.48) shows that when the QL and QT approximations apply, the refractive indices are only slightly changed from those of the principal waves at 6 — 0 and 90J, respectively. Furthermore, the same approximations applied to the polarization 1.0 1.5 Electron density (up/o:)2 = n[nc FIG. 1.19 Refractive indices for characteristic waves at various angles 0; a>5/co=0.5, v/d) = 0. The shaded regions indicate the domains of the ordinary and extraordinary waves. relation (1.4.44) show that the QL waves are very nearly left and right circularly polarized,13 and the QT waves nearly linearly polarized (wave E in the plane of B0 and y for the ordinary wave, E perpendicular to this plane for the extraordinary). Except for a small range of conditions such 13 The handedness of the circular polarization is defined with respect to ihc magnetic field direction, not the wave propagation direction. 1.4 Wave propagation with magnetic field 41 that \p\~ 1, the oblique waves may thus be understood as having properties closely similar to the principal waves discussed in Section 1.4.9. Graphs of the refractive index as a function of electron density are shown in Figs. 1.19 and 1.20 for 1, neglecting collisions (vjto ~>0). 1.5 2.0 Electron density (up/u)2 = n/nc FIG. 1.20 Same as tig. 1.19, but v>J_a = \.S. Curves for propagation angles other than those shown lie within the cross-hatched regions, and can be estimated by interpolation. An interesting anomaly appears at ct>p2/to3— 1. The 0" curves are seen in interchange. The curve for the left-hand wave drops down abruptly and become? the right-hand wave; the right-hand wave goes through a resonance to become the left-hand wave. This purely mathematical dilliculty arises from the fact that at oj//uj2 = I the QL condition |p2|«l -_w2 = 2.0 45.? f¥ \\ \0.5 \\l \ i 90" | 1 /If / I / ,// i / V! fsf}.5 , H ir i 1 1 90° 3 2 1 0 1 2 3 (b) FIG. 1.21 Polar plots of refractive index (index or slowness surfaces), (a) (u6/iu = 0.5, (6) wr,/ai= 1.5. Ordinary wave, dashed curves; extraordinary, solid; no collisions. FIG 1.22 Polar plots of reciprocal refractive index (phase-velocity or wave-normal suiTaces), (a) mbjai = 0.5, (A) m,,la> = 1.5. Ordinary wave, dashed curves; extraordinary, solid; no collisions. r 44 Electromagnetic wave propagation in a cold plasma Chap, 1 1,4 Wave propagation with magnetic field 45 cannot be obtained without collisions. There is thus a critical angle defined by jS&f^ (L4.U6) |e©S%[i| wp such that when 8 > 8crit the curves for the ordinary and extraordinary QT waves are continuous across the boundary j,2/aja = 1 (as in Figs. 1.19 and 1.20 for (f-£Q), but when 8<8c.rit the curves for the left and right circular QL waves are continuous (in Figs. 1.19 and 1.20 only for 0 = 0). When 0=8„lh the two curves intersect at u>v2l and the vector propagation coefficient P = Y/j". One may express the vector p in terms of its components in either cartesian (x, y, z) or spherical (j3, 8, ld0)e, one obtains15 15 The dispersion relation /(<", J3, #) = 0 is mathematically similar to an equation of state. Appropriate methods of handling the partial derivatives are developed in thermodynamics texts. See, for example, Sears (1953) and Crawford (1949). — Left-hand wave -Right-hand wave (1.4.119) Electron density (Wp.fb>)2 = »/",• IIG. 1.25 Contour map of real part of refractive index for the left- and right-hand Circularly polari/ed waves propagating along the magnetic field (cyclotron waves) (*/«•= 10-°). 48 Electromagnetic wave propagation in a cold plasma Chap. 1 where vre„ has the magnitude (1.4.118) and the direction implied bv (1.4.117). The problem of energy flow in an anisotropic, dispersive, lossy medium becomes exceedingly complicated. The reader is referred to treatments of 1.4 Wave propagation with magnetic field 49 the magnetic field (Helliwell and Morgan, 1959; Rao and Booker, 1963), discussed further in Section 6.5.3. 1.4.12 Refractive index contour maps. The zeros and poles of the refractive index were shown in the uip2-cob plane as a function of propagation angle in Fig. 1.18. For a particular angle of propagation, iines of constant refractive index then make up a contour map. The cases of Electron density (Wp/w)* = n/nc FIG. 1.26 Contour map of real part of refractive index for the extraordinary wave propagating across the magnetic field (9 = 90°; */eb= I0"3). The ordinary wave is represented by the (<*>p/u)2 axis (that is, a>,,/ = 0). 0 = 0 and 90° are shown in Figs. 1.25 and 1.26. The ridge of high index in the vicinity of resonance and the valleys of depressed index approaching cutoff are apparent. Cross sections at constant o>p2 yield the real parts of the curves in Fig. 1.8 for 0 = 0, and those in Fig. 1.14 for 6 = 90°. Cross sections at constant oih yield the real parts of the curves in Figs. 1.11 and 1.15, respectively. Contour maps for propagation at other angles are shown in Figs. 1.27 and 1.28. Here the "ordinary" resonance is seen emerging in the upper SO Electromagnetic wave propagation in a cold plasma Chap. 1 j 1 v mr I fa n I-r * 1 o II 3 ü E right corner. It does not extend to low density for any magnetic field, so that the only possibility of coupling to it from outside a plasma is' by evanescent waves through thin cut-off regions or by mode conversion (see Section 4.2.3). 1.5 Ion motion effects Our discussion thus far has neglected motion of the heavy positive ions. In the high-frequency domain, which is our principal interest, (he ion 1.5 Ion motion effects 51 current is very small relative to the electron current, on account of the greater inertia of the ions. However, at lower frequencies (for example, near the ion cyclotron frequency) the ion current can be dominant. Coincidcntally, it usually happens that those frequencies for which ion motion is significant imply wavelengths comparable to the size of laboratory plasmas and, hence, boundary conditions must be considered simultaneously (Stix, 1962). In contrast, our basic viewpoint for high-frequency waves is that of such small wavelengths that the laboratory plasma can be treated as an infinite medium. 1.5.1 Conductivity with ion motions. The contribution of ion currents may be evaluated by a straightforward extension of the methods of Sections 1.4.7 and 1.4.8 (Astrom, 1950; Allis, Buchsbaum, and Bers, 1963). The equation of motion of the kth particle species is dv tnk=?ä(E+ v,(. x B0) - mkvkyk, (1.5.1) where mh is the mass, qk the charge, and vk an effective collision frequency or damping term for the Ath species; E is the wave electric field and B(1 the static magnetic field (as usual, neglecting the wave magnetic field). The corresponding current density is (1.5.2) where nk is the density of the /cth species. As in Section 1.4.7, the relations (1.5.1) and (1.5.2) can be expressed as a tensor Ohm's law J(; = ÖVE. The total current density is then and the total conductivity16 *=;>>*. k The corresponding total tensor dielectric constant is * 2** ft=l-/—=1-7-*— coe0 ojc0 (1.5.3) (1.5.4) (1.5.5) (1.5.6) "' This argument is often phrased in terms of mobility instead of conductivity. See footnote 3 in Appendix A. 52 Electromagnetic wave propagation in a cold plasma Chap. 1 In the circularly polarized coordinate system and notation of Section 1.4.8, the elements of the diagonalizcd dielectric constant tensor, replacing (1.4.77) to (1.4.79), are ^-2 ui\Oi — 0}, I-file) h -jvk) where the generalized plasma and cyclotron frequencies are m,. (1.5.7) (1.5.8) (1.5.9) (1.5.10) (1.5.11) 1.5.2 Principal waves including ion motions. For the important special case of a two-component electron-ion plasma with equal densities ne=nv = n and negligible collision rate ve — vi—0, kt— 1 KT M2 KT (1.5.12) (1.5.13) (1-5.14) where /)2 + wl + wf, o(4) special densities; «C'K = hilw) _(l-tt>g,K)(l-a&K) The refractive indices for the principal waves are 8 = 0: u , (in — ea-jXp) + o)3) _n± — n ih ~((d4-wbe)(ui — mM) n1 (to + ■,)(tu — tuj) _n2 — n (ui-oibe)((ji + wbi) n2 q>2-(mp')2 _nc'— n nc' 0 - 90": (n1-n)(n2-n) (l-üj^w/o.3)^,,-«) (1.5.16) 1.5 Ion motion effects 53 Special frequencies: wiK x M2+col - <+«$ - o{<*thl>d (1.5.17) (1.5.18) (1.5.19) (1.5.20) (1.5.21) (1.5.22) (1.5.23) (1.5.24) (1.5.25) (1.5.26) (1.5.27) There is a new resonance for the left-hand wave at the ion cyclotron frequency, as we would expect intuitively. Jn addition, there is the new lower hybrid resonance for the extraordinary wave at wlh, where mlh and the modified oivh are the two positive solutions of the equation (Auer, Hurwitz, and Miller, 1958) »* - IM2 + a>(,.a.l!J[(co?')s-r-0, ' (1.5.28) Wave frequency a FIG. 1.29 Qualitative variation of refractive index with frequency for principal waves, including ion motions. Compare l-ig. 1.16. 1.5 Ion motion effects 55 r-1 k FIG. 1.30 Characteristic shape of phase-velocity surfaces for various regions oT density and field, including ion motions with r = ui^Jut,,,. Compare with Fig. 1.23. The labeling of O and X waves changes across the dashed line, which corresponds to the condition kiK,=x±K(i. 56 Electromagnetic wave propagation in a cold plasma Chap. 1 approximate solutions of which are given in (1.5.18) and (1.5.19), neglecting terms of order ^/^j, and higher. At high densities, bebi (1.5.29) which is the well-known d-e dielectric constant of a collision-free plasma perpendicular to a magnetic held (Spitzcr, 1962, §2.4), if two or more ion species (having different qjm ratios) are present, additional hybrid-type resonances are obtained (Buchsbaum, 1960). Ion resonances are of particular importance in connection with the problem of heating thermonuclear plasmas (Stix, 1962). Note that ion-motion effects are significant only at high fields and low frequencies such that (ujbt,juj)2 becomes comparable to or greater than m,/me»l. Furthermore, we have neglected collisions in the above discussion. When the electron and ion collision frequencies are included [by using (1.5.7) to (1.5.9) with Appleton's equation (1.4.81) in a straightforward manner], at frequencies below the electron collision frequency the electron motion again is dominant and, indeed, swamps the ion resonances when v>cu6i. 1.5.3 Oblique propagation with ion motions. For propagation at arbitrary angles, cutoffs and resonances may be found from the "tan20" Appleton equation (1.4.81). Cutoffs, independent of angle, occur when frh or *r vanish. Resonances occur when tan20 = — k[jk1 = — 2*r;i/(k|+ *,-)- For the two-component, collision-free case the resonance condition is (I oj2 (1 -a.ilGa>tJl/a>2)sinae-r-(l ~ oj^Jw2)(l - w2,/ojz)cOS2(f ;>a)bl,»t». CHAP T E R 2 Collision processes 2.1 Introduction The preceding chapter developed the propagation characteristics of electromagnetic waves in a uniform ionized medium from the equation of motion of a single, "typical" electron. This analysis would be rigorous if the ionized medium were to consist of nothing more than free electrons which (I) do not interact with the background of neutral atoms and charged ions, and (2) possess thermal speeds negligible with respect to the phase velocity of the wave. Both of these qualifications refer to processes that permit exchange of energy between the electron gas and the electromagnetic wave. Therefore, it was physically reasonable to anticipate these effects by including a simple viscous damping term in the equation of motion. Qualification (/), at least, is approximately accounted for by assuming that the background gas, of neutrals and ions, can be represented by a continuous, stationary, charged fluid through which the electrons move with a drag force proportional to the velocity. The present chapter justifies the identification of the damping term with a collision frequency and investigates its physical meaning in terms of the discreteness of the background particles. Our principal interest is in the difficult problem of electron-ion (coulomb) "collisions." Although mention of collision processes implies particle motion and hence a nonzero temperature, the role of electron temperature in collision processes is essentially different from that implied in qualification (2) above. It is appropriate to describe as "cold" those plasmas for which qualification (2) is well satisfied. The cases of "warm" and "hot" plasmas are considered in Chapter 3. We again assume that the positive ions are so massive that they do not move under the action of the wave field. 58 Collision processes Chap. 2 2.2 Elementary considerations of collision processes 2.2.1 Collision cross sections and frequencies. Consider the interaction of a test particle (an electron, let us say) with a group of field particles (atoms and molecules, or ions). Initially, we assume that the mean free path is sufficiently long and the interparticle force sufficiently short-range that a collision can be treated as a discrete two-body interaction. This assumption is notably troublesome in the case of the long-range coulomb force. For a specific central-force law between two particles there exists, in general, a relation between the impact parameter b, the relative velocity v, and the scattering angle of the test particle (see Fig. 2.1). The analysis in the general case is most readily carried out in center-of-mass coordinates and then transformed to the laboratory system. We shall be principally interested in the case where the field particle has large mass and small velocity compared to the test particle. In this case, i' and are simply the speed and scattering angle of the test particle in laboratory coordinates. The probability per unit path length for the test particle to be scattered FIC. 2.1 Geometry of a collision. The impact parameter b is the distance of closest approach if there were no interparticle force. The scattering angle '/> is (he deflection of the orbit asymptote. 2.2 Elementary considerations of collision processes 59 through an angle between and is fib dP^=nf 2tt b(v, ) db=nf 2w b(v, , (2.2.1) where nf is the number of field particles per unit volume. It is customary to write this probability in terms of a differentia! cross section qa{v, ), having dimensions of area per unit solid-angle, such that dP) 2rt sin d (2.2.2) where 2-n sin d = dii is the element of solid-angle. The total cross section qt is obtained by integration over all angles: qfv) = 2v [ qn(v,d, Jo (2.2.3) an integral which, in some cases, diverges at the lower limit. The total probability of collision per unit path length is1 m= f dPlS, = nfqt(v). Hi (2.2.4) For test particles having a speed i>, the total probability of collision per unit time, synonymous with the total collision frequency vt, is The mean free path is v,(v)=n, qfv) v. w \ 1 v 1 P{v) vt(v) nfqt(v) (2.2.5) (2.2.6) Next consider the energy and momentum exchange in a collision. For simplicity, assume that a test particle of mass in and velocity v 1 In classical gas discharge terminology, it is customary to define a normalized probability of collision P, as the probability per unit path in the field gas at I mra-Hg pressure and 0"C (Brown, 1959). Thus d k PsP{t+27V' where L is Loschmidt's constant (2.7 x 1010 cm-3), and p and Tare the actual pressure and absolute temperature of the field gas. This formulation is not useful in the case of highly ionized plasmas where particle densities and temperatures, rather than pressures, are the experimentally important quantities. 60 Collision processes Chap. 2 impinges upon a stationary field particle of mass M? From conservation of momentum and energy, one obtains for an elastic collision (Symon, 1960, pp. 171-75), Fractional energy 2r[l -eos^fl -r7- sin2 1 +r (2.2.7) (2.2.8) where cj> is the angle through which the test particle is scattered and r — m/M. In the interesting case of m«M: Fractional energy lost 2m M (1 —cos^) Fractional forward momentum lost 1—cos: I ^)(1 — cost/>)sin<£ d Mn(pi ^)sin<^ d (2.2.11) In problems of energy and momentum transfer, the important collision cross section is the cross section for momentum transfer qjv) = 2tt J qa(D, £)( 1 - cos<£)sin^ ^(t')- (2.2.12) Relations equivalent to (2.2.4) to (2.2.6) exist for the momentum-transfer cross section. In particular, there is the collision frequency for momentum transfer, vm=nfqm(v)v, (2.2.13) which is generally the physically important collision frequency in problems of wave propagation. Indeed, for convenience, we shall normally omit the subscript m hereafter for this quantity. The functional forms qQ{v, <£), or equivalently b(v, <£), may be computed, in principle, from the dynamics of the specific interparticle force law. As an example, consider an electron (regarded as a mass-point) colliding with 2 A more general discussion of'eollision geometries, using cenler-of-mass coordinates and allowing for field particle velocity, is given by Aliis (1956, Section lb). The neglect of target particle veloeily is reasonable except in unusual eases where (he ion temperature exceeds the electron temperature by a factor of i he order of (Mlm)%. 2.2 Elementary considerations of collision processes 61 FIG. 2.2 Geometry of the "electron-molecule" hard-sphere collision. a massive hard-sphere molecule of radius R. If the electron impinges with an impact parameter b, it is easily seen (Fig. 2.2) that it is deflected by an angle for which , 2lf- COSi/> = R* I. Thus, we obtain for the three cross sections: b cbm sn\ 4 2 Jo COS^)sill(^ dff> = TrR2 =2m/9(=l (2.2.14) (2.2.15) (2.2.16) (2.2.17) (2.2.18) Both total cross sections are equal to the geometrical cross section, independent of velocity, and the average energy lost per collision is 2m! M, a well-known result. When the problem is treated by quantum mechanics, the total cross section is found to be greater by a factor of two to four, depending on velocity, on account of diffraction of the electron wave (Mott and Massey, 1949, pp. 38^40). For short-range electron-molecule (or electron-atom) collisions, with forces falling off more rapidly than \jrs, it has been shown by Mott (Massey and Burhop, 1952, p. 3) that the total cross section qt (2.2.3) is bounded on account of quantum effects. Furthermore, for most common atoms or molecules and electrons of low to moderate energy (~1 eV), scattering is approximately isotropic, and the total and momentum-transfer cross sections do not differ appreciably. 62 Collision processes Chap. 2 As a second example, in the case of coulomb scattering by a massive ion of charge Ze, the well-known Rutherford formula gives (Symon, 1960, pp. 135-38) Ze2 from which one obtains; 1 /_ZfM 1 *5) Dipole U/ra) Coulomb (Mr*) Velocity dependence Cross Collision Mean section frequency free path q v = nqv A= \jnq I l/r2 1/p* 1 1/c 1 In reality, interparticle forces are not, in general, simple inverse powers of distance. Furthermore, the situation is complicated by resonances and quantum mechanical effects, especially at energies such that the dcBroglie wavelength of the electron is comparable to the dimension of the cross section (more precisely, to the product of impact parameter and scattering angle, b&) (Mott and Massey, 1949; Massey and Burhop, 1952; and Vogt and Wannier, 1954). Thus, one must in general resort to 64 Collision processes Chap. 2 experiment to determine cross sections empirically, including the dependence on electron velocity and on scattering angle (Rose and Clark, 1961; Brown, 1959). Thus far, monocnergetic test particles have been assumed. If they have some velocity distribution, not necessarily Maxwellian, cross sections and collision frequencies must he suitably averaged over velocities. For instance, the average momentum-transfer collision frequency is — I <7m0!) vf(v) d3v vm=nfqmv=nfi- - \f(v) dsy (2.2.28) where /"(;■) is the velocity distribution function for the test particles and the bar over a quantity denotes this particular form of average. Tt should be noted, however, that this direct average of the collision frequency is not the only one with useful physical significance. Section 2.4.3 discusses other forms of velocity averaging which are more important for our purposes. 2.3 Effect of collisions on electron motion Since mw, where v is the momentum-transfer collision frequency (2.2.13), represents the average rate of change of forward momentum due to collisions, it is possible to write the equation of motion for an "average''1 electron in the form mi~ -m^-e£„ exp yW, (2.3.1) identical to (1.3.12) with E—E0 exp_/W. This model, of an electron moving through a viscous medium, follows directly in the limit where the electron makes a large number of collisions during one oscillation of the wave held (that is, co«v). However, we wish to consider the full range of conditions, including oj> v, for which the validity of (2.3.1) is not obvious. To do this, we again consider an average electron, but distinguish between the ordered velocity component produced by the wave lield and the random component of thermal motion (Appleton and Chapman, 1932). The equation of motion for the ordered component, during the time interval between collisions, is m£ = ~eE0 expjcot. (2,3.2) We assume the boundary condition that this ordered velocity is zero at the instant of collision, /x. Thus eE £=./™ [expO'oj/O-expC/WO] eE' m~t CX P*'/W)[ 1 ~ eXp^ ~ ^]' (2.3,3) 2.3 Effect of collisions on electron motion 65 where t — X — tv. This equation is, of course, valid only for the interval lx < t < /a, where t2 is the time of the next collision. In order to find the average ordered velocity, it is necessary to know the statistical distribution of the time intervals t, in the past, at which the electrons made their last collision. This distribution is entirely equivalent to that of the time intervals t, in the future, when the electrons make their next collision. Consider a group of electrons, N0 in number, at an arbitrary instant of time. The number N of these that have not undergone a collision after an interval r is obtained from dN dl = -v/V, giving N=Na exp(-vr). (2.3.4) (2.3.5) Therefore, the fraction making the previous collision within the interval t to r+dr, in the past, is dN --v exp( — vr) dr, (2,3,6) the familiar Poisson distribution. Combining (2.3.3) and (2.3.6), we obtain for the statistical average of the ordered velocity component at time t (a eEn f"° =j j^— exptjVu?) J [1 — exp(— Jo)t)]v exp(—yr) d-r eE0 exp jwf. (2,3.7) But if we had written the equation of motion (2,3,2) with a damping term g£, m'i= -g£-eE0 expjW, (2.3.8) we obtain f =--r~-— exp /tuf. g+jmv> Comparison of (2.3,9) with (2.3.7) leads to the identification g = mv. (2.3.9) (2.3.10) It should be noted that (2.3.7) assumes that % is either independent of velocity or that the random velocity is large compared to the ordered component. In the latter case, a common one, v is an appropriate average over the electron velocity distribution (Section 2.4.3). Comparison with (2.3.1) shows that the assumption that the ordered velocity is zero after a collision is equivalent to the identification of v as the momentum-transfer 66 Collision processes Chap. 2 collision frequency. It should be noted that collision frequency k compares with the radian wave frequency u>, not the cyclic wave frequency at/27r. 2.4 Analysis of particle interactions The dynamics of a macroscopic system (such as a plasma) consisting of a large number of interacting microscopic systems (electrons, ions, and neutral molecules) can be analyzed at several levels of detail. The most important of these are the orbit, kinetic, and hydroniagnetic theories. It is obviously impossible to follow the exact dynamics of each of the microsystems, a hopeless many-body problem. One may, however, follow the behavior of a "typical" microsystem and infer therefrom the behavior of the macrosystem. This approach, often known as orbit theory, is essentially that employed in Chapter 1 (Rosenbluth and Longmire, 1957). It is useful when the interactions among the microsystems are either (/) weak enough to be negligible in some sense, or (2) strong enough to be represented by a self-consistent interaction between the average microsystem and a fluid representation of the rest of the microsystems. In either case, the statistical fluctuations of the interactions are necessarily ignored. The statistical features oT the problem are preserved in the kinetic theory approach (Jancel and Kahan, 1955; Delcroix, 1960). We consider explicitly the particle distribution function/, defined such that at time t there are f(r, v, t) dx dy dz dvx dvv dtz microsystems of a given class (electrons, let us say) located in the element of phase space between (r, v) and (r + rfr, v + d\). In the absence of collisions, at a time t + dt precisely these same particles, and none others, will occupy the element of phase space at (r + \dt, v + ac/0, where a = F/m is the acceleration imposed by external forces (common to all particles). By a Taylor expansion f(r + vdt, v + atff, t + dt)=f(r, v, /)+ V i\ ¥- di+ V a, | /rH est, fri o dt+^-dt. m ct (2.4.1) Thus, it follows that the total rate of change of /along a stream line is df _ 8f _ df df 2-2 * £+2 «£+5 = v.V,/+a-Vt,/+ It (2.4.2) 2.4 Analysis of particle interactions 67 where % and V„ are respectively the gradient operators in configuration and velocity space.3 If no collisions (or other processes such as ionization and recombination) occur, w;e have the kinetic or Vlasov equation (Vlasov, 1938) f=f+v.Vl/+a.Vs/=0) (2.4.3) which is closely related to the Liouville theorem on the conservation of density-in-phase of statistical mechanics. It may also be regarded as a generalized equation of continuity. If collisions or other statistical processes do occur, then there may be a net change (Sfl&t)^ in the number of particles in the given element of phase space, and f+y.WJ+a-VJ- 01 (2.4.4) which is the Boltzmann equation. In general, it is macroscopic quantities, corresponding to integrations over the distribution function, that are of physical interest. It is often possible to carry out these integrations without specifying the exact nature of the distribution function, thereby obtaining a set of hydroniagnetic equations in the macroscopic variables (such as velocity of mass motion, pressure, and current density). These equations are often known as Boltzmann transport equations, although this modifier is sometimes used for (2.4.4). The hydroniagnetic description will be used in Chapter 3 (Spitzer, 1962; Bernstein and Trehan, 1960). 2.4.1 Boltzmann equation. Since the collisionless Boltzmann, or Vlasov, equation (2.4.3) implies the particle dynamics (Bernstein and Trehan, 1960), it is entirely equivalent to orbit theory in cases where collisions may legitimately be neglected. However, some problems may be mathematically more tractable in the kinetic theory treatment. An example is the "warm" plasma case of Chapter 3, for which electron thermal speeds are no longer assumed negligible with respect to the phase velocity of the wave. A major problem in the use of the Boltzmann equation is the evaluation of the collision term (cfjdt)call, which is in general an integral not only over /(r, v, /) for the class of particles in question, but also over the corresponding distribution functions for all other classes of particles with which interactions occur (Allis, 1956; Bernstein and Trehan, 1960). Thus, strictly speaking, we have an //'/egro-differential equation. A common simplification is to assume that, in the absence of external " Wo call attention lo the common use of the notation S/Br and dj, is usually taken to be the reciprocal of the momentum-transfer collision frequency v; thus, we substitute in the Boftzmann equation (2.4.4) •ccu= -"(«) (f-fd- (2.4.6) A second major problem is the mathematical complexity of solving the Boltzmann equation. It is usually necessary to expand the distribution function in a series, and retain only low-order terms. Two such expansions are commonly used. The Chapman-Enskog technique (Chapman and Cowling, 1951) assumes only smail departure from local thermodynamic equilibrium due to some perturbing agent of strength measured by a parameter a; thus where fir, v, 0=/c(r, v, t) + a%(r, v, 0 + /0(r,^) = K(r)(^)%xp(-^), (2.4.7) (2.4.8) the Maxwell distribution. This expansion does not converge rapidly in problems where the perturbed distribution is anisotropic. Where there exists a preferred direction in space, it is useful to expand in spherical harmonics in velocity space (Allis, 1956): f(r, v, 0=2-Mr< v, 0^(c°s^) =/o(t'Wi(r, Ocos0+/2(r, u\ ,) 3 cos^ 1 4- (2.4.9) where f0 is assumed isotropic but not necessarily Maxwellian. The volume element of velocity space, often written in the alternate notations, ds\=dav = dv = dvx dv-j di\,, (2.4.10) becomes explicitly in this case d3v = vasinBdvd8d (2.4.11) where v, S, are the spherical coordinates in velocity space. Most physical problems are symmetric with respect to the azimuthal angle ; the trivial integration then yields the simplification4 da\ -> 2ttv2 sin8 du d0= — 2ttv2 dv rf(cosf3). (2.4.12) It is useful to evaluate some moments of the distribution function /for the spherical harmonic expansion. Recalling that spherical harmonics are orthogonal, we obtain the normalization condition J/(r, t) rf3r= j 1 ./'(r, v) Ittv2 d(cos6) dv =4tt J^/oCr, v2) v2 dv = n(r) (2.4.13a) where n(r) is the particle density. Frequently it is more convenient to separate the space and velocity dependence, replacing /(r, v) by /7(r)/(v) with the normalization v2dv=l. (2.4.13b) The average value of the velocity component in the preferred direction, (2.4.14) vv = v cost 7j^= J j (v cos0)/(r; v)2-7re3 ^(cosfl) dv = y ^Ut,v?)v*dv. (2.4.15) 2.4.2 Elementary Boltzmann theory of plasma conductivity. Consider a spatially uniform plasma subject to a weak oscillating electric field E0 exp>W in the x direction (Burnett, 1931; Margenau, 1946, 1958; Sampson and Enoch, 1963). There is no static magnetic field. The Boltzmann equation for the electron distribution function becomes df gg0exp,;W c/ /QA Si* m W/co(t (2.4.16) Note that the magnetic field associated with the oscillatory electric field is neglected, and that the electric field is assumed uniform in space (infinite wavelength). In the absence of the electric field the equilibrium distribution, maintained by unspecified agencies, is/0(t'), assumed isotropic 1 11 tin- ;i,iniiii lull single i/i is significant, we make the expansion (2.4.9) in the spherical harmonies /V"(eosW)exp]»!•(>, P<" being the associated Legenclre function. 70 Collision processes Chap. 2 but not necessarily Maxwellian. Using the spherical harmonic expansion (2.4.9) to first order, we have fit, 0=/^ + ^^exp>r, (2.4.17) in which the time dependence of/: is shown explicitly and cos6 is written as vxjv. Evaluating the collision term in the form (2.4.6) and substituting (2.4.17) in (2.4.16), we may solve for eEolm 8f0 (2.4.18) v(v) + ju> dv where the small nonlinear term in d(vxfilv)ldvx has been discarded. The fact that/j is complex signifies that the electron velocity perturbation is not in phase with the electric field, an entirely reasonable result. The resulting current density for electrons of charge — e is J= —nev — ne j vxf(v, I) d3\ An f 4mie2 3 m Ee exp(/W) 4fa ,. V dv (2.4.19) v) 4-ycu dv using (2.4.15) and the alternative normalization (2.4.13b). Equation (2.4.19) is equivalent to the complex conductivity 3 rn Jo v\ 4tt ne'2 3 m (r)+jw dv dv (2.4.20) the latter form obtained by an integration by parts. The corresponding complex dielectric constant is k=\-j- <=0oj f-1 (2.4.21) e0/?to 3 J cxi— jv(v) dv Two important special cases oT (2.4.20) and (2.4.21) may be obtained readily, as follows. Collision frequency independent of velocity: _ ne2 1 2 I ne" = 1 — cfinm u»—jv w(ui—jv) (2.4.22) (2.4.23) 2.4 Analysis of particle interactions 71 independent of fa(v), results which coincide with the Lorcntz conductivity (1.3.14). This illustrates the well-known Maxwell condition under which the transport properties of a gas arc independent of the distribution function. No such simplification exists for other dependencies on velocity. Maxwellian velocity distribution: Assuming one obtains for (2.4.20) 3 V He* m I v(u) +jc «* exp( — u2) du, where u is the velocity normalized to the most probable speed, u = v'i{2kTjm)'/-. (2.4.25) (2.4.26) In the case of a static magnetic field, a similar analysis summarized by Allis(1956) yields for the elements of thediagonalized (rotating coordinate) conductivity tensor (1.4.71): r i d/M ti3 dv 3 /« Jo v(v)+jo) dv 4tt ne2 t7l.r=--5" 1 3 m Jo ^u)+y() which may be used directly ill the Lorentz formula. This approach is easily followed in the limits of n Tho electron temperature T may here be regarded more generally as any appropriate scaling parameter of the velocity distribution, specifying the distribution even in non-Maxwellian cases. 72 Collision processes Chap. 2 very low and very high frequencies, with the results summarized in Table 2.2. In both limiting cases the frequency dependence drops out. Table 2.2 Effective collision frequencies for extreme wave frequencies Direct-current limit, Radio-frequency limit, co»v Loren tz conductivity, Eq. (2.4.29) Boltzmann conductivity, Eq. (2.4.20) Effective collision frequency «£f 1 m v není dm , t!3 dv - Wne^C J_ 3m J v(v) dv 3 J v(u) dv 4mic'2 3 m J lw~ to] dv 47t f ) 3 In particular, we shall be interested in the effective collision frequency in the high frequency limit, a useful parameter which we identify by the special notation <"> = v(v) —,— v1 dr dv (2.4.30) where ./'=/o is the unperturbed electron velocity distribution function. This is, in fact, a kind of average collision frequency, like (2.2.28), but obtained with a different weighting of the velocity distribution. For a Maxwellian distribution (2.4.24), the general form (2.4.30) reduces to *>-i^)>*«H-&)"*- <"•*> Assuming a simple power law [see (2.2.27)] v(v)=--Cvl, (2.4.32) we may write (2.4.31) in terms of gamma functions (Jahnke and Emde, 1945) as (2kT\"* r(~) I (2.4.33) 2.4 Analysis of particle interactions 73 For the same assumptions, the direct average collision frequency (2.2.28) is f- J\{r)/(r)4-r <7r Maxwell distribution /2A-7"\''2 I 2 J Thus (2.4.34) (2.4.35) special cases of which have been quoted in the literature (Molmud, 1959; Phelps, 1960). For each /, there will exist, some characteristic velocity for which the velocity-dependent collision frequency equals one or the other of the averages (2.4.33) and (2.4.34). Figure 2.3 shows these characteristic velocities Vi and v2 defined such that (2.4.36) 1 1 1 1 1 _ v (v\) - < V >■ ..------- - — rms — ^Average ■Most probable - / 1 1 S 1.5 Ě 1.0 0.6 -3 -2 -1 0 1 Collision-frequency exponent I. l(v) equals the high-frequency effective collision frequency <» or the sinpple average collision frequency 9, as a function of I he power-law exponent /. 74 Collision processes Chap. 2 3.0 2.0 1.0 0.5 / = -2 -1 0.3 I 0.01 100 ltd + £d;,|/o > FIG. 2.4 Correction factors g and /i for electron-molecule collision frequencies varying as v'; (a) exponent / negative, (b) exponent / positive. They are not far different from the usual velocities characteristic of a Maxwell distribution, namely (Sears, 1953): root-mean-square average most probable [ff (2.4.37) (2.4.38) (2.4.39) This mathematical analysis in terms of an assumed power law (2.4.32) is physically meaningful so long as the empirical collision frequency approximates the power law over the range of velocities for which the integrands of (2.4.30) and (2.4.34) are large. For practical purposes, this range may 2.4 Analysis of particle Interactions 75 \'j>±m\l ±^t,\K"» A*,0-»A(|«±eu»|/ is that in the limit of high frequencies g and h go to unity. In the two extremes (I (2.4.43) (2.4.44) g(co) = A(oo)=l. The integrations from which g and h are calculated in simple power-law cases have been carried out explicitly by Molmud (1959) and by Sen and Wyller (1960), using slightly different notation. An equivalent formalism has been developed by Gurevich (1956). The integrations are also reducible to functions discussed and tabulated by Dingle et al. (1957). The resulting g and h factors are given in Fig. 2.4 for several power-law exponents. The expression of the conductivity in the form of (2.4.40) and (2.4.41) has the power that the g and /( factors can be generalized to include the effects of electron-ion and electron-electron collisions, as discussed in Section 2.5.3. Extensive calculations of various cases for partially ionized gases have been made by Shkarofsky (1961). It is also possible to use the g- and //-factor formalism for non-Maxwellian velocity distributions. We now summarize the procedure for computing conductivity and hence propagation constants, as functions of o>, co„, v(v), and f(T, v). In general, one must perform the integrations (2.4.27) and (2.4.28). In practice, one can usually make use of one of the special cases given in Fig. 2.4 for simple power-law velocity dependence and a Maxwellian distribution. To do the latter, one computes 0(7", /)> from (2.4.33) and evaluates three g's and three h's as functions of the respective arguments w/\v>, («j+«j6)/>, and |co — a>(,|/. For given a and aib, the conductivity tensor elements may then be computed from (2.4.40) and (2.4.41), replacing the simple Lorentz elements (1.4.72) to (1.4.74). The corresponding dielectric constant tensor elements follow trivially from (1.4.76) and (1.4.80). The propagation constants may then be found from the Appleton equation (1.4.65) or (1.4.81). 2.5 Coulomb interactions 2.5.1 Debye shielding. We wish to investigate the manner in which the long-range coulomb forces in a plasma act to maintain charge neutrality, a problem similar to the behavior of ions in an electrolyte (Debye and Huckel, 1923). Consider a uniform plasma with n electrons and njZ positive ions, of charge Z, per unit volume to which a single additional ion of charge Q is added at the origin. Unlike charges will be attracted, like repelled. We wish to find a self-consistent solution for the electric potential ifi(r) at radius r in the vicinity of this excess ion. According to Boltzmann statistics, the plasma ions and electrons will now be distributed with the respective densities (we assume Tr,= Ti) nt(r) =- exp(-Ze>plkT), ne(r)=ri exp(eipjkT). The resulting net charge density is (except at the origin) p(r) = — ne[exp(eiljjkT) — exp( —Zeifi/kT)], which in the high-temperature limit Ze p(r) ~> -■ kT (2.5.1) (2.5.2) (2.5.3) The potential and charge density are related by Poisson's equation VV=-P,V (2.5.4) Writing the Laplacian operator in spherical coordinates and substituting the approximation (2.5.3), we have r'2 dr 1 Zd>p\(\+Z)ne24-V dr) e0kT the appropriate solution of which is Q exp(-A-/A,j) 4irec where (\+Z)ne2 (2.5.5) (2.5.6) (2.5.7a) For many applications in plasma physics, involving dynamic processes at frequencies of the order of -{ne*) Wem sy cm, (2.5.7b) as plotted in Fig. 2.5. The significance of this result is that the plasma shields any local excess charge so that at a distance of a few dehyc lengths its effect is no longer felt. The debye length is a measure of the thickness of the sheath formed at the boundary between a plasma and a conductor. The debye length may also be thought of as the magnitude of charge separation in a plasma for which the resulting electrostatic energy density equals the particle thermal energy density (Spitzer, 1962, §2.1). Hence, significant departures from electrical neutrality are not to be expected over distances larger than r\D. Note also that V'2 XD = v, (2.5.8) 1000 100 I 10 4/ i y y +•* y 1 y y ^ ^- y . y y - ^" — y____■ ___^7 4/ y ^ •y y y ■ ^ y+** y ^ i l / y \\ y^*~"' ___■ ~y i i / y b. y _ y 101 10* 10 Electron density n [cm-3] 10L 101 FIG. 2.5 The debye shielding length (2.5.1b), solid curves; and the logarithm of Spilzer's culotT ratio (2.5.18) for Z= 1, dashed curves. (See also Fig. 5.20.) Note corrections to In ASp in Table 2.3. where vmv is the most probable electron speed (2.4.39) and ujp is the plasma frequency. The debye length A0 and plasma frequency to„ represent two fundamental, interrelated scaling parameters of a plasma, measuring the distance and time scales over which the plasma establishes electrical neutrality, A necessary condition for the validity of the debye shielding picture is that the number of electrons ND in a sphere of radius equal to the debye length be large, where lte$&* (kTf* 3 6Vtt \ e J n> = 1.7 ■ JO' (2.5.9) («[cm ;3])s/j 2.5.2 Coulomb collisions. We now consider the coulomb interaction between electrons and ions as a collision process. The differential cross section was given in (2.2.20). It was observed that the total cross section (2.2.21), and even the momentum-transfer cross section (2.2.22), diverge for small scattering angles. We wish to estimate the relative importance of discrete close encounters as against the cumulative effect of many distant encounters in deflecting a test particle. When a light test particle impinges on a stationary massive target, the cross section for a deflection of at least is, from (2.2.19), and the corresponding collision frequency is -2 cot3^ Va — -nnZ i—) (2.5.10) (2.5.11) In particular, the collision frequency for close collisions with ^^90° is TTlZ \-.- -=i (2.5.12) which is a measure of the frequency of large-angle scattering. We now estimate the rate at which 90° deflections are produced by multiple small-angle scattering. The collision frequency for deflections between and + d is ' 3 COS^ip j\Att^w} t,asin3-2^ which becomes in the limit of small deflections '2 M «4 (2.5.13) (2.5.14) 80 Collision processes Chap. 2 2.5 Coulomb interactions 81 In a time r there are t collisions resulting in deflections between and , and the expectation value of the resulting total deflection is 2=l2+4>z2 + J ml --t %wHz(-t——V X, In 4>T. (2.5.15) where min and nax represent the range of small deflections considered. Since the limits enter only logarithmically, they may be rather grossly approximated. Let max = 90°, which does not greatly violate the small-angle approximation of (2.5.14). For „in we take the angle for which the impact parameter equals the debye length AD, since for b>\D the shielding effectively masks the deflecting ion (Cohen, Spitzer, and Routly, 1950)e; from (2.2.19) hmin_ Ze2 2 \27T€nkTXD 2V Zn'K (2.5.16) in which the kinetic energy \mv2 has been replaced by its mean thermal value fkT. The argument of the logarithmic term becomes ^w = 3Vy (A^kT\^ 1 where ASp is a parameter defined by Spitzer (1962, §5.2) as the ratio of the debye length to the mean impact parameter for a 90" collision h90. Specifically /477<,:/>'r. : SP bs~2Vn\ ez 1 Zn^V2^v\Z2JiJ = 1.55.101 Z(«[cm-3])'i (2.5.18) Under common laboratory conditions ln/lSj,~10 (see Fig. 2.5). Note that the number of electrons in a "debye sphere" (2.5.9) is lZASl>. The time required to achieve a cumulative deflection of 90° is, from (2.5.15), 7t lAire0my 32 V e2 J Zn In/lap (2.5.19) 6 The debye length cutoff is preferable to setting the impact parameter equal to the mean interionic spacingn''/», for which case the logarithmic term is ln|(.»» '/!(>)'' ■ \„VS\. and the number of times per second a deflection by 90° is achieved is 1 32 I e \2Zn\nASv 1..... 32 / #V 00 ~ Tao 77 \47rc0m/ 32 laASB '■-2- ''90 + - (2.5.20) Thus, the deflection rate (collision frequency) due to cumulative small-angle deflections is, typically, of the order of thirty times larger than that due to individual large-angle deflections. A more rigorous analysis in terms of velocity-space diffusion theory, in which velocity changes rather than angular deflections are considered, has been carried through by Chandrasekhar (1943) and by Spitzer (1962, §§5.2-5.3). In particular, Spitzer defines a "90° deflection time," the reciprocal of which is 2Ztt la/1,* (2.5.21) this is to be regarded as an improved version of (2.5.20). In this theory the argument of the logarithm emerges as the ratio between maximum and minimum impact parameter and the definition AXp = XnjbB0 given by (2.5.18) follows logically. However, these particular limits must be modified under a variety of circumstances, discussed in Section 2.5.4. In the following section we use the general symbol A to denote the appropriate ratio. In summary, we can divide electron-ion coulomb collisions into three classes (Bernstein and Trehan, 1960). (1) Impact parameter b< impact parameter for 90" deflection b90. In this case of close encounters, the electron-ion interaction may validly be considered as a discrete two-particle collision and treated by the usual collision integrals of kinetic theory (Allis, 1956; Dcsloge and Matthysse, 1960). (2) bnfi I) this case 2 is strongly dominant over case /. Case / may be included by a small change in integration limits. 82 Collision processes Chap. 2 2.5 Coulomb interactions 83 (3) b> kD. The effect of the debye shielding is to eliminate statistical (uncorrected) particle encounters. Particle motions on this scale represent correlated processes such as the propagation of waves. 2.5.3 Effective coulomb collision frequencies. It remains to be found how to incorporate into the conductivity formalism (2.4.40) and (2.4.41) the effects of electron-ion and electron-electron collisions. From the Fokker-Planck equation, or directly from the Boltzmann equation, it can be shown that an electron-ion collision frequency7 \4wenmJ 2 \2ZnlnA (^[cm/sec])'* (2.5.22) plays exactly the same role in the calculation of conductivity as the more straightforward electron-molecule momentum-transfer collision frequency 1 vD, defined in (2.5.21), is the reciprocal of Spitzer's "90° deflection time." 1000 100 P. 10 i y | i y | i y ' i 1 ^y X <$X X y y*- ~X y^ y — X x ^x - X XX X^ ií^X^ X y i 1 iy i i \y i i i 101 1012 1013 Electron density n [cm-31 101 101 FIG. 2.6 The effective electron-ion collision frequency (2.5.23). The numbers shown arc /7wZ In/1. The appropriate value of In/1 must be round from (2.5.30) and Fiys. 2.8 and 2.9. The 2tt normalization facilitates comparison with the wave eye lie-frequency = <^řn> + , (2.5.25) the latter form following because of the linearity of the operation defined by (2.4.30). The critical degree of ionization above which coulomb collisions are dominant is given by = (vy, which of course depends on the velocity dependence of the particular electron-molecule interaction and is usually a strong function of temperature. At room temperature the critical ionization for common gases in low-pressure discharges may be as small as ~ 10~7; at IcT— 1 eV, —TO-4 (Lin et al., 1955; Anderson and Goldstein, 1955). Using (2.5.23) and (2.4.43) in the d-c limit of the conductivity (2.4.40), we obtain £(0) = 3tt/32 and _ne2 1 _4v/2 (47re0)a (kryÁ tt% mAe2 Z\nA = Zlň/T" mll°/cm, (2.5.26) B For /= —3, it may be seen from (2.4,30) that in the general case off(p) not necessarily Maxwellian = (4W3)C/(0) where/(0) is the value of the electron velocity distribution function for v = <3. Thus, the high-frequency effective collision frequency is insensitive to departures from the Maxwellian distribution affecting only the high energy end. " If the velocity dependence of AS!,, A(v)= ^(^/bsoU')^^, is retained in the integration, we obtain (2.5.23) with the logarithmic term replaced by ln[(2/3y)/IKJ ft! In(0.37A/;,,), where y = 1.78 is Euler's constant and where ASv = XDjbS0 is Spitzer's average defined by (2.5.18). Other refinements are discussed in Section 2.5A 84 Collision processes Chap. 2 2,5 Coulomb interactions 85 which is precisely the result obtained by Spitzer (Cohen et ah, 1950; Spitzer, 1962, §5.4; and Maecker et al., 1955). The density dependence cancels out except in ln/1. Our discussion of coulomb collisions has thus far assumed a completely ionized plasma with infinitely massive, stationary (cold) positive ions. The electrons have been assumed to interact with each other only in a long-range, collective sense. We now consider the randomizing effect of electron-electron collisions, which effect is quite different from that of collisions between electrons and other species. It is clear, on physical grounds, that like-particle collisions cannot extract momentum or energy from this component of the plasma. They do, however, cause a diffusion in velocity space, which modifies the perturbation part of the velocity distribution function. From a mathematical point of view, since the electron distribution function also enters for the scatterer, the Boltzmann equation is inherently nonlinear. The effect depends on the ionic charge Z since the relative density of ions to electrons varies as l/Z and the cross section as Z2. Neglecting electron-electron collisions is equivalent to the limit of large Z. Numerical calculations of special cases have been undertaken by a number of authors (Spitzer and Härm, 1953; Hwa, 1958; and Kelly, 1960). Shkarofsky (1961) has shown that it is possible in principle to retain the formalism (2.4.40) and (2.4.41) involving the g and h correction factors (2.4.42). Extensive calculations, based on an expansion of the collision term in generalized Laguerre polynomials, have been tabulated. The g and h factors become functions not only of jcu + w„\j(v') but also of ionic charge Z and the degree of ionization of the gas. Figure 2.7 shows the correction factors for a fully ionized gas with and without electron-electron collisions, that is, forZ= 1 and oo, respectively. In the case of no magnetic field (o>b=Q), Spitzer and Härm (1953) and Kelly (I960) have used a numerical factor yB to express the ratio of the conductivity including electron-electron collisions to that for a Lorentz gas 2.0 YMl(y), Z) o(o>/<», CO) yt.(0, 1) = 0.582, yB(oo, Z) = I. (2.5.27) (2.5.28) The ratio remains of order unity at intermediate frequencies. As shown by Hwa (1958) and Kelly (1960), the major physical effect of electron-electron collisions on wave propagation is to broaden and damp the cyclotron resonance somewhat. Electron-electron collision effects may safely be neglected at high frequencies such that |«»±«>(,|/»l. 1.0 0.5 0.3 X Lorentz (Z= ~) Z= 1 Z= 1 0.01 100 FIG. 2.7 Effect of electron-electron collisions on the correction factors g anci h for a fully ionized gas with no magnetic field. The Lorentz gas calculation ignores electron-electron collisions; they are included in the curves for singly charged ions. 2.5.4 The logarithmic term. Most formulations of the electron-ion interaction process involve an integration over the impact parameter b that yields a result proportional to \n{bm!lxlbmi^). The ratio A = bmx;bmtr. and its logarithm are divergent unless we invoke physical arguments to cut off the range of integration (Theimer, 1963). The arguments sketched in Section 2.5.2 led to the identification of bmax with the debye length AD (because of shielding) and of bmin with the mean impact parameter for a 90° deflection h$0 (because of the relative ineffectiveness of close encounters). The argument of the logarithmic term then becomes Spitzer's 4fj>= ^d/^so, as given by (2.5.18). Aside from numerical refinements to ASp of order unity, there are two general effects which alter the form of A, one depending on electron temperature and one on wave frequency (De Witt, 1958). We discuss these in turn. The temperature effect pertains to the lower limit bmin. At electron temperatures above about 80 eV the deBroglie wavelength fijnw exceeds the 90" impact para meter />,„)-=Zr!/4ir«„M«fa and replaces it as the limit 86 Collision processes Chap. 2 bmin. That is, the electron is diffracted rather than classically deflected by the ion, and the interaction must be treated by wave mechanics (Marshak, 1940). The classical and quantum limits have been discussed extensively by Oster (1961b). Numerical calculations of the transition have been made by Greene (1959) and Oster (1963a). Meanwhile the frequency effect pertains to the upper limit bmax. Above the plasma frequency o>p, the distance traveled by the electron in one period vthloj replaces the debye length \0xvihj) is weak, and a good approximation to \nA(T) is obtained simply by the substitution for v of a thermal velocity of order (/c!T/m)^. The significance of more careful analysis is clouded by the fact that the physically invoked cutoffs (for example, bmax ~ XD) are uncertain within a numerical factor of order unity. However, in certain cases, it is possible to choose a limiting procedure that does not explicitly require an ad hoc cutoff (Oster, 1961b; Kihara and Aono, 1963). Table 2.3 summarizes the asymptotic values of lnA(T) for the various limiting cases, to be used in the effective electron-ion collision frequency (2.5.23) assuming a Maxwellian distribution. The numerical coefficients shown arc the results of refined calculations, but some may still be in error by small factors (Oster, 1961b; Dawson and Oberman, 1962). The low-frequency, low-temperature result is seen to differ from Spitzcr's approximation by a factor of about one fourth. The most important limit for Tam.f 2.3 Limiting forms of InA * 2.5 Coulomb interactions 87 Low frequency (high density) o)v (n»«t,) High frequency (low density) uj»cup (n«nc) Low temperature kT<^ZvRu ( 8_V kT t kT \-[eyV Hwp \Z*rJ = ln ^it(-)2AS! 3e - \yl In (0.25)AP , , , 4 kTt kT ln/LQ = In — = In (0.45) High temperature kT»Z2R„ \ eyl 8\,/* kT kT . 4 kT kT 7— = ln (1.29) j.— In -T-=ln (2.24) rr- * e = 2.1 \ 8; y= 1.781 (Euler's constant); R„ = 13.6 cV (Rydberg energy constant); ■ASp = ha/bt,Q, see (2.5.18). common microwave propagation experiments is the high-frequency, low-temperature result (see Fig. 2.8) (Scheuer, I960) = 6.2-104^ (kT[eV]f (2.5.30) Z w/27r[Gc] v where y= 1.781 is Euler's constant and R,,= 13.6 eV is the Rydberg energy constant. Figure 2.9 shows the corrections to be applied to (2.5.30) in the transition regions between the limiting cases of Table 2.3. The temperature dependence is taken from Greene's (1959) revised calculations. The frequency electron-density dependence is taken from Dawson and Oberman (1962). The "bump" in the neighborhood of ;), are shown in Fig. 2.10. To summarize the effect of coulomb collisions on microwave propagation in a plasma without magnetic field, we recall that the formalism of (2.4.40) is used with the effective coulomb collision frequency (2.5.23). The g and h correction factors are given to good approximation by the /= -3 case of Figs. 2.4 and 2.5. The In/1 term appearing in (2.5.23) is evaluated using the A0 of (2.5.30) and Fig. 2.8 modified if necessary by 88 Collision processes Chap. 2 1000 2.6 Nonlinear effects 89 10 100 iooo Wave frequency w/Z/rZ^ [Gc] FIG. 2.8 The logarithm lnA> of the high-frequency, low-temperature cutoff ratio (2.5.30), for ions of charge Z. The temperature correction of Fig. 2.9 is shown here by the dashed curves; the density correction must still be obtained from Fig. 2.9. the two independent corrections given in Fig. 2.9. For most practical purposes one may use for A the asymptotic limits of Table 2.3 throughout the range of parameters, changing from one limit to the other at the points where the asymptotic forms are equal, namely, kT=y*Z2/^ = (77 eV)Z2 = (890,000° K)Z2 --Y n '2e' ^0.328^^ (2.5.31) (2.5.32) Finally, we note that further modifications of A are required in strong magnetic fields for which the gyroradius is smaller than the debye length, that is, when a)„>«)„ (Sweet, 5959). The situation is complicated not only by excitation of longitudinal plasma waves but also by the fact that the cutoff conditions are no longer isotropic in a magnetic field, so that -0.5 \ \\ \ V \ X I ,1 , \ 1 0.01 10 100 Electron temperature kT/Z2 [eV| 0.1 1 Electron density n/ne = (o}p jm)2 1000 10 FIG. 2.9 Quantum and shielding correction terms to be applied to the low-temperature, high-frequency In A, of (2.5.30). The value of each correction subtracts from In A, which may be read from the solid lines of rig. 2.8. The two corrections are independent. From Greene (1959, revised) and Dawson and Obcrman (1962). different forms of A enter the collision frequency terms of the various elements of the dielectric constant tensor (Silin, 1962; Eleonskii et ah, 1962; and Oberman and Shure, 1963). However, for most practical purposes the field-free values may be used. 2.6 Nonlinear effects A number of nonlinear processes occur in plasmas, leading, for example, to harmonic generation and interaction between two waves (Ginzburg and Gurevich, 1960). The best known, the so-called Luxembourg effect, arises from a change in the effective collision frequency as a result of electron heating by the electromagnetic wave. Thus, it vanishes for the special case of collision frequency independent of velocity. Other types of nonlinearities arise directly from the electron dynamics and the non-transverse electric fields in a nonuniform plasma or a plasma in a magnetic field. Further discussion of nonlinearities can be found in Chapter 6. 2.6.1 Criterion for linearity. A rough criterion for the validity of the linearized theory, already noted in Section 2.4, is that the velocity increment due to the electric field be small compared to the random thermal velocities; that is, e2E2 kT —m « — (2.6.1) 90 Collision processes Chap. 2 1000 •ICC ^ 10 \ \io-£S \ \ f \ = 10_scps-cm3 \ \ \ \ \ \ \ 10 100 Wave frequency o)/27r [Gc] 1000 FIG. 2.10 Solid curtes. Effective electron-ion collision frequency for a hydrogenic plasma (Z = l), as a function of frequency and temperature, assuming no shielding (uj»tu,,) but including the temperature correction of Fig. 2.9. The numbers given on the contours arc to be multiplied by the electron density n in cm"3 to obtain the cyclic (not radian) frequency v\1tt in cps. Compare Fig. 2,6. Dashed curves. The corresponding power attenuation coefficient 2a = 2x">jc in the high-frequency limit of (1.3.30). The numbers given arc to be multiplied by (»[cm"'])! to obtain 2a in decibels/cm. Attenuations derivable from this graph arc in general very small because of the restriction on electron density implied by the condition ci>p. For plasmas with ionic charge Z use the sealing relations: K«, kr)]z=z-*\H«>iz-\ kTiz^h,, ir)]z=Z'i[«(«/^2, kT/z2)h,l This criterion permits the discard of the nonlinear term in the Boltzmann equation expansion (2.4.18), but is not sufficiently restrictive to prevent significant heating of the electron gas. In each collision with heavy particles an electron's coherent motion is randomized. The heating effect, although small, is thus cumulative. From a calorimctrie viewpoint 2.6 Nonlinear effects 91 the electrons are heated by the electric field and cooled by collisions with heavy particles. The power dissipated per unit volume in the electron gas by the electromagnetic wave is ^o>£2, where ar is the real part of the conductivity and E the peak amplitude of the wave field. From (1.3.14) in the limit of high frequencies (oi2»vz; no magnetic field) the power dissipated per electron is then ve2E2j2m^\ v being the effective collision frequency for momentum transfer. The electrons lose energy to the molecules and positive ions at the rate {ImjM^vi^kTe—\kT0\ the factor 2mjM conning from (2.2,9) and Te and T0 being the electron and heavy-particle temperatures. In the steady state, these two terms are equal. Thus, if the wave is to produce negligible heating c2E2 6mlcT m2u>2 M m ' (2.6.2) which is more severe than (2.6.1) by the mass ratio miM. With a magnetic field, or is resonant at the cyclotron frequency for some modes and (2.6.2) must be modified accordingly. The condition (2.6.2) may be put in a crude but handy form for propagation experiments in the laboratory by noting that the intensity of a wave is E%\2rt, where t]X377 ohms is the wave impedance of the plasma medium, and by furthermore assuming arbitrarily that an area of dimension one wavelength is illuminated (Section 4.6.2). Then the total power radiated is of order P~£EAE/2ij, and the criterion for negligible heating is U-!72m2c2kT Me2?} .90 ^ watts, (2.6.3) A being the atomic number of the molecules or ions involved. Note especially that this criterion fails near both the plasma and cyclotron frequencies, where the nonlincarity is enhanced, 2.6.2 Breakdown. The production of a gas-discharge plasma by high-frequency breakdown is well known. This topic has been discussed at length by many authors, including Brown (1956) and Francis (1960, Chapter 4). Since our principal concern is the use of microwaves for diagnostic purposes, we omit further discussion. 2.6.3 Luxembourg effect. The Luxembourg effect is the process whereby an amplitude-modulated high-power wave (the "disturbing" wave) causes a corresponding modulation of the collision frequency and hence the attenuation coefficient of the plasma medium. Thus, this modulation is transferred to a second, iow-power wave (the "wanted" wave) as il passes 92 Collision processes Chap. 2 through the disturbed region. The effect was first encountered in connection with radio-broadcast waves in the ionosphere (Bailey and Martyn, 1934). It is necessary not only that the electron-heavy particle collision frequency be velocity dependent (as it is for most real gases) but also that the modulation be slow enough that the electron temperature can follow it. Extending the calorimetric argument of Section 2.6.1, we see that the temperature variation is given by the differential equation d(kTe) e1 2m (2.6.4) dt 3mw2 v e> w M E(t) being the modulated amplitude of the disturbing high-frequency field. If the functional dependence of v (the effective collision frequency for momentum transfer, Section 2.4.3) on Te is specified, (2.6.4) may be integrated to give the time variation of v and hence of the amplitude attenuation coefficient « (Anderson and Goldstein, 1955). If the "disturbing" wave is suddenly turned on or off, v and « relax to their equilibrium values with the time constant [(2m/Af>]"1 (Spitzer, 1940). Or, if the disturbing wave is sinusoidally modulated such that the attenuation varies over the range a±Aa, a wanted wave passing through a disturbed region of length d emerges with an amplitude-modulation index -\ Aa d. When the plasma is in a magnetic field, the real conductivity is resonant at the cyclotron frequency for certain modes. The evaluation used for u.r in (2.6.1) is no longer correct, and significant heating is caused by a nearly resonant disturbing wave at power levels well below the criterion of (2.6.2) and (2.6.3). Experimental techniques exploiting the Luxembourg effect have been used extensively to measure the velocity dependence of collision frequencies (Rao et al., 1961). It might also be noted that modifications of plasma properties by means other than a disturbing electromagnetic wave (for instance, electron-stream-driven longitudinal plasma oscillations) can modulate a wanted probing wave (Rosen, 1960; Baranger and Mozer, 1961). This aspect is discussed further in Section 2.6.5 and Sections 6.6 and 6.7. 2.6.4 Other nontirtearities. The presence of an electromagnetic wave changes the electron velocity distribution function f(v) which, in turn, may alter the propagation characteristics. These matters may be investigated in detail by kinetic theory methods (Epstein, I960, 1962; Chen, 1962; and Sodha and Palumbo, 1963). We have already discussed the change of collision frequency due to electron heating, the Luxembourg effect. However, in addition, the a- V„/(v) term of the Boltzmann equation (2.4.4) is inherently nonlinear in time, a feature discarded in the expansion (2,4.18). Consequently, if the criterion (2.6.1) is violated, a single wave propagating in such a medium generates harmonics (Rosen, 1961), 2.6 Nonlinear effects 93 We have assumed throughout that the oscillating electric field, and consequently the electron velocity distribution function, are uniform in space. This assumption is invalid when a-c spacecharge is associated with the wave, as is generally the case for the nontransverse-electric waves in plasmas in magnetic fields and for inhomogeneous plasmas (Sections 1.4.5 and 4.4), The assumption also obviously ignores the wave nature of the electric field, a matter of importance even in the linear theory, and discussed at length in Chapter 3. When E and/(v) are space dependent, for one or more reasons, additional terms in the Boltzmann equation are nonlinear (Ginzburg, 1959; Wetzel, 1961; Whitmer and Barrett, 1961, 1962; and Baird and Coleman, 1961). The density modulation, for example, may cause phase modulation of a probing wave, analogous to the Luxembourg amplitude modulation. Finally, we note that the vxB force arising from the wave (a-c) magnetic field is inherently nonlinear, but may usually be neglected for nonrelativistic plasmas. 2.6.5 Incoherent scattering. In conclusion, we discuss briefly the process of incoherent scattering, which arises from thermal fluctuations rather than nonlinearities in the usual sense. Under the influence of an electromagnetic wave, a free electron oscillates and reradiates, a process known as Thomson scattering, the classical analog of the Compton effect (Heitler, 1954, §4j5, 22, and 33). This reradiation from an assembly of initially stationary electrons is coherent and produces the change in phase velocity described macrosc opt call y by the refractive index (RatclifFe, 1959, Chapter 3). At a finite temperature, however, thermal fluctuations in the electron density give rise also to an incoherent scattering of the incident wave. The total power radiated into unit solid angle per unit volume of the scattering medium is ysin3© (2.6.5) where r0 = e3l4?reQmc2 = 2.8 ■ 10"1S meter is the classical electron radius, n the average density of scattering electrons, / the intensity (watts/m2) of the incident wave, and & the angle between wave polarization and direction of observation (Fejcr, I960). There is no frequency or temperature dependence of the total power so long as the incident wavelength is large compared to the debye length. The corresponding amplitude attenuation coefficient of the primary wave, a^^r02n, is negligible for most laboratory plasmas (Sampson, 1959). The frequency spectrum of the scattered radiation is temperature dependent, and may be obtained from detailed calculations, which are especially complicated by a magnetic field (Renau etal., 1961; Farley etal., 1961) . Harmonics of the incident frequency are generated (Vachaspati, 1962) . Experimental measurements of backscatter from the ionosphere 94 Collision processes Chap. 2 have been used to determine electron density and temperature as a function of altitude (Bowles, 1961; Bowles et al., 1962). Modifications in the scattered intensity occur when the plasma is in a nonequilibrium state. For instance, the electron and ion temperatures may differ (Salpeter, 1963) or electron-stream-driven plasma oscillations may be present (Drummond, 1962). A laboratory experiment has been performed by Kino and Allen (1961). The presence of nonthermal fluctuations in the plasma also influences the low-frequency conductivity (Yoshikawa, 1962). A more complete discussion is given in Section 6.7. C HAPT ER 3 Waves in warm plasma 3.1 Introduction The preceding chapters have assumed a cold plasma in which electron thermal motion could be neglected.1 Specifically, the wave nature of the electromagnetic field has been ignored; that is, the phase velocity and wavelength of the wave have been assumed infinite. We shall use the term "warm" to designate the case in which the temperature is considered explicitly but for which nonrelativistic mechanics is still appropriate. The term "hot" will be reserved for the relativistic case. Several new phenomena, described below, appear when the plasma is assumed warm. (I) Spatial variations (gradients) in density and temperature over the wavelength drive particle currents, which are in addition to those driven directly by the electric field. This effect, in addition to modifying the propagation of electromagnetic waves, provides a new class of waves variously known as plasma, electrostatic, spacecharge, or eleciroacoustical waves. Tn suitable limits these waves are longitudinal, analogous to sound waves in un-ionized gases. In this class are the modes commonly referred to as "plasma oscillations" (Bohm and Gross, 1949). In the presence of a magnetic field or density gradients, spacecharge waves may couple to electromagnetic waves, as discussed in Chapter 5. 1 Even in a cold plasma, however, the electron velocity must be greater than the velocity increments produced by ihe electromagnetic field (Section 2.3) if the presence of the wave is not to distort the distribution function significantly. In recognition of this lower bound on velocity, the term "temperate plasma" has been used (Allis, Buchsbauni, and Bcrs, 1963). In a sense, however, this is more an upper bound on the electric field than a lower bound on thermal velocity. 95 96 Waves in warm plasma Chap. 3 (2) A group of electrons having thermal speeds close to the wave phase velocity can exchange energy with the wave by the processes of Landau damping and Cerenkov radiation, processes which on a macroscopic scale take place in the linear accelerator and traveling-wave tube. For electromagnetic waves, a slow phase velocity (high index of refraction) is found only when a static magnetic field is present. The clectroacoustic waves may be slow even without the field. (J) The presence of a static magnetic field introduces a new scale of length, the gyroradius, which may be comparable to the wavelength. Under this condition the relation between current and electric field (that is, the conductivity) is in general no longer a function of a point in the plasma, but depends upon the spatial variation of the field and the past history of the particles reaching that point (Drummond, 1958; Drummond et al., 1961). This chapter summarizes the results of theoretical analyses which take these effects into account. 3.2 Magnetic permeability of a plasma To calculate propagation constants in a cold plasma, we have considered all particle motions explicitly. Thus the plasma represents a continuous medium having the dielectric constant and magnetic permeability of free space and the complex tensor conductivity a defined by J = ct-E. Alternatively, it is possible, for a particular frequency to, to regard the plasma as a dielectric medium having zero explicit conductivity, the permeability of free space, and the complex dielectric constant = 1+- JU)C0 (3.2.1) With the introduction of finite electron temperature in the presence of a static magnetic field, we recognize that the gyrating particles possess magnetic moments and exhibit diamagnetism (Astrom, 1958; Neufeld, 1963). An intuitive approach to including the effects of finite temperature would be to compute an appropriate magnetic permeability, which could then be used in addition to the cold-plasma dielectric constant in the dispersion relation obtained from Maxwell's equations. The magnetic moment of an electron orbiting in a magnetic field is ■IA WW 2 J-mt;, B (3.2.2) where 1 and A arc the current and area generating the magnetic moment, 3.2 Magnetic permeability of a plasma 97 r,, = 27TvJu>h is the gyroradius, and v± is the velocity component perpendicular to the magnetic field.2 The magnetization vector (magnetic moment per unit volume) for a thermal distribution of electrons is thus M=- nkT ' B2 B. (3.2.3) By definition H=—— M / nkT\B (3.2.4) where the coefficient in parentheses is formally the reciprocal relative permeability. Clearly, no simple proportionality exists between H and B, so that permeability is not in general a valid concept. However, in the special case where the static magnetic field iJ0 is large compared to the wave magnetic field, and the two field components are spatially orthogonal, as in propagation along the static field, the relative permeability «m is essentially constant: nkT\-1 I, nkTY ■E b I kT The parameter 2fi0nlcT ~~ B02 (3.2.5) (3.2.6) represents the ratio of material to magnetic field pressure, and occurs frequently in the theory of plasma confinement (Glasstone and Lovberg, I960, p. 52).3 Note that j3 is the square of the ratio of the gyroradius to the wavelength of a free-space wave at the plasma frequency. For plane, transverse waves in a medium with zero conductivity, Maxwell's equations give the dispersion equation (3.2.7) - In passing, we note that the magnetic moment of a gyrating particle is an adiabatic invariant of the motion, a feature exploited in magnetic mirror devices (Glasstone and Lovberg, I960, p. 337; Lenard, 1959). This is essentially a consequence of the conservation of angular momentum. " This parameter, conventionally represented by the ambiguous symbol |9, is not to be confused with the rclativi.slie velocity ratio v/c, nor with the phase propagation coefficient) 98 Waves in warm plasma Chap. 3 3.3 Hydromagnetic calculation of plasma waves 99 where [j. is the index of refraction and tc the dielectric constant, infer in this special case, using (1.4.20), (to>p/q>)a F2= 1 1 + /o^Y kT_ \o>J inc3 Hence we (3.2.8) i + k/w). This naive argument cannot be expected to have the validity of a treatment which considers explicitly the effect of the wave on the electron velocity distribution—that is, considers particle motion and thus conductivity rather than dielectric and diamagnetic properties. In particular, the permeability argument ignores resonance effects when w ~ wh. The correct treatment is outlined in Section 3.4, from which it may be seen that the temperature correction in (3.2.8) is of the right order of magnitude. 3.3 Hydromagnetic calculation of plasma waves The kinetic (Boltzmann) theory for waves of finite wavelength in a warm plasma is mathematically difficult. To provide insight we first attack the problem using the hydromagnetic approximation, mentioned in Section 2.4. This treatment necessarily excludes the phenomenon of Landau damping; therefore, we must require that the value of the electron velocity distribution function be small at the wave phase velocity. In addition, we shall assume only high-frequency oscillations so that the ion motion may be neglected. We neglect collisions and all nonlinear terms. 3.3.1 Moment equations. Given the Boltzmann equation (2.4.4) M dt -r-V-VJ+a-V, 1J KLr' (3.3.1) we multiply through by any function A(y) and integrate over all velocity space. The integrals for the left-hand side are (Spitzer, 1962, appendix) where 0 ¥d3v = St 1« (3.3.2) V.v" V ~ (nvxA) ox (3.3.3) a7- r/Jv = 8vx (3.3.4) «(r, ř)= /(r, v, t)dav (3.3.5) and the bar denotes an average over the velocity distribution as, for instance, JAv)/(r, v, 0 <*3v «(r, /) (3.3.6) If first A(y) is taken to be unity, we obtain the equation of continuity |-ivr.J = 0 (3.3.7) for electrons of charge — e and the macroscopic current density J=~ne\. (3.3.8) The integral (3.3.4) is zero for electromagnetic forces; the integral over the collision term is also zero, since collisions cannot alter the local density. Second, we take A(\) to be the momentum my, and assume that the acceleration arises only from electromagnetic fields E and B, to obtain the equation of momentum transport* --^ + V.*l»-)-fieE-JxB = 0, e dt (3.3.9) in which * is the pressure tensor nwivv arising in the integral (3.3.3), and the collision term has been neglected. Maxwell's equations provide further relations between E, B and «, J. The remaining task, characteristic of the hydromagnetic formulation, is to evaluate the pressure tensor One approach is to approximate it by a scalar pressure For high frequencies, pressure changes will occur adiabatically so that we would expect p = (r,01-'kT)n\ (3.3.10) and therefore V-*^ Vp = ykTVn, (3.3.11) where y is the ratio of specific heats and «0 and Tthe equilibrium electron density and temperature. For strictly longitudinal plane waves the compression is essentially one-dimensional, implying y = 3 (Spitzer, 1962, §3.2). This approximation leads to the correct dispersion relation for the space-charge waves in the low magnetic field limit but does not indicate any temperature perturbation of the electromagnetic waves (Allis, Buchsbaum, and Bers, 1963). A more accurate treatment of the problem is to form an additional moment equation, which may then be rather crudely approximated. Integrating the Boltzmann equation multiplied by the dyadic nivv, we obtain the equation of motion of the pressure tensor. Neglecting 4 In the more general treatment, the corresponding equation for ions is also obtained. The sum of these two equations yields the macroscopic equation of motion; the difference, the generalized Ohm's law relation (Spitzer, 1962, §2.2), The treatment given here is appropriate for high frequencies when ion motion can be neglected, and for weak fields when a linearized equation is adequate. 100 Waves in warm plasma Chap. 3 all nonlinear terms and those involving the magnetic field, one obtains (Bernstein and Trehan, 1960) [V.J1 + VJ + (VJ)'] = 0, the divergence of which is 1(V.*)-^[2VV-J + V.¥J]=0. (3.3.12) (3.3.13) The approximations made in this equation are essentially the assumptions of low temperature and low magnetic field.5 3.3.2 Hydromagnetic dispersion relations. We now assume a plane wave traveling in the direction of the complex vector propagation constant y, and expand the variables in the form J = Ji exp(jwt — y-r) E=Ei exp(y'cuf -y-r) B = B0. (3.3.14) The products of first-order quantities, and of all higher-order quantities, will be neglected. Thus the magnetic component of the wave field may be ignored. Equations (3.3.9) and (3.3.13) yield -J 2? J: -j % (2fiMi + y-yJO*W&k + B0x J, = 0. (3.3.15) e eoi Rearranging, we have where e0m eB (3.3.17) The coefficient of Jj is a tensor resistivity, similar to a'1 in (1.4.28). It may be inverted to obtain the conductivity and dielectric constant tensors. A medium for which the dielectric constant is a function of the propagation constant is said to exhibit spatial dispersion (Neufeld, 1961). Without loss of generality, we take the propagation constant to lie in the x-z plane and the magnetic field along the z axis; then ab = (0, 0, (ob) (3.3.18) 5 We have neglected the magnctostatic field terms, which are or first order, simply because of algebraic complexity. Therefore the solution will be correct only to first order in temperature for infinitesimal field and, alternatively, for any field at zero temperature. Terms involving the product of temperature and magnetic field will not in general be correct. This restriction does not apply to the kinetic theory results quoted in Section 3.4. J.J Hydromagnetic calculation of plasma waves 101 where ji is the complex index of refraction and £, £ are the direction cosines of y. The coefficient of Jx in (3,3.16) becomes in matrix form 1 where -s(i+2n JY -JY -5 0 -mm " o 1-8(1+2£2). (3.3.19) (3.3.20) Y=ea„lot 8=fj.2kTlmc2. It is to be noted that the parameter S is essentially the square of the ratio of electron thermal velocity to wave phase velocity, which we require to be small (see footnote 7). The reciprocal of (3.3.19) is the conductivity -J 2h (l-3)2(l-35)- K2[1-S(1+2h] 1-5(2+20 ;y[i-s(i+20] + S2(l+2h -/r[l-S(l+2£2)] 2^8(1-8) 2^8(1-8) 1-4S + 3S2 J2Y&& -./2 ms l-3(2 + 2|2) + S2(1+2p)- Yz (3.3.21) where the coefficient is understood to multiply each clement of the matrix. The corresponding dielectric constant is k = 1 + JU)€0 (3.3.22) These expressions reduce to (1.4.56) to (1.4.60) in the limit 8 —> 0. The propagation constant is determined by substitution of (3.3.21) in the electromagnetic wave equation VxVxE+ 1 t)2E (3.3.23) which for a wave of the form exptjW — y • r) in a homogeneous medium reduces to o)2 yX(yxE)-^ii.E=0. (3.3.24) This is a system of homogeneous equations; the determinant of the coefficients must vanish to ensure a nontrivial solution. For y in the x-z plane, as given by (3.3.18), this condition is ťtt-S. 1-M2Č2-.v, =0, (3.3.25) 102 Waves in warm plasma Chap. 3 3.3 Hydromugnetic calculation of plasma waves 103 where the elements sxx, sxy,... are the respective elements of the matrix portion of (3.3.21) multiplied by (cjpH2 (l-S)z(l-3§)- X'![l-S(J+2£2)] For propagation along the field (i = 0, £=1), we obtain from (3.3.25) the dispersion relation kw2 (uip/w)2 (i-«)-K/w)J 1- (3.3.26) where S = p.2kTjmc2. The first two factors are the circularly polarized electromagnetic waves; they reduce to the results obtained in Section 1.4.1 in the limit 8^0. To first order in temperature and magnetic field (see footnote 5) 1- (o>„l)2]S + 382)^0. (3.3.31) For small S this equation is of the form (A + a)nz-(B + b) = 0, (3.3.32) " Since S contains /t2, this bracket is in fact a cubic in ji2, one of whose roots violates [lie assumption S« I and is extraneous. 104 Waves in warm plasma Chap. 3 where a«A and b«B, and has the formal solution „ Bl, a b \ (3.3.33) ^ mm The perturbed electromagnetic wave thus has the index f[i-(M>Msf-(MtH1if,_KA2 XT where we have ignored magnetic field in the temperature term in accord with footnote 5. Since the index of refraction for the spacecharge wave is relatively large (although bounded by the requirement S = ^2(kTjmc2)« 1), we obtain an approximation to the dispersion relation by equating the coefficient of ^ to zero. Since (a>„/cu)2« 1, this coefficient, in the form given in (3.3.29), may be rewritten within our approximation (i-8xi-3o)[i--KHa]' whereupon the dispersion relation becomes (see Fig. 3.1) (l-3S)[l- km8] - km2 sl-38-km2 - km2=o; . 1-km2-km2 r1 or 3Á-T//WC2 3kT 12-n (3.3.35) Mathematical subtleties are involved in obtaining this dispersion relation in the limit where both temperature and magnetic field are taken to be small quantities (Gross, 1951 ; Bernstein, 1958). We note that this space-charge wave occurs only for frequencies near {m2 + w2yA xH^-P (1 -cos Ys)-H2^2 h = p,\kTlmcz) f^mjm C=$in0 £ = cosO 106 Waves in warm plasma Chap. 3 As in Section 1.4.8 by using the unitary matrix transformation where á' = -XJaU' "I -j o - u - ' V2 ' I 1 0 " u = -U V2 1 j 0 _0 0 \/2_ ./ —j o .0 o (3.4.7) (3.4.8) we transform to a rotating coordinate system in which the conductivity tensor is symmetrical and is diagonal for the special case of propagation along the field. The result is (Bernstein, 1958): At rr - ~^- ±>*V = J0 [ 1 - y2 (i - COS ft)J xexP[-y (i + (i-cosy*)-iw] * P (1 -t2oS2) expMs)] ds <*> Jo ct„.= 0>,= V2 (3.4.9) The conductivity elements can readily be converted to dielectric constant elements by the usual relation k = 1 + 6jjw€0. The propagation constant is obtained from the dispersion relation (3.3.24) and may be written in the form A*p.i-B*,i2 + C* = 0, (3.4.10) where is the complex refractive index and A*=Kxxe+k,.y+2KXZa C*-l>) (3.4.11) with |*| denoting the determinant of the dielectric constant tensor. It must be remembered that the refractive index is contained implicitly in the 3.4 Kinetic (Boltzmann) theory of waves 107 dielectric constant elements, so that (3.4.10) is in general transcendental. Expanded to first order in the quantity 8=fi.'-'(kTjmcz), the elements of the dielectric constant tensor are, in fixed coordinates (Sitenko and Stepanov, 1957) :7 oj-JV w (co —jv)2 — 0)b2 «.»[(«,-»» +3a.,,'] , [(« -»2-)2 + 8co,,2] ai to —_/v 3c -a+ř2s (-ji>)2-wb cu (a) — 7V)2 — co,,2 K^-»2-^2]2 2cocop2(co —/v) (ío-/V)2-4coí, _ coa)pacO),[3(cu -»2 - )a[(w _>)a _ oj)j2]2 (3.4.12) The integrals of (3.4.6) are in general complex, even when collisions are neglected (v -» 0). However, the first-order expansions (3.4.12) are, neglecting collisions, pure real or imaginary in such a manner that propagation is either unattenuated or evanescent. Noneollisional damping has again been excluded. We now catalog indices of refraction for the principal electromagnetic waves. ''The expansion parameter 8 = ^(kT/mc1), which appears in both hydromagnetic ami kinetic theories of warm plasmas, using nonrelativistic mechanics, is directly proportional to the square of the ratio of thermal velocity to that of light. If a lirsl-order expansion of a nonrelativistic theory is to be valid, we must require />-'»l. This is of course precisely the condition under which the warm plasma results are significantly different from the cold case. 108 Waves in warm plasma Chap. 3 3.4 Kinetic (Boltzmann) theory of waves 109 3.4.1 Propagation along the field (£=0, £ = 1) (Platzman and Buchsbaum, 1962, 1963; Willctt, 1962). tf. r = (lii + fliuvjc j - =; J" dsexpi-j\l±Y)s] j" =j (" ř/.vexp[-y(l± Y)s-{SSz] where i>„, if. Ihn)' • and /«'•'/&/." 110 Waves in warm plasma Chap. 3 3.4 Kinetic (Boltzmann) theory of waves 111 where/(X) is the distribution of electron velocities in the axial direction (not necessarily Maxwellian). For small velocities (low temperatures), the integrand can be expanded to obtain J eo±wb+{i±to,,) (to + ojbY L to ± tob — top2/co \ \co) „=0, T^O: kT to(to + u>b) (to + to,,)3 mc2 (3.4.16) We note that the corrections are in opposite directions. In the most important case, near cyclotron resonance in the right-hand mode (Fig. 3.2), approximate cancellation occurs when 3 /v\2_ a IcT 4 W mc2' (3.4.17) Assuming electron-ion collisions, using (2.5.23), and approximating p2 by to„2/to(to„-to)»l, we find the finite temperature effect dominates when the inequality to (/c7'[eV])4 > 10- (3.4.18) o)b — an «[cm"3] is satisfied. Changes in the wave attenuation are discussed in Section 3.5. 3.4.2 Propagation across the field (f =1, £=0). Etf || B0 (Dnestrovskii and Kostomarov, 1961): A2=(H:5i!)c=0 ■l^i f " exp[ - /(1 ->/a>)s - (8/ Y2){\- cos Ys)] ds o>" Jo -/^exp(-S/y2) ^ U^Y*) SKP[-j(l-nY-jvjHin(s/ni WW" A (1->H2-«272J -^-exp(-S/r2) < f !»(«-» ( 1 + AT (fO-»2-0)(, [(to->)2-to/][(to->)2-4to, » 1 1 + !-yV)2- W] me2 J In(x) = I _ n(X) is the modified Bessel function of the first kind, and use has been made of the identity oo exp(xcosj')= 2 MN exP((3.4.20) Er/ ±BV: p2_{ kxxk!iv "b Kq\ „ f[l-(top/to)a32-(tob/to)2-l I ~o\ i-K/tof-KH2 // 1 + to/[(to2 - to,2)2 + o>b2{lto2 - 4to/) - 8to, *] kT_ ,.2 (3-4.21) (to2—4to„2)(to2—top2 — to2)2 mc The simplification of ignoring collisions in the latter form is made because of algebraic complexity. The ordinary (parallel polarization) wave is no longer independent of magnetic field but, rather, shows resonances at all integral harmonics of the cyclotron frequency. Likewise, the extraordinary wave shows both the expected upper hybrid resonance at to2 = a>b2 + top2 and additional resonances at harmonics of toh. The results obtained here may be compared with those obtained from the hydromagnelic calculation of Section 3.3.2. Further physical insight into the origin of the high-temperature modifications may be found in the discussion of Drummond (1958). 112 Waves in warm plasma Chap. 3 3.4.3 No magnetic field (dib — 0): F2 = (*)*>,,= Q =i v r o(to-» L 1 + kT {u>—jv)2 *l mc2 ' (i «ŕ i /kry -jV)4 ^ ^mc3,) [J._^2ľ_|/[l+^ L ^(^-jľjj/ L (to—j ■ *s —a (3.4.22) Since for this case the refractive index never exceeds unity, the correction is numerically significant only for temperatures requiring relativistic mechanics, as discussed in Section 3.6. 3.4.4 Plasma or electrostatic waves. For propagation along the field the dispersion relation is I +vjj™0 = 0. (3.4.23) From (3.4.6): (*«)<=<>= (5)" (1 -Si3) exp[-y(l-j^-Us^j ds to(w-y^) L (to-^)3 (to-y^)1 1 — L (<*>-jv) J cu(u) — /jj) Therefore to lowest order in temperature 3/cT" 1 — »1 (3,4.24) (3.4.25) in agreement with (3.3,26). There is no magnetic field dependence, since all coherent particle motions are parallel to the field. A corresponding calculation for propagation across the field is algebraically complicated. Bernstein (1958) has shown that the dispersion relation for the plasma wave in an arbitrary direction can be put in the form 1 +(to/c0j,)E§ = / J exp[<£(.v)] ds n ■= — oj x J" exp[-; ("^^^tf) ds, (3.4,26) 5,5 Landau damping and wave absorption 113 where 8=jx2(kTjmc2), X Kil° K( to c 2rc ftia«Kra 2k/* c The condition x2«r2 15 a fortiori satisfied if k^oc,2. (3.5.3) 114 Waves in warm plasma Chap. 3 The integrals of (3.4.6) are in general complex. This may be seen most easily in the formulation (3.4.14) in which the integrand has a simple pole at V;= — ( f (to ± u>b)c (3.5.4) where the one-dimensional distribution function is in the Maxwellian case m)=(^^'\M-nu/j2kT). (3.5.5) Thus, for low temperatures, the attenuation constant is, from (3.5.3) (Sagdeyev and Shafranov, 1958; Cullen, 1960), where < ] to(co + to,,)J cxp [ ("±Q2 (mc°-\ [ 2toV I kTj (3.5.6) This case has been studied in detail by Scarf (1962), Willett (1962), and Platzman and Bucbsbaum (1962, 1963). This result is an example of noncollisional damping known gcnerically as phase mixing or line scale mixing (Gershman, 1960). In the context of plasma oscillations, the phenomenon is usually called Landau damping. It is closely related to the inverse process of Cerenkov radiation. Physically, it arises because of synchronism between particle and wave velocities. Electrons in one region of the wave move, as a result of thermal motion, into adjoining regions where the phase of the wave is different. On the average, particles riding with the wave extract more energy from the wave (linear accelerator effect) than they give to it (traveling-wave tube effect); this energy is then gradually shared with nonsynchronous particles by collisions. Thus, thermal energy is increased at the expense of wave energy (Dawson, 1961). In the presence of a magnetic held, the effect may also be considered as arising from the doppler shift of the wave frequency as seen by the moving electron, and is frequently called cyclotron clamping. This dissipative process was inherently excluded in the hydromagnetic calculation of Section 3.3.2 and discarded in the first-order expansions of (3.4.12). The application of theory to experiment is complicated by the fact that the strong interaction between wave and synchronous electrons distorts the distribution function near the wave velocity, whereas collisions 3.6 Relativistic plasmas 115 tend to restore Maxwellian distribution. Hence, the magnitude of noncollisional damping depends inherently upon the collision rate (Platzman and Buchsbaum, 1961). In cases where a highly non-Maxwellian velocity distribution is somehow maintained, such that there are more fast particles than slow in the neighborhood of the wave velocity (implying a "double-humped" velocity distribution), then wave amplification or growth can occur, as calculated in Section 5.6. Examples are the traveling-wave tube and various plasma instability processes (Drummond and Pines, 1961). Most theoretical treatments of noncollisional damping have been concerned with the longitudinal plasma waves, rather than the transverse electromagnetic waves (Landau, 1946; Bohm and Gross, 1949; and Kildal, 1961). Furthermore, the more common problem has been to assume an initial disturbance and follow its decay in time; the frequency to, rather than the propagation constant y, is assumed complex. The problem of determining a complex y inherently requires consideration of driven waves. To contrast with (3.5.6), we may compute the ordinary collisional attenuation, in the special cases of propagation along the magnetic field (left and right circular polarizations) or across the field (ordinary mode, putting co« = 0). From (1.4.21), X°>, C 2c(a> ± iob)''-{(x) ±v2j(kT)%, according to (2.5.23). For given co, a>r, and tab, there will exist some critical temperature below which collisional dissipation dominates and above which phase mixing dominates. Because of the exponential term in (3.5.6), this critical temperature is in the kilovolt region for electromagnetic waves in most laboratory plasmas, except near the gyrorcsonance. It is worthwhile to note, in passing, that the real and imaginary components of the propagation constant are linked by general considerations of causality, the mathematical formulation of which is known as the Kramers-Kronig dispersion relations (Kittel, 1958; Leontovich, 1961; and Pradhan, 1962). 3.6 Relativistic plasmas Since the expansion parameter 8 = p.\kTjmc2) which appears in the kinetic treatment of "warm" plasmas, using nonrelativistic mechanics, is of order (i%vjc)9, it may be necessary to employ relativistic mechanics if one is to retain terms of order (kTjmc2) for fi*~ 1 (Silin, 1960-1962; Graben, 1963). Also, synchrotron radiation processes imply that the interaction between a wave and plasma at multiples of the gyrofrcquency should be 116 Waves in warm plasma Chap. 3 enhanced for a relativistic plasma (Beard, 1959). Imre (1962) has considered the problem of electromagnetic wave propagation in relativistic plasmas in detail. As an example, he obtains for propagation along the field, to first order in kTjmc2, 1 - a»(oi + to,,) kT' 2(a> ± cufc) mc'2\ 1 + kT (to ± cob)a mc2 (3.6.1) which is to be compared with (3.4.13). Johnston (1962) has developed weakly relativistic expansions, obtaining, for example, for electromagnetic waves in a plasma with no magnetic field > iop, the plasma frequency (except for special techniques exploiting a static magnetic field or a detailed independent knowledge of the density profile). Thus, the beam technique is most readily analyzed when .(c/Z))a (4.1.1) where D is the dimension of the plasma. The first of these two independent conditions permits convenient simplifications in the analysis by avoiding the plasma resonance^ the second is essentially a diffraction condition which permits reducing the problem of propagation of a finite beam of electromagnetic waves through a finite plasma to a one-dimcnsional, plane-wave problem, as a first approximation. The propagation constant of a microwave beam in a plasma has been 117 118 Wave propagation through bounded plasmas Chap. 4 shown, in Chapter 1, to depend upon the magnetic field, electron density, and collision frequency, and indirectly upon the temperature. The following basic arrangements, sketched schematically in Fig. 4.1, are useful in the case of high-temperature, highly ionized plasmas (that is, f«cop). (7) Simple transmission or reflection. For electron densities nnc it is opaque and totally reflecting, where nc = (^r,mle2)ajz is the critical density.1 The transition between 1 In the presence of a magnetic field, the effective critical density may be altered. However, the situation is qualitatively unchanged. / Source 1 \ Detector lfaj. Transmission Detector Source < 1(6). Reflection Source Phaseshifter Attenuator 2. Phase shift FfG. 4.1 Elementary microwave observation schemes. 4.1 Introduction 119 these conditions is sharp. Thus, in principle, this elementary technique indicates whether the plasma density is above or below the critical value. Measurement at a given frequency is capable of determining only one value of density. The sharpness of the transition implied by the sudden change in the attenuation coefficient is not realized in practice because of the following factors. (a) For densities below but approaching critical, the dielectric constant discontinuity at a sharp boundary produces an increasingly strong surface reflection*(and corresponding reduction in transmission). (b) If the plasma is only a few wavelengths thick, interference effects occur between the surface reflections. (c) Inhomogeneous density distributions are not averaged in a simple manner. (d) Refraction and scattering by the plasma occur because of inadequacies in the one-dimensional, plane-wave approximation. If the plasma density is far above critical, an impinging signal is strongly reflected at the boundary. Therefore, motions of the effective boundary produce doppler shifts in the frequency of the reflected signal. (2) Phase shift (microwave bridge or interferometer). If the signal from an auxiliary transmission path, with adjustable amplitude and phase elements, is balanced against the primary transmission signal to give a null in the absence of plasma, the output signal of the waveguide (hybrid) junction is a measure of the attenuation and phase shift in the primary path due to the plasma. In the fully transparent region of electron density, where n«nc, a detected signal represents only phase shift which, in turn, is essentially a function of electron density only. Since the shift in phase can be calibrated, one has a continuous measurement of density between the upper limit of serious amplitude effects in the transmission path, and the lower limit of detector sensitivity. This technique is ideally suited to the observation of density as a function of time. The propagation of the microwave beam through the bounded plasma is most readily analyzed in two limiting cases: first, the gradual boundary, with density varying slowly over a wavelength, to which an adiabatic analysis may be applied; and, second, the sharp boundary which can be attacked as a boundary-value problem. A formally similar situation occurs in quantum mechanics, in which the first case is known as the WKB approximation (Bohm, 1951). The usual geometrical optics limit partakes of both the above lirnTts. It neglects reflections at the "sharp" boundaries which separate regions of different propagation characteristics and, thus, can be self-consistent only for plasmas large compared to a Wavelength. The models of plasma geometry that are most useful for 489 120 Wave propagation through hounded plasmas Chap. 4 4.2 Simple adiahatic analysis of a plasma slab 121 analytical purposes are the plane slab and the cylinder. We recognize, however, that most experimental plasmas will fail to conform exactly to these limiting cases and simple geometries. 4.2 Simple adiabatic analysis of a plasma slab While most experimental situations approximate cylindrical symmetry, it is often possible to treat the plasma as a slab illuminated by plane waves, and thereby reduce the problem to one dimension. We can further simplify the situation by assuming that plasma properties vary slowly near the boundaries so that reflection and interference effects are negligible, the adiabal ic approximation. 4.2.1 Average electron density. For a high-temperature, highly ionized plasma, for which v2«u>2, dissipative attenuation is small. For simplicity, we here neglect magnetic field effects; they may be included readily by substituting the appropriate propagation formulas from Chapter 1. The phase constants for vacuum and plasma are, respectively, ft 2rr 2VÁ 2n A A (4.2.1) The phase advancement introduced by the plasma in a transmission path is then, in the adiabatic approximation, A=- j(B9-fi0)dx -JH'-fDT*-. <«-2> wherethe integration is carried out along the direct path from transmitting to receiving antenna. To (irst order in njnc, (4.2.2) becomes A n«nc Art, «2 jii(.x) dx jn(x) dx. (4.2.3) 2eamcw Thus, for n«nc, the phase shift is linearly proportional to the electron density averaged along the propagation path. For a path length L we can write the average density « = ľ m J n dx 2<ž0mc wA /7[cm-3] = 118.4 e" L t»/27r[cps] zl[rad] L [cm] (4.2.4) Assuming this first order approximation, we may evaluate the dynamic range of average densities which can be measured. Because of dissipative and nonlinear effects, (4.2.5) where f is a fraction which in a practical case might be 1 /3. The minimum measurable density is 2e0mc u> Amin --7- (4.2.6) "min —" wherezl^,njJ1 is the minimum detectable phase shift, which depends upon the detector noise level and system stability. Hence 1 (4.2.7) The range of measurement scales with . 4.2.2 Adiabatic measurement of density profile. It is clear that in order to evaluate an electron density distribution from "free-space" phase-shift measurements we must know or assume two of the following three parameters. (/) The shape function or profile for the density distribution along the transmission path (for example, rectangular, trapezoidal, cosinusoidal, etc.). (2) The index of width (thickness) of this shape function. (3) The numerical density coefficient of the shape function. In the first-order binomial expansion, given above, we must assume from other considerations a characteristic thickness of the plasma; we then obtain a legitimate average electron density over this thickness. If, 1.0 0.5 1 ! 1 1 4 t 1 J-—r Linear___ / approx.' / Jr / / / / / / / / / / / / — // — // // // / - / / i i i i i i i 1 1 0 0.5 1.0 IK;. 4..t Universal density/phaseshifl curve for uniform density. See Fig. 6.20 for phaso-shifl curves for several spatial distributions. 124 Wave propagation through bounded plasmas Chap. 4 however, we do not restrict our consideration to this first-order case, but still retain the adiabatic approximation, we can (/) expand the Atf> integrand to higher orders, in which case the integrals obtained arc higher-order averages of the distribution function (for example, \na(x) dx); or (2) integrate A directly using an appropriate distribution function. In either case, a meaningful average electron density is not obtained without an independent knowledge of the distribution function, since the phase shift is not linear with density, and the method becomes less useful for the quantitative measurement of even average densities. If, for example, we assume a constant electron density (that is, a rectangular profile which, incidentally, is somewhat contradictory to the adiabatic assumption), the integration is trivial, and we obtain a parabolic dependence of density on phase shift n \L 2tt) \L lit) (4.2.10) Figure 4.3 is a universal graph of this relation. Since the phase shift introduced by the plasma sample is, in general, a nonlinear function of electron density, we obtain information on the distribution of density (profile) by making simultaneous measurements at different frequencies and/or with different polarizations with respect to a magnetic field. We can expand the integrand in (4.2.2) (assuming no magnetic field), 277 dx 1 pn{x) I p2ri\x) 1 p3n3(x) 5 p*n*(x) ~ -5— + 7~ -1--f ~ -r--1-- 16 128 + ...]=* (4.2.11) where p = e2je0m. Note that the series does not converge rapidly. As an example, consider two measurement frequencies for which w2 = 2o>, 4* =| £ jm dx+\g jn*ix) dx + ± g \m ^* • ■ £ H dx+U dx+TT2£? J"3W dx+ Therefore ^,-2^ = 24, \>^x)dx + ^ jn3(x)dx+. (4.2.12) (4.2.13) 4.2 Simple adiabatic analysis of a plasma slab 125 and we obtain for the first two averages of the distribution £ j**=4{[^a-i (^-24«] £8 J> dx+...} (4.2.14) 5 J* dxJT {^-2A^-iTe $ J* *- • • •} (4-2-15) Cco° The usefulness of this approach is limited by the accuracy of the differential measurement A2. When this quantity can be successfully measured, (4.2.14) provides a refined evaluation of the average density and (4.2.15) an estimate of the mean-square density. Procedures for obtaining profile information have been developed by Motley and Heald (1959) and by Wharton and Slager (I960). Wharton and Slager use only the magnctic-lield-independent parallel-polarization case. Their data-reduction procedure is to calibrate the peak electron density by means of the cutoff of a "low-frequency" wave, and obtain information from the simultaneously observed phase shift of a "high-frequency" wave. Motley and Heald, using different polarizations, calibrate the average density with the high-frequency wave, infer profile from the low-frequency wave. Because of the greater phase-shift non-linearity of the perpendicularly polarized wave near cyclotron resonance, the multiple polarization technique, when applicable, is somewhat more sensitive. The Wharton and Slager technique provides profile information only at the instants of time for which cutolf occurs; the Motley and Heald technique is limited to situations where the cyclotron frequency is comparable to the plasma frequency and is accurately known. Both methods benefit from additional phase-shift data channels at other frequencies and/or polarizations, at the expense of instrumentation and data-reduction complexity. Neither method is able to distinguish a hollow discharge from a peaked one. Experimental applications of these principles are discussed in Sections 6.4 and 6.5. 4.2.3 Reflections from cutoffs and resonances. Cutoffs, at which the index of refraction fx —=► 0, and resonances, at which /i -*■ 00, occur for certain combinations of frequency, density, and magnetic field. When a wave propagating in an inhomogeneous plasma impinges upon regions having these special characteristics, reflection and absorption must be considered even in the adiabatic approximation. Near the cutoff, the wavelength grows large, while near the resonance the wavelength becomes small. In both cases, the groyp velocity goes to zero. The analysis of this situation is formally identical to that resulting in the so-called turning-point connection formulas of the quantum-mechanical WKB approximation (Schilf, 1955). It can be shown that in the case of a cutoff the wave is 126 Wave propagation through hounded plasmas Chap, 4 4.3 The slab with sharp boundaries 127 reflected from the anomalous region with little dissipation (Denisov, 1958; Stix, 1960). The external behavior is thus very similar to that of a sharply bounded, high-density (n > nc) plasma. In the case of a resonance, however, the wave is largely absorbed. This distinction is of considerable significance for both reflection-type microwave probing measurements Vacuum ■•Bi Hi Er" fllr x = 0 Plasma ke) / Vacuum Ei Iii 1 = 0 (a) Plasma m E2 Vacuum m m FIG. 4.4 Reflection and transmission at sharp boundaries, (ft) Vacuum-plasma interface. (6) Plasma slab. and thermal radiation measurements, as well as for the nondiagnostic question of plasma heating by electromagnetic radiation. It will be noted from the graphs of Chapter I that, in general, for a given magnetic field, cutoff occurs at a lower density than the resonance. Thus, characteristically, waves entering the plasma from outside are reflected before reaching the resonance. The resonance may, in some cases, be made accessible by allowing the wave to enter the plasma in a region of high magnetic field (generally such that the cyclotron frequency «>„>«)) which then decreases spatially within the plasma, so that the resonance is approached from the high-field side. In Section 6.5.4 an experiment using this technique is described. A situation of this sort has been exploited in the "magnetic beach" geometry for the dissipation of ion-cyclotron waves (Stix, 1958). If the regions of cutoff and resonance are close together within the plasma, relative to a wavelength, it may be possible for a sort of "tunnel effect" to occur in which the resonance region extracts energy from the evanescent wave passing through the cutoff. Tunneling or "bridging" may also take place by mode conversion processes (Ratclilfe, 1959, Chapter 17). Stix (1960, 1962) has shown that at a resonance high-temperature and ion-mass effects may reduce absorption, increase reflection, and excite other plasma modes. 4.3 The slab with sharp boundaries We again consider the interaction of a plane wave with a slab plasma. However, in contrast with the adiabatic case of Section 4.2, we now assume a homogeneous plasma with sharp boundaries, that is, the transition between vacuum and uniform plasma occurs over a distance much less than a wavelength. There exists a well-defined reflection coefficient at each interface, and reflection and transmission coefficients are determined by boundary conditions on the wave fields at the interfaces. Consider first the single interface of Fig. 4.4a. Waves traveling to the right are represented by the phase factor exp(/a>/ — yx), and waves to the left, by cxp(j(vt + yx) where y — a+j^={jwjc)k 1/5 is the complex propagation constant. In the case of a plasma, the complex dielectric constant H, and hence y, are known functions of electron density, collision frequency, magnetic field, etc., as developed in Chapter I, In accordance with Maxwell's equations, the magnitudes of the electric and magnetic wave fields are related by the wave impedance (4.3.1) with respective polarizations as shown in the figure. The wave impedance t) is, in general, complex on account of S, Since the waves arc transverse 128 Wave propagation through bounded plasmas Chap. 4 4.3 The slab with sharp boundaries 129 and there are no surface currents at the interface, the boundary conditions require that E and /-/ are continuous across the interface. Therefore the wave amplitudes, in the notation of Fig. 4.4a, are related by El+ET<=>Ei Ei-Er = ii^El. (4.3.2) It follows that the (complex) amplitude reflection and transmission coefficients are, respectively, „ Er 1-P Et 2 (4.3.3) Et l+ffl ý + rio where rj0 = 377 ohms is the wave impedance of free space. Note the significance of the wave impedance that there is no reflection when the impedances of the two media are equal.2 The single-interface power reflection and transmission coefficients are, respectively, (l-^+x2 (i+m)2+x2 t=l-r = p.\Ť\2=- 4p. "(l+,*)2 + x2 where p,—jx = ><''*= ~jycjo>, and the voltage standing-wave ratio is3 (4.3.4) VSWR = 1-1 1 (4.3.5) 2 In a more general (nonplasma) case with the relative permeability k,„ different from unity and perhaps also complex, then T? = (K,nft0/«„)!». Reflection at the interface between two media is suppressed so long as the ratio k„Jk is the same for both media, even though k and kn themselves change by large factors. This effect is exploited in the design of microwave-absorbing wall coatings in which both k and k,„ have imaginary (lossy) components (see Chapter 10). 11 When the imaginary component of k is negligible, the VSWR = I7kV4 = 1/^=i)/i)0. The positive sense of polarization of the reflected wave has been chosen arbitrarily for the ease *.<■!, If k> I, the sense of £V is reversed and VSWR = kxA = n = i)0/r). We note, in passing, a convenient procedure for calculating the maximum transmission loss due to reflection. From standard transmission-line theory the maximum VSWR from two discontinuities is the product of the respective VSWR's (and the minimum, the quotient). Thus, the maximum transmission loss due to reflection from a slab can be obtained from standard charts assuming a single discontinuity with VSWR — i ri<\ This procedure applies only if I here is no dissipative loss between discontinuities. The situation of practical interest is that of the slab of Fig. 4.46. By setting up boundary conditions similar to (4.3.2) at the two interfaces (or, alternatively, summing the infinite series of internally reflected waves), one finds (Stratton, 1941) the amplitude reflection and transmission coefficients _-exp(-2y/c)rf] 'M, \~'ŕ exp(-2ýrf) (4.3.6) (4.3.7) md the power reflection, transmission, and absorption coefficients are R- r{[l -exp(-2« is the phase angle of p— \p\ exp(y'f) and 2x ) (4.3.8) (4.3.9) (4.3.10) rv- cosi/i = 1-m2 (4.3.11) It is to be%noted that the coefficients (4.3.8) to (4.3.10) are oscillatory functions of slab thickness d (or of frequency w) as a result of interference of internally reflected waves. Likewise, the phase of the transmitted wave, which may be calculated from (4.3.7), is perturbed by interference. As a simplification, we may assume that the reflected waves are incoherent, thereby suppressing interference effects, and obtain4 1 Interference is suppressed by considering only power relations. The fraction r of the incident wave is reflected at the lirst surface of the slab, the fraction a(\—r) [where p/aj)2 = n/nc ® FIG. 4.5 Power reflection, transmission, and absorption coefficients, from (4.3.8) to (4.3.10), for a homogeneous slab plasma four wavelengths thick (>/i0H (4.4.1) VxH=yW0k-E (4.4.2) V.(ft-E)=0 (4.4.3) V-H=0. (4.4.4) Taking the curl of (4.4.1) and using (4.4.2), we obtain the wave equation for E in the form . .2 VxVxE-^ k-E. 0. The vector identity Vx VxE=V(V-E)-equation for E as (4.4.5) v"2E allows us to rewrite the 2 V2E + ^ ic-E = V(V.E), (4.4.6) 4.4 Inhomogeneous plasmas 133 where in general the term on the right-hand side cross-couples the three components of E. Similarly, multiplying (4.4.2) by R"1, then taking its curl and using (4.4.1), we obtain the corresponding wave equation for H in the form VxIk-^VxH)]—g H = 0. (4.4.7) Thus, either anisotropy or in homogeneity causes the wave equations for E and H to be different and to contain terms which cross-couple the scalar field components. In the nonhomogeneous case, the difference arises physically from the fact that the wave impedance (that is, the ratio of £ to //) changes even when the Poynting vector (the product of E and H) is approximately constant. From (4.4.6) and (4.4.7) one can deduce the nature of initially plane waves for various assumed forms of dielectric constant, directions of inhomogeneity, and directions and polarizations of the waves (Bachynski, 1960). The results are summarized in Table 4.1. For instance, even in the absence of a magnetic field, a wave propagating perpendicular to the density gradient is no longer transverse electromagnetic (TEM). 4.4.1 Isotropic inhomogeneous plasmas. In the special case with no magnetostatic field and consequently an isotropic, scalar dielectric constant ii, (4.4.6) becomes V2E + ^r kE-I-V m = 0. (4.4.8) If, furthermore, we assume that the wave is initially plane and transverse and a changes only in the direction of propagation, then (V£)-E=0 and the wave equation reduces to d*E dx2 -t^k(a-)E=0. (4.4.9) Indeed, the adiabatic approximation of Section 4.2 is simply a first-order solution of (4.4.9). For the same special case, (4.4.7) reduces to r/2H to2 „. . 1 dk dli ,. . 1^~VT2K{X)^W)TX^ (4A10) the magnitudes of E and H being related by (4.4.1) as W±AM, (4.4.11) (i)fi0 ax If an effective propagation constant y(x) is defined such that (Ostcrberg, 1958) " , ?<*)=4f* (4A12) 134 Wave propagation through hounded plasmas Chap. 4 4.4 Inhomogeneous plasmas 135 then (4.4.9) requires that y satisfy the Riccati differential equation5 (4.4.13) 6 In a homogeneous medium (4.4.13) gives the familiar result y= ±JKVtI) = 0.25 - 0.64 - 1.44 - 4.0 - ---- x = L 0.5 0.5 l/x 1.0 1.0 1 1 1 1 - \ \k (*>!) = 0.25 V \ - - \ 4.0 ___ - N/No^ ---^ i.m\\ 1 1 1 1 FIG. 4.7 Amplitude reflection and transmission coefficients for a linear-ramp variation of electron density n, as a function of ramp length L for real dielectric constants of the form k = I —///»,.; A is free-space wavelength. (Reproduced from Albini and Jahn, 1961, by courtesy of the Journal of Applied Physics.) 136 Wave propagation through bounded plasmas Chap. 4 may be solved directly in terms of Airy functions and computations made for linear ramp or trapesoidal profiles (Albini and Jahn, 1961; Wort, 1962). Figure 4.7 illustrates the dependence of reflection coefficient on ramp length and dielectric constant. Numerical calculations for other simple profiles have been made by Taylor (1961), Klein et al. (1961), and Hain and Tutter (1962). Interference effects, arising between reflections from the two sides of an inhomogeneous slab appear to be much more pronounced in the amplitude and phase of the reflected wave than for the transmitted wave. A somewhat similar problem has been considered in connection with tapered waveguides (Johnson, 1959). 4.4.2 Anisotropic inhomogeneous plasmas. In more general cases it is usually easier to deal with the magnetic vector, since it is always solenoidal. Once H is found from (4.4.7), E may be obtained from (4.4.2). Consider as a somewhat more general special case an inverse dielectric tensor in the form 0 0 0 0 (4.4.15) which is appropriate to a cold plasma in a magnetic field directed in the z direction. Further assume that the elements of ft"1 are functions of x only and that propagation is in the x direction with H-polarization alternatively in the y or z direction (ordinary or extraordinary waves, respectively). Expansion of (4.4.7) indicates that the magnetic field remains transverse for both cases, whereas (4.4.6) indicates that the electric field is transverse only for the ordinary wave. Assumption of a space-dependent, effective propagation constant analogous to (4.4.12) „, , 1 dll leads to the differential equation for y(x) analogous to (4.4.13) 4 (^y) + rc-^ + ^ = u, dx (4.4.16) (4.4.17) where «c"1=Kn~1(.r) or *ci_1(x) for the ordinary and extraordinary wave, respectively. Numerical calculations for this anisotropic case have been made by Hain and Tutter (1962). The problem of an inhomogeneous cylindrical plasma is again more complex, since the wave equation must be dealt with in cylindrical coordinates. With a plane wave incident upon a cylindrical plasma, it is possible in principle to calculate the phase and amplitude of the scattered 4.5 The geometrical optics of a uniform cylindrical plasma column 137 wave as a function of scattering angle (King and Wu, 1959). Since these quantities are readily measurable as a function of angle, the inverse problem of deducing the profile from scattering data provides an interesting technique for measuring plasma profiles (Shmoys, 1961; Kerker and Matijevic, 1961). 4.5 The geometrical optics of a uniform cylindrical plasma column A very approximate but useful model of common laboratory plasmas assumes a homogeneous cylindrical plasma several Tree-space wavelengths (of the probing microwave) in diameter, and yet neglects reflections at the boundary—the geometrical optics limit. The basic parameters of this geometry are defined in Fig. 4.8. The problem is assumed two-dimensional, the elements being of infinite extent normal to the paper. If the plasma is distant by at least a wavelength from the antenna, induction effects can be neglected and the situation treated as a radiation problem. If Aj A»l geometrical optics is a valid approximation, and we can talk in terms of rays which, except for refraction, travel in straight lines. 4.5.1 Transmission loss by refraction. We now consider the effect of refraction (Heald, 1959a; Wort, 1963). Since the index of refraction of the plasma (no magnetic field, or parallel polarization) is /« = (1 — «/«c)I/s < 1 the plasma column constitutes a divergent cylindrical lens. With the help of Fig. 4.9 we compute the refraction of rays in the geometrical-optics limit for a homogeneous plasma with sharp boundaries. The exit Transmitting antenna Cylindrical plasma Receiving antenna FIG. 4.« Micmwuve beam geometry for a cylindrical plasma. 138 Wave propagation through bounded plasmas Chap. 4 V 1 \ 0e = 6i +2(02-0!) / 1 \ 1 Ui - Sr> FIG. 4.9 Cylindrical refraction. angle $e is given in terms of the incident angle 9t and the entrance ordinate P\2 by the following simultaneous equations: sin(A-(?()=| sinf?!^^ sinfl2 (4.5.1) If now the exit ray is to strike the edge of the receiving aperture, at cartesian coordinates (R, A\l) with respect to the center of the cylinder cross section, we have the following condition on Sf and Pj2 for the most divergent ray accepted by the receiving aperture, tanfie = A-Ds,m(ei + 26.2-$1) 2JR-Dcos(ö, + 2ö;i-ei)" (4.5.2) In many cases of practical interest it is reasonable to make small angle approximations. We obtain from (4.5.1) and thus (4.5.3) Setting m = (l/F)-], from (4.5.2) (2R-D) 2m~+(2m+])di =A - D (2m +1) ~ + 2(m+ 1)0,]. (4.5.4) 4.5 The geometrical optics of a uniform cylindrical plasma column 139 -UnrefracEed ray -Refracted ray FIG. 4.10 Effect of refraction in geometrical optics approximation. Solving for PjD, we have P A-[2(2m+])R + D}el D AmR + D (4.5.5) For p ,„ a\ or for small angles '2(L + R)-Dcos(81-ei) ft* P 1 2(L + R)-D Eliminating 3; in (4.5.5) and rearranging, we have finally P A[(L+R)-DI2] D AmR(L + R)+ D(L+2R) (4.5.7) (4.5.8) (4.5.9) We recall that Pjl\ the largest entrance ordinate of rays that pass into the receiving aperture. Therefore, when PjD« \ the cylinder is equivalent to 140 Wave propagation through hounded plasmas Chap. 4 a slab of thickness D. With the above evaluation of 6\ the small angle approximations will be self-consistent if from (4.5.6) or using (4.5.9) P 2{L + R)-D D 2(2m+l)(L + R) + D A« 4mR{L + R) + D(L + 2R) (2m+l)(L + R) + D>2 (4.5.10) (4.5.11) Neglecting dissipation in the plasma, we obtain a reduction in amplitude at the receiving aperture because of the loss of highly refracted rays. This (power) transmission ratio is D(L + 2R) Pfo=i) 4mR{L+R) + D(L + 2R) (4.5.12) 15 10 - ---Refraction ■-Dissipation ----Reflection (max.) 0.5 Electron density (nfnj 1.0 FIG. 4.11 Loss of transmitted amplitude from refraction 4RH + R)l D(L 4- 2/0 = 3.6, dissipation vDjw'A-QA, and reflection. 4.6 The antenna problem 141 where we recall that 1 ■ m=--1 1 .-TA. f (1-«K)* 2«c Figure 4.11 shows this transmission loss as a function of electron density for the particular case of 4R(L+R) D(L+2R) = 3.6. 4.5.2 Other sources of loss. For comparison, we compute the dissipative loss in the plasma due to collisions. From (1.3.30), for n<«c and low dissipation (that is, k2«cuP2 < aP), this transmission loss is given in decibels by T[dB]= -8.686 aD= -?r(8.686) , ^ (4.5.13) (1 -~njna)'2 2ttC for rays passing near the center of the plasma. Figure 4.11 shows this relation for the numerical case 2ttc tu A For the numerical cases chosen, the refraction loss dominates except very close to the critical density. Finally, we consider the question of interference effects due to reflections at the sharp plasma-vacuum interfaces. In the usual case of PID«1 the only rays received are those which pass near the center of the plasma, and we can regard the plasma as a slab of thickness D. For the case of lossless slabs, the transmission ratio (4.3.9) becomes 1 which varies between (4.5.14) as the relative phasing of the reflections changes. This maximum transmission loss is also shown in Fig. 4.11, 4.6 The antenna problem The observed interaction of an electromagnetic wave with a plasma of finite size necessarily implies a "beamed" wave of finite extent and thus depends upon the antenna system used to radiate and receive the wave (Beard et al., 1962). The plane-wave model, which has been tacitly 142 Wave propagation through bounded plasmas Chap. 4 FIG. 4.12 Geometry of the physical optics of (a) an aperture, and (6) a microwave horn antenna. assumed in the preceding discussion, is a mathematical idealization which oversimplifies the practical situation, especially when the wavelength is not much smaller than the plasma sample. Therefore, it is useful to review some ofthe basic principles of diffraction. 4.6.1 Fresnet zones. Consider a circular aperture, of diameter A, in an opaque screen illuminated with waves from a point S at a distance L to the left, as in Fig. 4.12a. We wish to investigate the nature of the radiation field in the vicinity of an observation point P on the axis a distance R to the right. In accordance with elementary Huygens-Kirchhoff-Fresnel diffraction theory, we may divide up the wave front in the aperture into Fresnel half-period zones, such that the radiation passing from S to P travels an additional half wavelength for each zone (Andrews, 1960). Specifically, the «th zone is a circular strip, the radius rn ofthe outer edge of which is defined such that (L2-I-rn*)* + (R* + r*)* =L + R + n ~ (4.6.1) 4.6 The antenna problem 143 Setting rn = Ajl and assuming A«L and R, (4.6.1) may be expanded binomially to obtain A2/I 1\ n=Tx[L+R)- (4.6.2) If the aperture is uniformly illuminated, the contributions of adjacent zones are out of phase and of approximately equal amplitude and, thus, tend to cancel. Insight into the intensity distribution at various observation points (not necessarily on the axis) may be obtained by investigating the number of zones and the fractional area of each zone exposed by the aperture. For instance (Fig. 4.13), the intensity at P on the axis is a maximum for an aperture exposing 1, 3, 5,.. . zones, and a minimum for 2, 4, 6,... zones. The intensity at a point off the axis is small if roughly equal areas of odd and even numbered zones are exposed. To a first approximation the radiation pattern of a horn antenna, or diameter A as in Fig. 4.126, may be described by this analysis (Silver, 1949), We are here interested in only a qualitative description and, therefore, will not be concerned with the modifications required by a rectangular rather than circular aperture, by the polarization of an electromagnetic (transverse) wave, and by nonuniformity of illumination ofthe horn aperture. However, in passing, it may be noted that for the rectangular aperture with waveguide feed the diffraction field depends upon two factors each of which depends, in turn, on only one of the aperture dimensions—that is, the two dimensions are uncoupled (Schel-kunoff and Friis, 1952, Chapter 16). We take A to represent the dimension controlling the radiation pattern of interest (for example, in the plane perpendicular to the axis of a cylindrical plasma as in Fig. 4.10), and FIG. 4.13 Fresnel zones in a circular aperture; to) on axis, (ft) slightly off axis, and (c) I'iii i.IV avis (enlarged scale). (See also 1%. '>.?.<*.i S. K O II 12 n I > B <=■» 3 S s d. ? O O -> — a; H. "O w. -1 s b! ry < V ° P S g. § ó? 2. 1 ° _ 3 3 3 o- O. t: n ~ 3 Íl a o S1 3 — C o — ■ J- TO H n> i—i 1 > era s» tv j*0 o po —> 3 c °" <« fa f-* ^ o n BO 2 o o 3 cr» 3) 3 ^.2. 5^ a -3 o g. S n 3 i? er o o =5 f o £ O C « o B-r TI 1 I 2 V! S e. O Dl 0. o g' cr- 1. ŕ 2 g ^ 5-' S- O. N -■ O P 3 ft f w g' — B* t*1 Sa B- 2 O g" £ c O 3" tí » a » 3 3 (I vi o* 3 o. -n ř"S 3 -ŕ- -j os 3" o o J c. Vi os 2L C 3 1 e* 5" i a* 0 £ 1 p* n 146 Wave propagation through bounded plasmas Chap. 4 4.6 The antenna problem 147 If n is very large, exclusion of induction fields requires R>\ and therefore A>2nv-X»X, and the intensity distribution near P is essentially that of geometrical optics; that is, uniform intensity falling sharply to zero in the geometrical shadow of the aperture. If a lens of focal length R is inserted at the aperture, a Fraunhofer diffraction pattern is obtained in the plane containing P, as in the familiar problem of the astronomical telescope. If, on the other hand, n is much less than unity, a Fraunhofer diffraction pattern is obtained atP even without a lens. This is the familiar far-field case of conventional microwave antenna theory. To the extent that nzA2j4\R«l, we have R»A2j4X. This is equivalent to the well-known rule for the far (Fraunhofer) field of an antenna, which is usually written7 R>A2jX (4.6.4) and signifies that the maximum phase differential between "rays" is less than A/8, or that the aperture is less than one-fourth of the first Fresnel half-period zone (Montgomery, 1947). The total angular width of the central maximum of the Fraunhofer diffraction pattern is 2XJA. Therefore, the spatial width of the central maximum falling on a plane in the far field is (2XjA)R>2A. Thus, if A»X the intensity distribution in the vicinity of P is quite smooth over distances of the order of a wavelength, as in the high n case but in contrast to the 1 A2jX without the use of collimating lenses as normally required in the optical region. That is, the "far field" of a radiation aperture, or an obstacle, is a much closer distance, in wavelengths, than for similar apertures in the optical case. Therefore, in many situations, far-field theory can be used to describe the microwave field. Meanwhile, the use of lenses becomes less powerful since the focal length F of the lens m ust be F2A2,'K corresponding to A/16 or one-eighth zone. Amplitude errors due to interference are then about 2% as opposed to 5% for the criterion given above. if the focusing effect of the lens is to influence the diffraction pattern appreciably. The so-called/number of the lens is then f=FIA 1 for practical lenses. The far-field region can be effectively extended somewhat closer to the antenna aperture by using a lens to partially overcome the diffraction spreading (Sherman, 1962). The angular half-width of the central maximum of the Fraunhofer diffraction pattern is XjA. In geometrical optics a ray leaving the edge of an aperture of width A at this angle appears to originate at a point located a distance A2j2X on the source side of the aperture, and therefore the insertion of a lens of focal length F=A2j2X will render this extreme ray parallel to the axis. The/number of such a lens is f=Aj2\, (4.6.8) agreeing closely with the upper limit of (4.6.6). Table 4.2 summarizes the characteristics of the radiation field for various regimes of the parameters. The best collimation (~ A) is obtained Table 4.2 Field patterns and collimation of antennas ■Number of zones in aperture Small aperture ^/A~l Large aperture Aj\»\ n» 10 1 A) in the AjX~\, ri<[ case (antenna far field) and the AjX»l, n»\0 case with lens (geometrical optics). The latter, however, is a strongly 148 Wave propagation through bounded plasmas Chap. 4 converging wave passing through a focus. It appears best to design the experiment so as to avoid the low-order region (1 D, appreciable energy passes around the plasma, reducing sensitivity and severely complicating interpretation. With D>A>A, in order to avoid induction field effects and Fresnel-zone interference effects, we must have the plasma located in the far (Fraunhofer) field of the antennas, R>AzjX. It is a well-known rule of antenna engineering that for a pair of antennas to be located in the far-field region, by the usual A2jX criterion, the minimum insertion loss is of the order of !6dB (Montgomery, 1947). Since only about two per cent of the radiated power is received, the probability of interference from spurious reflected signals is high. We are, therefore, interested in pushing as close to the near field (Fresnel zone number «~ 1) as possible without encountering severe amplitude and phase disturbances from interference. This leads to the alternative of small (nondirective) antennas relatively close to the plasma or large (directive) antennas farther back. 4.6 The antenna problem 149 The concentration of rf energy produced in the field of a horn antenna depends upon two factors: the width of the wavepacket launched, and the angle of spread of the wave. Empirical plots of intensity contours in the field of millimeter horn antennas indicate that one half of the energy is confined within a beam width \2 /k\B\2-\ IbXRV' (4.6.9) where a and b are correction factors depending on geometry and aperture illumination and departing only slightly from unity. For a given A and R, this is minimized when (4.6.10) giving 2b A W„lin = {ab\Rp. (4.6.11) This condition corresponds to an aperture of about one half a Fresnel half-period zone at R. The insertion loss between two such antennas spaced 2R apart is about 8dB, depending upon the other dimension of the antenna aperture. Sometimes mechanical constraints of the apparatus will prescribe R, in which case A is determined by (4.6.10). If both A and R are at the experimenter's disposal, it is necessary to consider the role of the diameter D of the plasma column. The relative beam size W\D varies as R'^jD, whereas the relative spreading of the field over the plasma 1) ldW\ wUr)^ varies as DjR. Since we wish W«D«R, we arbitrarily take D = {WminR^MabXR^. (4.6.12) Recapitulating, given D and A and assuming a = b= 1, we choose ,1 (2A//A»l, R«D2jX and, therefore, the plasma approximates an infinite slab as far as diffraction is concerned. By reciprocity and symmetry arguments, we conclude that transmitting and receiving antennas should be identical. The preceding discussion has been based on the assumption of simple horn antennas without lenses. It has also assumed a "long" horn {L>A2jX) and L>2R, which may be impractical. Thus, two uses for lenses emerge: (1) to permit a less-than-long horn, in accord with conventional practice; and (2) to focus the beam or at least oveivcollimate to compensate partially for diffraction. If a horn is "long," its far-field (R>A2jX) pattern cannot be appreciably narrowed by addition of a lens. However, in the previous section we have discussed the use of a lens to focus the energy at a distance R/A~I, the so-called Gouy phase anomalies in the vicinity of the focus (Fig. 4.15) could severely complicate the interpretation (Linfoot and Wolf, 1956; Bekefi, 1957; and Farnell, 1958). In this case, it appears that ir lenses are to be used they should be focused at the opposite antenna or beyond (Christian and Goubau, 1961). Further discussion of the practice of using lenses can be found in Sections 6.4 and 9.3. Because of the perturbation that thick dielectric windows (glass, mica, quartz, etc.) make on the field of a millimeter-wave antenna (Redheffer, 1949), it is often useful to locate the vacuum seal at a convenient point back in the waveguide so that the antennas are wholly within the vacuum system. Such window design follows standard practice as used in microwave tube output windows and waveguide pressurizing windows; examples are given in Section 9.6. Alternatively, care must be taken to provide matching structures at the vacuum walls (Jahn, 1962). 4.6.4 Validity of the geometrical-optics, slab model. Since the geometrical-optics, plane-slab model is particularly convcnienl to analyze, 4.6 The antenna problem 151 -270 -180 4 -90 +90 1 1 1 1 / i i i i i y 0° ^ Attenuation _ yr i r I l 1 —1— ^-~"^T ~"^F>i 1-1 ^ \ \ A 1yi1i1 1 ^ 1 1 1 1 1 20, 10 : < Diameter D/\ (in paraffin) FIG. 4.16 Phase shift and allenualion as a function of cylinder diameter in paraffin analog experiment (Rosen, 1949). Wave E-tield parallel to cylinder axis. Dimensions normalized to wavelength in the medium of Ihe antennas: H-plane aperture of horn 3.1 ; E-plane aperture 2.6; axial length of horn (to apex) 6.8; horn separation 13.4. Measurement frequency 35 Gc. it is of interest to investigate the validity of this model for the more practical case of a cylindrical plasma. In addition, since the nearness of the antennas, as well as of extraneous objects such as vacuum system walls, severely complicates theoretical analysis, it is often most effective to perform an analog experiment (Warder, Brodwin, and Cambel, 1962; lams, 1950; and Lashinsky, 1963). A plasma, with dielectric constant less than unity, can be simulated by cutting holes in a large block of low-loss dielectric, in which a scaled antenna system is imbedded. In one such experiment the phase shift and insertion loss were measured for various size cylindrical holes cut in FIG. 4.17 Propagation through a dielectric cylinder, small compared to effective microwave heam. 152 Wave propagation through bounded plasmas Chap. 4 paraffin (/c = 2.25) (Rosen, 1959). This dielectric constant ratio corresponds to a plasma of 0.56 critical density. The antenna system was chosen for a plasma diameter of about four wavelengths in accordance with the design criteria of Section 4.6.3, the prototype system having an insertion loss of approximately 8dB in vacuum. Typical results are shown in Fig. 4.16. For cylinder diameters greater than about three wavelengths, the observed phase shift differs negligibly from what would be expected for a plane slab, except for the loss of a full wavelength. This latter effect can be explained as the result of interference between the wave passing through the cylinder and the wave passing around it. Using the notation of Fig. 4.17, we regard the wave entering the receiving aperture as composed of two components: (a) the wave passing through the cylinder of amplitude (PjW)Vi phase 2tt(ji— l)D/X (neglecting internal interference effects), where ji< 1 is the refractive index of the cylinder (air) relative to the paraffin, and A is the wavelength in the paraffin; and (b) the unperturbed wave passing around the cylinder of amplitude {\~ DjW)Vi phase 0. W is the effective beam width at the plasma. and from (4.5.9) v L+2R ' A[(L+R)-Dj2] ° 4^-- IJR(L+R)+D(L+2R) (4.6.14) (4.6.15) The resultant wave is then P_ W exp[/2w0*-l)D/A] I exp(j'0) = Cexp(j'J<£) (4.6.16) where C and A are the amplitude and phase shift of the resultant wave. Wc have C= If + J^ = tan ([-w)+2{wY(] ~wY «p-fr-w cos[27r(/t-l)D/A] + (4.6.17) (4.6.18) -270 & -180 153 2 3 Diameter D/S (in paraffin) FIG. 4.18 Results of simple geometrical optics theory for the conditions of Fig. 4.16, exhibiting "loss" of 360°. -90 - -180 - -450 2.25 2.0 1.8 1.6 1.4 Dielectric constant k 1.0 0 0.1 0.2 0.3 0.4 Simulated electron density (nfaj 0.5 FIG. 4.19 Phase shift im a function of dielectric constant of cylinder in paraffin-analog experiment, simulating plasma of varying density. Cylinder diameter 4.6 wavelengths. 154 Wave propagation through bounded plasmas Chap. 4 The theoretical C and Aj> are plotted in Fig. 4.18 for parameters corresponding to the experimental conditions of Fig. 4.16. In order to obtain these relations for C and A, several small-angle approximations have been made, diffraction was completely neglected, and interference effects inside the cylinder were disregarded. In spite of the crudity of the geometrical optics analysis, the numerical agreement is reasonably good. This "lost-wavelength" effect could cause misleading results in a plasma experiment in the uncommon situation in which the plasma is created with a small diameter (<2A) which subsequently grows larger.8 More commonly, the plasma is created with a relatively large diameter (>3A) and then grows denser (due to increased ionization) or smaller (due to some form of magnetic compression). In these cases, the transition from the vacuum (no plasma) phase-shift condition, as the plasma develops, appears to be unambiguous. By inserting rods of various known dielectric constants in a fixed diameter hole in the paraffin environment, the data of Fig. 4.19 was obtained, simulating varying plasma densities (Rosen, 1959). On the basis of this study, we conclude that the slab analysis is satisfactory for a cylindrical plasma diameter of at least three wavelengths, provided the antenna system is chosen judiciously. n The expanding-diameler situation coukl occur during a plasma decompression event or an expanding cylindrical shock. CHAP T E R 5 Guided wave propagation 5.0 Introduction The effects of finite plasma dimensions on wave propagation were discussed in Chapter 4. The boundaries were found to cause reflections and refraction of transmitted waves and, in some cases, to affect the radiation patterns of antennas. In most cases, the boundaries led to problems, rather than being beneficial to the propagation experiments. In the present chapter, we discuss another class of bounded plasmas; in this case, boundaries arc essential to the wave propagation. Resonant cavities and waveguides have metallic walls that carry currents and, thus, set up propagation modes. The electromagnetic fields penetrate the enclosed plasma, whose conductivity, in turn, affects the mode cut-off frequency. Measurements of wave phase shift or resonant frequency and loaded Q then can be related to the plasma properties. Plasmas having vacuum or dielectric boundaries can support space-charge-wave modes and, thus, can act as waveguides. Certain space-charge-wave modes propagate along the plasma surface (surface waves), while others are carried within the plasma (body waves). When a magnetic field is*present, the waves tend to be a combination of both types. Electromagnetic waves and spacecharge waves may propagate simultaneously along the same bounded plasma. Under certain conditions, the different wave types may couple to one another but, in general, the coupling coefficients are rather small. 5.1 Measurements on, plasmas contained in resonant cavities The resonance properties of a cavity containing a lossy dielectric can be staled in terms of Q 155 156 Guided wave propagation Chap. 5 5.1 Measurements on plasmas contained in resonant cavities 157 Q- oj0 (energy stored) _ 2tt (energy stored) average power loss energy lost per cycle (5.1.1) The stored energy is calculated from a volume integral of the fields contained in the cavity; the power is dissipated in both the wall losses and the dielectric losses. When the lossy dielectric filling the cavity is a dilute plasma, of complex conductivity o-, the change in Q and the shift in resonance frequency due to introducing the plasma are given by a perturbation equation (Slater, 1946) (ßi Go) i (r)E2(r) dV (5.1.2) dV where the 0 subscript represents unperturbed values and the 1 subscript the perturbed conditions, that is, with plasma present. The intrinsic or unloaded Q, Qu, is not measurable, however, since we must couple to the cavity through some kind of impedance. The equivalent coupling circuit, representing effects of the coupling orifice or loop, the wall losses, dielectric losses, and coupling line or p G„ <^VvV\A- Yt = 1/z,. P' ,1s: 1 T F St t M/WV (a) FIG. 5.1 (a) Lquivalent circuits for an empty resonant cavity, and {!>) one containing a plasma. The subscript 1 denotes plasma parameters. Other symbols are defined in the text. FIG. 5.2 Standing-wave ratios, expressed in decibel format, in the feed line for (a) an empty and, {/>) a plasma-filled resonant cavity in the vicinity of resonance, showing detuning. Symbols are defined in the text. Subscript 1 denotes plasma conditions. waveguide losses, is sketched in Fig. 5.1. The reflection coefficients are shown in Fig. 5.2. Yc is the matched characteristic transmission line admittance at /'P'(Kruno and Whinnery, 1953). The network Xs represents the coupling reactance, and Gs the coupling losses. Xs is generally negligible and, in the remainder of the analysis, will be omitted (Slater, 158 Guided wave propagation Chap. 5 1946). The cavity across 77' is represented by parallel reactances L and C and conductance G. The cavity is coupled to a transmission line by an admittance ratio A2:l. In the reduced equivalent circuit, the quantities are normalized to Y.: ?5 = G,/FC g=AaGjYe l=LYcjAz c=CAz!Yc. (5.1.3) The conductance g0 across PP' at resonance, when the susceptance of l-c is zero (reactance is infinite), is obtained from go (5.1.4) The dimcnsionless coupling coefficient j8s represents the ratio of energy stored in the cavity to that coupled into a matched transmission line. A system for which j3c<1 is undercoupled, while one having /3C>] is over-coupled. The Q of the system loaded by the coupled transmission line is the loaded Q go (5.1.5) For critical coupling, that is, ft;= I, Q,=\Qv The loaded Q is related to the width of the resonance curves of Fig. 5.2 ßt=f- Oft» (5.1.6) The measurable or external Q is defined in terms of the coupling factor j8e and the unloaded Q Qu QqQl ßc Qv-Ql (5.1.7) 5.1.1 Measurement of plasma admittance. The effects of the plasma on the cavity resonance properties are conveniently determined by observing the complex reflection coefficient from the coupling window or the phase and standing-wave ratio in the feed waveguide (Rose and Brown, 1952). If the orifice and window are electrically thin and lossless, we are justified in neglecting Xs and Gs. Measurements of the shift in resonance frequency Aa>, and of the loaded Q, QL, for the two conditions (with and without plasma) then permit the calculation of the plasma admittance y1=gi+jb1. This is best done with an impedance chart, such as the Smith chart (Southworth, 1959), since explicit calculations tend to be laborious. 5.1 Measurements on plasmas contained in resonant cavities 159 5.1.2 Measurement of plasma density and collision frequency. For low density plasmas (wp2«co2) having low collision rates (v«w) and no external magnetic field, (5.1.2) may be written in two parts: one dealing with frequency shift, and the other with change in Q (Bucfisbaum and Brown, 1957), 1 1 we ~2 1+(»/«)" J («pa/«W dV Qi So \QI Í dV K>W dv f V Jv dV (5.1.8) (5.1.9) The spatial distributions of both the electron density and the cavity electric fields must be known. The field configurations of cavity modes, in general, are known. If the plasma density is uniform across the cavity, the evaluation of the integrals in (5.1.8) and (5.1.9) is straightforward (Buchsbaum and Brown, 1957). If the plasma consists of a small diameter 1000 500 200 100 - 10'° Plasma column Cylindrical resonant cavity B Field i2rn 2a- 10n 101 Electron density [cm-3] 10' FJG. 5.3 Frequency sAift of TMIIM cylindrical cavity with a plasma post; r/o= 1/10, = 5555 Megacycles. (Courtesy S. C. Brown, M.I.T. Research Laboratory for Electronics,) 160 Guided wave propagation Chap. 5 column, much smaller than the cavity diameter, it may be looked upon as a lumped admittance, shunting the cavity. As an example, consider a cylindrical cavity excited in the TMnao (linear accelerator) mode, containing a plasma post of diameter 1 /1 Oth that of the cavity. The curve for frequency shift vs. electron density is shown in Fig. 5.3 (Brown, 1958). Typically, Aeo/u) of 10"4 is easy to measure by observing an oscilloscope trace. The minimum detectable density then is about one per cent of that detectable with a transmission interferometer having a path length of 10 wavelengths and a resolution Aj, of 10'. Even greater sensitivity is obtainable by using heterodyning or frequency counting techniques to measure Aw. When the electron density is nonuniform inside the plasma, as is nearly always the case, the frequency shift may be expressed in terms of an average density n and a geometry factor a, — an. (5.1.10) This operation is valid only for low densities, that is, when the plasma can be treated as a perturbation in the cavity. The separation of (5.1.8) into the two parts represented by (5.1.10) is especially useful when the plasma density is to be studied as a function of time. The factor a. constant in time, then is calculated for various geometrical conditions, and the results tabulated or plotted (Oskam, 1957; Buchsbaum and Brown, 1957). A cavity mode that is particularly useful as well as easy to treat mathematically is the TM0io mode in a cylindrical cavity. The electric field is entirely axial (Eg) and the magnetic field azimuthal (Ho). The frequency shift is given by Brown (1958) 1 1 +"jo (5.1.11) This equation applies also when there is an axial magnetic field, since E* II B,. Other modes are useful as well. When an axial magnetic field is applied, the plasma becomes anisotropic. The cavity electric fields can be represented as right-hand and left-hand circularly polarized fields, much as described in Section 1.4. The cavity frequency response then splits into two peaks, instead of the one shown in Fig. 5.2. Some cavity modes are degenerate, with two (or more: see Fig. 9.19) modes occurring at the same frequency. The presence of the magnetized plasma in the cavity removes the degeneracy and the different modes appear at different frequencies. As an example consider the TM1U and TE011 modes. For wl,jw« 1 the frequency shifts are given by Brown (1958): 5.1 Measurements on plasmas contained in resonant cavities 161 \—(A)bl\ a ty! \W-~2 co2 (!+«.„ (1-«,,/<'>)*+("/»)* (TMni) Aco a ok a>0 2 o>2 l-K/o)5 (TEau) (5.1.12a) (5.1.12b) (5.1.12c) A study of the frequency dependence of the three modes yields information about the density spatial distribution. For higher densities and collision rates, the frequency shift for the 1 b011 mode must be written in more complete form a (».. w0~2 co2 [v2 + (w + w0)2][v2 + (^-^bf] (5.1.13) An interesting feature of (5.1.13) is that when «yW + "2.there is no frequency shift A /due to plasma density variations. The cavity Q also is approximately minimum for this condition. This effect can be used to measure the collision frequency v (Hirshfield and Brown, 1958). FIG. 5.4 Frequency,hi,t of a TEW1 mode cavity for high p.asma densi.u.with and without magnetic held applied. (Courtesy S. C. Brown, M.l.T. Rcsca.ch Laboratory for Electronics.) 162 Guided wave propagation Chap. 5 5.2 Waveguides containing plasmas 163 5.1.3 Special cavity modes for high density plasmas. The TE011 mode is useful for high densities (co„/w> 1) when no static magnetic field is applied and the electron density gradient is radial (and, thus, perpendicular to the electric field). The circumferential electric fields induce no spacecharge fluctuations in the plasma (Persson, 1957). When a magnetic field is applied, the anisotropy permits radial currents and fields and the frequency-shift relations become complicated, as shown in Fig. 5.4. At very high densities and low collision rates, where the skin depth is much smaller than the cavity diameter, a high mode TE01M cavity may be used, with M as large as perhaps 8 or 10. The smaller the "active" volume of cavity, that Pulsed power klystron Pulser Master pulse and trigger generator Time mark generator Frequency meter {© detector Coupling orifice lor the ionizing signal -0 Indien toi Absorption frequency meter Level set >VW attenuator 10 dB Coupling orilice lor measuring signal Marker ' input Reflection signal ((j) detector 6 dB Continuous signal generator ® Frequency stabilizing circuit Video amplifier Oscilloscope Reflection -1 signal input FIG. 5.5 Microwave cavity excitation and measurement system for studies of plasma contained in a resonant cavity. The ionizing signal couples to a different cavity mode, at a different frequency, from the measuring signal. Either the reflected signal (as shown) or VSWR may be observed. is, that part of the cavity fields perturbed by the plasma, the smaller will be AujIAn and the less sensitive will be the measurement. It is important to note that these techniques for high-density plasmas, with the skin depth much less than the plasma diameter, require an independent knowledge of the plasma density (and temperature) profile if anything more than the plasma surface properties are desired. 5.1.4 Experimental techniques for cavity measurements. Often a cavity can be excited in two modes at once, at widely separated frequencies, such that there is no coupling between the two inputs. It is then possible to use one mode to break down a gas and heat the plasma with a high power source, and to use the other mode for diagnostics. A sketch of a typical system is shown in Fig. 5.5. The excitation source may be steady, or it may be pulsed, with the diagnostics done in the afterglow period. Ordinarily, the frequency of the signal source is swept back and forth over the resonant frequency of the cavity. The reflection coefficient or the standing-wave ratio is observed on an oscilloscope, and the shift in resonance frequency and cavity de-Q'mg are recorded for various plasma conditions. If the plasma is transient, the timing of the sweep may be delayed to sample various parts of the plasma transient event (Biondi, 1951; Sexton et al., 1959). A second technique is to fix the frequency of the signal source and wait until the cavity resonance frequency sweeps through, as the plasma density decays (Biondi and Brown, 1949). 5.2 Waveguides containing plasmas Waveguides also have normal modes and their cut-off frequencies will be shifted, much as the resonance frequencies of cavities are shifted, by the presence of a plasma inside. To find the effects of a plasma on the propagation characteristics, we employ some of the methods of cavity techniques and some of free-space propagation. When waves propagate along dielectric or conducting walls, they are no longer strictly TEM waves. The currents flowing in the walls lead to components of electromagnetic field along the direction of propagation, even when tro magnetic field is present. As a clean example, consider a TE wave propagating in a plasma-filled rectangular waveguide with no external magnetic field present, as sketched in Fig. 5.6a. The wave equation can be separated into two parts, one describing the variations across the waveguide and one the variations along it. The wave equation becomes VaH = V(2H + y2H=-cUVo?H= — *H (5.2.1) 164 Guided wave propagation Chap. 5 Plasma (a) (b) FIG. 5.6 Plasma-filled waveguides; (a) rectangular, (b) cylindrical. where ? = e9/c is the equivalent complex permittivity ofthe system and y is the propagation coefficient or complex wave number, which depends on the waveguide mode. In rectangular coordinates, the transverse term (V,3) is ox cy2 (5.2.2) For TE waves, no E, component exists, and it is convenient to solve (5.2.1) for the H, component %"Ířo|c0S ^~ xj(cos y yj exp(/W-ýz) (5.2.3) 5.2 Waveguides containing plasmas 165 where m is the mode number across x (or a) and n is the mode number across y (or b). From (5.2.1), (5.2.2), and (5.2.3), where ^)'-(t)^*--^--^ <"-4> /3c = mode cutoff wave number Ac = mode cutoff free-space wavelength o)t. = mode cutoff frequency. Here we neglect any losses in the waveguide walls. Solving (5.2.4) for the propagation coefficient, we have y9 = &a-^*r+y^«<- (5.2.5) The value of k, calculated in Section 1.3.2 for a Lorcntz plasma, is CO2 l+v2l2 1+^/w2 To obtain the attenuation and phase constants, we take the square root of (5.2.5) as in Section 1.3.3 (5.2.7) /3 = { + | (as above) + \[(as above)2 + (as above)2]'4}1 (5.2.8) where w,. = c/3t. is the mode cutoff frequency in absence of plasma. These equations look similar to (1.3.20) and (1.3.21), except for the presence of the mode cutoff wave number |8e. This factor tends to increase the plasma CUtoff frequency (now >eu,) or, at fixed frequency, to cause the transmission io cut off at a lower density than for the free-space case. 166 Guided wave propagation Chap. 5 5.3 The c/u = 0.9j 0.8/ 0.62/ 0.5/ 0.37/ / = 0.0 -J\ ! J Ii, l (free space) — A 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Normalized electron density (up/co)2 = FIG. 5.7 Wave attenuation for a plasma-filled waveguide as a function of plasma density, for various mode cutoff wave numbers. Attenuation in dB is dB = 8.69«; the independent variable, (1.3.20) and (5.2.8) may be rewritten 1 + (free space) (in waveguide) (5.3.1) (5.3.2) where aia = cfia is the mode cutoff frequency and the approximation is due to setting v — 0. The cutoff frequency, u)c„, of a "hot mode'" (that is, when plasma is present) is higher than that for a "cold mode," u>,.. From (5.2.8), a)cp is seen to be (5.3.3) Figures 5.9 and 5.10 show cu-jS diagrams normalized in two ways for the plasma slab and for the rectangular waveguide, where the collision frequency v is very small, the density is uniform, and no magnetic field is present. The slope of the radius vector to each point on the diagram is seen to be proportional to oi//S, which is the wave phase velocity %. The slope of the tangent at each point is proportional to dwjdfi, which is the wave group velocity vg. 5.4 Nonuniform plasma in a waveguide When we considered that the plasma was uniform across the waveguide diameter, we implied that it had no effect on the magnitude of the microwave fields, that is, that E and H were unperturbed. That, or course, is not true for dense plasmas, when wp approaches w. Let us take the case for a rectangular waveguide again, in which the plasma has a diffusion-limited density distribution (Brown, 1959) n(x, y) = Mo^cos I J| COS 7 V J by) (5.4.1) where nQ is the axial density. The corresponding dielectric constant is 2H. d2H, c ear r2+-^ >y) ^ = [«(Vk-1)x(VxH)]3 (5.4.3) in accord with (4.4.7). For sufficiently small gradients, the right-hand side, which cross-couples //- to Hx and Hv, may be neglected. Even so, j_I_I_I_I_I_l FIG. 5.9 Dispersion (cu-ß) diagram, normalized to the plasma frequency o>p, for a uniform plasnuf in waveguide; u>c=cold mode cutoff frequency, uCP=(wcz + u>ps)Vz = hot mode cutoff frequency. the simplified equation is not separable and cannot be solved explicitly for y> as before. It resembles Mafftieu's equation r% +(/> + c sin -.\W=0. d. (5.4.4) 170 Guided wave propagation Chap. 5 5.5 Spacecharge waves 171 FIG. 5.10 Dispersion (»-/3) diagram for electromagnetic waves, with propagation phase constant normalized to the mode cutoff wave number, for a plasma filling a waveguide; aj,. = £„r = cold mode cutoff frequency. Various solutions to (5.4.4) exist, and series and numerical approximations are also possible (Madelung, 1943). 5.5 Spacecharge waves In Chapter I, it was pointed out that spacecharge fluctuations in a cold, stationary, infinite, uniform plasma were nonpropagating. Any disturb- ance or departure from equilibrium is confined to the location of its origin, within the order of a debye length. There are conditions, however, under which these fluctuations can propagate and transfer wave energy away from the source. (/) The plasma electrons have a drift velocity, vQ. (2) The plasma is finite and possesses normal modes. (5) The electron temperature is finite. Let us now consider the characteristics of some of these propagating plasma waves under various plasma conditions. 5.5.1 Spacecharge waves in a cold, drifting plasma. Let the electrons have a drift velocity v0. If plasma oscillations (at toZiojp) were present, they would be convected along at the drift velocity, and would constitute a convective wave. If no oscillations are present, but rather an externally applied electric field varying at frequency m is present, we can also excite plasma waves arising from the resulting velocity fluctuations. First from a naive point of view, we recall that the spacecharge oscillation in a cold stationary plasma, discussed in Section 1.2, occurs at the plasma frequency op independent of its spatial variation. Thus we may choose a standing wave or arbitrary wavelength A, which in turn may be regarded as the superposition of two traveling waves with phase velocities vp 'slow spacecharge wave" (5.5.1b) Now on a»more analytical level, we seek the dispersion relations for these waves under the action of an applied electric field. We rewrite the equation of motion of a Lorentz plasma, (1.4.1), in Eulerian rather than Lagrangian coordinates (that is, we watch the plasma stream flow past our fixed coordinate system, rather than use a coordinate system in which the plasma is at rest); the force equation now is (Marcuvitz, 1958) '■I +y-Vv + v(v-v0) = in (E + vxB). (5.5.2) 172 Guided wave propagation Chap. 5 5.5 Spacechargc waves 173 Let us now assume the following. (J) The drift velocity v0a is constant, in the +z direction. (2) The induced a-c velocity variations are much smaller than the drift velocity. (J) The driving electric field is of the form Elz exp(/W-yz), that is, a longitudinal wave propagating in the z direction. (4) No magnetic lield effect is present (vxB —> 0). (5) We neglect collisions (v—>0). The electron velocity is of the form '-',(2, 0^'^ + ^expOW-yz) (5.5.3) where the subscript 0 represents the steady-state component and the subscript 1 the complex perturbation component. The second term of (5.5.2) may be evaluated to first order as (5.5.4) where is the perturbation part of (5.5.3), in the z direction. Substituting in (5.5.2) and canceling the phase factor sxp(jwt-yz), wc obtain (J<*>-YVo)Vis= —— Els (5.5.5) In contrast to the case with electromagnetic (transverse) waves, the a-c current density Jls depends on the oscillating spaeecharge as well as the oscillating velocity component; thus Jz=^J0s + Jiz exp(jwt-yz) = -/lo^oz-efwo^+flii^) expfjW-yz), (5.5.6) neglecting higher than first-order terms. To evaluate the perturbation component nl of the electron density, n = na + n1 expf/W-yz), (5.5.7) we invoke the equation of continuity (3.3.7) oz tit which becomes • yJ-.z=jue>h-Substitution of (5.5.9) in (5.5.6) yields 4 or "dV-.z ■—- V. me (5.5.8) (5.5.9) (5.5.10) (5.5.11) which in turn permits us to use (5.5.5) to find the (complex) conductivity {w+jyv^f (5.5.12) In problems dealing with waves, it is useful to translate the conductivity into the equivalent dielectric constant Kzz=i -■■]'—■— 1 ~~t—r~--W (5.5.13) The presence of the v Vv term in (5.5,2) makes the drifting plasma, when viewed from a stationary position, behave like an anisotropic electromagnetic medium, and the dispersion equation (A.74) is applicable. Since wc arc here concerned with longitudinal (spaeecharge) waves propagating in the z direction (Trivelpiece, 1958), (A.74) reduces to simply km=0; from (5.5.13) w+jy1'oz= or y=jtt,±0) exp(-7j8z) expfjW) (5.5.22) where n is frie order of the angular mode number. Writing (5.5.21b) in cylindrical coordinates gives for the radial function CP 77r('fH*.-^-° W which is Bessel's equation for imaginary argument. The solution for 0X is of the form * 0X = [C ln((3r)+ D Kn(Ml exp/(«i/-«0-iSz), (5.5.24) 176 Guided wave propagation Chap. 5 where L. and Kn are the modified Bcssel functions of the first and second kinds. We cannot have infinite fields at r = 0 so the constant D must be zero within the plasma column. The field components within the plasma column (r) -j (5.5.25a) (5.5.25b) (5.5.25c) E10 = Cj~ 10?) } expj(t»t-/t6-pz). (5.5.26) Elz = Cjpln(pr) The permittivities for the two regions (Fig. 5.12) are, using (1.3.5), a, which simplifies the solution for the constant C. The tangential electric fields and Eie) must also be zero along the conducting wall at r = b, but continuous across the boundary at r — a. The potentials inside and outside the plasma column are then found to be MM r„ ln(pr)K,CSft)-IBQ3fc)KB(pr) (5.5.27) (5.5.28) FIG. 5.13 Potential variations in radius Tor the lowest-order spaccelittrge surface wave in zero magnetic field. 5.5 Spacecharge waves 177 where the Tn and K„ are again the modified Bessel functions of the first and second kinds. The potential variation for the lowest order mode (n = 0) is sketched in Fig. 5.13, It is immediately apparent from Fig. 5.13 that the electric field is indeed confined very near to the surface or the plasma, hence the name surface waves. Demanding continuity of the normal displacement Dt = eEr at r = a leads to the dispersion equation of surface wave propagation, when no magnetic field is present (Trivelpiece, 1958) 1 -u>*\o? In'(JSa)_In'(^)Kn(^)-l„(^)Kn'(^) (5.5.29) The w-j3 diagram Tor the lowest order mode is shown plotted in Fig. 5.14. Only the + fa curves are shown, the diagram being symmetrical about the oi axis for -j3a (that is, waves traveling in the opposite direction). The difference in curves for &/ oo is seen to be quite small, indicating that the presence of metal walls farther than about one column 0.8 0.7 0.6 0.5 3 5" 0.4 5 0.3 0.2 0.1 1 1 1 1 III] ki, - 1 (vacuum) Transition case of thin- — walled dielectric (x = 4) S with air space between - S, it and the metal wall — /lb/a = 2 " __ ff-b/a = ^^^-^— k{, = 4 (glass) /r*~bja = 2 i i I i I'll 4 5 6 7 Propagation constant {la 10 FIG. 5.14 The diagram for propagation of surface waves on a plasma surrounded by a vacuum interspace (t,,-en) between the column and the wall and for a material of dielectric constant « = 4 filling the space (sec Fig. 5.12 for geometry). 178 Guided wave propagation Chap. 5 diameter from the plasma has negligible effect. The propagation extends Irom zero frequency up to a cutoff frequency of ^ = (TT^ (5.5.30) where kb is the dielectric constant of the medium surrounding the plasma. o.t i 0.7 =>\i>/« = - b/a = 2 0.5 - 0.3 0.2 0.1 Kb = 4 b/a = 2 4 6 8 Propagation constant j8a 10 12 FIG. 5.15 Phase characteristics for a surface wave mode of one angular variation (Invelpiece, 1958). 5.5 Spacechurge waves 179 In Fig. 5.15 the tuco. An interesting analogy between the electromagnetic fields and space-charge-wave fields can be drawn. We pointed out that the magnetic component of the surface wave fields can be neglected. But, for a wave to propagate, the energy stored by the electric field must be interchanged with something and, in this case, it is the mass motion of the charges. For this reason, spacecharge waves are sometimes called electromechanical waves. The charges move back and forth in such a way that the electric field lines terminate on them, which in part explains the small penetration of the field outside the column and the difficulty of experimentally coupling to these waves without perturbing them. 5.5.3 Spacecharge waves in a plasma column in a magnetic field. The presence of a magnetic field changes the propagation characteristics considerably. The medium is now anisotropic, and the dielectric permittivity € becomes a tensor, €->£ = e„k. The permittivity derived for electromagnetic waves in Chapter 1 is perfectly general and applies also to spacecharge waves (Pines, 1960); (1.4.60) where the components for a collision-free plasma are (Section 1.4.7) Kj_ = 1 — K± ->* 0 " k = +>* 0 0 0 fC II _ 1 -cu„2/cu2 _(«vV"2)k/«0 k v — l< I = 1--- -2 2 To find the wave solutions for the magnetic-field case, then, we must rewrite (5.5.18) to include the anisotropic permittivity V-D^V-e-E^coV-k-E^O (5.5.31) 180 Guided wave propagation Chap. 5 from which (5.5.20) becomes V.e.(-V*1)=«oV-K.(-Vd>1)=0. (5.5.32) The form of (5.5.32) in cylindrical coordinates is analogous to (5.5.23) but with the permittivities included and has solutions similar to (5.5.27) and (5.5.28), except that £2 is now replaced by (S^i/ki). The electric field in the plasma is given by the gradient of the potential i'.'[^fe)T <5-5-35' The azimuthal field is similarly obtained E1B = —~L = — 301_ n ßElz dB ~ßr dr Hm (5.5.36) The last three equations are seen to be analogous to (5.5.25). They point out that all electric field quantities are derivable from the axial field. Tnvelpiece (1958) finds a propagation equation similar to (5.5.29), but with the permittivity components included, («|k± + "** = V(ßB)KJßb) - ln(ßb)Kn'(ßa) ßa«t Ußa)Kn(ßb)-Ußb)KK(ßa) ' (5.5.37) Substituting in the permittivity components or (1.4.60) leads to a complicated expression for the cutoff frequency which involves both w, and 5,5 Spacecharge waves 181 (Wp2+ Ub'2)* FIG. 5.16 Phase characteristics of spaeechargc waves (to-ß diagram) for a plasma column bounded by a dielectric and metal, in an axial magnetic field of various magnitudes. p. The to-/3 diagrams in Fig. 5.16 show the trends for various relationships of w„ to oi„. When wb is very large, most of the wave energy is carried by charge accumulation within the plasma column, with little surface rippling. These waves are called body waves, and have a cutoff frequency approaching toH, as do one-dimensional spacecharge waves. Conversely, for wb —> 0, the wave energy is carried by surface rippling alone, with little spacecharge bunching within the plasma. The cutoff frequency for these surface waves then approaches a>Pl(l + The surface waves cannot propagate if the dielectric space (vacuum or material) is absent between the plasma and the metal wall since, then, no charge accumulation can occur. The usual electromagnetic waves can propagate, of course, and a mixing of electromagnetic and spacecharge waves is usually present. For values of aitrjoip between 0 and oo, and at various values of fia, the wave mechanisms and wave types undergo smooth transitions. At large (3a the electric field is almost pure Ez and a mixed electromechanical and TM-typc wave ensues. For smaller fia, the wave goes from TM to mixed TE to pure TE (Bevc and Everhart, 1961). The upper branch shown in Fig. 5.16 is a backward wave whose characteristics are not influenced by the geometry and whose cutoff frequency is w => (tv0+«op2)y* (Smullin and Chorney, 1958). This frequency is recognized as the transverse resonance or upper-hybrid frequency a>uh 182 Guided wave propagation Chap. 5 (see 1.4.53) for propagation across the magnetic field, from the Appleton equation (1.4.40). This wave is due apparently to a mixture of longitudinal and transverse electric fields, leading to electric polarizations both along and across the magnetic field. The passband of this wave corresponds to just the cutoff band of the electromagnetic extraordinary wave. The angularly dependent spacecharge modes can exhibit Faraday rotation in certain circumstances. Considerable confusion may exist in experimentally distinguishing between these modes and whistler-mode waves, since both waves are slow and may have similar passbands. The whistler mode, of course, is a TEM wave and has no Ez component. Nearly any antenna in a finite plasma column has fringing fields that could excite the spacecharge wave, however, and since both modes require close antenna coupling to the plasma they could be equally excited. The propagation characteristics of the two waves depend very differently upon electron collision frequency and variations in geometry; some care in selecting the desired one is required. 5.5.4 Surface spacecharge waves on a drifting plasma column. The dispersion relations for a finite plasma column drifting along its own axis (z direction) with a velocity v03 are found by applying the nonrelativistic equivalent of a Lorentz transformation to the dispersion relation ai'-w±^v0 (5.5.38) U 0.6 0.5 // /^/v0/c = 0.02 (lOOeV) -// X / 0.3 A /X \ 1 1 A/ 1 1 \ Ji -ft f 1 1 1 I 1 1 -6 -5 -3 -2 -1 0 / -0.1 1 2 3 4 5 6 0a FIG. 5.17 Phase characteristics of surface waves on a drifting plasma column. Example shown is c/ui!,a = 5 (for instance, Ez exp(yW-yz), we may expand the velocity function as f{r, v, f) • >/:,(v) +/; (v) exp(./W - yz), (5.6.3) where fc is the equilibrium distribution function, and (5.6.2) becomes to first order (5.6.4) The divergence equation relates the electric field and the charge density, V-eoE= - toYEc= ~ne J/: dsv, (5.6.5) where the uniform positive-ion background has been assumed to cancel the equilibrium electron density. Substituting/! from (5.6.4) into (5.6.5) and canceling the arbitrary amplitude Ek, we have «ea [(ofjduz) dvx dvy — e0y —— - i -;-—-- m J jw-yvz dv. The integration over the transverse velocity components vx and j| may be carried out formally, to yield for the dispersion relation l^iff (5.6.6) where f0s(vs) is the one-dimensional equilibrium distribution. 1 The adiabatic assumption allows the gas temperature to fluctuate due to density fluctuations. The neglect of this effect (the isothermal approximation) leads to a Tactor of 1/3 discrepancy in the thermal term in the final dispersion relation (Bernstein and Trehan, 1900). 5.6 Spacecharge waves in a warm plasma 185 For generality we may assume that both the frequency w and the propagation coefficient y are complex, (5.6.7a) & — ; am«a> 1iw rt>« The dispersion equation then becomes i=^[l + 3(ßaf]+jA( Ol (5.6.11a) or by binomial expansion 3 1 ^ ~\ +^(ßaY+j^^A<äy (5.6.11b) Since fivtlJw and Am are in general small, the frequency at which wave damping or growth is maximum is only very slightly above coP. We note that when vfh = kTJm, that is, for a Maxwellian distribution of electron velocities, then ax\D the debye length, which is AjQ = %/cu!,= (e0kTjne2)Yi. The damping is found to increase rapidly at plasma wavelengths (A = 2W/3) shorter than about one debye length, that is, when the phase velocity is about equal to the mean thermal velocity. For the general, non-Maxwcllian case, the damping depends on the slope of the distribution function as. seen in (5.6. lib). From (5.6.1 lb) the growth or damping rate in time is ' 2/S2 \dvjv (5.6.12) In terms of space, the corresponding growth or damping rate in (5.6.7b) is v9~ 2/3 \dvjv where v„ is the group velocity. 5.6 Spacecharge waves in a warm plasma 187 -1000 -500 0 500 ß [radians/meter) 1000 l-'IG. 5.19 The w-p diagram for spacecharge waves in a warm, infinite plasma for different values of debye length Au. In the steady-state condition (that is, df-, dr ■ 01, the damping vanishes, and we recover the form of dispersion relation obtained in Section 3.3, or m (5.6.14a) (5.6.14b) Equation (5.6.14) is plotted as an to-ß diagram in Fig. 5.19. In certain theoretical cases the damping does not have the exponential dependence shown here, but rather follows an inverse power law, a much weaker dependence (Weitzner, 1963). In Fig. 5.20, the debye length for various electron temperatures is plotted against plasma density to show the orders of magnitude involved in experiments. Figure 5.19 shows that the propagation phase velocity is very slow for values of debye length typical of many laboratory plasmas. The group velocity, dcojdß, is also slow, tending to zero as to > Si ........ ...... \ i i i i i li 1 "i>UIII J>KIM i rrnin 10B 109 10'° 10" 1012 Electron density |cm_3| IO13 101 FIG. 5.20 Debye length in a plasma vs. electron density, as a function of electron temperature (see also Fig. 2.5). 5.6.2 Spacccharge waves in a warm, magnetized plasma. To derive the wave dispersion relation in this case, the Boltzmann equation (5,6.1) again is used but the vx B term must be included in the electric field l+v.V^-^E + vxBJ.^d)^ (5.6.16) The Vlasov form of (5.6.16) (that is, omitting the collisional term) has been solved (Berstein, 1958) in the small signal case by perturbation analysis, using Laplace transforms. The waves can no longer be treated as one-dimensional because the medium is anisotropic. Again, as in the discussion in Chapter 1 of electromagnetic waves at arbitrary angles to the magnetic field, we assign vector properties to the propagation coefficient y( &;p). The frequency will be complex to allow for wave growth or damping.2 After the Laplace and Fourier transforms have been applied to the distribution function and Maxwell's equations substituted in, the longitudinal wave part of (5.6.16) becomes where íú — w+ja0 2ß.E + E.Q-|S = ß-a (5.6.17) This linear treatment of course describes only the initial few e-foldings of any growing waves present, and not the final, saturation, equilibrium conditions of wave propagation. The latter would have to be arrived at with a nonlinear treatment (Drummond and Pines, 1961 ; Sturrock, 1961). 5.6 Spacecharge waves in a warm plasma 189 Q = a triple integral equation similar to (5.6.7) but dyadic and involving 0, the angle between p and B, and including the electron gyrofrequency u>b. a = a vector quantity involving E, B, and |i, and the triple integral of the velocity distribution function over velocity space, evaluated at ( = 0. The transverse wave part of (5.6.16) has already been discussed in Section 3.4, at which point Landau damping was introduced. Solutions of (5.6.17) cart be found that yield either wave growth or damping, depending upon the slope of the tail of the distribution function (for example, a double-humped distribution leads to growth). The analogous situation for transverse waves may not occur, that is, the growth modes seem to be absent. The coupling between the longitudinal and transverse waves is, in general, small (Sturrock, 1961; Bevc and Everhart, 1961). The general solution or (5.6.17) is exceedingly complex. Some simplified cases yield convenient results, however, and we shall discuss these. For a small magnetic field and a Maxwellian distribution at low temperature, the dispersion relation is found by asymptotic expansion of (5.6.17). For the conditions «)2»aj(2, P2) sin30 is really the parameter, the curves also apply to other angles such that the products are 0, tVj 1» and 4. The imaginary part of -p diagram for spacechargc waves in a warm, magnetized plasma. The curves are for constant ((«„/(«„) sin 6 but are labeled in tt/2, that is, propagation across B, the Landau damping approaches zero, and within the passbands there is no wave attenuation. There are stop bands, however. For particular values of co„ and pt. there are gaps in the propagation spectrum as fizXD2 is varied. For /32AD2«1 the spectrum for longitudinal waves propagating across the B field is given by Bernstein (1958) as q-2 a P Pe a>* (y7e2j32 sin2fi) 1 +- 1 W I \°>b 1 (5.6.23) Poles are seen to exist for u> — uib and its harmonics. A plot of a>jwh vs. fP\Dz is shown in Fig. 5.22. The poles are clearly evident bounding the passbands and stopbands. 3 Cil 2 1 ! ^^^^ 0 II -2 -1 0 mi FIG. 5.22 Typical m-fi diagram for spacecharge waves propagating across the magnetic held (6 = 90"). CHAPTER 6 Microwave propagation experiments 6.1 Transmission-attenuation and reflection experiments One of the simplest microwave diagnostic experiments is a transmission-attenuation measurement (Wharton et al., 1955) in an isotropic plasma. Two radiators are arranged so that the path between them passes through the region to be studied, as sketched in Fig. 6.1a. When the plasma density in the path reaches a value high enough that (a) Horn antenna Signal detector (receiver) r ' Video output signal Reflection signal detector Pad attenuator Video output, reflection signal Calibrated I i Impedance attenuator i i Directional transformer coupler (ret lee to meter) Frequency meter Signal source Video output, transmission signal PIG. 6.1 Block diagrams of basic microwave transmission experiments in a plasma, i liagram h shows the addition of a directional coupler i.o permit observing reflections from the plasma. coefficients. Two additional components added to Fig. 6.1a allow us to measure the over-all reflection from the plasma: a directional coupler and an additional detector. A typical composite circuit is sketched in Fig, 6.\b.1 The complete circuit includes a source of microwave power 1 The microwave components or "hardware" are described in some detail in Chapter Al this point, vjjc treat components as blocks having certain properties, in building up the circuits. Readers wishing to familiarize themselves with the waveguide del ails are referred to Chapter 'J and to the references listed there. 194 Microwave propagation experiments Chap. 6 (usually, a klystron or backward-wave oscillator), a frequency meter, a calibrated attenuation standard, a directional coupler (which couples only the reflected signal into the "reflection signal detector"),an impedance-matching transformer by which the fixed reflections are canceled, a vacuum window and transmitting antenna, a receiving antenna and vacuum window, a level-setting pad attenuator, and a microwave detector. 6.1.2 Transient plasmas. Often the plasma to be studied is of transient duration. Shock waves and controlled-fusion containment experiments (Post, 1958) are examples. The electron density rises rapidly and then 1,50 - -wp mt*& (110 Gc) .Electron density n FIG. 6.2 Transient plasma event. The microwave transmission and reflection signals at a frequency of 90 Gc are shown, in relation to assumed density and collision-rate temporal variations (a), (b) Microwave transmission signal, detected by a silicon diode. Transmission path L through plasma is 6 inches, (c) Microwave reflection signal, detected by a silicon diode. The term 100% refers lo reflection from a copper cylinder placed at the location of the plasma. 6.1 Transmission-attenuation and reflection experiments 195 mm ■in ■■■■ 9 ■HI 111 ■■■■ ■■■■ ■ II ■ ■ rnmmrn ■■ BIG, 6.3 Reflection and transmission signals for microwave propagation through a transient plasma. The frequency is 90 Gc. The amplitude of the reflection spikes corresponds to about 25% reflection. Time scale is 20 microseconds per division. decays more slowly. As an illustrative example, let us consider the use of Ihe circuit of Fig. 6.16 to diagnose a transient plasma having no magnetic field applied.2 A typical event is sketched in Fig. 6.2a. The dashed line represents the density at which o>p We estimate, from other measure- ments, a cosine spatial distribution of density, 23 wavelengths across at 90 Gc. A peak central density of 1.5-1014 cm-3 corresponds to a cutoff, frequency of 110 Gc. This plasma column has density gradients steep enough to give small external and internal multiple reflections. The electron temperature in the event chosen reaches only about 2 eV, so that vjio is around 0.02; the internal reflections, thus, are partially damped by collision losses. The received signals, as detected by a square-law detector (silicon crystal diode) are sketched in Fig. 6.2b. The small fluctuations, due to multiple reflections, are evident. The reflected signal will depend upon the steepness of the gradient, since the signal must penetrate a lossy layer of plasma to reach the reflecting, cutoff plane and then pass through the layer again after being reflected. The magnitude of the fluctuations, then, gives a qualitative estimate of the steepness of the gradient. A typical reflection signal is sketched in Fig. 6.2c, where 100% corresponds to the reflection from a sheet of copper at 1 The following data were taken by one of us (C13W) at the University of California, I awrencc Radial ion ^Laboratory, on a high-density pulsed plasma experiment (Hester Bnd Reagan, I960), in the Controlled Fusion Program. We thank the I hiivcisily of California for releasing (his unpublished data. 196 Microwave propagation experiments Chap. 6 6.2 Frequency diversity 197 the location of the plasma. The over-all reflection coefficient is seen to be less than 25% for these data. This is typical of many plasma experiments, including some controlled-fusion experiments; the plasma looks either transparent or surprisingly "black." Oscilloscope traces of reflection and transmission are shown in Fig. 6.3. The spikes of reflection correspond to about 25% and, presumably, arise from steep gradients due to turbulence. The rise rate of density is very fast compared to the decay rate. 6.2 Frequency diversity The simple reflection-transmission experiment, just discussed, gives information only when the electron density is near cutoff, that is, ajpZaj. Additional temporal information can be gained by using more than one mm-'m i _ -'■mmmmm^r::- mm. I'lG. 6.4b Photograph of discharge chamber. Side arms containing the microwave horns and vacuum windows are shown. The two 70 Gc (4-mm band) horns arc mounted in place. One of the 90 Gc (3-mm band) horns is shown removed. I he black teflectionlcss coating inside the chamber can be seen in the vicinity of the horns. The chamber internal diameter is 3 inches. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) 198 Microwave propagation experiments Chap. 6 6.2 Frequency diversity 199 -tep = o> at 110 Gc Plasma density as a function of time oip = oi at 90 Gc — ü>p = oi at 70 Gc FIG. 6.5 Transient plasma event. Microwave transmission and interferometer responses, using square-law (silicon diode) video detectors, at 70 and 90 Gc arc shown in relation to the assumed electron density variation in time. frequency, so that cutoff is reached at different times. Let us take the transient event again, but employ two identical transmission paths, one using 90-Gc and the other 70-Gc equipment, simultaneously. The arrangement of radiators used is shown in Fig. 6.4a.5 The horn radiators were \ inch in diameter, separated 3^ inches. The 4-inch dimension, calculated from (4.6.4), avoids Fresnel interference problems but gives poor coupling between the horns. Radiators of 1-inch diameter were 3 Sec footnote 2. FIG. 6.6 Transmission amplitude signals for microwave propagation through a transient plasma. Frequencies of 90 Gc (top trace) and 70 Gc (bottom trace) were used. tried, giving much closer coupling, but the Fresnel interferences were excessive. Since the walls of the chamber are only a few wavelengths away from the transmission path, stray scattering of the diverging portions of the wave leads to spurious interferences. It was found necessary in this experiment to coat the walls of the chamber with a nonreflecting material4 to eliminate these interferences. A photograph of the chamber, showing one of the vacuum windows and the absorbing coating, appears in Fig. 6.4/3. The idealized density variation is sketched in Fig. 6.5cr, with the cutoff conditions shown by dashed lines. The resulting transmission signals, for square-law detectors, are sketched in Fig. 6.5b. Figure 6.6 shows an oscilloscope trace of measurements on a transient plasma event, recorded at 70 and 90 Gc. Small fluctuations (< 5%) are seen on the trace, evidence of internal reflections. The over-all reflection signal is fairly small. 6.2.1 Frequency diplexcrs. A feature that is immediately evident is that unless a plasma in the configuration of Fig. 6.4 is symmetrical about the axis (it often is symmetrical) the two transmission paths do not traverse the same thickness of plasma. To ensure the same plasma sample L for both frequencies, the two paths may be incorporated into one pair of antennas by frequency diplexers.h The two signals are combined into one ' Bee Section 9.6.4 for descriptions of such materials. n See Section 9.2.3. 200 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 201 waveguide, transmitted through the single path and, then, separated to two detectors. 6.2.2 Polarization Diplexers. Polarization diplexers, such as fin-line couplers? can also accomplish the function of frequency diplexing, not by using frequency-selective filters but by orienting the two wave polarizations normal to. each other. In an isotropic medium, free of nonlinear effects, the waves will not generally couple to each other, and will travel through the same effective path length. In an anisotropic plasma, the two waves of different polarizations will couple to different propagation modes. These effects are discussed in Section 6.5. 6.3 Phase-shift measurements Without much additional complication, the circuit of Fig. 6.1 can be expanded to become a phase-measuring interferometer, analogous to the Mach-Zehnder interferometer used in optics. A null path is necessary to provide a phase and amplitude reference with which to compare the transmitted signal (or the reflected signal). 6.3.1 Microwave interferometer. The interferometer or phase-bridge circuit (Wharton and Gardner, 1959), sketched in Fig. 6.7, is particularly useful in measurements on transient plasmas. The reference path and transmission path are kept the same electrical length to avoid differential phase changes if the klystron frequency drifts. The circuit shows three detectors. Only the one labeled "phase detector" gives phase-shift information. The "transmission-amplitude detector" plays the role of the detector in Fig. 6.1. In typical operation with transient plasmas the reference path is adjusted to null with the signal path in the absence of a plasma; the phase shift and amplitude then are observed as the plasma fills and then leaves the test path. The signal levels required depend upon the kind of detectors and video amplifiers used but, ordinarily, are in the microwatt-to-milliwatt range.7 An interferometer that is initially nulled will produce a maximum signal when the angle of the aggregate transmission coefficient has shifted 180°, and will return to a null at 360°, repeating indefinitely as the angle rotates through successive values of tt and 27r. A square-law detector will yield a sinusoidal variation as the plasma density changes. 0 See Section 9.2.3 for this and other polarization diplexers. 7 Co mm only, the detectors are silicon diodes. Up to about 100 microwatts, a silicon diode has a fairly faithful square-law response, that is, the output voltage is proportional to input power. The input-output characteristic then begins to straighten out, until at a few milliwatts the characteristic is nearly linear. Most microwave crystals are damaged by power levels in excess of 10 milliwatts. Microwave source (klystron, etc.) vVV^ Output attenuator f Power divider NULL PATH 0-360° Phase shifter Pad attenuator Crystal detector (phase) 6-dB 1 Coupler %\\^~^ 1 t J1 / 1 [^comparator Crystal detector (reflection) /vV} Frequency Kl>> meter vWi£4 I 6-dB tf\ Coupler Reflection signal output Receiving horn antenna .Vacuum window in wave guide SIGNAL PATH Video p ream p. 0-5 Mc Interferometer output signal Calibrated attenuator Crystal detector [(amplitude) Direct transmission output signal FIG. 6.7 Microwave interferometer and rcflcctometcr for plasma diagnostics. ■ 202 Microwave propagation experiments Chap. 6 ■ ■i FIG. 6.8a Microwave interferometer responses to a transient plasma. Frequencies of 70 Gc (top) and 90 Gc (bottom) were used. Sweep time was 100 (tsec/cm. Small calibration pulses at 750 jiascc indicate that the bridges were slightly off null at the time the traces were made. Oscilloscope traces of the interferometer responses to the same event as recorded in Fig. 6.6 are shown in Fig. 6.8. Cutoff for the two frequencies is reached at different times as the density falls from maximum to zero. A different number of variations or fringes8 are noted for the two frequencies, due to the different number of effective wavelengths in the path. The attenuation of the signal is evidenced by the amplitude envelope of the fringes. In cases where the collision rate is high, so that signal cutoff is reached at a density somewhat below nc (see Section 1.4), the observed number of fringes is less than expected for a given path length and density distribution. Such a situation is shown in Fig. 6.8A. The top set of records of this figure was made from a discharge between plane electrodes in helium at a pressure of 80 microns. The responses at the bottom of the figure were made by adding 200 microns (partial pressure) of argon, to increase the collision rate from a value of ~0.001 to ~0.01 to at the time of (iip = a>. The damping is evident. The presence of the plasma under certain conditions increases the coupling between the horns above the vacuum level (not to be interpreted as amplification!), and introduces multiple interferences from stray scattering around the plasma. These effects are especially noticeable when the horns are poorly aligned with each other, when the plasma column is only a few wavelengths across, or when there is an impedance mismatch. G Analogous to the fringes observed in an optical interference pattern on a Mach-Zchnder or Fabry-Pcrot interferometer. 6.3 Phase-shift measurements 203 FIG. 6.8b Microwave interferometer responses to a transient plasma, showing the elfect of low and high collision frequencies. Transmission frequencies of 70 Gc (top) and lJn Gc (botiQjn) were used. The top pair of records was made in a helium discharge al a pressure of 80 microns. The bottom pair was made after 200 microns of argon had been added. 204 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 205 iiii; I 19 il It H i I SI Fill Ifi aw 'Si M if? SI ■ s n ■ m i ■ s FIG. 6.8c Microwave interferometer response to a transient plasma showing the effect of horn misalignment. The top record was made with the 70-Gc horns misaligned by 20° in a discharge chamber 8 centimeters in diameter. The 90-Gc horns (bottom record) were properly aligned. The effect of misalignment is shown in Fig. 6.8c to cause a distortion of the shape of the fringe envelope. Photographs of a 7^-band (25 Gc) interferometer and a 3-rom band (90 Gc) interferometer are shown in Figs. 6.9 and 6.10. Components are arranged on panels for rack mounting. Coiled lengths of waveguide or flattened copper tubing9 are mounted behind the panel to compensate the length of the null path (or the plasma path, whichever is longer) to avoid differential phase changes if the klystron frequency drifts. 6.3.2 if Modulation envelope. The interferometer circuit of Fig. 6-7 may be used with an rf modulated carrier. The klystron is amplitude modulated, at say 500 kc to 30 Mc, and the amplifiers following the various detectors are tuned to the modulation frequency. The system then resembles a superheterodyne receiver, with the i.f. frequency being that used to modulate the klystron; the name pseudosuperhet is sometimes applied to this system. The i.f. modulation envelope may be demodulated but, for frequencies below about 2 Mc, is better viewed directly on the oscilloscope. The transmission amplitude signal will appear as shown in 9 See Section 9.1. FIG. 6.9. X-band (22 to 27 Gc) panel-mounted microwave interferometer for fringe-shift (zebra-stripe) presentation. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) Fig. 6.11 (Buser and Buser, 1962). The interference fringes will appear as shown in Fig. 6.12. The Q of the tuned amplifier must not be too high if rapid fluctuations are to be followed. The ampliiier band width must be at least 2/tm where rm is the time from crest to crest of an interferometer fringe. A useful preamplifier circuit is shown in Fig. 9.50. The no-plasma signal level is easy to see with the modulation present, even without d-c amplifiers; this makes a system that is easy to keep in adjustment. If the two arms of the interferometer bridge are approximately the same length, the incidental microwave frequency excursion of the klystron caused by the modulation will not cause problems with differential phase shift. The modulation envelope will be symmetrical about the base line if the amplifiers are linear. Stray pickup, which sometimes can shock-excite tuned circuits, wilf'be asymmetrical and will cause the base line to bow; pickup is thus easy to distinguish from the real signal. 206 Microwave propagation experiments Chap. 6 6.3 Phase-shift measurements 207 I! FIG. 6.10 A 3-mm band (89 to 95 Gc) panel-mounted microwave interferometer. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore, Calif.) 6.3.3 Fringe-shift or zebra-stripe interferometer. A more sophisticated interferometer is the "fringe-shift" interferometer (Wharton and Gardner, 1959; Heald, 1959c). In this type of data presentation, the phase shift is plotted directly on the oscilloscope, and the effects of amplitude variations FIG. 6.11 An rf modulation envelope presentation of attenuation due to transmission through a plasma. The notches are time markers. (Photograph courtesy of R. Buser, Ft. Monmouth, N.J.) FIG. 6.12 The rf modulation envelope presentation of interferometer fringes for the same plasma event as that in Fig. 6,11. (Photograph courtesy of R. Buser, Ft. Monmouth, N.J.) Microwave source (swept-freq. klystron, etc.) Reseller signal Ferrite isolator 10-rJB Coupler Sawtooth generator (a a These values are tabulated in Table 6.1 and plotted in Fig. 6.20. The plot of a trapezoidal distribution with b = dj6 fits within the experimental points, and probably represents a good approximation to the true distribution. A rough check with Langmuir probes showed the profiles to be flat-topped, giving further correlative evidence with the microwave data. Profile information from plasmas in a magnetic field may be obtained by methods discussed in Section 6.5.1. FIG. 6.21 Photograph of the Little Fig diagnostics correlation experiment. At the left is a 24 Gc (K-band) radiometer; right is a crystal video receiver for the interferometer. The vacuum flange on the left end carries one of the cathodes and an ion gauge. The front flange has a waveguide vacuum window, a voltage isolation section, and a Langmuir probe. The top flange contains a movable (Wilson) vacuum seal to permit the radiation-receiving antenna to be moved in and out of the plasma chamber. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livcrmore, Calif.) 216 Microwave propagation experiments Chap. 6 ii FIG. 6.22 Zebra-stripe data display of a transient plasma at four frequencies simultaneously. The frequencies are: 18 Gc, tower left; 22.6 Gc, upper left; 24.6 Gc, lower right; and 33 Gc, upper right. At top right is the anode current trace, one 6.4 Density distribution: profile measurements 217 9h ampere maximum. At lop left is the plasma radiation display, as received with a 24-Gc superheterodyne receiver. Time scale is 1 millisecond per major division. Phase-shift data at 1 = 3 msec is given in Table 6.1 and plotted in Fig, 6.20. 218 Microwave propagation experiments Chap. 6 Table 6.1 Tabulation of data from a plasma sample at five frequencies* /[Gc] 17.6 18.0 22.6 24.6 33 Xjd 0.200 0.206 0.1 64 0.151 0.112 folic 1.00 0.96 0.61 0.51 0.28 N 4.2 3.6 1.8 1.5 1.0 qy 0.11 0.26 0.70 0.77 0.89 * Repeatability of measuring N is ±i Iringe. The values of N arc taken from Fig. 6.22 at / = 3 milliseconds. Plasma diameter f/=S.l cm. FIG. 6.23 A seven-beam, 70-Gc focused probing system. Seven transmitting horns (/)), fed by dividing the power from a single klystron, are focused on the plasma by a lens (F). The seven receiving antennas (B) and split reference paths (C) feed signals to the waveguide junctions (D), which in turn supply the video detectors and preamplifiers (E). The on-axis vertical resolution of each beam is ~ 1 cm. (Photograph courtesy of R. 1. Primich, General Motors Defense Research Laboratory, Santa Barbara, Calif.) 6.5 Magnetic field effects 219 Geometrical scanning can be used to obtain profiles if the microwave beam can be focused to a diameter considerably smaller than that of the plasma. Either Abel integration or trapezoidal approximations of the transverse variations in depth of plasma can be used. A multiradiator system, developed by Primich and Hayami (1963), provides focusing in the transverse plane by feeding a common 10-inch diameter, long focal length lens with seven horns. The resolution obtained on axis ( — 10 dfi beamwidths) of each beam was 0.5 inch at 35 Gc and 0.25 inch at 70 Gc. A photograph of the apparatus is shown in Fig. 6.23. The multiple horns and the interferometer couplers appear at the right. The resolution is improved by having E-field polarizations of adjacent beams at 90° to each other. 6.5 Magnetic field effects When a magnetic field is present in the plasma, there are additional quantities to measure. When we propagate a wave across the field lines (# = 90°), we are able to couple to either the ordinary or the extraordinary wave. By using a square horn and waveguide and separating the two waveguide waves (E-fields at right angles with each other) with a polarization diplexer, we can transmit both waves simultaneously and make observations on a dual beam scope. When we propagate along the field lines (6 = 0°), we can measure the Faraday rotation (Section 1.4.2) either by rotating the receiving horn or, again, by using the fin-line coupler to compare the relative magnitudes of the two (x and y) components. When the phase shift and Faraday rotation are combined, the bridge output shows amplitude fluctuations superimposed on the phase fringes. A 180° phase reversal also occurs with each half rotation. Losses in the plasma affect the two wave types differently, leading to ellipticity. An accurate measurement of Faraday rotation is thus difficult, since both the relative and absolute amplitudes of the x and y components are changing. When one of the circular polarizations is attenuated to cutoff, the Faraday rotation ceases, and both the x and y components behave alike, since there is only one wave remaining. If circularly polarized antennas are used to study these waves, the two counterrotating waves can be studied independently. If the walls of the discharge chamber are close to the plasma, so that currents can flow on them, some ellipticity will result and true right-hand and left-hand waves do not exist. Also, when density and magnetic-lield gradients are present, the modes are no longer clean. We observed in Fig. 1.19 that at o> = a>v, when 0 is not quite 0° and the collision rate is very low (Section l?4.10), the left-hand wave jumps down to become the right-hand one (which is then cut off) and the right-hand wave jumps up 220 Microwave propagation experiments Chap. 6 6.5 Magnetic field effects 221 out of cutoff to become the left-hand one. We must, therefore, use caution, as pointed out in Section 1.4.4, when labeling waves as "left-hand" and "right-hand." 6.5.1 Ordinary and extraordinary waves: density profiles. The ability to propagate two wave types at the same frequency over the same path allows us to obtain considerable additional data. For particular experimental conditions, the appropriate frequency (or frequencies) is obtained by reference to Figs. 1.2 and 1.13?, The values of refractive index vs. plasma density and magnetic field for the ordinary and extraordinary waves are given in Figs. 1.6 and 1.15, respectively. These values, of course, apply only at one particular location within the plasma; if density and/or magnetic field gradients are present (as they invariably are in real experiments), the aggregate transmission properties must be obtained by the methods discussed in Sections 4.2, 4.3, and 4.4. In the large-plasma case, where the variations of density are gradual in the space of a wavelength in plasma, the adiabatic analysis applies to the total integrated transmission coefficient of the extraordinary wave as well as to the ordinary wave (Section 6.4). If the collision rate is low enough to be ignored, the phase shift for the extraordinary wave in a plasma path of extent d, may be approximated by where, from (1.4.50) 2tt f 1 —(üp2jw2 — cob2jo)'- (6.5.1) (6.5.2) If, in addition, we are content to operate in the range in which tojm < 1 and cojoj < 1 — i0bla>, we may write the phase shift in the form of (4.2.2) zJ0; -27 n. ij dx (6.5.3) in which the cutoff hc is no longer specified by ui2\m2-± 1, but rather, on account of (1.4.52), by wp2jta2—^ [± -1 -2 A/2tt (parallel) (jf-= 70 Gc) -3 FIG. 6.25 Phase-shift data plotted on theoretical curves for transmission through a plasma of trapezoidal density profile across a magnetic field. The values arc for phase shift of the extraordinary wave (perpendicular polarization) plotted against phase shift of the ordinary wave (parallel). Experimental points arc from aStellarator discharge at Princeton University; to„jio = 1.25. (Courtesy of R. Motley and M. Heald, Princeton University, Princeton, K.J.) obtained from the B-l Stellarator (Coor et ah, 1958) at Princeton. The frequencies chosen were 35 Gc for the extraordinary wave (K J_ B), and 70 Gc for the ordinary (E || B), to optimize the sensitivity and measurable density range. The magnetic field intensity, and thus o>b, were known very accurately. The fringe-shift (zebra stripe) interferometer prcsenta- 6.5 Magnetic field effects 223 tions are shown in Fig. 6.24. The phase reversal as the refractive index goes through unity is apparent in the 35 Gc trace at 1200 /usee. No phase reversal is seen in the 70 Gc trace, since the index for the ordinary wave is always less than unity. Plots of the calculated phase shift for the two waves at various values of hid, together with experimental points, are shown in Fig, 6.25. The points are seen to follow the plots of bjd—1 (uniform distribution) very closely. This is a reasonable conclusion for the Stellarator, since the plasma is defined by orifice plates at the diameter d to which these data were normalized. 6.5.2 Faraday rotation. When the ends of a magnetized plasma column are available so that antennas can be mounted with their radiation patterns looking along field lines as shown in Fig. 6.26, the circularly polarized waves can be studied, Electrodeless discharges (rf or pulsed) (Lisitano and Tutter, 1961) and plasma compression or confinement experiments (Consoli et ah, 1961) usually lend themselves to such arrangements. Occasionally, discharge experiments are of a configuration that permits small radiators to be inserted directly in the discharge electrodes (Mahaffey, 1963) without upsetting the plasma uniformity. If no access from the ends can be arranged, it is sometimes possible to insert curved dielectric rod antennas from the sides, as shown in Fig. 6.27. In low-collision plasmas, having small electron orbit sizes, the rods cast a "shadow" along the field lines, but in dense plasmas, or plasmas whose ions or electrons have gyroradii larger than the probe diameter, the rods seem to give little perturbation. Rods made of boron nitride or glass-bonded mica have low loss and low vapor pressure, are highly directive, are refractory, and are easily machined. An example of the use of such rods with horn radiators is shown in Fig, 6.27. The propagation and radiation characteristics of dielectric rods are given in Section 9.3. To confine the radiation pattern to the desired volume, the curvature of the Transmitter horn (^-oriented) x/2 Flat./ quarlz window o o o o d o o r^^P^//; Plasma ^^^^^ o o V 0 o Fin-line ■~~~^coupler j-Component detector x- Component detector FIG. 6.26 Faraday rotalion experiment in rf-excited plasma in steady magnetic field. 224 Microwave propagation experiments Chap, 6 6.5 Magnetic field effects 225 Ware-la unc hing ' horn Vacuum window in waveguide Dielectric rod— Vssium__■„ chamber 111 FIG- 6.27 Curved dielectric rods used to launch waves along the magnetic tield (or at an angle to it) in a plasma. bend must be gradual, and the diameter must be held large enough that most of the fields are within the rod until the taper at the end begins. Some photographs of curved dielectric radiators are shown in Fig. 9.37. Because the waves transmitted through the plasma seldom emerge with purely circular polarization, it is difficult to make quantitative measurements of Faraday rotation. Nevertheless, when differential attenuation of the two counterrotating waves is not excessive, a measurement of the total rotation gives information about the plasma density in a path along the experiment axis. If the magnetic field is known as a function of position, such as in low £ plasmas11 confined by external fields, the spatial distribution along the axis may be estimated by techniques similar to those described in Sections 4.2, 6.4, and 6.5.1. The total rotation is obtained by integrating (1.4.24). (6.5.5) ] rd 1 rd In the case of low-collision-frequency plasmas the approximation of the phase constant $ given in (1.4.20) can be used (6.5.6) The Faraday rotation then becomes m rr n(z) l nc \ + B{z)jB 1 «, 1- B(z)iB. -1 dz (6.5.7) where nc is the density to give cutoff without a magnetic field and BR is the magnetic field to give gyroresonance. 11 ß, here, refers not to the phase constant, but is the ratio of the plasma kinetic pressure to magnetic pressure, ß = nkTj(B'2j2nB), a notation commonly used in con trolled-fusion research. 1.0 / Balmer series limit Phase shift Faraday rotation 1.5 2.0 2.5 3.0 Anode voltage of RF power oscillator, kv 3.5 FIG. 6.28 Electron density in rf discharge, measured by three methods, as functions of rf level. (Compiled from data presented in Lisitano and Tutter (1961) and von Gierke et al. (1961),) Equation (6.5.7) is difficult to integrate, but if B is constant or may be approximated by a stairstep function or a trapezoid, the same concept of an effective cutoff density as explained in Section 6.5.1 may be used for ü)„/ü)< 1 and a»6/«>1 (see Figs. 1.25 to 1.28). Both the density and magnetic lield increase, at first, but then the field falls to the value at the center of the chamber. The angle 8 has been changing slightly along the path due to field curvature. At the center, 8 is the smallest although the density is (usually) high, so that resonance will be achieved here first as the density rises and last as the density falls. The paths at 0 = 0" and 90° yielded resonances at (di,/tu=l in their respective regions; the 8=30° path gave a resonance at a lower field strength. The foregoing experiment was far from a clean, quantitative one. 228 Microwave propagation experiments Chap. 6 Pulsed field coils 0° —C^1 Microwave horn Magnetic field lines 90° Horn FIG. 6.30 Propagation at angle to the magnetic held lines in a mirror-compression experiment. Nevertheless, the principles involved may be applied to certain classes of experiments where the parameters are under more precise control of the experimenter, in which case the results will be meaningful. Coupling to the ordinary resonance of Figs. 1.27 and 1.28 can occur only through wave coupling or by "tunneling" of evanescent waves through cutoff regions. The resonance in either case would be quite small and easily masked by other spurious fluctuations. If it could be detected, however, the measurement of its position in frequency would extend the measurable plasma density a factor of 4 or 5 at angles near 60°. 6.5.5 Doppler-shifted gyrofrequency in drifting plasmas. When the electrons in a plasma are drifting along magnetic field lines, the electron gyrofrequency is doppler-shifted in the laboratory frame of reference. A circularly polarized wave, directed along the field lines, will then have a different transmission coefficient in one direction than in the other, especially noticeable if the frequency is near the electron gyrofrequency. The coupling rods shown in Fig. 6.27 allow transmission from left to right or from right to left. If the plasma being studied is drifting in either direction, the measurement of transmission coefficient will be non-reciprocal. The amount of nonreciprocity will depend upon the derivative of the curve of refractive index vs. frequency,12 the maximum effect occurring at frequencies slightly below gyroresonance. The eifect, which is somewhat analogous to the Fizeau effect, is not an easy one to measure in practice unless the magnetic field and frequency are 12 For example, see Fig. 1.9, obtained from (1.4.17). 6.6 Propagation through fluctuating plasmas 229 well known and constant during the measurement. The change in wavelength in the moving medium is AX= ± Xfiv0jc (6.5.8) where A is wavelength, and p is the refractive index, and v0 the drift velocity. The phase velocity in the moving frame is then c c dp .. c cX dp va (6.5.9) ft p CIA p p UA C The phase shift observed in path length L from the left or right sense is thus 2nLp_1 radians. (6.5.10) As an example, consider that the plasma electrons are drifting with 10 eV of directed velocity (c0 = 2 108 cm/sec). Near gyroresonance, X dfijdX may be between 10 and 100, and ft may be between 2 and 10, respectively. The differential phase shift for a given path length LjX may, thus, be between 0.07 (2ttL/A) and 23 (2ttL/A) for the example chosen. 6.6 Propagation through fluctuating plasmas A plasma whose electron density is fluctuating periodically may phase-modulate and amplitude-modulate an electromagnetic wave propagating through it. These effects are in addition to the Luxembourg effect (due to temperature fluctuations), discussed in Section 2.6. For example, an examination of the frequency spectrum of a signal transmitted through an rf-excited plasma invariably reveals side bands at frequencies displaced by the rf-excitation frequency and its harmonics (von Gierke et al., 1961). Figure 6.31 sketches such a spectrum, where fm is the rf driving frequency. The harmonics, if present, presumably arise from nonlinearities (Dreicer, 1961). /() - 2/m fa - fm fo ft) +fm f0+ 2fm FIG. 6.31 Frequency spectrum of wave propagated through a plasma fluctuating at frequency /„,. 230 Microwave propagation experiments Chap. 6 Suppose that the electron density is fluctuating sinusoidally with a small amplitude a, at frequency wmj2-n- h(/)=«0(1 +acosatmt)=n0+n1(l), where a«l. Ignoring damping, for the moment, the refractive index is then (6.6.1) (o2 im0u)2 «i(0 = /V-^i2(0 (6.6.2) where the subscript 0 represents steady-state quantities and the 1 represents first-order perturbation quantities. If the frequency of fluctuation cura is slow, the wavelength of the disturbance in the plasma will be large, and the electromagnetic wave as a whole will be equally affected.1:5 There is not the limitation on modulation frequency here as we found in the Luxembourg effect, since here we do not have a relaxation time to consider. For l/^xl smaller than about 0.2^n, the time-varying index may be approximated by expansion ~/*oO-\a coswn0- (6.6.3) The perturbed electric field is then obtained from a perturbation solution of the wave equation, E{() = E0 exp exp [piaf* a cos«jmr (6.6.4) Tf a (or «0 is not small, these linearizations are not valid and the calculation must begin with substitution of nx into the Maxwell equations, either in terms of a fluctuating conductivity or a fluctuating dielectric constant. The Poynting flux calculated from solutions of Maxwell's equations then will give the frequency spectrum and the component magnitudes. To find the frequency spectrum of (6.6.4), we must find its Fourier transform, E{o>). The transformation is of the form exp(/'.v cos0) = ^ ./"JnC*) cos(n0). (6.6.5) 13 If tam is large, so that the wavelength of the plasma disturbance is comparable to the electromagnetic wavelength and to the extent of the region being probed by the microwave beam, the present analysis will not be valid. The reader is referred to Sections 6.7.1 and 6.7.2 for the latter case. 6.6 Propagation through fluctuating plasmas 231 Expanding (6.6.4) according to (6.6.5) and taking the real part (since we are interested in the transmitted power) yields E(w) = E0 exp -j 0(^f> cosc 2wm)r] + .A. J 3"m)0 (6.6.6) Equation (6.6.6) is seen to have a frequency spectrum such as that sketched in Fig. 6.31, with the amplitudes of the side bands given by the Bessel coefficients. An examination of (6.6.6) shows that the fundamental frequency component vanishes if the argument of J0 is such that Jo-^0; that is, TrL/ioC/A equals 2.40; 5.52; 8.654-mr. Likewise, the first side band vanishes for the Jt zeros, etc. In fact, however, since we have used a perturbation analysis, tacitly assuming that E remains essentially unchanged, the Bessel arguments must remain much smaller than unity. For larger arguments, the spectral appearance will be qualitatively similar but the relative magnitudes of the side bands will not be correctly given by (6.6.6). In warm plasmas a collisionless Luxembourg effect (see Section 2.6.3) may occur, when a wave suffering Landau damping propagates through a lluctuating plasma. If the fluctuations are in density, as in (6.6.1), we see from (3.5.6) that changes in Wj, influence the attenuation exponentially, leading to possibly large amplitude modulation for small density changes. If it is the slope of the velocity distribution, 8f0,l8v, that is fluctuating, due to some nonthermal effect for example, the attenuation again is affected exponentially, as shown by (5.6.13). The amplitude modulation introduced by fluctuations in the collisionless damping produces sidebands similar to those shown in Fig. 6.31, although the spectrum may be asymmetrical, due to heavier damping at shorter wavelengths. The upper limit in modulation frequency, imposed by the relaxation time in collisional Luxembourg modulation, is not present in the collisionless case, since I here is no analogous relaxation phenomenon. A wave propagating through a fluctuating plasma then can suffer both amplitude and phase modulation simultaneously. 232 Microwave propagation experiments Chap. 6 Collisionless damping occurs for spaeecharge waves as well, so that all of the above effects apply qualitatively to spaeecharge wave propagation. One might expect, for example, to see an electron spaeecharge wave modulated by a low frequency ion wave. If the fluctuations are at random frequencies (that is, noise fluctuations), the side bands will tend to run together into a continuous noise spectrum. The phase sense of the transmitted wave then becomes ambiguous. The ambiguity becomes worse for /x-uVo increasing, which occurs as the density approaches cutoff. This effect often is called phase-sense scrambling, but should be distinguished from phase mixing, associated with Landau damping (Section 3.5). 6.7 Microwave scattering experiments The foregoing section has discussed a subject that could also be handled by treating the transmission as coherent forward scattering. Although scattering cross sections for free electrons are in general very small, in the case of a scattering angle 6=0 the phases of all the scattered wavelets are the same, and the properties of the scattered wave are identical to those discussed up to this point in terms of propagation through plasmas (Ratciiffe, 1959). For angles other than 6 = 0, however, we must specify the scattering cross sections and sum up all the wavelet components to find the scattered intensity in a given solid angle dQ. The total intensities are generally small, and stray reflections from walls and obstacles, as well as plasma radiation, may contribute as much (or more) power to the detector as the plasma scattering, unless great care is taken. Typically, the background must be down by at least 90 dB and often 120 dB to be able to detect the scattered signal with a good, broad-band superheterodyne receiver. If coherent detection is used,1'1 the background level can be considerably higher before the scattered signal becomes undetectable. With coherent detection, of course, the response time is necessarily long, which may be a disadvantage. 6.7.1 Incoherent scattering. particles, namely, electrons, Thomson cross section is The scattering from individual charged is called Thomson scattering. The total 9i = y/y^O.66-10"28 m2 (6.7.1) 14 The scatterer is modulated at a low frequency, for instance, 1000 cps, and a phase-sensitive detector compares the received signal with a sample of the modulating signal. Further elaboration is given in Section 9.5. 6.7 Microwave scattering experiments 233 FIG. 6.32 Coordinate system for microwave scattering. where r0 = e%l4TK0mc1 = 2.%-10 15 meter, the classical electron radius. The differential cross section is defined as qa^dQ dq _ energy scattered/unit solid angle-unit time incident energy flux/unit area-unit time = rt)2sin2<9 (6-7.2) where 0 is the angle between the electron acceleration a and the direction of observation r, as shown in Fig. 6.32. In the absence of an external magnetic field (isotropic medium) the electron acceleration will be along E; in an anisotropic medium, this may not be the case. The angular distribution is given by (Jackson, 1962): sin20=l -sin2fJ cos^^-f). (6.7.3a) If the incident radiation is randomly polarized, the distribution is obtained by averaging over 6 sin2 0=i(l-l-cos20). (6.7.3b) The rate of incoherent reradiation (scattering) per unit solid angle, dQ, assuming that the electron displacement is small compared to wavelength, for s electrons and dQ solid angle is (Fejer, 1960) Pn = sl0 dqjdQ = n VIQ dqjdQ = \nVI0ra2 sin28 (6.7.4) where V is the scattering volume IQ is the incident intensity (watts/m2), // is the electron density in V. If the incident radiation is randomly polarized, the scattered power received is one half that given in (6.7.4). 234 Microwave propagation experiments Chap. 6 As an example, consider the following case. n= 1012 electrons/cm3 V= 1 cm3 I0 = 25 watts/cm2 dQ— 1 steradian. The scattered power is 2-10~12 watts, which is just above the detectable threshold at /=90 Gc. Further discussion of incoherent scattering appears in Section 2.6.5. 6.7.2 Scattering from plasma fluctuations of any wavelength. Two kinds of fluctuation scattering can be observed, both due to collective electron interactions. In both cases, the scattering cross sections are considerably enhanced over the Thomson cross section by the degree of coherency. For complete coherency, the scattered-signal intensity would be proportional to n2 instead of n. as shown in (6.7.4). The first type that we consider is incoherent backscattering; it has a cross section that is roughly proportional to the potential energy associated with plasma waves of any wavelength; the scattered frequency is doppler-broadened in proportion to the ion velocities (Rosenbluth and Rostoker, 1962). The second type (Drummond and Pines, 1961) also has a scattering cross section proportional to the potential energy of plasma waves, but is sensitive to angle and has scattered-frequency components in side bands spaced at multiples of the frequency of the plasma oscillations giving the scattering, similar to the case discussed in Section 6.6. In a given experiment, both kinds of scattering may occur at once, the scattered side bands being broadened by the doppler shift. The plasma fluctuations may be driven by an external frequency source, in which case the plasma column may oscillate in the dipolc resonant mode as well (sec Section 5.5). Or the oscillations may be due to an electrostatic instability, in which case the amplitude will be a function of position. Nonlinear effects can lead to the generation of harmonics of both the fundamental frequency and of the side bands. The scattered intensity is a function of angle and of frequency. The probing frequency to be scattered should be much higher than the frequency of plasma oscillations, perhaps an order of magnitude or so. For instability waves arising from electrons streaming through a plasma, the density fluctuations due to the plasma spacecharge waves are traveling essentially with the electron drift velocity, v0. The wave numbers (phase constants) of the incident wave, spacecharge wave, and scattered wave will 6.7 Microwave scattering experiments 235 'ft FIG. 6.33 Wave-number vectors for scattering from fluctuations in a plasma column. be related vectorially by the Bragg relationship, as sketched in Fig. 6.33 (6-7.5) where px = incident wave number, p2 = ai2\0jv0z — fluctuation wave number, p0 = final (scattered) wave number. For a wave vector incident at angle 6 to the plasma wave, the scattering angle is given by ft, cos(^-fl)=j8a-j31 cosfl (6.7.6a) or, since '03|=>|&l (6.7.6b) costft cosö + sin^ sinr?==^2 —cost?. In a plasma that is only weakly unstable—that is, the e-folding growth coefficient is only slightly larger than the damping coefficient—the plasma wave amplitude will be essentially constant over several wavelengths of the electromagnetic scattering wave. The scattered waves will contain side bands, much as in the case described in Section 6.6, the frequencies being tox and o}^±r,2+ß22 3kTejme (6.7.7) The frequency of oscillation of such waves is near and lies between wp and V2o>p, depending on the density distribution, temperature, and magnetic field. If the instability is due to counlcrstreaming (lows, such w s n — T2 >* CD Cß o 3 -i a li a & n 3' •3. o V 3 = s - t< lg II ** 2 a er od g 3 § s - g. e B Č? o < o —. a Cl Cl Cl n> o o* CS 03 < c — ft VÍ r-k p 3" n re —. O t? O 5 — (I Cl re re CT* 3 íS „< rre ja H CT O ft 3 i. et S o o g 3 ft re ~ ES. lX 3 tra 3 SL £ a. ST" O ft C T ca " o š' o ü o ™ 3 =f 1 o o ct n ft r*- Cl cř i—! P í £1 ST £ O" _ _. ft 3- 3 5 5 e M 3 y O- o % i 5 §.» | £ So re re ct p p 2. »' cr c 13 C_ p Ti as C CfQ Cl .«° cr rt re_ re o ft o. ". £ S3 1+ rt to CL Ě ta cr o 3" a ct Q VI ES. p "to II en O Cl p" g >- 1 fj ll II e to to JO ■a ir « II to ]| II kj s oc kO 5 -^j T ■a *. 3 — II oo s bs ll oj o II 3 3 □ to o 3 3" _ n ET p r- - s h ° řO =i !l CjJ L/l to ft < Cl < tt o o o ft < 3" re re a 3 05 re < re oo ° S & í ö p -i & ■ 3 p 2 ü s 3 > re =• 3 p 3 C- a c 3 3 00 in ö fí ? I f "H. cí 9 p » S [o 3 i 3 '< w „ i|ij re p cS o 3 CL 3" 2 p ^—. re re f 0 rj~ 35 p p OB 3 p CO re ft —! —. p O o? 3 3 p • p < CT T3 re 3- era Li. Transmitter; 25 W. CW klystron Power-atíenuater Plasma Receiver bandpass response Second (AFC) local oscillator * AFC mixer r/=30 Mc 30 Mc AFC amplifier 30 Mc AFC discriminator AFC control on-off FIG. 6.34 Transmitter-receiver system used to study the scattering from an oscillating plasma. The receiver is tuned to one of the side bands and locked there by automatic frequency control (AFC), the difference frequency being determined by the second local oscillator. 238 Microwave propagation experiments Chap. 6 II II m FIG. 6.35 A 35-Gc microwave scattering system for incoherent scattering from plasmas. The microwave source is a 25-watt c.w. klystron. The receiver is a sensitive superheterodyne, capable of either broad-band or coherent detection. The block diagram is shown in Fig. 6.34. (Photograph courtesy of General Atomic, San Diego, Calif.) The angle of scattering for the example assumed is obtained from (6.7.6), assuming |&| = |frl- L<=t 0=45°. Equation (6.7.6) reduces to I:)- cost? (6.7.6c) tf)-0 = 86° -H13T. 6.7 Microwave scattering experiments 239 A block diagram of a typical scattering experiment is shown in Fig. 6.34. A photograph of the 8-mm wavelength equipment used to observe scattering from plasma turbulence is shown in Fig. 6.35. 0.20 0.15 0.10 0.05 \ 1 1 1 1 Scattering from a dielectric - "rod of dielectric constant two - \ Plasma column, (o)p/6o)2 = 10; - V~— toh/w = 1; vjta = 0.1 - h Plasma - 1- column, — - td[,/tO — 0 1 -1 - 36 72 108 Scattering angle [degrees] (a) 144 10" 10- 10" Differential scattering cross section for 8 = 90° wbfw = 0.2; p/u = 0.01 0.5 0.8 1.0 Plasma density (uj/w)J m 180 1,2 I'll',. 6.36 Two-dimensional differential scattering cross section, per unit length and niiiI radius, as a function of (a) scattering angle and (/>) plasma density for 90" scattering; InujX = i. (From data compiled from Plal/.man and Ozaki, I960, and Smythe, 1950). 240 Microwave propagation experiments Chap. 6 6.7.3 Scattering from small plasma columns. When the scattering plasma has a diameter comparable to a free-space wavelength, it may have a scattering cross section that is highly frequency-sensitive. Under certain circumstances, a resonance condition between plasma waves and external waves may occur, leading to a manyfold enhancement of the scattering cross section (Herlofson, 1951). The resonance condition occurs for a plasma cylinder only when the incident electric vector is Discharge tube (a) FIG. 6.37 Microwave scattering from a resonant plasma column, (a) The small discharge tube mounted across an S-band waveguide. (/>) Relative power transmitted through the waveguide vs. normalized plasma current (Dattner, J957). 6.7 Microwave scattering experiments 241 perpendicular to the axis of the cylinder z, in which case the electric field couples to the dipole resonance mode and the scattered magnetic vector H lies entirely in the z plane (Smythe, 1950; Boley, 1958). The scattering cross section as a function of angle between the incident wave vector and the plasma cylinder axis may also exhibit maxima and minima. Calculations for the differential cross section of a magnetized plasma cylinder (Platzman and Ozaki, 1960) for j80a= 1 as a function of angle are shown in Fig. 6.36a. The cross section as a function of plasma density for "perpendicular resonance" co = (wp2 +iob2)ys is sketched in Fig. 6.36/;. The maximum and minimum shown are due to interferences of the electromagnetic wave, matched to the boundaries. Even when the plasma column is very small compared to a wavelength, maxima and minima are observed, in some cases having multiple peaks (Romell, 1951). An example of the transmission vs. plasma density, through a discharge tube mounted in the waveguide is shown in Fig. 6.37 (Dattner, 1957). Multiple peaks of reflection (scattering) also are found. An explanation for the multiple peaks is that the dipole resonance of plasma waves gives enhanced scattering whenever a particular plasma wave mode is excited (Battocletti, 1963; Hershberger, 1960; Crawford et al., 1963). Since the plasma waves are very slow, a number of modes can fit into the diameter of the plasma column. The mode spectrum, for normal incidence, is given by 2N 2N+1: Nff- (6.7.10) Other investigators have been unable to find the multiple peaks in their experiments, but, instead, find only one scattering peak (Shapiro, 1957; Kaitel, 1956) at the dipole frequency co = wjV2. (6.7.11) A possible reason for the absence of fine structure in some experiments is that the column is many debye lengths in diameter and, with density gradients or collisions present, the resulting closely spaced resonance peaks would tend to run together. Noise fluctuations also may wash out the resonances. C HAPTER 7 Microwave radiation from plasma 7.1 Introduction We can, in general, distinguish between incoherent and coherent radiation from a plasma. The former arises from uncorrclated radiation processes of single particles and can be interpreted statistically in terms of a "radiation temperature." The latter arises from collective motions of a large number of individual particles and has little or no connection with such statistical concepts as temperature. From a diagnostic point of view, the measurement of thermal (incoherent) radiation provides information on the electron temperature of the plasma, while observation of nonthermal (coherent) radiation indicates the presence of instabilities and other cooperative processes. The interest in thermal radiation at microwave frequencies, specifically, arises because of the availability of microwave superhet receivers which are sensitive, have a well-defined band width, and can easily be built to have fast transient response for use with pulsed discharges. Furthermore, a plasma most closely approximates a blackbody at low frequencies and especially near the plasma and cyclotron frequencies. The question of radiation necessarily involves two problems: (/) the mechanism by which the plasma generates electromagnetic waves; and (2) the transport of such waves through the plasma and across the boundary (Bekefi and Brown, 1961a). 7.2 Strict blackbody radiation The energy density (joules/m3) of radiation in thermodynamic equilibrium with matter at temperature T, in the frequency band between to and w + dw, is (Planck, 1914; Richtmycr et al., 1955) U,,doj = ^i-,.. l.„,—Tdw, (7.2.1) e.\p(/)«j/AT)-l 7.2 Strict blackbody radiation 243 where h = h\2-n, h = Planck's constant. This energy density represents an isotropic flux of electromagnetic energy flowing at velocity c. Thus, the radiation intensity (watts/m2) into the solid angle dil is cUjdQ{4w) da,, and, eU. 1 (7.2.2) the so-called Planck function, is the intensity per unit solid angle per unit radian frequency interval. In the Rayleigh-Jeans limit {hw«kT), which is generally appropriate for microwaves, (7.2.2) reduces to kTa (7.2.3) ' 4ir3c2 The prescript 2 denotes that two polarizations are included. The intensity in one of the two available polarizations is lBa=l*Ba. (7.2.4) Meanwhile, the directive properties of a transmitting antenna (in the far field) are specified in terms of a gain function dVjdQ <7(6\,£) = - (7.2.5) where dxP is the power radiated into the solid angle dQ in the direction (9, $) from the antenna and SF is the total power radiated. The angles 9 and j> are those of a conventional spherical coordinate system with origin at some convenient point within the antenna and, as usual, dQ = sinfl dO d4>-By definition (±}j>G(0, 1BiOS(0,)dQda (7.2.8) 242 244 Microwave radiation from plasma Chap. 7 a well-known result, valid whenever a receiver is matched to a blackbody noise source (Nyquist, 1928; Knol, 1951).1 If the absolute power is measured with a receiver of known band width, the temperature of the emitter is readily obtained, according to (7.2.8). However, a number of conditions must be satisfied if this simple analysis is to be applicable. First, the emitted radiation must be in radiation equilibrium with the emitting medium; that is, the body must be truly "black" at the frequency band in question. To ensure this, the following conditions must be met. CO The depth of the medium must be large with respect to the absorption length et~*. A partially transparent medium is often referred to as a "gray body," although in some usage this term also carries the implication that the optical thickness is independent of frequency. (2) The radiation must be able to escape freely from the medium and dissipate itself in the detector. If the nature of the medium is such that significant reflection occurs at the boundary, we may speak of a "silvery body." These considerations are implicit in the absorptivity or emissivity coefficient of Kirchhoff's law in radiation theory. In the case of a plasma for which the collision frequency is much less than the plasma frequency (V2«cojj2), these conditions are often hard to meet. This question is considered further in Section 7.5. Second, there are conditions on the receiving antenna. With the notation 7\ = distance from surface of blackbody to antenna /f = linear dimension of the antenna aperture Z> = transverse dimension of the blackbody we have three additional requirements. (3) The antenna must be beyond the induction field of the blackbody2 R£A. (7.2.9) (4) The blackbody must be in the far (Fraunhofer) field of the antenna R>A2jX. (7.2.10) 1 It is important to note the distinction between the specific intensity (per unit solid angle) of blackbody radiation (7.2.3), which is proportional to «j2, and the power received by an antenna (7.2.8), which is independent of frequency because of the Xs term in (7.2.7). The fact that the gain function G is, in general, frequency dependent is of no consequence because of the normalization (7.2.6). 3 Under other conditions, we may wish to sample fields of modes that are trapped inside the plasma and have no radiation field (Dawson and Oberman, 1959). 7.3 Bremsstrahlung in a transparent medium 245 (5) The antenna must "see" only the blackbody (2XIA)R) exp( /W) d) - a(t) exp(-joiř) dt W J - m i*" {a(t)?dt = Tt I"" KH|2<<" J — tc JfJ (7.3.3) (7.3.4) (7.3.5) The total energy emitted jn the encounter in the frequency band w to ca + dit) is thus W.Jui) duj - e'2p 677e0c; Tr\c>a(w)\2 doj. (7.3.6) For an electron in the coulomb field of an ion of charge Z, the acceleration is in general F(t) Ze2 1 *(/) = - m AiriQm [r(l)f (7.3.7) The dominant contribution to low-frequency radiation involves distant encounters with small deflections. In this limit we may approximate the acceleration by transverse and longitudinal components in the form «,(/) = Ze2 4ttc,/« [b2 + (vt)2]% "i;(0 = Ze2 vt (7.3.8) 4^am [b2 + (vif]% 3 Quantum mechanical effects can be neglected when livi«imo;i< K„ = 13,6 eV; relativists elfecls, when \m 510 keV, 7.3 Bremsstrahlung in a transparent medium 247 where b is the impact parameter and the time of closest approach is taken as ( = 0. The Fourier transforms (7.3,4) are 1 Ze2b Ze2 2tu 47reum ttí;2 f™ exp(-jwt)dt [(blv)2 + t2]% 1 Ze2b p fexp(->f)tft IMS 7t 4-TTt-^mV2 Ze2 (7.3.9) 47renm -rnr where the K's are modified Bcssel functions of the second kind.4 The spectrum is 2 i ^ ■ m^ s, i a [ * fS**J k2«+ic0« í—) \47T€0(.7 m2 irb2v'" (7.3.10) As a given electron passes through the plasma, the number of ions per second lying with impact parameters between /; and b + db is n-jftnb db; the total power emitted per electron at frequency oj, assumed small, is 2Pm(v) dcu = ntv H'Jb. v) 2irb db dw A (/Y^g- L^lnH do, (7.3.11) Thus, the result of this simplified theory contains the same divergent factor, ]n(bmax)bmH), as was obtained in the theory of coulomb collisions, Section 2.5. It is necessary to invoke physical arguments to provide appropriate cutoffs on the maximum and minimum impact parameters. We return to this point in Section 7.3.5. i K^x} = (TTl2]i" + tH„"-\jx) where H,,"1 is the (iirst) Hankel function (Jahnke and Emde, 1945, pp. 133-36, 236-42). 7=1.78 lim Kc(x)=ln(2jyx) x lim Kn(x) ac-tO (/,-[)! 12 /r>0 lim Kn{x) = (~yÁ cxp{-x) 00. 248 Microwave radiation from plasma Chap. 7 Meanwhile, the number of electrons per unit volume having speed v is )ief(v)4TTV2 dv; the total power emitted per unit volume at low frequencies is 2pJT)dco = n, \ Jo 16tt / e2 \ WJp) f(v, T) 4itv2 dv dto 3nenlz2 r r VI J f(v, T) 4m; dv do, Maxwc distribution f°° T V3 b * i., "ma \fj.-In -— Jo L w nmiJ 16/2»A*/ e2 \3 »^Z2 r V3 W (7.3.12) 7.3.2 77ie Gaunt factor. The calculation of free-electron bremsstrahlung was originally undertaken by Kramers (1923), using a classical analysis similar to the above. However, he was principally concerned with higher frequencies, the dominant contribution to which comes from close encounters (large electron deflections). In a limit of moderate frequencies and low initial electron velocities (parabolic orbits), one obtains results identical to (7.3.11) and (7.3.12), except for the absence of the quantity in square brackets. This is a "white" spectrum, independent of frequency. It is possible, then, for arbitrary frequency and velocity (temperature), to write the results of any calculation in the forms: Power radiated per electron of velocity v: *Pjfi>, v) da»=^t jx^— ] ^i" a 9(a), »)„„„, such that 4»»,2,i„ ~luo, rather than from zero. Since the velocity integration was oT the form [neglecting here the logarithmic velocity dependence of the Gaunt factor S(t>)] 2A J exp( — Ati2) v dv = 1, the modified integral is 2A | exp(-A^2) vdv = 2X r cxp( — Ať2) v dv -2A Jo exp( — Av2) v dv = l-[l-exp(-/to//<7,yj =exp(-#«/fcr). (7.3.20) That is, the general expression (7.3.14) is to be multiplied by exp( — fiaijkT} [note also (7.4.28)]. It is customary to write this factor explicitly in (7.3.14), rather than including it within the Gaunt factor S(«kl\ Quantitatively, this limit is ^[Gc]«2.4 105/rT[eV], (7.3.21) This inequality is usually well satisfied for microwave frequencies in laboratory plasmas. Parenthetically, for the case of high frequencies Jiio>kT, the electron orbits are highly perturbed and the contribution of transitions to bound states is important, so that a classical analysis is inadequate. One can obtain the bremsstrahlung cross sections from the exact (nonrelativistic) theory of Sommerfeld or the more tractable Born-Elwert approximation (Sommerfeld, 1951; Elwert, 1939). Brussard and van de Hulst (1962) argue that the effect of free-bound transitions can be approximated by omitting the exponential factor while keeping the S calculated for free-free transitions alone. Extensive numerical computations have been made for the high-frequency case (Greene, 1959; Karzas and Latter, 1961). The relativistic case is generally not of interest in laboratory plasmas (Heitler, 1954, §25; Koch and Motz, 1959). Meanwhile, when one treats the dynamics of the collision process itself by quantum mechanics, in analogy to Section 7.3.1, one obtains in the low-frequency limit (Gaunt, 1930) V3, \4(jmv2)~\ 9f>, v) ^ infill. (7.3.22) 252 Microwave radiation from plasma Chap. 7 This result is most readily obtained by a calculation using the Born approximation, which is valid in the limit of high temperatures and low frequencies (Sauter, 1933). Comparison with (7.3.16) indicates that the impact parameter bao has been replaced by A/y where h = ftjrnv is the reduced de Broglie wavelength. Velocity averaging of (7.3.22) gives SO, T) /if.)«IcT tt * \J4kT\ \yhw } (7.3.23) The quantum mechanical result can be expected to apply when A>/3H0 (a relation depending only on temperature and atomic number), since then the deflection caused by diffraction of the electron wave is greater than that of the classical collision process (Marshak, 1940). The crossover from the classical to quantum form can be estimated from the condition for equality between (7.3.17) and (7.3.23); we obtain kTxyaZ2Ry = (77 eV)Zz = (890,000 °K) Z3, (7.3.24) where R1/=\3.6e\ is the Rydberg energy constant. Note that the criterion may be stated in terms of the relation between kT (-electron energy) and the ionization potential for the (fully stripped) ion. Note further that the criterion (7.3.24) for the applicability of classical vs. quantum mechanics is of a different form from the Rayleigh-Jeans criterion (7.3.21). Thus, a purely classical analysis is justified only for temperatures such that hui«kT«y3Z2R„. (7.3.25) 7.3.4 Electron shielding. A second important correction to the Kramers theory concerns the proximity of other particles when the plasma is not infinitely dilute. At frequencies of the order of o>P and below, the electron cloud is able to adjust itself so as to shield the scattering ion (Chang, 1962a). The radiating electron no longer "sees" ions farther away than the debye length AD. Qualitatively, we may say that the radiation contributions from individual scattering ions arc no longer independent and uncorrclated. The result of this effect appears only in the argument of the logarithm of (7.3.17), which is then multiplied by a factor of approximately (o/cop at frequencies less than wp (Burkhardt, Elwert, and Unsold, 1948; DeWitt, 1958; Ostcr, 1964). Consideration of shielding requires that we amend the use of the term "low frequency" as used in the important limiting case given by (7.3.17). To avoid shielding but satisfy ">« t>Mi/^no> \«oJlcop«A!>v = XDjbBD. (7.3.26) As noted in Section 2.5.2, Spil/.er's ratio ASp is approximately the number 7.3 Bremsstrahlung in a transparent medium 253 of electrons in a sphere of radius equal to the debye length, and is normally quite large (~104) in common laboratory situations. 7.3.5 The Gaunt factor and InA. The simple bremsstrahlung theory given in Section 7.3.1 led to results, (7.3.11) and (7.3.12), containing a term of the form ln(bmaxlbmin). A term of this same form arose in the theory of the electrical conductivity of an electron-ion plasma discussed in Section 2.5, where the generic notation InA was used. These two theories have rather different points of view. Bremsstrahlung is concerned with incoherent radiation by a (thermal) electron, accelerated in the field of an ion. The conductivity theory is concerned with the loss of directed (wave-induced) momentum by the electron, in being deflected by the ion. However, the two effects are very closely related (Theimer, 1963). The quantitative connection, invoking the principle of detailed balance, is made in Section 7.4.4. Comparing (7.3.11) and (7.3.12) with (7.3.13) and (7.3.14), we may make the identification Q = ^3 [n/t (7.3.27) where the Gaunt factors and the InA term may be thought of as equivalent -0.2 I "5—0.4 3 kb -0.6 -0.8 1 0.01 10 100 Electron temperature kTJZ2 [eV] 0.1 1 Electron density njnc = (up/uf FIG. 7.2 Corrections to the unshielded, classical Gaunt factor So of (7.3.17) to take account of quantuiwand shielding effects. The two corrections are independent and are to be applied simultaneously. This figure represents the same data as Fig. 2.9. 254 Microwave radiation from plasma Chap. 7 correction factors (slowly varying functions of plasma parameters) arising in the bremsstrahlung and conductivity theories, respectively. In Section 2.5.4, we discussed at length the form of the impact parameter ratio A = bmaxjbmin appropriate for various regimes of electron density and temperature. The same arguments and calculations apply here. For convenience, Fig. 7.2 reproduces Fig. 2.9 (Greene, 1959; Dawson and Oberman, 1962), labeling the scales as corrections to the high-frequency, low-temperature Gaunt factor (7.3.17). The two corrections are independent of each other in the Rayleigh-Jeans limit (liai«kT) (Oster, 1963a). The analytic form of the Gaunt factor in various limiting cases of electron density and temperature may be obtained from Table 2.3, using (7.3.27). 7.3.6 Summary of microwave bremsstrahlung. To summarize the theoretical results for bremsstrahlung emission, we first recognize that three independent variables are involved: frequency, electron density (plasma frequency), and electron temperature. These may most conveniently be packaged in the following normalized parameters: ImjkT, which measures the importance of bound atomic states and the validity of the Rayleigh-Jeans approximation; tajto, which measures the importance of shielding; and kTiRy, which determines the applicability of classical vs. quantum mechanics.13 For microwave frequencies, where ho>jkT9 E-h 1000 Wave frequency cjfejvZ2 |Gc] FIG. 7.3 Bremsstrahlung power radiated per unit volume in one polarization by a plasma of electron density n and ionic charge Z, as a function of frequency and temperature, assuming no shielding (to » a),) but including the temperature correction of Fig. 7.2. The numbers given on the contours are to be multiplied by {(/.[cm-3])2 ■t/cu/2jr[Mc]} to obtain the emitted intensity in watts/cm3. For the corresponding absorption coefficient see Fig. 2.10. with S given by (7.3.17) and Figs. 7.1 and 7.2. This power density is shown in Fig. 7.3. The experimental problem of relating the emission from a transparent plasma to the power measured by the microwave receiver is very complex, since it requires numerical volume integration of the source emissions over the antenna pattern. This is especially troublesome when nearby apparatus provides danger of reflection of radiation into the antenna as well as resonance or interference effects. It is to be noted that, at a giveiAnicrowave frequency, the emission goes clown with increasing temperature. 256 Microwave radiation from plasma Chap. 7 7.3.7 Atom bremsstrahlung and total radiation. Our discussion of bremsstrahlung emission has been limited to encounters between electrons and bare nuclei, the dominant process in highly ionized, low atomic number plasmas. Historically, much interest has centered on bremsstrahlung from electron-atom encounters, in which case the atomic electrons screen the nucleus in a manner analogous to, but quantitatively different from, the dcbye-type shielding by other free electrons considered above (Nedelsky, 1932; Hettner, 1958; and Koch and Motz, 1959). For high-frequency radiation, requiring large accelerations, the major contribution is made by electrons which penetrate close to the nucleus, and atomic screening is not important; the system radiates like a bare nucleus of charge Ze. For very low frequencies, mainly produced in distant encounters, an A'-fold ionized ion can be expected to radiate like a simple nucleus of charge Xe. Radiation from electron-electron encounters is very small for nonrelativistic electrons, since no change of dipolc moment is involved (Joseph and Rohrlich, 1958; Stickforth, 1961). The total power radiated in all frequencies is an important quantity in determining power balance and energy transfer in many applications of plasma physics. Integration of the spectrum (7.3.14) yields a total radiation proportional to Z2T'A, whereas the emission at any (low) frequency is proportional to Z2jTv* (neglecting Gaunt factor variation). The spectrum (7.3.14) is integrated from zero to u>max = kTlh to obtain where <*{T)~l is the Gaunt factor averaged over both electron velocity and frequency (Greene, 1959). The plasma is assumed transparent throughout, ignoring self-absorption and nonpropagation at low frequencies. For interesting temperatures, frequencies well above the microwave region contribute most of the radiated power. For moderate temperatures where the high-Z impurities are only partially ionized (Z-fold, for instance) they contribute approximately as Z2 to the total bremsstrahlung but only as X2 to the microwave bremsstrahlung. It follows, then, that the total power loss is sensitive to relatively small concentrations of high-Z impurities in a low-Z plasma. Furthermore, the presence of bound electrons permits enhanced radiation at the frequencies of the positive-ion's discrete spectrum, which may greatly increase the total radiation power loss but does not greatly affect the microwave radiation (Post, 1961). Likewise, the presence of a magnetic field introduces cyclotron radiation or "magnetic bremsstrahlung," which can be the dominant radiative loss mechanism for a hot plasma. This point is discussed further in Section 7.6. 7.4 Radiation transport and the gray body 257 7.4 Radiation transport and the gray body To handle situations between the two limits of blackbody radiation and transparent-medium bremsstrahlung, we must consider the effects of stimulated absorption and emission. Furthermore, we do not wish to be restricted to only the limiting cases of isotropic radiation and infinite plane waves, or to homogeneous media. We shall, however, consider in detail only those situations which can be satisfactorily described by ray concepts rather than detailed diffraction analysis. 7.4.1 Energy flow in an inhomogeneous medium. Let Im(6,<{,)d<1>dQ (7.4.1) be the radiation power per unit perpendicular area flowing into the element of solid angle dQ in the (f?, ) direction, in the angular frequency band 21,0 dft 8s \p2j (U2 3s p3 ds (7.4.4) where s is the spatial coordinate measured along the ray path in the direction of energy flow. If the total spontaneous emission power per unit volume8 is ^p^dw, assumed isotropic, and the net space-dependent 1 A monochromatic plane wave is an idealized limiting ease. Relations for the total intensity / (watts/ma) of a monochromatic plane wave may bo obtained from relations given for specific intensity by the formal substitution du> —> 1. However, relation (7.4.3) is no longer valid for a plane wave. " Many texts use the notation /., = pJ4* lor the emission power per unit volume per unit Solid tingle. See Section 7.6. 258 Microwave radiation from plasma Chap. 7 ft 1 FIG. 7.4 Refraction of a pencil at an interface. amplitude attenuation coefficient is «(.5), the change in intensity along a ray path from all effects (except reflection at a discontinuity) is -aJ-^i+4^-2aJ- <7A5> Using (7.4.4), we obtain finally It is convenient to introduce a new distance coordinate, the optical depth {"observer Hio)= 2 W\). Emission of a photon of energy Iioj= W2 — is associated with a transition from level 2 to level 1, where ň = h/2ir = 1.05-10~34 joule-sec is the reduced Planck's constant. The probability per unit time per unit solid angle that a system in level 2 will undergo a spontaneous transition to level 1 is denoted by A21. If the number of systems per unit volume in level 2 is n2, the total radiated power per unit volume is 4nn2A21ňío; (7.4.10) this is the emission considered in Section 7.3.9 However, the presence of radiation of frequency w will cause induced or stimulated emission and absorption. We assume that the probabilities of these induced processes are proportional to the intensity of the ambient radiation at the proper frequency. The probabilities (per unit time-volume-solid angle) for induced emission and absorption may then be written B2JB> and B^f,,,, respectively. The A and B coefficients represent the detailed interaction processes of the physical system with the radiation field and have the same value whether or not there is thermal, or even kinetic, equilibrium. However, for the special case of matter and radiation in thermodynamic equilibrium, we can make the following assumptions. (/) The relative population of the two states is given by the Maxwell-Boltzmann distribution G2 exp(- nx GiCxpf- W2(kT) G2 wjkrrG;^-^1^ (7.4.11) The first term on the right represents the contribution from the intensity where Gi and C2 are the statistical weights or degeneracies of the respective energy levels, relevant only when dealing with bound atomic states. " In the case of the continuous energy levels of interest in free-free transitions, we consider frequencies in the hand cu to v> + du>. The coefficients A and li then contain implicitly the frequency interval du>. 260 Microwave radiation from plasma Chap. 7 (2) The specific intensity is given by the Planck function (7.2.2), generalized by the theorem (7.4.3) (Oster, 1963b), I 4tt3c3 exp(/i<7.4.12) are substituted into (7.4.13), we obtain An G., 4tt3c2 " (^i~B2l exp(-fe//Cr). (7.4.14) But, since the A and B coefficients may or may not depend upon temperature, while the equation must hold for all temperatures, it follows that both sides of (7.4.14) may be equated to zero, with the result 4iřAC2 3— ~q~ ai.l- (7.4.15) In the case where the material system is itself in kinetic equilibrium, described by temperature T, but not in thermal equilibrium with the radiation field, we may use the general relations to obtain for the net emission power density (watts/m3) 4md 1 = ^[1~?JX (7A16) where the specific intensity Iuy is no longer restricted to the Planck distribution (7.2.2) but is assumed unpolarized. Comparison of (7.4.10) and (7.4.16) shows that the effect of induced transitions is a net absorption,10 Expression (7.4.16) establishes, in principle, the connection between the transparent-medium case of Section 7.3, for which Ia is effectively zero, and the blackbody case of Section 7.2, for which Im has the Planck value (7.4.12). However, it remains to be found how to evaluate Im in the intermediate case. 10 On the contrary, a net emission is obtained in a noncqutlibrium situation such that the level populations are inverted f/i2>fli)> which may be described by a negative temperature in (7.4.11), This is the basic principle of the mascr and laser, 7.4 Radiation transport and the gray body 261 7.4.3 The partially transparent plasma. We wish first to relate phenom-cnological Einstein A. and B coefficients to the specific processes for a highly ionized gas. In Section 7.3, we calculated the emission power density from bremsstrahlung; (7.3.13) and (7.4.10) may be identified as the same quantity, «2 -t'._ da) = 4-rrhojn2A21. (7,4.17) The summation over all available initial (upper) states corresponds to integrating over the Maxwellian electron velocity distribution; we obtain 2Pwdm=4whoynAzl (7.4.18) where A21^ \ A21f(\) d3v. Meanwhile, under conditions where the spontaneous emission may be neglected (for example, an externally generated signal propagated through a relatively transparent plasma), the observable amplitude attenuation constant a, defined by (see footnote 7) — = -2al ds is related to the Einstein B coefficients by11 2a.la do> = liconl,,, j [% exp^j - #21] /(v) daY (7.4.19) — hom I hoi \ .1 4tPc [kř)~ Jwv 4& ,,2 2 ^21 • B, (7.4.20) Canceling out Ta and using (7.4.18), we find that the ergiebigkeit parameter appearing in (7.4.8) and (7.4.9), has the value12 (7.4.21) the Planck function (7.2.2), Thus, under all conditions in which the matter is in local kinetic equilibrium (not necessarily in equilibrium with the radiation), (7.4.9) can be written as (k\ =IS exp(-r)+f 2JB„[rW]exp(-r)^. (7.4.22) 11 The velocity integrations in (7,4.18) and (7.4.20) are both written in terms of the upper-stale donsilyfr/a, so that they are fully comparable. u Closer examination revea'.S (hat this general relation is only valid for relatively low-lost media, such Ihul x'J«VJ (RytOV, 1953). 262 Microwave radiation from plasma Chap. 7 If the electron distribution function of the matter is not Maxwellian, the cvc,{l\ 21 = 2rr3C2fm^ 4 /2\'V e2 \* nZplnA , Comparison of (7.4.23) and (7.4.25) shows that the two theoretical models lead to identical results except for the respective correction factors §=^ln/l. 77 (7.4.26) Indeed, as already shown in Section 7.3.1, the simple classical "straight-line" model yields a S of precisely the form (vf/w) In/I, where A is a 7.5 Radiation front a slab and Kirchhoff s law 263 ratio of impact parameters. The evaluation of these factors, for various regimes of frequency, electron density, and temperature, has been discussed at length in Sections 2.5.4 and 7.3.5. The Einstein-coefficients argument enables us to infer the emission and absorption for high frequencies at which the Rayleigh-Jeans approximation breaks down. We obtain * tIt) ml *tjexp(-^)dw {1A21) 2a = 1677 3 (3) (4**0 n2Z m%ch(kTyA, a>p, T), for this case, must usually be obtained by numerical calculations (Greene, 1959; Karzas and Latter, 1961; and Brussard and van de Hulst, 1962). 7.5 Radiation from a slab and Kirchhoff's law As an introduction to the thermal radiation transport problem, we restrict ourselves to the case of a dilute plasma having the propagation properties of free space; that is, we assume a>2»a>ri2>u2. I-'1 ' .' 'if ' FIG. 7.5 Geometry of slab radiation. 264 Microwave radiation from plasma Chap. 7 7.5.1 Effect of antenna gain. Consider an infinite homogeneous slab of plasma of thickness d and uniform kinetic temperature (Allen and Hindmarsh, 1955). We wish to calculate the intensity of thermal radiation received by an external antenna. The radiation emitted in a unit volume per unit solid angle by spontaneous transitions is &$J&, T) dm\ATT [watts/ m3-sterad], given by (7.3.14). As it passes a distance Ax, through the plasma, the intensity is attenuated by the factor exp(-2a Ax), where the connection between a, and 2pa is given by (7.4.21). Therefore, we may compute the power entering a polarized antenna according to the geometry of Fig. 7.5 by integrating the quantity (-^irY''2 M ^){^P\--Mr-X sec 0)]}pj#] (7.5.1) over the volume of the plasma. The factors represent, respectively, the polarized emission power density per unit solid angle, volume element, attenuation factor, and solid angle subtended by the antenna using the effective-area formalism of (7.2.5) and (7.2.7). The integration over r, between the limits R sec 8 and (R+d) sec 8, yields 1Ba(a>, T)[l -exp(- 2nd sec 8)]S(8, ^) sinfJ d8 (7.5.2) where 1B0y = ^Bo is the polarized Planck function (7.2.2). The antenna distance R drops out, although we must assume that the plasma lies in the 1 0.1 0.3 Slab thickness 2ad FIG. 7.6 Emissivity of dilute slab for the antenna types of Table 7.1; « = amplitude absorption coefficient, d = slab thickness; w' » wpa > v*, 7.5 Radiation from a stab and Kirchhoff's law 265 far field of the antenna (R>Smax!X). To proceed further, analytically, we must make some assumption as to the antenna pattern, the effective area S(8, 4) being proportional to the gain function G(8, = 10 dB Gain, G0, $ 4 cost? 0 0. For a low-loss moderately thick (d»cj2v) plasma, we have lad (2coJ3(//c»l) where the refractive index is p=* [1 -(<*vfaYXA and the critical frequency oj0(d, Wf, T) is defined by (7.5,5). The thermal radiation spectrum of a plasma slab may be presented in two formats, shown in Fig. 7.7. The 10 This approximation is valid for a plasma which is nearly optically thick, d> 1/2«, or when a wide enough frequency band Ato is accepted to average out the resonances, d£2we/u Am. It may also be reasonable when the change in propagation characteristics at the boundary, is "smeared out" or "bloomed" over an appreciable fraction of a wavelength, instead of being perfectly sharp, but then r is considerably reduced. Otherwise, our result, will give only an average result which suppresses resonances (.Hekcil and Brown, 1961a). 1.0 0.5 1 \ 1 1 ! 1 \ j*. Blackbody radiation a - NX0.5 / °\\ \l.O -A 1.5 1 1.0 2.5 i.O 1.5 2.0 Frequency tu/cjo m FIG. 7.7 Emission spectrum of plasma slab of low loss (v3 « «jp2) and of thickness » cju>p, (a) Specific intensity normal to slab, (b) Power received by high-gain antenna. The curves forming the lower limits of the shaded areas assume a sharp plasma boundary, according to (7.5.12). The upper limits assume a diffuse boundary, according to (7.5.15). The critical frequency . (7.5.13) where zBa is given by (7.2.2) or (7.2.3). The power received by a polarized, high-gain antenna is (Rayleigh-Jeans limit) ■ kr 2tT (7.5.14) In both cases, the emissivity e measures the fraction of the corresponding blackbody radiation level emitted. By way of contrast, it may be assumed that surface reflection is negligible (that is, the gradual boundary of Section 4.2). Then the emissivity is the high-gain case of Table 7.1 %— 1 — exp( — lad). This condition is also shown in Fig. 7.7. (7.5.15) 7.5.3 Kirchhoff's law. The experimentally measurable quantities for a slab are the (power) transmission and reflection coefficients, T and R, for an externally generated test wave. With the same assumptions made above as to incoherency of the internally reflected waves (see footnote 15), we obtain (Section 4.3) (1 -rf exp{-2ad) i — ŕ exp(—4c£(/) /■[!+(!-2r) exp(-4arf)] 1 — r2 exp(-4ad) The corresponding absorption coefficient or absorptivity A is A = ]-T-R- (1 -f)[l -exp(-2ad)] 1 — r exp( — lad) (7.5.16) (7.5.17) (7.5.18) This identification of the absorptivity (of a test wave directed from observer to the plasma sample) with the emissivity (for thermal radiation from the sample to observer), which we have here demonstrated in a highly idealized case, is in fact a general principle, known as Kirchhoff's radiation law (Planck, 1914). We can argue, thermodynamically, that the fraction of a test wave directed at the plasma that is absorbed in the sample is equal to the emissivity, independent of details of propagation characteristics and plasma geometry, so long as the propagation characlcristics are fully reciprocal. The formal statement may be made by identifying the emissivity with the quantity j ReíJ-E*)^ Re(E()XH0*) dS (7.5.19) where V and S are the volume and projected area of the sample, J and E are the a-c current and electric field existing inside the sample, and E0 and H0 are the test wave fields in the absence of the sample (Levin, 1957; Bekefi, Hirshlield, and Brown, 1959; and Bekefi and Brown, 1961a). Note that J may be related to E by the well-established conductivity theory. This formulation holds generally for homogeneous isotropic samples; caution must be used in extending it further. An important feature is that it is valid for boundary-value problems, so that we no longer need demand that ray optics be applicable. Thus, experimentally, we can establish a power balance whereby we measure R and T directly, in an auxiliary calibration experiment, and infer e = A = 1 -R-T therefrom. While this technique automatically accounts for internal reflections and diffraction effects within the plasma, in practical situations other than infinite plane slabs it is often difficult to measure dependably all nonabsorbed power, because of scattering, refraction, and diffraction. In many cases where the interface reflection coefficient /■ and the optical thickness 2v, it is possible to approximate the emissivity by (Bekefi, Hirshfield, and Brown, 1959; Hirshfield and Brown, 1961) <-Xl-r«\. (7.5.20) However, we note that under these conditions a small experimental uncertainty in r makes a much larger uncertainty in e. Also, since the received radiation comes only from the surface layer of the plasma sample, it may not be at all representative of the physical state of the interior. Wort (1964) has given a simple model for computing the emissivity of a turbulent plasma. In the presence of a magnetic field, which we have thus far ignored, the anisotropic propagation characteristics and opportunity for mode coupling and nonreciprocity require caution in the use of Kirchhoff's law (Martyn, 1948; Rytov, 1953; Bunkin, 1957; and Hirshfield and Brown, 1961). In terms of microscopic processes, in addition to ordinary bremsstrahlung from electron-ion encounters, we also have cyclotron radiation, which we discuss in Section 7.6. In the case of non-fvlaxwellian electron velocity distributions, the~K.iivhholI radiation law must be suitably reinterpreted (Bekefi, Hirshlield, anil Brown, 1961a; fields, Bekeli, and Brown, 1963). 272 Microwave radiation from plasma Chap. 7 7.6 Cyclotron radiation 273 7.6 Cyclotron radiation When a plasma is immersed in a steady magnetic field, the individual charged particles execute orbits which are in general helical. The particles are accelerated and radiate electromagnetic energy at the cyclotron frequency. Relativistic effects cause radiation at harmonics of the cyclotron frequency; this extension is often called synchrotron radiation (Schwinger, 1949; di Francia, 1959). The total cyclotron radiation is also known as magnetic bremsstrahlung (Trubnikov and Kudryavtsev, 1958). We note that ordinary bremsstrahlung occurs only during collisions, whereas cyclotron radiation occurs during the time interval between collisions. 7.6.1 Total radiation. The relativistic equivalent to (7.3.1) for the power radiated by an accelerated charge is (Panofsky and Phillips, 1962) * 6Jre0C3(l-je2)2 (7-6-l:) where fi = vjc and the index of refraction of the surrounding medium is assumed to be unity. The acceleration of an electron resulting from the Lorentz (magnetic) force is e|vxB| evxB where eB is the cyclotron frequency defined for the electron rest mass m0 total rate of radiation for one electron is then = ^i(l-^)'4, (7.6.2) (7.6.3) The ectron is then p=6™0c*(i-m' °-6A} If the electron velocity distribution is isotropic and Maxwellian, the total power radiated per unit volume is The prescript 2 signifies that the receiving antenna is assumed unpolarized. The total power from ordinary and magnetic bremsstrahlung may be compared; from (7.3.29) and the leading term of (7.6.5), where Jiy, the Rydberg energy constant, equals 13.6 eV and 8~1 is the averaged Gaunt factor. In many practical situations the magnetic Held (ccw„) and the particle pressure (~nkTazw/kT) are correlated by confine- ment considerations such that 2\o>b2 — ajb)2 -\-v2 (7.6.9) where y is the effective collision rate. This result may be obtained either from a single-particle analysis or from the macroscopic absorption coefficient invoking Kirchhoff's law (Oster, 1960; Birshfield and Brown, 1961). The full width at half maximum is Sw=2v. (7.6.10) The factor (7.6.9) assumes v independent of velocity. By the use of macroscopic arguments, it follows that the formalism of Section 2.4.3, involving the correction factors g and h, can be used when v = i(v) (Kelly, Margenau, and Brown, 1957). In the case of highly ionized plasmas, the effect of electron-electron collisions must be included in the correction factors (Hwa, 1958). (2) If the gyrating electrons stream rapidly through the sensitive region of the antenna pattern in a time t, the length of the received coherent wave train is again limited. This effect may be termed transit-time broadening, and is significant when t<[jv. The line is again Lorentz-shaped, as (7.6.9) with v replaced by 1 jr. (5) The component of motion of the electron parallel to the field produces a doppler shift which, of course, depends upon the direction of observation; ; -f °!\-* (7.6.11) l-(%e] cost? * ' For a Maxwellian distribution, the resulting doppler broadening is given by (7.6.7) multiplied by the gaussian shape factor mm mm^^lmw^im (7-6-l2) The full width at half maximum is I 8(ln2)/ťn'^ -a— cosii/c and 3± = vxic, with ^Z=^iz+^A2, the generalization of (7.6.15) to include the resulting doppler shift is (Trubnikov, 1958) wm%im:-,$8i r ...... /cnnO-s^2- ňíí/ň-8*r%c(l-ft cosfl)" r + \ sint? J *>^J^ spL sinfl •I- l-j8|i cosi (7.6.16) 276 Microwave radiation from plasma Chap. 7 7.6 Cyclotron radiation 277 In the special case of observation across the field (0 = Tr/2), the term in Js' gives the intensity in the extraordinary polarization (Er/ J B), which is dominant, while the term in Js gives the intensity in the ordinary polarization (Er/ |j B). Along the field (6=0), only the first harmonic remains, and the radiation is circularly polarized. The intensity ratio of successive harmonics may be obtained from (7.6.16) by expansion of the Bessel functions for the limit valid in the barely relativistic case, BL« 1. We obtain %^~%XZ-T) r 3W=Tj (i-/3:lcost?)2' (7'6J7) which, by hypothesis, is very small; the intensity decreases monotonically with increasing order. In particular, the ratio of second harmonic to fundamental, observed perpendicular to the field, is (7.6.18) For high harmonics (s»\) in this weakly relativistic limit the envelope of the harmonic intensities may be seen from (7.6.17) to fall off exponentially, at 20 log, 0(2/e£) decibels per harmonic (e = 2.718). In the highly relativistic case, on the other hand, the intensity increases somewhat with harmonic number up to a broad maximum near s~(\ - ft2)-*'* (Landau and Lifshilz, 1962, §74). The radiation of a single relativistic electron consists of the series of harmonics where s= 1, 2, 3,... is the harmonic number. 7.6.5 Spectrum (relativistic). For a group or electrons, with some assumed velocity distribution, the spectrum is a series of shifted, broadened lines. The shift is produced by the relativistic mass-increase (time-dilation) factor (I -/32)'4 and, for a Maxwellian distribution, amounts to a 1 /W\ (7.6.20) The broadening arises from the same effects mentioned in thenonrelativistic case, plus the dispersion of the velocity distribution in the relativistic shift. The shape factor for the latter effect alone is approximately 2 /wcV J j^^-gjj^ (7.6.21) where y=\ -(cu/jcoh)>0; the full width at half maximum is >«l.'s( kT\ /wt"7 (7.6.22) This is normally much smaller than the doppler width (7.6.13) except very close to perpendicular observation. To obtain the complete spectrum in the relativistic case it is necessary to integrate (7.6.16) over the electron velocity distribution/(p). This integration, which is very complicated in the general case, may be stated formally as 1 -ft cost? J /(P, T) d3p dw dQ (7.6.23) where jas is the power radiated per unit volume, solid angle, and frequency interval in the sth harmonic, p = v/c, and S(x), the Dirac delta-function, is simply a formal way of stating the resonance condition between w, p, and 8.ls The total radiation spectrum is then obtained by summing over harmonics jjtn, T) dto=^j(as(a>, T) doj, (7.6.24) and the total power radiated (per unit volume and solid angle) by integrating over frequency The integration (7.6.23) may be carried out numerically or in certain limiting cases analytically (Hirshfield, Baldwin, and Brown, 1961; Motley, Lustig, and Sanders, 1961; Beard and Baker, 1962; and Trubnikov and Yakubov, 1963). Figure 7.8 shows an example of such a spectrum. Cyclotron radiation may also be calculated from the macroscopic absorption coefficient, invoking Kirchhoff s law in analogy to the bremsstrahlung 10 The Dirac delta-function is defined by the properties (Dirac, 1947) B(x)\ (,~*oc x = 0, J* b(x) dx = \, JF(x) &(,x-a)dx = F(a) where F(.v) is an arbitrary function and the limits of integration extend over the entire range of the variable x. It is somewhat analogous to the Kronecker delta defined by [=1 l=m where / and in are Integral indices. 278 Microwave radiation from plasma Chap. 7 10 FIG. 7.8 Cyclotron emission spectrum from transparent plasma slab of temperature kT = 50 keV, and thickness d, integrated over angles. The emission intensity la is modified by reabsorption and other effects when the condition Ia « Ba (the black-body intensity level) is violated. (Figure reproduced by courtesy of the American Journal of Physics; Uekefi and Brown, I9fiia.) treatment of Sections 7.4.4 and 7.5.3 (Beard, 1959). An important practical problem is radiation from plasmas with non-Maxweliian electron velocity distributions (Oster, 1961a; Beketi, Hirshfteld, and Brown, 1961b). 7.6.6 Effect of collective electron motion. Except for the mention of collisional broadening and diamagnettc shifting and broadening, we have thus far assumed that individual electrons radiate independently of each other. When a significant electron density is present, any existing cyclotron radiation exerts a synchronous force on neighboring electrons (Chang, 1962b; Obcrman and Shure, 1963). The result is to displace the 7.6 Cyclotron radiation 279 [Degrees! FIG. 7.9 Coefficient measuring displacement of cyclotron resonance for finite electron density but negligible collision rate, as a function of propagation direction. Resonant frequency is (tt>t,a + resonance line, to an extent depending upon the direction and polarization of the wave considered (and also to broaden it if the plasma density is inhomogeneous). Specifically, for the right-handed-extraordinary wave the resonance frequency, from (1.4.81) with ^ -» co, may be written in the form otf^ + W (7.6.26) where the coefficient r, is given as a function of aiv and 9 in Fig. 7.9. The coherent motion of neighboring electrons also broadens the line asymmetrically toward the cutoff frequency (7.6.27) 280 Microwave radiation from plasma Chap. 7 These results are obtained by adopting a macroscopic point of view like that mentioned in Section 7.5.3. However, when the density is high enough for these effects to be substantial, the radiation is further complicated, both theoretically and experimentally, by the problems of self-absorption, refraction, and boundary reflection (Hirshfield and Brown, 1961). Another important effect, introduced by the presence of a plasma medium, is Cerenkov radiation, discussed in the next section. Electrons making distorted orbits, such as those making collisions with sheaths and walls, have their radiation frequencies shifted slightly above the electron cyclotron frequency, since they complete their orbits in a slightly shorter time than normally (Simon and Rosenbluth, 1963). The radiation spectrum also will be rich in harmonics. If, in addition, the orbiting electrons couple coherently to the sheath, the radiation intensity at the fundamental frequency, and the harmonics as well, may be quite large (see also Section 8.4.2). 7.7 Cerenkov radiation Radiation occurs when a perturbing particle transverses a medium with a velocity greater than the phase velocity of a wave in the medium. Although the phenomenon is quite general (including the bow wave of boats), the usual case identified with Cerenkov concerns a charged particle, such as an electron, producing electromagnetic radiation (Jelley, 1958; Ginzburg, Direction of radiation Direction of particle FIG. 7.10 Geometry of Cerenkov radiation. 7.7 Cerenkov radiation 281 1960; and Lashinsky, 1961). In a nondispersive, isotropic medium, the Huygens wavelets excited by the moving particle interfere constructively in the particular direction 6C (Fig. 7.10) such that where p, is the index of refraction and v=j3c is the particle speed. Clearly, this effect is possible only when jS> 1/ju.; that is, when the particle velocity exceeds the wave velocity. The radiation is emitted in the directions which form the elements of a cone of half-angle 0C. The radiation from a single particle is received as a short pulse (delta-function) with, consequently, a broad-band frequency spectrum. When the medium is dispersive, so that p = ^), the condition for constructive interference involves the phase velocity and (7.7.1) remains valid, with the characteristic angle 6C depending on frequency as well as particle speed. The pulse radiated by a single particle is of finite duration. The frequency spectrum is dependent on the particular variation of p{oi). When the medium is further complicated by anisotropy, so that p. depends also on the direction of propagation, the radiation pattern is no longer conical, except in the special case of particle motion along the optic axis. In the absence of a magnetic field, the refractive index of a plasma for electromagnetic waves is less than unity; Cerenkov radiation cannot occur. When a field is present, the index is greater than unity for certain frequency bands. We shall consider only the special case in which the electron motion is parallel to the magnetic field. Thus, the Cerenkov angle 8C coincides with the angle 6 of propagation (with respect to the field) and (7.7.1) becomes cos# = - 1 (7.7.2) p p(p> a) For low plasma temperatures, the index is given by the Appleton formula (1.4.40), so that the Cerenkov condition is (neglecting collisions) 1 p2 cos2e l - l- \a>-'2 sin20 1 Aoy3 sinW | qj,2 cos2flT [\ 2 j at2 J This relation may be solved for cos20, yielding (Kolomcnski McKensie, 1963) (7.7.3) 1956; cos* =2f?%«>2 - - "WhT + (l - £>TJ2]} *{2(V-oV)2f32 -cVP^2 + (l-p>Pa] (7.7.4) 282 Microwave radiation from plasma Chap. 7 Insight into this relation for low (nonrelativistic) velocities may be obtained from (1.4.81) subject to the condition that y? -» oo (cyclotron resonance); Cerenkov radiation can thus occur near the angles given by 7.7 Cerenkov radiation 283 tana0=--' = - (7.7.5) 0 1 2 (a) FIG. 7.11 Cerenkov angles for an electron moving parallel to the magnetic field; (a) nonrelativistic, (b) v/c = 0.5, (c) vjc = 0.707. FIG. 7.11 (continued) Figure 7.1 Iff shows this condition. The relativistic case is Tar more complicated; numerical illustrations are shown in Figs. 7.116 and 7.11c. It should be noted that for all cases the radiation has one or the other of the two states of polarization, characteristic of propagation in the given direction. When the radiating electron is a representative member of the thermal distribution, other electrons of similar velocity are present to support the inverse ahsorplion"* process, which may be recognized as the Landau damping of Section 3.5. Hence, the emission of Cerenkov radiation 284 Microwave radiation from plasma Chap. 7 \ j V 1 / 0/ / /° \ // fl = 20 degrees // 30 \ NNM~~\45 \ \ - 90\\ \ \ 1 i > | FIG. 7.11 {concluded) cannot exceed the blackbody limit. Also the relation (7.7.5) shows that the conditions for Cerenkov radiation by nonrelativistic electrons are precisely those for cyclotron radiation. Tints, the processes of cyclotron, Cerenkov, and inverse Landau radiation are largely indistinguishable, if not synonymous, in a nonrelativistic, thermal (that is, approximately Maxwellian) plasma (Kihara et al., 1961; Pakhomov et al., 1962). Together they determine the plasma emissivity in the vicinity of the cyclotron frequency. There are even close relations to eollisional bremsstrahlung (Lawson, 1954; Butler and Buckingham, 1962). 7.8 Coherent radiation 285 The more distinctive role of Cerenkov radiation is that associated with fast, superthermal electrons. Such particles are produced, for instance, by the runaway process which occurs when a static electric field exists parallel to the magnetic field and the electron energy gain per mean-free-path exceeds the average energy lost per collision (Dreicer, 1959-1960; Harrison, 1960). The fast electrons may be relativistic even though the main plasma is "cold." The radiations produced are nonthermal in the sense that they are not characteristic of the temperature of the main plasma. Since the coulomb cross sections decrease with increasing velocity, heat transfer to the thermal component of the plasma and eollisional bremsstrahlung are not important energy loss mechanisms for these fast electrons. The more significant radiative losses are synchrotron and Cerenkov radiation. The former is most dependent on the velocity component transverse to the magnetic field; the latter, by contrast, on the longitudinal component. The spectrum and the energy loss Tate have been calculated by Eidman (1958). Johnson (1962) gives a brief numerical example. Another form of Cerenkov-like radiation in a plasma is the generation of electrostatic or spacecharge waves (plasma oscillations) of the sort discussed in Sections 3.4.4, 5.5, and 5.6 (Cohen, 1961; Eidman, 1962). Since the phase velocity of these waves is of the order of the electron thermal velocity, the Cerenkov condition is fulfilled for a superthermal electron even in the absence of a magnetic field. In many experimental situations involving a magnetic field and superthermal electrons, Cerenkov radiation and excitation of plasma oscillations are the dominant known processes by which the fast electrons are degraded. 7.8 Coherent radiation Under nonequilibrium and irreversible conditions, coherent radiation may be produced. A cooperative process, driven by instability of confinement or a directed macroscopic current, may be energetically capable of producing a radiation intensity greater than that for a blackbody at the plasma temperature. Such processes are intrinsically complex and difficult to study, both experimentally and theoretically. Conversely, it is notoriously difficult to produce a plasma in the laboratory that is not subject to some instability or nonequilibrium process, known or unknown. The most tractable simple theoretical model is that of a high-speed electron stream, or beam, traversing an equilibrium plasma (Sumi, 1959; Stepanov and Kitsenko, 1961). Such streams may be injected from outside the plasma or produced within the plasma by the runaway phenomenon mentioned in the preceding section. If the beam density is large enough, beam bunching may occur, as in a klystron, so that beam electrons interact coherently and therefore vastly more efficiently than as individual particles. Alternatively, 286 Microwave radiation from plasma Chap. 7 the theoretical model may assume interpenetrating plasmas or a "double-humped" velocity distribution (Rukhadze, 1962; lchimaru, 1962). Further discussion is given in Section 8.4. Since longitudinal plasma oscillations become slow traveling electrostatic waves, either in the finite temperature case or for a bounded plasma, a relatively fast electron stream can synchronize with the wave in a manner closely related to Cerenkov radiation and to the conventional traveling-wave tube (Kompfuer, 1952). This process can be an important energy loss from the beam. The theory and relevant experiments have been discussed by Boyd, Gould, and Field (1961) and Emeleus and Mahaffey (1961). Other experiments have been described by Kharchenko et al. (I960) and Targ and Levine (1961). The longitudinal electrostatic waves generated in a plasma do not couple efficiently to electromagnetic waves external to the plasma. Several coupling mechanisms have been studied, such as the presence of a magnetic field and of density gradients or boundaries (Field, 1956; Wyld, 1960; Oberman, 1961; Majumdar, 1961; and Tidmau and Boyd, 1962). Electromagnetic radiation, which may be due to plasma oscillations, has been observed from astronomical objects such as the sun and Jupiter (Kuiper, 1953; Field, 1959), from the aurora (Forsyth et al., 1950), and from Stellarators in the presence of runaway electrons (Bernstein et al., 1958; Ffeald, 1956). A common feature of the experimental observations is the occurrence of the radiation in short intense bursts. It also appears that two colliding spacecharge waves can produce electromagnetic dipoie radiation at fj) = 2u)j„ the oscillating currents being maintained by momentum conservation (Aamodt and Drummond, 1963) The conversion efficiency is small, but still the radiation intensities can be very much larger than blackbody levels. Microwave amplification by a nonequilibrium plasma has been proposed (Bekefi, Hirshfield, and Brown, 1961a). Amplification at frequencies other than those near the cyclotron and plasma frequencies has been postulated on the basis of a parametric amplification theory with the beam-driven plasma oscillation serving as pump (Coor, 1961; Kino, 1960). CHAPTER 8 Plasma radiation experiments 8.1 Radiation from dense plasmas: blackbody radiation When a plasma is in thermal equilibrium, or at least in radiative equilibrium, the radiation spectrum is determined by the absorption spectrum (Rytov, 1953). The radiation should be maximum when the plasma is "black." A meaningful radiation experiment, therefore, must be coupled with an absorption measurement and, perhaps, also a density determination. The absorptivity A cart be calculated from (7.5.19) for complicated geometries, or from (7.5.18) for plane plasma configurations. For small power reflection coefficients r, such as achieved with diffuse boundaries, A can be identified with e, the emtssivjty, shown graphically in Fig. 7.7/j. In addition, for a high-gain antenna and a uniform plasma of depth d, approximating a one-dimensional case, A reduces to A = e=l-cxp(-2ad). (8.1.1) The radiation power in the frequency interval dm, radiated from an equilibrium plasma, is given by Kirchhoff's law when the absorptivity is large dw S (8.Í.2) where S is the radiating surface area, and A is the wavelength. The power received by a high-gain antenna is given by (7.5.4), or in terms of (8.1.1) Pn,da =/cre~[l-exp(-2ar/)]. 2-Tt (8.1.3) For incomplete opacity, wall reflections become troublesome and must be eliminated (sec Section 9,6,4 for absorbers). For complicated geometries 287 288 Plasma radiation experiments Chap. 8 an integration over the plasma volume must be performed, taking account of wall reflections, ir present (Wort, 1964). A quantitative determination of the radiation temperature of a dense plasma can be made with the radiometer circuit of Fig. 9.44, together with the transmission circuit of Fig. 6.7 (or that of Fig. 6.13). A microwave FIG. 8.1 A 3-mm (90-Gc) radiometer, including the klystron local oscillator the power supply and the 30-Mc i.f. circuit. The i.f. circuit can be operated with direct detection video output or with coherent detection and d-c output. (Photograph courtesy of the University of California Lawrence Radiation Laboratory, Livermore Calif.) 8.1 Radiation from dense plasmas: blackbody radiation 289 FIG. 8.2 A coaxial line, UHF swept radiometer, for use with an octave bandwidth, swept-frequcncy local oscillator. A band-pass filter in the signal input line is required, since otherwise the coaxial line passes extraneous noise and signals. (Photograph courtesy of General Atomic, San Diego, Calif.) system is pictured in Fig. 6.21 having only one klystron source; part of the signal provides local oscillator drive for the radiometer and part of the signal is used for the interferometer. No cross coupling between the two circuits exists, since the radiometer receiver is tuned to a frequency displaced by the i.f. frequency from the interferometer. Also the horns are cross-polarized, giving 20 dB of geometrical decoupling. A 3-mm radiometer (90 Gc) is shown in Fig. 8.1, including a 30-Mc i.f. amplifier and second detector that permits either direct video output or coherent detection with a long averaging time (see Section 9.5.5). A coaxial-line, swept-frequency radiometer, operating in the uhf bands with a swept-frequency local oscillator, is shown in Fig. 8.2. A band-pass filter is required in the signal input line to reject spurious signals, since a coaxial line does not have a lower cut-off frequency like a waveguide. Further discussion of radiometers and the associated hardware can be found in Section 9.5. Figure 8.3 demonstrates the relationship between opacity and radiation intensity. The lower trace is the transmission attenuation signal (the same event as recorded in Figs. 6.3 and 6.6) at 90 Gc. The upper trace shows the detected noise output from the radiometer of Fig. 8.1. Enhanced noise is seen to occur early in time, just as the hot, central core of the plasma is reaching cutoff, and later, just as it is coming out of cutoff. The noise level is low in intermediate times, since the cut-off, radiating shell is al a large radius, where the plasma is relatively cold. Also, when (u„»to, the plasm*is no longer "black" but "shiny" (see Sections 7.2 and 7.5). 290 Plasma radiation experiments Chap. 8 8.2 Radiation from a plasma in a magnetic field 291 FIG. 8.3 Plasma radiation intensity correlated with opacity. In the top trace the 90-Gc radiometer shows radiation signals during the times that the attenuation measurement in the bottom trace indicates a>pKui. The radiometer of Fig. 8.1 was used. Similar evidence is presented in Fig. 8.4, where the cutoff is indicated by the vanishing of the interference fringes (see also Wharton, 1961, p. 326). Three events are recorded, of consecutively higher-peak densities from top to bottom, all showing the increase in noise near cutoff. The data were made with the apparatus shown in Fig. 6.21, at 24 Gc. The peak amplitudes correspond to blackbody temperatures of 2 to 5 eV. A signal compression is inherent in superhet radiometers, since the output signal is proportional to the input voltage and thus to the square root of the noise temperature TN. The vertical scope deflection then is Dcc(TN)vK The peak noise signal of Fig. 8.3 thus corresponds to a blackbody (electron) temperature of ~ 10 eV, which agreed within a factor of 2 to the value obtained spectroscopically.1 In general, noise temperature (blackbody radiation) measurements made in dense, high-collision-rate plasmas yield electron temperatures in good agreement with those obtained by Langmuir probes (Knol, 1951; Easley and Mumford, 1951) or spectroscopically (Harding et al., 1958). Even in high-temperature plasmas, where the collision rates are too low to provide the thermalizing mechanism for electrons, nevertheless the electrons often are found to have a Maxwellian distribution (Gabor et al., 1 Unpublished data of C. B. Wharton, .1. E. Katz, and D. Reagan, University of California, Lawrence Radiation Laboratory, Livermorc, Calif., 1961. FIG. 8.4 Plasma radiation intensity correlated with opacity. The radiometer response is maximum just before the plasma goes to cutoff, as indicated by the disappearance of the interferometer fringes. The apparatus of Fig. 6.21 was used. 1955), and in many cases the electron temperatures inferred from microwave radiation intensities compare favorably with temperatures measured by other methods (Dellis, 1958; also see Section 8.2.1). Occasionally, however, some obviously nonthermal radiation is observed, emanating from plasmas. Evidence for this type of radiation was reported in Section 7.8, and further discussion is given in Section 8.4. 8.2 Radiation from a plasma in a magnetic field An electron spiraling about a magnetic field line will radiate, due to its acceleration, as demonstrated in Section 7.6, where radiation intensities and frequency spectra were calculated. In transparent plasmas, the 292 Plasma radiation experiments Chap. 8 8.2 Radiation from a plasma in a magnetic field 293 maximum radiation intensity was found to be in the plane of the orbits. In dense plasmas having radial density gradients (as real plasmas do), however, the cyclotron radiation is inhibited from escaping radially across the magnetic field, since it eventually reaches a cut-off region as the density falls toward zero at the edge (see Section 4.2.3). But, if the cut-off region is not extensive, some of the radiation generated in the hot interior can "tunnel" out and escape the plasma. Experimental evidence of this tunneling has been reported (Wharton, 1959; Motley et ah, 1961). 8.2.1 Magnetic mirror radiation experiments. Cyclotron radiation emitted along the field lines is often observed. This can be explained in terms of Kirchhoff's law. The trapped magnetic bremsstrahlung is strongly absorbed at the QL resonance frequency (see Section 1.4.10) so that the plasma reaches an anisotropic radiative equilibrium (Beard, 1961). The radiation is able to diffuse along the magnetic lines or in directions in which the field increases in the appropriate manner to avoid the cut-off' condition as the density decreases. The radiation escaping directly along magnetic field lines should be right-hand circularly polarized, since that is the wave that exhibits the large absorption. Measurements made on magnetic mirror machines at Livermore, Calif. (Wharton, 1958, 1959, and 1961), however, failed to show this polarization effect, presumably because of polarization coupling or wall reflections. Microwave receiving antenna - FIG. 8.5 Geometry for observing radiation from the end of a magnetic mirror plasma experiment. L is the radiation absorption length of the volume V. The region to the left of Kiscut off, that to the right is transparent. Cis the central region, M the mirror regions. S is the mirror separation. The radiation geometry for mirror or cusp experiments is sketched in Fig. 8.5. The radiation absorption length L denotes the distance in which radiation is effectively absorbed and reradiated in random phases. It is important that L be smaller than or comparable to the dimensions of the plasma for at least two reasons: (/) to establish quasi-blackbody conditions, and (2) to quench any coherent radiation by phase scrambling during the absorption-reradiation process. The phase scrambling2 is enhanced by a magnetic field gradient, such as found in a magnetic mirror, with a resulting action much like that of a magnetic beach (Stix, 1962). Absorption lengths typical of many controlled fusion experiments are from 0.5 to 5 cm. If the plasma extends for some distance into the magnetic mirror regions, M, it is this part of the plasma and not that in region C that will be sampled at a frequency giving cyclotron resonance in M. Region C can be probed by a second receiver, tuned to the lower frequency corresponding to the cyclotron frequency of that location. Waves can propagate in regions having higher magnetic fields, allowing the radiation from region C to pass unimpeded out through M. 8.2.2 Absorption-radiation experiment in a pulsed mirror machine. Many magnetic mirror experiments have pulsed fields to provide magnetic compression and heating of the plasma (Post, 1958). Fixed-frequency radiation receivers thus can be in tunc with the cyclotron frequency twice each pulse, once during the rise and once during decay. Even though the intrinsic cyclotron resonance may be sharp (Drummond, 1958; Hayakawa et al., 1958), the received signals tend to be broadened by the magnetic field fluctuations and gradients. Typical variations of the fields in the Tabletop TI experiment at Livermore (Post et al., 1960) with time are sketched in Fig. 8.6. The upper curve pertains to the field strength in the mirror region M, the lower curve to the field strength in the central region C. The low-/S plasma, of course, is confined in the region within the mirrors, that is, between the curves of Fig. 8.6. The Tabletop II experiment had a chamber 15 cm in diameter and 1.5 meters long, with a mirror separation S of 50 cm (see Fig. 8.5). The peak pulsed field was programmable up to 50,000 gauss. The plasma was provided by injection from titanium hydride sources into the evacuated chamber (Coensgen et al., 1958), the peak injected density before magnetic compression rising to about 101Z electrons/cm3. To determine the plasma opacity, an absorption experiment also was performed. The equipment is shown in Fig. 8.7. The receiver is a ■ To be distinguished from the phase mixing of Landau damping, which may also conlributc lo the establishment of radiative equilibrium, but not radiation, except through the inverse process, CVicnkov radiation (see Sections 6.6 and 7.7). 294 Plasma radiation experiments Chap. 8 T-1-1-1-1-1-71-Tl-[ C 3 o o o " ö o S o so 5 .5 ň S f tí; ä rfl f... „ "5, <1> >i ŕif.iii:.!i-:iLL"[? asi a u ind;no E 3 uj t-; c = OJ c U) g o E I .g O 3 01 ň O u superhet, having a 10 Mc band width and 0.1 micro-microwatt threshold sensitivity, with detected noise viewed directly on an oscilloscope. Amplitude and geometry calibration were made by moving a standard 15.2 dB thermal noise source (with a small horn radiator attached) about inside the experiment chamber before it was evacuated (see Section 9.5.6). The transmitter klystron is modulated by random noise lo produce a microwave spectrum some 200 Mc wide. The effects of high-order Load isolator 8.2 Radiation from a plasma in a magnetic field 295 Horn-. I Z ~ '. [.x-Horrii-- 1 num-. Plasma Compressioi -~r*-n-rrT77777r/////77777777Tn Experiment ^ 24 Gt : :0 i Klystron oscillator Random noise generator Power supply Energy storage bank ~10e joules n /Momf ?.4 Gc Klystron local oscillator Power supply Trigger Load isolator 0-100 d B Pad Balanced mixer 30 Mc if amplifier (10 Mc B.W.) 5 Mc Video Oscilloscope FIG. 8.7 Microwave equipment used for a 24-Gc microwave gyro resonance absorption-radiation experiment on a pulsed magnetic mirror machine, (See Figs. K.6 and 8.8 tor experimental results.) waveguide modes and reflections in the vacuum chamber are thus minimized. The frequency stability of the receiver and transmitter are good enough that no AFC system is required. The transmitter power level is a few milliwatts; the receiver output is set to a steady arbitrary signal level by adjusting the 0 to 100 dB attenuator in the input. Two such systems were used simultaneously, one in the K-band (23 to 28 Gc) and one in the 8-mm band (32 to 40 Gc). All plasma radiation (of order !0"u watts maximum) is masked by the transmission signal (of the order of 10-r! watts). A A"-band reflection experiment at the transmitter end was performed, showing plasma reflection coefficients less than 1% under all conditions, when the plasma was both opaque and transparent. Absorption lengths are of the order of a centimeter or two (many plasma wavelengths, at resonance). When the radiation was to be viewed, the transmitters were turned off and the input attenuators turned to zero. The plasma experiment was operated again, exactly as before, but now the receivers viewed the plasma radiation. Typical receiver output responses are shown in Fig. 8.8. flie limes of resonance arc also shown on the lickl plots of Fig. 8.6, 296 8.2 Radiation from a plasma in a magnetic field 297 (a) FIG. 8.8 Plasma microwave radiation data. Responses of radiometers to the cyclotron radiation from a magnetic mirror machine, («) at /= 35 Gc, (/>) at/= 32 Gc, and (c) at /=24 Gc. Trace ((/) is the absorption response, with a ,M Gc noise-modulated transmitter propagating through the plasma. The response limes arc shown In 11 iu. H.d 1Kb1 1 Im ali"(S«ll fit Rio 9 7 I together with a sketch of the receiver response to indicate the interpretation of the results. The received radiation is seen in this experiment to correspond to radiating regions between the central and mirror portions of the chamber, somewhat closer to the central region during times of maximum compression. The trace in Fig. 8.8a was made at 35 Gc, that in Fig. 8.%b at 32 Gc, and that in Fig. 8.8c at 24 Gc. The trace in Fig. 8.8rf is the attenuation response with the noise-modulated transmitter on (at 24 Gc), showing sharp absorption at the same times that radiation resonances occurred. During times between resonances (B>BC), the transmission signal strength is seen to be twice as high as for the vacuum case, indicating enhanced coupling when the cyclotron frequency is above resonance, as would be expected for ducted cyclotron wave propagation (low-density case of whistler-mode propagation). It is highly unlikely that the enhancement is due to any kind of wave growth or amplification. The amplitudes of the first radiation peaks (at r~400 ^sec), in many of the events, corresponded to blackbody temperatures of 15 keV (170 million degrees Kelvin). A direct energy analysis of escaping electrons (Ellis and Parker, 1958), assuming adiabatic trapping, yielded an average plasma electron energy of 17 keV at that time. The velocity distribution was not distinguishable from Maxwellian. The ions did not have a Maxwellian distribution, but their average energy was estimated at between 500 and 1000 eV. The average energy of escaping X-rays was within the range 10 to 100 keV, as estimated from absorber measurements. 8.2.3 Absorption-radiation measurements in a waveguide or cavity. The interactions between plane waves and plasma slabs can be simulated under certain conditions by enclosing the plasma in waveguides or resonant cavities (Buchsbaum et al., 1960). The boundary conditions can be satisfied, but the plasma ordinarily does not look "optically thick," so that local equilibrium is not established, except in the cases of high collision rates or strong resonances. Also, the sharp boundaries may permit charge separation, with attendant coupling between spacechargc and electromagnetic waves, or may allow wave tunneling through otherwise cut-off regions. For two cases, the radiation from bounded plasmas can be studied by applying corrections to the free-space relationships (Hirshfield and Brown, 1961): (/) weakly absorbing, tenous plasmas, utilizing a perturbation analysis, and (2) highly absorbing plasmas, such that the absorption length is much smaller than the plasma dimensions. In case / the plasma critical frequencies are shifted by the mode cutoffs (see Section 5.2) and I lie total wave absorption is decreased by the increase in guide wavelength. In case 2, when Ihe absorption is large, the effects of boundaries 298 Plasma radiation experiments Chap. 8 8.4 Radiation of nonthermal origin 299 are small, and the resonance occurs at the tree-space frequency given by (1.4.111) o.t3(l-mpacosBfl) where 6 is the angle in respect to the magnetic field. It is usually not convenient in waveguide experiments to observe radiation at angles very far from 0 = 0" (along B) or 0 = 90° (across B). Solenoidal magnetic fields can be made very uniform, so that u>b is well denned. If the density along the plasma column is also uniform, u>p and the transverse resonant frequency, b>B=(aips+wb^, are also well defined. The presence of the reflecting walls may increase the effective emissivity by multiple internal reflections, very much as in an optical hohlraum or integrating sphere (Wort, 1964). The total radiation transmission coefficient Trad analogous to (4.3.13), is obtained by summing over all of the S single-interface reflection coefficients r, for a system of dimension d TTaá = V-rf J {rf-'t^i-lSad) =(i-02 exp( — lad) 1 — rz exp(—4ad) The total radiated power, compared to the blackbody power, is (I-/•)[!-exp(-2b for the known magnetic field. The conclusion was that either the magnetic field was not as uniform as they thought (0.5%), the electron temperature exceeded 4 eV, or some additional mechanism was contributing. The radiation intensities for blackbody radiation at densities from 1010 to 1012 cm"3 and for cyclotron radiation for densities between 10s1 and 1010 gave a noise temperature of 2 eV. At low densities (109 down to 107 electrons/cm3), the noise temperatures for cyclotron radiation rose to 7 or 8 eV. The data were all obtained at a fixed frequency by sweeping the magnetic field and the electron density. 8.3 Swept-frequency radiometers In some experiments the plasma conditions change radically as either the magnetic field or electron density arc varied. The swept-frequency radiometer pictured in Fig. 8.2 avoids some of the difficulties by providing a voltage-tunable octave band. Both side bands are received, since it would be difficult to sweep a tunable rejection filter in exact synchronism with the local oscillator. The balanced mixer must be carefully matched over the band to achieve optimum performance. A coaxial-line ferrite isolator and a band-pass filter in the input minimize the spurious responses. If single-sideband reception is required then the band-pass filter must be tunable and tracked with the local oscillator. If an integrating detection radiometer is used, the frequency sweep rate must be slow enough that the response can follow. Typical response times of Dicke radiometers are from 0.1 to 10 seconds. A swept intermediate frequency also can be used to scan over a spectrum. This permits a fixed-frequency local oscillator, with a high-Q cavity to stabilize it and remove much of its noise contribution to the mixer circuit. Single-ended mixers can then be used, instead of the more critical and expensive balanced mixers. For narrow frequency scanning, such as looking at resonance line profiles, the i.f. amplifier can be a conventional low-noise i.f. strip, followed by either a second mixer and swept local oscillator or a sweeping filter (Long and Butterworth, 1963). For wide frequency scanning, a low-noise traveling-wave tube or distributed amplifier, in conjunction with a swept-frequency tuned amplifier, can be used (Colin et al., 1963). These techniques are particularly useful at millimeter wavelengths, where mixers are critical, especially if harmonic mixing is required (see Section 9.5). 8.4 Radiation of nonthermal origin It is often difficult to distinguish nonthermal from thermal radiation. True, when ihe radiation is observed in intense bursts or has an abnormally 300 Plasma radiation experiments Chap. 8 high harmonic content, the generation is clearly not of purely thermal origin. But the radiation emanating from a plasma having a non-Maxwellian velocity distribution or containing plasma waves or electrostatic instabilities may not appear abnormal. It is important to understand the basic radiation processes, then, to be able to attach much importance to various features of the emission. 8.4.1 Instability-generated radiation. Among early observations of nonthermal radiation were bursts in the VHF bands of fairly short-duration emanations from the sun, presumably associated with plasma oscillations in solar prominences (Kuiper, 1953). Similar large bursts at 8 mm wavelength have been observed in Stel-larators (Heald, 1956), presumably due to instabilities driven by runaway electrons. The intense "generation" of waves by the Stellarators was of sufficiently high power to endanger crystal detectors in microwave interferometers used for diagnostics. An electron beam-plasma interaction experiment was set up at Liver-more in 1959 to simulate the runaway electron interaction, but with controlled electrons from a gun. The gun was pulsed with 5 to 20 /xsec pulses 30 times per second, firing into the steady-state P-4 plasma (Hall and Gardner, 1961) as shown in Fig. 8.9. Radiation pulses, of microwatt intensity (that is, about 10e times the blackbody level) were detected during the electron pulses, in a narrow frequency band centered about the plasma frequency, at 34 to 36 Gc,3 depending on the electron density at the location of the beam. The radiation intensity varied smoothly with variation of the pulse current, the detectable threshhold occurring when the peak current was about 5 to 10 milliampcres. Two orientations were used for the pickup horns, as shown in Fig, 8.9, both using square cross-section horns and fin-line couplers to resolve the two polarizations. With the horn looking along the column (actually looking up at the plasma with a 20° angle from the axis), the radiation intensity was found to be about 5 times as large from up stream (radiation coming from the gun end) as from down stream. With the horn looking at 90° to the plasma, the radiation polarized along the direction of the beam was 6 times as intense as that polarized across the beam. The intensities measured at three ports along the column were essentially the same, although the frequencies varied by about 10%. The signals from two ports detected simultaneously with two wide-band 8-mm radiometers were uncorrelated, when fed into a coincidence circuit (<2% coincidence over 10 seconds). a Here, the word "radiation" is used in the strictest sense, that is, far-field, electromagnetic reception, as distinguished from "noise," which may be picked up by a probe or antenna immersed in a plasma, and subject to near-field induction or spacechargc field fluctuations. 8.4 Radiation of nonthermal origin 301 Waveguide to radiometers m ' Rotation drive to rotating langmuir probe Insert, showing details of mounting waveguide horns in the vacuum ports Break indicates 10 feet of chamber omitted Reverse field coil for tile burial chamber ~J&ma_c^lBa_;_, Microwave!, horn I Wilson J_L S6a' TT Waveguide inside round pipe through U Wilson seal FIG. 8.9 Electron beam-ptasma interaction experiment. The electron gun injects 20 fisec, l-amp pulses of 25 keV electrons into the steady P-4 plasma column. The P-4 plasma was 99% ionized, with 1013 electrons/cm3 at 10 to 20 eV temperature. (From unpublished data of C. B. Wharton and A. L. Gardner, University of California, Lawrence Radiation Laboratory, Liver mo re, Calif.) When a Langmuir probe was inserted in the plasma in the space in front of the horn, the radiation power increased a hundredfold, and the frequency emitted became a function of the radial position of the probe.4 Presumably, the enhancement was due to conversion of some of the space-charge wave energy into currents on the probe which, in turn, radiated electromagnetic waves as a dipole. Since the beam-induced plasma oscillations are very nearly at the plasma frequency (Stepanov and Kitsenko, 1961; also see Section 5.5), a measurement of the radiation frequency gives a measurement of the local electron density. Correlative measurements of saturated ion current of the Langmuir probe (see '' Actually, the probe was swept through the plasma with a 15% duty cycle to prevent its burning up (OardriK el al., 19(51), so that the radial positioning was actually done by timing the firing of Ihe electron beam pulse with a variable delay. 302 Plasma radiation experiments Chap. 8 8.4 Radiation of nonthermal origin 303 Section 10.12) and of microwave phase shift at 64 Gc gave values of electron density and density profile, all agreeing remarkably well. Radiation was also viewed with a swept-frcquency S-band radiometer (Katz, 1959) to look for cyclotron radiation. Signals at a noise temperature of about 5 to 10 eV were received at a frequency of 2.6 Gc, which is the gyrofrequency in a magnetic field of 930 gauss (P-4 had Bx950 gauss). The signal was enhanced about twofold during about 25% of the electron beam pulses. We concluded that the coupling of the instability to cyclotron radiation was not large for those conditions. 8.4.2 Nonthermal cyclotron radiation. The cyclotron radiation spectrum, discussed in Sections 7.6.4 and 7.6.5, contains harmonics, whose relative intensities are strong functions of the plasma electron velocities. A typical harmonic spectrum for high energy electrons is shown in Fig. 7.8. For thermalized electron temperatures typical of most plasma experiments, including controlled fusion, harmonic numbers of 3 or 4 are about the theoretical limit. In several plasma experiments, nevertheless, harmonics as high as the 24th have been detected, with relative intensities that have little to do with conventional cyclotron radiation theories. In the experiment of Landauer (1961, 1962), up to the 24th harmonic of tab was observed in a plasma generated by a P.I.G. discharge, a type of discharge that is notorious for various instabilities. In the experiment of Bazhanova et al. (1961), a spectrum of 10 harmonics of the ion-gyrofrequency were detected in the Ogra machine (Artsimovich, 1958; Golovin, 1959) having a high-energy plasma of density about 10s ions/cm3. The observations of 8 or 10 harmonics of o)b by Fields et al. (1963) and Bekefi et al. (1962) were made in the positive column of a hot cathode arc discharge having copXo>. All of these observations were made by holding the frequency of the receiver constant and varying the magnetic field. Landauer used two frequencies at once, 34 Gc and 10 Gc, and found that in all cases the peaks occurred at slightly higher frequencies than multiples of to,,, rather than being shifted downward as given by (7.6.19). A typical spectrum is shown in Fig. 8.10a, in which it is seen that the intensities of the first 18 peaks are essentially the same. Observations made with the antenna oriented with E along the magnetic field even showed several harmonics, but with reduced amplitude. Landauer (1962) offers several qualitative explanations, involving "quasirelativistic" electrons and high, nonlinear refractive index, that partially explains the observed effects. A theory given by Pistunovich and Shafranov (1961), also involving a very large refractive index due to resonance, partly explains La r.da tier's and also Bazhanova's results on Ogra. A recent theory by Simon and Rosenbluth (1963) provides a reasonable fit to Landauer's results and, in part, to Bekefi's (1962) findings. Simon and Rosenbluth calculate the harmonics and line shapes generated by i-1 i r 10 9 m = 5 (a) 'Ml 1000 1500 2000 2500 3000 Magnetic field Igauss] £2.2 i .6 0.25 0.50 Magnetic field 0.75 1.0 FIG. 8.10 Cyclotron harmonic radiation spectra for nonthermal radiation, (a) Some results of Landauer (1962); pressure= 1.5-10"2 torr.; in is the harmonic number. (/)) and (f) Some results of Bekeli et al. (1962), showing the effect of pressure. In (/)) the pressure was 5.7-10 :l lorr; in (c), 0.6 torr. The peak intensities are orders of magnitude-above blackbody. 304 Plasma radiation experiments Chap. 8 particles making cyclotron orbits and collisions with walls and sheaths simultaneously. The broadening and shifting of the low harmonic number peaks and the variations with plasma frequency agree well with Landauer's data. The observations by Bekeli (1962) were made in argon and mercury discharges at S-band and .Y-band, at several angles in respect to the magnetic field. Typical results are sketched in Fig. 8.10ft. It is interesting to note the absence of a radiation peak at the fundamental frequency, that is, (jib — ca. This effect was also observed by Landauer, but is not explained by the Simon-Rosenbluth theory. It may simply be evidence that the extraordinary wave is trapped by cutoff and cannot escape from the plasma across the magnetic field. The higher harmonics are not trapped and can escape. Landauer's results give further evidence that this may be the case, since the fundamental component was present when radiation was viewed along the Held, although its magnitude was small. From Fig. 1,10, we note that the extraordinary or Tight-hand wave for propagation along field lines is not cut off by density gradients, as it is for propagation across the field lines. Another explanation is offered by Tanaka et al. (1963), who find that the resonance frequencies shift progressively up in harmonic number {N=\ -> N=2, N=2->N = 3, etc.) as -»l. C HAP T ER 9 Microwave hardware and techniques 9.1 Transmission lines and fittings The ranges of electron densities, magnetic fields, and plasma dimensions of many laboratory plasma experiments require the use of microwave frequencies in the region of 3 to 90 Gc for free-space (beamed) transmission experiments. Fortunately, this frequency interval includes several of the bands that have been developed extensively. Some experiments call for higher frequencies, the hardware for which is still largely experimental. Table 9.1 lists some of the bands and the present standard waveguides and flanges of each. Plasma experiments involving spacecharge wave transmission and hybrid ion resonance effects are conveniently done at lower frequencies, for example, 500 to 5000 Mc, using coaxial cable components. Extensive lines of components are available, employing type N, C, BNC, TNC, and other well-matched fittings and covering octave band widths. 9.1.1 Waveguide considerations. For millimeter wavelengths, the nominal skin-depth in metals is given by s= 1 meters, (9.1.1) where/is the frequency, pa is the permeability, and a is the conductivity, mhos/meter; S is of the order of 2.5 to 5- 10"s cm (0.25 to 0.5 micron). The wave attenuation in a waveguide therefore is influenced by surface roughness, oxidation and chemical deposits, and work-hardening (Thorp, 1954). A quantity that describes the surface condition is the surface resistivity, 306 Microwave hardware and techniques Chap. 9 Table 9.1 Standard waveguide bands 9.1 Transmission lines and fittings 307 Frequency Band-center Designations Waveguide Flanges range, Gc wavelength JAN EIA 2.6 3.95 10 cm S RG-48 WR 284 UG 53, 54A 8.2-12.4 3 cm X, Xs RG-52 WR 90 UG 39,40A 12.4-18 20 mm Ku, P RG-91 WR 62 UG419, 541 18-26.5 12 mm K RG-53 WR 42 UG425, 595, 596 26.5-40 8.6 mm Ka, R, U, V RG-96 WR 28 UG381,599, 600 33-50 6.8 mm Q RG-97 WR 22 UG383 50 15 4.3 mm M, V, W RG-98 WR 15 UG385 60-90 3 mm E RG-99 WR 12 UG387 75-110 3 mm — — WR 10 ' 90-140 2.2 mm F RG-138 WR 8 EIA > standard flanges* 110-170 140-220 170-260 2 mm 1.5 mm 1.2 mm G RG-136 RG-135 RG-137 WR WR WR 7 5 4 220-325 1 mm — RG-139 WR 3 j * A U.N. commission, the International Electro-Technical Commission, is currently studying waveguide standards problems and is making recommendations for standard flanges in the millimeter bands. These will become EIA standard llanges. J?5 has the same value as the d-c resistivity of a plane conductor of thickness S and conductivity a. Surface roughness and porosity increase Rs. For example, a microscopic roughness of depth and spacing equal to 2S will increase Rs about 50% (Lending, 1955; Morgan, 1949). High-conductivity, nonoxidizing materials commonly are used for waveguide fabrication, including pure copper, coin silver (90% silver, 10% copper) and pure silver laminated on copper or bronze. At the shorter wavelengths, gold, chromium, or iridium often are plated on exposed surfaces to impede oxidation. Figure 9.1 shows skin depths for several materials at various frequencies. Waveguide attenuation is dependent on the frequency, the dimensions of the waveguide, and the type of transmission mode. For example, the attenuation due to wall losses for the TE10 mode in rectangular waveguide of width a, height b is adB' 8.686J?s(e0//j.0)'^ t dB/metei" (9-L3) 3 4 5 6 8 10 20 30 40 50 60 80 100 Frequency [Gc| 200 300 FIG. 9.1 Skin depth In metals of various conductivities as a function of frequency, d-c Conductivities of common metals in units of 108 mhos/meter: silver 64; copper 59; gold 41; chromium 38; aluminum 35; magnesium 22; iridium 16; brass 15; platinum 9; soft solder 7; mercury 1.1. where A is the free-space wavelength. The attenuations of the common RG-type waveguides in the TE10 mode are shown in Fig. 9.2. In addition, the values for some high-mode waveguides are shown, including some RG waveguides of extreme oversize (Valenzuala, 1963). In oversize or high-mode waveguides, mode purity is sometimes difficult to maintain (Lewin, 1959). Usually, bends, junctions, or obstructions lead to mode conversion, with attendant frequency sensitivity and attenuation. A common practice is to use conventional-waveguide-size components for circuits, short runs, sharp bends, etc., and taper up to the oversize or high-mode guide with a long, electroformed transition for long straight runs, where excessive attenuation in the conventional waveguide would normally occur. Round copper tubing that has been slightly flattened 3 4 5 6 7 8 10 20 30 40 50 60 80 100 200 300 Frequency [Gc] FIG. 9.2 Waveguide attenuation as a function of frequency for standard RG waveguides and for some special waveguides. 310 Microwave hardware and techniques Chap. 9 9.1 Transmission lines and fittings 311 between rollers to prevent polarization rotation is a convenient method to cover distances up to about 50 feet, cheaply, with propagation in the oversize TEn mode. Bends having radii of curvature of about 20 wavelengths or more can be tolerated with little mode conversion. Typical 50-foot runs of flattened f-inch tubing at 90 Gc have a loss of 7 to 10 dB, including the two transitions at the ends, as compared to 70 to 80 dB for RG-99/U. Even lower losses are obtainable with TE0l round waveguide (Miller and Beck, 1953), at the expense of complicated mode transitions and the necessity for inserting mode filters at frequent intervals along the run. Figure 9.3 shows some TE01-mode circular waveguide components (Lanciani, 1954). Figure 9.4 shows views of several other mode transitions, filters, and bends, as used at the University of California Lawrence Radiation Laboratory. Highly-oversize, special-mode components are available commercially, although they tend to be somewhat costly. 9.1.2 Open waveguide transmission lines. Other interesting types of wave-guiding structures utilize surface waves on metallic (Sobel et ah, 1961), dielectric-coated metallic (Diament et al., 1961), or purely dielectric E-Field lines E-Field lines H-Field lines r-4 V VI -Metal plates _ Dielectric center strip FIG. 9.5 Sketches of field patterns of open waveguide transmission lines. The fields are seen lo fringe some distance outside the line, (a) Dielectric rod, HEn dipole mode, (b) Dielectric rod, TM0i mode, (c) Trough, (rf) W-Guide. surfaces and rods (Chandler and Elsasser, 1949; Weiss and Gyorgy, 1954). Dielectric rod waveguides of teflon, polystyrene, etc., are particularly interesting, since they are nonconductors and can be used to bridge across high voltage environments or to operate in a corrosive atmosphere. The fields extend some distance outside the rod, however, as sketched in Fig. 9.5, and coupling can occur between rods coming close to each other or from the rod to support structures. Also, radiation will occur at sharp bends or regions of nonuniform cross sections. Dielectric rod radiators, discussed in Section 9.3.3, are made by tapering the end of a rod. Various open waveguide or surface-wave propagation structures have been devised. Among the more useful ones are H-line, V-line, trough-line (Tischcr, 1956, 1958), and dielectric-coated wire or G-line (Goubau and Schvvering, 1961). The attenuation at millimeter wavelengths of these lines can be several orders of magnitude lower than that of conventional waveguide, the attenuation decreasing with frequency. Wave launching FIG. 9.6 Free space transmission path utilizing dielectric lenses. Path length of I feel has a loss of I dll al 70 Ge. (Courtesy TRG, Inc.) 312 Microwave hardware and techniques Chap. 9 is generally difficult and the structure tolerances are often extreme. They are useful Tor very long runs, but have little advantage for short ones. 9,1.3 Free-space radiation. Free-space radiation links are possible at millimeter wavelengths because large-aperture antennas (in wavelengths) are easily realizable. The minimum insertion loss is limited in practice by diffraction and interference conditions, as pointed out in Section 4.9. Large apertures can be obtained with lenses, Fresnel zone plates (van Buskirk and LHendrix, 1961) and parabolic reflectors. An example of a transmission link using lenses is shown in Fig. 9.6. Transmission over distances of 15 to 20 meters between zone-plate radiators with attenuation as low as 10 dB at 140 Gc is obtainable. Even lower loss is obtained if the cross-sectional phase distribution is corrected at periodic intervals (Goubau and Schwering, 1961) by inserting long-focal-length lenses in the transmission path. Attenuations as low as 2 dB over path lengths of a kilometer have been achieved. 40 50 60 80 100 Frequency [Gc] 200 300 400 FÍG, 9.7 Approximate atmospheric attenuation of electromagnetic waves for horizontal propagation as a function of frequency. The water vapor density was 7.5 grams/meter3 for the upper curve and 1.0 gram meter3 for the low-cr curve, (Compiled from data of Straiton and Tolberg, 1960; Dieke et a'.., 1946; and Thcissing and Caplan, 1956.) 9.2 Special components 313 In free-space transmission, atmospheric attenuation becomes important. Except over very long paths, however, the air attenuation is practically negligible. Figure 9.7 shows the attenuation due to molecular resonances and the atmospheric "windows" between (Dicke et al., 1946; Theissing and Caplan, 1956; Straiton and Tolbert, 1960). 9.2 Special components Many techniques and component designs can be carried over from the highly perfected waveguide bands into millimeter wavelengths by simple scaling laws. There are, of course, some problems: (1) Dimensional tolerances become exacting, (2) Losses increase, due to small skin depth and small component size. (3) Oscillators and amplifiers become less efficient (and more expensive) because of greater beam density and heat dissipation requirements, (4) . Crystal detectors become less efficient because of increased internal impedance, and have lower power handling capabilities. The cost of waveguide systems, in general, increases with frequency in a nearly linear fashion above JC-band. For example, a 4-mm interferometer costs about twice as much as an 8-mm one, and a 2-mm interferometer FIG. 9.8 Simple waveguide bending tool and samples of some elbows fabricated in RG-9(i/U waveguide. {Courtesy General Atomic, San Diego, Calif.) 314 Microwave hardware and techniques Chap. 9 9.2 Special components 315 about twice as much again. The choice of the frequency at which to do diagnostics, then, clearly contains an economic factor as well as physical factors. Nearly always, some kind of compromise is necessary. Often, the budget can be eased by fabrication of components, both waveguide and electronic, within the laboratory shops. Elbows, twists, phase shifters, terminations, etc., are particularly easy and inexpensive to make. As an example, a simple waveguide bending tool and some elaborate bends made in RG-96/U waveguide are shown in Fig. 9.8. No reflections or attenuation due to these bends were measurable in the systems where they were employed. The flange faces were finished off in a lathe after soldering. The same techniques have been used successfully up to 90 Gc. 9.2.1 Phase shifters. Waveguide phase shifters change the electrical length of a section of waveguide, either physically (as with a line stretcher) or by varying the cut-off frequency and thus the phase velocity. Commonly, phase shifters are made with a dielectric vane oriented in the E-plane and either lowered through a slot in the top wall or moved back and forth across the broad dimension of the guide. Both or these types must be used with considerable caution. The first type tends to radiate badly, unless the vane is well shielded and, even then, may introduce standing waves. Both types can support higher modes when the vanes are well in, or when the frequency is near the top of the band; frequency FIG. 9.9 Squeeze-section phase shifter in R.G-96/U waveguide. Maximum phase shift «360° at 35 Gc. (Courtesy General Atomic, San Diego, Calif.) sensitivity and standing waves then result, leading to significant errors in phase measurements. A phase shifter whose cut-off frequency is decreased by operation (no higher modes arc then possible) is the squeeze section. A longitudinal slot is cut through both top and bottom walls of the waveguide along the center line for about ten guide wavelengths. A captive screw is then soldered to one side wall, and a manipulator attached that can both spread and squeeze together the slit. Phase shifts of 2tt to 4?r are easily obtained, with essentially no radiation or reflections. A squeeze-section phase shifter is shown in Fig. 9.9. A more elegant type of phase shifter is formed by rotating a half-wave plate, in circular waveguide carrying a circularly polarized wave, between two quarter-wave plates that transform from circular to linear polarization (Fox, 1947). The phase is shifted 2tr with each rotation. If the center section is rotated at constant velocity with a motor, a frequency adder (or single-side-band modulator) thus results. The rotation velocity can be made very large if it is performed electrically by means of a ferrite section driven by a varying or rotating magnetic field (Cacheris, 1954; and Fox et al., 1954). Such devices are useful for serrodync detection systems. The line-stretcher types of phase shifters include a short-slot hybrid with ganged shorting plungers (Barnett, 1955) and a circulator with a shorting plunger. 9.2,2 Hybrid junctions. Hybrid junctions are used as power dividers, phase comparators, and in balanced mixers (Jones, 1961). The matching posts and irises used in the conventional "magic tee1' at frequencies up to A!-band become difficult to fabricate in the millimeter range. The hybrid ring or "rat-race" is often used, since no matching is required. It may be milled, dye-cast or electroformed, even in small waveguide sizes. Typical band width, determined by the geometry of the ring, is about 6%. The basic design, shown in Fig, 9.10, is straightforward. The isolation is obtained by arranging the coupling arms A/4 apart, so that a signal entering arm 1 divides equally between arms 2 and 4, the waves that travel around the ring in opposite directions reinforcing. The waves reaching arm 3 just cancel. The junction ports are reciprocal, of course. Isolation between opposite arms is usually about 20 dB within the 6% band. Short-slot hybrids and multihole directional couplers having 3 dB coupling can also be used as hybrid junctions over fairly wide band widths, although directional couplers tend to be somewhat larger and not as well balanced as rat-races (Riblet, 1952). Fin-line couplers and turnstile junctions (Sections 9.2.3 and 9.2.5) also are useful for hybrid junctions and balanced mixers (Loth, 1956). 316 Microwave hardware and techniques Chap. 9 FIG. 9.10 Plan of rat-race hybrid junction, showing the lengths of the ring segments. The E-plane dimension of the ring groove is typically 1/V2 that of the side arm grooves. Coaxial-line hybrid junctions use hybrid rings, A/4 couplers, etc., and may have octave band width. 9.2.3 Polarization and frequency diplexers. A simple and useful polarization diplexer is the fin-line coupler (Robertson, 1956), sketched in Fig. 9.11 and shown in Fig. 9.4. By installing two side arms at 45° on either side of the plane of symmetry in the cylindrical form, a 3-dB coupler is obtained. Two arms, installed at 90° to each other, make a polarization diplexer, with the arms decoupled from each other. By properly phasing the drive to the two arms, a circularly polarized wave results. A square-cross-section coupler is made either by electroforming walls around a copper plate that has had a slot etched in it (see Fig. 9.4) or by machining and soldering. One way of fabrication is to mill off the top walls of two pieces of waveguide, fit them together with the iins sandwiched between and with the side arm positioned by a jig. The assembly is then soldered. The cross-coupling between two waves having their electric vectors oriented at 90" to one another in either type of fin-line coupler is typically down by 20 to 40 dB. The coupling loss to each of the polarizations is typically 0.5 to 1.5 dB, most of the loss being in reflections and in excitation of resonances in the structure. Resonances also lead to cross-coupling. Fin-line couplers are useful at frequencies between about 3 and 70 Gc. 9.2 Special components 317 Side-arm waveguide Resistive sheet to damp spurious mode responses FIG. 9.11 Sketch of a Jin-line coupler in cylindrical waveguide in a cut-away view, to show the internal construction of the coupling vanes or fins and the resistive mode suppressor. The fin-line coupler can also be used as a frequency diplexer if the frequencies are not too widely spaced. For example, a 25-Gc wave can be propagated in one polarization and a 35-Gc wave in the other, with little cross-talk. An electroformed waveguide having a square cross section for carrying the two polarizations is also shown in Fig. 9.4. A square-cross-section vacuum window and horn antenna complete the dual-polarization system. Some measurements utilizing such a system are described in Section 6.5. Frequency diplexers or multiplexers ordinarily use filters to separate the channels. A waveguide itself is a high-pass filter; a squeeze section of waveguide then becomes a variable-cutoff high-pass filter. A simple diplexer can be made using a series tee junction, containing a squeeze section in one arm for the high frequency signal and a phase shifter and low-pass 11 Iter or an impedance transformer in the other arm for the low frequency wave. Tlíc phase shifter and squeeze section are adjusted for matching at the junction at the frequencies in use. Isolation in excess of 318 Microwave hardware and techniques Chap. 9 Detected low-frequency signal Detected high-frequency signal Impedance transformer section to reject the high frequency Series T junction mm frequency input Tapered squeeze section having variable cut-off position for the low frequency FIG. 9.12 Dual frequency diplexer-detector for frequencies 30% to 80% apart. 30 dB between 25 Gc and 35 Gc signals propagating in RG-96/U, with little signal loss and reflection, have been obtained with such a diplexer. An advantage of this type of diplexer over that using fixed-frequency filters is that it is easy to adjust for changes in operating frequency, and not very sensitive to incidental frequency changes. The unit is sketched in Fig. 9.12. 9.2.4 Filters. Waveguide filters commonly arc made up of inductive or capacitive iris-coupled sections. The susccptance of the iris and the spacing between them determines the cutoff characteristics (Conn, 1957). Low-pass, high-pass, and band-pass configurations can be made, the design following standard transmission line theory (Guillemin, 1948). Typical six-section filters have an increase in insertion loss beyond cutoff of 36 dB per octave. J-Tigh-g cavity filters of the band-pass or band-rejection type may have a 30-dB change for a 1% frequency change (Riblet, 1958b). For example, the band-rejection filter used with the scattering experiment described in Section 6.7, consisting of two orifice-coupled tunable waveguide cavities, has an insertion attenuation of ~ 50 dB over a 20-Mc band and >20 dB over a 50-Mc band, but falls to 1,5 dB outside 9.2 Special components 319 < ; '■iuiHfflil FIG. 9.13 Band rejection filter. Two-section orifice-coupled cavity filter, tunable between 30 and 40 Gc with peak rejection >50 dB. (Courtesy General Atomic, San Diego, Calif.) /0±100 Mc. A photograph of the filter is shown in Fig. 9.13. The orifice-coupled waveguides are Xs!2 cavities, tuned by conventional shorting plungers. Low-pass filters find application in rejecting harmonics from mixers and from magnetrons or klystrons. Band-pass and high-pass filters remove unwanted noise or image frequencies in superhet receivers. EMgh-fJ, band-pass filters are usually a transmission cavity, that is, two waveguides orifice-coupled through a resonant cavity. Minimum insertion loss is usually at least 2 to 6 dB. A common construction method for iris-coupled waveguide filters is to saw thin slots part way through the waveguide, insert copper shim stock to the depth giving desired performance, and solder the shims in place. Alternatively, the iris vanes (shim stock) may be inserted in slots in an aluminum mandrel and the walls then electroformed around them. Either method gives satisfactory performance, using graphical design data to obtain the spacings (sec Microwave Engineers Handbook and Cohn, 1957). 9.2.5 Circular polarizers. A basic arrangement for transforming between linear and circular polarization is a quarter-wave plate oriented at 45° with respect to t he linear polarization. In waveguide terms, a quarter- 320 Microwave hardware and techniques Chap. 9 2,2 2.1 11 2.0 1.9 1.8 ( Odd CjiT Ever 0 0.1 0.2 0.3 0.4 0.5 0.6 Eccentricity e FIG 9.14 Mathieu-function roots for TEn-Iike modes in elliptical waveguide, as functions of eccentricity. wave plate is a section of waveguide capable of supporting two orthogonal modes having different phase velocities such that the respective electrical lengths differ by one-quarter wavelength. A simple technique is to deform circular waveguide (TEU mode) into an elliptical cross section. The cut-off wavelength in elliptical waveguide is given by (K)lm — —— ' Im (9.2.1) where a is the semimajor axis and rim is a Mathieu function root. These roots are given for TEn-like modes in Fig. 9.14 as functions of eccentricity. The length of a quarter-wave section is given in Fig. 9.15 as a function of wavelength, perimeter, and eccentricity (Blati and Heald, 1959). In practice, quarter-wave sections of this type can be made by electroforming on a precut mandrel or by mechanically deforming a section of circular waveguide, with gradually tapered transitions from circular to elliptical cross section to suppress reflections. End effects from the transitions usually require a linal empirical adjustment by further deformation. The useful frequency band width is about 4%. 9.2 Special components 321 1Q0 50 20 10 OS 0.2 1.6 0.1 e 0.2 03 04, / 2.0 2.5 3.0 3.5 4.0 F[G. 9.15 Dependence of length L on wavelength A for waveguide quarter-wave sections in elliptic waveguide of eccentricity e and perimeter p. Waveguide quarter-wave plates may also be made by dielectric or ferrite loading. Wide-band designs have been developed (Ayres et al., 1957), as well as designs in which the quarter-wave plate can be rotated 45° to provide the option of circular or linear polarization. Such a polarizer, used as a horn feed, is shown in Fig. 9.16. A system for transmitting simultaneously in the two counterrotating circular polarizations can be made by using a polarization diplexer with the quarter-wave section. Figure 9.17 illustrates such a system in which both modes arc fed from a single power source. The transmitting antenna launches a linearly polarized wave which is equivalent to two counter-rotating circularly" polarized waves of equal amplitude. The receiving antenna analyzes the clfcct of the plasma region between the horns in 322 Microwave hardware and techniques Chap. 9 FIG. 9.16 Components used to launch circularly polarized waves. Bottom, rectangular to circular transition; center, rotatable quarter-wave section; top, circular horn antenna. (Courtesy UeMornay-Bonardi Corp.) changing the relative phases, amplitudes, and polarizations of these waves. Another interesting circular polarizer is the turnstile junction (Montgomery et al., 1948; Meyer and Goldberg, 1955). The basic junction, sketched in Fig. 9.18«, is a six-terminal network, consisting of four Transmitting horn Rectangular guide -Or Receiving hom> Quarter-wave s ^section ^ ^ Fin-line polarization di plexer Circular (or square) guide TP Ttansistions back to rectangular guide F[G. 9.17 System for diplexing circularly polarized waves from a common source, using a polarizer and On-line coupler. 9.2 Special components 323 rectangular waveguide arms and one circular waveguide arm excited in two orthogonal TEn modes. Two pins in the junction, concentric with the circular waveguide axis, are used to match the junction. Tn the matched condition, and with all arms properly terminated, power entering arm 1, denoted by Elt will divide, half of it creating EA in the circular arm and half of it splitting between E3 and Et. No power reaches arm 2. Likewise, power entering arm 3 splits between arms 1 and 2 and creates E„ in the circular guide. The junction is thus useful as a polarization diplexer, as well as for a number of other applications; we shall discuss, here, its use as a circular-polarization analyzer. To generate a right-hand circularly polarized wave, we short-circuit arms 3 and 4 at 5Ag/8 and 7AJ8, respectively, and drive the junction from input 1, as shown in Fig. 9.186. The wave emitted will have a clockwise rotation sense. As a receiver, a circularly polarized wave having a counterclockwise rotation sense will be coupled into the side arm 2 and one having a clockwise sense into arm 1. If matched detectors are put on the two arms, we now have a direct measurement of an elliptically polarized wave entering the junction. (a) PIG, 9.18a Basic turnstile junction, showing the four rectangular waveguide arms and the circular wavegttide. The matching post is used to isolate opposite arms and to match impedtmces. 324 Microwave hardware and techniques Chap. 9 9.2 Special components 325 Arm 3 B shorted f .:1. . . fe m:..... FIG. 9.18b Turnstile junction connected to transmit or receive right-hand or left-hand circularly polarized waves. The addition of a conical horn to the circular junction completes the instrument for polarization analysis (Allen and Tompkins, 1959). 9.2.6 Resonant cavities. Cavities are used in a number of applications. Calibrated wave meters measure the frequency; coupled cavities are used in filters; low-g cavities are employed in some waveguide high-mode couplers; and evacuated resonant cavities are used to heat and measure density of plasmas, as discussed in Section 5.1. A common configuration is the right circular cylinder, sketched in Fig, 9.19. The wavelength of the resonance frequency for TM modes is (Moreno, 1948) 2/i [S2 + (l,lanP„J^ (9.2.2) where h is the height, a the radius, S is the integral number of nodes along h, Pmn is the nth root of lm(P) = 0, m is the number of circumferential maxima, and n is the number of radial maxima. 1.9-1,8-1.7- 1.6-1.5- ■U-l I 1.3 1.2-1 1.1- 1.0-0.9- 1% "OTM, 0.8 0.7 0.6 0.5-J TM, 2>TM0 oTM021 o 8TE411 o TM 022 oTE]22 °TM212 °TEQ11,TMm oTMon oTEm °TE3 °TE0 °TE212 oTM0I2 °TEiis TM023 §TE,23,TE4]3 TM213 oTE3]3 °TEoi3, TMj,3 8TM013 0 0.5 1.0 1.5 Example' 2o/\ LOB TM 2a/h 1.5 ö 2.0 2a Given: 2afh 1.5 Find: 2a/X 1.08 -2.5 PIG. 9*19 Mode chart Fot resonant cavity modes in a right circular cylinder. (( ourtesy R. N. Brucewell, Stanford University, Stanford, Calif.) I FIG. 9.21 Gain parameters Gm and Gs for linear rectangular horn antennas. (From SchelkunofT and Friis, 1952, courtesy of Bell 5 Telephone Laboratories.) u, 328 Microwave hardware and techniques Chap. 9 32 FIG. 9.22 Gain of conical horn antennas. (From King, 1950, courtesy of the Proceedings of the Institute of Radio Engineers.) The Q of a cavity depends upon the ratio of skin depth S to wavelength (see Fig. 9.1). For TE modes, the Q is generally maximum when the cavity height and diameter are about equal, as shown in Fig. 9.20. Further extensive data on resonant frequencies, mode configurations, Q, re-entrant end walls, and coupling techniques can be found in several texts (Montgomery, 1947; Wilson et al., 1946-1947; and Moreno, 1948). Cavities that must be evacuated require vacuum seals for electrode leads and waveguides. These considerations are discussed in Section 9.6. 9.3 Antennas and radiators 329 9.3 Antennas and radiators 9.3.1 Horns. One of the most useful microwave antennas is the pyramidal horn. A horn, gradually flared over several wavelengths, matches the wave impedance of a waveguide (typically 450 to 500 ohms) to that of free space (377 ohms) and provides directivity to the radiated wave. The directivity, expressed as gain over an omnidirectional antenna, depends primarily upon the height and width (in wavelengths) of the final aperture, but also secondarily upon the length. The gain of a horn flared in both the E-plane and H-plane is given as (Schelkunoff and Friis, 1952) t CmA CCA ^32 /; a (9.3.1) where Gm and Ge are gain factors given in Fig. 9.21. Gain data for circular (conical) horns are given in Fig. 9.22 (King, 1950). For both types of horns, at a particular length, there is a particular aperture for which maximum gain is achieved. A horn having these dimensions is called an optimum horn. Conversely, for given aperture dimensions, the gain does not increase appreciably with increasing axial length beyond the long horn length of abj\. In most plasma diagnostics experiments, where aperture diameters must be restricted, the long horn is preferred in order to achieve reasonably good gain and minimum radiation field curvature. The curvature can be further corrected by using a lens in front of the horn (see Section 4.9). The long horn also has a larger effective area for radiation, the area S being about 80% of the geometrical area. The optimum horn has S about 50% the geometrical area. The angle off-axis at which the radiation intensity has dropped 3 dB is related to the gain flaaB-P-^) degrees. (9.3.2) Several of the horn-design parameters are summarized in Fig. 9.23. We note in passing that, at millimeter wavelengths, horns having modest gains (for example, 10 to 15 dB) are not physically large. As pointed out in Section 4.9, the maximum permissible coupling between two horns is limited by Fresnel interferences to about —6 dB. The presence of the plasma, because of its refractive index, may cause refraction elfects that change the effective coupling. Often, the coupling is seen to improve by a decibel or so when a dilute plasma is present in the transmission path. The effect is not signal amplification, but is due simply to enhaocea coupling or better impedance matching, unless there be some amplifying mechanism also present. 330 Microwave hardware and techniques Chap. 9 fW X ' b = 0-3la M ' í 's rj.í,Onŕ) *H p$ e3dB(E)« 53 [deg| jj| "m. Ssá0.52-^2 4 0MB(E) as 60 X/d [deg], : 03dB(H)Sí7O X/d r ; |Long (iäa2/^ ĎVX> d2/x>! S = O.Slo/j 1S = 0.84 2^-1 i 4 1 H WE) «51 X/6 [deg] H FIG. 9.23 Summary of horn-design parameters. 9.3 Antennas and radiators 331 Waveguide horn feed FIG. 9.24 Metal plate (artificial dielectric) microwave lens, parallel to the metal plates. The electric field is '1.3.2 Lenses. Two modified forms of the ordinary horn find applications in many experiments: the horn-lens and the horn-fed dielectric rod. Microwave lenses are classed as dielectric, artificial-dielectric, or metal-plate. The metal-plate or "vcnetian-blind" lenses are made or parallel strips of metal of varying width, supported parallel to the incident electric field, as shown in Fig. 9.24. The phase velocity of the wave between the plates is greater than c (equivalent refractive index less than unity), so that a concave lens focuses (Kunz, 1954). Short-focal-length, stigmatic lenses that bring the waves to a line focus on the axis of a plasma column are straightforward to make by this method. The properties, including focal length, are frequency-sensitive, since it is the plate spacing that determines the equivalent refractive index (Brown, 1953). Dielectric lenses may be simply convex, as the common optical lens, or made up of zones (van Buskirk and Hendrix, 1961). The relation between a lens and Fresnel zone plates is shown in Fig. 9.25. The simple /one plate rejects half of the incident power because alternate zones are clear and opaque, as shown in Fig. 9.26. The phase-reversing zone plate is made by arranging the thickness of dielectric material in adjacent zones to give a phase difference of n, The full transmission then results. 332 Microwave hardware and techniques Chap. 9 Simple zone plate 15 Phase-reversing zone plate Quarter-period zone plate Fresnel lens Simple lens FIG. 9.25 Relationship between a Fresnel zone plate and a microwave lens. (Courtesy Electronic Communications, Inc.) 9.3 Antennas and radiators 333 mĚĚĚmĚmĚĚĚĚĚĚĚĚttĚĚKĚmmmm I ■ FIG. 9.27 Dielectric microwave lens, using impedance matching grooves. (Courtesy TRG, Inc.) FIG. 9.26 Fresnel zone plate for use as a microwave lens. (Courtesy Electronic Communications, Inc.) Wave launching horn 7-/-rr f i \X ^-Dielectric lens O-Ring vacuum seal Wave receiving system FJG. 'J.2K Microwave lens used as a vacuum window in a plasma experiment (see • il..... if.. (-.:'.!). (Gardner, 1lx..'.> 334 Microwave hardware and techniques Chap. 9 9.3 Antennas and radiators 335 Horn-lens combinations ordinarily require far-field illumination, or at least as long a horn as possible. They are useful for either focussing the waves at some point in the plasma experiment, or to correct for wave-front curvature. The horn-lens shown in Fig. 9.27 has lineal grooves, which are A/4 deep, but evenly spaced, to reduce reflections from the surface of the dielectric convex lens. The lenses may serve as vacuum seals, as shown in Fig. 9.28. In vacuum systems that do not require baking, the flat side of the lens is sealed with an O-ring, the convex side being illuminated by a horn. At 4 mm wavelength, a lens of 4 inches diameter is found to be quite satisfactory (Gardner, 1962). Some reflections, due to the dielectric interface, are present; a matching layer (analogous to coated optics) could be added to correct for these at the expense of band width, or the grooves shown in the lens of Fig. 9.27 could be used. 9,3.3 Dielectric rod antennas. The waves propagating in a dielectric rod (see Section 9.1.2) tend to excite radiation fields when discontinuities are present in the line. To intentionally make a radiator, then, one shapes the dielectric radiator cross section to achieve both the desired radiation pattern and a good impedance match to the feed waveguide (Mueller and Tyrell, 1947). Dielectric rods can support various modes, as shown in Rectangular 'waveguide (a) (b) FIG. 9.29 Dielectric rod antennas, showing launching mechanisms. Dimensions arc discussed in the text in Section 9.3.3. Fig. 9.5. One mode that closely approaches the circular TEU waveguide mode is the dipole or HEn mode (Brown and Spector, 1957). The mode may be launched by simply inserting a tapered dielectric rod into an open waveguide (Harvey, 1963), as shown in Fig. 9.29«. A circularly polarized wave at 35 Gc, launched by a teflon Tod having íí= 1.0 inch, b = 1.0 inch, = 1.125 inch, rf=0.328 inch, and e = 0.125 inch, produced a nearly symmetrical radiation pattern having E- and H-plane widths of 25°. A slightly better impedance match is obtained by flaring the waveguide into a small horn, as sketched in Fig. 9.29b (Elsasser, 1949). Rectangular cross-section rods can be excited by inserting them into a rectangular TE10 mode waveguide, as shown in Fig. 9.29c. The taper is not very critical in determining beam width, but does affect the impedance match. An average rod thickness between 0.2A and 0.4A seems to give optimum results (Watson and Horton, 1948). A curve of beam half-widths as a function of rod length is shown in Fig. 9.30. The curve also applies reasonably well to rods of circular cross section. Part of the radiation field arises directly from the launching structure, part from the end of the rod, and part from the sides along the rod. Tapered rods of low dielectric constant material, longer than about 2 wavelengths, fed from a horn that is not more than a wavelength in diameter, are usually free of side lobes. If the feed horn is too broad, the dielectric constant too high, or the taper too abrupt, the pattern may develop side lobes or even split into two major lobes. Long rods (6 to 18 wavelengths) of low dielectric constant material produce the cleanest, narrow beams. The beam width is not very sensitive to frequency (that is, '0 123456789 10 Length of dielectric rod in wavelengths (L/\) I'flG. 9.30 Beam width of radiation pattern of rectangular dielectric-rod antennas. 0 is the half wklLh at which the radiation intensity is down 10 dB from that at the center. 336 Microwave hardware and techniques Chap. 9 the radiator is broad-band), the dependence of long rods being approximately (Watson and Horton, 1948) e^óOÍA/L)1^ degrees (9.3.3) where L is the length. Typical rod antennas are useful over a 2-to-l frequency range. Narrow beams are obtained also by feeding the rod from double-ridge guide. The rod is inserted between the double ridges, filling the guide in the H-plane. Beam widths as narrow as 12° have been obtained by such an antenna rod 20 wavelengths long, tapered to a point at the end (Parker and Anderson, 1957). A rod, of course, is an insulator and can be inserted into some plasmas with little perturbation. Plasma measurements, utilizing this feature, were described in Section 6.5. Fused quartz, boron nitride, teflon, and sapphire are particularly good for such applications. Boron nitride is easily machinable, and retains its dielectric properties up to high temperatures. 9.3.4 Reflecting antennas. The gain and, thus, directivity of an antenna system can be increased greatly by adding a reflector of appropriate shape. For example, a 90" corner reflector placed A/2 from a half-wave dipole increases the dipole's gain by 10 dB (Harris, 1953; Cottony and Wilson, 1958). Parabolic reflectors, fed from the focal point, in theory generate strictly plane waves. At microwave frequencies, however, where the wavelength (a) m (c) FIG. 9.31 Some common feed systems for parabolic reflectors. («) Horn-feed. (/») Cassegrain. (c) Hog-horn. 9.3 Antennas and radiators 337 1 ?í FIG. 9.32 Small paraboloidal reflector fed by a horn, for operation in the 4-mm band, 68 to 75 Gc. (Courtesy of TRG, Inc.) 100 FIG. 9.33 Gain over a dipole for paraboloidal reflectors as a function of diameter, 0(X, The gain figures assume a 55% aperture efficiency (RETMA standard). I lie beam width is aflhe half-power (-3 dB) level, curves arc al the peaks. The first and second side-lobe 338 Microwave hardware and techniques Chap. 9 is appreciable, the operation is modified by diffraction effects, with subsequent beam divergence and the generation of side lobes. The side-lobe intensities may be decreased by illuminating the reflector nonuniformly, the usual taper being 10 dB from the center down to the edge (Crompton, 1954). Feed systems include the dipole, the horn (see Fig. 9.32), the Cassegrain, and the offset or hog-horn. Some of these systems are sketched in Fig. 9.31. The Cassegrain or double-reflector system (Hannan, 1961) uses a parabolic contour for the main reflector or "dish" and a hyperbolic or elliptical contour for the subrcflector. The subreflector is fed by a horn or dipole primary feed as sketched in Fig. 9.31b. The Cassegrain is finding wide application in the millimeter and submillimeter wavelength region, especially for transmission path links (Section 9.1.3). A full paraboloid produces a pencil beam; a parabolic cylinder produces a fan beam. Extreme fan beams, cosecant beams, and other specially shaped beams can be formed by feeding the reflectors off focus, or by shaping the reflector contours. Figure 9.33 gives the gain over a dipole for paraboloidal reflectors as a function of diameter. 9.3.5 Slot radiators. A slot cut in a conducting sheet illuminated by an electromagnetic wave can radiate. The radiation fields are derivable from the scattered magnetic induction, much as the radiation fields from a dipole are derived from its driving electric field. The slot and dipole are thus complementary elements; their interrelation is described by Babinet's principle (Schelkunoff and Friis, 1952; Booker, 1946). The electric field in the radiation pattern of a slot is perpendicular to the long axis (that is, E lies across the slot) in contrast to the dipole, in which the electric field is parallel to the length. The input admittance of a waveguide containing a slot is modified by both conductance (dissipative) and susceptance (reactive) effects, depending upon the size and orientation of the slot or slots. A slot whose perimeter is about one wavelength (length roughly A/2) is resonant, and thus offers only conductance. An array of resonant slots whose conductances (or radiation resistances) add up so as to just terminate the wave impedance of a waveguide constitutes a matched antenna. To avoid tapering illumination as the wave progresses down the guide, the far end is usually shorted and the reflected wave also radiates. Figure 9.34 shows a number of slot configurations. In general, slots lying on planes of current symmetry do not radiate; the radiative coupling, and thus the waveguide loading, increases with asymmetry in a complicated manner (Oliner, 1957). The radiation patterns of slot arrays depend upon several factors (Stevenson, 1948). The wavelength in the waveguide invariably is longer 9.3 Antennas and radiators 339 RS FIG 9.34 Radiation and nonradiating slots in rectangular waveguide, propagating the TEI0 mode. NR indicates nonradiating; RSh, radiating shunt slots; and RS, radiating series slots. than the free-space wavelength, unless the guide is filled with dielectric material. In order to drive the slots in phase, then, they must be spaced more than a free-space half-wavelength apart, leading to possible diffraction effects. Aside from these effects, the design criteria for ordinary dipole radiator arrays may be applied directly to the complementary structure, the slot, with the interchange of the E and H fields. Slot radiators Waveguide supported near wall 1 Detail of slot radiators FIG. 9.35 Slot antennas installed in a plasma experiment chamber. 340 Microwave hardware and techniques Chap. 9 1 HH i^dMHH ■ }' J; 1 FIG. 9.36 Twelve-clement slot-radiator array in a waveguide, giving 7° beam width With 20-dB side lobes. (Courtesy TRG, Inc.) Slot arrays are particularly useful for antennas in cramped quarters, such as inside a vacuum system accessible only from the end, as shown in Fig. 9.35. Photographs of some slotted waveguide antennas are shown in Figs. 9.36 and 9.37. Several waveguides, each containing slots, may even be stacked up in a curved broadside array to give focussing, analogous to a lens. Slot arrays may be fabricated by milling, by drilling and sawing the slots in a waveguide, or by electroforming the waveguide around a FIG. 9.37 Several microwave radiators useful for plasma experiments. {Courtesy of University of California, Lawrence Radiation Laboratory, Livcrmore, Calif.) 9.3 Antennas and radiators 341 mandrel containing fins at the location of the intended slots, as the unit shown in Fig. 9.36 was made. Several antennas useful for plasma diagnostics are shown in Fig. 9.37. Thin-wall, stainless steel horns are useful in pulsed magnetic field systems. 9.3.6 Antenna pattern measurements. Antenna radiation patterns (that is, far-field measurements) are usually plotted by rotating the radiator about an axis and recording the intensity measured by a distant, isolated, pickup antenna. Radiation patterns may be displayed on polar or cartesian coordinate paper. The field strength may be calibrated relative to the maximum signal or absolutely by a field-strength meter or comparison against a known standard antenna (Lawson, 1948). Often, it is inconvenient to monitor the radiation at the large distances required for far-field measurements. Measurements made within the Fresnel zone can be related to the far fields if the radiator can be re-I'ocussed (Woonton and Carruthers, 1950). Indeed, it may be the Fresnel /one or even the near-field zone fields that are of interest in the case of radiators used in plasma experiments. Often the field configurations of the antennas in situ (that is, mounted in the same position as they are to be I'M;. 9.38 field plotting probe for near-field amplitude and phase contour measure-moiiIs. Typical plots"are shown in Fig. 'J,39. (Courtesy of University of California, Lawrence Radial inn Laboratory, Livcrmore, Calif.) 342 Microwave hardware and techniques Chap. 9 9.3 Antennas and radiators 343 used) are desired, to determine the effects of walls, magnet coils, and other diagnostic apparatus. The radiator is then fixed in place, and the fields probed either by a scatterer or a small receiving probe (Richmond and Tice, 1955; Buser and Buser, 1962). An example of a field plotter, using a small pickup probe, is shown in Fig. 9.38. The probe, shown at the right of center, is a section of under-size waveguide, electroformed around a dielectric rod. The probe is Pyrex vacuum chamber Center line of the 9-inch diameter pyrex vacuum chamber FIG. 9.39a Electric field contour plots of a horn radiator in a pyrex vacuum chamber. Zero dB refers to the indicated signal when the probe is inserted inside the throat of the horn radiator. large signal loss. The small probes have 90° wide radiation patterns (at —3 dB), and thus introduce little angular error in most measurements. The scattering technique ordinarily employs a small rotating dielectric or resistive rod, mounted on a large x-y positioning frame. The rod is motor-driven to produce modulation in the scattered signal (Buser and Buser, 1962). Some systems use a semiconductor diode, supported by slightly conducting thread (silk painted with Aquadag), through which an Impedance matching plate Pyrex vacuum chamber Equal amplitude contour lines Absorbing coating painted on the pyrex walls Center line of the 9-inch diameter pyrex. vacuum chamber FIG. 9.39b Electric field contour plots of a horn radiator in a vacuum chamber of pyrex, coated with absorbing materials to damp internal reflections. surrounded by absorbing material to prevent reflections and mounted on an x-y motion carriage, in this case a converted drafting machine. A solenoid-actuated stylus, controlled by a foot switch, marks the paper to allow direct plots of either phase or amplitude contours (Klapper, 1962). The probe shown is rather large, and can cause serious perturbations in enclosed systems. Very small probes, made by electroforming thin walls around high dielectric constant cores, reduce the perturbation but cause audio current is passed to modulate the conductivity of the diode. These systems require complicated servo-systems to indicate the position of the scatterer. Sonic sample field contour patterns of a 35-Gc horn r;uli;i(or, made with n pickup probe inside a 9-inch diameter pyrex vacuum chamber, are shown in Fig. 9.39. In Fig. 9.39«, the chamber walls were bare glass; multiple internal reflections are very apparent in the ridges of the contour 344 Microwave hardware and techniques Chap. 9 lines. In Fig. 9.39*, the glass had been coated with a special Sauereisen-base absorber (Section 9.6.4); the internal reflections are hardly discernible. 9.4 Signal sources A variety of signal sources are available, at prices that are steeply ascending functions of frequency. To study a time-varying plasma requires either a CW source or a pulsed source having a very rapid repetition rate. Powers of a few milliwatts or more are usually necessary to provide adequate signal-to-noise ratios in the systems normally employed. Millimeter and submillimeter-source development is currently an active field. An extensive current tabulation of klystrons and other microwave tubes, of both United States and foreign manufacture, is given in the annual Microwave Engineers'" Handbook (edited by T. Saad, Horizon House, New York). 9.4.1 Klystrons. Reflex klystrons are convenient signal sources for transmitters and local oscillators over much of the millimeter band. Available power outputs range from 5 to 500 milliwatts at frequencies from 2 to 120 Gc, with the upper limits being extended year by year. Reflex klystrons are generally mechanically tunable, with built-in resonant cavities, although some lower frequency tubes require external cavities. They are electrically tunable about the cavity frequency by about £%, or typically 50 to 100 Mc, by varying the repeller voltage. Operating voltages, available from a number of commercial klystron power supplies, range from 350 to 3500 volts and must be ripple-free and regulated to 0.1% for most applications. Floating drift tube and multicavity klystrons are generally medium-to-high power tubes, useful for scattering experiments or plasma heating. Power outputs range from 0.5 watts to 50 kilowatts CW, and several megawatts pulsed, at frequencies from 1 to 80 Gc. Many types are mechanically tunable over a 5 to 10% frequency band and electrically tunable over 10 to 50 Mc. 9.4.2 Traveling-wave oscillators and amplifiers. Backward-wave oscillators (BWO) are another class of voltage-tunable tubes, many of them having output over a full frequency octave (2:1). BWO's, also known as Carcinotrons, are not mechanically tuned. Power outputs range from a milliwatt to a few watts at frequencies from 0.5 to 220 Gc. These tubes tend to be more expensive than klystrons, require fancier and more expensive power supplies, and are quite heavy, due to their focusing magnets. However, if there is need for their performance, then the choice is clear. Many types have a control grid to permit power leveling or amplitude modulation. 9.4 Signal sources 345 In this class may be included the M-type or cross-field oscillators (as distinguished from lineal beam or O-type), although often they are referred to as voltage-tunable magnetrons. Aside from slightly higher noise content, their performance is comparable to that of the O-type BWO's. Traveling-wave amplifiers of both O-type and M-type can be used to increase the available power from a transmitter or to serve as a preamplifier in a receiver. They are broad-band, some covering an octave and, at frequencies up to X-band, many O-types have low noise figures rivaling superheterodyne receivers. 9.4.3 Magnetrons. Magnetrons are normally intended for high-power, pulsed operation, at levels up to a few megawatts from 500 Mc to 100 Gc. They are useful for plasma heating, scattering, or transmitting over long distances, such as for studying plasmas in the upper atmosphere or in space. Voltages from 5 to 60 kV in pulses from 0.1 to 10 nscc are required. 9.4.4 Tunnel diode oscillators. Tunnel diodes (Esaki, 1958) are negative resistance elements and will oscillate at low power levels when placed in resonant structures. All-solid-state, tunable signal sources of miniature construction, having fair frequency stability, are available at the lower microwave frequencies. 9.4.5 Harmonic generators. Nonlinear elements, such as semiconductors, ferrites, and electron tubes, produce harmonics when driven at high levels (Page, 1958). Commercial harmonic generators for CW millimeter waves, such as that shown in Fig. 9.6, ordinarily use a crystal diode or a varactor (Bloom and Chang, 1957), operated at 10 to 100 milliwatts input power. Conversion efficiency to the second harmonic may be as high as -20 dB, yielding 0.1 to 1 milliwatt of harmonic power (Johnson et al., 1954). Harmonic mixing, discussed further in Section 9.5.2, may also be accomplished with the unit shown in Fig. 9.6. Ferritc harmonic generators must be operated at kilowatt levels and, thus, are pulsed at a low-duty rate to permit cooling. Conversion I'lliciencies of —10 dB at 20 kilowatts input level yield pulsed output powers as high as 50 watts at 150 Gc (Ayres et al., 1957). Gas discharges, field-emission rectifiers, arcs and relativistic electron b«ams have also yielded harmonic output (Coleman and Becker, 1959). Microwave tubes, such as klystrons, BWO's and especially magnetrons, may contain usable harmonic content in their outputs, especially if the Operating conditions are optimized. Frequency-multiplier klystrons and carcinotrons are available commercially; in these tubes the harmonic i onicnl is extracted directly from the electron beam and excites a resonator. 346 Microwave hardware and techniques Chap. 9 9.5 Signal detection The detector is one of the most critical parts of the transmission-reception system. At the millimeter wavelengths, especially, where the transmitted power is small, the receiver sensitivity must be good, while the hand width must be kept large enough to allow the output to follow the rapid fluctuations imposed by the plasma variations. /Generally, point-contact crystal diodes arc used for the detector; the crystal material may be silicon, germanium, gallium arsenide, indium antimonide, etc. (Smith, 1959; Torrey and Whitmer, 1948). Thermal detectors, such as calorimeters, Golay cells, and bolometers (Byrne and Cook, 1963) may be made to have great sensitivity, but have very slow response time, typically a second or more. 9.5.1 Video crystal detectors. A diode that rectifies the microwave current, producing a video output proportional to the impressed wave envelope, is the simplest form of receiver. The limit of sensitivity is governed by the Johnson noise, due to temperature-induced field fluctuations in the junction (Johnson, 1928), or to shot noise, due to current fluctuations ir the junction is biased (Petritz, 1952). The shot noise and flicker noise are, ordinarily, the most serious for video detectors operating at low modulation frequencies, since the noise in the output is proportional to l//(van der Ziel, 1950). It is well to avoid bias (that is, allowing direct current to pass through the junction) if possible, for this reason. A justification for using bias is that most diodes have nearly square-law response at low power levels (10~7 to 10~4 watts); the application of a forward current of 10 to 100 microamperes both lowers the internal impedance (Stanirorth and Craven, 1960) (thus reducing the output RC time constant and improving the video frequency response) and raises the operating point to a steeper slope region on the 1-V curve. Unless the load resistance is decreased, however, the increase in noise, due to the current flicker offsets the increase in current sensitivity, and no improvement in signal-to-noise ratio results. A second and perhaps more vital reason to avoid using externally introduced bias is that unwanted stray pickup (see Section 9.7) may enter the circuit in this low-level part of the receiver. The sensitivity and video impedance of millimeter crystals vary widely with type and particular sample; typical sensitivities are 300 microvolts/ microwatt, and impedances are 1000 to 10,000 ohms at a level of 5 microwatts. Short-circuit current sensitivity is between 0.2 and 10 microamperes per microwatt (Richardson and Riley, 1957). The threshold sensitivity of a crystal video microwave receiver is expressed in detectable power below 1 milliwatt, or —dBm. Figure 9.40 gives curves of threshold 9.5 Signal detection 347 i I I Imgp] Jamod 3|qi=p3)ap UJnUltUJW £> 3 1 P "O 6m M ° TJ £ 0 -' tn 01) 1 * t| ■ a > Tg O ti 3? o X 6 o a I 9 4 — □ Cfl 2 3 3 f 1 § g — 2 I ^ o Ó °, 348 Microwave hardware and techniques Chap. 9 sensitivity vs. receiver band width for various crystal figure-of-merit values. The figure of merit (FM) is obtainable from the manufacturer. Typical sensitivities vary from -57 dBm at 2.5 Gc (FM=200) to -40 dBm at 300 Gc (FM=4), for 10 Mc video band width. For 100-kc video band width, the sensitivities are improved to -67 dBm and -50 dBm, respectively. Further improvement can be realized at the lower microwave frequencies by placing a low-noise traveling-wave amplifier before the detector (Taussing, 1962; Wade, 1961). The crystal detector often is mounted directly on the video preamplifier-chassis to provide minimum capacitive shunting and to lessen the likelihood of stray pickup. If the preamplifier uses vacuum tubes, the detector must be mounted well away from them to avoid heating the crystal; temperature increases both degrade the noise figure and lower the power handling capabilities (typically <5 milliwatts CW). Video preamplifiers should be designed to have minimum input capacitance and a smaller noise contribution than the detector itself. Circuits, such as the cascode, both for vacuum tubes and transistors, are ideal from both standpoints. FIG. 9.41 Crystal video detector and video preamplifier. Amplifier voltage gain is 400 over a broad video band of 100 cps to 2 Mc or 1000 to 2000 over narrow, Iuned bands at 10 kc, 100 kc, and 1 Mc (see circuit diagram in Fig. 9.49). (Photograph courtesy of General Atomic, San Diego, Calif.) 9.5 Signal detection 349 D-c heaters for vacuum lubes, and ripple-free, well-regulated power supplies are necessary, since the signals being amplified are as small as a microvolt. An example of a crystal video receiver is shown in Fig. 9.41. The cascode triodes, followed by a pentode and a cathode follower, have an over-all voltage gain of 400, over a band from 100 cps to 2 Mc. The output can then be fed directly to an oscilloscope. To obtain the full video band width, the crystal detector may have to be shunted with a 5000-ohm or smaller resistor to reduce the RC time constant. This compromises the figure of merit, but improves the relative response above a megacycle or so. <>.5.2 Superheterodyne receivers. The microwave superheterodyne has considerably increased sensitivity over the simple crystal video detector (Harvey, 1963). The threshhold signal magnitude is related to the noise figure of the receiver, as shown in the curves of Fig. 9.42. Typical noise figures for balanced mixers at centimeter to millimeter wavelengths lie between 10 and 18 dB, yielding sensitivities of -90 dBm to -100 dlim for usual video band widths, and -120 dBm for very narrow band widths and coherent detection (Smith, 1951; also see Section 9.5.5). The minimum detectable signal power may also be expressed as an equivalent noise temperature Tn of the receiver, the conversion factor being derived from (7.2.8) jti _J mm (9.5.1) For ^»=1 picowatt (-90 dBm) and ůf= 10 Mc band width, the equivalent noise temperature is r„ = 7250 °K = 0.625 eV. The greater sensitivity of the superhet over the simple video detector is iluc mainly to its higher intermediate frequency. Since, for a given video band width, the shot noise is roughly proportional to Ijf, the choice of a high intermediate frequency reduces the over-all noise figure. T.f. amplifiers, however, have worsening performance above 20 Mc or so, requiring ii compromise in intermediate frequency; the optimum value, for normal hand widths from 2 to 10 Mc, is between 30 and 60 Mc (Rennie, 1957). The over-all receiver noise figure depends also on the mixer conversion efficiency, which improves with higher crystal current (Pound, 1948). Since the shot noise becomes worse with higher crystal currents, a compromise again is required; the optimum crystal current, usually, is found It) lie between 0.2 and 0.6 milliamperes, corresponding to a local oscillator powei of about a milliwatt for a balanced mixer (2 crystals). The current flowing in mixer crystals raises their operating point on the / V characteristic cTuve well out of the square-law region. For small Signals* tlM output voltage is thus nearly linear with input signal, rather 9.5 Signal detection 351 than quadratic as in direct video detection. For this reason, superheterodyne systems have much greater dynamic range than crystal video systems, besides having considerably better sensitivity (Tauband Giordano, 1954). Harmonic mixing provides a useful detection method for millimeter wavelengths where local oscillators are not available. The local oscillator is operated at a subharmonic of the frequency at which detection is desired. The pulses in the rectified crystal current contain harmonics that mix with the incoming signal to produce beats at the intermediate frequency (Johnson, 1954). Silicon and gallium arsenide crystals seem to give best results, yielding over-all noise figures intermediate between straight superhct receivers and crystal video receivers. Harmonics as high as the fifth and sixth give useful mixing action. The superhet has some disadvantages, however, the chief one being the need for a local oscillator. For transmission and scattering experiments, in order to keep the receiver tuned to the transmitter, it is usually necessary to provide automatic frequency control or AFC (Jenks, 1947), requiring still further circuit and operational complication. The reflex klystron is a convenient local oscillator for use with AFC because it is voltage-tunable over a small band. Changing the transmitter frequency over a large excursion requires retiming the local oscillator. A backward-wave oscillator is voltage-tunable over a wide range and, with proper adjustment, can be made to track a transmitter over its entire tuning range. Ordinarily, the mixer assembly requires peaking for each particular frequency, although some mixers retain low noise figures over a 6% band. A swept-frequency superhet can be made by sweeping the local oscillator electrically or mechanically. A well-matched, broad-band mixer, such as a short-slot hybrid or rat-race, is necessary for this type of service. An application requiring a swept receiver was mentioned in Section 8.3 for examining the electron gyro-frequency harmonics. 9.5.3 Parametric mixer-amplifiers. The parametric amplifier (Bloom and Chang, 1957), in some respects, resembles a superheterodyne receiver; I he pump is analogous to the local oscillator, and the nonlinear reactive clement is analogous to the mixer. The idler frequency appears at the sum and difference frequencies, for down-mixing, just as does the intermediate frequency of a superhet. In its component form, the microwave parametric amplifier uses a nonlinear capacitor, often a silicon p-n junction varactor with reverse bias so that it forms a voltage-variable, capacitive depletion layer, but draws no current. The resistive component of the "mixer" is thus practically absent, and the shot and Johnson noise contributions are very*small (van der Ziel, 1959). Equivalent noise tempera-11lies of unils operated under ambient conditions may be as low as 300 11K 352 Microwave hardware and techniques Chap. 9 and, when cooled to liquid nitrogen temperature (77 CK), the noise temperature may be reduced to about 80 °K. In addition, since the nonlinear element has little loss, the over-all conductance can be made negative by proper choice of operating conditions. Instead of "conversion loss," as in a diode mixer superhet, the unit then exhibits "conversion gain" over a narrow frequency spectrum. The varactor usually is mounted in a resonant structure, which makes tuning difficult. A typical A"-band parametric amplifier, pumped at 10.15 Gc, exhibited a gain of 50 dB over a band width of 2.4 Mc at an input frequency of 8.5 Ge, with an equivalent noise temperature of 600 °K (Bossard et ah, 1960). Generally speaking, parametric amplifiers, including traveling-wave and distributed-line units (Mount and Begg, 1960), have the widest application at the UHF and low microwave frequencies in fixed-tuned operation. Typical output powers are between I and 500 milliwatts. As components and techniques improve, the upper cut-off frequencies are being raised, and amplification at frequencies as high as A>band or the 8-mm band (35 Gc) has now become possible (de Loach, 1960). Other nonlinear Teactive elements, such as ferrites, electron beams, garnets, and various solid-state, crystalline substances, are also finding application (Mount and Begg, 1960). 9.5.4 Miscellaneous receiver systems. Quantum-mechanical amplifiers or masers arc useful for very low-level amplification at particular frequencies corresponding to the energy gaps in the molecules of various solids, liquids, and gases. To allow the levels to be populated to high enough concentrations to give spontaneous emission, the element must be cooled, often to liquid helium temperatures. In. principle, at 0 °K, the maser should be capable of detecting individual microwave photons (Weber, 1959). To achieve level-splitting or paramagnetic resonance, a magnetic field is required. The maser, thus, is not a simple amplifier to operate but, for some critical applications, such as radio astronomy, it provides amplification at noise temperatures of a few degrees Kelvin and frequency stability unmatched by other devices (Gordon et al., 1955). Tunnel diodes (Esaki, 1958) will oscillate if placed across a tuned circuit because they possess negative resistance. When isolated by nonreciprocal elements, such as a circulator, however, they act as amplifiers, having noise figures as low as 7 dB. Gallium arsenide diodes have been used as Esaki amplifiers as high as 26 Gc (Holonyak and Lesk, I960). The diodes are low-power elements, operating at levels up to a milliwatt or so. Thermionic diodes are useful as low-level detectors up to about S-bancl (3 Gc) before transit-time effects degrade the efficiency. At higher power levels (100 watts to 100 kW pulsed power), coaxial diodes, in which the 9.5 Signal detection 353 thermionic cathode is the center conductor, have been constructed with waveguide inputs for frequencies from 3 to 35 Gc (Hawkins et al., 1958). Commercially available units have large enough voltage outputs to deflect a cathode ray tube directly, with 0.1 microsecond response times and, thus, may be used directly for power oscillator monitoring. 9.5.5 Microwave radiometers. Receivers intended for reception of thermal or quasithermal radiation in the microwave spectrum are called radiometers (Harris, I960). The received signals have the characteristics of noise, generally over a broad frequency band, leading to the name white noise. The signals may be steady, slowly fluctuating, or transient. Receiving antenna 'irS^Vacuum i window Cal.brated . attenuator >vVV Balanced miser Ferrite chopping switch i - i i ľ? tow noise balanced i.f. amplifier '-- 3D Mc Nnlse source (Hm discharge tube et Iml lead in : ■ ■ ■ I: I-.- J Direct wine-hand video output Long mc-cons(ant rk output lUi. 9.43 Microwave radiometer with coherent detection. The input is switched between the signal and a noise standard cither mechanically or electrically. The detected, chopped signal is correlated with the reference, Uncorrclated signals myitiikc to zero, Wilhoul chopping, a direct, wide-hand signal is obtained. 354 Microwave hardware and techniques Chap. 9 The steady or slowly fluctuating noise signals can be measured by the use of coherent detection (Dicke, 1946) with narrow band widths and long averaging times. The block diagram of a typical coherent-detection radiometer (also called phase-sensitive detection) is shown in Fig, 9.43. The signal modulator may be a motor-driven attenuator or phase shifter (see Section 9.2.1) or a ferrite modulator. Or the radiating source itself may be modulated at a low frequency and the detected signal compared with the modulating signal, in the coherent detector. Crystal and plasma modulators ordinarily have too high noise temperatures to be useful for low-level modulation. An isolator must be mounted between the modulator and the mixer; local oscillator leakage back up the input line may be reflected back into the input and appear as a false signal. Ferrite modulators, either Faraday rotation or circulator types, have isolation inherent in their operation, with forward-to-reverse attenuation ratios of 18 to 30 dB. The reference signal, with which the detected signal is correlated, may be obtained from a photo diode, a magnetic pickup, or a miniature alternator (a-c generator) in the case of mechanically driven modulators, or from the a-c driving signal in the case of electrically operated modulators. The reference signal is usually reshaped, either by clipping to give a square wave or by passing it through a tuned circuit to give a sine wave. The radiometer system shown in Fig. 8.1 uses a motor-driven mica-disk attenuator having three attenuating regions to produce 90 cps modulation, thus avoiding problems with 60-cps pickup. The reference signal is obtained from a light beam that passes across the waveguide through the mica disk, the chopped beam being picked up by a photo transistor. Coherent detection compares the wanted signal (which has been chopped) with the reference signal in additive phase. The noise of the amplifiers (which has not been chopped) is added in random phases and, with sufficient averaging time, adds to zero. In practice, both the band-width and integrating times are finite (~2 cps and ~10 seconds, respectively) and the random noise rejection is not complete. Nevertheless, an improvement of 20 dB in the threshhold sensitivity of a supcrhet is relatively easy to obtain, and 26 dB can be achieved with some care. When the received signals are transient, such as those from plasma radiation experiments, the long averaging time cannot be afforded. Direct, wide-band detection is then necessary, and the added sensitivity given by coherent detection is not possible. Fortunately, the signals from most transient plasma experiments are large enough that direct detection with a sensitive superhet receiver is possible, and the detected video signal can be displayed directly on an oscilloscope. Some transient plasma experiments operate at a fast repetition rate (for example, 60 pulses/sec). If the radiation intensity is reproducible 9.5 Signal detection 355 from pulse to pulse, the time-average noise temperature can be obtained by switching the radiometer input back and forth between the radiation input and a standard noise source with a fast-operating ferrite switch, as shown in Fig. 9.43 (Bekefi et al., 1960). The amplitude variation in time during the radiation pulse can be measured by making the sampling pulse width much shorter than the radiation pulse width and varying the sampling delay back and forth over the duration of the radiation pulse. The noise temperature may be obtained either by calibrating the output against a known noise standard (White and Greene, 1956) or by reading a calibrated attenuator in the input as it is varied to achieve a null in the output signal. Signal sensitivities equivalent to a Dicke-type radiometer are obtainable, even with a transient discharge, by this technique. ^.5.6 Measurement of receiver performance. Hie performance of a crystal video receiver is best evaluated by plotting the output response against the input, obtained from a calibrated signal generator or a klystron whose output has been calibrated with a power meter. The receiver input is ordinarily fairly broad-band, and not very sensitive to any adjustments. The performance of a sensitive superhet receiver is somewhat more complicated to measure. Ordinarily, some kind of white noise source is used, often a gas discharge tube inserted in a waveguide (Mumford, 1949). flic noise temperature of these sources is expressed as excess noise power above that of a blackbody at 290 °K (room temperature). For argon discharge tubes, the excess is typically 15.2 dB (10,100 °K) and, for neon lubes, 18 dB (18,600 °K). The noise figure of the receiver (see Fig. 9.42) is defined as the ratio of available signal-to-noise power ratio at the input lo that at the output. The threshhold signal is one that is barely discernible. The tangential signal is one that just doubles the output over the noise background. The Y-factor of a receiver is the ratio of noise power in Ihe output, when the noise source is connected, to that without. Noise figures are generally expressed in terms of the Y-factor (Lebenbaum, 1956), assuming that band widths are constant (1W290)-1 Y-\ (9.5.2) where Z^s is the effective noise temperature of the noise source. The muse liguie is usually expressed in decibels, thus, NFaB - 10 log10(^-l) -10 log10( Y-1). (9.5.3) The y-factor is obtained directly from a calibrated attenuator inserted In iween the noise*source and the receiver. The attenuator is increased mil it Ihe effect on the out put signal of turning the source on and off just 356 Microwave hardware and techniques Chap. 9 25 20 a 15 10 J; I II 1 II II \ Neon discharge _ Vt—Noise temperature \ V T„ = 18 dB II 1 1 - \ \ - - Argon discharge Jf" - ~i i : 15.2 dB 1 1 1 1 1 1 1 1 MM" 10 Y|dB] 15 20 FIG. 9.44 Receiver noise figure vs. measured T-factor for argon and neon gas-discharge noise sources. A graph of noise figure vanishes; this setting is Y, expressed in decibels, vs. Y factor is given in Fig. 9.44. 9.6 Vacuum system considerations The vacuum system is a necessary part of a plasma experiment; still, the vacuum walls impose some difficult problems in making diagnostic measurements. Microwave radiators and probes must be inserted through vacuum seals, often buried deep inside a maze of magnet coils, outgassing heater windings, pulsed electrodes, water, air, and liquid nitrogen lines, and all the rest of the control and diagnostics connections. In many experiments the radiators and probes must be movable or, at least, removable. 9.6.1 Vacuum materials. Many materials commonly used in waveguide components are completely unsatisfactory in a vacuum environment. Most organic materials have a high vapor pressure (Dushman, 1962), and deteriorate under plasma bombardment. Materials that are fibrous or contain voids usually trap air or moisture and take excessively long to 9.6 Vacuum system considerations 357 pump out. Screws tapped into blind holes are exceedingly bad in this respect; all threaded members should be slotted, drilled, or filed to allow rapid pump-out. In general, metals, ceramics, glasses, some synthetic rubbers, and some plastics, such as teflon, polyvinyl chloride, PTFE, etc., are useful in systems having ultimate vacuums down to 5xl0~7 torr (Kohl, 1960). To go below that level, most systems must be baked, perhaps to 400 "C, and then the troubles begin. Plastics and rubber are out of the question, and even metal alloys that contain low-melting temperature components, such as brass, most solders, and sealing compounds, can be used only with care (Strong, 1938). 9.6.2 Transmission-line windows. In glass vacuum systems it is possible to mount horn antennas outside the glass and couple to the plasma region inside (see Fig. 9.39). Unless the glass is A/2 thick, or its mismatch can be matched at both surfaces, however, its presence in the near field may distort the radiation pattern or lead to frequency sensitivity due to multiple internal reflections. For reliable measurements, it is usually best to mount the antennas inside the vacuum and run the waveguides out to some convenient point before sealing them (Bendcrly and Kilduff, t962). Waveguide and coaxial-line windows for continuously pumped, non-hnkable vacuum systems present no special problems. In regions where magnetic materials are permissible, commercially produced kovar-sealed windows may be attached by soft solder, epoxy, or silver chloride, or by .lamping against an O-ring. Such windows are available in waveguide sizes down to about WR 28 (RG-96/U). At higher frequencies, it is usually necessary to "do it yourself." A simple window may be made by clumping a thin (~0.001 inch) teflon, mylar, or mica sheet against an ()-i ing between a choke-and-cover flange assembly. If insulating screws (such as nylon) are used, the window section serves also as an insulation break to suppress ground-current loops. Such a window is somewhat risky for long-term operation, and an improvement is to cement a 0.001-inch thick mica disc on either a choke flange or in a recessed area cut with n lathe in a contact flange, as shown in Fig. 9.45. Windows for waveguides AS small as WR 12 (RG-99/U), having standing-wave ratios smaller than 1.2 and losses of 0.5 dB, can be made by this technique. Some examples lire shown in Fig. 6.4 for operating frequencies of 70 and 90 Gc. Square Dross-section waveguide windows are made by the same techniques, allowing polarization diplexing (see Sections 9.2.3 and 6.2.2) or simply changing from one polarization to the other without opening the vacuum system. The simple O-riflg and cemented seals, mentioned above, are not suitable I hi ultrahtgh-vacuum syslems, operating at base pressures perhaps as low 35S Microwave hardware and techniques Chap. 9 Waveguide soldered to flange choke /mm j -S" groove 1 -■■» ■ jív: VňVjíÄ-aT-Ji-." -v-"-J.-.:ä..:-íľ.íi5:.- :; Jwí Mica sheet S Choke flange, "^^^gK'H^ffi showing groove /^%|JflP^ to be filled / ^fc-r with epoxy or silver chloride Round waveguide flange, with recessed surface FIG, 9,45 Waveguide vacuum windows for use on nonbakable vacuum systems. The windows are cemented in place with a Iow-vapor-pressure epoxy, vacuum wax, or silver chloride. as 10"10 torr. To attain such low pressures, it is necessary to exclude all high vapor-pressure materials and bake out the entire vacuum system to temperatures as high as 400 to 450 "C. Transmission-line windows Tor these conditions follow standard practice for microwave tube oulput 9.6 Vacuum system considerations 359 Nose assembly removed at bakeout Gold seal for assembly to system Waveguide and horn of OFHC copper BT brazed 70 Gc Microwave window assembly 0.003 -v J| Strain relief groove 0.007 -0.009 y^^P^Kovar T i CL|P 0.007 Rectangular waveguide inside dimensions Design frequency tr rad. b c d dia. 0,280 x 0.140 (RG-96/U) 0.280 x 0.280 0,148 x 0.074 (RG-98/U) 0.080 X 0,040 (RG-138/U) 35 Gc 35 70 140 0.160 0.203 0.086 0.076 0.026 0,032 0.012 0.020 0.089 0.086 0.044 0.023 0,203 0.345 0.109 0.047 FIG. 22 dB decrease) are obtained by making the lattice members with random widths or by letting points protrude a wavelength or so. Still other types of absorbers consist of an array of absorbing spears, set side by side, or bundles of short lengths of glass tubing, on which metal has been evaporated or colloidal graphite painted. 9.7 Circuitry considerations 9.7.1 Electronic circuits. Most of the electronic circuitry used in the transmitters, receivers, control and timing chassis, and power supplies required for microwave diagnostics are more or less standard. Power 364 Microwave hardware arid techniques Chap. 9 supplies, generally, must be well regulated and completely free of ripple and noise (. The circuit sketched in Fig. 9.49 includes a switch to select one of several tuned frequencies or broad-band video amplification. Power supplies for klystrons and for crystal video receivers often require regulator on top of regulator to eliminate the hum and noise. D-c heater supplies for the microwave sources and preamplifier tubes are generally desirable. 9.7.2 Circuit interference and stray pickup. Stray pickup around plasma experiments is a constant source of concern. The large pulsed currents (kiloamperes to megamperes) used in controlled fusion experiments make ground-current loops a major problem. Fortunately, a waveguide run can be broken up by inserting insulating films between choke llanges or by using short lengths of dielectric rod. Video crystal mounts especially must have insulation breaks, since even small currents flowing in the low level video-current return paths can induce serious interference. An example of a voltage isolation section in WR-28 (RG-96/U) waveguide is shown in Fig. 9.50. Mylar film 0.002 inches thick separates the flanges, which are fastened with nylon screws. The assembly holds off 1500 volts dry and 3000 volts when the holes through the mylar are filled with a silicone grease. The VSWR is 1.05 at 35 Gc, and the loss is less than 0.5 dEt. To hold off higher voltages requires external clamps (no holes in the insulating sheet) and a thicker sheet. Flange alignment then becomes 9.7 Circuitry considerations 365 FIG. 9.50 Waveguide voltage isolation section. Loss is 0.5 dB and VSWR is 1.05 at 35 Gc. Hold-off voltage is 1500 V. (Courtesy of General Atomic, San Diego, Calif.) :i problem and with sheets thicker than about 0.005 inches there is some leakage as well as reflection. Screen-cage enclosures are generally helpful for the diagnostics system although the main requirement to eliminate pickup is simply to make all ftround returns have low inductance paths. Running all waveguides, nigger cables, etc., so that they are well grounded as they pass through a huge metal sheet or a metal wall usually suffices. In addition, the power lines, oscilloscope trigger lines, and timing signal lines may need decoupling lilter networks or transformers. Coordination with the other diagnostic instrument leads is also necessary. Direct radio-frequency interference by electromagnetic radiation is seldom encountered, but occasionally spark gaps radiate "hash" (broad-band rf noise) and power tubes used in modulators, etc., may have parasitic or Barkhausen oscillations that radiate to sensitive circuits. CHAPTER 10 General plasma diagnostic techniques 10.1 Tabulation of some useful diagnostic techniques Several diagnostic techniques yield information similar to that obtained by microwaves, that is, the plasma electron properties. Many of the same experimental requirements also apply, such as shielding against stray pickup, multiple-channel simultaneous measurements, and the need for simplified, automatic data presentation. Some measurements involve internal probing, such as Langmuir and magnetic probes, which may perturb the plasma, while other external measurements, such as optical diagnostics, are nonperturbing. Some techniques are useful only in dense plasmas or plasmas containing a strong magnetic field. To sort out the measurements useful for a given kind of determination, brief tabulations of several techniques, grouped according to applications, will be helpful. Techniques useful for ion diagnostics and other plasma measurements are also included. The ranges of applicability shown are not necessarily exclusive, but are intended only as qualitative guides. Elaboration on a few of the techniques will be given in subsequent sections. A much more extensive discussion may be found in Huddle-Stone and Leonard, ed., Plasma Diagnostic Techniques (Academic Press, New York, 1965). electron density and distribution (a) Microwave interferometer; 1010<«e< 1014 cm-3. (b) Microwave cavity perturbation; 108<«e< I012 cm-3. (c) Rf-conductivity probes; 10s<«e<1015 cm-3; for high collision rates. (d) Microwave scattering; 1012<«e< 101* cm"3; sensitive to instabilities. 366 10.1 Tabulation of some useful diagnostic techniques 367 (e) Optical interferometer; 10141016 cm-3 for 10,000 gauss. (g) Optical spectroscopic intensities; «c>1013 cm-3; equilibrium plasmas. (h) Optical scattering; 1014<«e<1018 cm"3. (i) Optical Balmer series limit; 10130.1 eV; stable plasmas. (h) Doppler broadening of cyclotron radiation line; Te>50 eV. (c) Infrared and optical intensities; Te>l0 eV; equilibrium plasmas. (d) X-ray intensities; 2"e>6 keV; wall problems. (e) Relative intensities of spectral lines; l5 eV. (A) Langmuir probes; 0.1 < Te< 1000 eV; moderate densities. kin density and distribution (a) Stark broadening of spectral lines; «(> 1015 cm-3. (/?) Langmuir probes, single and double. (r) Electron, ion, neutral atom, or neutron beam probes; nx> 1014 cm-3. (d) Diamagnetic effect (requires knowledge of temperature). (e) Alfven and sound wave propagation; dense plasmas. (/') Calorimetry (requires knowledge of temperature). (g) Radioactive gas tracers and collimated detectors. (A) Charge-exchange neutral detectors. ION TEMPERATURE AND ENERGY (a) Calorimetry; total energy and momentum. (h) Doppler broadening of spectral lines; T{>5 eV. (<■) Externa] energy-momentum analyzer; samples escaping ions. ( i ■. 111 poliiri/alion prism. (Courtesy of W. E. Quinn, G. A. Sawyer, and F. L. Ribe, i u; Alamos Scientific Laboratory, University of California, Los Alamos, N.M.) 372 Generat plasma diagnostic techniques Chap. 10 FIG. lfl.2b Mach-Zehntler interferometer presentation as in Fig. 10.2«, but showing the turbulence due to a flute instability, developing at high compression ratios. (Courtesy of W. E. Quinn, G. A. Sawyer, and F. L. Ribe, Los Alamos Scientific Laboratory, University of California, Los Alamos, N.M.) A similar optical arrangement is used for Schlieren photography, except that changes in refractive index are recorded as modulations of light intensity, rather than interference fringes. An experimental arrangement for studying shock waves is shown in Fig. 10.4 (Lovberg, 1963). The light source is a spark between tungsten electrodes in nitrogen. The accurate timing necessary to follow the fast front is obtained by a Kerr-cell light shutter. A typical photograph is shown in Fig. 10.5, showing the sharp electron density gradient in the "snow plow" front. From Table 10.1, it is apparent that greater sensitivity for plasmas of medium density is obtained in the far infrared. Golay cell and bolometer 10.2 Optical and infrared probing 373 ITG. 10.3 Streak interfere gram of the refractive index changes vs. time in an electromagnetic shock tube. The plasma is generated by a conical pinch driver (Seott and Josephson, 1957) in hydrogen at 2 torr. The Mach number was 20, lit 18 inches down stream from the driver. The initial positive deflection of fringes { — Au) is due to an increase in neutral density by compression and dissociation, followed by the negative deflection ( — Zlfi) due to the electrons in the plasma. Peak electron density was 5.7x 10lc cm"3. (Courtesy of A, F. Klein, Aerospace Corp., I (is Angeles, Calif.) detectors are useful at wavelengths all the way across the infrared to the microwave band, although their response times arc ofthe order of a second. Nevertheless, some plasma experiments have been done at infrared wavelengths (Brown, 1962; Harding et al., 1961) with results that compare favorably with microwave results. Optical Faraday rotation can be used to study dense plasmas in strong magnetic fields (Dougal, 1963). The total rotation is given by (6.5.7), where the collisionless linear approximation is easily justified for these frequencies. Equation (6.5.7) then reduces to 180° v " A Hg 1-SW (10.2.1) where B is the magnetic field and BK is the magnetic field necessary to give |>.yroresonance, Bu = aim\e. As an example, the plasma density in a l()-cm path with 20 kilogauss applied, necessary to give 90° of rotation at ,\ 2,6 microns is 4.36 x 10' cm The sensitivity increases in direct 374 Generat plasma diagnostic techniques Chap. 10 Y t Camera FIG. 10.4 Experimental arrangement for studying fast current sheets in a parallel-plate plasma accelerator by Schlieren photography. (Courtesy of R. Lovbcrg, General Atomic, San Diego, Calif.) proportion to the density, the path length, and the magnetic field applied. Crossed polarization plates, sensitive to rotations as small as 5°, are experimentally feasible. 10.3 Conductivity probes Plasmas having high collision rates have appreciable real components of conductivity, as can be seen from (1.3.14). A small if coil immersed in such a plasma will induce currents in the plasma. The current is complex, the real part extracting power, and the imaginary part changing the coil's effective inductance by diamagnetic effects. Figure 10.6 shows two methods to measure the effect of the plasma on the coil. In Fig. 10.6tf, the voltage across the coil is a measure of its impedance, and the phase angle in respect to the input voltage gives the relative magnitudes of ./< to In Fig. 10.6/3, the unloaded coil is resonated at the drive frequency f0. The plasma current then both detunes and dc-Q's the circuit, much as in the resonant cavity cases in Section 5.1. Equation (5.1.2) may he used to calculate the loading as a function of conductivity. When 10.3 Conductivity probes 375 IK, 10,5 Schlieren photographs of a current sheet, traveling at 7 cm/fisec between pmalk'l plates in a plasma accelerator. (Courtesy of R. Lovbcrg, General Atomic, Nan Diego, Calif.) 376" General plasma diagnostic techniques Chap. 10 FIG. 10.6 Conductivity probes for high collision rate plasmas. In (a) the changes in coil impedance are measured. In (h) the changes in Q and fa of a resonant circuit arc measured. C2 > 10Ci; It X t»L. v»a> and wp»w, the major effect of the plasma on the coil is in lowering its Q. If the coil is immersed in the plasma so that a is uniform in space, (5.1.2) may be approximated as 1 1 i l 5, and 14 Mc are shown. (After Wharton and Hawke, 1962.) I In- center frequency f0 was held constant by trimming Cx slightly as the .....cluctivity was varied. Q0 was 105 for the 5-Mc coil and 85 for the II Mc coil, with distilled water in the chamber. A Faraday screen inside i In roil helped reduce electrostatic effects between the coil and the solution, mid the coil and the plasma, when later the assembly was used to study Ini• 11 di'ir.ily plasmas. 378 General plasma diagnostic techniques Chap. 10 10.4 Langmuir probes A conducting probe immersed in a plasma will emit or collect current, depending upon the voltage impressed (Tonks and Langmuir, 1929; Chen, 1965). The technique has been extensively used and in many cases the quantities measured compare very well with those obtained by other means (Schulz and Brown, 1955; Talbot et al., 1963). 10.4.1 Single Langmuir probe, no magnetic field. Typical characteristics for single probes, whose dimensions are small compared to electron and ion mean-free-paths, are shown for a plasma, in the absence of magnetic rJ (Current density) in II, vJ -30 -20 — 10i/ 1 1 ij + 10 +20 Volts 1 1 I J+ + 10 Volts -J (a) -40 -30 -20 -10 —I_! I I T 4i + 10 +20 + 30 +40 Volts J_LJ_J_L (e) FIG. 10.8 Langmuir probe V-l characteristics, (a) Linear characteristic. (/>) Logarithmic characteristic, (c) Double floating probe characteristic. 10.4 Langmuir probes 379 Reference probe (or wall) Glass Probe tip. Movable vacuum seal Straight probe Glass .90° (bent) probe Disc- Disc probe Double floating probe ■ FIG. 10.9 Some typical Langmuir probes. Glass capillary tube Bold, in Fig. 10.8. Sketches of some typical probes and their geometry m- riven in Fig. 10.9. The probe potential is measured in respect to some convenient, fixed-l" Initial point, such as the anode or walls of a discharge tube or a floating w.iII probe," which has an area at least 50 times as large as the probe Itielf. The requirement that the potential difference between the plasma ,ii n I lite reference point remain fixed often excludes the use of the anode or i number walls for this purpose because of current fluctuations. We shall pii",tune, for our fTurposcs, that the probe potential can be specified in n i-nl lo the plasma potential, and potential V refers to that value. 380 General plasma diagnostic techniques Chap. 10 When V\s made very negative, all electrons are repelled and only ions collected. The random ion current passing through an area A in the plasma is related to the ion density and the velocity where /., is the random ion current, amp, y+ is the random ion current density, amp/m2, Ap is the area of the probe, m2, m+ is the ion mass, 1.67 x 10~27 kg for protons, n+ is the ion density, no./m3, ulft = (2/r77m+)1'/2 = mean kinetic ion velocity, m/sec. Equation (10.4.1) would be valid also for ion current collected by a probe if the presence of the probe caused no perturbation in the surrounding random plasma currents. The probe does perturb the plasma, however. The volume which the probe occupies provides an energy sink for all particles which strike it and the fringing fields extend for a considerable distance into the surroundings (Genevalov, 1959; Bernstein and Rabinowitz, 1959). As a result, the collected ion current density seems to be more a function of the electron temperature than of the ion temperature. This effect is due to the formation of a positive sheath around the probe. The extent of the sheath's influence is determined by the electron temperature. For TtV) + A.JQ exp(^F) (10.4.8) where Ali2 are the areas of the respective probes, JQ is the random electron space current density, V^_.2 are the plasma-to-probe potentials, VD=V2-VU Id — Ipi ^el= f*Si Ip2r _e_ _J_ 11,600 *"fcrrre[eV] re[°K]' The logarithm of (10.4.8) is ■4>VD+la At (10.4.9) Hie slope of (10.4.9) plotted against VL yields the electron temperature, jusl as in Section 10.4.1. 10.4.6 Double probes in a magnetic field. Double probes are not influenced as much by magnetic fields as single probes. The collected i hi rent is governed by ion mobility, and it is not until the magnetic field in ngth is very large (several thousand gauss) that the ion gyroradius is as small as the probe size. When the negative voltage is sufficiently large that the sheath thickness i. luge compared to the probe diameter, the theory of Langmuir and Motl-Smith (1924) applies, even in magnetic fields up to several thousand is, The current-voltage relationship then follows a square law (10.4.10) ./, is linear with ion density, as shown in (10.4.1). The ion density, far mil in region I, then is found to be (Langmuir and Mott-Smith, 1924) l2rtm + \ n J+ A = 3.32x 10u i (10.4.11a) (10.4.11b) 386 General plasma diagnostic techniques Chap. 10 where S is the slope of/2 vs. V expressed in amperes and volts, and A is the effective collecting area of the probe sheath. Gardner (1962) expresses the relationship in terms of current density, not specifying A, but assuming that it remains constant as V varies dV 2e:J (10.4.12) The circuit of Fig. 10.10, in the switch position marked "Square," gives direct readings of n + z, when the cursor slope has been calibrated. 10.4.7 Double probes: miscellaneous effects. Although floating probes represent less of a perturbation than a single probe in an equilibrium discharge, when the ion temperature is well above the neutral gas temperature any kind of an object in the plasma leads to severe energy loss. The probe element suffers from bombardment and, even though it floats, it possesses capacitance, which tends to hold its potential fixed long enough to expose it to arcing, which of course melts the elements away. 10.5 Plasma wave and resonant probes When plasma waves or oscillations are present, they may be detected with probes. Spacecharge waves may also be launched with probes, but this method of launching tends to excite all modes. Langmuir-typc probes, having coaxial shields brought up near the collecting surface, are adequate for many measurements up to frequencies of 1000 Mc (Bailey and Emeleus, 1955). A pair of small disc probes has been used successfully to measure cop in dilute plasmas (Yeung and Sayers, 1957) and, in fact, wire and disc probes were used in experiments that probably were the first microwave diagnostic measurements (van der Pol, 1920). When a current-collecting probe simultaneously has a large rf voltage applied to it, the nonlinear sheath characteristics cause rectification of the rf signal. The d-c current is thus altered a small amount. The rf electric field around the probe is maximum at a frequency near the plasma frequency, leading to an increase in the rectified direct current as the applied frequency is swept through the local plasma frequency (Takayama et al., 1960; Ikegami and Takayama, 1963). The probe, of course, perturbs the plasma and thus the frequency measured is slightly below the true plasma frequency. Resonant probes, fed by transmission lines (Levitskii and Shashurin, 1961), permit measurement of the plasma impedance, plasma density, and guide wavelength (see Sections 5.5 and 5.6). Probes loosely coupled to a tunable filter, such as a motor-driven coaxial resonator (Malmbcrg ct al., 10.7 Ballistic probes 387 1963), permit rapid analysis of the frequency spectrum of oscillations picked up in the plasma, or as a means to filter received signals. The movable probes shown in Fig. 9.47 were used for these purposes. 10.6 Magnetic probes Small inductive probes immersed in the plasma will have voltages induced in them by changes in the local magnetic field, dBjdt (Glasstone and Lovberg, 1960; Colgate et al., 1958). Field sensitive elements, such as I lall current probes, measure the instantaneous magnitude of the magnetic field B. Magnetic probes may be made as small as 1 mm in diameter and grouped in x-y-z arrays to measure three-dimensional field configurations (ToHock et al., i960). Current density contours and the presence of hydromagnetic instabilities in dense plasmas are measured by a linear array across current channels. The data can be displayed by rapid sampling. The output voltage of the coil-type probe may be integrated to yield the magnitude of field. The resulting signals are very small (depending on the integration time), and care must be used to avoid stray pickup. Hall probes have outputs of a volt or so, response times up into the megacycles, and are easily calibrated with a standard magnet. They are somewhat temperature sensitive. Another kind of coil assembly that measures rates of change in enclosed current channels is the Rogowsky loop or girdle (Golovin et al., 1958; < ooper, 1963). The assembly consists of two sets of coils, one around the mi ire experimental region and the other around only the current channel or a part of it. The difference in induced voltage represents the currents not enclosed, such as wall currents. The coils may be segmented, with leads brought out separately, to indicate current profiles. Low inductance coils can also be used to pick up high frequency fluctuations, such as those associated with ion-wave instabilities or ion-c vi lotron-frequency instabilities. These frequencies are typically from 10 kc to 10 Mc. 10.7 Ballistic probes I he measurement of thrust, shock intensity, and momentum transfer by plasmas is possible by using ballistic probes (Marshall, 1958). Considerable care is required to avoid electrostatic force deflections and thermal [hock deflections. Piezo-electric transducers (Stern and Dacus, 1961) are particularly useful, since they can be driven by insulated pickup probes and produce an output voltage having very fast time response. These transducers me also sensitive to thermal shock produced by intense light pulses |li-i example, laser beams) Or microwave pulses (While, 1962). 388 General plasma diagnostic techniques Chap. 10 10.8 Optical spectroscopy Spectroscopy is a very large subject in itself, having wide and wet developed applications to nearly all categories of plasma research. At best, we can only hope to give a few of the highlights in this section, indicating where fuller treatments of the various spectroscopic techniques may be found. 10.8.1 Constituent identity and state. Identification of radiating species has been studied perhaps longer than any other of the spectroscopic techniques, and numerous tables of line wavelengths, intensities, and related information exist (Chemical Rubber Handbook, 1962; M.I.T Tables, 1955; Kelly, 1959; and A.I.P. Handbook, 1963). A photographic plate exposed in a calibrated spectrograph can reveal immediately a great deal of qualitative information about the degree of ionization, whether molecular dissociation is occurring, what impurities are present, and even some knowledge of the electron temperature by observing the highest ionization potentials excited. The one-dimensional extent of the various species is obtained from the lengths of the lines. The presence of a continuum between the lines indicates the presence of recombination or bremsstrahlung. A quantitative examination of the lines, such as with a line splitter (Scott et al., 1962) or microdensitometer, then reveals detailed information about plasma properties (Wulff, 1959). Of the three types of line broadening, only the Stark and Doppler broadening give measurable effects in most normal plasmas. Collisional or pressure broadening is generally a very small effect. Zeeman splitting of lines, if observable in laboratory plasmas, gives information on the local magnetic field strength. Field strengths of 50 to 100 kilogauss are required to be able to see the effect at all, in most cases. 10.8.2 Stark broadening. Tn high-density, low-temperature plasmas («> 1015cm"3, Te <4 eV) the spectral lines are shifted and broadened by the electric fields due to ions and electrons. Tn the Holtsmark theory (Griem, 1960), the line profile is described by a slowly varying function S(a) S(ct)da=l (10.8.1) where a=AAjFa jr70 = 2.6l en54 = the Holtsmark normal field strength n — electron and ion density, number/cm3 A A = displacement from unperturbed line, A e = electron charge 10.8 Optical spectroscopy 389 S(a) is a complicated function, and it is convenient to use either approximations or tabulated values (Griem, Kolb, and Shen, 1959; Underbill and Waddell, 1959). Several modifications to the Holtsmark theory have been made to account for correlations among ions (Margenau, 1932, 1951) and for nonadiabatic effects. The density dependence, nevertheless, does not depart much from the n%, especially in the wings of the Balmcr and Lyman series lines. Some lines exhibit linear Stark effect and others quadratic effects; the total broadening is calculated by summing over all of the Stark coefficients. Typical measurements of Stark broadening range from a fraction of an angstrom for the N IV 3479 A line due to a 5% N2 impurity in a 1017 enr3 D+ plasma (Lukyanov and Sinitsin, 1958) to tens of angstroms in dense H2 plasmas having n ~ 10ie cm"3 (Wilcox et al., 1961). The Stark shift, which is generally quite small, has a slight temperature dependence imd a density dependence ~nl22 (Margenau, 1951). 10.8.3 Doppler broadening. The relative velocity, v, of a radiating atom in respect to the observer leads to a doppler shift of the radiation frequency Av = v — va = v0vjc. (10.8.2) If the emitting atoms are in kinetic equilibrium, this leads to a gaussian line shape for the doppler broadening SD(y)=^— exp The doppler half-width in frequency or wavelength is SAD (10.8.3) (10.8.4) (ln2)^0~(ln2)»Ao ({n2)\Mc2) where M is the ion mass. For example, the doppler broadening of the Palmer HB line is approximately SA ^^4-10-5(^+[eV])'-. (10.8.5) In terms of the intensity at the line center, /„, the intensity I(v) is '<*>-'° exp[-g(>vf]- G»*« typical line half-widths are between 0.3 to 30 A for hydrogen ion u tnperatures between I0B to 109 °K. Transient plasmaSfoften arc not reproducible from pulse to pulse, and it desirable to observe the line profile completely with each shot. A line 390 General plasma diagnostic techniques Chap. 10 V 7 FIG. 10.12 Optical line splitter assembly, showing the exit slit with the cylindrical lens A, the deflection mirror B, the photomultiplier array C, and the light-tight entrance tube D. {Courtesy of F. R. Scott, General Atomic, San Diego, Calif.) splitter (Scott et al., 1962) permits a sampling of a broadened line at six intervals over a narrow wavelength band. A cylindrical lens, attached to a 250^-wide slit on a 500-mm JACO monochromator, allows a variation in the dispersed band from 0.04 A/mm to 18 A/mm. Seven 1P21 photo-multipliers are used as detectors. The instrument is shown in Fig. 10.12. Besides giving the profile shape, if the detector outputs are observed for signal time correlation, one can determine if the broadening is true doppler (thermal) broadening or due to mass motions in the plasma. 10.9 Bremsstrahlung and recombination continuum 391 10.9 Bremsstrahlung and recombination continuum The radiation from free-free and free-bound transitions (see Section 7.3) also may be analyzed in the X-ray, ultraviolet, optical, or infrared wavelengths as a diagnostic technique. In the optical spectrum, this radiation appears as a continuum between the emission lines. The continuum is made up of bremsstrahlung radiation (due to the acceleration of electrons during collisions), from high, overlapping harmonics of synchrotron radiation (due to acceleration of electrons between collisions), and from recombination radiation (due to the radiative capture of electrons into the hound states of atoms). The contributions of these three depend, in different manners, on the electron density and temperature and on the wavelength at which observations are made. For example, in equilibrium hydrogen plasmas (electrons having a Maxwellian velocity distribution), the difference in background intensities of the Balmer continuum (that is, the background intensity between Balmer series lines) and that just beyond the series limit (A < 3642 A) is strongly dependent on the electron tempera-lure, but not very sensitive to density (McWhirter, 1965). The absolute intensity of the continuum, however, is proportional to nenh with a square-root dependence on temperature. At short wavelengths, then, where effects of impurity radiation are not too serious, the electron density can be measured by absolute intensity measurements of the continuum (Glasstone and Lovberg, 1960; McWhirter et al., 1959). The frequency dependence of the continuum intensity (that is, the shape of the spectral intensity curve) is a strong function of temperature in other portions of the spectrum away from the Balmer series limit as well, especially if there is self absorption. For example, in the far infrared the bremsstrahlung continuum merges into blackbody radiation. In experiments at Harwell, England, by Harding and Roberts (1961), the gap between optical and microwave radiation was bridged by the use of a special far infrared spectrometer, covering wavelengths of 0.1 to 2.0 mm. The radiation intensities were found to follow blackbody curves quite well up in about the plasma frequency and then to saturate, becoming insensitive In further changes in frequency. The experimental curves had the general lhape of Fig. 7.7«. 10.8.4 Doppler shift. If the plasma as a whole is moving (mass motion), the emission lines will be shifted in wavelength. Tn a rotating plasma, the spectral lines originating in that part of the plasma moving toward the observer will be shifted to shorter wavelengths and vice versa. If the entrance slit to the spectrometer spans the plasma diameter, the lines will be tilted, leading to the slant effect (Dickerman and Morris, 1961). APPENDIX A Review of electromagnetic wave propagation The purpose of this appendix is to review the well-known arguments of electromagnetism leading to the concepts of complex conductivity and dielectric constant and to wave propagation in a general lossy medium. The final sections consider microscopic fields and the Lorentz-term paradox, and propagation in anisotropic media. For a more extensive discussion of these matters, sec von Hippel's Dielectrics and Waves (Technology Press/Wiley, New York, 1954), and also the standard treatises on electromagnetism by Panofsky and Phillips, Jackson, Stratton, and others. A.l Basic relations for a linear medium Maxwell's equations for a macroscopic medium are, in rationalized mks units, VD = P (A.1) V-B=0 dt VxH=J+ dt (A.2) (A.3) (A.4) where p is the volume density of free charge and J the explicit current density. The field vectors E, D, B, and H are related by the constitutive relations D = eoE + P (A.5) ho (A.6) 392 A.l Basic relations for a linear medium 393 where P and M are the polarization and magnetization (net dipole moments per unit volume) and e0, pQ are the usual dimensional constants of mks units.1 In the case of a linear, isotropic medium P=%eE M = f,„H (A.7) (A.8) where % and Wm, the electric and magnetic susceptibilities, are constants characteristic of the medium.2 For this linear case, the constituitive relations (A.5) and (A.6) reduce to D = eE = /ie0E P km[l0 (A.9) (A. 10) where e and p. are the permittivity and permeability of the medium and ,<=l + We (A. 11) *m=l + lI'm (A 12) are the dielectric constant (or relative permittivity) and relative permeability. I ikewise a linear, isotropic medium obeys Ohm's law, J = oE (A. 13) where a is the conductivity.3 Thus the electromagnetic properties of a linear, isotropic medium are specified by the three independent constants h, km, and a. 1 The mks units of these various quantities are E volt/meter D, P coulomb/meter2 B weber/meter2 = 104 gauss H, M ampere/meter ji coulomb/meter3 .1 ampere/meter2 |/4irf0 = 10-,c2«9-10l> meter/farad (c = velocity of light) (i(1/47r= ]0"7 henry/meter. 1 We follow convention here in writing M proportional to H, rather than to the Average microscopic field B. 1 Many authors use mobility, defined as the ratio of average electron velocity to electric Reld, rather than conductivity in discussing ionized gases. Thus, mobility equals i. II more than one species of charged particle contributes to the current, the Conductivity is where nH, qKt and .,1ik are the density, charge, and mobility nf the kiU species. 394 Review of electromagnetic wave propagation Appendix A From an energy point of view, the constants k and iK£0)E. (A. 14) (A. 15) (A. 16) If the medium exhibits dispersion, a, k, and km are functions of frequency. The form of (A. 16) makes it convenient to incorporate one of the two constants a, k in the other by means of complex notation. This technique is directly analogous to the familiar representation of the resistive and reactive processes of circuit theory by a complex impedance. In the case of ordinary dielectrics and wave propagation, it is customary to define a complex dielectric constant such that or a+ja>ice0 —> joji (A. 17) (A. 18) where we use the symbol U to denote a complex quantity and the subscripts r and i to indicate real and imaginary parts.4 The a-c k{ may include hidden dissipative currents (such as dielectric hysteresis loss) in addition to currents resulting from ohmic conductivity. Some texts refer to K( = o-/e0to as the loss factor of the medium. The ratio of conduction current to displacement current is often called the loss tangent, tan§=- (A. 19) The power factor of the medium is sinS which, for small losses, is nearly equal to tanS. Some of these quantities, for various common dielectric materials, are listed in Table A.l, for comparison with plasma characteristics. 4 Note the choice of sign such that k,>0 in dissipative media. A.l Basic relations for a linear medium 395 Table A.l Dielectric constant and loss tangent i or common materials at 253C and 25 Gc Materia! tan 3 Glass, tCovar sealing 7052 4.85 0.011 Neoprene 4 0.03 Paraffin wax 2.2 < 0.001 Polyethylene 2.26 < 0.001 Teflon 2.08 < 0.001 Water 34 0.26 Wood ~1.7 -0.02 In the case of ionized gases, it is customary to employ a complex conductivity such that or ä = ov-p/o-i= a+jct)(i<— l)e0 = u+jcoWe€0, (A.20) (A.21) where We is the susceptibility from (A.7). It is to be noted that (A. 18) and (A.21) are alternative, equivalent formalisms, which are not to be used '.imultaneously.5 The interrelations are i -—)- / (—) «j£o \ wf0/ \a>e0; a = or +>i =Ju>€0(k — i)= [kt +j(kr — l)]a>e0 (A.22) (A.23) 11| her authors make use of a complex susceptibility or a complex mobility. The complex dielectric constant formalism (A. 18) eliminates a and, hence, the explicit current density J in Ampere's law (A.4). It also automatically eliminates the explicit charge density p in Gauss's law (A.l), winch may be seen by considering the equation of continuity of charge bt (A.24) 11 Depending upon the microscopic models used, it is sometimes useful to ascribe PQIliplex properties simultaneously to two or three of the constants, a, «, and k„, 11- re each represents a different physical effect. For instance, a might describe the mi phase and out-of-phasc components of explicit currents, while the imaginary Bltrtd o(i< and in dcsciTlic ilipole hysteresis loss nol counted in f-y#) (A.33) where y is the propagation coefficient, and5 r = «+7)3 (A.34) where « = attenuation coefficient (nepers/meter) 13= phase coefficient (radians/meter). Substituting in (A.29) and (A.30), and following the procedure leading to (A.32), we obtain and hence (A.35) (A.36) I he sign of the square root is to be chosen such that fi = ]m(y) is positive. The reciprocal of the attenuation coefficient a is the attenuation length S, (he distance in which field amplitudes diminish by a factor of e. The phase coefficient /3 is related to the wavelength A and phase velocity Vj, by 8=—=—-A 1)0 (A.37) The refractive index \i is conventionally defined as the ratio of the vacuum velocity of light c to the phase velocity in the medium; c c = P — V4, oj (A.38) " While our notation is that commonly found in electrical engineering literature, most theoretical physics authors use the complex angular ware number k = —jy. Often these authors put k = a+j$, in which case the definitions of a and /3 are interchanged with respect to our notation! A second common variation is the choice of sign in Hi, phase factor; although arbitrary, this choice controls the sign of the imaginary iHi'. 01 S, k, y, etc. Often, but not always, these two notational conventions are Ignaled by the use of I or j for v'-|. Thus, the notations exp(/<«/ — fx) and < p ilk v -nit) are most common. 398 Review of electromagnetic wave propagation Appendix A Accordingly, it is useful to define a complex refractive index fx, or normalized propagation coefficient,7 M = P-JX = -# ~ = ("Km)%> (A.39) where u-(real) refractive index X = attenuation index, and the sign of the square root is taken such that ft is positive. Thus a=x - c (A.40) (A.41) If fx varies with frequency, the medium is said to exhibit dispersion. The group velocity of a wave packet in a loss-free dispersive medium is 5 d^ 7 Note that some authors use the form where £ = x!f- is called the extinction coefficient. (A.42) -(1 0 +p FIG. A.l Example of an «j-,3 diagram. See also Fig. 1.5, and Section 5.3. A.l Basic relations for a linear medium 399 The dispersion relation between frequency and wavelength may be shown graphically (Fig. A.l) as an -/J diagrams is made in Chapter 5. In most cases of practical interest in plasma physics, the relative permeability k,„ is unity.3 With this simplification we have A = P-JX = fci = («r -M)Vi-Squaring and equating real and imaginary parts, we obtain from which '=1 4 2™ = ** j 2X 2x 2fj, (A.43) (A.44) (A.45) (A.46) (A.47) The coefficients 8 and jS are readily obtained from (A.40) and (A.41) A number of approximations of (A.46) and (A.47) are useful; Kr«Ki (ohmic conductor) ic1,ll»ici3; kr>0 (low-loss dielectric) 4*15) k,'-*»ki2; Kj<0 (cut-off low-pressure plasma) (A.48) (A.49) (A. 50) (A.51) (A.52) (A. 53) 11 In Section 3,2 we make use of the permeability to approximate high-temperature '.....i in a plasma, 400 Review of electromagnetic wave propagation Appendix A From (A.35) and (A.36), it follows that the wave impedance -r), defined as the ratio of E to H, is (A.54) Thus, E and H are in phase in a dielectric {#? real), 45° out of phase in a conductor (H imaginary). In vacuum ij = (p-oha)^ ~377 ohms. By applying electromagnetic boundary conditions at the interface between two homogeneous regions, one can discuss reflection and transmission of waves at discontinuities. The combination of incident and reflected waves represents a standing wave in the first medium. The depth of penetration of the wave into the second medium is known as the skin depth, equal to the attenuation length S = 1/«. There is no reflection when the wave impedances of the two media are equal. These questions are considered in more detail in Chapters 4 and 9. A.2 Microscopic relations The preceding discussion has been concerned with macroscopic fields, averaged over dimensions large compared to the mean interparticle separation Clearly, the local, microscopic fields vary violently in the neighborhood of particles which carry net charge or possess a dipole moment. In order to relate the macroscopic constants to atomic properties, one must investigate the relation between the macroscopic fields and the effective average microscopic field experienced, for instance, by a free electron or polarizable molecule (VanVleck, 1932; Rosenfeld, 1951).9 It is shown in most modern electromagnet ism texts that E and B are the macroscopic space averages of the microscopic electric and magnetic fields, respectively, inside a uniform medium. H is the space-average magnetic field which would exist if all (true) atomic magnetization currents were replaced by (fictitious) magnetic dipoles. A similar direct visualization for D is more awkward, since it requires replacing the (true) electric dipoles by (fictitious) magnetic currents consisting of circulating poles. However, for the special case of free electric charge imbedded in a uniform medium, D is the electric field which would exist if the polarization (bound) charge were simply ignored. A familiar operational specification for the fields is given by Kelvin's arguments involving needle- and disc-shaped cavities cut in the medium. e The references cited in this appendix are included in the cumulative reference list on pp. 412-438. A.2 Microscopic relations 401 '.:/..I The microscopic field in a dielectric. To relate the macroscopic dielectric constant «= DhaE or susceptibility ¥/e=/)/eQ£ to molecular properties, one proceeds to calculate the effective field at the molecule Ecf/ by decomposing it into three component fields. (/) The externally applied field, identical with the macroscopic field E. (2) The average field E: of nearby molecules, treated as individuals. (3) The average field Ea oT more distant molecules, treated as a continuum. The distinction between nearby and distant molecules is made formally by imagining a spherical boundary of radius large compared to intermolecular distances but small on a macroscopic scale. By symmetry arguments, one can show that in most cases E^O. The polarization charge appearing at the surface of the spherical cavity yields, in a simple calculation, Ey —P;3f0. which is often called the Larentz polarization term (Lorentz, 1915). Thus Eef / — E-r-Ei + Ea — E + ; (A. 55) An equivalent argument is to say that the effective held experienced by a molecule is the uniform macroscopic field E diminished by the molecule's own field Emo!. The field of a dipole averaged over a sphere of volume l/« is Emu,= — P/3e0, from which (A.55) follows. IT the dipole moment p of a molecule is directly proportional to Ee// by the constant polarizability a, then the polarization P resulting from n molecules per unit volume is given by P = říp = rt«e0E whereupon na e0E \—\na £0E 1 — J«o 3(«-l) ——77-=na. k+2 (A. 56) (A.57) (A.58) (A. 59) Relation (A.59), which says that the quantity (k — 1 )/(«.+ 2) is directly proportional to molecule density, is known variously as the Clausius-Mossotti or the Lorentz-Lorenz formula, and has been verified experimentally for polariuablc gases over a wide range of pressures (Boudouris, 1963). 402 Review of electromagnetic wave propagation Appendix A A.3 Propagation in an anisotropic medium 403 A.2,2 The microscopic field in a plasma. To derive the electromagnetic properties of an ionized gas, we consider directly the dynamics of the electrons (and positive ions, also, if significant). At high frequencies, it is well known that a plasma behaves macroscopically like a dielectric. The question then arises as to what effective local electric field should appear in the force equation for the electron's motion. The answer is that the effective field in the case of unbound charge is simply the applied field E, and the Loreutz polarization term P/3e0 is omitted from (A.55). Thereupon (A.57) to (A.59) take the simpler form *P"e — na k — l + na k — l — na, (A. 60) (A.61) (A. 62) the latter relation being known as the Sellmeier formula. The difference between (A.59) and (A.62) is of practical interest only when k differs significantly from unity, but this is an important case in plasma physics. Extensive controversy has surrounded this apparent paradox whereby the formal treatment for free electrons (in a plasma) differs from that for bound electrons (in polarizable molecules), even though both media may be described macroscopically by a dielectric constant (Ratclilfe, 1959, Chapter 15). Indeed, the basic relation for high-frequency propagation in a magnetized plasma, commonly known as the Appleton-Hartree formula, was derived by Appleton and by Hartree on models which omitted and included the Lorentz term, respectively. Hartree's version does not permit the low-frequency "whistler" mode, for example, in contradiction with experiment (Storey, 1953; Jackson, 1954; and Buchsbaum and Brown, 1962). The distinction between the bound and free electron cases is most clear when the plasma is represented by the electron gas approximation, with the ions treated as a smeared-out, continuous background charge (a Lorentz plasma). Then it is clear from the i sot ropy of the electron's monopole field (in contrast to the polarized molecule's dipole field) that no modification to the effective local field arises. Further, if we look into the meaning of the polarization P in this case, we see that within a uniform medium there is no unique definition of the origin of P since a displacemenl of the electrons yields a new situation indistinguishable from the old. (Nevertheless, the time variation of P is well defined in terms of the electron velocity, cY;'ůt— —new, and it is this variation which appears in Maxwell's equations.) An additional distinction is that the molecular dipoles store potential energy in the stretched bonds, whereas the free electrons do not, The problem becomes more subtle when the discreteness of the positive ions is included. Even though the ions and electrons are energetically dissociated, it is not obvious that there are no statistical correlations between them which would give more direct meaning to the polarization. The dilemma can be posed more concretely by using the classical model of an electron elastically bound in a massive molecule with one or more natural (resonant) frequencies 0, and it is not clear at what point in the limiting process the Lorentz term is dropped (Unz, 1963), Theoretical explanations of the distinction between the two cases have been sought in terms of the amplitude of the electron motions and of the dynamics of electron-ion collisions (Booker and Berkner, 1938; Darwin, 1943; and Theimer and Taylor, 1961). An interesting application of these theories is to the case of propagation through shock waves in dense gases, in which both dielectric polarization of neutrals (and positive ions) and the dynamic polarization of free electrons contribute simultaneously to the observed dielectric constant (Alpher and White, 1959; Ascoli-Bartoli et al., 1960; see also Fig, 10.3). By way of summary, we note that the fundamental approach for treating wave propagation in an ionized gas is to consider all particle motions explicitly, expressing the results in terms of the conductivity or mobility coefficients. The formalism of dielectric polarization and susceptibility coefficient is often useful (and that of magnetization, occasionally), on a macroscopic level. However, these macroscopic concepts are treacherous when dealing with questions of local field and energy storage in the medium. A.3 Propagation in an anisotropic medium The electromagnetic properties of an anisotropic medium depend upon the Orientation of the fields with respect to the medium. A plasma in a niagnetostatic field is anisotropic, as are many crystals and ferritcs. ('urrents are not, in general, parallel to the applied electric field. Likewise, the polarization P and magnetization M are not necessarily parallel in !he applied E and B. Formally, this situation is handled by treating Hie constants u, *, km as tensors (see Appendix B). From (A.9), (A.10), and (A.13), J = o-E (A.63) D = c0k.E (A.64) tt = finKmV-trjH yxH = -jWqK-E (A.68) (A.69) (A.70) (A.71) Comparison of the vector properties of (A.69) and (A.71) shows that the vectors D = enk-E, H = B;Vo> and y form a mutually perpendicular right-handed set. However, E may have a longitudinal component. By crossing y into (A.70) and using (A.71), we obtain the wave equation for an anisotropic medium, yx(yxE)-?k.E = 0 (A.72) or, in terms of the complex vector refractive index (i= —jycfw from (A.39), jix(jixE) + k.E = 0. (A.73) Equation (A.73) is a set of three linear homogeneous equations for the field components (Ex, Ey, E3). A nonzero solution exists only if the determinant of the coefficients vanishes: *xx fry2 Kyx + P-yPx "zx + P-zfix Kxy ~r l^xPy » -13 v s 411 412 References Chapters 1-8 413 Delcroix. J. L. (1960). Introduction to the Theory of Ionized Gases, translated by M. Clark et al. Interscience, "New York. Longmire, C. L, (1963). Elementary Plasma Physics. Interscience, New York. Montgomery, D, C, andD, A. Tidman (1964). Plasma Kinetic Theory. McGraw-Hill, New York. Spitzer, L., Jr, (1962). Physics of Fully Ionized Gases, 2nd ed, Interscience, New York. Thompson, W. B. (1962). An Introduction to Plasma Physics. Pergamon, New York. Uman, M. A. (1964). Introduction to Plasma Physics. McGraw-Hill, New York. Controlled fusion Glasstonc, S., and R. H. Lovberg (I960). Controlled Thermonuclear Reactions. Van Nostrand, Princeton, N.J. Rose, D. J., and M, Clark, Jr. (1961). Plasmas and Controlled Fusion. M.I.T, Press and Wiley, New York. Compilations of tutorial research-level papers Dmmmond, J. E., ed. (1961). Plasma Physics. McGraw-Hill, New York. Huddiestone, R., and S. L. Leonard, cd. (1965). Plasma Diagnostic Techniques. Academic Press, New York. Kunkel, W., ed, (1965). Plasma Physics, in Theory and Application. McGraw-Hill. New York. Landshoff, R., ed. (1958). The Plasma in a Magnetic Field. Stanford University Press, Stanford, Cal. Milchner, M., ed. (1961). Radiation and Waves its Plasmas. Stanford University Press, Stanford, Cal. Microwave techniques and hardware Harvey, A. F. (1963). Microwave Engineering. Academic Press, New York. M.I.T, Radiation Laboratory (194S). Scries of 32 volumes, especially Vols. 8-12, McGraw-Hill, New York. Moreno, T. (1948), Microwave Transmission Design Data. Dover, New York. Saad, T., ed. (1964). The Microwave Engineer's Handbook. Horizon House, New York. CUMULATIVE REFERENCE LIST FOR PREFACE, CHAPTERS 1-8, AND APPENDICES The numbers appearing in brackets after an item indicate the pages on which that item is cited in the text. The corresponding reference lists for Chapters 9 and 10, dealing with somewhat different material, are given separately below. These listings serve as an author index, although cross-referencing of co-authors has been omitted for brevity. Aamodt, R. E,, and W. E. Drummond (1963). Nonlinear coupling of plasum oscillations to transverse waves. Plasma Phys. (.1. Nuclear Energy C) 6, 147, [286] Phys. Rev. 71, 777. Andrews, C. L. (1950). microwave region. Andrews, C. L. (1960). Albini, F. A., and R. G. Jahn (1961). Reflection and transmission of electromagnetic waves at electron density gradients. J. Appi. Phys. 32, 75. [135, 136] Allen, J. E., and W. R. Hindmarsh (1955). The brcmssl.rahlung radiation from ionized hydrogen. Rept. A.E.R.E. GP/R-J761, U.K. Atomic Energy Authority. [264] Alter, L. H. (1953). Astrophysics: Atmospheres of the Sun and Stars, Ronald Press, New York. Pages 251-256. [275] Allis, W. P. (1956). Motions of ions and electrons. Handbuch der Physik 21, 383 (Springer-Vertag, Berlin); preprinted as M.I.T. Tech Rept. 299 (1956). [60,67, 68, 71, 81, 83] Allis, W. P., S. J. Buchsbaum, and A. Bers (1963). Waves in Anisotropic Plasmas. M.I.T. Press, Cambridge, Mass. [x, 26, 45, 51, 56, 81, 95, 99, 105, 405] Alpher, R. A., and D. R. White (1959). Optical rcfractivity of high-temperature gases. Phys. Fluids 2, 153, 162. [403] Anderson, J. M., and L. Goldstein (1955). Interaction of electromagnetic waves of radio-frequency in isothermal plasmas: collision cross section of helium atoms and ions for electrons. Phys. Rev. 100, 1037. [83, 92] Andrews, C. L. (.1947). Diffraction pattern of a circular aperture al short distances. Li 44, 145] Diffraction pattern in a circular aperture measured in the J. Appl, Phys. 21, 761, [145] Optics of the Electromagnetic Spectrum. Prentice-Hall, Englewood ClitTs, N.J. Chapter II, [142] Appleton, E, V. (1932). Wireless studies of the ionosphere. J. Inst. Elec. Engrs, (London) 71, 642. [vii, 24] Appleton, E. V., and P. W. Chapman (1932), The collisional friction experienced by vibrating electrons in ionised air, Proc, Phys. Soc, (London) 44, 246. [64] Arlsimovich, L. A. (1958). Research into controlled thermonuclear reactions in the U.S.S.R, Proc. 2nd U.N. Conf. Peaceful Uses of Atomic Energy (Geneva) 31, 6. [302] Ascoli-Bartoli, U., A. De Angelis, and S. Martellucci (1960). Wavelength dependence of the refractive index of a plasma in the optical region. Ntiovo ctmento 18, 1116. [403] Astrom, E, (1950). On waves in an ionized gas. Arkiv Fysik 2, 443. [31, 51] Astrom, E. (1958). Waves in a hot ionized gas in a magnetic field. In B. Lehnert, ed., Electromagnetic Phenomena in Cosmical Physics. University Press, Cambridge. Page 81. [96] Auer, P. L., H. Hurwitz Jr., and R. D. Miller (1958). Collective oscillations in a cold plasma. Phys. Fluids I, 501. [53] Itachynski, M. P. (I960). Nature of electromagnetic waves in n on homogeneous, anisotropic plasmas. Rept. 7-801,7, RCA Victor Research Laboratories, Montreal. [133, 134] Itailey, V. A., and D. F. Martyn (1934). ionosphere. Phil. Mag. 18, 369. [92] Baird, J. R., and P. D. Coleman (1961). discharge. Proc. I.R.E. 49, 1890. [93] Baranger, M., and B. Mozer (1961). Light as a plasma probe. Phys. Rev. 123, 25. [92] Itiilloeletli, J. H., and W. D. He^shberger (1962), Resonances in the positive column of a low-pressure arc discharge. ./. Appl. Phys. 33, 2618. [241] The influence of electric waves on the Frequency conversion in a microwave 414 References Bazhanova, A. E., V. T. Karpukhin, A. N. Karkhov, and V. I. Pistunovich (1961). Cyclotron and thermal radiations of plasma in Ogra. Nuclear Fusion 1962 Supplement (Salzburg Conf.), Part 1, 227. [302] Beard, C. 1., T. H. Kays, and V. Twersky (1962). Mid-field forward scattering. J. Appl Phys. 33, 2851. [141] Beard, D, B, (1959). Cyclotron radiation from magnetically confined plasmas. Phys. Fluids 2, 379. [116, 278] Beard, D. B. (1961). Optical properties and emission of relativistic plasmas at cyclotron resonance. Plasma Phys. (./. Nuclear Energy C) 2, 94. [292] Beard, D. B., and J. C. Baker (1962). Synchrotron radiation losses from energetic plasmas. Phys. Fluids 5, 1113. [277] Bekefi, G. (1957). Studies in microwave optics. Eaton Electronics Research Lab. Tech. Rept. 38, McGill Univ., Montreal. Chapter 3. [150] Beketi, G., and S. C. Brown (1961a). Emission of radio-frequency waves from plasmas. Am. J. Phys. 29, 404. [242, 267, 270, 271, 278] Bekefi, G., and S. C. Brown (19616). Microwave measurements of the radiation temperature of plasmas. J. Appl. Phys. 32, 25. [262] Bekefi, G., J. L. Hirshtield, and S. C. Brown (1959). Incoherent microwave radiation from plasmas. Phys. Rev. 116, 1051. [271] Bekeii. G., J. L. Hirshlield, and S. C. Brown (I96I«). Kirchhoff's radiation law for plasmas with non-Maxwcllian distributions. Phys. Fluids 4, 173. [271, 286] Bekefi, G., J. L. Hirshtield, and S. C. Brown (1961/j). Cyclotron emission from plasmas with non-Maxwellian distributions. Phys. Rev. 122, 1037. [278] Bekefi, G., J. D. Coccoli, and E. B. Hooper (1962). Microwave emission and absorption at cyclotron harmonics of a warm plasma. Phys. Rev. Letters 9, 6. [302, 303] Bernstein, I. B. (1958). Waves in a plasma in a magnetic field. Phys. Rev. 109, ID. [104, 105, 106, 112, 1 13, 183, 188, 190] Bernstein, 1. B., and S. K. Trchan (I960). Plasma oscillations. Nuclear Fusion 1, 3. [67, 81, 100, 184] Bernstein, W„ P. F. Chen, M. A. Heald, and A. Z. Kranz (1958). "Runaway" electrons and cooperative phenomena in B-l Stellarator discharges. Phys. Fluids 1, 430. [2861 Bcvc, V., and T. E. Everhart (1962). Fast waves in plasma-filled waveguides. Electronics Research Lab. Rept. Scries 60, No. 362, University of California, Berkeley, Calif. Condensed version in J. Electron. Control 13, 185. [167, 181, 1891 Biondi, M, A., and S. C. Brown (1949). Measurements or anvbipolar diffusion in helium. Phys. Rev. 75, 1700. [vil, 163] Biondi, M. A. (1951). Measurement of the electron density in ionized gases by microwave techniques. Rev. Sit. Insir. 22, 500. [163] Bohm, D. (1951). Quantum Mechanics. Prentice-Hall, Engelwood Cliffs, N..I. Chapter 12. [119] Bohm, D., and E. P. Gross (1949). Theory of plasma oscillations. Phys. Rev. 75, 1851, 1864. [95, 103, 115] Boley, F. 1. (1958). Scattering of microwave radiation by a plasma column. Nature 182, 790. [241] Booker, H. G. (1935). The application of the magneto-ionic theory to the iono> sphere. Proc. Roy. Soc. {London) A1S0, 267. [38] Chapters 1-8 415 Booker, H. G., and L. V. Berkner (1938). An ionospheric investigation concerning the Lorentz polarization correction. Terrest. Magn. Atmos. Elect. 43,427. [403] Boudouris, G. (1963). On the index of refraction of air, the absorption and dispersion of centimeter waves by gases. /. Research Natl. Bur. Standards 67D, 631. [401] Bowles, K. L. (1961). Incoherent scattering by free electrons as a technique for studying the ionosphere and exosphcre: some observations and theoretical considerations. ./. Research Nail. Bur. Standards 65D, I. [94] Bowles, K. L., G. R. Ochs, and J. L. Green (1962). On the absolute intensity of incoherent scalier echoes from the ionosphere. J. Research Natl. Bur. Standards 66D, 395. [94] Boyd, F. E. (1950). Converging lens dielectric antennas. Rept. 3780, U.S. Naval Research Laboratory, Washington, D.C. [150] Boyd, G. D., R. W. Gould, and L. M. Field (1961). Interaction of a modulated electron beam with a plasma. Proc. l.R.E. 49, 1906. [286] Branch, G. M., and T. G. Mihran (1955). Plasma frequency reduction factors in electron beams. l.R.E. Trans. ED-2(2), 3. [130] Brandstatler, J.J. (1963). An Introduction to Waves, Rays and Radiation in Plasma Media. McGraw-Hill, New York. [48] Brown, S. C. (1956). Breakdown in gases: alternating and high-frequency fields. Handbuch der Physik 22, 531 (Springer-Verlag, Berlin); preprinted as M.l.T. Tech. Rept. 301 (1955). [91] Brown. S. C. (1958). Microwave studies of gas discharge plasmas. Proc. 2nd Intl. Conf. Peaceful Uses of Atomic Energy (Geneva) 32, 394. [160, 331] Brown, S. C. (1959). Basic Data of Plasma Physics, technology Press, M.I.T., and Wiley, New York. [59, 62, 64, 168] lirussard, P. L, and H. C. van de Hülst (1962). Approximation formulas for non- relativistic brcmsstrahlung and average Gaunl factors for a Maxwcllian electron gas. Revs. Modern Phys. 34, 507. [250, 251, 263] Uuchsbaum, S. J. (1960). Resonance in a plasma with two ion species. Phys. Fluids 3, 418. [56] Buchsbaum, S. J., and S. C. Brown (1957). Microwave measurements of high electron densities. Phys. Rev. 106, 196. [132, 159, 160] Uuchsbaum, S. J., and S. C. Brown (1962). The dielectric coefficient of a plasma. /. Electron. Control 13, 573. [402] Buchsbaum, S. J.. L. Mower, and S. C. Brown (I960). Interaction between cold plasmas and guided electromagnetic waves. Phys. Fluids 3, 806. [297] Buddcn, K. G. (1961). Radio Waves in the ionosphere. University Press, Cambridge. [45, 48] Uunkin, F. V. (1957). Thermal radiation from an anisotropic medium. Soviet Phys. J FTP 5, 665. [271] Burkhardt, G., G. Elwert, and A. Unsold (1948). Über das langwellige Ende des kontinuierlichen Röntgenspektrums und die Theorie der Radiofrequenzstrahlung der Milchstrasse und der Sonne. Z, Astrophys. 25, 310. [252] Burnett, D. (1931). The propagation of radio waves in an ionized atmosphere. Proc. Cambridge Phil. Soc. 27, 578. [69] Unser, R., and W. Buser (1962). Determination of plasma properties by free-space microwave techniques. ./. Appl. Phys. 33, 2275. [205] Untier, S. T.i and M, .1. Buckingham (1962). Energy loss of a fast ion in a plasma. Phys. Rev. 126, I. |284| 416 References Chandrasekhar, S, (1943). Stochastic problems in physics and astronomy. Revs. Modem Phys. 15, 1. [81] Chang, D. B. (1962a)', Bremsstrahlung from a plasma. Phys. Fluids 5, I 5.58. [252] Chang, D. B. (1962ft). Plasma correction to single-particle cyclotron radiation. Phys. Fluids 5. 1564. [2781 Chapman, S., and T. G. Cowling (1952). The Mathematical Theory of Won-Uniform Gases. 2nd ed. University Press, Cambridge. Chapter 7. [68] Chen, F. F. (1962). Radial electric field in a reflex discharge. Phys. Rev. Letters 8, 234. [213] Chen, K-M. (1962). Interaction of a high-intensity EM field with a low-density plasma. l.R.E. Trans. AP-10, 31. Nonlinear electrical conductivity of a fully ionized gas. l.R.E. Trans. AP-10, 43. [92] Christian. J. R,, and G. G on ban (1961). Experimental studies on a beam waveguide for millimeter waves. l.R.E. Trans. AP-9, 256. [150] Cillic, G. (1932). The hydrogen emission in gaseous nebulae. Mon. Not. Roy, Aslron. Soc. 92, 820. [251] Clemmow, P. C, and R. F. Mullaly (1955). The dependence of the refractive index in magneto-ionic theory on the direction of the wave normal. In The Physics of the Ionosphere. Physical Society, London. Page 340. [45] Coensgen, F. H., F. C. Ford, and R, E, Ellis (1958), Pyrotron plasma heating experiments. Proc. 2nd U.N. Conf. Peaceful Uses of Atomic Energy (Geneva) 32, 266. [293] Cohen, M. H. (1961). Radiation in a plasma: Cerenkov effect. Phys. Rev. 123, 711. [285] Cohen, R. S., L. Spitzer Jr., and P. M. Routly (1950). The electrical conductivity of an ionized gas. Phys. Rev. 80, 230. [80, 81, 83, 84] Cohn, M., F. E. Wentworth, and J. C. Wiltse (1963). High-sensitivity 100 to 300 Gc radiometers. Proc. l.E.E.E. 51, 1227. [299] Condon. E. V., and G. H. Shortley (1951). Theory of Atomic Spectra. University Press, Cambridge. Chapter 4. [259] Consult, T., R. Legardeur, and L. Slama (1961). Interaction d'ondes electromag- netiques á polarisation circulaire avec un plasma. Saclay Rept. 1995, Commissariat á 1'Energie Atomique, Paris. [25, 2231 Coor, T., S. Cunningham, R. Ellis, M. Hcald, and A. Kranz (1958), Experiments oil the ohmic heating and confinement of plasma in a Stellarator. Phys. Fluids 1, 411. [222] Coor, T. (1961). Plasma diffusion in Stellarators. Plasma Phys. (J. Nuclear Energy C) 2, 81. [286] Crawford, F. H. (1949). Jacobian methods in thermodynamics. Am. J. Phys. 17, I. [47] Crawford, F. W., G. S. Kino, and A, B. Cannara (1963). Dipole resonances of a plasma in a magnetic field. J. Appl. Phys. 34. 3168. [130, 241] Cullen, A. L. (I960). Propagation of microwaves through a magneto-plasma, and tl possible method for determining the electron velocity distribution. J. Research Natl. Bur. Standards 64D, 509. [114] Darwin, C. G. (1943). The refractive index of an ionized medium. Proc. Roy. Soc. (London) A182, 152. [403] Dattner, A. (1957). The plasma resonator. Ericsson Technics 13, 309. [240, 2411 Davies, L. W., and E. Cowchcr (1955), Microwave and meter wave radiation from the positive column of a gas discharge. Australian J. Phys. 8, 108. |2f)2] Chapters 1-8 417 Dawson, J. (1959). Nonlinear electron oscillations in a cold plasma. Phys. Rev. 113, 383. [4] Dawson, J. (1961). On Landau damping. Phys. Fluids 4, 869. [114] Dawson, J., and C. Oberman (1959). Oscillations of a finite cold plasma in a strong magnetic held. Phys. Fluids 2, 103. [130, 244] Dawson, J., and C. Oberman (1962). High-frequency conductivity and the emission and absorption coefficients of a fully ionized plasma. Phys. Fluids 5, 517. See also Oster (1964) and A. Ron and N. Tzoar, Phys. Rev. 131, 12 (1963). [86, 87, 89, 254] Dawson, J,, and C. Oberman (1963). Effect of ion correlations on high-frequency plasma conductivity. Phys. Fluids 6, 394. Sec also C. Oberman, A. Ron, and J. Dawson, Phys, Fluids 5, 1514 (1962). (77, 86] Debye, P., and E. Hiickel (1923). Zur Theorie der Elektrolytc. Physik Z. 24, 185. English translation in The Collected Papers of Peter J. IV. Debye. Interscience, New York, 1954. Page 217. [77] Delcroix, .1. L. (I960). Introduction to the Theory of Ionized Gases, translated by M. Clark et al. Interscience, New York. [66, 183] Dcllis, A, N. (1957). The measurement of electron temperatures by microwave methods. Rept. A.E.R.E. GP/R2265, U.K. Atomic Energy Authority. [270, 291] Dcllis, A. N., and J. M, Weaver (1964). Whistler-mode propagation in a laboratory plasma. Proc. Phys. Soc. (London) 83, 473. Sec also Nature 193, 1274 (1962). [226] Denisov, N. G. (1958). Resonance absorption of electromagnetic waves by an inhomogeneous plasma. Soviet Phys. JETP 7, 364. [126] Desloge, E. (1963). Fokker-Planck equation. Am. J. Phys. 31, 237. [81] Desloge, E. and S. W. Matthyssc (I960). Collision term in the Boilzmann transport equation. Am. J. Phys. 28, I, [81] De Witt, H, (1958). The free-free absorption coefficient in ionized gases. Lawrence Radiation Lab. Rept. UCRL-5377, Univ. of California, Livermore, Calif, [85, 252] di Francia, G. T, (1959). Introduction to the theory of synchrotron radiation. In Solar Radioast I onomy, Varenna Summer School (Zanichelli, Bologna), page 414. [2721 Dingle, R. B., D. Arndt, and S. K. Roy (1957). The integrals <£p(x) and ty.r) and their tabulation. Appl. Sei. Research 6a, 245. [76] Dirac, P. A. M. (1947). The Principles of Quantum Mechanics, 3rd ed. Clarendon Press, Oxford. Section 15. [277] Dnestrovskii, Yu. N,, and D. P. Kostomarov (1961). Dispersion equation for an ordinary wave moving in a plasma perpendicular to an external magnetic field. Soviet Phys.—JETP 13, 986. [Ill] Dreicer, H. (1959-1960). Electron and ion runaway in a fully ionized gas. Phys. Rev. 115, 238 and 117, 329. [229, 285] Dreicer, H. (1961). Incoherent scattering of microwaves by plasma vibrations. Nuclear Fusion 1962 Supplement (Salzburg Conf,), Part 3, 1141. [229] Drummond, J. E. (1958). Basic microwave properties of hot magnetoplasmas. Phys. Rev. 110, 293. See also Phys. Rev. 112, 1460 (1958). [96, 105, 111, 293] Drummond, J. E., R. A. Gerwin, and B. G. Springer (1961). The concept of conductivity. Plasma Phys. (J. Nuclear Energy C) 2, 98. [96, I 32] Drummond, W. E. (1962), Microwave scattering from unstable plasma waves, Phys. Fluids 5, 1133. [94] 418 References Chapters 1-8 419 Drummond, W. E., and D. Pines (1961). Non-linear stability or plasma oscillations. Nuclear Fusion 1962 Supplement (Salzburg Conf.), Part 3, 1049. [115, 186, 188, 234,236] Drummond, W. E., and M. N. Rosenbluth (1963), Cyclotron radiation from a hot plasma. Phys. Fluids 6, 276. [273] Easley, M, A,, and W. W, Mum ford (1951). Electron temperature vs. noise temperature in a low pressure Hg-A discharge. J. Appl. Phys, 22, 846. [290] Eidman. V. Ya. (1958). Investigation of the radiation of an electron moving in a magnetoactive medium. Soviel Phys.—JETP 1, 91. Note corrections in Soviet Phys.—JETP 9, 947 (1959). [285] Eidman, V. Ya, (1962). Radiation of a plasma wave by a charge moving in a magnetoactive plasma. Soviel Phys. JF.TP 14, 1401. [285] Eiconskii, V, M., P. S. Zyryanov, and V. P. Silin (1962). Collision integral for charged particles in a magnetic field. Soviel Phys.— JETP 15, 619. [89] Ellis, R. E., and N. L. Parker (1958). Electron energy distribution. Lawrence Radiation Lab. Rept. UCRL-5117, Univ. of California, Livermorc, Calif. [297] Elwert, G, (1939). Verschärfte Berechnung von Intensität und Polarisation im kontinuierlichen Röntgenspektrum. Ann. Physik 34, 178, [251] Emeleus, K.. G„ and D. W. MahalTey (1961). Plasma oscillations, Plasma Phys. (J. Nuclear Energy C) 2, I 17. [286] Epstein, M. (1960). Nonlinear behavior of the electrical conductivity of a slightly ionized gas. Phys. Fluids 3, 1016. [92] Epstein, M. (1962). Electromagnetic-wave propagation in a plasma with nonlinear electrical conductivity. Phys. Fluids 5, 492. [92] Farley, D. T., J. P. Dougherty, and D. W. Barron (1961). A theory of incoherent scattering of radio waves by a plasma. Proc. Roy. Soc. (London) A263, 238. [93] Famcll, G. W; (1958). Measured phase distribution in the image space of a microwave lens. Con. J. Phys. 36, 935. See also G. W. Famell, "Phase distribution in the image space of a microwave lens system," Eaton Electronics Research Lab. Tech. Rept. 39 (1958), McGill Univ., Montreal. [145, 150] Fejer. J, A, (I960). Radio-wave scattering by an ionized gas in thermal equilibrium. J. Geophys. Research 65, 2635. [93] Field, G. B. (1956). Radiation by plasma oscillations. Astrophys. J. 124, 555. [286] Field, G. B. (1959). The source of radiation from Jupiter at decimeter wavelengths. J. Geophys. Research 64, 1169. [286] Fields, FL. G. Bekcfi, and S. C. Brown (1963). Microwave emission from non-Maxwellian plasmas. Phys:, Reo. 129, 506. [271, 302] Forsyth, P. A., W. Petric, and B. W. Currie (1950). On the origin of ten centimeter radiation from the polar aurora. Con. J. Research A28, 324. [286] Francis, G. (I960). Ionization Phenomena in Gases. Academic Press, New York. [62,91] French, t. P., G. G. Cloutier, and M. P. Rachynski (1961). The absorptivity spectrum of a uniform anisotropic plasma slab. Can. J. Phys. 39, 1273. [ 1.1(1, 192, 270] Fried, B. D„ and S, D, Conle (1961 J, The Plasma Dispersion Function. Academic Press, New York. [105] Gabor, D„ E. A. Ash, and D. DracoLl (1955), Langmuir's paradox. Nature 176, 916. [290] Gallet, R., J, M. Richardson, B. Wieder, G. D, Ward, and G. N. Harding (1960). Microwave whistler mode propagation in a dense laboratory plasma. Phys. Rev. Letters 4, 347. [226] Gardner, A. )_., W, L. Barr, R. L. Kelly, and N. L. Oleson (1961). Diagnostic measurements on the P-4 steady-state plasma. Lawrence Radiation Lab. Rept. UCRL-6562, Univ. of California, Livcrmore, Calif. [301] Gaunt, J. A. (1930). Continuous absorption. Phi!. Trans. Roy. Soc. London A229, 163. [248,251] Gershman, B. N. (I960). Nonresonance absorption of electromagnetic waves in a magneloaclive plasma. Soviet Phys. J FTP 10,497. Gyromagnetic absorption of electromagnetic waves in a plasma. Soviet Phys.- -JETP 11, 657, [114] Ginzburg, V. L. (1944). On the absorption of radio waves and the number of collisions in the ionosphere. J. Phys. (U.S.S.R.) 8, 253. [83] Ginzburg, V. L. (1959). Nonlinear interaction of radio waves propagating in a plasma. Soviet Phys.—JETP 8, 1100. [93] Ginzburg, V, 1(1960). Certain theoretical aspects of radiation due to supcrluminal motion in a medium. Soviet Phys.— Vspekhi 2, 874. [280] Ginzburg, V. L., and A. V. Gurevich (I960). Nonlinear phenomena in a plasma located in an alternating electromagnetic field. Soviet Phys.—Uspekhi 3, 115, 175. [89] Glasstone, S., and R. H. Lovberg (1960). Controlled Thermonuclear Reactions. Van Nostrand, Princeton, N.J. [97, 273] Goldstein, L. (1958). Nonreciprocal electromagnetic wave propagation in ionized gaseous media. f.R.E. Trans. MTT-6, 19. [20] Golovin, I. N. (1959). Studies of trapping fast charged particles in a constant magnetic field, conducted in the U.S.S.R. according to the program of controlled thermonuclear reactions. Atomic Energy Institute Rept. 23, U.S.S.R. Academy of Sciences, Moscow. Reprinted in Proc. Inst. Fleet. Engrs. 106, Pi. A, Suppl. No. 2, p. 95 (1959). [302] Graben, H. W. (1963). Relativislic corrections lo microwave hUcrfcromctry in plasmas. Phys. Fluids 6, 1659. [115] Graf, K., and M. P. Bachynski (1961). Transmission and reflection of electromagnetic waves at a plasma boundary for arbitrary angles of incidence. Can. J. Phys. 39, 1544. See aiso Can. J. Phys. 36,456 (1958). [130] Greene, J. (1959). Bremsstrahlung from a Maxwel'.ian gas, Astrophys. J. 130, 693 (Fig. 3 is in error; corrected data appears in our Figs. 2.9 and 7.2). An additional appendix is included in U.S. Atomic Energy Comm. Rept. NYO-7905 "(1958). [86, 87, 89, 251, 254, 256, 263] Gross, E. P. (1951). Plasma oscillations in a static magnetic Meld. Phys. Rev. 82, 232. [104] Gurevich, A, V. (1956), On the etfect of radio waves on the properties of plasma (ionosphere). Soviet Phys.—JETP 3, 895. [76] Hain, K., and M. Tutter (1962). Durchgang von Mikroweiien durch ebene Plasma- schichten. Z. Naturforsch. 17a, 59, [136, 192] 11 all, L. S., and A. L. Gardner (3961). P-4, a steady-state plasma system. Lawrence Radiation Lab. Rept, UCRL-6561, Univ. of California, Livermore, Calif. [300] Harding, G., A. Dcllis, A. Gibson, B. Jones, D. Lees, R. McWhirler, S. Ramsden, and S. Ward (1958). Diagnostic techniques used in controlled thermonuclear research al Harwell. Proc. 2nd U.N. Conf. Peaceful Uses of Atomie Energy (Geneva) 32, 365. [2901 420 References Chapters 1-8 421 Harrison, E. R. (I960). Runaway and suprathermal particles. Plasma Phys. (J. Nuclear Energy C) 1, 105. [285] Hayakawa, S., N. Hokkyo, Y. Terashima, and T. Tsuneto (1958). Cyclotron radiation from a magnetized plasma. Proc. 2nd U.N. Conf. Peaceful Uses of Atomic Energy (Geneva) 32, 385. [293] Heald, M. A. (1956). Microwave generation in B-l Stellarator. U.S. Atomic Energy Commission Rept. TID-7520, Vol. 1, 202. [vil, 2S6, 300J Heald, M. A. (1958). Microwave measurements in controlled fusion research. I.R.F.. Nail. Conv. Record, Part 9, 14. Heald, M. A. (1959a). The application of microwave techniques to Stellarator research. U.S. Atomic Energy Commission Rept. MATT-17. [137, 148] Heald, M. A. (1959/j). Experimental measurements in controlled fusion research. l.R.E. Trans. NS-6(J), 33. Heald, M. A. (1959c). Engineering design of microwave electron density measuring systems. Project Mattcrhorn Tech, Memo 78, Princeton Univ., Princeton, N.J. [206] Heald, M. A. (I960). An attempt of observe whistler mode propagation in Stellar-ators. Project Mattcrhorn Tech. Memo. 109, Princeton Univ., Princeton, N.J. [226] Heitler, W. (1954). Quantum Theory of Radiation, 3rd ed. University Press, Oxford. [93, 251] Helliwell, R. A., J. H. Crary, J. H. Pope, and R. L. Smith (1956). The nose whistler— a new high-latitude phenomenon. J. Geophys. Research 61, 139. [225] Helliwell, R. A., and M. G. Morgan (1959). Atmospheric whistlers. Proc. l.R.E. 47, 200. [19, 49, 225] Herlofson, N. (1951). Plasma resonance in ionospheric irregularities. Arkh Fysik 3, 247. [240] Hershbergcr. W. D. (1960). Absorption and reflection spectrum of a plasma. J. Appl. Phys. 31,417. [241] Hester, R., and D. Reagan (I960). Hose instability beam experiment. Lawrence Radiation Lab. Rept. UCID-4192, Univ. of California, Livermore, Calif. [195] Hettner, G. (1958). Zur Thcorie der Bremsstrahlung in Plasmen hoher Temperature Z. Physik 150, 182. [256] Mines, C. O. (1951). Wave packets, the Poynting vector, and energy How. ,/. Geophys. Research 56, 63, 197, 207, 535. [45] Hirshfield, J. L., D. E. Baldwin, and S. C. Brown (1961). Cyclotron radiation from a hot plasma. Phys. Fluids 4, 198. [277] Hirshfield, J. L., and S. C. Brown (1958). Microwave method for measuring the probability of elastic collision or electrons in a gas. J. Appl. Phys. 29, 1749. [161] Hirshfield, J. L., and S. C. Brown (1961). Incoherent microwave radiation from a plasma in a magnetic field. Phys. Rev. 122, 719. [271, 274, 275, 280, 297, 298] Hu, M-K. (1961). Frcsnel region fields of circular aperture antennas. J. Research Natl. Bur. Standard!,651), 137. [146] Hwa, R. C. (1958). Effects of electron-electron interactions on cyclotron resonances in gaseous plasmas. Phys. Rev. 110, 307. [84, 274] lams, Hi (1950). A method of simulating propagation problems. Proc. l.R.E. 38, 543. [151] Ichimaru, S. (1962). Wave properties of a plasma with a doubly humped velocity distribution. Phys. Fluids 5, 1264. [286] Ichtchenko, G. (1962). Propagation d'une onde clcctromagnetiques parallclcment á on champ magnetique dans tin gas ionise. Thesis, Faculte des Sciences, Universitě de Paris. [226] Imrc, K. (1962). Oscillations in a relativistic plasma. Phys. Fluids 5, 459. [116] Jackson, J. D. (1962). Classical Electrodynamics. Wiley, New York. [233] Jackson, J. E. (1954). Measurements in the E-layer with the navy Viking rocket. J. Geophys. Research 59, 377. [402] Jahn, R. G. (1962). Microwave probing of ionized-gas flows. Phys. Fluids 5, 678. [150] Jahnke, E., and F. Emde (1945). Tables of Functions, 4th ed. Dover, New York. [72, 247] Janeel, R., and R. Kahan (1955). Examen critique de la théorie des plasmas basée sur le libré parcours moyen, á la lumiére de la méthode fondée sur la fonction de distribution solution de ['equation de Boltzmann. /. phys. radium 16, 824. [66] Jelley, J. V. (1958). Cerenkov Radiation. Pergamon Press, New York. [280] Johnson, P. S. (1962). Cerenkov radiation spectra for a cold magnctoactive plasma. Phys. Fluids 5, 118. [285] Johnson, R. C. (1959). Design of linear double tapers in rectangular waveguides. l.R.E. Trans. MTT-7, 374. [136] Johnston, T. W. (1962). Waves in warm quiescent plasmas. Can. J. Phys. 40, 1208. [105, 116] Joseph, J., and F. Rohrlieh (1958). Pair production and bremsstrahlung in the field of free and bound electrons. Revs. Modern Phys. 30, 354. [256] Karzas, W. J,, and R. Latter (1961). Electron radiative transitions in a Coulomb field. Astrophys. J., Supplement Series 6, 167. [251, 263] Katz, J. E. (1959). Instabilities excitation experiment diagnostics. Electronics Dept. Rept. LE-319, Lawrence Radiation Lab., Univ. of California, Livermore, Calif. [302] Kcitcl, G. H. (1956). On the dipole resonant mode of an ionized gas column. Australian J. Phys. 9, 144. [241] Kelly, D. C. (1960). Microwave conductivity of a plasma in a magnetic field. Phys. Rev. 119, 27. [84] Kelly, D. C, H. Margcnau, and S. C. Brown (1957). Cyclotron resonance: method for determining collision cross sections for low-energy electrons. P/iys. Rev. 108, 1367. [274] Kerker, M., and E. Matijevič (1961). Scattering of electromagnetic waves from concentric infinite cylinders. J. Opt. Soc. Am. 51, 506. [137] Kerns, D. M.. and E. S. Dayhoff (1961). Theory of diffraction in microwave interferometry. J. Research Natl. Bur. Standards 64B, I. [130] Kharchenko, I., Ya. Fainberg, R. Nikolaev, E. Kornilov, E. Lutsenko, and N. Pcdcnko (1960). Interaction of an electron beam with a plasma. Soviet Pbys.—JETP 11, 493. [286] Kihara, T., and O. Aono (1963). Unified theory of relaxations in plasmas: basic theorem. /. Phys. Soc. Japan 18. 837. [86] Kihara, T., O. Aono, and R. Sugihara (1961). Theory of Cerenkov and cyclotron radiations in plasma. Nuclear Fusion 1, 181. [284] Kihara, T., M. H. Taylor, and J. O. Hirschfelder (1960). Transport properties for gases assuming inverse power intermolecular potentials. Phys. Fluids3,715. [62] Kildal, A. (1961), "Energy absorption in a longitudinal wave in collisionless, magnetized plasma. Plasma Phys. (,/. Nuclear Energy C) 3, 256. [115] 422 References Chapters 1-8 423 King, R. W. P., and T. T. Wu (1959). The Scattering und Diffraction of Waves. Harvard Univ. Press, Cambridge, Mass. [137] Kino, G. S. (1960). Parametric amplifier theory for plasmas and electron beams. /. Appl. Phys. 31, 1449. [286] Kino, G. S., and M. A. Allen (1961). The effects of fluctuations on propagation through a plasma medium. Proc. 5th Intl. Conf. Ionization Phenomena in Gases (Munich), 602 (North Holland Publ. Co., Amsterdam, 1962). [94] Kittel, C. (1958). Elementary Statistical Physics. Wiley, New York. Section 44. [115] Klein. M., H, Grcybcr, .). King, and K. Brucckncr (1961), Interaction of a nonuniform plasma with microwave radiation. Planetary and Space Sei. 6, 105. [136] Knol, K. S. (1951). Determination of the electron temperature in gas discharges by noise measurements. Phillips Research Rept. 6, 288. [244, 266, 290] Koch, II. W., and J. W. Motz (1959). Bremsstrahltmg cross-section formulas and related data. Revs. Modern Phys. 31, 920. [251, 256] Kolomenskii, A. A. (1956). Radiation from a plasma electron in uniform motion in a magnetic held. Soviet Phys.—Doklady I, 133, 1281] Kompfner, R. (1952). Travelling-wave tubes. Repts. Progr. Phys. 15,275, [286] Kramers, H. A. (1923). On the theory of x-ray absorption and of the continuous x-ray spectrum. Phil. Mag. 46, 836, [248, 250] Kuipcr, G. P., ed. (1953), The Sun. Univ. of Chicago Press, Chicago. Chapter 7. [286, 300] Landau, L. D, (1946). On the vibrations of the electronic plasma. J. Phys, (U.S.S.R.) 10, 25. [115, 185] Landau, L. D., and E. Lifshitz (1962). Classical Theory of fields, 2nd ed., translated by VI. Hamcrmcsh. Addison-Wcslcy, Reading, Mass. [248, 275, 276] Landauer, G. (1961). Magnetfeldabhängige Mikrowellenstrahhmg aus einer He- ß-äsent ladung. Proc. 5th Intl. Conf. Ionization Phenomena in Gases (Munich), 389 (North-Holland Publ. Co., Amsterdam, 1962). [302] Landauer, G. (1962). Generation of harmonics of the electron-gyrofrequency in a Penning discharge. Plasma Phys. (J. Nuclear Energy C) 4, 395. [213, 302, 303] Lashinsky, H. (1961). Ccrenkov radiation at microwave frequencies. Advances Electronics and Electron Phys. 14, 265. [2811 Lashinsky, H. (1963). Analogous propagation modes in inhomogeneous plasma and tapered waveguides. Plasma Physics Lab. Tech. Memo 177, Princeton Univ., Princeton, N.J. [151] Lawson, J. D. (1954). On the relation between Ccrenkov radiation and brems- strahhmg. Phil. Mag. 45, 748. [284] Lenard. A. (1959). Adiabatic invariance to ail orders. Ann. Phys. 6, 261. [97] Leontovich. M. (1961). Generalization of (he Kramers-Kronig formulas to media with spatial dispersion. Soviet Phys.—JETP 13, 634. Sec also V. L, Ginzburg and N. N. Mciman, Soviet Phys.—JETP 19, 169 (1964). [115] Levin, M. L. (1957). Thermal radiation of good conductors. Soviet Phys.......J EVP 4,225. [271] Lewis, R, M., and J. B. Keller (1962), Conductivity lensor and dispersion equation for a plasma. Phys. Fluids 5, 1248. [105] Lin, S.-C, L. I... Rcsler, and A. Kantrowitz (1955). Electrical conductivity of highly ionized argon produced by shock waves. ./. Appl. Phys. 26, 95. [83] Linfool, E. H., and E. Wolf (1956). Phase distribution near focus in an aberralion- free diffraction image. Proc. Phys. Soc. (London) B69, 823. [145, 150] Lisitano, G. (1962). Mikrowellen Interferometer mit l^s Zeitablosung, Report IPP 2/15, Institut für Plasmaphysik, Garching b. München, Germany. [209] Lisitano, G. (1963), The application of the sinusoidal interference method in the design of a reflectometer and Polarimeter in the range of millimetric waves. Nachr. tech. Z-CJ, No. 3, 103, English version of Nachr. tech. Z. 15, 446 (1962). [210] Lisitano, G., and M, Tutter (1961). Mikrowellcnmessungen an einer hochfrequenzerregten Gasentladung. Z. Naturforsch. 16a, 692. T210, 223, 225] Long, M. W., and J, C. Buttcrworth (1963). New technique for microwave radt-ometry. IEEE Trans. MTT-1I, 389. [299] Lorentz, H. A. (1915). The Theory of Electrons, 2nd ed. Dover, New York, 1952. Sections 117-19. [401] Madelung, E. (1943). Die Mathematischen Hilfsmittel des Physikers, 3rd cd. Dover, New York. [170] Maeckcr, H., T. Peters, and IL Schenk (1955). Ionen und Atomquerschniltc im Plasma verschiedener Gase. Z. Physik 140, 119. [84] MahafTey, D. W. (1963). Microwave propagation through a plasma in a magnetic field. Phys. Rev. 129, 1481. [108, 223] Majumdar, S, K. (1961). Radiation by charged particles passing through an electron plasma in an external magnetic field. Proc. Phys. Soc. (London) 77, 1109. [286] Marcuvilz, N. (1958). General electronic waveguides. Proc. Symp. Electronic Waveguides, 63 (Polytechnic Inst, of Brooklyn). [171] Margenau, H. (1946). Conduction and dispersion of ionized gases at high frequencies. Phys. Rev. 69, "508. [oil, 69] Margenau, H. (1958). Conductivity of plasmas to microwaves. Phys. Rev. 109, 6. [69] Marshak, R. E. (1940). The internal temperature of white dwarf stars. Astropliys. J. 92, 321. [86, 252] Martyn, D. F, (1948). Solar radiation in the radio spectrum. Proc. Roy. Soc. (London) A193, 44. [86, 262, 271] Massey, H. S. W-, and E. H. S. Burhop (1952). Electronic and Ionic Impact Phenomena. Clarendon Press, Oxford. [61, 63] McKensie, J. F. (1963). Cerenkov radiation in a magneto-ionic medium, Phil. Trans. Roy. Soc. London A255, 585. See also H. S. Tuan and S. R. Seshadri, IEEE Trans. MTT-11, 462. [281] Milne, E. A. (1930). Thermodynamics of the stars. Handbuch der Astropliys. 3(1), 65 (Springer, Berlin). [257] Mitra, S. K.(I952). The Upper Atmosphere, 2nd ed., The Asialic Soc, Calcutta, [oil] Molmud, P. (1959). Langevin equation and the a,c. conductivity of non-Maxwellian plasmas. Phys. Rev. 114, 29. [73, 76] Montgomery, C. G,, ed. (1947). Technique of Microwave Measurements, M.l.T, Radialion Lab. Series Vol. 1 I. McGraw-Hill, New York, Chapter 15. [146, 148] Molley, R., and M. A. t-lcald (1959). Ose of multiple polarizations for electron density profile measurements in high-tcmpcraturc plasmas, Proc. Symp. Millimeter Waves, 141 (Polytechnic Inst, or Brooklyn). See also Project Matterhorn Tech. Memo 83 (1959), Princeton Univ. [125, 221) Motley, R„ C. D. Lusiig, and S. Sanders (1961). Synchrotron radiation from runaway electrons in the Stollarator. Plasma Phys, (J, Nuclear Energy C) 3, 17. [277, 292] 424 References Conductivity of a warm plasma. Phys. Rev. 116, 16. [105] (1949). A broadband microwave noise source. Bell System [266] Radiation from slow electrons. Phys. Rev. 42, 641, [256] Mot l, N. F., and H. S. W. Massey (1949). The Theory of Atomic Collisions, 2nd ed. Clarendon Press, Oxford. [61, 63] Molt-Sniith, H. M. (I960). Collision trajectories for inverse power intermolecular potentials. Phys. Fluids 3, 721. [62] Mower, L, (1956). Propagation of plane waves in an electrically anisotropic medium. Rcpt. MP1..-I, Sylvania Microwave Physics Laboratory, Mountain View, Calif. [35J Mower, L. (1959). Mumford, W. W Tech. J. 28, 608. Nedelsky, L. (1932). Neufeld, J. (1961). Space dispersive properties of plasma. Phys. Rev. 123, 1. [1001 Neufeld, J. (1963). Constitutive equations for a plasma-like medium. /. Appl. Phys. 34, 2549. [96] Nicodemus, F. E. (1963). Radiance. Am. J. Phys. 31, 368. [257] Nyquist, H. (1928). Thermal agitation of electric charge in conductors. Phys. Rev. 32, I 10. [244] Oberman, C. (1961). On the radiation from co-operative phenomena in plasmas. Plasma Phys. (J. Nuclear Energy C) 2, I 54. [286] Oberman, C, and F. Shurc (1963). High-frequency plasma conductivity in a magnetic field. Phys. Fluids 6, 834. See also C. Oberman and A. Ron, Phys. Rev. 130, 1291 (1963). [89, 278] Osborne, F. J. F. (1962). A multiple-probe microwave system for plasma studies. Can. J. Phys. 40, 1620. [210] Oskam, H. J. (1957). Microwave investigation of disintegrating gaseous discharge plasmas. Thesis, Univ. of Utrecht, Holland. [160] Oster, L. (I960). Effects of collisions on the cyclotron radiation from relativistic particles. Phys. Rev. 119, 1444. [2741 Oster, L. (1961c). Cyclotron radiation from relativistic particles with an arbitrary velocity distribution, Phys. Rem 111,Ml. [278] Osler, L. (I961A). Emission, absorption, and conductivity of a fully ionized gas at radio frequencies. Revs. Modern Phys. 33, 525. [86, 249] Oster, L, (1963«), Note on thermal radio radiation, Astrophys. J. 137, 332. [86, 87, 254] Oster, L. (1963^). Radiative transfer in dispersive media. Astrophys. J. 138, 761. See also R. P. Mercier, Proc. Phys. Soc. (London) 83, 811 (1964). [260] Oster, L. (1964). Bremsstrahlung cross sections in the neighborhood of the plasma frequency. Phys. Fluids 7, 263. Sec also R. P. Mercier, Proc. Phys. Soc. (London) 83,819 (1964). [252] Osterbcrg, H. (1958). Propagation of plane electromagnetic waves in inhomo- geneous media. J". Opt. Soc. Am. 48, 513. [133] Pakhomov, V. 1„ V. F. Aleksin. and K. N. Stcpanov (1962). The radiation from an electron moving in a spiral in a magnetoactive plasma. Soviet Phys.—Tech. Phys. 6, 856. [284] Panofsky, W. K. H., and M. Phillips (1962). Classical Electricity and Magnetism. 2nd ed. Addison-Wesley, Reading, Mass. Chapter 20. [272, 275] Papoular, R., and J. Wegrowe (1961). Utilisation de lenliltcs focalisanles pour le diagnostic micro-onde des plasmas. Proc. 5th Intl. Corn", Ionization Phenomena in Cases (Munich). 1456 (North-Holland Publ. Co., Amsterdam, 1962). [ 150] Chapters 1-8 425 Cyclotron radiation by ions in cold Dover, New York, 1959. [242, 270] Effect of collisions on the Landau Parzen, P., and L. Goldstein (1951). Current fluctuations in the direct current gas discharge plasma. Phys. Rev. 82, 724. [245] Pcrsson, K. B. (1957). Limitations of the microwave cavity method of measuring electron densities in a plasma. Phys. Rev. 106, 191. [162] Phelps, A. V. (1960). Propagation constants for electromagnetic waves in weakly ionized, dry air. J. Appl. Phys. 31, 1723. [73] Pines, D. (I960). Plasma oscillations of electron gases. Proc. Intl. Congress Many-Panicle Problems (Utrecht). [1791 Pistunovich, V. ]., and V. D. Shafranov (1961). plasma. Nuclear Fusion 1, 189. [302] Planck, M. (1914). Theory of Heat Radiation. Platzman, P. M., and S. J. Buchsbaum (1961) damping of plasma oscillations. Phys. Fluids 4, 1288. [109, 115] Platzman, P. M., and S. J. Buchsbaum (1962). Wave propagation along a magnetic field in a warm plasma. Phys. Rev. 128, 1004. [108, 114] Platzman, P. M., and S. J. Buchsbaum (1963). Transmission of electromagnetic waves through plasma slabs. Phys. Rev. 132, 2. [108, 114] Platzman, P. M., and H. T. Ozaki (1960). Scattering of electromagnetic waves from an infinitely long magnetized cylindrical plasma. /. Appl. Phys. 31, 1597. [130, 239, 241] Posener, D. W. (1959). The shape of spectral lines: tables of the Voigt profile. Australian .1. Phys. 12. 184. [275] Post, R. F. (1958). Summary of UCRL Pyrotron (mirror machine) program. Proc. 2nd U.N. Conf. Peaceful Uses of Atomic Energy (Geneva) 32, 245. [194, 293] Post, R. F. (1961). Impurity radiation losses from a high temperature plasma. Plasma Phys. (J. Nuclear Energy C) 3, 273. [256] Post, R. F., R. E. Ellis, F. C. Ford, and M. N. Rosenbluth (1960). Stable confinement of a high temperature plasma. Phys. Reu. Leiters 4, 166. [293] Pradhan, T. (1962). Causality and the dispersion formulas for waves in a plasma. Ann. Phys. 17, 418. See also B. Gourary, ./. Appl. Phys. 28, 283 (1957). [115] Primich, R. I., and R. A. Hayami (1963). Millimeter wavelength focussed probes and focussed resonant probes for use in studying ionized wakes behind hypersonic velocity projectiles. Proc. Millimeter and Submillimeter Conf. of the IEEE (Orlando, Fla.). See also General Motors Defense Research Lab. Report TR63-2I76 (ARPA Order No. 347-63), Santa Barbara, Calif., and IEEE Trans. 1V1TT-12, 33 (1964). [219] Ramachandran, G. N., and S. Ramaseshan (1961). Crystal optics. Handbuch der Physik 25/1, I (Springer-Verlag, Berlin). Sec also A. Sommerfeld, Optics, Academic Press, New York, 1954, Chapter 4; and M. Born and E. Wolf, Principles of Optics, Pergamon Press, New York, 1959, Chapter 14. [48] Ramo, S„ and J. R. Whinnery (1953). Fields and Waves in Modern Radio, 2nd ed. Wiley, New York. Chapters 1 and 7. [157] Rao, K. V. N., J. T. Verdeyen, and L. Goldstein (1961). Interaction of microwaves in gaseous plasmas immersed in magnetic fields. Proc. I.R.E. 49, 1877. [92] Rao, M. S. V. G., and H, G. Booker (1963). Guiding of electromagnetic waves along a magnetic field in a plasma. J. Geophys. Research 68, 387, [49] Ratclill'e. .1. A. (195^. The Magneto-Ionic Theory and Its Applications to the Ionosphere. University Press, Cambridge. [24,45,93,127,402] 426 References Rcdhcffcr, R. M. (1949). Microwave antennas and dielectric surfaces. J. Appl. Phys. Iii, 397. [130, 1501 Renan, J., J. Camnitz, and \V. Flood (1961). The spectrum and total intensity of electromagnetic waves scattered from an ionized gas in thermal equilibrium in the presence of a static quasi-uniform magnetic field. J. Geophys. Research 66, 2703. [93] Richtmycr, F. K., E. H. Kennard, and T. Lauritsen (1955). Introduction to Modem Physics, 5th ed. McGraw-Hill, New York. Chapter 4. [242J Romell, D. (1951). Radio reflexions from a column of ionized gas. Nature 167, 243. [241] Rose, D. J., and S. C. Brown (1952). Measurement of discharge admittance and electron density. J. Appl. Phys. 23, 719, 1028. [158] Rose, D. J., and M. Clark, Jr. (1961). Plasmas and Controlled Fusion. Wiley, New York. Chapter 3. [64] Rosen, B. B. (1959). The "inverted universe" plasma analog for microwaves. Project Mattcrliorn Tech. Memo 85, Princeton Univ., Princeton, N.J. [151, 152, 154] Rosen, P. (1960). Scattering of electromagnetic waves by longitudinal plasma waves. Phys. Fluids 3, 416. [92] Rosen, P. (1961). Generation of the third harmonic by an electromagnetic signal in a plasma. Phys. Fluids 4, 341. 192] Rosenbluth, M. N., and C. L. Longmire (1957). Stability of plasmas confined by magnetic fields. Ann. Phys. 1, 120. [661 Rosenbluth. M. N., and N. Rostoker (1962). Scattering of electromagnetic waves by a non-equilibrium plasma. Phys. Fluids 5, 776. [234] Rosenfeld, L. (1951). Theory of Electrons. Intcrscience, New York. [400] Rukhadzc, A. A. (1962). Electromagnetic waves in interpenetrating plasmas. Soviet Phys.—Tech. Phys. 6, 900. [286] Rukhadze, A. A., and V. P. Silin (1962). Line shape of cyclotron resonance absorption in a plasma. Soviet Phys. Tech. Phys. 7, 307. [275] Rytov, S. M. (1953). Theory of Electric Fluctuations and Thermal Radial ion. Translation TR-59-162. Air Force Cambridge Research Center, Bedford, Mass., 1959. [261, 271, 287] Sagdeyev, R. S., and V. D. Shafranov (1958). Absorption of high-frequency electromagnetic energy in a high-temperature plasma. Proc. 2nd U.N. Conf. Peacefnl Uses of Atomic Energy (Geneva) 31, 118. [114] Salpeter, E. E, (1963). Density fluctuations in a nonequilibrium plasma. ./. Geophys. Research 68. 1321. [941 Sampson, D. II. (1959). The opacity at high temperatures due to Compton scattering. Astrophys. J. 129, 734. [93] Sampson, D. H., and .1. Enoch (1963). Electron distribution function and electrical conductivity of a slightly ionized gas. Phys. Fluids 6, 28. [69] Sautcr, E. (1933). Zur unrelativistischen Theorie des kontinuierlichen Röntgenspektrums. Ann. Physik 18, 486. See also Ann. Physik 20, 404(1934). [252| Scarf, F. L. (1962). Landau damping and the attenuation of whistlers. Phys, Fluids 5, 6. [114,226] Schclkunoff, S. A., and H. T. Friis (1952). Antennas: Theory and Practice. Wiley, New York. [143, 144, 243] Scheuer, P. A. G. (1960). The absorption coefficient of a plasma at radio frequencies, Mon. Not. Roy. Astron. Soe. 120, 231. [87, 249] Chapters 1-8 427 Schiff, L. J. (1955). Quantum Mechanics, 2nd ed. McGraw-Hill, New York. Section 28. [125] Schlüter, H. (1961). Untersuchungen an Balmer-Spektrcn bei einer Hochfrequenzentladung. Z. Naturforsch. 16a, 972. [225] Schott, G. A. (1912). Electromagnetic Radiation. University Press, Cambridge. Page 109. [275] Schwinger, J. (1949). On the classical radiation of accelerated electrons. Phys. Rev. 75, 1212. [272, 275] Sears, F. W. (1953). Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics, 2nd ed. Addison-Wesley, Reading, Mass. [47, 74] Sen, H. K., and A. A. Wyller (1960). On the generalization of the Appleton-Hartree magneto-ionic formulas. /. Geophys. Research 65, 3931. [76] Sexton, M. C, J. 1. Lcniion and M. J. Mulcahy (1959), Mircowave method or investigating the afterglows of pulsed gaseous discharges. Brit. J. Appl. Phys. 10,356. [163] Shapiro, H. (1957). Electromagnetic scattering properties of a resonant plasma. Ph.D. Thesis, California Inst, of Technology, Pasadena, Calif. [241] Sherman, J. W. (1962). Properties of focused apertures in the Ercsncl region. l.R.E. Trans. AP-10, 399. [147] Shkarofsky, 1. P. (1961). Values of the transport coefficients in a plasma for any degree of ionization based on a Maxwcllian distribution. Can. J. Phys. 39, 1619. [75, 76, 83, 84] Shkarofsky, I. P., I. B. Bernstein, and B. B. Robinson (1963). Condensed presentation of transport coefficients in a fully ionized plasma. Phys. Fluids 6, 40. [75] Shklovsky, I. S. (1960). Cosmic Radio Waves, translated by Rodman and Varšavsky. Harvard Univ. Press, Cambridge, Mass. [273] Shmoys, J. (1961). Proposed diagnostic method for cylindrical plasmas. J. Appl. Phys. 32, 689. [137] Silin, V. P. (1960). Kinetic equation for rapidly varying processes. Soviet Phys.— JETP It, 1277. [86] Silin, V. P. (1960-1962). Electromagnetic properties of a relativistic plasma. Soviet Phys.—JETP II, 1136 (I960); 13, 430 (1961); 14, 115 (1962). [115] Silin, V. P. (1962). High-frequency dielectricconstantofa plasma. Soviet Phys.— JETP 14, 617. [89] Silver, S. (1949). Micro wave Antenna Theory and Design, M.l.T. Radiation Lab. Series. Vol. 12. McGraw-Hill, New York. [143] Simon, A., and M. N. Rosenbluth (1963). Single particle cyclotron radiation near walls and sheaths. Phys. Fluids 6, 1566. [280,303] Sitenko, A. G., and K. N. Štěpánov (1957). On the oscillations of an electron plasma in a magnetic field. Soviet Phys.—JETP 4, 512. [105, 107] Slater, J. C. (1946). Microwave electronics. Revs. ModernPhys. 18,441. [156, 157] Slater, J. C, and N. H. Frank (1947). Electromagnctism. McGraw-Hill, New York. Chapter IX. [403] Smerd, S. F., and K. C. Westfold (1949). The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere. Phil. Mag. 40, 831. [259] Smullin, L. D., and P. Chorney (1958). Properties of ion filled waveguides. Proc. l.R.E. 46, 360. [181] Smyth, W. R. (1950). '"Static and Dynamic Electricity. McGraw-Hill, New York. Chapter 14. [239, 2411 428 References Chapters 1-8 429 Snyder, W., and R. A. Hclliwell (1952). Universal wave-polarization chart for the magneto-ionic Iheory, J. Geophys. Research 57, 73. [25] Sodha, M. S., and C. J, Paiumbo (1963). Nonlinear propagation of electromagnetic waves in magnetoplasmas. Can. J. Phys. 41, 2155. Sec also Can. J. Phys. 42, 349 (1964). [92] Sommerfeld, A. (1951). Atombau und Spektrallinien, Vol. It (Wellenmechanik). Frederick Ungar, New York. Chapter 7. [251] Southworlh, G. C. (1959). Using the Smith diagram. Microwave J. 2(1), 25 and 2(2), 24. [158] Spitzer, L., Jr. (1940). The stability of isolated clusters. Man. Not. Roy. A stran. Soc. 100, 396. [92] Spitzer, L., Jr. (1962). Physics of Fully Ionized Gases, 2nd ed. Interscience, New York. [56, 67, 78, 80, SI, 84, 98, 99] Spitzer, L., Jr., and R. Härm (1953). Transport phenomena in a completely ionized gas. Phys. Rev. 89, 977. [84] Stepanov, K. N., and A. B. Kitsenko (1961). Excitation of electromagnetic waves in a magnctoactive plasma by a beam of charged particles. Soviet Phys.—Tech. Phys. 6, 120. [285, 301] Stickforth, J. (1961). Zur Theorie der Bremsstrahlung in Plasmen hoher Temperatur. Z. Physik 164, 1. [256] Stix, T. H. (1958). Generation and Ihcrmalization of plasma waves. Phys, Fluids 1, 308. [127] Stix, T. H. (I960). Absorption of plasma waves. Phys. Fluids 3, 19. [126, 127] Stix, T. H. (1962). The Theory of Plasma Waves. McGraw-Hiil, New York. [45, 47, 48, 51, 56, 105, t08, 127, 293] Storey, L. R. O. (1953), An investigation of whistling atmospherics. Phil. Trans. Roy. Soc. London A246, 113. [225, 402] Stratum, J. A. (1941). Electromagnetic Theory. McGraw-Hill, New York. Sections 9.10-9.12. [129] Sturrock, P. A. (1957). Non-linear effects in electron plasmas. Proc. Roy. Soc, (London) A242, 277. See also Plasma Phys, (J. Nuclear Energy C) 2, 158 (1961). [188, 189] Sumi, M. (1959). Theory of spatially growing plasma waves. J. Phys. Soc. Japan 14, 653. [285] Sweet, P. A. (1959). Coulomb scattering in a magnetic held. Phil. Mag. 4, 1155. [88] Symon, K. R. (I960). Mechanics, 2nd cd, Addison-Wesley, Reading, Mass. [60, 62] Tanaka, S., K. Mitani and H. Kubo (1963). Microwave radiation from a plasma in a magnetic field. Institute of Plasma Physics Rept. 1PPJ-13, Nagoya University, Nagoya, Japan. [304] Targ, R., and L. P. Levine (1961). Backward-wave microwave oscillations in u system composed of an electron beam and a hydrogen gas plasma. J. Appl. Phys. 32, 731. [286] Taylor, L. S. (1961). Electromagnetic propagation in an exponential ionization ' density. FILE. Trans. AP-9, 483. [136] Theimer, O. (1963). Collective aspects of bremsstrahlung emission and electricitl conductivity in a plasma. Ann. Phys. 22, 102. [85, 253, 262] Theimer, O., and L. S. Taylor (1961). On the index of refraction in the ionosphere. J. Geophys. Research 66, 3157. [403] Thoncmann, P. C. et al. (1958). Controlled release of thermonuclear energy. Nature 181, 217. [226] Tidman, D. A,, and J. M. Boyd (1962). Radiation by plasma oscillations incident on a density discontinuity. Phys. Fluids 5, 213. [286] Tonks, L., and I. Langmuir (1929). Oscillations in ionized gases, Phys. Rev. 33, 195, 990. [vii, 2] Trivelpiece, A. W. (1958). Slow wave propagation in plasma waveguides. Tech. Rept. 7, California Inst, of Technology, Pasadena, Calif. [173, 175, 177, 178, 180] Trubnikov, B. A. (1958). Plasma radiation in a magnetic field. Soviet Phys.— Doklady 3, 136. [275] Trubnikov, B. A,, and V. S. Kudryavtsev (1958). Plasma radiation in a magnetic field. Proc. 2nd U.N. Conf Peaceful Uses of Atomic Energy (Geneva) 31, 93. [272] Trubnikov, B, A., and V. B. Yakubov (1963). Cyclotron radiation of electrons having a two-dimensional Maxwcltian distribution. Plasma Phys. (J. Nuclear Energy C) S, 7. [277] Turner, C. H. M. (1954). Birefringence in crystals and in the ionosphere. Can. J. Phys. 32, 16. [31, 33] Unz, H. (1963). The magneto-ionic theory for bound electrons. /. Atmosph. Terr. Phys. 25, 281. [403] Vachaspati (1962). Harmonics in the scattering of light by free electrons. Phys. Rev. 128, 664. [93] van der Pol, B. (1920). De invlocd van een geionisecrd gas op bet voortschrijden van electromagnetische golven. Thesis, Univ. of Utrecht. Reprinted in van der Pol, Selected Scientific Papers, Vol. I (North-Hoi land Publ. Co., Amsterdam, 1960). [vi] Van Vleck, J. H, (1932). The Theory of Electric and Magnetic Susceptibilities. University Press, Oxford. [400] Vlasov, A. A, (1938). Theory of vibrational properties of an electron gas and its applications. /. Exptl. Theoret. Phys. (U.S.S.R.) 8, 291. See also J. Phys. (U.S.S.R.) 9, 25, 130 (1945). [67] Vogt, E,, and G. H. YVannier (1954). Scattering of ions by polarization forces. Phys. Rev. 95, 1190. [63] von Gierke, G., L. Lisitano, G. Müller, H. Schlüter, M. Tutter, and H. Wulff (1961). Vergleich speklrosckopischc Untersuchen mit Mikrowellen- und Sonden, Messungen. Proc. 5th Intl. Conf. Ionization Phenomena in Gases (Munich)-380 (North-Holland Publ, Co., Amsterdam, 1962). [225, 229] Warder, R., M. Brodwin, and A. B. Cambel (1962). Sources of error in the microwave diagnostics of plasmas. J. Appl. Phys. 33, 2868, [151] Watkins, D. A. (1958). Topics in Electromagnetic Theory. Wiley, New York, [167] Wcitzner, H. (1963). Plasma oscillations and Landau damping. Phys. Fluids 6, 1123. [187] Westfold, K. C. (1950). Refractive index and classical radiative processes in an ionized gas. Phil. Mag. 41, 509. [248, 262] Wetzel, L. (1961). Wave interaction in plasma inhomogeneities. /. Appl. Phys. 32, 327. [93] Wharton, C. B. (1957). Microwave diagnostics for controlled fusion research. Lawrence Radiation Lab. Rept. UCRL4836(Rev.), Univ. of California, Livermore, I 'tilif. 430 References Wharton, C. B. (1959). Microwave radiation measurements of very hot plasmas. Proc. 4th Intl. Conf. Ionization Phenomena in Gases (Uppsala), 737 (North-Holland Publ. Co., Amsterdam, 1960). [213, 226, 292] Wharton, C. B. (1961). Microwave diagnostics for controlled fusion research. In J. E, Drummond, ed., Plasma Physics. McGraw-Hill, New York. Chapter 12. [290, 292] Wharton. C. B., and A. L. Gardner (1959). Microwave circuits and horns for plasma measurements. U.S. Patent No. 2,971,153. [200, 206] Wharton, C. B,, R. F. Post, and T. Prosser (1955). Microwave diagnoslics in arc research. Lawrence Radiation Lab, Rept. UCRL-4477, Univ, of California, Livermore, Calif, [oii, 192] Wharton, C. B., and D. M. Slager (I960). Microwave determination of plasma density profiles. J. Appl. Phys. 31, 428. [125,212,214] Whitehead, J. D. (1952). The quasi-transverse (QT) approximation to Appleton's magneto-ionic equation. ./. Almas. Terrest. Phys. 2, 361. [38] Whitmer, R. F., and E. B. Barrett (1961, 1962). Nonlinear interaction of an electromagnetic wave with a plasma layer in the presence of a static magnetic field. Phys. Rev. 121. 661 (1961); 125, 1478 (1962). [93] Willctt, J. E. (1962). Effects of electron random motion on microwave propagation through a plasma parallel to a magnetic field. /. Appl. Phys. 33, 898. [108, 109, 1141 Woolley, R. R,, and D. W. N. Stibbs (1953). the Outer Layers of a Star. Clarendon Press, Oxford. Chapter II. [259] Wort, D. .1. H. (1962). The emission of microwave noise by plasma. Plasma Phys. (J. Nuclear Energy C) 4, 353. [136, 266] Wort, D. J. H. (1963). Refraction of microwaves by a plasma cylinder. Rept. CLM-R27, U.K. Atomic Energy Authority. [137] Wort, D. J. H. (1964). The microwave emissivity of turbulent plasma. Plasma Phys. (J. Nuclear Energy C) 6, 237. [271, 288, 298] Wyld, H. W., Jr. (I960). Radiation by plasma oscillations in a bounded plasma in a magnetic field. Phys. Fluids 3, 408. [286] Yamada, K. (1962). General expressions of absorption coefficient and radiation intensity in plasma and KirchhofFs law. Progr. Theoret. Phys. (Kyoto) 28, 599. [262] Yoshikawa, S, (1962). Electrical conductivity of a turbulent plasma. Phys. Fluids 5, 1272. [94] REFERENCES FOR CHAPTER 9 Allen, P. J., and R. D. Tompkins (1959). An instantaneous microwave Polarimeter. Proc. I.R.E. 47. 1231. [324] Ayres, W. P., P. H. Vartanian, and J. L. Mclchor (1957). Frequency doubling In ferritcs. J. Appl. Phys. 27, 188. [321,345] Barnett. E. F. (1955). A new precision X-band phase-shifter. Trans. 4th Conf. on High Frequency Measurements, IRE, AIEE, NBS and URS1 (Washington, D.C.), 150. [315] Bekcfi, G., J. C. [ngraham, and .1. J. McCarthy (1962). A transient micro wo V( radiation pyrometer. Research Lah. of Electronics Quarterly Report (March), M.T.T., Cambridge, Mass. [355] Chapter 9 431 Benderly, A. A., and T. J. Kilduff (1962). A teflon microwave window. Microwave J. 4(3), 101. [357] Blau, R., and M. A. Heald (1959). Mathieu function roots for TE„ modes in elliptic waveguide. Project Matterhorn Tech. Memo 79, Princeton Univ., Princeton, N.J. [320] Bloom, S., and K. K. N. Chang (1957). Theory of parametric amplification using non-linear reactances. R.C.A. Rev. 18, 578. [345, 351] Booker, H. G. (1946). Slot aerials and their relation to complementary wire aerials. /. Inst. Elec. Engrs. (London) 93, Part II1A, 620. [338] Bossard, B. B., E. Frost, and W. Fish be in (I960). X-band super-regenerative parametric mixer. Proc. I.R.E. 48, 1329. [352] Brown, J. (1953). Artificial dielectrics having refractive indices less than unity. Proc. Inst. Elec. Engrs. (London) 100, Part IV, 51. [331] Brown, J., and J. O. Spector (1957). The radiating properties of end-fire aerials. Proc. Inst. Elec. Engrs. (London) 104B, 27. [335] Buser, R., and W. Buser (1962). Determination of plasma properties by free-space microwave techniques. J. Appl. Phys. 33, 2275. [205, 342, 343] Byrne, J. F.. and C. F. Cook (1963). Microwave type bolometer for submillimeter wave measurements. IEEE Trans. MTT-11, 379. [346] Cacheris, J. (1954). Microwave single-sideband modulator using ferrites. Proc. I.R.E. 42, 1242. [315] Chandler, C. H. (1949). An investigation of dielectric rod as waveguide. J. Appl. Phys. 20, 1188. [311] Cohn, S. B. (1957). Direct-coupled resonator filters. Proc. I.R.E. 43, 187. [318, 319] Coleman, P. D., and R. C. Becker (1959). Present state of the millimeter wave generation and technique art. I.R.E. Trans. MTT-7, 42. [345] Cottony, H. V., and A. C, Wilson (1958). Gains of finite-size corner reflector antennas. I.R.E. Trans. AP-6, 366. [336] Crompton, J. W. (1954). On the optimum illumination taper for the objective of a microwave aerial. Proc. Inst. Elec. Engrs. (London) 101, Part III. 371. [338] Diament, P., S. P. Schlesinger, and A. Vigants (1961). A dielectric surface wave structure: the V-line. I.R.E. Trans. MTT-9, 322. [310] Dicke, R. H. (1946). The measurement of thermal radiation at microwave frequencies. Reo. Sei. Instr. 17, 268. [354] Dicke, R. IL, R. Beringer, R. L. Kyhl, and A. B. Vane (1946). Atmospheric absorption measurements with a microwave radiometer. Phys. Rev. 70, 340. [312, 313] Dushman, S. (1962). Scientific Foundations of Vacuum Technique, 2nd ed. Wiley, New York. [356] Elsasser, W. M. (1949). Attenuation in a dielectric circular rod. J. Appl. Phys. 20, 1193. [335] Esaki, L. (1958). New phenomenon in narrow Ge p-n junctions. Phys. Rev. 109, 603. [352] J-iskc, M. D. (1946). Resonant windows for vacuum seals in rectangular waveguides. Rev. Sei. Instr. 17, 478. [360] Fox, A. G. (1947). An adjustable waveguide phase changer. Proc. I.R.E. 35, 1489. [3151 Fox, A. Si, S. Ii. Miller, and M. T. Weiss (1954). Behavior and applications of ferrites in the microwave region. Bell System Tech. J. 34, 5. [315] 432 References Gardner, A. L. (1962). Diagnostic measurements of a highly ionized, steady-state plasma. In Engineering Aspects of Magnetohydrodynamics, C. Mannal and N. Mather ed., Columbia University Press, p. 438 [333, 334, 386] Gordon, J. P., H. J. Zeiger, and C. H. Townes (1955). The maser—new type of microwave amplifier, frequency standard and spectrometer. Phys. Rev. 99, 1264. [352] Goubau, G., and F. Schwering (1961). On the guided propagation of electromagnetic wave beams. l.R.E. Trans. AP-9, 248. [311, 312] Guillemin, E. A. (1948). Communication Networks. Wiley, New York. [318] Guthrie, A., and R. K. Wakerling (1949). Vacuum Equipment and Techniques. McGraw-Hill, New York. [360] Hannan, P. W. (1961). Microwave antennas derived from the cassegrain telescope. l.R.E. Trans. AP-9, 140. [338] Harris, D. 13. (1960). Microwave radiomelry, Microwave J. 3(4), 41 and 3(5), 47. [353] Harris, E. F. (1953). An experimental investigation on the corner reflector antenna, Proc. f.R.E. 41, 645. [336] Harvey, A. F. (1963). Microwave Engineering. Academic Press, New York. [349, 361] Hawkins, P. O., H. J. Curnow, and R. Redstone (1958). The coaxial-line diode: a rectifier of microwaves. Proc. Inst. Eke. Engrs. (London) 105B, Sup. No. 12, 886. [353] Holonyak, N., and l. A. Lesk (I960). Gallium arsenide tunnel diodes. Proc. l.R.E. 48, 1405. [352] Jasik, Henry, ed. (1961). Antenna Engineering Handbook. McGraw-Hill, New York. [329] Jcnks, F. A. (1947). Simplified microwave A.F.C. Electronics 20(77), 120, and 10(12), 132. [351] Johnson, C. M. (1954). Superheterodyne receiver for the 100 to 150 kMc region. l.R.E. Trans. MTT-2(3), 27. [351] Johnson, C. M., D. M. Slager, and D. D. King (1954). Millimeter waves from harmonic generators. Rev. Sei. Instr. 25, 213. [345] Johnson, J. B. (1928), Thermal agitation of electricity in conductors. Phys. Rev. 32, 97. [346] Concerning hybrids. Microwave J. 4(10), 98. [315] The radiation characteristics of conical horn antennas. Proc. [328, 329] Microwave field plotter. Electronics Dept. Rept. LER 579, Lawrence Radiation Lab., Univ. of California, Livermore, Calif. [342] Kohl, W. H. (I960). Materials and Techniques for Electron Tubes. Reinhold, New York. [357] Kunz, K. S. (1954). Propagation of microwaves between parallel doubly curved conducting surfaces. ./. Appl. Phys. 25, 642. [331] Lanciani, D. A. (1954). HO] mode circular waveguide components. l.R.E. Trans. MTT-2(3), 45. [310] Lawson, J. D. (1948). Some methods of determining the power gain of microwave aerials. J. Inst. Elec. Engrs. (London) 95, Part 111, 205. [341] Lebenbaum, M. (1956). Measurement of noise figure. Chapter XI in Handbook of Electronic Measurements. Polytechnic Institute of Brooklyn Press, Brooklyn. N.Y. [355] Jones, C. W. (I96J). King, A. P. (1950). l.R.E. 38, 249. Klapper, H. (1962). Chapter 9 433 McGraw-Hill, New Dover, New York. Lending, R. D. (1955). New criteria for microwave component surfaces. Proc. Nat. Electronics Conf II, 391. [306] Lewin, L. (1959). A note on quasi optical methods at millimeter wavelengths. Proc. of Symp. on Millimeter Waves, Polytechnic Institute of Brooklyn Press, Brooklyn, N.Y. [307] de Loach, B. C. (1960), 17 and 30 kMc parametric amplifiers. Proc. l.R.E. 48, 1323. [352] Loth, P. A. (1956). Recent advances in waveguide hybrid junctions. l.R.E. Trans. MTT-4, 268. [315] Malmberg, J., N. W. Carlson, C. B. Wharton, and W. II. Drummond (1963). A collisionless plasma for wave propagation studies. Proc. 6lh Intl. Conf. on Ionization Phenomena in Gases (Paris) 4, 229. [361] Meyer, M, A., and I-I. B. Goldberg (1955). Applications of the turnstile junction. l.R.E. Trans. MTT-3((J), 40. [322] Miller, S. E., and A. C. Beck (1953). Low-loss waveguide transmission. Proc. l.R.E. 4L 348. [310] Montgomery, C. G. (1947). Technique of Microwave Measurements. M.I.T. Radiation Lab. Series, Vol. II. McGraw-Hill, New York. [328] Montgomery, C, G., R. H. Dicke, and E. M. Purcell (1948). Principles of Micro-Have Circuits. M.I.T. Radiation Lab. Scries, Vol. 8. York. [322, 328] Moreno, T. (1948). Microwave Transmission Design Data. [325, 328] Morgan, S. P. (1949). Effect of a surface roughness on eddy current losses at microwave frequencies. J. Appl. Phys. 20, 352. [306] Mount, E., and B. Begg (1960). Parametric devices and mascrs: an annotated bibliography. l.R.E. Trans. MTT-S, 222. [352] Mueller, G. E., and W. A. Tyrrell (1947). Polyrod antennas. Bell System Tech. J. 26, 837. [334] Mumford, W. W. (1949). A broadband microwave noise source. Bell System Tech. J. 28, 608. [355] Oliner, A. A. (1957). The impedance properlies of narrow radiating slots in the broad face of rectangular waveguides. l.R.E. Trans. AP-5, 4, 12, [338] Page, C. H, (1958). Harmonic generation with ideal rectifiers. Proc. l.R.E. 46, 1738. [345] Parker, C. F., and R. J. Anderson (1957). Constant beamwidth broadband antennas. l.R.E. National Convention Record, Vol. 5, Part I, 87. [336] Petritz, R. L, (1952), On the theory of noise in p-n junctions and related devices. Proc. l.R.E. 40, 1440. [346] Pound, R. V. (1948). Microwave Mixers. M.I.T. Radiation Laboratory Scries, Vol. 16. McGraw-Hill, New York. [349] Rennie, J. C. (1957). Design considerations in a wideband microwave mixer and i.f. preamplifier. l.R.E. Trans. CS-5, 221. [349] Riblet, H. J. (1952a). The short-slot hybrid junction. Proc. l.R.E. 40, 180. [315] Riblet, H. J. (19526). Synthesis of narrow band direct-coupled filters. Proc. l.R.E. 40, 1219. [318] Riblet, H. J. (1958). A unified discussion of high-Q waveguide filter design theory. l.R.E. Trans. MTT-6, 359. [318] Richardson, .1. M., andTc B, Riley (IL'57). Performance of 3-mm harmonic generators and crystal detectors. l.R.E. Trans. MTT-5, 131. [346] 434 References Richmond, J. H., and T. E. Tice (1955). Probes for microwave ncarfield measurements. l.R.E. Trans. MTT-3(J), 32. [342] Robertson, S. D. (1956). Recent advances in finline circuits. I.R.E. Trans. MTT-4, 263. See also The ullra-bandwidlh finline coupler, Proc. l.R.E. 43, 739 (1955). [316] Schel k u n o IT, S. A., and H, T. Friis (1952). A Mennos: Theory and Practice. Wi I ey, New York. [327, 329, 338] Severin, H. (1956). Nonreflecting absorbers for microwave radiation. l.R.E. Trans. AP-4, 385. [362] Smith, R, A, (1951). The relative advantages of coherent and incoherent detectors. Proc. Inst. Ehe. Engrs. (London) 98, Part rv, 43. [349] Smith, R. A. (1959). Semiconductors. University Press, Cambridge. [346] Sobel, F., F. L. Wentworth, and J. C. Wiltsc (1961). Quasi-optical surface waveguide and other components for the 100 to 300 Go region. I.R.E. Trans. MTT-9 512. [310] Staniforth, A., and J. H. Craven (1960). Improvement in square law operation of crystals. I.R.E. Trans. MTT-8, 111. [346] Stevenson, A. F. (1948). Theory of slots in rectangular waveguides. J. Appl. Phys. 19. 24. [338] Strailon, A. W., and C. W. Totbert (I960). Anomalies in the absorption of radio waves by atmospheric gases. Proc. l.R.E. 48, 898. [312, 313] Strong, .1. (1938). Procedures in Experimental Physics. Prentice-Hall, Bnglewood Ctiffs, N.J. [357] Taub, J., and P. J. Giordano (1954). Use of crystals in balanced mixers. I.R.E. Trans. MTV-2(2), 26. [351] Taussing, N. W. (1962). Some design factors affecting pulse and CW sensitivity of crystal video receivers employing RF amplification. Microwave J. 5(2), 94. [3481 Theissing, H. H., and P. J. Capiat! (1956). Atmospheric attenuation of solar millimeter wave radiation, J. Appl. Phys. 27, 538. [312, 313] Thorp, J. S. (1954). Rf conductivity in copper at 8 mm wavelength. Proc. Inst. E/ec. Engrs. (London) 101, Part III, 357. [3051 Tischer, F. J. (1956). The H-guide, a waveguide for microwaves. I.R.E. National Convention Record, Part V, 44. [311] Tischer, F. J. (1958). Properties of H-guide at microwaves and millimeter waves. I.R.E. Wescon Convention Record, Part I, 4. [311] Torrey, H. C, and C, A, Whitmer (1948). Crystal Rectifiers. M.I.T. Radiation Lab. Series, Vol. 15. McGraw-Hill, New York. [346] van Buskirk. L. F., and C. E. Hendrix (1961). The zone plate as a radio frequency focusing element. l.R.E. Trans. AP-9, 319. [312, 331] van der Ziel, A. (1950). On the noise spectra of semiconductor noise and of flicker effect. Physica 16, 359. [346] van der Ziel, A. (1954). Noise. Prentice Hall, Englewood Cliffs, N.J. [346] van der Ziel, A. (1959). Noise figure of reactance converters and parametric amplifiers. J. Appl. Phys. 30, 1449. [351] Valenzucla. G. R. (1963). Millimeter transmission by oversize and shieldcd-beain waveguides. IEEE Trans. MTT-11, 429. [307] Wade, G. (1961). Low noise amplifiers for centimeler and shorter wavelength!) Proc. I.R.E. 49, 880. [348] Watson, R. B., and C. W. Horton (1948). The radiation patterns of dielectric rods—experiment and theory. Appl. Phys. 19, 661, [335, 336] Chapter 10 435 Weber, J, (1959). Masers. Revs. Modern Phys. 31, 681. [352] White, W. D., and J. G. Greene (1956). On the effective noise temperature of gas discharge noise generators. Proc. I.R.E. 44, 939. [355] Wilson, I. G., C. W. Schramm, and J. P, Kinzcr (1946-1947). High Q resonant cavities for microwave lesling. Bell System Tech. J. 25, 408 (1946), and 26, 31, 410 (1947), Reprinted in Radar Systems and Components (Van Nostrand, New-York, 1949), pp. 909-1020. [326, 328] Woonton, G. A., and .1. A. Carruthers (1950). Indoor measurements of microwave antenna radiation patterns by means of a metal lens. J. Appl. Phys. 11, 428. [3411 REFERENCES FOR CHAPTER 10 Alpher, R., and D. White (1959). Optical refractivity of high temperature gases. Phys. Fluids 2, 153, 162. [369] Ascoli-Bartoli, 1J., and A. dc Angelis (1963). Interferomctric study of a rf discharge. Proc. 6th Intl. Conf. on Ionization Phenomena in Gases (Paris) 2, 475. [369] Ascoli-Bartoli, U„ A. de Angelis, and S. Martellucei (1960), Wavelength dependence of a plasma in the optical region. Niwvo cimenio 18, 1116. [369] Bailey, R. A,, and K. G. Emeleus (1955). Plasma electron oscillations. Proc. Roy. Irish Acad. 57A, 53. [386] Bennett, F., D. Shear, and H. Burden (1960). Streak interferometry. J. Opt. Soc. Am. 50, 212. [370] Bernstein, I. B., and I. N. Rabinowil/ (1959). Theory of electrostatic probes in a low-density plasma. Phys. Fluids 2, 112. [380] Billings, B. H. (1951). Narrow band optical interference fillers. Phot. Eng. 2(2), 45. [369] Böhm, D., E. H. S. Bui hop, and H. S. W. Massey (1949). Use of probes for plasma exploration. In The Characteristics of Electrical Discharges in Magnetic Fields, edited by A. Guthrie and R. K. Wakerling, McGraw-Hill, New York, Chapter 2. [380, 384] Boyd, R. L. F., and N. D. Twiddy (1959). Electron energy distribution in plasmas. Proc. Roy. Soc. (London) A250, 53. [381] Brown, S. C. (1963). Far infrared studies of high density plasmas. Bull. Am. Phys. Soc. 8. 174. See also S. C. Brown, G. Bekefi, and R. E. Whitney,/, Opt. Soc. Am. 53, 448 (1963). [373] Chen, F. F. (1965). Electrostatic probes. In Plasma Diagnostic Techniques, edited by R. Huddlesion and S. Leonard, Academic Press, New York. Chapter 4. [378] Colgate, S„ J. Ferguson, and H. Furth (1958). A toroidal stabilized pinch. Proc. 2nd U.N. Conf. on Peaceful Uses of Atomic Energy (Geneva) 32, 129, 140. [387] Cooper, J. (1963). On the high-frequency response of a Rogowsky coil. Plasma Phys. (J. Nuclear Energy C) 5, 285. [387] Dickerman, P. J., and J. C. Morris (1961). Experimental studies of the temperature in a field-free plasma. In Optical Spectroscopic Measurements of High Temperatures, P. J. Dickerman, ed. Univ. of Chicago Press, p. 170. [390] Dougal, A.A. (1963). Optical maser probing theory for magneloplasma diagnostics. Proc. 4th Syn'posittm on Engineering Aspects of Magnctohydrodynamics (Berkeley), p, 1. [373] (J 436 Referencea Elmore, \V. C, E. M. Little, and W. E. Quinn (1958). Neutrons from plasma compressed by an axial magnetic field (Scylla). Proc. 2nd U.N. Conf. on Peaceful uses of Atomic Energy (Geneva) 32, 337. [370] Gardner, A. (1962). Diagnostic measurements of a highly ionized, steady-stage plasma. In Engineering Aspects of Magnetohydrodynamics, C. Mannal and N. Mather, ed., Columbia University Press, New York, p. 438. [386, 333, 334] Generalov, N. P. (1959). The theory of probes. Plasma Phys. (J. Nuclear Energy C) 9, 148. [380] Glasstonc, S., and R. H. Lovberg (1960). Controlled Thermonuclear Reactions. Van Nostrand, Princeton, N.J. [387, 391] Golovin, I, N., D. P. Ivanov, V. D. Kirilov, D. P. Petrov, K. A. Razumova, and N. A. Yaulinsky (1958). Stable plasma column in longitudinal magnetic field. Proc. 2nd U.N. Conf on Peaceful Uses of Atomic Energy (Geneva) 32, 72. [387] Griem, H. R. (1960). Stark broadening of higher hydrogen and hydrogenlike lines by electrons and ions. Astrophys. J. 132, 883. [388] Griem, H. R., and A. C. Kolb (1959). Advances in the theory of Stark broadening. Proc. 4th Intl. Conf. on Ionization Phenomena in Gases (Uppsala), North-Holland Publishing Co., Amsterdam, p. 808. [389] Griem, H. R., A. C. Kolb, and K. Y. Shen (1959). Stark broadening of hydrogen lines in plasma. Phys. Rev. 116, 4. [389] Harding, G., and V. Roberts (1961). Spectroscopic investigation of plasma in the wavelength range 0.1 to 2.0 mm. Proc. 5th Intl. Conf. on Ionization Phenomena in Gases (Munich), p. 1977. [391] Harding, G. N., M. F, Kimmitt, J, H. Ludlow, P. Porteous, A. C. Prior, and V. Roberts (1961). Emission of sub-millimetre electromagnetic radiation from hot plasma in ZETA. Proc. Phys. Soc. (London) 77, 1069. [373, 391] Harp, R. S. (1963). Circuit for display of Langmuir probe characteristics. Rev. Sci. Instr. 34, 416. [382] Ikegami, H., and K. Takayama (1963). Resonance probe. Institute of Plasma Physics Report 1PPJ-J0, Nagoya University, Japan. [386] Javan, A., W. R. Bennett, Jr., and D. R. HerrioU (1961). Inversion and continuous optical mascr oscillation in a gas discharge containing a Ho-Ne mixture. Phys. Rev. Letters 6, 106. [369] Johnson, E. O., and L. Matter (1950). A floating double probe method for measurements in gas discharges. Phys. Rev. 80, 58. [384] Kelly, R. L. (1959). Vacuum ultraviolet emission lines. Lawrence Radiation Lab. Rept. UCRL 5612, Univ. of California, Livermore, Calif. [388] Klein, A. (1963). Some results using optical interferometry for plasma diagnostics. Phys. Fluids 6, 310. [371] Langmuir, I., and H. Mott-Smith (1924). Studies of electric discharges in gases at low pressures. Gen. Elec. Rev. 27, 449, 538, 616, 762, and 810. [385] Lengyel, B. A, (1962). Lasers. Wiley, New York. [369] Levitskii, S. M., and I. P. Shashurin (1961). Resonant microwave probe for measuring charge density in a plasma. Soviet Phys.—Tech. Phys. 6,315. [386] Lovberg, R. H. (1963). Acceleration of plasma by displacement currents resulting from ionization. Proc. 6th Intl. Conf. on Ionization Phenomena in Gases (Paris) 4, 235. [372] Lovberg, R. H. (1964). Measurement of plasma density in a rail accelerator by means of Schlieren photography. IEEE Trans. NS-11(/), 187. [372| Chapter 10 437 Lukyanov, S. Yu., and V. I. Sinitsin (1958). Spectroscopic research of high temperature plasmas. Proc. of 2nd U.N. Conf. on Peaceful Uses of Atomic Energy (Geneva) 32, 358. [389] McWhirtcr, R. W. P. (1965). Spectral intensities. In Plasma Diagnostic Techniques, edited by R. Huddleslone and S. Leonard, Academic Press, New York. Chapter 5. [391] McWhirter, R. W. P., W. G. Griffin, and T. J. L. Jones (1959). The interpretation, in terms of atomic collision processes, of a measurement of the absolute intensities of some of the Balmer spectral lines, as emitted by a deuterium discharge in ZETA. Proc. 4th Intl. Conf on Ionization Phenomena in Gases (Uppsala), North-Holland Publ. Co., Amsterdam, p. 833. [391] Malmbcrg, .1. H., N, W. Carlson, C. B. Wharton, and W. E. Drummond (1963). A collisionlcss plasma for wave propagation studies. Proc. 6th Intl. Conf. on Ionization Phenomena in Gases (Paris) 4, 229. [386] Margenau, H. (1932). Theory of pressure effects of foreign gases on spectral lines. Phys. Rev. 40, 387. See also Phys. Rev. 48, 755 (1935). [389] Margenau, H. (1951). Statistical theory of pressure broadening. Phys. Rev. 82, 156. [389] Marshall, J, (1958). Acceleration of plasma into vacuum. Proc. of 2nd U.N. Conf. on Peaceful Uses of Atomic Energy (Geneva) 31, 341. [387] Mcdtcus, G. (1956). Simple way to obtain the velocity distribution of the electrons in gas discharge plasmas from probe curves. J. Appl. Phys. 27, 1242. [381] Mott-Smith, H., and I. Langmuir (1926). Theory of collectors in gaseous discharges. Phys. Rev. 28, 727. [381] Pollock, IL, L. Goldman, and W. Westendorp (1960). Multiple magnetic probe measurements in compressed deuterium plasma. Bull. Am. Phys. Soc. 5, 337 [387] Ramsden, S., and E. McLean (1952). Optical refractivily of free electrons. Nature 194, 761. [370] Schawlow, A. L., and C. H. Townes (1958). Infrared and optical masers. Phys. Rev. 112, 1940. [369] Schulz, G. J., and S. C. Brown (1955). Microwave study of positive ion collection by probes. Phys: Rev. 98, 1642. [378] Scott. F. R., and V. Josephson (1957). Apparatus for producing high velocity shock waves and gases. U.S. Patent No. 2,923,852 (appl. October, 1957, granted February, 1960). [373] Scott, F. R., E. Dacus, and R. G. Tuckfield (1962), Optical emission line profile analyzer. Rev. Sci. Instr. 33, 1001. [388, 390] Serchuk, A. (1962). Commercially available optical masers. Microwaves 1, 54. ~ [369] Stern, M. O., and E. Dacus (1961). Piezoelectric probe for plasma research. Rev. Sci. instr. 32, 140. [387] Takayama, K,, H. Ikemagi, and S. Miyasaki (I960). Plasma resonance in a radio-frequency probe. Phys. Rev. Letters 5, 238. [386] Talbot, L., J. Katz, and M. Brundin (1963). Comparison between Langmuir probe and microwave electron density measurements in an arc-heated low-density wind tunnel. Phys. Fluids 6, 559. [378] Tonks, L., and I. Langmuir (1929). Theory of the arc plasma. Phys. Rev.34,876. [378] Underbill, A., and W. Waddell (1959). Optical wavelength tables. Natl. Bur. of Standards Circulars No, 603 and 604 (U.S. Govl. Printing Office). [389] 438 References van der Pol, B. (1920). De invloed van een geioniseerd gas op hel voortschrijden van electromagnetischc golven. Thesis, Utrecht University, Holland. Reprinted in van der Pol, Selected Scientific Papers, Vol. 1 (North-Holland Puhl. Co., Amsterdam, 1960). [376, 386] Wharton, C. B., and R. Hawke (1962). Calibration of rf conductivity probes by immersion in ionic solutions. Lawrence Radiation Lab. Llectronics Engineering Report LLL, Univ. of California, Livermore, Calif. [376] White, R. M. (1962). Elastic wave production by electromagnetic wave absorption or particle bombardment. General Electric TWT Rept. T1S-R62EIM232, Palo Alto, Calif. [387] Wilcox, 3. M., A, W. De Silva, W. S. Cooper, and F. I. Boley (1961). Experiments on Alfven-wave propagation. In Radiation and Waves in Plasmas, edited by M. Mitchncr, Stanford Univ. Press, Stanford, Calif., p. 138. [389] Wullf, H. (1959). Plasma diagnostics by speclroscopical means. Nuclear Instr. and Methods, 4, 352. [388]. Vcung, T. IL Y., and J. Saycrs (1957). An if probe technique for the measurement of plasma electron concentrations in the presence of negative ions. Proc. Phys. Soc. (London) 70, 663. [386] Absorbers, microwave, 196, 197, 343, 361 Absorption coefficients, 129, 130, 131, 270; see also Absorptivity; Attenuation coefficient Absorption length, 244, 293, 297, 298; see also Attenuation length Absorptivity, radiation, 270, 287 relation to emissivity, 270, 287, 289 Accelerated charge, radiation by, 246, 272 Adiabatic analysis, 120, 123, 133, 220 Admittance, transmission line, 157 measurement of, 158 plasma equivalent, 156, 158 Amplification, see Growth Amplifiers, i.f., 295, 299, 349, 353 maser, 352 parametric, 351 video, 201, 205, 346, 363 Anisotropic medium, 45, 160, 179, 403 Antennas, calibration of, 294, 342 Cassegrain, 336 circularly polarized, 223, 226, 322 dielectric rod, 224, 309, 334, 340, 360 effective area of, 243 far field of, 146, 341 for free-space beams, 141, 311 for spacecharge waves, 182, 360, 386 Frcsnel zone, 332 gain of, 243, 264, 265 horn, 143, 197, 204, 224, 237, 292, 322, 329, 333, 360 Subject index Antennas, lens, 146, 150, 218, 311, 331, 333 measurement of field pattern, 145, 341 near field of, 341 optimization of, 148 parabolic reflector, 336, 337 phase anomalies of, 145, 150 slotted waveguide, 223, 338 Appleton (-Hartrec) equation, 23, 24, 31 including ion motions, 56 "tan26" form, 31, 32 Area of antenna, effective, 243 Atmosphere, wave attenuation in, 226, 312 Attenuation, atmospheric, 226, 312 by internal reflections, 128, 140, 141 by refraction, 137, 140 collisional, 115, 121, 140, 141, 202, 226 collisionless, 113, 186, 226 in finite plasma, 129 in plasma-filled waveguide, 163, 166 measurement of, 122, 192, 200, 210, 287, 293 waveguide, 306, 308 Attenuation coefficient, 5, 397; see also Transmission coefficient for electron-ion collisions, 90, 263 for incoherent scattering, 93 for Landau (collisionless) damping, 114, 186, 190 Attenuation index, 7, 165, 398 439 440 Subject index Attenuation index, for cyclotron wave, 16, 18, I 15 for extraordinary wave, 27, 28 for ordinary wave, 11 in warm plasma, 114 Attenuation length, 8, 10, 195, 307, 397; see also Skin depth; Absorption length Automatic frequency control (AFC), 237, 295, 351 Averages, collision frequency, 64, 73 electron density, 120, 125, 160, 221 velocity, 74, 261 Backward-wave oscillator, 289, 299, 344; see also Microwave sources Beam, free-space microwave, 117, 141 Beam-plasma interactions, 185, 188, 285, 300; see also Instability Bessel functions, 113, 176, 180, 247, 275 Beta (fi), of magnetically confined plasma, 97, 224, 275 Blackbody radiation, 242, 266, 269. 287, 297 conditions for, 244 see also Radiation Boltzmann (Maxwell-Bolt zmann) distribution, 259 Boltzmann equation, 67, 105 collision term, 67, 188 in deriving spacechargc wave propagation, I 12, 183, 188 Boltzmann theory of conductivity, 69 Boundaries, cutoff regions due to, 125, 191, 292, 297, 304 diffuse, 120, 297 effects on spacecharge waves of, 155, 174 finite gradient, 130, 134, 136 internal reflections from, 129, 131 ramp profile, 135 refraction by, 137 sharp interface, 127; see also Inho-mogeneous plasmas Breakdown, 91, 163 Bremsstrahlung, 245 correlation of theory with conductivity theories, 262 Bremsstrahlung, from atoms, 256 magnetic, 256, 272 power radiated as, 248, 254, 255, 263, 268, 272 summary of, 254 total, 256, 391 Bridge, microwave, I 19, 200; see also Interferometer Brfflouin (w-y) diagram, 10, 11, 102, 167, 177, 187, 398, 399 Broadening, collisional, 274, 298 doppler, 274, 298, 367, 389 due to collective effects, 279, 304 due to inhomogencity, 275, 293, 299 of cyclotron radiation, 274, 279, 298, 303 Stark, 367, 388 Calortnietric viewpoint, 90, 92 Cavities, coupling coefficient of, 158 frequency of, 325 frequency shift due to perturbation in, 159, 297 high mode number, 162 measurements using, 163 modes of, 155, 159, 325 plasmas in, 155, 158, 297 Q of, 156, 326 reflection coefficient of, 157 Cerenkov radiation, 96, 114, 280, 282, 286, 293; see also Landau damping; Cyclotron radiation Chapman-Enskog technique, 68 Characteristic velocities, 73 Characteristic waves, 24, 38; see also Principal waves Charge separation in a plasma, 78 Cillie radiation, 251, 263, 266, 269 Circularly polarized waves, 12, 40, 108, 160, 223, 292 handedness, 14, 15, 40 see also Elliptically polarized waves Classical electron radius, 93, 233 Clausius-Mossotti formula, 401 Coherent detection, 232, 298, 353; see also Detectors; Radiometer Coherent radiation, 242, 285; see also Collective interactions Cold plasma, 1, 57, 171 Subject, index 441 Collective interactions, 235, 242, 278, 285, 299 Collimation of microwave beam, 147, 219, 331, 336 Collisional broadening, 274, 298; see also Broadening Collisional damping, 6, 64, 121, 140, 141, 203, 226; see also Damping Collision frequency, 6, 57, 66 average, 64, 73 coulomb, 82, 87, 90 dependence on velocity, 63, 92 effective, 71, 72, 73, 82, 83, 90 measurement of, 161, 192, 203, 374 momentum transfer, 60, 64, 65 total, 59 Collisionless (Landau) damping, in fluctuating plasma, 231 of electromagnetic waves, 96, 114, 226, 231, 275, 283, 293 of spacecharge waves, 185, 190, 232 Collisions, attenuation by, 121, 140, 141 coulomb, 62, 79, 81, 87. 110 clastic, 60 electron-electron, 76, 84, 85, 256 electron-ion, 62, 76, 79, 82, 90, 110 electron-molecule, 61, 63 energy lost in, 60, 91 hard sphere, 61 inelastic, 62 ionization by, 62, 91 of electrons with sheaths, 280, 304 Collision term in Boltzmann equation, 67 Complex conductivity, see Conductivity Complex dielectric constant, see Dielectric constant Complex refractive index, see Refractive index Components, laboratory fabrication of, I 313-328 Compton effect, 93 Conductivity, 393 complex, 70, 71, 395 d-c, 83, 94 Lorcntz, 6, 71 measurement of, 374, 378, 387 of metals, 307 Conductivity, summary of computation, 76 tensor, see Conductivity tensor warm plasma, 96, 105 Conductivity probes, 374, 386 calibration of, 377 Conductivity tensor, 30, 71, 75, 101, 105, 404 including ion motions, 51 in rotating coordinates, 32, 34, 1Ü6 Conductivity theory, correlation with bremsstrahlung theory, 262 Continuity, equation of, 67, 99, 395 Cooperative processes, 235, 242, 285, 299 Coordinate systems, for propagation in magnetic field, 20 rotating, 13, 31 Correction factors, g and h, 74, 75, 85, 274 tB, 84 Correlation, between microwave and other diagnostic techniques, 216, 225, 290, 297, 373, 378 of in st ability-generated radiation, 299, 300 Cotton-Mouton effect, 29 Coulomb collision frequency, 82, 87, 90; see also Collision frequency Coulomb collisions, 62, 79, 81, 87, 110 Coulomb cross section, 62; see also Cross section Coulomb force, 63 Coulomb logarithm, see In A Couplers, directional, 193, 201, 315, 322 iin-h'ne, 200, 219, 223, 300, 316, 322 Coupling, between antennas, 148, 198, 218, 329 mode, 182, 227, 307, 309 resonant cavity, 158, 318 spacecharge wave to radiation field, 189, 286, 300 wave-type, 1.03, 127, 181, 189 Critical angle, 45 Critical electron density, 12, 118; see also Cutoff conditions Cross modulation, see Modulation 442 Subject index Subject index 443 Cross section, coulomb, 62 dependence on velocity, 62, 63 differential, 59 momentum transfer, 60 Thomson, 232 total, 59 Cutoff conditions, 18, 28, 34-39, 44, 53-56, 125, 179 Cutoff density, 118, 220, 224; see also Critical electron density Cutoff regions, 125, 191, 292, 304; see also Tunneling Cutoffs on impact parameter, 85 Cyclotron damping, 114, 225; see also Cyclotron wave; Damping Cyclotron frequency, 15, 52, 189, 224, 226, 272, 292 shift of, 228, 304 spatial variation of, 228, 292, 293 Cyclotron radiation, 272, 284 anisotropy, 273 at harmonics of w!r 191, 304 effects of collective motion, 278, 279 line shape, 274, 278, 298 power radiated as, 272 relalivistic, see Synchrotron radiation Cyclotron resonance, 18, 84, 91, 92, 110, 279 Cyclotron wave, attenuation of, 16, 1 14 Faraday rotation of, 19, 224 in warm plasma, 102, 108, 116 refractive index for, 15, 35, 47, 109 spacechargc mode, 181, 183, 191 whistler mode of, 19, 48, 54, 182, 225, 297 Cylindrical plasma, inhomogeneous, 136, 174, 179, 212, 222 sharply bounded, 130, 137, 138, 175 sec also Boundaries Damping, collisional, 6, 64, 91, 121, 140, 141, 203, 226 collisionless (Landau), 96, 114, 185, 190, 226, 231, 275, 283, 293 cyclotron, 1 14, 225 of spacccharge waves, 185, 190, 232 deBroglie wavelength, 63, 85, 252 Debye length, 78, 80, 86, 93, 186, 188, 236 Debye shielding, 76, 82, 256, 380 Debye sphere, 79, 80, 236 Delta functions, 265, 277 Density, see Electron density Detailed balance, principle of, 260 Detectors, bolometer, 372 coherent, 232, 298, 353 crystal-video, 195, 200, 218, 346, 348 diode, 200, 346, 352 noise in, 346, 349 sensitivity of, 346, 350 Golay cell, 372 mixers, 349 harmonic, 351 parametric, 351 performance of, 355 square-law, 198, 200 superheterodyne, 204, 232, 295, 349, 353 Diagnostic techniques, catalog of, 366-368 phase shift, 200-223 radiation as, 287-299, 391 resonant cavity, 155-163 Diamagnetism of plasma, 96, 275 measurement of, 275, 374 Dielectric constant, 393, 395 complex, 5, 70, 176, 179, 394 d-c, 56 Lorentz, 6, 71 summary of computation of, 76 Dielectric constant tensor, 30, 71, 101, 106, 179, 404 including ion motions, 52 in rotating coordinates, 32 Differential cross section, 59 Diffraction theory, 142 Diffuse boundaries, plasma with, 120, 297; see also Boundaries Diplexer (Duplexer), frequency, 199, 316, 318 polarization, 200, 219, 223, 315, 316, 322, 324 see also Filters; Polarization Dipole resonance, 178, 181, 235, 240, 386 D-c conductivity, 83, 94 D-c dielectric constant, 56 Dispersion, 46, 281, 398 spatial, 100, 405 Dispersion equation, 46, 47, 168, 225, 404 Böhm, 103, 186, 187, 235 for spacccharge wave, 177, 180, 184, 186, 189, 190 Disturbing wave 91, 229 Doppler effect, broadening by, 274, 298, 367, 389 frequency shift due to, 110, 114, 119, 228, 274, 275, 390 in drifting plasmas, 171, 182, 228 shift of cyclotron frequency by, 228, 299 Druyvesteyn distribution, 262 Effective collision frequency, 71, 72, 73, 82, 90 Einstein A and B coefficients, 86, 259, 261 Elastic collisions, 60 Electroacoustical waves, see Plasma waves; Spacecharge waves Electron density, average, 121, 160, 222 catalog of measuring techniques for, 366 critical, or cutoff, 12, 118, 194, 198, 214 errors in determination, 212, 214, 218, 222 fluctuations of, 229 measurement by phase shift, 200, 214, 225 measurement by probes, 376, 380, 385, 386 measurement by resonant cavities, 155, 162 measurement by spectroscopy, 225, 388 range of measurable, 121, 122, 370 spatial distribution of, 123, 161, 212, 1 214, 218, 222 special, 35, 53 wave propagation as function of, 11, 28, 35, 37, 47-50, 56, 194, 214, 225 see also Ion density Electron-electron collisions, 84, 85, 256 Electron energy, analysis of, 297 average, 74 Electron energy, distribution of, 68, 1 14, 184, 189 measurements of, 297 related to noise temperature, 290, 302 see also Velocity; Electron temperature Flectron-ion collisions, 62, 79, 82, 87, 110 Electron radius, classical, 93, 233 Electron temperature, catalog of measuring techniques for determining, 367 measurement by probes, 380, 385, 387 measurement by radiation, 243, 254, 255, 261, 265, 269, 287, 290, 294, 297, 302, 391 Elliptically polarized waves, 24, 29, 219, 227, 273; see also Circularly polarized waves Emissivity, 264, 265, 267, 269, 270, 271, 284 effect of antenna gain on, 264, 265 effect of surface reflection on, 266, 269 relation to absorptivity, 270, 287, 289 see also Absorptivity Energy, electron, 74 loss by bremsstrahlung, 256 loss by collision, 60 loss by cyclotron radiation, 272 spacecharge wave, 187, 234, 236 Ergiebigkeit, 259, 261 Eulcr's constant, 87 Evanescent wave, 8; see also Tunneling Extinction coefficient, 398; see also Attenuation index Extraordinary wave, 26, 34, 48, 53, 134, 161, 220 in warm plasma, 104, 111 Faraday rotation, 19, 223, 322, 373 Far field of antenna, 146, 341; see also Antennas Field particle, 58 Field strength, measurement of, 367, 387, 388 Filters, band rejection, 237, 299, 319 frequency diplexer, 316, 318 mode, 177, 307, 310 444 Subject index Subject, index 445 Filters, resonant cavity, 299, 325 waveguide, 229, 319 Fine-scale mixing, 114 Fizeau effect, 228 /-Number of lens, 147 Fokker-Planck equation, 81, 186 Force laws, interparticle, 62, 63 Fourier analysis, 230, 246, 302 Fraunhofer diffraction, 146, 341; see also Antennas Free-free transitions, 245; see also B rem sstrah lung Free-space beams, 117, 141; see also Antennas Frequencies, special, 34, 53 Frequency, collision, see Collision frequency complex, 185, 188 control of, 351; see also Automatic frequency control critical (for sell-absorption), 266 cyclotron, harmonics of, 191, 276, 302, 304 cyclotron (or gyro), 15, 52, 272 hybrid, 35, 53, 181, 220 Larmor, 15 plasma, 3, 52 propagation as function of, 9, 10, 35, 36, 54, 56 resonant cavity, 157 Frequency bands, EIA, 306 Frequency diversity, 197, 199, 216 Frequency spectrum, due to fluctuating plasma, 229, 231 for cyclotron radiation, 191, 228, 304 for synchrotron radiation, 276, 278 of scattered wave, 93, 235 Fresnel zones, 142, 143 antenna using, 332 interferences of, 198 g and h Correction factors, 74, 75, 85, 274 Gain of antenna, 243 dependence of emissivrty on, 264, 265 Gamma functions, 72 Gamma rays, see X-rays 7^ Correction factor, 84 Gannt factor, 86, 248 connection with In A, 253, 262 high frequency, low temperature, 249 quantum mechanical form, 252 shielding correction, 252, 253 temperature correction, 249, 253 Gaussian shape factor, 274 Geometrical optics limit, 119, 146, 150 Geometrical optics of cylindrical plasma, 137 Gray body radiation, 244, 257, 261; see also Blackbody radiation Group velocity, 5, 11, 46, 168, 178, 187, 398 Growth, wave amplitude, 185, 190, 236, 260, 286; see also Damping Gyration, sense of, 15 Gyro frequency, see Cyclotron frequency Gyroradius (Larmor radius), 88, 96, 97, 190 Hall current probes, 367, 387; see also Field strength; Magnetic probes Handedness of circular polarization, 14, 15, 40 Hankel functions, 249 Hard sphere interaction, 61, 63 Harmonic generation, in plasma, 92, 93, 302, 345 with ferrites, 345, 352 with semiconductors, 345, 351 Heating, plasma, 91 "High." frequencies, 9, 72, 87 Horn antenna, 143, 197, 204, 215, 237, 292, 322, 329, 360 as feed to dielectric rod, 224, 309, 334, 340 as feed to lens, 218, 311, 333 as feed to reflector, 336, 337 circular waveguide, 322, 324, 328 Hot plasma, 95, 115 Huygens wavelet, 45, 281 Hybrid resonance frequency, lower, 53 upper, 35, 53, 181, 220, 298 Hydromagnetic theory, 66, 67, 98 Impact parameter, 58 classes in coulomb collisions, 81 Impact parameter, cutoffs, 85 for 90° collision, 80, 85 Incoherent radiation, 242; see also Radiation Incoherent scattering, 93, 232, 366; see also Scattering Index, attenuation, 7, 165, 398 Faraday rotation, 19, 224 growth, 186, 236 refractive, see Refractive index Induced absorption and emission, 259 Inelastic collisions, 62; see also Collisions Infrared probing, 368, 373; see also Optical probing; Radiometer Inhomogcneity broadening, 275, 293, 299; see also Broadening fnhomogeneous plasmas, cutoff regions in, 125, 191, 292, 304 nonlinearities in, 93 propagation in, 130, 135 see also Boundaries Instability, driven by electron beam, 285, 300 electrostatic, 185, 188, 235, 286, 300 hydromagnetic, 226, 372, 387 two-stream, 189, 300 velocity-space, 190, 302 see also Turbulence; Spacecharge waves Intensity, specific, 257, 270; see also Radiation Interaction, electron beam-plasma wave, 185, 189, 190, 235, 300; see also Growth; Beam-plasma interaction; Instability Interference, due to internal reflections, 129, 131, 240, 334, 342, 361 Fresnel, 145, 198, 332 fringes, 198, 202, 207, 216, 371, 373 stra^, see Pickup Interferometer, Fabry-Perot, ■ 369, 389 fringe-shift, 206 Mach-Zehnder, 200, 367, 369, 371 microwave, 119, 200 polar-plot display, 210 streak, 370, 373 zebra stripe, 206 Intermediate frequencies, 8 <* Intermediate frequencies, of super he t receiver, 295, 299, 349, 353 Internal reflections, 126, 129, 131 attenuation by, 141 interferences from, 129, 131, 240, 334, 342, 361 Interp article force laws, 62, 63 Ion density, measurement of, 367, 380, 386, 389; see. also Electron density Ion motions, conductivity with, 51, 380 dielectric constant with, 52 oblique propagation with, 56 principle waves with, 52 Ion temperature, measurement of, 367, 387, 389 Ion waves, see Spaceeharge waves Ionosphere, cross-modulation in, 92 propagation in, 225, 312 scattering from, 93, 232 Isotropic medium, propagation in, 133, 396 Johnson noise, 245, 346, 349 lunction, directional coupler, 193, 201, 315, 322 fin-line coupler, 200, 219, 223, 317, 322 hybrid, 315, 316 magic tee, 315 turnstile, 315, 323, 324 Kinetic (Vlasov) equation, 67, 184 Kinetic theory, 66, 104 Kirchhoff's law, 244, 270, 274, 277, 292 Klystron, 201, 206, 208, 218, 237, 295, 344; see also Transmitter Kramers-Kronig dispersion relations, 115 Landau (collisionless) damping, in fluctuating plasma, 231 of electromagnetic waves, 96, 114, 226, 231, 275, 283, 293 of spaceeharge waves, 185, 190, 232 Langevin equation, 12 Langmuir probe, 378 circuits for data reduction using, 379, 382 446 Subject index Subject index 447 Langmuir probe, current-voltage curves of, 378, 383 double probe, 384 floating potential of, 380, 384 magnetic field effects on, 383, 385 saturation current of, 380, 382, 385 single probe, 301, 360, 378 typical configurations of, 215, 360, 379 Laplacian operator, 77, 188 Larmor frequency, 15; see also Cyclotron frequency Larmor radius (gyroradius), 88, 96, 97, 190 Laser, 369 emission wavelengths, 370 interferometer using, 371 I.egendrc functions, 69 Lenses, microwave, 146, 150, 218, 311, 331, 333: see also Antennas Linear electromagnetic medium, 393 Linearity, criterion for, 89 Linearly polarized wave, 26, 40 Lines, optical emission, 368, 388 resonance, see Resonance; Broadening transmission, see Transmission lines; Waveguide Liouville theorem, 67 In A, 85, 87 connection with Gaunt factor, 253, 262 high frequency, low temperature, 87, 88 quantum correction, 86, 88, 89 shielding correction, 86, 88, 89 Spilzer's, 78, 80, 85, 86, 266 ■ ■ Lorcntz conductivity, 6, 71 I.orentz dielectric constant, 6, 70, 179 Lorentz-Lorenz formula, 401 Lorentz plasma, 1, 6, 12, 85 Lorentz polarization correction, 24, 401 Lorentz shape factor, 274 Loschmidt's constant, 59 Loss factor, 394 Loss tangent, 394 "Low" frequencies, 8, 72, 87 Lower hybrid frequency, 53; see also Hybrid resonance frequency Luxembourg effect (cross-modulation), 91, 229 collisionless, 231 Magnetic beach, 127, 293 Magnetic bremsstrahlung, 272; see also Cyclotron radiation Magnetic field, measurement of, 367, 387, 388 propagation as function of, 16, 17, 27, 109 propagation in, 12, 18, 27, 40, 44, 55, 76, 88, 106, 219, 223, 226, 228, 293 Magnetic mirror, 97, 292 propagation through, 223, 228, 295 radiation from, 292, 295, 297, 302 Magnetic moment, 96 Magnetic probes, 367, 387 Hall current, 387 miniature coil, 387 Rogowsky loop, 387 three dimensional, 387 see also Field strength Magnetrons, 345; see also Transmitters; Microwave sources Magnetization vector, 97, 393 Matrix form of tensor, 22, 29, 31, 105, 407 Maxwell-Boltzmann population distribution, 259 Maxwell condition, 71 Maxwellian velocity distribution, 68, 71 Maxwell's equations, 392 Mean free path, 59; see also Collision frequency Microscopic fields in a medium, 400 Microwave components (hardware), 201, 205, 207, 211, 215, 218, 237, 238, 288, 305-344 antennas and radiators, 196, 207, 218, 224, 226, 292, 301, 327 couplers, 193. 201, 207, 237, 300, 315, 317, 323 .niters, 237, 318 frequency diplcxcr, 199, 316 hybrid junction, 218, 315, 323 mixer, 207, 237, 288, 289, 295, 299, 315, 351, 353 Microwave components (hardware), phase shifter, 201, 207, 212, 314 polarization diplexer, 200, 223. 300, 316, 322 polarizer, 319, 322, 324 resonant cavity, 155-163, 297, 324 waveguide, 204, 207, 305-312 Microwave sources, millimeter waves, 344, 345; see also Klystron; Magnetron; Transmitter; Backward-wave oscillator; Traveling-wave tubes Mirror, magnetic, 97, 223, 228, 292, 297, 302 parabolic, 336, 337 Mixers (frequency converters), balanced, 237, 295, 299, 315, 349 for swept frequency receiver, 289, 299 harmonic, 349 parametric, 351 receivers using, 237, 288, 295, 299, 353 Mobility, 393 Mode cutoff, 165, 169, 179, 297, 310; see also Cutoff conditions Modulation, collisionless Luxembourg, 231 cross, 91, 229 frequency spectrum due to, 229, 231 Luxembourg, 89, 91 of scattered waves, 235 phase, due to plasma fluctuations, 93, 229 Moments of velocity distribution function, 69, 98 Momentum lost in collision, 60 Momentum transfer, collision frequency for, 60, 64 cross section for, 60 Momentum transport, equation of, 99 Monochrometer, 369, 388, 390; see also Optical spectroscopy Motion, equation of, 6, 12, 64, 99 Neutrality, departure of plasma from, 78 90" Deflection time, 81 Noise temperature, 290, 349, 355 of plasma radiation, 242, 290, 299, 353 of standard source, 244, 266, 355 Nonlinear effects, 89, 229, 234 Non reciprocity, due to electron drift, 171, 182, 228 due to Faraday rotation, 20, 223 Normalization of velocity distribution function, 69 Oblique propagation to magnetic Held, 38^11, 49, 50, 190, 227 with ion motions, 56 Ohm's law, 21, 51, 99, 393 Omega-beta diagram (Brillouin), 10, 11, 102, 167, 177, 187, 398, 399 Opacity, plasma, 197, 244, 258, 293, 298 Optical depth, 258, 265, 292, 298; see also Absorption length; Attenuation length Optical (infrared) probing, 368 by Faraday rotation, 373 by scattering, 232, 367 interferometers for, 369, 374 light sources for, 370 measurement of refractive index by, 369 using lasers, 369 Optical spectroscopy, 388 atom identification by, 388 Balmer series of, 367, 391 contmua in, 388, 391 doppler broadening in, 367, 389, 390 measurement of line shape in, 388 Stark broadening in, 367, 388 Zeeman line splitting in, 388 Optimization of antennas, 148; see also Antennas Orbit, theory, 66 Ordered component of velocity, 64 Ordinary wave, 11, 26, 34, 49, 53, 220 in warm plasma, 103, 111 Oscillator, backward-wave (BWO), 289, 299, 344 local, in superhet, 237, 295, 344, 351, 353 tunnel diode, 345, 352 448 Subject index Subjecl index 449 Oscillator, voltage-tuned, 289, 299, 302. 344, 351 see also Microwave sources; Transmitters Parscval's theorem, 246 Permeability, magnetic, 96, 393 Permittivity, 179, 393; see also Dielectric constant Phase anomalies, at focus of antenna, 145, 150 due to multiple transmission paths, 132, 152, 204 in Faraday rotation measurement, 219 Phase coefficient, 397 of electromagnetic wave, 5, 165 of spacecharge wave, 173, 177, 181, 185, 190, 235 vector, 46, 188, 235 see also Propagation coefficient Phase factor, for plane wave, 21, 397, 404 Phase mixing, 114, 232, 293; see also Landau damping Phase scrambling, 232, 293 Phase shift, 118, 119, 121, 123, 126, 129 at optical frequencies, 369, 373 in moving medium, 174, 182, 229 in nonuniform medium, 214, 220 measurements of, 166, 200, 206, 210, 221, 229 Phase shifter, electronic, 168, 315 line stretcher, 315 waveguide, 314, 318 Phase space, 66 Phase velocity, 5, 11, 45, 397 calculated from ta-jS diagram, 10, 174, 187 of spacecharge waves, 171, 177, 181, 185, 187, 235 relation to ray velocity, 46 Phase-velocity surface, 43-46, 55 Phasor, 13 Photon emission, 259 Pickup, stray, 205, 300, 348, 354, 364 Planck function, 243, 260, 261 Plasma, confinement of, 97, 210, 221, 226, 272, 292 Lorentz, 1, 6, 12, 85 production of, 194, 213, 221, 223, 226, 293, 301, 302, 360 Plasma analogs, 151 Plasma frequency, 3, 52, 178, 186 Plasma oscillations, 4, 86, 87, 92, 94, 95, 102, 103, 112, 170, 175, 185, 235, 285, 286, 300; see also Spacecharge waves Poisson distribution, 65 Poisson's equation, 77 Polarizability, 401 Polarization coefficient, 24 Polarization correction, Lorentz, 24, 401 Polarization diplexer, 200, 223, 300, 316, 322 Polarization of waves, linear, 24, 40 circular, 12, 40, 108, 160, 223, 292 elliptical, 24, 29, 219, 227, 273 rotation of, 19, 224 Polarization vector, dipolc, 393 Power factor, 394 Poynting vector, 45, 133, 230 Pressure tensor, 99 Principal waves, 34, 36, 37 including ion motions, 52, 54 see also Characteristic waves Probability of collision, 59 Probes, ballistic, 387 conductivity, 374, 386 electromagnetic, 226, 234 Langmuir, 215, 301, 360, 378 magnetic, 367, 387 movable, 341, 360 optical beam, 368 resonant, 386 spacecharge wave, 182, 360, 379, 386 waveguide, 224, 334, 340 Profile, measurement of electron density, 123, 212, 219, 221, 224 Propagation, angle of, 21, 38, 49, 227 as function of electron density, II, 28, 35, 37, 47-50, 56. 194, 214, 225 Propagation, as function of frequency, 9, 10, 35, 36, 54, 56 at angle to magnetic field, 38-41, 49, 50, 56, 190, 227 in drifting plasma, 228 in inhomogeneous plasma, 130, 212 in Lorentz plasma, 6 in magnetized plasma, 12, 18, 27, 40, 41, 44, 55, 76, 88, 106 in nonmagnelizcd plasma, 4, 87, 112 in waveguide, 155, 163, 297 through fluctuating plasma, 229 Propagation coefficient, 5, 397, 404 for spacecharge waves, 177, 185, 190 summary of computation for, 76, 87 vector, 21, 46, 188, 235, 404 QL and QT approximations, 38, 292 Radiance, 257 Radiation, at angle to magnetic field, 227, 298 blackbody, 242, 266, 269, 287, 297 bremsstrahlung, 245, 248, 254, 256, 263, 391 by electron-electron collisions, 256 Cerenkov, 96, 114, 280, 282 Cillie, 251, 263, 266, 269 coherent, 242, 285, 299 cyclotron, 272, 274, 278, 284, 292, 297, 302 cyclotron-frequency harmonic, 191, 275, 304 determination of Tt. by, 293, 298 from finite plasma, 263, 264, 269, 278, 287, 298 impurity, 256, 388 grey body, 244, 257, 261 incoherent, 242 instability-generated, 286, 300 measurement of, 289, 292, 297, 353, 391 nonthermal. 242, 285, 299, 302 recombination, 388, 391 relation to absorption, 270, 287, 289 specific intensity of, 257 synchrotron, 272, 275, 276, 278, 292, 302 Radiation, thermal, 242 see also Bremsstrahlung; Cyclotron radiation; Emissivity Radiation measurements, 287 from finite, free-space plasma, 263, 264, 269, 278, 287, 298 in resonant cavity, 297 in waveguide, 295, 297 see also Radiation; Radiometers Radiation transport, 257 Radiators, see Antennas; Probes Radiometer, calibration of, 290, 355 coherent detection, 298, 353 Dicke, 298, 353 direct detection, 288, 295, 300, 353 infrared, 373 swept frequency, 289, 295, 299, 302 Range of measurable electron densities, 121, 122, 366, 370 Ray surface, 45, 46 Ray velocity, 46 Raylcigh-Jeans approximation, 87, 243, 251, 254, 265 Receivers, crystal-video, 195, 200, 218, 346, 348 microwave, 289, 295, 346 noise in, 346, 355 performance measurements of, 355 radiometer, see Radiometer superheterodyne, 204, 232, 288, 295, 349, 353 Reduced mass, 52 Reflection coefficient, cavity, 157, 162 plasma, 128-131, 194, 199, 267, 270, 287 Reflections, from small plasma column, 239, 240 internal, multiple, 126, 129, 131, 204, 240, 298, 342 measurement of, 193. 195, 289, 298, 361 see also Scattering Refraction, attenuation by, 137, 140 Refractive index, 5, 7, 93, 398 as function of propagation angle, 23, 38, 42, 45, 49, 56 complex, 7, 398 contour maps of, 47-50 for characteristic waves, 23, 34, 52 450 Subject index Subject index. 451 Refractive index, for cyclotron wave, 15, 35, 47, 53, 108, 110 for extraordinary wave, 26, 35, 48, 53, 111, 220 for Faraday rotation, 19, 219, 224 for ordinary wave, 26, 35, 48, 53 in fluctuating medium, 230 in moving medium, 229 in warm plasma, 108, 109, 111, 116 Refractive-index surface, 42, 45 Relativistie effects, 115, 275, 284 Relaxation time constant, 68, 92 of tuned circuit, 156, 205, 364 relation to cross-modulation, 92, 230, 231 Resistivity, surface, 305, 361. tensor, 21, 29 waveguide, 305, 307 see also Conductivity Resonance, cavity, 157 cyclotron, 18, 84, 91, 92, 110, 279 dipole (perpendicular), 178, 241 hybrid, 34, 53, 181, 220, 298 Resonance conditions, 18, 28, 34—39, 44, 53-56, 125, 279 Resonance frequency, shift of, as function of propagation angle, 38, 42, 49 Rogowsky loop, 368, 387; see also Field strength; Magnetic probes Rotating coordinates, 31 Rotating unit vectors, 13, 31 Runaway electrons, 285, 300 Rutherford scattering formula, 62 Rydberg energy constant, 87, 252 Scattering, angle of, 58, 233, 235 Bragg, 235 cross section for, 93, 232, 233 enhanced, 94, 236 experiments on, 232, 237, 239, 241, 366 Irom plasma column, 235, 240, 241 from plasma fluctuations, 234, 236 incoherent, 93, 232 optical, 367 stray, 195, 204, 227, 232, 236, 361 Thomson, 93, 232 SchHeren photography, 372, 374 Sellmeier formula, 402 Sharp boundaries, cylindrical plasma with, 130, 137, 138, 175 slab plasma with, 127 Sheath, between plasma and conductor, 78, 386 effects on Langmuir probes, 380, 384, 386 effects on plasma radiation, 302 oscillations in, 300, 386 Shielding, correction to Gaunt factor, 252, 253 correction to In A, 86, 89 debye, 76, 82, 256, 320 Shock waves, measurement of intensity of, 387 measurement of velocity of, 368, 373, 374 Skin depth, 400 in metals, 305, 307 in plasma, 8, 10 see also Attenuation length Skin effect, 8, 306 Slab plasma, diffusely bounded, 120 propagation through, 120, 126, 127, 131, 212 radiation from, 263, 264, 269, 278, 298 sharply bounded, 127 transmission coefficient for, 129, 131, 298 validity of mode) for, 150, 151 Slowness surface, 42, 45 Snell's law, 257 Sources, plasma, 293, 301; see also Plasma, production of Spaccchargc, 2, 132, 172 Spacechargc waves, 170 body, 155, 181, 235 coupling to electromagnetic radiation, 182, 227, 286, 300, 302 dipole resonance frequency of, 179, 235, 241 dispersion equation for, 177, 184 fast and slow, 171 group velocity of, 174, 187 growth of, 185, 188, 236 in bounded plasma, 175 Spacecharge waves, in drifting plasma, 171, 182 in fluctuating plasma, 229, 232, 241 in plasma in magnetic field, 179, 188, 191 in warm plasma, 183, 189 Landau damping of, 185, 189, 232 a-ß diagram for, 174, 177, 181, 190 phase velocity of, 185, 236 probes for, 360, 379, 386 propagation coefficient for, 173, 177, 182, 185, 187, 190 related to electrostatic instabilities, 185, 189, 234 surface, 155, 176, 182 .fee also Plasma oscillations Spatial dispersion, 100, 405 Spatial distribution, of electron density, 123, 161, 212, 214, 218, 222 measurement of, 123, 212, 219, 221, 224 Special electron densities, 35, 53 Special frequencies, 34, 53 Specific heats, ratio of, 99 Spectroscopic measurements, see Optical spectroscopy Spectrum, frequency, see Frequency spectmm Spherical harmonics, 68 Spitzer's In A, 78, 80, 85, 86, 226 Spontaneous emission, 259 Standing wave, 129, 157, 162, 400 Standing-wave ratio, 128, 157 Stimulated absorption and emission, 259 Superthermal electrons, 285, 300, 302 Surface reflection, effect of gradient on, 195, 199 effect on emissivity, 197, 266, 269 Susceptibility, 393, 401 Synchrotron radiation, 272, 275, 292, 302 power radiated as, 275 spectrum of, 276, 278 see also Cyclolron radiation; Radia-I ion Temperature, blackbody, 243, 298 electron, see Electron energy; Electron temperature noise, 290, 299, 349, 353, 355 radiation, 242 Tensor, hermitian, 407 inverse or reciprocal, 21, 29, 33, 408 symmetric, 407 unit, 23, 30, 408 Tensors, uses of, 22, 29, 31, 51, 105, 407 Test particle, 58 Thermal radiation, 242; see also Radiation Thermal velocity, 74, 86; see also Electron energy; Velocity Thomson scattering, 93, 232; see also Scattering Three frequency regions, 7 Threshold signal sensitivity, 346, 347; see also Receivers Time constant, relaxation, 68, 92, 230, 231 tuned circuit, 156, 205, 364 video, 346, 364 Total collision cross section, 59; see also Cross section Total collision frequency, 59; see also Collision frequency Transconductance, see Interaction; Modulation Transit-time broadening, 274; see also Broadening Transmission, ducted, 48, 182, 225, 297 free-space beam, 117, 141 see also Propagation Transmission coefficient, 128-131, 267, 270 measurement of 192-212 of plasma in waveguide, 155, 163, 298 of plasma slab, 129, 131, 298 Transmission lines, 305 attenuation in, 166, 306, 308 free-space link, 311, 312 special waveguide, 307, 320 waveguide, 165, 3C5 see also Waveguides 452 Subject index Transmitter, microwave, 193, 212, 237, 295, 344 frequency control of, 208, 237, 295, 351 modulation of, 204, 206, 208, 295 see also Microwave sources Transparent medium, radiation from, 245, 254, 263; see also Brems-strahlung; Radiation Traveling-wave tubes, 344, 345, 348; see also Backward-wave oscillator; Microwave sources; Transmitter Tunnel diodes, as amplifiers, 352 as oscillators, 345 Tunneling, of radiation, 127, 292 through cut-off regions, 127, 227. 228 Turbulence, instability-generated, 197, 234, 372 measurement of, 227, 234, 366, 372, 387 Unit vectors, rotating, 13, 31 Unitary matrix transformation, 33, 106 Upper hybrid frequency, 35, 53, 181, 220, 298 Vacuum considerations, 356 ultra-high, 357 Velocity, electron thermal, 74, 86 group, 5, 11, 46, 168, 178, 187. 398 ion, 51, 367 ordered component of, 64 phase, 5, I 1, 45, 171, 187, 397 see also Electron energy Velocity dependence of collision frequency, 63, 72, 92 Velocity distribution function, 64, 66, 92, 104, 183 double-humped, 185, 189, 234, 286 Maxwellian, 68, 71, 186, 189 measurement of, 297, 367 moments of, 69 normalization of, 69 Viscous damping, representation of collisions by, 6, 64; see also Col-lisional damping Vlasov (kinetic) equation, 67, 184 Voight line shape, 275, 298; see also Broadening; Optical spectroscopy Voltage standing-wave ratio (VSWR), 128, 157 Wanted wave, 91 Warm plasma, 95, 109 Wave, characteristic, 24, 38 cyclotron, 15, 35, 114, 181, 224 evanescent, 8 extraordinary, 26, 34, 48, 53, 111, 134, 161. 220 in warm plasma, 108, 111, 116 ordinary, II, 26, 34, 49, 53, 111. 220 principal, 34, 52 spacecharge, 170, 175, 181, 188, 191, 235 surface, 155, 176, 182 velocity of, 5, II, 45, 168, 178, 187, 397 whistler, 18, 48, 182, 225, 297 Wave equation, 396, 404 Wave impedance, 91, 127, 133, 400 Wave-normal surface, 42, 45, 55 Wave polarization coefficient, 24 Wave propagation, see Propagation Waveguide, attenuation in, 165, 166, 306, 308 beyond cutoff, 8 dielectric, 224, 310, 311, 334 elliptical, 320 materials for, 306, 307 modes of, 165, 307-310 w-p diagram for, 167, 170 plasmas in, 163, 168, 297 propagation coefficient of, 165, 298 radiation measurements in, 297 spacecharge wave, 155, 174, 178, 235 special modes of, 309-312 standard bands, 306 see also Transmission lines Wavelength, free-space, 397 guide, 165-167 Wave number, 7, 397 Whistlers (whistler mode), 18, 48, 182, 225, 297 damping of, 114, 185, 226 Windows, for atmospheric propagation, 312 waveguide vacuum, 197, 333, 357 WKB approximation, I 19, 125 X-rays, measurement of, 297, 367, 391