using Plots
using Printf

Prednaska 8 - vady osove symetrickych systemu

V minule přednášce jsme uvedli, tvar aberačního polynomu pro osově symetrické systémy \begin{align} \Delta w = M(C_cw_o^{\prime}\kappa+C_3w_o^{\prime 2}\bar w_o^{\prime}+2K_3w_o^{\prime}\bar w_o^{\prime}w_o+\bar K_3w_o^{\prime 2}\bar w_o+ F_3w_o^{\prime}w^2_o\bar w_o+A_{f3}\bar w_o^{\prime}w_o^2+D_3w^2_o\bar w_o) \end{align}

Efekt jednotlivych vad na trajektorie castic

Efekt aberací je partný více méně jen v blízkosti roviny obrazu, protože v její blízkosti je relativně malý vliv paraxiální aproximace.

Chromaticka aberace

Trajektorie elektronu, ktery v predmetu vychází z osy ($w_o = 0$): $$ u(z) = h(z) w_o^{\prime}-g(z)C_c\kappa w_o^{\prime} - h(z)c_c\kappa w_o^{\prime}$$ což v blízkosti obrazove roviny lze psát $$ u(z) \approx h(z_i+(z-zi)) w_o^{\prime}-g(z_i)C_c\kappa w_o^{\prime} - h(z_i)c_c\kappa w_o^{\prime}\approx M_a(z-z_i) w_o^{\prime} - MC_c\kappa w_o^{\prime}$$

Cc = 5e-1; Ma = -10; M=-0.1
th = (-1:0.2:1)*1e-3
Ph0 = 1e4
kappa = [-0.3,0,0.3]/Ph0
col = [:red,:black,:purple]
dz = (-1:0.1:1)*1e-6
p=plot()
for i=1:length(kappa)
    print("dE = ", kappa[i]*Ph0, " eV: dz = ", M/Ma*Cc*kappa[i]," m\n")
    plot!(p,dz,Ma*dz*th'-M*Cc*ones(size(dz))*th'*kappa[i],color=col[i],label=false)
end
plot(p)

dE = -0.3 eV: dz = -1.5e-7 m
dE = 0.0 eV: dz = 0.0 m
dE = 0.3 eV: dz = 1.5e-7 m

Sfericka aberace

Obdobným způsobem jako v předchozím případě dospějeme ke tvaru trajektorií v blízkosti roviny obrazu $$w = M_a(z-z_i)w_o^{\prime}+MC_3w_o^{\prime\,2}\bar w_o^{\prime}$$

Ma = -10; M=-0.1;
C3 = 5e-3/M^4;
th = (-1:0.1:1)*1e-3
Ph0 = 1e4
dz = (-2:0.1:1)*1e-6
#print("dE = ", kappa[i]*Ph0, " eV: dz = ", M/Ma*Cc*kappa[i]," m\n")
p=plot(dz,Ma*dz*th'.+M*C3*ones(size(dz))*th'.^3,color=:black,label=false)

Kaustiky

Jedna se o množiny v nich je nulovy Jacobian zobrazeni (mezi množinou parametru a vyslednymi polohami trajektorii). Zobrazeni \begin{align} w = m(z-z_i)w_o^{\prime}+MC_3w_o^{\prime\,2}\bar w_o^{\prime}\\ \bar w = m(z-z_i)\bar w_o^{\prime}+MC_3\bar w_o^{\prime\,2} w_o^{\prime} \end{align} ma Jacobian \begin{align} \mathrm{det}\begin{pmatrix}\frac{\partial w}{\partial w_o^{\prime}}& \frac{\partial w}{\partial \bar w_o^{\prime}}\\ \frac{\partial \bar w}{\partial w_o^{\prime}}& \frac{\partial \bar w}{\partial \bar w_o^{\prime}}\end{pmatrix}= M_a^2(z-z_i)^2+4MM_aC_3(z-z_i)w_o^{\prime}\bar w_o^{\prime}+3M^2C_3w_o^{\prime\,2}\bar w_o^{\prime\,2} \end{align} Pokud polozime Jakobin = 0 a vyřešíme pro (z-z_i) dostaneme dvě akustiky \begin{align} (z-z_i) &= -3\frac M{M_a}C_3w_o^{\prime}\bar w_o^{\prime}&w &= -2MC_3w_o^{\prime\,2}\bar w_o^{\prime}\\ (z-z_i) &= -\frac M{M_a}C_3w_o^{\prime}\bar w_o^{\prime}&w&=0 \end{align}

dz2 = -3*M/Ma*C3*th.^2
x = -2*M*C3*th.^3
plot!(p,dz2,x,color=:red)

Coma

Zde se budeme zabyvat pouze deviaci v obrazove rovině $$\Delta w_i = 2MK_3w_o^{\prime}\bar w_o^{\prime}w_o+\bar MK_3w_o^{\prime 2}\bar w_o$$

th = (0.1:0.1:1)*1e-3
alpha = 0:0.01:2*pi
wo = 1e-5
K3 = -1.e3
p=plot()
for i=1:length(th)
    dwo = th[i]*exp.(im*alpha)
    dwi = 2*M*K3*dwo.*conj(dwo)*wo+M*conj(K3)*dwo.^2*conj(wo)
    plot!(p,real(dwi.+M*wo),imag(dwi.+M*wo),aspect_ratio=:equal,label=:false)
end
plot(p)

K3=[2e2,-2e2,2e2im,-2e2im]
xo=(-1:0.25:1)*1e-5; yo = copy(xo)
ph = (0:0.02:1)*2*pi
p1=Any[]

for ip=1:length(K3)
    p=plot(legend=false,aspect_ratio=:equal)
    for i=1:length(xo)
        for j=1:length(yo)
            for i1=1:length(th)
                wo=xo[i]+yo[j]*1im
                dwo = th[i1]*exp.(1im*ph)
                wi = M*wo.+2M*K3[ip]*wo*conj(dwo).*dwo+M*conj(K3[ip])*conj(wo).*dwo.^2
                plot!(p,real.(wi),imag.(wi),color=:black,title="K3 = "*string(K3[ip]))
            end
        end
    end
    push!(p1,p)
end
plot(p1...,layout=(2,2),size=(1024,1024))

Field curvature (sklenutí pole)

V blíkosti fokusu můžeme psát trajektosii ve tvaru $$ w \approx Mw_o + M_a(z-z_i)w_o^{\prime} + MF_3w_o\bar w_o w_o^{\prime}=Mw_o + (M_a(z-z_i) + MF_3w_o\bar w_o) w_o^{\prime}$$ Ze vztahu je patrné že k fokusu dochazí v jiné rovině, než v ronine obrazu. Jeji poloha je různa pro různe polohy predmetu. Jak se liší?

$w = g(z) w_o+h(z)w_o'=g(z_i+dz)w_o+h(z_i+dz)w_o'\approx (g(z_i)+g'(z_i)dz)w_o+(h(z_i)+h'(z_i)dz)w_o'$

$w =Mw_o+M_adzw_o'$

wo = (-1:0.25:1)*1e-5
th=(-1:0.25:1)*20e-3
dz = (-1:0.1:0.2)*1e-6
F3=4e5
p=plot()
for i=1:length(wo)
    plot!(p,dz,M*wo[i].+Ma*dz.*th'.+M*F3*wo[i]*conj(wo[i])*ones(size(dz))*th',label=:false,color=:black)
end
plot(p)

xo=(-1:0.25:1)*1e-5; yo = copy(xo)
ph = (0:0.02:1)*2*pi
p=plot(legend=false,aspect_ratio=:equal)
for i=1:length(xo)
    for j=1:length(yo)
        for i1=1:length(th)
            wo=xo[i]+yo[j]*1im
            dwo = th[i1]*exp.(1im*ph)
            wi = M*wo.+M*F3*wo*conj(wo).*dwo
            plot!(p,real.(wi),imag.(wi),color=:black)
        end
    end
end
plot(p)

Astigmatizmus

V blíkosti fokusu můžeme psát trajektosii ve tvaru $$ w \approx Mw_o + M_a(z-z_i)w_o^{\prime} + MA_{f3}w_o^2 \bar w_o^{\prime}$$ poud zvolime $w_o = x_o$ a $A_{f3} = \bar A_{f3}$ dostaneme \begin{align} x &= M x_o + M_a(z-z_i)x_o^{\prime}+MA_{3f}x_o^2x_o^{\prime}\\ y &= M_a(z-z_i)y_o^{\prime}-MA_{3f}x_o^2y_o^{\prime} \end{align}

Tj. paprsky, ktere jsou v rovine zx jsou fokusovane v jine rovine nez paprsky, ktere jsou v rovine zy. Velikost defokusi zavisi na pocatecni poloze v objektu.

A3 = 1e5im
xo=(-1:0.25:1)*1e-5; yo = copy(xo)
th=(-1:0.25:1)*20e-3
ph = (0:0.02:1)*2*pi
dz = [0,0,4,-4,8,-8,12,-12,18,-18]*1e-8
p1=Any[]
for ip = 1:length(dz)
    p=plot(legend=false,aspect_ratio=:equal)
    for i=1:length(xo)
        for j=1:length(yo)
            for i1=1:length(th)
                wo=xo[i]+yo[j]*1im
                dwo = th[i1]*exp.(1im*ph)
                wi = M*wo.+M*A3*wo^2*conj(dwo).+Ma*dz[ip]*dwo
                plot!(p,real.(wi),imag.(wi),color=:black,title=@sprintf("dzi = %g",dz[ip]))
            end
        end
    end
    push!(p1,p)
end
plot(p1...,layout=(5,2),size=(1024,5*512))

dz

10-element Vector{Float64}:
  0.0
  4.0e-8
  8.0e-8
  1.2000000000000002e-7
  1.6e-7
  0.0
 -4.0e-8
 -8.0e-8
 -1.2000000000000002e-7
 -1.6e-7

Distorze

Budeme zkoumat pouze pozice v rovine obrazu $$ w(z_i) = M w_o + MD_3w_o^2\bar w_o$$ Tato aberace meni polohy obrazu v obrazove rovine

function meshgrid(x, y)
    X = [i for i in x, j in 1:length(y)]
    Y = [j for i in 1:length(x), j in y]
    return X, Y
end

xo = (-1:0.2:1)*1e-4;
yo=copy(xo)
Xo,Yo = meshgrid(xo,yo)
plot(M*Xo,M*Yo,color=:black,label=:false,aspect_ratio=:equal)
plot!(M*Xo',M*Yo',color=:black,label=:false,aspect_ratio=:equal)
D3=-5e6im
Wo = M*(1 .+D3*(Xo.^2+Yo.^2)).*(Xo .+im*Yo) 
plot!(real.(Wo),imag.(Wo),color=:red,label=:false)
plot!(real.(Wo)',imag.(Wo)',color=:red,label=:false)

Aberacni koeficienty v parametrizaci pomoci polohy v predmetu a aperture