ZernikeLogAiryImage.jpg
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Summary
| Description |
English:
A flat-top beam being imaged by a lens. The beam is under the effect of the first 21 Zernike polynomials. The Log scale helps visualize the minima in intensity. For example, the top plot represent is a perfect beam (no aberration), whose image is a Bessel function
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| Date | |
| Source | Own work |
| Author | Rolancito |
(*https://mathematica.stackexchange.com/questions/33574/whats-the-\
correct-way-to-shift-zero-frequency-to-the-center-of-a-fourier-transf*)
fftshift[dat_?ArrayQ, k : (_Integer?Positive | All) : All] :=
Module[{dims = Dimensions[dat]},
RotateRight[dat,
If[k === All, Quotient[dims, 2],
Quotient[dims[[:w:k]], 2] UnitVector[Length[dims], k]]]]
ZernikePoly =
Table[Table[
If[m < 0,
Sqrt[(2 (n + 1))/(\[Pi] (1 + KroneckerDelta[0, m]))]
ZernikeR[n, -m, \[Rho]] Sin[m \[Theta]],
Sqrt[(2 (n + 1))/(\[Pi] (1 + KroneckerDelta[0, m]))]
ZernikeR[n, m, \[Rho]] Cos[m \[Theta]]], {m, -n, n, 2}], {n,
0, 5}] /. {\[Rho] -> Sqrt[x^2 + y^2], \[Theta] -> ArcTan[x, y]} //
FullSimplify
(*The flat-top beam size (HeavisideTheta) has radius 1*)
(*The Fourier plane is taken to be much larger than the aperture size*)
(*The Fourier transform is the result of the imaging setup*)
(*Each Zernike polynomial is simulated with a coefficient of 0.5*)
Column[Map[
ArrayPlot[
Log[Abs[fftshift[
Fourier[Table[
HeavisideTheta[1 - Sqrt[x^2 + y^2]]/Sqrt[\[Pi]]
E^(-I 0.5 #), {y, -20, 20, 0.051}, {x, -20, 20,
0.051}]]]]^2],
PlotRange -> {{393 - 60, 393 + 60}, {393 - 60, 393 + 60}},
PlotTheme -> "Scientific"] & , ZernikePoly, {2}], Center]
Export["ZernikeLogAiryImage.jpg", %]
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