Image plane of a flat-top beam under the effect of the first 21 Zernike polynomials. The beam starts in the so-called Fourier plane, which is imaged onto the image plane by an ideal lens.

A lens of focal length f is a Fourier-transform device between the place at distance f before it and the plane at distance f after it. These planes are called "Fourier" and "Image" plane respectively.

(*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 2*)  Column[Map[   ArrayPlot[     Abs[fftshift[       Fourier[Table[         HeavisideTheta[1 - Sqrt[x^2 + y^2]]/Sqrt[\[Pi]]           E^(-I 2 #), {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["ZernikeAiryImage.jpg", %]

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