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Airy pattern

  • To: mathgroup at smc.vnet.net
  • Subject: [mg94751] Airy pattern
  • From: Max Ulbrich <mulbrich at berkeley.edu>
  • Date: Tue, 23 Dec 2008 06:58:03 -0500 (EST)
  • Organization: University of California, Berkeley

Hi,

I am trying to calculate the Airy pattern and also the pattern I get 
when not in focus, similar to what is shown in the middle image here:
http://en.wikipedia.org/wiki/File:Spherical-aberration-slice.jpg

I though this pattern should result from integration over wavefronts
that come from different angles of a sector of a sphere.

I tried with the integral

amp[x_, z_, ap_] :=
NIntegrate[Sin[ph]*Cos[(x*Sin[th]*Sin[ph] + z*Cos[ph])], {th, 0, 2Pi}, 
{ph, 0, ap*Pi/180}]^2

where x,z are the positions in the image, and ap the aperture.
It looks ok in focus, but different from the image on wiki when
out of focus.

A similar pattern for 2D can be generated by

amp[x_, z_, ap_] := NIntegrate[Cos[(x*Sin[a] + z*Cos[a])], {a, -ap*Pi/
         180, ap*Pi/180}]^2
DensityPlot[amp[x, z, 30], {x, -20, 20}, {z, -20, 20},
      Mesh -> False, PlotPoints -> 100];

which is much faster to calculate but gives qualitatively similar 
images. In particular I think there should not be z values with almost
zero intensity, e.g. compare these two

Plot[amp[x, 0, 45], {x, -15, 15}, PlotRange -> {0, 3}];
Plot[amp[x, 1.744, 45], {x, -15, 15}, PlotRange -> {0, 3}];

Does anyone know what is wrong with the integral?
Or what is the explanation for the deviation from the picture
shown on Wiki?

Thanks,
Max



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