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
