Fourier-Bessel transform / FT in polar coordinates

• To: mathgroup at smc.vnet.net
• Subject: [mg76279] Fourier-Bessel transform / FT in polar coordinates
• From: Mathieu G <ellocomateo at free.fr>
• Date: Sat, 19 May 2007 04:29:05 -0400 (EDT)

```Hello,
How can I compute a Fourier Transform in polar coordinates?

Here is where I am so far, but it seems the CircularFourierTransform
functions are badly defined??? Or is it with the Beam and Hole functions?:

(* **********Code following********** *)
Clear["Global`*"];

HoleSize = Rationalize[1.5 1*^-6 /2];
BeamPower = 81*^-3;

(*2D Gaussian*)
Gaussian2D[r_, Radius_: 1, Amplitude_: 1] :=

(*Disk shaped hole*)
DHole[r_?NonNegative, HoleSize_: 1] := Boole[r <= HoleSize];
(*Test function*)THole[r_?NonNegative, Dummy_: 1] := DiracDelta[r];

Hole[r_?NonNegative] := THole[r, HoleSize];
RevolutionPlot3D[Beam[r], {r, 0, 4 BeamRadius}, Mesh -> All,
PlotRange -> All, ColorFunction -> "Rainbow"]
RevolutionPlot3D[Hole[r], {r, 0, 2 HoleSize}, Mesh -> All,
PlotRange -> All, ColorFunction -> "Rainbow"]

(*Using Fourier transform definition*)
CircularFourierTransform[f_, \[Omega]r_] :=
2 \[Pi] Integrate[
r f[r] BesselJ[0, 2 \[Pi] r \[Omega]r], {r, 0, \[Infinity]}];
InverseCircularFourierTransform[f_, r_] :=
2 \[Pi] Integrate[\[Omega]r f[\[Omega]r] BesselJ[0,
2 \[Pi] r \[Omega]r], {\[Omega]r, 0, \[Infinity]}];
TFBeam = CircularFourierTransform[Beam[r], \[Omega]r]
TFHole = CircularFourierTransform[Hole[r], r]

```

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