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]; BeamRadius = Rationalize[3.81 1*^-6]; BeamPower = 81*^-3; BeamAmplitude = BeamPower/(2 \[Pi] BeamRadius^2); (*2D Gaussian*) Gaussian2D[r_, Radius_: 1, Amplitude_: 1] := Amplitude Exp[-1/2 (r/Radius)^2]; (*Disk shaped hole*) DHole[r_?NonNegative, HoleSize_: 1] := Boole[r <= HoleSize]; (*Test function*)THole[r_?NonNegative, Dummy_: 1] := DiracDelta[r]; Beam[r_?NonNegative] := Gaussian2D[r, BeamRadius, BeamAmplitude]; 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]