       Re: Fourier-Bessel Transform

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1768] Re: Fourier-Bessel Transform
• From: siegman at ee.stanford.edu (A. E. Siegman)
• Date: Wed, 26 Jul 1995 00:54:17 -0400
• Organization: Stanford University

```In article <DC1EGD.48y at wri.com>, jt12799 at meibm9.cen.uiuc.edu (Jehan Tsai) wrote:

>  I was wondering if there was any code in existence that implements the
>  Fourier Bessel Transform.  I have not been able to get the code shown below
>  to run correctly-- either giving me Indeterminate answer or nothing at all.
>
>
>  Integrate[2 Pi r BesselJ[0, 2 p Pi r] (some input function),
>                  {r,0, Infinity}]
>
>  where r is the radius in the space domain and p is rho (radius) i n the
>  frequency domain.
>
>  Does anybody know of a way to properly implement this?  Thanks.
>
>  J.T. Tsai

Haven't tried this myself in Mathematica, but the Fourier-Bessel
transform of a Laguerre-gaussian function in r is a Laguerre-gaussian of
the same order in rho.  Maybe mma knows this; maybe it doesn't.

A numerical algorithm I once devised for doing fast numerical
calculations of Fourier-Bessel transforms on discrete data points, which I
believe is still the fastest such approach if properly programmed is
described in:

1)  A. E. Siegman, "Quasi fast Hankel transform," Opt. Lett. 1, 13-15
(March 1977).  New method of calculating Hankel or Fresnel transforms
using an FFT approach, and its application to optical resonators with
cylindrical symmetry.

2)  S.-C. Sheng, Studies Of Laser Resonators and Beam Propagation Using
Fast Transform Methods, Ph.D. dissertation, Department of Applied Physics,
Stanford University (March 1980).  Describes "lower end corrections"
needed to be applied to the FHT algorithm.

If you want to do a _lot_ of Hankel transforms, with max speed or
limited computer power, these could be useful to you.

```

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