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5.8. Hankel Transform                  279

       In this case, Kj -> oo, g l = 1; we let the scale parameter s l in the input be
       chosen freely. The fractional order is determined by Eq. (5.30) as a function of
       the propagation distance z:

                                an/2 = arctan(Az/A'?)                 (5.32)

       and s 2, R 2 also can be computed from Eq. (5.30). Hence, from the aperture
       z — 0 to z -> (X) at every distance z, the complex amplitude distribution on the
       reference surface of radius R 2 is the fractional Fourier transform of the
       transmittance t(x), computed according to Eqs. (5.30) and (5.31) where the
       fractional order a increases with the distance z, from 0 to 1, according to (5.32).
       At the infinite distance, z -> oo, we have a -> 1 according to Eq. (5.32), the
       Fresnel diffraction becomes the FraunhorfF diffraction, and the fractional
       Fourier transform becomes the Fourier transform. Notice that if we measure
       the optical intensity at distance z, the phase factor associated to the reference
       spherical surface with radius R 2 has no effect.



       5.8. HANKEL TRANSFORM


          Most optical systems, such as imaging systems and laser resonators, are
       circularly symmetrical, or axially symmetrical. In these cases, the use of the
       polar coordinate system is preferable, and the 2D Fourier transform expressed
       in the polar coordinate system will lead to the Hankel transform, which is
       widely used in the analysis and design of optical systems.


       5.8.1. FOURIER TRANSFORM IN POLAR COORDINATE SYSTEM

          The Hankel transform is also referred to as the Fourier-Bessel transform.
       Let /(x, y) and F(u, v) denote a 2D function and its Fourier transform,
       respectively, such that


                     F(u, v} =  f(x, y) exp( — i2n(ux + vy}) dx dy.


       The Fourier transform is then expressed in the polar coordinates, denoted by
       (r, 0) in the image plane, and by (p, r/>) in the Fourier plane, respectively, as
                           2
                          f * f*
                 F(p, </>) =     /(r, 0) exp(-/2rcrp cos(# - <p))r drdO.  (5.33)
                          Jo Jo
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