Page 296 - Introduction to Information Optics

5.8. Hankel Transform 281

expansion; that results in

exp(-zm</>)d</> X f n(r)exp(in9)
o J 0 J 0 n = oo
(5.34)

dr,

where the Bessel function of the first kind and of order m is represented by

J m(x) = — exp(/ma) exp(/x cos(a)) da

with a = 6 — (f).
The second equality in Eq. (5.34) is the definition of the nth order Hankel
transform. Hence, a 2D input function and its Fourier transform are related in
such a way that the Hankel transform of the circular harmonic function of the
input function is the circular harmonic function of its Fourier transform.

5.8.2. HANKEL TRANSFORM

Mathematically, the Hankel transform of a function f(r) is expressed as

H m(p)= f(r}J m(2nrp}r dr. (535)

If m > —1/2, then it can be proved that the input function can be recovered
by the inverse Hankel transform as

H m(p)J m(2nrp)pdp. (536)

All the optical systems that perform the optical Fourier transform, as described
in Sec. 5.3, perform the Hankel transform as described in Eq. (5.35), where f(r)
is in fact the mth order circular harmonic function of the input image, and
H m(p) and is the mth order circular harmonic function of the Fourier transform
of the input image.