Page 295 - Introduction to Information Optics

280 5. Transformation with Optics

If the input image is circularly symmetric, f(r, 8) is independent of 6; we have

POO f2n /*oe
F(p, <£) = f(r)rdr \ exp(-i2nrp cos(6 - $))d0 = f(r)J 0(2nrp)rdr,
Jo Jo Jo

where J 0 () is the Bessel function of the first kind and zero order. Hence, the
Fourier transform F(p, </>) is the Hankel transform of the zero order of the
input image. According to the zero order Hankel transform, the Fourier
transform F(p, (j)) of a circularly symmetric input image is independent of r/>
and is also circularly symmetric. An example is when an optical beam is passed
through a disk aperture; the Fraunhofer diffraction pattern of the aperture is
the Airy pattern, which is computed as the Hankel transform of the zero order
of the disk aperture.
In general /(r, 9) is dependent of 6. In this case, we can compute the circular
harmonic expansions of /(r, 0) and F(p, </>), which are the one-dimensional
Fourier transforms with respect to the angular coordinates, as

f m(r) = —- j /( r, 9) exp( - /m0) d0

and

where m is an integer. f m(r) and F m(p) are referred to as the circular harmonic
functions of the input image and its Fourier transform. Because both /(r, 9)
and F(p, <£) are periodic functions of period 2n, f m(r) and F m(p) are, in fact, the
coefficients in the Fourier series expansions as

/(', ») = Z /«(»•) expO'm 0)

and

Ofj
F(p, 0) = X ^m(p) exp(zw^).
— TC.
Hence, we can find the relationship between the circular harmonic functions
of the input image and that of its Fourier transform, which is obtained by
computing the angular Fourier transforms with respect to 4> in both sides of
Eq. (5.33) and replacing the input image /(r, B) with its circular harmonic