Page 94 - Introduction to Information Optics

2.4. Fourier Transform Processing /9

/(c, >/). By applying the Kirchhoff integral, the complex light distribution at
(a, /I) can be written as

where C is a proportionality complex constant, h t(c, rj) and ^-(x, v) are the
corresponding spatial impulse responses, and T(x, y) is the phase transform of
the lens.
By a straightforward but tedious evaluation, we can show that

</(«, j8) = C, exp

£,*/) exp - / - (a£ + /^)

which is essentially the Fourier transform of /(^, /;) associated with a quadratic
phase factor, where ju = ///. If the signal plane is placed at the front focal plane
of the lens; that is, / = /, the quadratic phase factor vanishes, which leaves an
exact Fourier transformation.

G(p, q) = C 1\\ /(& r,) exp[ - Hp£ + ^)] d£ </»/, (2.26)

where p = kz/f and q = kfi/f are the angular spatial frequency coordinates.

2.4. FOURIER TRANSFORM PROCESSING

There are two types of Fourier transform processors that are frequently used
in practice: the Fourier domain (filter) processor (FDP), and the joint trans-
form (spatial domain filter) processor (JTP), as shown in Figs. 2.9 and 2.10,
respectively. The major distinction between them is that FDP uses a Fourier
domain filter while JTP uses a spatial domain filter.

2.4.1. FOURIER DOMAIN FILTER

A Fourier domain spatial filter can be described by a complex amplitude
transmittance function, as given by

H{p t q) = \H(p, q)\ exppflp, q)l (2.27)