Page 305 - Introduction to Information Optics

290 5. Transformation with Optics

Input Plane Phase Mask Output Plane
( H, V )

Fig. 5.6. The optical system for implementing geometric transformation.

Bryngdahl was the first to analyze this system [25]. In the optical system
shown in Fig. 5.6, the Fourier transform of the product of the input and the
phase mask is exactly produced in the output plane as

G(w, v) = /(x, y)exp(j'4>(x, y)) exp -j2n — y \dxdv. (5.46)

The concept of the stationary phase [32] is useful when computing this Fourier
transform. We rewrite Eq. (5.46) as
<-*
I | f(x, y) exp | j — h(x, y, M, v) dx dy,
A
with
/0(X, >') U V
h(x, y u, v) = — — x - - v- (5.47)
2 7 1 / 7

In this type of the integral, as the wavelength A is several orders of magnitude
smaller than the coordinates u, v and x, y, the phase factor h can vary very
rapidly over x and y; as a result, the integral can vanish. The integral can be
fairly well approximated by contributions from some subareas around the
saddle points (x 0, y 0) where the derivatives of h are equal to zero and the phase
factor is stationary:

a/i a/j
(5.48)
5x ov