Page 140 - Introduction to Information Optics

2.7. Processing with Photorefractive Optics 125
transmission type. It is therefore apparent that the reflection-type filter is a
better choice for the application of the wavelength-multiplexed filter.

2.7.5. SHIFT-INVARIANT LIMITED CORRELATORS

In a Fourier domain-matched filtering system, a shift of the input target at
the input plane will cause a change in the readout angle at the Fourier plane
where the PR filter is located. When the change of the readout angle is large,
the readout beam intensity decreases rapidly due to high angular selectivity
(i.e., the Bragg diffraction limitation). In other words, the higher the angular
selectivity of the PR filter, the lower its shift tolerance will be. Thus, to optimize
the shift invariance in a thick PR filter, a minimum angular selectivity is
needed. However, from Fig. 2.40 we see that the wavelength selectivity for the
reflection-type filter has its highest value at 2a = 180° where the angular
selectivity is minimum. Therefore, a reflection-type wavelength-multiplexed PR
filter will be the best choice for two major reasons, large storage capacity and
optimum shift tolerance.
We now investigate the shift invariance of three commonly used PR-based
correlators, the Vandelugt correlator (VLC), the joint transform correlator
(JTC), and the reflection-type correlator (RC). First, let us consider the VLC
(shown in Fig. 2.41) in which a point light source located at position x 0
produces a plane reference beam. Within the PR crystal, this plane wave can
be described by vector k 0 = |k| cos az — jk| sin au, where a is the intersection
angle between wave vector k 0 and the optical axis inside the crystal, and u and
z are the transversal and the longitudinal unit wave vectors, respectively. By
referring to the well-known Snell's law, we have sin a = (sin 9)/n, and

2
1/2
cos a = (1 — sin a) = ( 1 —

crystal

q (x )
2 2
Fig. 2.41. A PR-based VLC.