Page 137 - Introduction to Information Optics

122 2. Signal Processing with Optics

where r represents the displacement vector between two points and

(2.106)
Ak = k 0 -k j +k, - k 3
is known as the dephasing wave vector and corresponds to the Bragg diffraction
(dephasing) limitation. In other words, if the reading beam is not matched with
the writing beam, the OPD will be nonzero. The field of scattered light from
the crystal, as a function of the (scattered) wave vector k,, can be expressed as

= exp(zAk-rMK (2.107)
J r

where the integration is over the entire volume of the crystal. We note that the
preceding equation is valid under the weak diffraction condition, such that
multiple diffraction within the crystal can be neglected.
In view of the preceding equation, we see that the performance of the PR
filter will be severely limited by the Bragg diffraction. Since the angular and
the wavelength selectivities of a multiplexed matched filter are limited by the
Bragg diffraction condition, the shift tolerance of the matched filter (related to
the Bragg mismatch) is affected by these selectivities, as will be discussed in the
following subsection.

2.7.4. ANGULAR AND WAVELENGTH SELECTIVITIES

Let us now consider the unslanted spatial filter synthesis for a transmission-
type and a reflection-type spatial filter, as depicted in Fig. 2.39. By using
Kogelnik's coupled-wave theory, the normalized diffraction efficiency can be
shown to be

1 + U.JU5)
2 2 2 1/ 2
sinh (v -f )
where the subscripts t and r denote the transmission-type and the reflection-type
holographic filters,

nAnd 2nnd sin 0 A
v, = - -- : , c, = ------------ A0,
r
' /cos0" ' /.
2nnd cos 0
, r — -- ,
/. sin 6 /,
r
n and An are the refractive index and the amplitude of its variation, d is the