Page 141 - Introduction to Information Optics

126 2. Signal Processing with Optics
where 6 is the intersection angle outside the crystal, and n is the refractive index
of the crystal. Under the paraxial approximation, and a « 1, wave vector k n
can be shown as
V 2
2 2
2n /
where / is the wavelength of the light source, and / is the focal length of the
lens. Similarly, the recording object wave vector k t , reading wave vector k 2 ,
and diffracted wave vector k 3 can be shown as

k2
k - ~

With reference to momentum conservation and the infinite extension in the
u direction, we have Ak u — 0. By substituting preceding wave vectors into Eq.
(2.106) and using the condition A/e u = 0, the dephasing wave vector for the
transversal and the longitudinal directions can be shown to be

Afc a = x t - x 0 + x 3 - x 2 = 0, (2.1 1 3)

X
!
z = " -2 ' 1 -^0 + ^3 ^3) ^ ""T^v-^l -^OA-^0 ~~ •^3)- (•^••" "•''')
n/j
If readout beam k 2 is shifted; that is, x 2 = x l — S, then we have

z 2 ]
n//' °

At the input plane x 1 the reference and object beams are represented by
x
x
x
4o( o) " ^( o ~ ^o) an d <?i( i)» respectively. If the crystal filter is read out by
x
a shifted object, represented by q 2(x 2) — 4i( i ~ $)•> tne output correlation
peak intensity can be calculated by Eq. (2.107) over all k 0 and k } with
weighting factors q 0(x 0) and q^x^, as given by
d/2
/(X 3 ) dx, du dz
J-oc. J-.//2 (2.116)
C/^AQ — yV Q^C/ i \A j — O^C/|^A i / t/A[J^t/Ali rjj .
• Ai V V ^/If *•* i V ^ I// / V ^ £* V tA/ I A I/ • 1" \ I ^