Page 192 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 173

T( ,y) = ( ), then at the output plane one obtains the WDF
∞

2
W(x, ) = W 0 (x 0 , ) dx 0 =|U 0 ( )| (5.17)
−∞
And consequently, since the signal u(x) can be recovered from its
WDF,

∞

x x
∗
u(x)u (0) = p ,x = W , exp (i2 x ) d (5.18)
2 2
−∞
we can rewrite Eq. (5.17) as the nonlinear mapping

∞ x x
∗ ∗
u(x)u (0) = u 0 x 0 + u 0 x 0 − dx 0 (5.19)
2 2
−∞
From Eq. (5.19) it is apparent that the equivalent optical processor is
useful for visualizing a correlation operation. Of course a similar re-
sult is obtained by setting T( ,y) = (y). For this latter example, at
the output plane, the WDF is
∞

2
W(x, ) = W 0 (x, 0 ) d 0 =|u 0 (x)| (5.20)
−∞
And now Eq. (5.19) becomes
2
∗
u(x)u (0) =|u 0 (0)| (x) (5.21)
The above results remind us that phase-space representations have
an inherent nonlinear nature, caused by using as input either the
product-space representation or the product spectrum representa-
tion. The nonlinear attribute is linked next to the Saleh bilinear
transformations. 13

5.4 Bilinear Optical Systems
A bilinear transformation relates the complex amplitude distribution
at the input to the irradiance distribution at the output. Hence, in
terms of the third term of a Volterra series expansion, a bilinear trans-
formation is defined as
∞
∞
2
∗
|u(x)| = V(x; x 1 ,x 2 )u 0 (x 1 )u (x 2 ) dx 1 dx 2 (5.22)
0
−∞ −∞