Page 188 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 169

horizontal axis. In mathematical terms, the second anamorphic pro-
cessor generates the AF, A( , y), by implementing over the product-
space representation the two-dimensional operation

∞
∞
A( ,y) = p(x, y 0 ) (y − y 0 ) exp (−i2 x) dx dy 0
−∞ −∞
∞

= p(x, y) exp (−i2 x) dx (5.6)
−∞
It is straightforward to extend the above results to similar cases. For
example, if we add a spherical lens to the anamorphic processors
in Fig. 5.3b, we find that the complex amplitude distribution of the
Fraunhofer diffraction pattern of A( ,y)is W(x, ). That is,
∞
∞
W(x, ) = A( ,y) exp [i2 (x − y)] d dy (5.7)
−∞ −∞
The above results are summarized pictorially in Fig. 5.4. This type
of diagram was introduced, by Brenner and Ojeda-Castaneda, 10 as a

y

x x x m
x
y n y y

p(x, y)
n y

x W(x, n) A(m, y) m

P(m, n)
n
m x m m
n n n y
m

FIGURE 5.4 Phase-space representation: a roadmap for coherent
illumination.