Page 191 - Phase Space Optics Fundamentals and Applications

172 Chapter Five

n
A(m, y)
W(x, n)
A (m, y)
0
y
x
n
0
m
(x ,n )
W 0 0 0
y 0
x 0
p (x ,n )
0 0 0
f
x 0
f T(m, y)
f
f
f

FIGURE 5.5 Optical setup for processing the Wigner distribution function.

If the complex amplitude transmittance of the pupil mask is T( ,y),
then just after the mask the complex amplitude distribution is the AF
A( ,y) = T( ,y)A 0 ( ,y). The above result can also be rephrased as
follows. If one uses a pupil mask with complex amplitude transmit-
tance T( ,y), then one generates the impulse response t(x, ), which
implements the mapping

∞
∞
W(x, ) = W 0 (x 0 , 0 )t(x − x 0 , − 0 ) dx 0 d 0 (5.16)
−∞ −∞

By comparison of Eqs. (5.14) and (5.16) we know that the impulse
response of the equivalent processor is W b (x, ; x 0 , 0 ). For an ideal
system, the impulse response is W b (x, ; x 0 , 0 ) = (x − x 0 ) ( − 0 ).
This WDF cannot be obtained from a product-space representation.
Hence, we comment on the two following approaches.
On the one hand, it is possible to use masks that can be expressed in
terms of the product spectrum representation, say, T( ,y) = P(y/ f, ).
And consequently, the impulse response is expressed in terms of the
product-space representation. And in this manner, the equivalent op-
tical processor effectively implements Eq. (5.9).
On the other hand, one can use masks that cannot be expressed
in terms of the product spectrum representation for implementing
optically nonconventional transformations in phase-space. For exam-
ple, if at the Fraunhofer plane of the equivalent processor one sets