Page 190 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 171

We note that several interesting features are apparent from Eq. (5.11).
First, independently of the value of x, the rays coming from the edge of
the pupil (marginal rays) can be considered as having zero amplitude,
W q (x, ) = 0. Second, in a lax manner, one can consider that all the
rayscomingtotheopticalaxis(x = 0)addconstructively.However,the
amplitude of the rays decreases as 4( −| |). Third, in a lax manner,
one can consider that outside the optical axis some of the rays add
destructively. Specifically, at the point x = 1/2 , the amplitude of the
rays varies as (2 / ) sin[2 (1 −| / |)].
According to Ref. 11, under the influence of wave aberrations, the
WDF of the diffraction-limited system W 0 (x, ) changes as follows:

[M/2] 2m+1
, ∂ d
W(x, ) = exp − C m ( ) 2m+1 W 0 x − f ,
∂x d
m=1
(5.12)
In Eq. (5.12) we employ the letter f for denoting the focal length of
the optical processor; ( ) denotes the wave aberration polynomial;
[M/2] = M/2 − 1if M is an even integer number; and [M/2] =
(M− 1)/2if M is an odd integer number. In the differential opera-
tor, the coefficients are

f 2m+1 d 2m+1
(5.13)
C m ( ) = 2m 2m+1
4 d
(2m + 1)!
It is apparent from Eq. (5.12) that the WDF W 0 (x, ) suffers from lateral
displacements, which are predicted by geometrical optics. In addition,
the WDF modifies its amplitude distribution, as predicted by wave
optics. 11
Next we discuss the use of an equivalent optical processor, as de-
picted in Fig. 5.5, for implementing filtering operations in the phase-
space representation. To that end, we represent the equivalent optical
image processor, by the superposition integral
∞
∞
W (x, ) = W 0 (x 0 , 0 )W b (x, ; x 0 , 0 ) dx 0 d 0 (5.14)
−∞ −∞
The impulse response W b (x, ; x 0 , 0 ) is the Bastiaans 12 double WDF.
By simple comparison between Eqs. (5.9) and (5.14), one finds that for
a space-invariant imaging system
W b (x, ; x 0 , 0 ) = W q (x − x 0 , 0 ) ( − 0 ) (5.15)

Next, at the pupil aperture of the equivalent optical processor,
we note that the complex amplitude distribution is the AF A 0 ( ,y).