Page 206 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 187

from focus error, and then the generalized pupil function is

3
3
S( ; ; W 2,0 ) = exp i2 (4
) exp i2

W 2,0 2

× rect exp i2 (5.61)
2 −| |
From Eq. (5.61) we find that, except for a normalization factor, the
MTF for the pair of quartic phase conjugates is
|H( ; W 2,0 ; )|

∞

12
2 W 2,0
= exp i2 + 2

2

−∞

× rect d (5.62)
2 −| |
From Eq. (5.62) we can see that again the integral for evaluating the
MTF is a Fresnel integral, as in the case of the wavefront coding tech-
nique. However, the phase “strength” of the element is now propor-
tional to the product
. Thus, by changing the lateral displacement
between the phase elements, we can choose the strength of the cubic
phase term of the wavefront coding technique.
In Fig. 5.11 we show four graphs of the MTF versus / and W 2,0 /
for lateral displacements / = 0.00, 0.02, 0.06, and 0.30, with
= 12.
It is apparent from Fig. 5.11 that the MTF has low sensitivity to focus
errors W 2,0 / with increasing values of the plates’ lateral displacement
/ .
Summarizing, we described the use of the optical anamorphic pro-
cessor for relating the product-space representation, the WDF, the AF,
and the product spectrum representation. We indicated that by adding
a rotating ground glass, for reducing the degree of spatial coherence,
the anamorphic processors are useful for linking the mutual inten-
sity with the WDF, the AF, and the mutual spectrum. We reported
two road maps for visualizing the basic integral transformations for
phase-space representations.
We explored the use of the WDF for linking geometrical optics and
wave optics. We discussed a method for analyzing the impact of wave
aberrations from the viewpoint of the WDF. Then we related bilinear
transformations with phase-space representations. We have shown
that for space-invariant systems, the Volterra kernel is the product-
space representation of the coherent impulse response.
We revisited the link between the OTF of an optical system that
suffers from focus errors and the AF of the pupil mask. We employed