Page 160 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 141

numerically (parts a and c) and experimentally (parts b and d) ob-
tained, for two different values of the obscuration b.
According to our previous discussion, the slices of the RWD for
x = 0 give rise to the SR for variable W 20 . These profiles are plotted
in Fig. 4.20 for three different pupils. From these results, it can be
observed that, as expected, annular apertures have higher tolerance
to defocus.
The knowledge of the SR is useful to characterize some basic fea-
tures of any optical system, such as the depth of focus. However, the
main shortcoming of the SR as a method of image assessment is that
although it is relatively easy to calculate for an optical design pre-
scription, it is normally difficult to measure for a real optical system.
Moreover, the quality of the image itself is better described through
the associated OTF. Fortunately, this information can also be obtained
from the RWD via its relationship with the AF established in Sec. 4.2.1,
since the AF contains all the OTFs H( ; W 20 ) associated with the opti-
cal system with varying focus errors according to the formula 42

2W 20 ( f + z)
H ( ; W 20 ) = A t − ( f + z) , 2 (4.86)
h
In this way, the AF of the pupil function t(x) can be interpreted as a
continuous polar display of the defocused OTFs of the system. Con-
versely,

2
x h
A t x , = H − ; W 20 =− (4.87)

( f + z) 2x

b = 0 mm
b = 0 mm
b = 1.3 mm b = 1.3 mm
1 b = 2 mm 1 b = 2 mm
S(z) S(z)

0 0
0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00
Fractional order p Fractional order p
(a) (b)

FIGURE 4.20 SR versus defocus for circular pupils with pupil function
t(x) = rect(x/h) − rect(x/b): (a) Computer simulation; (b) experimental
results. Again, the projection angle = p /2.