Page 197 - Phase Space Optics Fundamentals and Applications

178 Chapter Five

The zero-loci curves of Eq. (5.31) obey the relationship

n = 2 − W 2,0 (5.32)
2
where n = 1, 2, 3, 4, .... The zero-loci curves are useful for identifying
the points where the values of W 2,0 and have a MTF equal to zero,
and in this manner for setting tolerance values for the focus error, in
terms of W 2,0 . See Refs. 20 and 21. In Ref. 22, this approach is extended
to polychromatic illumination.

5.6 Tolerance to Focus Errors and
to Spherical Aberration
It is a misconception to assume that, under noncoherent illumination,
the phase distribution of the generalized pupil function does not in-
fluence the quality of an image. If the wave leaving the exit pupil has
departures from sphericity, the image quality is deteriorated. Hence,
one can expect that by modifying the phase of the generalized pupil
function, one can reduce the impact of aberrations on the MTF.
In other words, heuristically, one expects that by preventing an
image from becoming badly degraded (due to the presence of aber-
rations) its digitally restored version might have higher quality than
the restored picture of the nonpreventive image. 23−25 This approach
has found applications for extending the depth of field, of an optical
system, by using a suitable phase mask that preserves both lateral
resolution and light-gathering power.
There are several phase masks that are able to generate a MTF with
low sensitivity to focus error. 26−37 A suitable phase mask generates a
MTF that, inside its passband, does not have zero values for certain
amounts of focus error. However, inside its passband, the generated
MTF has reduced visibility.
Since one simultaneously records the images of planar scenes lo-
cated at different depths of the object field, by using a suitable phase
mask we ensure that each recorded image will suffer from virtually the
same amount of contrast reduction. For this reason, later on, by digital
processing, the image contrast can be simultaneously corrected for all
the recorded images. Next, we discuss a simple model for describing
this approach.
In Sec. 5.5 we indicate that for any pupil mask, its AF conveniently
contains all the defocused MTFs. Hence, an MTF with reduced sen-
sitivity to focus errors must be visualized as an AF with rotational
symmetry. If you will, the AF exhibits a “bow tie” effect, as depicted in
Fig. 5.8b. This type of AF was obtained early by using a parabolic FM,
in radar engineering. 38 However, for extending the depth of field, the