Page 202 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 183

that exhibits rotational symmetry. That is,
$ %

3

2 1
T( ) = exp i2
− circ (5.49)
2
If you will, in Eq. (5.49), the physical pupil mask has an annularly
distributed phase profile, which has odd phase variation provided
√
that (on the pupil aperture) we take the circle with radius = / 2
as the center of symmetry. The ambiguity function of the effective
pupil mask exhibits the bow tie effect. And consequently, the optical
system has low sensitivity to spherical aberration; see Ref. 33. Yet, the
ambiguity function of the effective pupil is not the OTF of the physical
mask.
Related to the previous discussion, we emphasize the following.
There is a difference between the use of radially symmetric phase
masks (axiconlike elements) for generating large axial irradiance
distributions 51−57 and the use of annularly distributed odd phase
masks for reducing the impact of aberrations on the MTF, as in Ref. 58.
So it may be useful to use the term high focal depth for describing axi-
conlike elements and the term extended depth of field for describing
optical elements that reduce the impact of focus error on the MTF.

5.7 Phase Conjugate Plates
An optical system with variable focal length is commonly desig-
nated as a varifocal, or zoom, system. For tuning the focal length,
one changes the longitudinalseparation between twoquadratic-phase
59
components,whichusuallyhaveopposite-signpowers.Alvarez and
Lohmann 60−62 independently proposed a varifocal system that con-
sists of a pair of cubic phase elements, which are laterally displaced.
This type of optical device is also useful for generating, in a tunable
fashion, wavefront aberrations. 63,64
In Fig. 5.10 we depict schematically the use of a pair of phase ele-
ments with opposite-sign powers. We assume that the pair is located
at the pupil aperture (Fourier domain) of a 4 f optical processor. The
pupil aperture has a rectangular shape. We use the Greek letters and
to represent the spatial frequencies along the horizontal axis and
the vertical axis, respectively, in the pupil aperture. For any phase
element, the cutoff spatial frequency is . The complex amplitude
transmittance of a single-phase plate is

Q( , ) = T( ) rect = exp [i
( )] rect rect
2 2 2
(5.50)