Page 330 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 311

this generalized sampling can be based entirely on the assumption of
a chirped signal.
It is also possible to deduce a more general interpolation formula.
This amounts to multiplying the signal samples by an appropriate
chirp function (the scale of the chirp is dependent on the LCT domain
in which the signal has finite support), followed by standard Shan-
non interpolation; and this is in turn followed by multiplying by the
conjugate of the fore-mentioned chirp. If a signal has finite support in
some LCT domain, the generalized sampling theorem predicts a finite
sampling rate, although this is associated with a signal that does not
exhibit a band limit in a classical sense. In this chapter we show how
phase-space diagrams allow us to understand and interpret general-
ized sampling in a most elegant manner as well as to calculate specifics
such as the most appropriate sampling rate for a given LCT bounded
signal. The generalized sampling theorem is of great importance for
digital holography. 22,34–42 It is of considerable interest to this research
area because it implies that one may place the object to be recorded
at a distance much closer to the camera than previously predicted by
the Nyquist-Shannon theorem. A shorter distance between object and
camera implies a greater numerical aperture, which in turn should
allow reconstruction of the object at a resolution previously thought
impossible and which is greater than the resolution of the recording
CCD. Furthermore the reduced camera-object distance implies that
a far greater range of three-dimensional perspectives may be recon-
structed as a result.
To the best of our knowledge, the Wigner distribution func-
tion (WDF), 27,43–47 was introduced to the optical community by
45
Bastiaans in 1979, and since then it has found application in describ-
ing numerous applications to which this book is a testament. The WDF
transforms a one-dimensional spatial signal into a two-dimensional
space–spatial frequency distribution. Besides being bilinear, the WDF
has a number of rich properties that are shared with both the spatial
representation of the signal and its Fourier transform. For example,
the integral projections of the WDF along the k and x axes yield the
space and frequency marginals, respectively. In the proceeding sec-
tion we review those properties of the WDF that are of interest in
the context of this chapter. One very useful method of visualizing the
WDF and conceptualizing its properties is by using Wigner charts or
phase-space diagrams 48–52 (PSDs). These PSDs were popularized by
Lohmann, Mendlovic, and Zalevsky in graphically describing signal
propagation through quadratic-phase systems (QPSs) and also super-
resolution systems. PSDs are plan view style diagrams of a signal’s
WDF. They do not include any information about the actual values
of the WDF other than the support in the xk plane. By endowing the
PSD with many of the properties of the WDF, such as the convolu-
tion property, we can conceptualize many optical processes. In some