Page 127 - Phase Space Optics Fundamentals and Applications

108 Chapter Four

closely related to the projections of the WDF in phase space and also
intimately connected with AF, as will be shown.
The general structure of this chapter is as follows. In Sec. 4.2, a gen-
eral overview of mathematical properties of the RWT is given, and
a summary of different optical setups for achieving it is presented.
Next, the use of this representation in the analysis of optical signals
and systems is developed in several aspects, namely, the computation
of diffraction intensities, the optical display of Fresnel patterns, the
amplitude and phase reconstruction of optical fields, and the calcula-
tion of merit function in imaging systems. Finally, in Sec. 4.4, a review
of design techniques, based on the utilization of the RWT, for these
imaging systems is presented, along with some techniques for optical
signal processing.

4.2 Projections of the Wigner Distribution
Function in Phase Space: The
Radon-Wigner Transform (RWT)
The RWT was first introduced for the analysis and synthesis of
frequency-modulated time signals, and it is a relatively new formal-
ism in optics. 1,2 However, it has found several applications in this field
during the last years. Many of them, such as the analysis of diffrac-
tion patterns, the computation of merit functions of optical systems, or
the tomographic reconstruction of optical fields, are discussed in this
chapter. We start by presenting the definition and some basic proper-
ties of the RWT. The optical implementation of the RWT which is the
basis for many of the applications is discussed next.
Note, as a general remark, that for the sake of simplicity most of
the formal definitions for the signals used hereafter are restricted to
one-dimensional signals, that is, functions of a single variable f (x).
This is mainly justified by the specific use of these properties that we
present in this chapter. The generalization to more than one variable is
in most cases straightforward. We will refer to the dual variables x and
as spatial and spatial-frequency coordinates, since we will deal mainly
with signals varying on space. Of course, if the signal is a function of
time instead of space, the terms time and frequency should be applied.

4.2.1 Definition and Basic Properties
We start this section by recalling the definition of the WDF associated
with a complex function f (x), namely,

W{ f (x),x, }= W f (x, )

+∞
x x

= f x + f ∗ x − exp (−i2 x ) dx (4.1)
2 2
−∞