Page 21 - Phase Space Optics Fundamentals and Applications

2 Chapter One

in Fourier-optical terms, the Wigner distribution will form a link to
such diverse fields as geometrical optics, ray optics, matrix optics, and
radiometry.
Sections 1.2 through 1.7 mainly deal with optical signals and sys-
tems. We treat the description of completely coherent and partially
coherent light fields in Sec. 1.2. The Wigner distribution is introduced
in Sec. 1.3 and elucidated with some optical examples. Properties of
the Wigner distribution are considered in Sec. 1.4. In Sec. 1.5 we restrict
ourselves to the one-dimensional case and observe the strong connec-
tion of the Wigner distribution to the fractional Fourier transformation
and rotations in phase space. The propagation of the Wigner distri-
bution through Luneburg’s first-order optical systems is the topic of
Sec. 1.6, while the propagation of its moments is discussed in Sec. 1.7.
The final Sec. 1.8 is devoted to the broad class of bilinear signal repre-
sentations known as the Cohen class, of which the Wigner distribution
is an important representative.

1.2 Elementary Description of Optical
Signals and Systems
We consider scalar optical signals, which can be described by, say,
˜
f (x, y, z, t), where x, y, z denote space variables and t represents the
time variable. Very often we consider signals in a plane z = constant,
in which case we can omit the longitudinal space variable z from the
formulas. Furthermore, the transverse space variables x and y are
combined into a two-dimensional column vector r. The signals with
˜
which we are dealing are thus described by a function f (r,t).
Although real-world signals are real, we will not consider these
˜
signals as such. The signals f (r,t) that we consider in this chapter are
analytic signals, and our real-world signals follow as the real part of
these analytic signals.
Throughout we denote column vectors by boldface lowercase sym-
bols, while matrices are denoted by boldface uppercase symbols;
transposition of vectors and matrices is denoted by the superscript t.
Hence, for instance, the two-dimensional column vectors r and q rep-
t
t
resent the space and spatial-frequency variables [x, y] and [u, v] ,
t
respectively, and q r represents the inner product ux + vy. Moreover,
in integral expressions, dr and dq are shorthand notations for dx dy
and du dv, respectively.