Page 31 - Phase Space Optics Fundamentals and Applications

12 Chapter One

diagonal matrix with positive diagonal entries

w x 0
W = (1.23)
0 w y
In subsequent sections, we will often work with these normalized
coordinates; it will be clear from the context whether normalization is
necessary.

1.4 Some Properties of the Wigner
Distribution
Let us consider some of the important properties of the Wigner dis-
tribution. We consider in particular properties that are specific for
partially coherent light. Additional properties of the Wigner distribu-
tion, especially of the Wigner distribution in the completely coherent
case, can be found elsewhere; see, for instance, Refs. 29 to 40 and the
many references cited therein.

1.4.1 Inversion Formula
The definition (1.14) of the Wigner distribution W(r, q) has the form
1
of a Fourier transformation of the cross-spectral density (r + r ,
2
r− r ) with r and q as conjugated variables and with r as a parameter.
1

2
The cross-spectral density can thus be reconstructed from the Wigner
distribution simply by applying an inverse Fourier transformation.
1.4.2 Shift Covariance
The Wigner distribution satisfies the important property of space and
frequency shift covariance: if W(r, q) is the Wigner distribution that
corresponds to (r 1 , r 2 ), then W(r − r o , q − q ) is the Wigner distri-
o
bution that corresponds to the space- and frequency-shifted version
t
(r 1 − r o , r 2 − r o ) exp[i2 q (r 1 − r 2 )].
o
1.4.3 Radiometric Quantities
Although the Wigner distribution is real, it is not necessarily non-
negative; this prohibits a direct interpretation of the Wigner distribu-
tion as an energy density function (or radiance function). Friberg has
shown 41 that it is not possible to define a radiance function that sat-
isfies all the physical requirements from radiometry; in particular, as
we mentioned, the Wigner distribution has the physically unattractive
property that it may take negative values.
Nevertheless, several integrals of the Wigner distribution have clear
physical meanings and can be interpreted as radiometric quantities.