Page 20 - Phase Space Optics Fundamentals and Applications

CHAPTER1

Wigner Distribution

in Optics

Martin J. Bastiaans
Technische Universiteit Eindhoven, Faculteit Elektrotechniek
Eindhoven, Netherlands

1.1 Introduction
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In 1932 Wigner introduced a distribution function in mechanics that
permitted a description of mechanical phenomena in a phase space.
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Such a Wigner distribution was introduced in optics by Dolin and
Walther 3,4 in the 1960s, to relate partial coherence to radiometry. A few
years later, the Wigner distribution was introduced in optics again 5–11
(especiallyintheareaofFourieroptics),andsincethen,agreatnumber
of applications of the Wigner distribution have been reported.
While the mechanical phase space is connected to classical mechan-
ics, where the movement of particles is studied, the phase space in
optics is connected to geometrical optics, where the propagation of
optical rays is considered. Whereas the position and momentum of a
particle are the two important quantities in mechanics, in optics we are
interested in the position and the direction of an optical ray. We will
see that the Wigner distribution represents an optical field in terms of
a ray picture, and that this representation is independent of whether
the light is partially coherent or completely coherent.
We will observe that a description by means of a Wigner distribu-
tion is, in particular, useful when the optical signals and systems can
be described by quadratic-phase functions, i.e., when we are in the
realm of first-order optics: spherical waves, thin lenses, sections of
free space in the paraxial approximation, etc. Although formulated

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