Page 59 - Phase Space Optics Fundamentals and Applications

40 Chapter One

1.9 Conclusion
WehavepresentedanoverviewoftheWignerdistributionandofsome
of its properties and applications in an optical context. The Wigner
distribution describes a signal in space (i.e., position) and spatial fre-
quency (i.e., direction) simultaneously and can be considered as the
local frequency spectrum of the signal, like the score in music and the
phase space in mechanics. Although it is derived in terms of Fourier
optics, the description of a signal by means of its Wigner distribution
closelyresemblestherayconceptingeometricaloptics.Itthuspresents
a link between Fourier optics and geometrical optics. Moreover, the
concept of the Wigner distribution is not restricted to deterministic
signals (i.e., completely coherent light); it can be applied to stochastic
signals (i.e., partially coherent light) as well, thus presenting a link
between partial coherence and radiometry.
Properties of the Wigner distribution and its propagation through
linear systems have been considered; the corresponding description
of signals and systems can be directly interpreted in geometric-optical
terms. For first-order optical systems, the propagation of the Wigner
distribution is completely determined by the system’s ray transfor-
mation matrix, thus presenting a strong interconnection with matrix
optics.
We have studied the second-order moments of the Wigner distribu-
tion and some interesting combinations of these moments, together
with the propagation of these moment combinations through first-
order optical systems. Special attention has been paid to systems that
perform rotations in phase space.
In the case of completely coherent light, the Wigner distribution is
a member of a broad class of bilinear signal representations, known
as the Cohen class. Each member of this class is related to the Wigner
distribution by means of a convolution with a certain kernel. Because
of the quadratic nature of such signal representations, they suffer from
unwanted cross-terms, which one tries to minimize by a proper choice
of this kernel. Some members of the Cohen class have been reviewed,
and special attention was devoted to the smoothed interferogram
in combination with the optimal angle in phase space in which the
smoothing takes place.

References
1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys.
Rev. 40, 749–759 (1932).
2. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv.
Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).