Page 298 - Phase Space Optics Fundamentals and Applications

CHAPTER9

Self-Imaging in Phase

Space

Markus E. Testorf

Dartmouth College, Hanover, New Hampshire, U.S.A

9.1 Introduction
There is little doubt that Fourier optics has shaped optical engineering
in ways only comparable to geometrical optics. Understanding wave-
fronts and optical hardware in terms of linear system theory has been
pivotal to integrating optical sciences with signal processing and nu-
merical computing. Topics such as diffractive optics design and com-
putational imaging are almost unimaginable without the theoretical
foundations of Fourier mathematics.
An emerging and fascinating alternative to Fourier optics is phase-
space optics. Forged by the marriage of joint time-frequency analysis
and the phase-space formalism of quantum mechanics, phase-space
optics is a platform for describing ray optics, radiometry, coherent
Fourier optics, and coherence theory with a single consistent frame-
work.
The phase-space interpretation is often perceived as a highly math-
ematical exercise with little additional information to complement the
standard treatment in terms of Fourier optics. This perception is per-
haps justified when looking at the mathematical properties of basic
phase-space tools, namely, the Wigner distribution function (WDF).
The WDF “inflates” the complex amplitude into an apparently redun-
dant multidimensional function with transverse spatial position and
spatial frequency as independent variables. In addition, the WDF is a
bilinear transformation, and hence the linearity of signal superposi-
tion is lost.

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