Page 82 - Phase Space Optics Fundamentals and Applications

CHAPTER3

Rotations in Phase

Space

Tatiana Alieva
Universidad Complutense de Madrid, Facultad de Ciencias F´ ısicas
Ciudad Universitaria s/n, Madrid, Spain

3.1 Introduction
The canonical (integral linear) transforms (CTs) are widely used in sig-
nal and image processing, optics, quantum mechanics, etc. 1−8 The
CTs are related to the affine transformations of the Wigner distribu-
tion, discussed in Chap. 1. The affine transformations in phase space,
defined by the position and spatial frequency (momentum) coordi-
nates, include scaling, shearing, rotation, etc. As we will see below, it
is a rotation that plays an important role for different applications in
information acquisition and processing, beam characterization, etc.
A well-known phase-space rotator is the Fourier transform (FT),
which produces arotation in the position(time)–spatial(temporal)fre-
quency plane of /2. The FT together with closely related convolution
and correlation operations 9,10 forms the basis for information process-
ing. The fractionalization of the FT 11−14 has opened new perspectives
in this field. Thus the fractional Fourier transform, which produces
the rotation in the position-frequency plane at arbitrary angle, has
been used for shift-variant filtering, noise reduction, chirp localiza-
tion, encryption, phase retrieval, etc. 4,6,15−19 The fractional FT is the
only possible phase-space rotator for one-dimensional signals. If the
dimension of a signal is larger than 1, there exist other phase-space
rotators.
In coherent optics the FT of a two-dimensional signal, associated
with complex field amplitude, can be performed by application of a

63