Page 107 - Phase Space Optics Fundamentals and Applications

88 Chapter Three

P input P out
L 1 L 2 L 3 L 4

Z Z Z Z

FIGURE 3.7 Scheme for the flexible setup performing the signal rotator.

been recently proposed. 54 It has been shown that four is a minimal
number of generalized lenses located in fixed positions needed to
preform the signal rotator. The optical scheme of the signal rotator at
angle is displayed in Fig. 3.7. If the distances between all elements
equal z, then the block matrices of the applied lenses are given by

1 3 + cos sin
G 1 ( ) = G 3 (− ) =
2z sin 5 − cos

4 00
G 2 ( ) =
z 01

2 1 + cos − sin
G 4 ( ) = (3.81)
z − sin 1 − cos
This scheme can be realized by implementation of analog generalized
lenses, which consist of three cylindrical lenses for L 1 and L 3 and one
cylindrical lens for L 2 and L 4 .
By using the optical setups performing the basic phase-space rota-
tions, other phase-space rotators can be constructed as their cascades.
Nevertheless there is no guarantee that the obtained setup is optimal.

3.7 Applications of Phase-Space Rotators

3.7.1 Generalized Convolution
As mentioned before, the well-known phase-space rotator—the
Fourier transform—plays a crucial role in signal and image process-
ing.Itformsabaseforshift-invariantfilteringwhichisusedforpattern
recognition, denoising, encryption, etc. Many good books are devoted
tothissubject;see,e.g.,Refs.9and10.Hereweconsidertheapplication