Page 93 - Phase Space Optics Fundamentals and Applications

74 Chapter Three

Laguerre-Gaussian mode is obtained. A detailed analysis of the gyra-
tor can be found in Refs. 30 to 33.

3.3.5 Other Phase-Space Rotators
The four uniparametric transforms—signal rotator, symmetric frac-
tional FT, antisymmetric fractional FT, and gyrator—are additive with
respect to their angle parameters and form the basis for the presen-
tation of other phase-space rotators. Thus, for example, the cascade
of the reflector, Eq. (3.31), and signal rotator produces a reflector with
rotation described by the matrix

− cos sin
= (3.36)
sin cos
U 1 ( ) = U r ( )U re f x
This transformation is not additive with respect to parameter , be-
cause U 1 ( )U 1 ( ) = U r ( − ) = U 1 ( + ).
The combination of this transform at angle /2 and the fractional FT
leads to the phase-space rotator described by the antidiagonal unitary
matrix

0 exp(i x )
U 2 ( x , y ) = U 1 U f ( x , y ) = (3.37)
2 exp(i y ) 0
This RCT for x = y has been considered in Ref. 34. It is also not
additive with respect to the angles U 2 ( x , y )U 2 ( x , y ) = U f ( x + x ,
y + y ) = U 2 ( x + x , y + y ).
The cascades of signal rotators and reflector correspond to all pos-
sible phase-space rotators with Y = 0. Nevertheless there exist phase-
space rotators with det Y = 0, but Y = 0. Since det Y = sin x sin y ,
it is easy to see that in this case x = n x or/and y = n y and n x,y
are integers. It means that for one coordinate, the fractional Fourier
transformer in the decomposition Eq. (3.32) acts as an identity or
rotation system. As an example of such a system, we mention one
considered in Ref. 35 and described by the unitary matrix

cos sin
U = (3.38)
−i sin i cos

3.4 Properties of the Phase-Space Rotators
In this section we consider the basic properties of the RCTs that are
useful for the application of these transformations for the signal pro-
cessing tasks and for the description of the related first-order optical
systems.