Page 106 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 87

A generalized lens constructed from two convergent cylindrical
lenses of the same power p rotated at angles 1 and 2 is character-
ized by

2 2 1
cos 1 + cos 2 − [sin(2 1 ) + sin(2 2 )]
2
G = p 1 2 2 (3.78)
− [sin(2 1 ) + sin(2 2 )] sin 1 + sin 2
2
which reduces to

1 − sin(2 1 )
G = p (3.79)
− sin(2 1 ) 1
for 2 =− 1 ± /2.ComparingEqs.(3.77)and(3.79),weconcludethat
in the gyrator setup every generalized lens L j ( j = 1, 2) is a combi-
nation of two convergent cylindrical lenses of equal focal distance z/j
rotated counterclockwise at angles ( j) = ( j) and ( j) =− ( j) ± /2
1 2
with respect to the OX axis. The gyrator at angle ϑ is achieved if
(2)
(1)
sin(2 ) = cot(ϑ/2) and 2 sin(2 ) = sin ϑ. We observe that this
setup is able to perform the gyrator for the angles from the in-
terval [ /2, 3 /2]. The experimental implementation of this optical
system has been demonstrated in Refs. 31 and 33 on the example
of orthosymplectic mode conversion. The experimental results are in
good agreement with theoretic predictions.
( j) ( j)
If the angles in Eq. (3.78) are chosen as 1 = + /4 and
( j) = ( j) − /4, we obtain the generalized lenses suitable for the
2
antisymmetric fractional FT setup.
( j)
j 1 − sin(2 ) 0
G j = ( j) (3.80)
z 0 1 + sin(2 )
Indeed, comparing Eqs. (3.76) and (3.80), we observe that these
lens combinations perform the antisymmetric fractional FT at angle
(1)
( , − ), where 2 sin(2 ) = cot( /2) and 2 (2) = . It is easy to see
from the last relation that this setup is able to perform the antisymmet-
ric fractional FT for the angles ∈ [ /2, 3 /2] that cover a interval
needed for the different applications, discussed in Sec. 3.7.

3.6.2 Flexible Optical Setup for Image
Rotator
Usually, Dove prisms are used for optical signal rotator realization.
But the diffraction effects during the propagation through the prisms
require additional optical elements for their compensation. Here we
consider the optical signal rotator based on the application of cylin-
drical lenses. 21, 22, 55
A flexible optical scheme performing a rotation at angle by only
the appropriate rotation of cylindrical lenses composing the setup has