Page 86 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 67

The separable fractional FT [ray transformation matrix T f ( x , y )]
described by the unitary matrix

exp(i x ) 0
U f ( x , y ) = (3.15)
0 exp(i y )

corresponds to rotations in the xp x and yp y planes through angles x
and y , respectively.
The gyrator transform associated with T g (ϑ)or

cos ϑ i sin ϑ
U g (ϑ) = (3.16)
i sin ϑ cos ϑ
produces twisting, i.e., rotations in the mixed xp y and yp x planes of
phase space.
23
It has been shown that any orthosymplectic matrix can be decom-
posed in the form
T O = T r ( ) T f ( x , y ) T r ( ) (3.17)

It means that R T O is a separable fractional Fourier transformer R T f
embedded between two rotators R . In particular for the gyrator
T r
matrix, we obtain T g (ϑ) = T r (− /4) T f (ϑ, −ϑ) T r ( /4).
Based on the modified Iwasawa decomposition Eq. (3.10) and
Eq. (3.17), we can write a general representation of the CT, which
is valid for any ray transformation matrix, including a singular sub-
matrix B, det B = 0. 23
t
T
f o (r o ) = R [ f i (r i )] (r o ) = (det S) −1/2 exp(−i r Gr o )
o
T f ( x , y ) −1
× R [ f i (X r ( ) r i )] X r (− )S r o (3.18)

3.3 Canonical Transformations Producing
Phase-Space Rotations
3.3.1 Matrix and Operator Description
The ray transformation matrix T O , which describes the phase-space
rotations, is symplectic, Eq. (3.8),
t
t
t
XY = YX t XX + YY = I
(3.19)
t
t
t
t
X Y = Y X X X + Y Y = I
t
and orthogonal, T O = (T ) −1 , and therefore it has only four free
O
parameters.