Page 100 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 81

From the fact that U f (
,
) commutes with any unitary matrix U
U
follows that the mode H m,n (r) is an eigenfunction for the symmetric
fractional FT with eigenvalue exp[−i(m + n+ 1)
]. Then the kernel of
the symmetric fractional FT, Eq. (3.27), can be alternatively presented
as a series of products of the orthosymplectic modes.

∞
, −1
K U f (
,
) (r i , r o ) = exp[−i(m + n + 1)
] H U (r 0 )H U (r i ) (3.66)
m,n
m,n
m,n=0
U
The orthosymplectic modes H m,n (r), modes obtained from the HG
ones H m,n (r) by the RCT associated with matrix U, have the following
generating function 40,42

2 t −1 t −1 √ t
exp −s U U s + 2s U r 2 − r r
∗
det U
∞ ∞ m+n 1/2
, , 2
U m n
= H (r) s s (3.67)
m,n m!n! x y
m=0 n=0
t
where s = (s x ,s y ). They can be expressed as 40,42
2
2
(−1) m+n exp[ (x + y )]
U
H m,n (r) = 1/2
2 m+n−1/2 ( m+n m!n! det U)
m
n
∂ ∂ ∂ ∂
× U ∗ + U ∗ U ∗ + U ∗
11 21 12 22
∂x ∂y ∂x ∂y
2
2
× exp[−2 (x + y )] (3.68)
where U jk ( j, k = 1, 2) are parameters of the unitary matrix U. Thus
for the separable fractional FT U = U f ( x , y ), this formula for any
angles x and y reduces to the HG functions up to the constant phase
1
1
exp[−i(m + ) x − i(n + ) y ].
2 2
The orthosymplectic modes satisfy the symmetry relations
U m+n U
H m,n (−r) = (−1) H m,n (r)
∗ −1
U U
H m,n (r) = H m,n (r) (3.69)
the derivative relations 39

t
∂ ∂ U √ √ U √ U t
, H m,n (r) = 2 U ∗ m H m−1,n (r), n H m,n−1 (r)
∂x ∂y
U t
−2 H m,n (r) [x, y] (3.70)