Page 94 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 75

In the case where det Y = 0, the RCT parameterized by U of function
f i (r i ) takes the form

∞
U
f o (r o ) = R [ f i (r i )](r o ) = F U (r o ) = (det iY) −1/2 f i (r i )
−∞

t −1 t −1 t −1
× exp i r Y Xr i − 2r Y r o + r XY r o dr i (3.39)
o
i
i
For X=0 the kernel reduces to a plane wave function, and the trans-
form can be denoted as a Fourier type. As it follows from Eqs. (3.32),
X = 0 only if x,y = /2 + k x,y , where k x,y is an integer.
If det Y = 0, the decomposition Eq. (3.18), which is also valid for
the nonsingular case, has to be used
U U f ( x , y )
f o (r o ) = R [ f i (r i )](r o ) = R [ f i (X r ( ) r i )][X r (− )r o ] (3.40)
where the parameters x,y , , and are defined from Eqs. (3.33) and
−1
(3.34). In particular, if Y = 0, then f o (r) = f i (X r). Since | det U|=
| det X|= 1, these RCTs correspond to signal rotation or signal rotation
with reflection and may be denoted as imaging-type rotators.

3.4.1 Some Useful Relations for
Phase-Space Rotators
Based on the analysis of the canonical integral transform performed
in Ref. 7, it is easy to formulate the main theorems for the phase-space
rotators.
The complex conjugation of the RCT parameterized by U of f (r i )
is equivalent to the RCT parameterized by U = X − iY of f (r i ), that
∗
∗
U
∗
U
is, {R [ f (r i )](r)} = R [ f (r i )](r).
∗
∗
As shown in Ref. 7, the gradient of the RCT for det Y = 0 can be
written as
U
∇ o f o (r o ) =∇ o R [ f i (r i )](r o )
t
t −1
T
= i2 (Y ) X r o f o (r o ) − R [r i f i (r i )] (r o ) (3.41)
t
where ∇= (∂/∂x, ∂/∂y) and therefore
t
Y
U t
R [r i f i (r i )] (r o ) = X r o + i ∇ o f o (r o )
2

t
U
t
R [∇ i f i (r i )] (r o ) = i2 Y r o + X ∇ o f o (r o ) (3.42)
The well-known Parseval theorem holds for the entire class of the
CTs, and therefore for the phase-space rotators

∞ ∞
∗ U U ∗ ∗
f (r i ) g (r i ) dr i = R [ f (r i )](r o ) R [g (r i )](r o ) dr o (3.43)
−∞ −∞