Page 96 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 77

3.4.3 Shift Theorem
A shift of the input function by a vector v, f i (r) → f i (r − v), leads
to a shift of the output signal by the vector Xv and to an additional
quadratic phase factor

U
t
U
R [ f i (r i −v)](r o ) = exp[−i (2r o −Xv) Yv] R [ f i (r i )](r o −Xv) (3.49)
where we have used the symplecticity conditions [Eq. (3.19)] and the
t
t
t
fact that v Zq = q Z v. This implies that the squared modulus of the
RCT, associated in optics with intensity distribution, does not change
due to a displacement by v, but is merely shifted by Xv:
U 2 2
R [ f (r i − v)](r o ) = |F U (r o − Xv)| (3.50)

U
Equation (3.49) reduces to R [ f (r i − v)](r o ) = F U (r o − Xv) and to
U
t
R [ f (r i − v)](r o ) = exp(−i 2r Yv) F U (r o ) for Y = 0 and X = 0,
o
respectively. The shift theorem underlines the position-variant nature
of signal processing in the phase-space domains if X = 0.
3.4.4 Convolution Theorem
Using the shift theorem, the RCT of the convolution between f and h

∞ ∞
C f,h (r) = ( f ∗ h)(r) = f (r − v)h(v) dv = h(r − v) f (v) dv
−∞ −∞
(3.51)
can be written in the form

∞
U t
R [( f ∗ h)(r i )] (r o ) = exp[−i (2r o −Xv) Yv] F U (r o −Xv) h(v) dv
−∞
(3.52)
t
−1
In the case where X = 0 (and thus also Y = Y ), it reduces to
U
R [( f ∗ h)(r i )] (r o ) = (det iY) 1/2 F U (r o ) H U (r o ) (3.53)
For imaging-type systems Y = 0, we have

∞
U
R [( f ∗ h)(r i )] (r o ) = F U (r o − Xv) h(v) dv (3.54)
−∞
3.4.5 Scaling Theorem
The scaling, as we discussed above, belongs to the class of CTs. There-
fore, as it follows from the additivity property of the CTs, the scaling
of the input function leads to a change of the parameterizing matrix.
Thus the phase-space rotation associated with matrix U of the scaled