Page 97 - Phase Space Optics Fundamentals and Applications

78 Chapter Three

t
U
function (det W) 1/2 f (Wr i ), R [(det W) 1/2 f (Wr i )](r o ) with W = W ,
. T
corresponds to the CT of f (r i ) itself, R [ f (r i )](r o ), parameterized by
the matrix

−1
−1
X Y W 0 XW YW
. = −1 (3.55)
T =
−Y X 0 W −YW XW
The scaling theorems for

w x 0
W = (3.56)
0 w y
can be formulated for important RCTs—signal rotator, fractional FT,
and gyrator transforms, relatively—as follows. 4, 29, 30

U r ( )
R [ f (Wr i )](r o ) = f (X r (− )Wr o )
U f ( x , y )
R [ f (Wr i )](r o )
$ %
1/2 2

cos x 2 cos x
= exp i x o 1 − cot x
cos x cos x
$ %
1/2 2
cos y 2 cos y
× exp i y o 1 − cot y
cos y cos y

U f ( x , y ) cos x cos y
× R [ f (r i )] w x x o , w y y o
cos x cos y
U g (ϑ)
R [ f (Wr i )](r o )

cos 2
cos
exp i2 x o y o 1 − cot ϑ
=
cos ϑ cos ϑ
cos
U g ( )
× R [ f (r i )] Wr o (3.57)
cos ϑ
where cot x,y = w 2 x,y , cot x,y , and cot ϑ = w x w y cot . Note that if
w x = w −1 = w, then R U g (ϑ) [ f (Wr i )](r o ) = R U g (ϑ) [ f (r i )](Wr o ).
y
The scaling property for the fractional FT has been used for the
analysis of fractal signals. 38
3.4.6 Phase-Space Rotations of Selected
Functions
Phase-space rotation of only a limited number of functions can be
expressed analytically. Among them there is the function

t t
f i (r) = exp i2 k r − r L i r (3.58)
i