Page 98 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 79

where L i is a symmetric matrix with nonnegative definite real part
and k i is a real vector. Following the calculations done in Refs. 7 and
42, one can find that the RCT of the function (3.58) takes the form

U
f o (r) = R [ f i (r i )](r) = [det (X + iYL i )] −1/2
t −1 t t
× exp −i k (X + iYL i ) Yk i + i2 k r − r L o r (3.59)
i o
t
t
where k = k (X + iYL i ) −1 and iL o = (−Y + iXL i ) (X + iYL i ) −1 .
o i
If L i =−iH i is imaginary, then L o =−iH o =−i(−Y + XH i )×
(X + YH i ) −1 is imaginary, too, which implies that f i (r i ) [Eq. (3.58)]
and f o (r) [Eq. (3.59)] are the generalized chirp functions, which in-
clude as particular cases the plane, elliptic, hyperbolic, and parabolic
waves.
t
For k i = 0, and thus f i (r) = exp(− r L i r), Eq. (3.59) reduces to

t
f o (r) = [det (X + iYL i )] −1/2 exp − r L o r (3.60)
2
2
A Gaussian beam exp[− (l 11 x + 2l 12 xy + l 22 y )] appears when L i is
real and positive definite.
t
For plane wave f i (r) = exp(i2 k r)(L i = 0) we obtain
i

t
−1
t
−1
−1
t
f o (r) = (det X) −1/2 exp −i k X Yk i + i2 k X r − i r YX r
i i
(3.61)
Then a plane wave remains a plane wave only under the imaging-type
phase-space rotations (Y = 0).
Equation (3.61) can be used for the calculation of the phase-space ro-
tations of periodic functions. Thus by representing a periodic function
f i (r) with periods p x and p y with respect to the x and y coordinates
as a superposition of plane waves,
∞
,
f i (r) = a mn exp (i2 k t mn r) (3.62)
m,n=−∞
with k t = (m/p x ,n/p y ) and using Eq. (3.61 ), we get after the RCT
mn
t
−1
f o (r) = (det X) −1/2 exp(−i r YX r)
∞
, t −1 t −1
× a mn exp −i k X Yk mn + i2 k X r (3.63)
mn mn
m,n=−∞
−1
If k t X Yk mn = j, where j is an even integer, then the generalized
mn
7
Talbot imaging is obtained

t
−1
−1
f o (r) = (det X) −1/2 exp −i r YX r f i (X r) (3.64)