Page 84 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 65

The canonical integral transform associated with matrix T is repre-
T
sented by the operator R
T
f o (r o ) = R [ f i (r i )](r o ) = F T (r o ) (3.3)
In the often used case det B = 0, the CT takes the form of Collins’
integral 1

∞
T
f o (r o ) = R [ f i (r i )](r o ) = (det iB) −1/2 f i (r i )
−∞

t −1 t −1 t −1
× exp i r B Ar i − 2r B r o + r DB r o dr i (3.4)
i
o
i
The kernel is a two-dimensional generalized chirp function since its
phase is a polynomial of second degree of variables r i and r o . For
A = D = 0 and B =−C = I, with I throughout denoting the identity
matrix, we obtain, apart from a constant phase factor exp(−i /2), the
Fourier transform F[ f (r i )](r o )

∞
t
F[ f (r i )](r o ) = f (r i ) exp(−i2 r r i ) dr i (3.5)
o
−∞
knowninopticsasanangularspectrumofthecomplexfieldamplitude
f . The matrix parameters A = D = I, C = 0, and B = zI correspond
to the Fresnel transform

1 ∞ ! 2 "
f o (r o ) = f i (r i ) exp i (r i − r o ) dr i (3.6)
iz z
−∞
which describes the propagation of the paraxial beams in homoge-
neous medium.
The case B = 0 corresponds to the generalized imaging condition
−1
−1
t
f o (r) = (| det A|) −1/2 exp(i r CA r) f i (A r) (3.7)
which includes a possible scaling and rotation of the input function
accompanied by an additional phase modulation.
T T
The CT is a linear transform: R [ f (r i ) + g(r i )](r) = R [ f (r i )](r) +
T
T 2
R [g(r i )](r). It is additive in the sense that R R T 1 = R T 2 ×T 1 . The ray
transformation matrix T is symplectic [see also Eq. (1.41)]
t
t
t
t
AB = BA t CD = DC t AD − BC = I
t
t
t
t
t
t
A C = C A B D = D B A D − C B = I (3.8)
and therefore it has only 10 free parameters. The inverse transforma-
−1
tion is parametrized by the matrix T , which, since T is symplectic,