Page 87 - Phase Space Optics Fundamentals and Applications

68 Chapter Three

If a complex field amplitude f (r) is canonically transformed with
the matrix T, then its Fourier spectrum is canonically transformed
t −1
with (T ) . It is easy to see that for the case of phase-space rotators
both transforms coincide.
For the description of phase-space rotations we can use the uni-
tary matrix U = X + iY instead of the ray transformation matrix T O .
Indeed, by introducing the following complex vector q = r − ip,it
is easy to check that the ray transformation equation, expressed in
dimensionless variables, Eq. (3.2), for the case of orthogonal matrix

r o X Y r i
= (3.20)
p o −Y X p i
can be rewritten as

(r − ip) o = (X + iY)(r − ip) i (3.21)
orinmorecompactformasq o = Uq i .Thispresentationunderlinesthe
similarity in the description of phase-space rotators and polarization
rotators of the monochromatic paraxial beams defined by correspond-
ing Jones matrices. 10
Further, the RCT operator associated with unitary matrix U will
U
be denoted as R . As well as for the ray transformation matrix T,
the additivity of the phase-space rotators is expressed as R U 2 R U 1 =
R U 2 ×U 1 .
Since matrix U has four free parameters, there are four uniparamet-
ric groups of phase-space rotators: symmetric fractional FT, Eq. (3.15),
x = y ; antisymmetric fractional FT, Eq. (3.15), x =− y ; gyrator,
Eq. (3.16); and signal rotator, Eq. (3.14). These transforms are often
written in the form of Hermitian operators 20,24−26
! "
1 2 2 2 2
J 0 =
. . x + . y + / p x + / p y
4
! "
1 2 2 2 2
J 1 = . x − . y + / p x − / p y
.
4
1
J 2 =
. . x. y + / p x / p y
2
1
J 3 =
. . x/ p y − . y/ p x (3.22)
2
where . x and . y, / p x =−i∂/∂x, and / p y =−i∂/∂y are position and mo-
mentum operators. The operators J 0 , J 1 , J 2 , and J 3 are associated with
. . .
.
symmetric and antisymmetric fractional Fourier transforms, gyrator,
and rotator, respectively. Note that the operator J 0 commutes with all
.
others and that [J i ,J j ] = iε ijk J k , where i, j, k = 1, 2, 3 and ε ijk is the
. /
.
totally antisymmetric symbol, normalized through ε 123 = 1.