Page 92 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 73

rotators 23 that can be expressed in matrix form as
U = exp(i
) U r ( ) U f ( , − ) U r ( ) (3.32)

where the parameters
, , , and are defined from the components
of the matrix U = X + iY by the following relations:
det U = exp(i2
)
det X + det Y = cos 2 (3.33)

and

X 11 + X 22 − Y 12 + Y 21 = 2 cos( + +
) cos
X 12 − X 21 + Y 11 + Y 22 = 2 sin( + +
) cos
−X 11 + X 22 + Y 12 + Y 21 = 2 sin( − +
) sin
X 12 + X 21 + Y 11 − Y 22 = 2 cos( − +
) sin (3.34)

Notethatdet X = cos(
+ ) cos(
− ) anddet Y = sin(
+ ) sin(
− ).
As we will see below, the fractional FT is widely used in signal and
image processing, phase retrieval, tomographic reconstruction of the
Wigner distribution, etc. More information about the fractional FT can
be found in Refs. 4 to 6, 13, 28, and 29.

3.3.4 Gyrator
As well as the signal rotator, symmetric and antisymmetric fractional
FTs, the gyrator defined by the unitary matrix U g (ϑ) with determi-
nant equal to 1 [see Eq. (3.16)] also forms a uniparametric group of
transformations. The kernel of the gyrator transform at angle ϑ has a
form of a hyperbolic wave

1 (x o y o + x i y i ) cos ϑ − (x i y o + x o y i )
U g (ϑ)
K (r i , r o ) = exp i2
|sin ϑ| sin ϑ
(3.35)
which reduces to (r i −r o ) for ϑ = 0, to (r i +r o ) for ϑ = , and to the
twisted FT kernel exp[∓i2 (x i y o + x o y i )] for ϑ =± /2. It is periodic
with 2 . The inverse transform is the gyrator at angle −ϑ. The gyrator
produces rotations in the twisted xp y and yp x planes of phase space
at angle ϑ.
The gyrator plays an important role in two-dimensional signal pro-
cessing, orbital angular momentum manipulation, and beam con-
version. Thus by applying the gyrator transform at angle ± /4
to the properly normalized Hermite-Gaussian beam, the helicoidal