Page 88 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 69

Due to these commutation relations and by analogy with spin an-
gular momentum, the operators J 1 , J 2 , and J 3 are often associated
.
. .
with orbital angular momentum (OAM) defined in phase space. Note
that only J 3 produces the rotation in configuration space (xy plane)
.
and relates to the beam OAM projection on the propagation direction.
Moreover, the OAM operators, as we will see below, provide an ele-
gant signal representation on the sphere, called, again by analogy with
polarization description, the orbital Poincar´ e sphere. This presentation
permits easy identification of the signal symmetry, its z-OAM projec-
tion, and defines the geometric phase accumulated by the Gaussian
beams during their propagation through the first-order optical sys-
tems, etc. Let us consider these basic transforms in detail.

3.3.2 Signal Rotator
The signal rotator transform associated with unitary matrix U r ( ),
Eq. (3.14); Y r =0, and

cos sin
X r = (3.23)
− sin cos
produces a clockwise rotation of f i in the xy plane and, correspond-
ingly, its FT (the angular spectrum) F i ( p x ,p y ) = F[ f (r i )](p i ) in the
p x p y plane at angle .

f o (x, y) = f i (x cos − y sin ,x sin + y cos )

F o ( p x ,p y ) = F i ( p x cos − p y sin ,p x sin + p y cos ) (3.24)

This transformation is additive with respect to angle parameter .
Thus R U r ( ) R U r ( ) = R U r ( + ) , and therefore the inverse transform for
R U r ( ) is a signal rotator at angle − . Note that det U = det X = 1.
The action of the signal rotator is easy to understand, and it is
demonstrated in Fig. 3.1, where the original signal (real image, pho-
tograph of Madrid street) is seen in Fig. 3.1a and its transformation
after the rotation at angle = /4 in Fig. 3.1b.
The signal rotator is an important tool for the study of signal sym-
metry.

3.3.3 Fractional Fourier Transform
We call the transform associated with ray transformation matrix T
separable if the block matrices A, B, C, and D are diagonal. The only
possible separable phase-space rotator is the fractional FT, as it is easy
to see from the orthosymplectic conditions [Eq. (3.19)]. Indeed for