Page 120 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 101

FT. The example of this system is a free-space propagation. Geomet-
ric phase accumulation requires a system with astigmatic elements,
and T O is similar to the antisymmetric fractional FT, as well as mode
asymmetry m = n.

3.8 Conclusion
In this chapter we have considered the phase-space rotators—the
transformations that produce the rotation of the Wigner distribution
in phase space. Several approaches to the description of phase-space
rotators by integral transforms, Hermitian operators, and ray transfor-
mationmatriceshavebeendiscussed.Therearefourbasicphase-space
rotators for two-dimensional signals: symmetric and antisymmetric
fractional FTs, gyrator, and signal rotator. The others can be obtained
as their cascades. The fractional FT certainly plays a main role in the
phase-space rotator description since it is associated with the diago-
nal unitary matrix, and any unitary matrix describing a phase-space
rotator is similar to it.
We have seen that the eigenfunctions for the phase-space rotators
are Gaussian functions modulated by the orthogonal polynomials,
with Hermite-Gaussian and Laguerre-Gaussian modes among them.
Thesemodesarewidelyusedforthedescriptionofopticalbeams.Dur-
ing the beam propagation through the optical system related to certain
phase-space rotators for which it is an eigenfunction, the Gouy phase
is acquired, because the eigenvalue is unimodular. We have stressed
that the phase-space rotators similar to the symmetric (antisymmet-
ric) fractional FTs are responsible for the accumulation of the dynamic
(geometric) phase, correspondingly.
The application of the phase-space rotators to the different signal
and image processing tasks such as filtering, pattern recognition, chirp
detection, and signal encryption has been discussed. We also note that
phase-space rotators play an important role in signal characterization,
orbital angular momentum manipulation, and beam conversion. Thus
the fractional FT is crucial for the phase-space tomography reconstruc-
tion of the Wigner distribution and therefore the complex field ampli-
tude or mutual intensity of the coherent or partially coherent beam
correspondingly.
The flexible optical setups for the experimental realization of ba-
sic phase-space rotators have been considered. We mention that the
systems constructed from the generalized lenses located at the fixed
position permit one to easily change the transformation parameters,
which is required in various applications of phase-space rotators.
In this chapter we have considered the application of phase-space
rotators to classical optical, beams, but they are also widely used in
quantum physics and signal processing.