Page 213 - Phase Space Optics Fundamentals and Applications

194 Chapter Six

passing them through the system, one can convert them back to their
proper location in the space domain. The process of converting to
and converting back of the degrees of freedom is also called encoding
and decoding or multiplexing and demultiplexing. The improvement of
resolution requires “payment.” The payment needed to improve the
resolution in the space domain is the devotion of other domain or
other subspaces into which the required spatial degrees of freedom of
the input signal can be converted. The conversion of spatial degrees
of freedom can be done to a single subspace or to a plurality of several
such subspaces.
To do this properly without losing the desired spatial information,
one needs to have a priori information on the signal. Having a priori
knowledge that a certain domain is not used by the signal may allow
one to designate it for the use of spatial resolution improvement. For
instance, knowing that the object does not vary in the time domain
may assist in using the time domain for the process of converting to
and converging back the degrees of freedom.
The pluralitiesofother domains that maybeusedforthistemporary
conversion of degrees of freedom are the time 11–15 wavelength, 16–18
polarization, 19,20 code, 21–25 gray levels, 26 field of view, 27–32 and even
light’s coherence 33–35 domain.
To better understand how this adaptation of degrees of freedom
may be done, one may describe the space-frequency distribution, i.e.,
a phase space, of the signal (SWI) and the one of the system (SWY) and
performtheadaptationtoallthedegreesoffreedomofSWIthatarenot
graphically overlapping with the SWY representation. 36−38 The phase
space that is simple for presenting the space-frequency distributions
of both the signal and the system is the Wigner transformation. 39,40
Although bilinear, this transformation has interesting properties of
representing basic optical modules as simple mathematical operations
in this domain. 40−42
In this chapter we provide a schematic description and explanation
for how the process of SR may be understood in the Wigner space. In
general, a more heuristic explanation for the SR process may involve
any other phase-space diagrams. However, the advantage of using the
Wigner as part of our mathematical description is related to the fact
that the Wigner, although it is a bilinear transformation, can be related
mathematically to the spatial degrees of freedom of a signal. In our
presentation we mainly focus on the diffraction-related limitation of
resolution.
The chapter is constructed as follows: In Sec. 6.2 we mathematically
define the space bandwidth (SW), i.e., the space-frequency distribution,
while separating the distribution of the signal from that of the system
(SWI versus SWY). In Sec. 6.3 we focus on five ways of performing SR
while explaining how those operations are represented in the Wigner