Page 320 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 301

A further variant is the Fourier dual case, where the input signal is
an array of discrete equidistant pulses and the system is composed of
only a parabolic lens. Then it is possible to describe the equivalent of
self-imaging in the frequency domain, which has been termed spectral
self-imaging, 40 which was shown to be of interest for time-domain
signals and optical fiber communication.

9.9 Another Path to Self-Imaging
So far we have assumed a periodic signal and explored the astound-
ing richness of Fresnel diffraction as a consequence of the signal’s
periodicity. We now change our perspective by using the self-imaging
phenomenon as the starting point. The task, then, is to identify those
signals for which self-imaging can be observed. In other words, in-
stead of analyzing a specific set of signals to discover self-imaging,
we now consider self-imaging as a given phenomenon and construct
signals with this property.
8
This design-oriented approach was suggested by Montgomery, in
1967 when seeking a necessary condition for signals to exhibit self-
imaging. The construction again can be carried out straightforwardly
in phase space.
We assume the signal to be composed of discrete frequencies. The
corresponding self-terms of the WDF all have the form of a line par-
allel to the x axis. This means that the information about permissible
spatial frequencies has to be obtained from the associated cross-terms,
all having the form

n + m
∗
W n,m (x, ) = 2a n a cos [2 ( n − m )x] − (9.41)
m
2
where n and m are integer numbers for labeling the spatial frequencies
and a n and a m are the associated Fourier amplitudes. Assuming m = 0
and 0 = 0, we can now use Eq. (9.41) to establish the condition for
self-imaging. The horizontal shift associated with Fresnel diffraction
has to be a multiple of the frequency n − m which modulates each
cross-term. Without loss of generality we assume m = 0tofind
n n
z M = (9.42)
2 n
from which follows the set of permissible frequencies

2n
n = (9.43)
z M
Equation (9.43) can be recognized as the Montgomery condition for
paraxial signals. Figure 9.10 illustrates the case of three discrete fre-
quencies and the associated cross-terms. It is a simple exercise to show