Page 306 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 287

n

1
d

x

S

H
d

FIGURE 9.4 PSD of a periodic function.

Eq. (9.16). The self-terms correspond to the WDF of individual discrete
frequency components and form lines without modulation located at

multiples of the base interval 1/d. All other terms n = n , in Eq. (9.17),
are cross-terms equivalent to the cross-term in Eq. (9.6). Note that the
choice of what term to identify as self-term depends on the particular
expansion we use to express the signal as a linear superposition of
signal components.
We can further observe that the modulation in x can again be in-
terpreted as a Fourier series; i.e., at each frequency 1/(2d) we find a
line, with periodic modulation in x. It is important to note that the
base frequency of this periodicity is d for terms m = (2m + 1)/(2d),
but is d/2 for m = m/d, with m being an integer number.
While this can be readily verified from Eq. (9.17), we can also in-
terpret this as an inherent property of the cross-terms. We can con-
struct any cross-term by considering all possible pairs of self-terms in
turn. For each pair we expect a cross-term to appear at half distance
in between the two self-terms. The cross-terms are generally modu-
lated periodically to ensure they do not contribute to the marginal if
we integrate the WDF along its axes. This modulation frequency is
proportional to the distance in phase space between the respective
self-terms.
In Fig.9.4 thecross-termsat m = (2m+1)/(2d),whichareinterlaced
with the self-terms, can only be constructed from self-terms separated
by an odd multiple of 1/d, and the base period of the modulation