Page 324 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 305

For a continuous incoherent source illuminating grating G 2 ,we
effectively integrate the phase-space distribution in the vertical direc-
tion, with the obvious result that no fringe pattern can be observed in
the far field. For a spatially inhomogeneous, but continuous source,
each shifted copy of the phase-space distribution is weighted with the
respective source strength, and we obtain as the far-field intensity the
expected convolution (or correlation) between the distribution for a
single source point and the source distribution.
For the case where G 1 consists of an array of equidistantly spaced
pinholes with period d, all copies again register perfectly in the vertical
direction, and we observe the Lau effect for quasi-monochromatic
illumination. In this context it is again emphasized that incoherent
signal superposition is a linear operation in phase space, and no cross-
terms are observed. This also means that the PSD of an array of point
sources (at G 1 ) corresponds to an array of vertical lines, rather than
the two-dimensional distribution of functions, in Fig. 9.6, which
represent an array of mutually coherent pulses.
While we will not attempt a more detailed analysis of the Lau effect,
we note that condition [Eq. (9.44)] is not the only configuration for
which far-field fringes can be observed. In fact it follows from our
analysis that the fringes can also be observed if grating G 1 is composed
of narrow slits with a period d/2.
A more general analysis was performed by using the formalism of
2
coherence theory predicting fringe patterns at z L = (M/N)d / , with
M and N being relative prime integer numbers. 25 Coherence theory
has a formal interpretation in phase space, 41,42 and it was shown that
this can also be used for a more rigorous derivation of the conditions
for observing the Lau effect. 27

9.11 Summary
Not all optical phenomena find a natural interpretation in phase space.
Self-imaging, however, is exceptionally well suited to be analyzed
with the help of the WDF and PSDs. In part, this is due to the strict
periodicity of the signals. As a consequence, the WDF is discrete in at
least one of its coordinates.
Not only did the use of phase-space optics allow us to find a qual-
itative interpretation, but more importantly we were able to perform
a quantitative analysis requiring only a minimum of mathematical
formalism. Pivotal to this analysis is the use of PSDs as mediators
between the mathematical formalism of the WDF and an intuitive
interpretation of physical optics.
The discussion presented in this chapter was aimed at recover-
ing well-known relationships. However, we never even came close