Page 321 - Phase Space Optics Fundamentals and Applications
P. 321
302 Chapter Nine
n
d 1,2
n 2
n 1,2
n 1
n 0,2
n 0,1
n
0
x
n 0,–1
n –1
S
FIGURE 9.10 Self-imaging based on the Montgomery condition.
that the condition in Eq. (9.43) automatically fulfills the self-imaging
condition for all other cross-terms with m = 0.
9.10 Self-Imaging and Incoherent
Illumination
Phase-space optics allows one to describe both coherent and partially
coherent signals with one consistent formalism. While it is not the
purpose of this discussion to explore partially coherent optics in phase
space, the study of self-imaging conditions allows us to take a sneak
peak at how problems involving incoherent signals can be addressed.
To this end we consider the setup in Fig. 9.11, which is a double-
grating configuration. The first grating G 1 is illuminated with inco-
herent quasi-monochromatic light. At distance z L a second grating is
located in the front focal plane of a Fourier lens f . This is one of vari-
ous systems for studying the Lau effect. 24 The first grating G 1 serves
as a periodically modulated incoherent light source, which is used to
illuminate a the second grating G 2 located at some distance z L along
the optical axis. For a discrete set of grating separations it is possible to
observe a pattern of high-contrast fringes in the Fraunhofer diffraction
plane of the 2- f system.
The Lau effect easily rivals the Talbot effect in terms of its mathe-
matical beauty and its potential for applications. Perhaps due to its
slightly higher complexity, however, the Lau effect has unquestion-
ably received much less attention than coherent self-imaging.