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Self-Imaging in Phase Space 285

The scope of self-imaging was dramatically expanded by the study
of Fresnel diffraction of periodic signals at rational fractions of the
Talbot self-imaging period. Namely, the work by Winthrop and
Worthington 10 identified Fresnel images, i.e., the diffraction patterns
at so-calledfractional Talbot planes, as cases, where the Fresneldiffrac-
tion integral can be expressed in simple analytic form. Subsequent
investigations further simplified the analytic expressions, 11–13 culmi-
nating in a discrete matrix formulation of near-field diffraction, which
relates the amplitudes of sampled periodic signals in different frac-
tional Talbot planes via linear transformations. 14–16 Interest in study-
ing the fractional Talbot effect largely increased by the invention of
the Talbot array illuminator, 17,18 a diffractive optical element to con-
vert a homogeneous wavefront to an array of high-intensity spots.
Today, a vast number of studies can be found in the literature that
describe design procedures, experimental work, and applications of
Talbot array generators (see Refs. 19–23 as only a small set of related
work).
A close relative of the Talbot effect is the Lau effect which is con-
cerned with incoherent periodic optical signals. 24–26 While not receiv-
ing the same attention as coherent self-imaging, perhaps due to its less
intuitive nature and a more difficult experimental implementation, the
Lau effect was shown to be useful for a number of applications and
remains a vivid member of the family of self-imaging phenomena.
Self-imaging in phase space was first studied by Ojeda-Casta˜neda
and Sicre 27 and applied to both the Talbot effect and the Lau effect.
The phase-space analysis was later extended to include the fractional
Talbot effect 28 and the design of Talbot array illuminators. 29
The remainder of the chapter, in part, is a review of previously
published results. In part, however, it contains original contributions
to highlight self-imaging as a phenomenon that is exceptionally suited
to be explored with phase-space optics.

9.4 The Talbot Effect
The setup to observe the Talbot effect is schematically depicted
in Fig. 9.3. An infinitely extended grating is illuminated with a
monochromatic coherent plane wave of wavelength . Figure 9.3
shows the simulated intensity pattern behind a Ronchi grating. After
some propagation distance z T we find the exact intensity distribution
which is observable immediately behind the grating, and we call z T
the self-imaging distance or Talbot distance. The fractional Talbot ef-
fect, which is discussed in detail in Sec. 9.6, is associated with Fresnel
diffraction at rational fractions M/N of the Talbot distance, where M
and N are integer numbers. Our discussion will exclusively assume