Page 308 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 289

grating, we indeed observe a self-image of the input grating pattern
at z T /2, but with the bright and dark lines exchanged.
For practical applications the transverse shift by d/2 may be neg-
ligible, and it seems almost justified to use z T /2 as the self-imaging
distance. In Sec. 9.6, however, we expand the notion of self-imaging to
include rational fractions of the Talbot distance. In this context it will
be more suitable to count only precise replicas of the input complex
amplitude as self-images. The Talbot image at half distance is then
identified as a Fresnel image, i.e., the Fresnel diffraction pattern at the
fractional Talbot plane z 1,2 = z T /2.

9.5 The “Walk-off” Effect
The Talbot effect is the result of the in-phase superposition of all plane
waves. The discrete nature of the spectrum guarantees equal phase
delays between adjacent frequencies; i.e., the in-phase condition can
readily be achieved.
This has to change as soon as we consider a finite grating aper-
ture. On one hand, the lines will be replaced by the WDF of a sinc
function along , and the quadratic propagator of the Fresnel diffrac-
tion transform is no longer sampled at the appropriate intervals only.
As a consequence the line shape of the Talbot grating image will be
perturbed.
On the other hand, in a quasi-geometrical sense, each discrete plane
wave mode is now truncated while propagating off-axis. This is ob-
servable as the so-called walk-off effect. 30–32 At the boundaries of
the propagating window a transition region develops, where no self-
images can be observed, which cuts into the domain of self-imaging
as a function of propagation distance. We can interpret this effect
in terms of truncated plane waves, which “walk off” the window
defined by the grating aperture until they no longer contribute to
the self-images. In close analogy to the Abbe theory of coherent
image formation, we need at least two interfering plane waves to
observe a self-image with structural information about the original
grating.
It is possible to estimate the maximum distance for self-imaging
with para-geometrical optics. The analysis is conveniently executed
with the help of the PSD in Fig. 9.5. We assume a periodic signal with
a finite bandwidth and thus a finite number of Fourier coefficients.
Thisdoesnotimposeanysevererestrictionssinceinpracticeallsignals
are essentially band-limited due to the frequency cutoff of systems for
signal generation and transport. We now assume a quasi-ray optical
perspective by further assuming a truncation of the periodic signal to
M periods without impact on its plane wave spectrum.