Page 310 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 291

effect is exceptional, because it defines a case in which the Fresnel
diffraction integral indeed has a trivial solution regardless of the grat-
ing’s groove shape.
A similar characterization applies to the fractional Talbot effect,
which also defines a set of cases in which the Fresnel diffraction in-
tegral can be expressed in simple closed form. The study of Fresnel
images, i.e., Fresnel diffraction patterns at rational fractions of the Tal-
bot distance, has revealed a formal structure of the fractional Talbot
effect, which appeals to the experimentalist as well as to the theoreti-
cian.
While the first systematic study of Fresnel images can be found in
the seminal paper by Winthrop and Worthington, 10 it is worth point-
ing out the extent to which the formulation of the fractional Talbot
effect was simplified during the past four decades. Formulating the
fractional Talbot effect in terms of phase-space optics allows us to
appreciate this progress in a particularly satisfying way.
Instead of postulating rational fractions of the Talbot distance as
diffraction planes worthy of our attention, the phase-space interpre-
tation effortlessly finds the fractional Talbot planes as the result of
searching for cases with interesting properties.
The analysis of the fractional Talbot effect requires some prepara-
tion, which also serves as a demonstration as to how problems can
be dissected for applying a phase-space analysis most effectively. For
Fresnel diffraction of periodic complex amplitudes, we have to ex-
press the Fresnel diffraction amplitude at distance z from the input
plane as

∞
,
u(x, z) = u p (x) ∗ (x − nx 0 ) ∗ h Fr (x, z) (9.20)
n=−∞

with ∗ denoting a convolution. This means that we formulate the
periodic input signal as the function describing a single period u p (x)
convolved with an infinite comb function. The propagation of the
signal corresponds to a second convolution with the pulse response
of free space, in Eq. (9.14).
The convolution operation is associative; i.e., we are at liberty to
interpret the output signal as a convolution of the comb function with
thepropagator,thenfollowedbyasecondconvolutionwiththegroove
shapeofthegrating.Itisthepropagationofthearrayfactorthatismost
conveniently discussed in phase space. This analysiscanbecarriedout
without explicitly considering the groove shape for which the WDF
may be hard or impossible to calculate analytically.
However, one more time, we need to extend our phase-space tool-
box to include the WDF of the comb function. Substituting the infinite