Page 313 - Phase Space Optics Fundamentals and Applications

294 Chapter Nine

for even numbers (1 + N)/2 and
(1 − N)N
2 = 2 (9.24)
2d
for even numbers (1 − N)/2. We can substitute back into the chirp
signal to obtain the phase of the chirp function
(1 ± N)N
2
(x) = 2 x = x 2 (9.25)
2d 2
This modulates a comb function which samples the chirp at equidis-
tant intervals d/N; that is, the sheared distribution of functions can
be expressed as
∞
,
z T 1 nd
u comb x, = √ exp[i (x)] x −
N N N
n=−∞
N−1 ∞
1 , , ld
= √ exp(i l ) x − nd − (9.26)
N N
l=0 n=−∞
where we used the fact that the samples

ld 1 ± N 2
l = = l (9.27)
N 2N
√
are N-periodic. The factor 1/ N in Eq. (9.26) can be deduced as a
consequence of intensity conservation. The result is equivalent to the
expressions given in Ref. 14.
We have arrived at a remarkable result, which recognizes the Fres-
nel diffraction amplitude of a periodic function as the superposition
of N replicas of the input function, each replica being laterally shifted
by a multiple of d/N and modulated with a constant phase factor.
√
Coefficients c l = exp(i l )/ N, called the Talbot coefficients, are ob-
tained as the samples of a chirp function. The importance of using
schematic, yet rigorous PSDs to study optical systems is perhaps best
illustrated by comparing the analysis of the fractional Talbot effect
given in this section with a rather formal application of the WDF to
33
the same problem. While the phase-space analysis in both cases pro-
vides a rather compact formulation of the fractional Talbot effect, the
analysis assisted by the PSD of comb functions at each step facilitates
the interpretation of the phase-space expressions in the signal domain
with simple and explicit relationships for both the complex amplitude
and the Talbot coefficients.
It is possible to extend the analysis 28 to include all fractional Talbot
planes, namely, those with M = 1 and even numbers N. However, in
what follows we will assume a slightly different perspective, which