Page 316 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 297

We now use the superposition principle and interpret the PSD of
the comb function after shearing, in Fig. 9.8, as the PSD of an input
signal with a 0 = 1 and a q = 0 for q = 1, ... ,Q − 1. Then the output
signal is the set of Talbot coefficients b q = c q . Next we investigate the
case a 1 = 1 and a q = 0 for q = 1. Thus the PSD in Fig. 9.8 merely has
to be shifted in x by d/Q. The same shift has to be considered for the
phase function of the modulating chirp, i.e.,

2
Q d
(x) = exp i x − (9.33)
d 2 Q
Sampling the shifted chirp again at x q = qd/Q results in a reordering
of the Talbot coefficients. We obtain the output coefficients b q = c q−1 ,
where the subscript of the Talbot coefficients needs to be evaluated
modulo Q.
This procedure can be repeated by considering all input samples
a q in turn. The solution of the diffraction problem is obtained as
a superposition of diffraction amplitudes associated with all input
coefficients, i.e.,
Q−1
,

b = c q −q a q (9.34)

q
q=0
where again the subscript q − q has to be calculated modulo Q.

This linear transformation is completely determined by the matrix
C ={c q −q }, and Eq. (9.34) can be used to obtain the Fresnel diffrac-

tion amplitude in any fractional Talbot plane (M, N). We can extend
the matrix description effortlessly to planes (M, 2Q), by applying the
M
linear transformation M times; i.e., the system matrix becomes C .
Planes defined by odd numbers Q are identical to fractional Tal-
bot planes (2M, 2L), with L = 2Q. This implies, however, that we
need to consider twice as many samples per period to cover this case
with the matrix formalism in Eq. (9.34). This is necessary because
for odd Q, the Q copies of the input signal are shifted laterally by
qd/Q + d/(2Q). 28,29,34 Doubling of the sampling frequency automati-
cally incorporates this shift. At least for formal explorations this seems
to be a small disadvantage if compared to the fact that all fractional
Talbot planes are covered with one single compact expression, while
other definitions need to distinguish carefully between different cases
(see, for example, Ref. 35).
Finally, we can also include planes specified by odd numbers N,
which were discussed in Sec. 9.6, by computing the transfer matrix
for fractional Talbot planes (4M, 4N). Again, we accept twice as many
samples as necessary to describe the diffraction problem in order to
apply the formalism without modification. Note, however, that the