Page 315 - Phase Space Optics Fundamentals and Applications

296 Chapter Nine

n
d
V, H
d/Q

x

FIGURE 9.8 Fractional Talbot in phase space: comb function at z T /8.

are sufficient to describe diffraction from one fractional Talbot plane
to another. Here, we again turn to phase-space optics for constructing
the transformation matrix.
We restrict our attention to the case of even numbers Q and argue
that this will allow us to obtain the transformation matrix for any
fractional Talbot plane. Figure 9.8 illustrates the shear of the comb
function for the case Q = 4. From this special case it is easy to verify
that we obtain exactly Q columns with only positive ’s if we shear
the WDF such that the row at = 1/d moves by d/(2Q). This shear
corresponds to the fractional Talbot plane

1
z 1,2Q = z T (9.30)
2Q
Wecanalsogeneralizethefactthatlines H and V (asdefinedinSec.9.6)
coincide; i.e., the chirp function which can be thought of modulating
the modified comb function is

Q 2
(x) = exp i x (9.31)
d 2
and the Talbot coefficients are samples of this chirp function at x q =
qd/Q,
2
1 q
c q = √ exp i (9.32)
Q Q