Page 312 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 293

n
V 2 V 1 H

1
d
x

d

FIGURE 9.7 Fractional Talbot effect of the comb function at z T /3.

straightforwardly calculated from the shearing operation, in Eq. (9.9),
with z M,N N/(2d) = Md,or
M 2d 2
z M,N = (9.22)
N
which we identify as the set of fractional Talbot planes.
For the present we restrict the discussion to cases M = 1 and odd
numbers N. The WDF in Fig. 9.7 does not strictly resemble a comb
function. However, instead of interpreting the distribution of points
as a Cartesian grid which was sheared horizontally, we arrive at the
same distribution by interpreting this as a (different) Cartesian grid
sheared in the vertical direction.
This vertical shear is not unique. In Fig. 9.7 line H corresponds to
the WDF of the point response of free space, while V 1 and V 2 illustrate
the WDF of two chirp functions which can be used to modulate a
comb function of period d/3 to obtain the distribution in Fig. 9.7. Any
suitable line V has to run through the origin and the location of a
function with positive sign at x = d/(2N). For N an odd integer, this
is only the case for multiples n of = 1/d.
2
We find n by determining the slope of H in Fig. 9.7 as N/(2d )
and seeking the discrete frequency for which the lateral shift caused
by the horizontal shear equals kd/2 + d/(2N), with k being an integer
multiplier. This allows us to find n = (1+ N)/4 and k = 1 for (1+ N)/2
being even, and n = (1 + N)/4, k =−1 for (1 − N)/2 being even.
This allows us to deduce the phase function of the chirp with the
help of Eqs. (9.12) and (9.13) by evaluating the slope of V. As solutions
we find
(1 + N)N
2 = 2 (9.23)
2d