Page 307 - Phase Space Optics Fundamentals and Applications

288 Chapter Nine

becomes d. In contrast, the cross-terms superposed with self-terms at
m = m/d can only be associated with pairs of self-terms separated
by integer multiples of 2/d, and as a consequence the modulation we
observe has a period of d/2, that is, one-half of what we obtain for
interlaced cross-terms.
Fresnel diffraction corresponds to a horizontal shear of the phase-
space distribution, and we see without difficulty that the PSD does
not change if the shear equals d at = 1/(2d). All other discrete
lines are automatically sheared by an integer multiple of the period
d, and we obtain the original phase-space distribution. The line S in
Fig. 9.4 corresponds to the tilt we would observe for a vertical line at
the input and is equivalent to the WDF of the associated point-spread
function of free space in Eq. (9.14). Using Eq. (9.15), the Talbot length
can now be deduced from d = z T /(2d), or

2d 2
z T = (9.18)

It is immediately clear from the PSD in Fig. 9.4 that self-images can
be found at any integer multiple of the Talbot distance. This is re-
lated to the fact that the WDFs (and not only their projections) be-
fore and after shearing are identical, which proves that self-imaging
recovers not only the intensity distribution, but also the complex am-
plitude of the input signal. Also note that once the WDF of a periodic
function is known, the self-imaging condition can be deduced with a
minimum of mathematical formalism. Furthermore, the quantitative
result can only be obtained correctly by including the cross-terms,
namely, the interlaced terms at half intervals, into our analysis. This
is of significance as PSDs are often constructed as heuristic notions of
phase space rather than from the results of a rigorous evaluation of
the WDF.
ThisrigorousanalysisoftheWDFalsoprovidesaccesstotheFresnel
diffraction amplitude at z T /2. The diffraction patter is often described
as an additional grating image which is reversed in contrast. 30 To un-
derstand this notion, we again turn to Fig. 9.4. If we consider the shear
associated with line H, which is only one-half the shear necessary for
Talbot self-imaging, we again recover the phase-space distribution of
the input signal, however shifted in x by d/2 compared to the distri-
bution in Fig. 9.4. By inspecting the points of intersection between line
H and the horizontal delta lines, we can verify that the lateral shift at
m = m/d in fact is a multiple of d, while it is an odd multiple of d/2 for
the interlaced frequencies. Thus the terms at even multiples of 1/(2d)
and at odd multiples of 1/(2d) only register because the base period
of the modulation at m = m/d is d/2, that is, one-half of what we
might expect intuitively for a phase-space distribution of a periodic
signal. For typical grating profiles, including, for instance, the Ronchi