Page 319 - Phase Space Optics Fundamentals and Applications
P. 319
300 Chapter Nine
The change of the self-imaging length, compared to the case of plane
wave illumination, can be expressed with the help of a magnification
factor
1
m R = (9.37)
1 − z T /R
For diverging waves R > 0, the self-imaging distance is increased,
while for converging wavefronts the self-imaging distance is reduced
compared to plane wave illumination. The Talbot distance can be re-
covered for R →∞.
The modified period of the first self-image can be deduced from the
new frequency spacing of the lines; i.e., we can write
1 1 d
= − (9.38)
2d 2d R
or
d = m R d (9.39)
Finally, the new radius of curvature at the first self-imaging plane
becomes
R = R + z p = m R R (9.40)
Additional self-imaging planes can be found by substituting R back
intoEq.(9.36).Foraconvergingwavefrontwefindself-imagingplanes
with increasing density along the optical axis, the closer they are lo-
cated to the focal point of the illuminating spherical wave. At the focal
point we expect to see self-imaging replaced by the Fourier spectrum
of the grating. In phase space this corresponds to a horizontal shear
which turns all lines vertical; i.e., the vertical projections will no
longer contain any information about the interference terms, but will
only show the distribution of discrete self-terms.
The case of self-imaging under spherical illuminations also serves
as an example to highlight other important generalizations of Talbot
self-imaging. For investigating diverging and converging wavefronts,
we had to drop the requirement of obtaining a perfect replica of the
complex amplitude. Instead, we accepted a scaled replica as a gener-
alized self-image. It should be mentioned that the geometric scaling
under spherical illuminations also applies to the fractional Talbot ef-
fect and was used to design Talbot array illuminators. 38
A further generalization is self-imaging in arbitrary ABCD optical
systems. Fresnel diffraction from a grating under spherical illumina-
tion is equivalent to plane wave illumination of the grating followed
by a system consisting of a parabolic lens and free-space propaga-
tion. In fact, conditions for obtaining self-images in arbitrary ABCD
systems have already been investigated. 39