Page 303 - Phase Space Optics Fundamentals and Applications

284 Chapter Nine

with the WDF reading
W ch (x, ) = ( − 2 x − ) (9.13)

which includes as a limiting case a line parallel to x for each off-axis
plane wave and a line parallel to as the representation of a single
point source.
The WDF of the chirp function also provides us with an alternative
interpretation of the affine transformations of phase space associated
with Fresnel diffraction and a thin lens. Fresnel diffraction can be
understood as a convolution of the complex amplitude distribution
with the point response function of free space

1 i 2
h Fr (x, z) = √ exp x (9.14)
i z z
This translates to

1 x
W h (x, ) = − = (x − z ) (9.15)
| z| z
which is a straight line in phase space. From this we obtain Eq. (9.9)
straightforwardly as the convolution in x between the input WDF and
W h (x, ). Similarly, convolution of the oblique line in with the WDF
of the input signal corresponds to the operation in Eq. (9.10).

9.3 Self-Imaging of Paraxial Wavefronts
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Self-imaging was first observed by Henry Fox Talbot in 1836 and the-
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oretically explained by Lord Rayleigh in 1881. In modern language,
the Talbot effect is concerned with Fresnel diffraction of a coherent
monochromatic wavefront that is strictly periodic in the transverse
direction. Then the physics of wave propagation ensures strict peri-
odicity along the axis of propagation z as well.
It was not until 1967 that Montgomery proved lateral periodicity to
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be a sufficient, but not a necessary, condition for self-imaging. In fact
it is possible to construct signals with a discrete plane wave spectrum,
which exhibit self-imaging not only within the bounds of paraxial
optics, but also for the nonparaxial domain of propagation.
With few exceptions the Talbot effect was ignored until affordable
coherent light sources became available and triggered a wave of re-
search related to coherent optical signal processing. Since then, the
Talbot effect has become a standard tool of Fourier optics. For a de-
tailed survey of the self-imaging phenomenon and its applications,
refer to the 1989 review by Patorski. 9