Page 317 - Phase Space Optics Fundamentals and Applications

298 Chapter Nine

oversampling of the signal can be avoided in this case by modifying
the expression for the Talbot coefficients in Eq. (9.32). 15
It may seem that the vector-matrix formulation of the fractional
Talbot effect carries a rather severe restriction: The base period has
to be expressed by a sampling expansion and the fractional Talbot
planes which we can access are linked with the number of samples per
period.However,theinterpretationasFresnelpropagationofsampled
functions dramatically extends the scope of the fractional Talbot effect
in general.
In particular, the matrix formalism provides the link between a
continuous description of Fresnel diffraction and discrete computa-
tions. It is a well-understood fact that numerical simulations of Fres-
nel diffraction have to be carried out with care, because the pulse
response is neither space- nor band-limited. The matrix expression
in Eq. (9.34) defines a set of cases where a numerical computation
of Fresnel diffraction can be carried out rigorously (see also Refs. 13
and 16). The restriction to periodic functions, in this context, is similar
to that of the discrete Fourier transformation which is also used to
approximate the corresponding continuous transform.
For numerical applications it is also interesting that C has the
structure of a unitary matrix. 15 This means, that the inverse Fres-
nel transform can be carried out without difficulty as well, which is
important for the performance of iterative methods, e.g., Gerchberg-
Saxton type of algorithms for phase retrieval and diffractive optics
design. 36,37

9.8 Point Source Illumination
So far we only considered self-imaging of periodic wave fields, which
can be interpreted as the result of illuminating a diffraction screen with
a coherent plane wave. If diffraction can be described with Kirch-
hoff’s approximation, the complex amplitude at the input plane is
completely equivalent to the transmission function of the diffractive
element.
A small yet important generalization is the illumination with spher-
ical wavefronts (or parabolic wavefront, if the paraxial approximation
is invoked). This means that the strictly periodic wavefront is mod-
ulated with the chirp function h Fr (x, R) in Eq. (9.14). With the sign
convention in Eq. (9.14) we obtain a diverging wave for R > 0 and a
converging wave for R < 0.
While the solution of this diffraction problem presents no funda-
mental difficulty if solved with standard Fourier optics (see, e.g.,
Ref. 9), it requires some bookkeeping effort, in particular when deal-
ing with the quadratic-phase terms of the Fresnel diffraction integral.