**ABSTRACT**
### Gabor's Signal Expansions in Optics

#### M.J. Bastiaans

##### Department of Electrical Engineering, University of Technology, Eindhoven, Netherlands

In this chapter some applications of Gabors signal expansion in the
field
of optics are considered. After a preparatory treatment of some necessary
optics fundamentals and the translation of relevant concepts of
time-dependent signals to signals that depend on spatial variables,
Gabors
signal expansion and its companion -- the Gabor transform -- are
introduced in the field of optics. Special attention is paid to
Gaussian
windows, which are related to the well-known concept of Gaussian light
beams in optics.

The case of critical sampling is considered, in particular in its relation
to the degrees of freedom of an optical signal and to the space-bandwidth
product of an optical system; to do this, the propagation of Gabors
expansion coefficients through optical systems is considered. The case of
integer oversampling is considered and it is shown how in that case the
Gabor transform can be transformed into a product of Zak transforms. It
is
demonstrated how this product of Zak transforms can form the basis of a
coherent-optical setup for generation of the Gabor transform and it is
shown how this setup can be used for an approximate generation of the
windowed Fourier transform.