Gabor’s signal expansion in optics

Abstract

In this chapter some applications of Gabor’s 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, Gabor’s 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 Gabor’s 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.