## Optical and acoustic Fourier-processors.(English. Russian original)Zbl 0843.43012

Commutative harmonic analysis III. Encycl. Math. Sci. 72, 129-175 (1995); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 72, 135-180 (1991).
This is an expository chapter written by a mathematician for mathematicians (and mathematical physicists), where the well known functions and operations related to the Fourier transform (Gaussians, convolution, correlation) are made flesh as optical signals and elements. Such monographs are very welcome because of the increasing prevalence of information processing by optical means in technology. The Fourier transform between object and image planes is realized by a simple lens array in the paraxial régime (small angles, thin lenses). Also free flight, pupils, phase plates and gratings are addressed, and the Fourier techniques are applied to holograms, acousto-optical transducers, cells and processors. For mathematics students, such setups are likely to clarify the context of otherwise abstract theorems. The treatment is function-theoretic. The choice of topics could have benefited had the author seen it fit to describe all paraxial systems as $$2 \times 2$$ matrices, followed coherent states in optical fibers, and perhaps pointed out to counterparts in quantum mechanics. The treatment is concise and succintly explains the prevailing technology. Reference to the early literature on optical Fourier transformation is most enlightening.
For the entire collection see [Zbl 0729.00011].

### MSC:

 43A99 Abstract harmonic analysis 78A40 Waves and radiation in optics and electromagnetic theory 94A12 Signal theory (characterization, reconstruction, filtering, etc.)