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Chapter 1:
First-Order Optical Systems for Information Processing
During the last decades, optics is playing an increasingly important role in acquisition, processing, transmission, and archiving of information. In order to underline the contribution of optics in the information acquisition process, let us mention such optical modalities as microscopy, tomography, speckle imaging, spectroscopy, metrology, velocimetry, particle manipulation, etc. Data transmission through optical fibers and optical data storage (CD, DVD, as well as current advances of holographic memories) make us everyday users of optical information technology. In the area of information processing, optics also has certain advantages with respect to electronic computing, thanks to its massive parallelism, operating with continuous data, possibility of direct penetration into the data acquisition process, implementation of fuzzy logic, etc. The basis of the analog coherent optical information processing is the ability of a thin convergent lens to perform the Fourier transform (FT). More than 40 years ago, Van der Lugt introduced an optical scheme for convolution∕correlation operation, based on a cascade of two optical systems performing the Fourier transform with filter mask between them, initiating an era of Fourier optics. This simple scheme realizes the most important shift-invariant operations in signal∕image processing, such as filtering and pattern recognition. Nowadays, the Fourier optics area has been expanded with more sophisticated signal processing tools such as wavelets, bilinear distributions, fractional transformations, etc. Nevertheless, the paraxial optical systems (also called first-order or Gaussian ones, which consist for example from several aligned lenses, or mirrors) remain the basic elements for analog optical information processing. In paraxial approximation of the scalar diffraction theory, a coherent light propagation through such a system is described by a canonical integral transform (CT). Thus starting from the complex field amplitude at the input plane of the system, we have its CT at the output plane. The two-dimensional CTs include, among others, such well-known transformations as image rotation, scaling, fractional Fourier and Fresnel transforms. We can say that the CTs represent a two-dimensional signal in different phase space domains, where the phase space is defined by the position and momentum (spatial frequency) coordinates. The signal manipulation in different phase space domains opens new perspectives for information processing. Indeed, several useful applications of the first-order optical systems for information processing have been proposed in the past decade. In particular first-order optical systems performing fractional Fourier transform have been used for shift-variant filtering, noise reduction, chirp localization, encryption, etc.
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