General theory of linear, coherent optical data-processing systems

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Abstract
This paper concerns a general theory of linear optical data-processing systems. Well-known basic ideas are critically reviewed, fundamental constraints upon the system behavior are investigated, and a new operational notation is proposed. Every system is considered as a bidirectional processor and every two-dimensional signal is equivalently described in the space and frequency domain. Without referring to field-theoretical notation, the constraints of “reciprocity” and “losslessness” are introduced. In rough terms, reciprocity states equality of transmissions A → B and B → A, where A and B are points in the two reference planes, losslessness states equality of signal energies in the two planes. “Symmetry” has to be independently defined in the space and the frequency domain, and “shift invariance” and “spreadlessness” appear to be dual constraints. The important constraint of “time reversibility” is obtained when a system simultaneously satifies reciprocity and losslessness. In the second part, the general theory is applied to lenses, sections of free space, Fourier transformers, and magnifiers. With the aid of a suitable shorthand notation, the main results of Fourier optics can be easily derived.

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