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Nonlinear laser optics. I: Duality of semisimple rings and phase matching. (English) Zbl 0625.20034

The breakthrough of representation theory of symmetric groups to applications on representations of general linear groups is due to I. Schur [Sitzungsber. Akad. Berlin 58-75 (1927). Gesammelte Abhandlungen, Band III, 68-85 (1973; Zbl 0274.01054)]. He recognized the fundamental duality which was elaborated by H. Weyl [Ann. Math., II. Ser. 30, 499-516 (1929). Gesammelte Abhandlungen, Band III, 171-188 (1968; Zbl 0164.301)]. - There has been considerable theoretical and experimental study of the nonlinear optical properties of anisotropic solids (crystals), liquids, and gases.
In the paper under review we calculate group theoretically the phase matching conditions of n-th order harmonic generation of high-power laser radiation by the multiplicity formula valid for the tensor bimodule \(E^{\otimes n}\). Here E denotes the two-dimensional complex vector space spanned by the phases of the pump fields. For more details and applications to nonlinear optical phase conjugation, see the forthcoming monograph entitled “Signal geometry” which continues the monograph “Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory”, Pitman Res. Notes Math. 147 (1986).

MSC:

20G45 Applications of linear algebraic groups to the sciences
78A60 Lasers, masers, optical bistability, nonlinear optics
22E70 Applications of Lie groups to the sciences; explicit representations
81V55 Molecular physics
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