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Extensions of the Heisenberg group and coaxial coupling of transverse eigenmodes. (English) Zbl 0723.22009

Optical fibers of rectangular or circular cross-section are classically modeled by a parabolic profile of index of refraction, so harmonic oscillator wavefunctions describe the transverse eigenmodes. These functions provide bases for irreducible unitary representations of the usual two-step nilpotent Heisenberg group, and for its semidirect product with solvable and compact subgroups of the one-dimensional real metaplectic group. The author studies the coaxial coupling coefficients between two eigenmodes in fibers of the same type, with different width parameters and separated by a flat interface. These are found by the integral (overlap) of their cross-ambiguity functions. Although predictably there are selection rules between eigenmodes with different angular momenta (circular case) and parities (rectangular case), it is remarkable that in the general case they are given in terms of values of the Krawtchouk polynomials.

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
43A80 Analysis on other specific Lie groups
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
22E70 Applications of Lie groups to the sciences; explicit representations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
78A05 Geometric optics
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
33E15 Other wave functions
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