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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 63, Iss. 9 — Sep. 1, 1973
  • pp: 1093–1094

Hermite-gaussian functions of complex argument as optical-beam eigenfunctions

A. E. Siegman  »View Author Affiliations

JOSA, Vol. 63, Issue 9, pp. 1093-1094 (1973)

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Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite-gaussian eigenfunctions ψn (x, z)consisting of a hermite polynomial of real argument Hn [√2x/w (z)] times the complex gaussian function exp[-jkx2/2q (z)], in which q (z) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψ⌃n(x, z) of the form Hn [√cx]exp [-c x2], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)½x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψ⌃n, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions φ⌃n (√cx). The new eigenfunctions ψ⌃n are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψ⌃n functions are not spherical for n > 1. However, they resonator and beam-propagation problems.

A. E. Siegman, "Hermite-gaussian functions of complex argument as optical-beam eigenfunctions," J. Opt. Soc. Am. 63, 1093-1094 (1973)

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  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

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