Abstract

Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite-gaussian eigenfunctions ψ<sub><i>n</i></sub> (<i>x, z</i>)consisting of a hermite polynomial of real argument <i>H<sub>n</sub></i> [√2<i>x</i>/<i>w</i> (<i>z</i>)] times the complex gaussian function exp[-<i>jkx</i><sup>2</sup>/2<i>q</i> (<i>z</i>)], in which <i>q</i> (<i>z</i>) 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⌃<sub><i>n</i></sub>(<i>x, z</i>) of the form <i>H<sub>n</sub></i> [√<i>cx</i>]exp [-<i>c x</i><sup>2</sup>], in which the hermite polynomial and the gaussian function now have the same complex argument √<i>cx</i> ≡ (<i>jk</i>/2<i>q</i>)<sup>½</sup><i>x</i>. The conventional functions ψ<sub><i>n</i></sub> are orthogonal in <i>x</i> in the usual fashion. The new eigenfunctions ψ⌃<sub><i>n</i></sub>, however, are not solutions of a hermitian operator in <i>x</i> and hence form a biorthogonal set with a conjugate set of functions φ⌃<sub><i>n</i></sub> (√<i>cx</i>). The new eigenfunctions ψ⌃<sub><i>n</i></sub> are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψ⌃<sub><i>n</i></sub> functions are not spherical for <i>n</i> > 1. However, they resonator and beam-propagation problems.

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