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

Abstract

Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite–gaussian eigenfunctions ψ_n (x,z) consisting of a hermite polynomial of real argument H_n [√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 ${\hat{\psi }}_n\left(x,z\right)$ of the form H_n[√cx]exp [−cx²], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)^1/2x. The conventional functions ψ_n are orthogonal in x in the usual fashion. The new eigenfunctions ${\hat{\psi }}_n$, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ${\hat{\varphi }}_n\left(\surd cx\right)$. The new eigenfunctions ${\hat{\psi }}_n$ are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ${\hat{\psi }}_n$ functions are not spherical for n > 1. However, they may still be useful as a basis set for other optical resonator and beam-propagation problems.