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

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 ψˆn(x,z) of the form Hn√cxexp −cx2, in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)1/2x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψˆn, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ϕˆn(√cx). The new eigenfunctions ψˆn are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψˆn 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.