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
of the form Hn[√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 ψn are orthogonal in x in the usual fashion. The new eigenfunctions
, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions
. The new eigenfunctions
are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the
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.
© 1973 Optical Society of America
Equations on this page are rendered with MathJax. Learn more.