Exact scalar nonparaxial laser resonator modes

It is well known that the modes of ideal stable laser resonators can be described by the solutions of the three-dimensional wave equation(Helmholtz equation) in the paraxial regime. When the resonator has circular cylindric geometry the laser modes are described by the Laguerre-Gaussian modes whereas the modes are Hermite-Gaussian when a rectangular geometry exists. In this paper we have analysed the spheroidal harmonics which are the exact nonparaxial modes of the wave equation. The Laguerre- and Hermite-Gaussian modes are characterised by their beam waist, beyond a critical value of the ellipticity parameter, there exists a relation with the beam waist of the paraxial modes.