Nonparaxial eigenmodes of stable resonators

A method to determine the nonparaxial eigenmodes of stable resonators is presented. The method is based on the perturbation theory of Lax et al. For calculating nonparaxial components of the electric field. A matrix formalism which uses a mode expansion into paraxial Hermite-Gaussian modes is applied to describe the nonparaxial propagation and the phase shift at a parabolic and a spherical mirror. Expressions for these matrices are derived analytically. Multiplying all matrices corresponding to a round trip, a matrix for the resonator is obtained. Eigenmodes of the resonator are numerically found by solving the eigenvalue problem. In the special case of paraxial propagation and parabolic mirror profiles, the standard Hermite-Gaussian modes result analytically. Nonparaxial modes of a given resonator are compared for different mirror profiles. It is found that, in the nonparaxial domain, spherical mirrors do not change the mode profile and the frequencies of the transverse modes, in contrast to parabolic mirrors which aberrate the beam profile and cause frequency shifts.