Optical resonators in the approximation of Gauss

A general theory of optical resonators based on the concept of complex point-eikonal is presented. The analysis is limited to the approximation of Gauss. The modes of resonance of open resonators formed by two spherical mirrors facing each other have been obtained in previous works by fitting the wavefronts of Gaussian beams to the mirror surfaces. This method becomes complicated when the resonator contains focusing elements or dielectric slabs. The approach proposed in this paper is more straightforward than previous approaches dealing with this problem and is applicable to resonators containing anisotropic media. The round-trip point-eikonal is first evaluated on the basis of the conventional laws of Gaussian optics. The presence of apertures with Gaussian transmissivity, and of lasers with quadratic transverse variation of the gain, are accounted for by introducing complex round-trip point-eikonals. The modes ψ
m
(x) of the resonator are obtained from a power series expansion of the Green function of a mode-generating system related to the round-trip point-eikonal S of the resonator. The resonance frequencies and the round-trip losses are given by simple and general formulas. The mode fields are described by Hermite-Gauss functions with complex arguments, explicitly in terms of S. For resonators that do not contain apertures, the wavefronts are the same for all the modes and they are plane at planes of symmetry. In reciprocal resonators, clockwise and counter-clockwise modes have the same losses and resonant frequencies, but different mode pattents. These modes are shown to be mutually adjoint. Adjoint modes, with < ψ
^†
m
, ψ
n
> = 0 if m ≠ n, are useful to evaluate the response of optical resonators to incident fields. Results applicable to resonators containing nonorthogonal astigmatic elements are also discussed.