Unstable open resonators: two-dimensional and three-dimensional losses by a waveguide analysis.

Previous calculations performed by a waveguide approach to unstable open resonators with cylindrical hyperbolic mirrors are further refined. By this approach, the resonator is regarded as a waveguide transverse to the resonator axis, and resonance is established by self-consistent reflection of a single waveguide mode, or of two selectively coupled waveguide modes, from the mirror edges and from the axial region. The refinement refers to improved values for the modal reflection coefficients that appear in the resonance equation. The theory is now shown to be capable of predicting the eigenmode loss behavior in resonators with large Fresnel numbers, as verified by comparisons with published numerical solutions of the resonator integral equation. The theory is also extended successfully to circular hyperbolic mirrors. Finally, the waveguide model is utilized to derive known equivalence relations, previously deduced from the integral equation, between resonators with symmetrical mirrors and those with other arrngements. Thus, the results in this paper lend further support to the validity of the coupled-mode waveguide model for unstable resonators.