Analytic approaches to unstable resonators

The marginally stable regime has been distinguished from the unstable regime both qualitatively and quantitatively. New asymptotic techniques for solving the Fresnel-Kirchoff equation for marginally stable resonators were obtained. The method is analytical in nature and yields excellent predictions for the modes and losses of such cavities. Numerical results for both marginally stable and marginally unstable cavities are presented. In addition, conditions for the suitability of using empty resonator modes to describe gain filled cavities have been derived. This deviation is based on manipulations of the differential equation describing gain filled cavities in Hankel transform space. The result is an ordinary, linear Fresnel-Kirchoff integral equation with gain renormalized Fresnel number and eigenvalue.