Eigenmodes diffraction losses of marginally unstable semispherical resonator

The theory of modes in open laser resonators, stable as well as unstable both has received much attention for many years, due to the usefulness of these devices. The open resonator eigenmodes may be determined by solving the free-space Maxwell equations with perfect conductor boundary conditions on the mirrors. In the practice usually the Fresnel approximation is made for formulating the problem in the form of the well-known Fresnel-Kirchhoff equation. The solutions to this equation are generally not obtainable analytically. Numerical methods are applicable in the practice only at relatively small Fresnel number. For larger Fresnel number asymptotic methods have been developed which are good in the highly unstable region. However this asymptotic technique fails for marginally unstable resonators. In the stable region the eigenmodes are well described by Gaussian-Hermite solutions, which also become less accurate as the resonator becomes marginally stable. Thus, there is an area between the stable and unstable regions where neither the Gaussian beam theory nor asymptotic solution is valid