Modes of stable and unstable optical resonators

A method is developed for the approximate determination of the normal modes of stable and unstable optical resonators and the associated resonant frequencies and power losses. The method is based on replacing the finite integration limits in the integral equation for the normal modes by infinite limits and, subsequently, finding a differential equation whose solutions coincide with or approximate the solutions of this integral equation. When the end reflectors of the resonator are conical surfaces, a differential equation is found which corresponds exactly to the integral equation with infinite limits. Moreover, the equivalent differential equation is found to be of the same form as the wave equation for a monochromatic transverse electric wave propagating in an inhomogeneous medium of infinite extent with the inhomogeneity being transverse to the direction of propagation, showing the correspondence between the modes of a homogeneously filled conical resonator and the eigenmodes of an infinite inhomogeneous mdium. For the stable, low loss (convergent) region the solutions of the differential equation are readily found. For the unstable, high loss (divergent) region the solutions are found by using the principle of analytic continuation. The specific example of parabolic end reflectors is treated in more detail, and solutions for the eigenvalues and eigenfunctions are given for the case of infinite strip and circular geometries.