Design of a high quality optical resonator

We consider resonance phenomena for the scalar wave in an inhomogeneous medium. Resonance can be described as a solution to the wave equation that is spatially localized while its time dependence is (mostly) harmonic except for decay due to radiation. The reciprocal of the decay rate is referred to as the quality of the resonator. We will discuss the problem of designing resonators which has high quality where the design variable is the index of refraction of the medium. We will start with an introduction to resonance and its computation, and describe a continuation approach that allows one to maximize quality. Numerical examples of high quality resonators will be presented.