Maximization of the quality factor of an optical resonator

We consider resonance phenomena for the scalar wave equation in an inhomogeneous medium. Resonance is a solution to the wave equation which is spatially localized while its time dependence is harmonic except for decay due to radiation. The decay rate, which is inversely proportional to the qualify factor, depends on the material properties of the medium. In this work, the problem of designing a resonator which has high quality factor (low loss) is considered. The design variable is the index of refraction of the medium. High quality resonators are desirable in a variety of applications, including photonic band gap devices. Finding resonance in a linear wave equation with radiation boundary condition involves solving a nonlinear eigenvalue problem. The magnitude of the ratio between real and imaginary part of the eigenvalue is proportional to the quality factor Q. The optimization we perform is finding a structure which possesses an eigenvalue with largest possible Q. We present a numerical approach for solving this problem. The method consists of first finding a resonance eigenvalue and eigenfunction for a non-optimal structure. The gradient of Q with respect to index of refraction at that state is calculated. Ascent steps are taken in order to increase the quality factor Q. We demonstrate how this approach can be implemented and present numerical examples of high Q structures.