Asymptotic and Numerical Techniques for Resonances of Thin Photonic Structures

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach very efficient. Convergence estimates are proved for the asymptotics. The second approach, based on the finite element method with a truncated perfectly matched layer, is not restricted to thin structures. We demonstrate the use of these methods in numerical calculations which further illustrate their differences. The asymptotic method finds resonance by solving a dense, but small, nonlinear eigenvalue problem, whereas the finite element method yields a large, but linear and sparse generalized eigenvalue problem. Both methods reproduce a localized defect mode found previously by finite difference time domain methods.