Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

Abstract

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.