Efficient method for lasing eigenvalue problems of periodic structures

Lasing eigenvalue problems (LEPs) are non-conventional eigenvalue problems involving the frequency and gain threshold at the onset of lasing directly. Efficient numerical methods are needed to solve LEPs for the analysis, design and optimization of microcavity lasers. Existing computational methods for two-dimensional LEPs include the multipole method and the boundary integral equation method. In particular, the multipole method has been applied to LEPs of periodic structures, but it requires sophisticated mathematical techniques for evaluating slowly converging infinite sums that appear due to the periodicity. In this paper, a new method is developed for periodic LEPs based on the so-called Dirichlet-to-Neumann maps. The method is efficient since it avoids the slowly converging sums and can easily handle periodic structures with many arrays.