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.

中文翻译：

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周期结构激光特征值问题的有效方法

激光特征值问题 (LEP) 是非常规特征值问题，直接涉及激光开始时的频率和增益阈值。需要有效的数值方法来求解 LEP，以便分析、设计和优化微腔激光器。现有的二维 LEP 计算方法包括多极法和边界积分方程法。特别是，多极方法已应用于周期性结构的 LEP，但它需要复杂的数学技术来评估由于周期性而出现的缓慢收敛的无限和。在本文中，基于所谓的 Dirichlet-to-Neumann 映射为周期性 LEP 开发了一种新方法。该方法是有效的，因为它避免了缓慢收敛的和，并且可以轻松处理具有许多数组的周期性结构。