Many photonic devices, such as photonic crystal slabs, cross gratings, and periodic metasurfaces, are biperiodic structures with two independent periodic directions, and are sandwiched between two homogeneous media. Many applications of these devices are closely related to resonance phenomena. Therefore, efficient computation of resonant modes is crucial in device design and structure analysis. Since resonant modes satisfy outgoing radiation conditions, perfectly matched layers (PMLs) are usually used to truncate the unbounded spatial variable perpendicular to the periodic directions. In this paper, we develop an efficient method without using PMLs to calculate resonant modes in biperiodic structures. We reduce the original eigenvalue problem to a small matrix nonlinear eigenvalue problem which is solved by the contour integral method. Numerical examples show that our method is efficient with respect to memory usage and CPU time, free of spurious solutions, and determines degenerate resonant modes without any difficulty.

翻译：许多光子器件，如光子晶体平板、交叉光栅和周期性超表面，均为具有两个独立周期性方向的双周期结构，并夹在两个均匀介质之间。这些器件的诸多应用与谐振现象密切相关，因此高效计算谐振模式对于器件设计与结构分析至关重要。由于谐振模式满足出射辐射条件，通常采用完美匹配层（PMLs）来截断垂直于周期方向的无界空间变量。本文提出一种无需使用PMLs的高效方法，用于计算双周期结构中的谐振模式。我们将原始特征值问题转化为一个由轮廓积分法求解的小型矩阵非线性特征值问题。数值算例表明，本方法在内存占用和CPU时间方面高效，无伪解产生，且能轻松确定简并谐振模式。