This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted.

翻译：本文针对横向磁极化和电极化两种偏振态下含多个多层腔体的电磁散射问题，提出了一种稳健的数值解法。通过在腔体开口处引入透明边界条件，将无界域问题转化为有界腔体问题。利用解的傅里叶级数展开，我们将原始边值问题简化为关于傅里叶系数的常微分方程两点边值问题。通过解析推导解的连接公式，我们构建了一个仅包含开口处傅里叶系数的小规模系统，从而简化求解过程。此外，我们提出了精确的数值求积公式，用于高效处理透明边界条件中出现的弱奇异积分。通过一系列数值实验，验证了所提方法的有效性与通用性。