The scattering of electromagnetic waves by three--dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, the fast, accurate, and reliable numerical simulations of these are indispensable for engineers and scientists alike. For this, High Order Spectral methods are frequently employed and here we describe an algorithm in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. This work extends our previous contribution regarding the Helmholtz equation to the full vector Maxwell equations, by providing a rigorous analyticity theory, both in deformation size and spatial variable (provided that the permittivity is, itself, analytic).

翻译：电磁波在三维周期结构上的散射对许多具有重要科学与工程意义的问题至关重要。由于这些波的复杂性及三维特性，快速、精确且可靠的数值模拟对工程师和科学家均不可或缺。为此，高阶谱方法被广泛采用，本文描述了一类此类算法。我们的方法本质上是微扰的，即将介电常数相对于常数值的偏差视为形变，并采用正则微扰理论。本研究将我们先前关于亥姆霍兹方程的贡献推广至完整的矢量麦克斯韦方程组，通过建立关于形变尺度与空间变量（前提是介电常数本身为解析函数）的严格解析性理论，得以实现这一推广。