A systematic and formal study of the global and elemental properties of the propagating-order scattering matrix of conically mounted and crossed gratings is presented. The most general formulation of the scattering matrix is established. Expressions of the global properties (reciprocity and unitarity) of the scattering matrix ($S$ matrix) in the general form previously not available in the literature are presented in the main text, and their full mathematical derivations are given in two appendices. The distinctive contribution of this work is an exposition of the elemental properties of the $S$ matrix. The elemental $S$ tensor and the elemental $S$ matrix, the latter being the linear-space representation of the former, for a pair of an incident plane wave and a diffracted order are defined and studied. The key results of the exposition are two sum rules of diffraction efficiencies and a dot-product-free, vectorial reciprocity theorem.

中文翻译：

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圆锥形和交叉光栅的传播阶数散射矩阵

提出了系统和形式化研究圆锥形安装和交叉光栅的传播级散射矩阵的整体和元素特性的方法。建立了散射矩阵的最通用公式。正文中介绍了以前没有文献中可用的一般形式的散射矩阵（$ S $矩阵）的全局属性（互易性和统一性）的表达式，并在两个附录中给出了它们的完整数学推导。这项工作的独特贡献是对$ S $矩阵的元素属性的说明。元素$ S $张量和元素$ S $定义和研究了一对入射平面波和一个衍射阶的矩阵，后者是前者的线性空间表示。博览会的主要结果是衍射效率的两个和规则以及无点积，矢量互易定理。