Propagating-order scattering matrix of conically mounted and crossed gratings

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.