Gratings are basic structures for a variety of photonic components. Depending on the size, characteristics, or function, a grating can also be referred to as a photonic crystal, a metamaterial, or a metasurface. The concept of a scattering matrix (S matrix) is key to understanding the electromagnetic theory of gratings. Lifeng Li investigated conically mounted and crossed gratings, and presented the most general formulation and the global and elemental properties of the propagating-order S matrix, which is related to all propagating incident and diffracted orders and can fully describe the grating's far-field properties. The author introduced the concept of the elemental S matrix, studied its eigensolutions and Hermitian products, and arrived at two sum rules of diffraction efficiencies and the vectorial reciprocity theorem of gratings. This deep and systematical understanding of the properties of the S matrix will advance the analysis and design of gratings for far-field applications including anomalies, perfect absorption, perfect blazing, and space reversal symmetry.

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