Theory of light refraction at the surface of a photonic crystal

A rigorous theory is developed for light refraction by a photonic crystal (PC) with arbitrary lattice-type and surface orientation. First, the refraction of a planar wave incident upon a photonic crystal surface is analyzed. We rigorously prove the equal partition of propagating PC modes by a surface under a general condition. The concept of surface-orientation-dependent mode degeneracy has been proposed and its relationship to quasi-periodic surfaces unfolded. With modes partitioned and the degeneracy properly recognized, a subset of the solved PC modes is identified to uniquely represent all PC modes that can be excited by an incident wave. The refraction problem can thus be rigorously solved in the plane-wave formulation. Essentially, we need to solve the field in only a single cell on the surface to solve the refraction problem. We further discuss the case where a Bloch wave illuminates the surface from inside a photonic crystal, which enables us to compute the transmissions along a complete light path through a series of interfaces. In addition, the transmission of a Gaussian beam is discussed, and the insertion loss formulas are presented. Other realistic beam profiles are discussed for designing photonic crystal devices. With all these issues solved, a complete theoretical framework of the photonic crystal refraction and transmission has thus been established. The theory has been applied to design a wavelength-division multiplexing demultiplexer that exhibits lower than 3 dB loss over a 25-nm spectrum. In examining the refraction by a quasiperiodic surface, a slight change of surface orientation is predicted to split one beam into an infinite number of beams.