Anomalous refractive properties of photonic crystals

We describe methods of investigating the behavior of photonic crystals. Our approach establishes a link between the dispersion relation of the Bloch m odes for an infinite crystal (which describes the intrinsic properties of t he photonic crystal in the absence of an incident field) and the diffractio n problem of a grating (finite photonic crystal) illuminated by an incident field. We point out the relationship between the translation operator of t he first problem and the transfer matrix of the second. The eigenvalues of the transfer matrix contain information about the dispersion relation. This approach enables us to answer questions such as When does ultrarefraction occur? Can the photonic crystal simulate a homogeneous and isotropic materi al with low effective index? This approach also enables us to determine sui table parameters to obtain ultrarefractive or negative refraction propertie s and to design optical devices such as highly dispersive microprisms and u ltrarefractive microlenses. Rigorous computations add a quantitative aspect and demonstrate the relevance of our approach. (C) 2000 Optical Society of America [S0740-3232(00)00306-9].