Extreme anisotropy of wave propagation in two-dimensional photonic crystals

We demonstrate that electromagnetic waves propagating in square and hexagonal photonic crystals can have fundamentally different anisotropy properties. The wave frequency and group velocity can be functions of the propagation direction even for vanishingly small wave numbers (near the Γ-point). This anisotropy, present in square but absent in hexagonal lattices, can be so extreme that the group velocity can be either parallel or antiparallel to the phase velocity depending on the propagation direction. An analytic explanation of this effect based on the k⃗∙p⃗ perturbation theory and group-theoretical considerations is confirmed by electromagnetic simulations. One manifestation of the extreme anisotropy is the divergent van Hove singularity in the density of photonic states at the Γ-point. New applications, including surface-emitting quantum cascade lasers, are proposed.