Splitting of triply degenerate refractive indices by photonic crystals

The refractive index is inherently triply degenerate in conventional crystals, but this degeneracy is lifted and the refractive index is split into a phase, a group, and a "fan" index by photonic crystals. These indices are explained as three different sections passing through the photonic band structure. We observed the fan index, which describes beam divergence and is defined by the local curvature of the equienergy section, range from near zero to almost infinity. This paper provides a unified description of photonic and conventional crystal optics, and may enable previously inconceivable optical designs.