Photonic crystals as topological high-Q resonators

It is well known that defects, such as holes, inside an infinite photonic crystal can sustain localized resonant modes whose frequencies fall within a forbidden band. Here we prove that finite, defect-free photonic crystals behave as mirrorless resonant cavities for frequencies within but near the edges of an allowed band, regardless of the shape of their outer boundary. The resonant modes are extended, surface-avoiding (nearly-Dirichlet) states that may lie inside or outside the light cone. Independent of the dimensionality, quality factors and finesses are on the order of, respectively, (L/λ)3 and L/λ, where λ is the vacuum wavelength and L >> λ is a typical size of the crystal. Similar topological modes exist in conventional Fabry-Pérot resonators, and in plasmonic media at frequencies just above those at which the refractive index vanishes.