Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals

In analogy to the Wannier equation for localized electronic impurity states in crystalline materials, a wave equation for the envelope of the resonant optical modes of local defects within two-dimensional periodic dielectric structures is derived. In the case of degenerate satellite extrema, this is generalized to a set of coupled Wannier-like equations for a multienvelope system. The localized Wannier envelope solutions are then used in conjunction with a group theoretical symmetry analysis to determine an approximate form for donor and acceptor modes in a hexagonal photonic crystal. For an effective harmonic potential formed by varying the filling fraction of the lattice, the localized resonant modes are explicitly calculated using the Wannier equation and symmetry analysis, and a comparison to exact numerically computed modes is presented.