Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions

I present a treatment of highly localized defect states in photonic band-gap crystals. Vector Wannier functions, which are expected to be highly localized about the atomic sites, are introduced as an orthonormal basis to expand the fields within the crystal. A Green’s function formalism is employed to take advantage of the fact that the defect occupies only a small fraction of the volume within a primitive cell and the localized nature of the vector Wannier functions. This suggests that the size of the matrices involved in determining the defect mode frequencies may be relatively modest.