Theory of Photonic Crystals and Microcavities :: Photonic Wannier Functions
Wannier functions are a form of Fourier transform of the Bloch functions: by summing over the extended Bloch functions, we obtain a set of localised Wannier functions. These can be used as an alternative localised basis for describing states in the band, and for doing perturbation calculations. Complications occur when several bands cross each other; it is then better to deal with Generalised Wannier Functions, which describe the set of crossing bands together, but cannot be identified with individual bands. We have successfully calculated photonic Wannier functions for a number of two dimensional lattices.
Generalised photonic Wannier functions for the first seven bands of a triangular lattice of holes - the second function is repeated five times at different orientations.
The calculated band structure, shown as dots, using these functions as basis states, is compared with the exact band structure for the lattice.