Construction of wannier functions in photonic structures

The modelling and simulation of periodic structures with defects define boundary value problems (BVPs) which are conceptually and numerically difficult to solve. Innovative and problem-tailored analysis methods need to be devised to solve defect problems efficiently and accurately. One possible attractive method is based on the ideas related to the construction of Wannier functions. Wannier functions constitute a complete sequence of localised orthogonal functions which are derived from associated periodic versions of defect problems. In this paper we review general properties of Wannier functions from a linear algebra point of view, introduce an easy-to-use symbolic notation for the diagonalisation of the governing equations and construct the Wannier functions for a variety of phononic devices. Using certain distinguished properties inherent in the wavenumber-dependence of the eigenvalues we prove the orthogonality and completeness of the Wannier functions in a conceptually novel way.