On the anatomy of Wannier functions in photonic structures

For high-performance analysis of photonic devices, with increasingly complex material and geometrical characteristics, physics-based, problem-specific modelling tools are indispensable. Thereby, the primary task is the determination of field solutions with controllable accuracy obtained in reasonable simulation times. To this end the decomposition of the a-priori unknown field functions in suitably chosen Hilbert space bases plays a significant role. Wannier functions have been found to posses excellent localization properties in spatial domain. In this work, choosing the most basic periodic boundary value problem in electromagnetics, we provide a comprehensive analysis of the distinctively superior properties of Wannier functions, including their localization properties in spatial and spectral domains, their inner and intraband orthonormality, and the nearly-optimal partitioning of the underlying Hilbert space. These properties make Wannier functions well suited candidates for a customised multiresolution analysis.