On the Customized Construction of Periodic Basis Functions: Towards Designing Maximally-locallized Wannier Functions

Periodic and Phased-periodic Boundary Value Problems (P-BVPs and PP-BVPs) occupy a prominent position in computational electromagnetics. Many BVPs in photonics involving the coupled system of Maxwell’s equations, and materials engineering problems utilizing the time-independent Schrödinger equation, lead to PP-BVPs. Bloch waves and the Floquet Theorem, when hybridized with other traditional numerical techniques are classical tools for solving P-BVPs and PP-BVPs. One important feature of the eigenfunctions for PP-BVPs is that they can be linearly combined to obtain localized and maximally-localized solutions of the underlying problems, leading to the so-called Wannier functions associated with the PP-BVPs. This paper scrutinizes the foundation of the governing equations and provides the first general recipe for the customized construction of periodic basis functions. It is hoped that the proposed procedure will further facilitate rigorous modeling of void and defects, in otherwise periodic structures.