On the nature of eigenpairs associated with wave propagation problems in photonic structures

We consider general spatially-periodic time-invariant anisotropic dielectric media. The periodicity may occur in one-, two- or three dimensions in space. We formulate phased-periodic electromagnetic wave propagation in these structures as a generalized eigenvalue problem. We state and prove a theorem on the fundamental properties of the corresponding eigenpairs. The theorem, in conjunction with four well-established results in spectral theory, enables a thorough understanding of a myriad of intricate properties of Wannier functions associated with photonic structures from first principles. The paper is a systematic collection of foundational ideas which can be followed as an independent exposition on the subject. The results are sufficiently general to be applied to sets of partial differential equations arising in mathematical physics or engineering problems describing arbitrary periodic systems subject to periodicity and time-invariance conditions stated above. In particular, as we can show, the presented conclusions are valid not only for photonic structures, but also for phononic and quantum electronic-based devices, and diffusion problems.