A rational framework for dynamic homogenization at finite wavelengths and frequencies

Abstract

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (with or without the source term) at finite frequencies and wavelengths spanning the first Brioullin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency (FW-FF) homogenization is pursued in~$\mathbb{R}^d$ via second-order asymptotic expansion about the apexes of "wavenumber quadrants" comprising the first Brioullin zone, at frequencies near given (optical) dispersion branch. We also consider the degenerate situations of crossing or merging dispersion branches with arbitrary multiplicity, where the effective description of wave motion reveals several distinct asymptotic regimes depending on the symmetries of the eigenfunction basis affiliated with a repeated eigenvalue. One of these regimes -- for whose occurence we expose a sufficient condition -- is shown to describe the so-called Dirac points, i.e. conical contacts between dispersion surfaces, that are relevant to the phenomenon of topological insulation. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the basis eigenfunction, (ii)~the reference cell of medium periodicity, and (iii)~the wavenumber-frequency scaling law underpinning the asymptotic expansion. We illustrate the utility of our analysis by several examples, including an asymptotic description of the Green's function near the edge of a band gap.