Homogenization of Irrational Metamaterials: Two-Scale Cut-and-Projection Method #

Sebastien Guenneau, Elena Cherkaev, Niklas Wellander, Frederic Zolla

10:30 Wednesday in 4Q05.

Part of the Metamaterial modelling and design session.

Abstract #

An asymptotic procedure based upon the two-scale expansion method is proposed for homogenization of wave equations in quasiperiodic inhomogeneous media. Partial differential operators (gradient, divergence and curl) acting on periodic functions with m variables in a higher-dimensional space are projected onto operators acting on quasiperiodic functions with n variables in the physical space (m>n). We replace heterogeneous quasicrystals (coined irrational electromagnetic metamaterials in the context of light waves) by effective media described by anisotropic tensors of permittivity and permeability, deduced from the resolution of annex problems of electrostatic type on a periodic cell in higher dimensional space. The main assumption being that the wavelength is much larger than the period of the higher dimensional unit cell. We derive in a heuristic way the homogenized Maxwell system via easily implementable two-scale asymptotic expansions, but two-scale cut-and-projection convergence can be applied to rigorously establish our asymptotic results. We point out that the two-scale cut-and-projection method can be applied to any physical problems in quasiperiodic structures governed by systems of linear partial differential equations. Our asymptotic procedure has been successfully applied also to irrational acoustic and mechanical metamaterials.