Joint work with Vadim Markel (University of Pennsylvania and Aix-Marseille Université, Institut Fresnel)
Most effective medium theories for periodic electromagnetic structures are asymptotic – i.e. valid in the limit of the lattice cell size vanishingly small relative to some characteristic scale. (For wave problems, this scale is the free-space wavelength; in statics, it is the scale of variation of the applied field and/or the size of the material sample.) It is now understood, however, that in this asymptotic limit all nontrivial effects – including, notably, magnetic response of intrinsically nonmagnetic structures – vanish.
We have developed non-asymptotic and nonlocal theories applicable to an arbitrary size and composition of the lattice cell. Interface boundaries play a critical role in the analysis and are an integral part of the methodology. Numerical examples demonstrate that nonlocal models can improve the accuracy of homogenization by an order of magnitude.
Effective medium theories of classical physics (Clausius-Mossotti, Lorenz-Lorentz, Maxwell Garnett) rely on simplification assumptions that work well for relatively simple mixtures but require extensions and enhancements in more complicated cases. While our perspective is very different from that of the 19th century physics, we show that classical theories fit nicely into the proposed framework.