Agnes Lamacz: Effective Maxwell's equations in a geometry with flat split-rings and wires

Propagation of light in heterogeneous media is a complex subject of research. It has received renewed interest in recent years, since technical progress allows for smaller devices and offers new possibilities. At the same time, theoretical ideas inspired further research. Key research areas are photonic crystals, negative index metamaterials, perfect imaging, and cloaking. 

The mathematical analysis of negative index materials, which we want to focus on in this talk, is connected to a study of singular limits in Maxwell's equations. We present a result on homogenization of the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that many (order \(\eta^{-3}\)) small (order \(\eta^1\)), flat (order \(\eta^2\)) and highly conductive (order \(\eta^{-3}\)) metallic split-rings are distributed in a domain \(\Omega\subset \mathbb{R}^3\). We determine the effective behavior of this metamaterial in the limit 
\(\eta\searrow 0\). For \(\eta>0\), each single conductor occupies a simply connected domain, but the conductor closes to a ring in the limit \(\eta\searrow 0\). This change of topology allows for an extra dimension in the solution space of the corresponding cell-problem. Even though both original materials (metal and void) have the same positive magnetic permeability \(\mu_0>0\), we show that the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response. Furthermore, we demonstrate that combining the split-ring array with thin, highly conducting wires can effectively provide a negative index metamaterial.