Homogenization of Split-Ring Arrays, Seen as the Exploitation of Translational Symmetry

Homogenization, which reduces the cost of numerical simulations in materials with repetitive structure, is a promising approach to the design of metamaterials. This cost reduction stems from the possibility to compute effective permeability and permittivity of an equivalent homogenized material by solving an auxiliary "cell problem" on the generating cell of the metamaterial. The first part of this paper is a tutorial, where the procedure is described in the context of the exploitation of symmetry via harmonic analysis, and justified by an appropriate asymptotic result when the size of the cell is small enough. The second part argues that this standard approach can fail, and explains how it does, when a second small parameter, besides the cell’s size, is present in the physical situation. This is precisely what happens in the case of an array of split rings, where the slit’s width competes, so to speak, with the cell’s size in the passage to the limit that leads to the cell problem. We show how this competition must be arbitrated in order to recover the negative effective permeability one may expect, on physical grounds, near some resonant frequency, in the case of a split-rings array. A simplified model, amenable to analytical computation, illustrates this "frequency dependent homogenization" procedure.