Homogenization of Negative-Index Composite Metamaterials: A Two-Step Approach

We present a two-step homogenization method for composite metamaterials. First, each layer of wires or resonators is homogenized as a slab with negative permittivity or permeability, respectively. Second, the single negative stack which results is homogenized to form the effective medium. Comparing the predictions of the first and second step can serve as a gauge of the homogeneity of the composite. We thus take a gradual approach to homogenization, asking not whether, but to what extent a composite metamaterial approaches the sought after effective medium. Our two-step approach can also capture phenomena which otherwise may be wrongly attributed to effective medium behavior. We illustrate by qualitatively reproducing and reinterpreting a set of experimental data from the literature.

Figure 1

First each monolayer making up the composite is homogenized giving a 1D stack composed of two anisotropic layers, each characterized by a permittivity and a permeability tensor. Second, the 1D stack is homogenized to give the effective medium. The three media are referred to below as (a) “the composite” (the 3D structure), (b) “the 1D stack”, and (c) “the effective medium”.

Figure 2

Left—transmission curves for the case of plasma frequency ($\sim 8.3\mathrm{GHz}$, wires and closed resonators—dashed line) below the magnetic resonating frequency (8.6 GHz, resonators alone—dot-dashed line). Right—the field profile for a stack of 40 periods, at $f=8.93\mathrm{GHz}$ ($\lambda =33.6\mathrm{mm}$).

Figure 3

Left—Transmission curves for the case of plasma frequency ($\sim 10\mathrm{GHz}$) above the magnetic resonating frequency (8.6 GHz). Right—The field profile for a stack of 40 periods, at $f=9.2\mathrm{GHz}$ ($\lambda =32.6\mathrm{mm}$). The period is $d=5\mathrm{mm}$.

Figure 4

Left—Transmission curves for the case of plasma frequency ($\sim 10\mathrm{GHz}$) above the magnetic resonating frequency (8.6 GHz). Right—The field profile for a stack of 80 periods, at $f=9.2\mathrm{GHz}$ ($\lambda =32.6\mathrm{mm}$). The period is $d=2.5\mathrm{mm}$.