Singular analysis to homogenize planar metamaterials as nonlocal effective media

Homogenization of metamaterials based on the analysis of their reflection and transmission spectra often leads to nonphysical values of the effective permittivity and permeability. Here we address this challenge by analyzing the scattering spectra in terms of perfectly emitting and absorbing modes considering nonlocal constitutive relations. The decomposition of the scattering spectra into even and odd modes permits us to quantify the nonlocal contributions and provides explicit formulas for the effective permittivity and permeability which always satisfy the passivity and causality constraints.

Figure 1

(a) Sketch of the fishnet structure; the boundaries of the elementary cell are shown as a dashed contour. (b) Reflection (blue), transmission (green), and absorption (shaded) spectra of the fishnet structure. The normal incidence is considered, with the electric field polarized along the $y$ axis. The results of the finite element method [21] are shown by filled circles, while the results of the analytical formulas [22] with parameters in Table 1 are plotted as solid lines. (c) The contributions of the even and odd modes to the absorption spectrum.

Figure 2

The effective parameters of the fishnet structure according to Eqs. (5) and (7).

Figure 3

The reflection and transmission spectra of the fishnet structure (filled circles) in comparison with the spectra of a homogeneous slab (solid lines). The effective parameters of the slab were found by assuming local (a) and nonlocal (b) response with $\alpha =0$ and $-{\varepsilon }_{\mathrm{eff}}{\mu }_{\mathrm{eff}}/\left(3G^2\right)$, respectively.