Effective optical response of metamaterials

We use a homogenization procedure for Maxwell’s equations in order to obtain in the local limit the frequency-dependent macroscopic dielectric-response tensor ${ϵ}_{ij}^M\left(\omega \right)$ of metamaterials made of a matrix with inclusions of any geometrical shape repeated periodically with any lattice structure. We illustrate the formalism calculating ${ϵ}_{ij}^M\left(\omega \right)$ for several structures. For dielectric rectangular inclusions within a conducting material we obtain an anisotropic response that may change from conductorlike at low $\omega$ to dielectriclike with resonances at large $\omega$, attaining a very small reflectance at intermediate frequencies which can be tuned through geometrical tailoring. A simple explanation allowed us to predict and confirm similar behavior for other shapes, even isotropic, close to the percolation threshold.

Figure 1

(Color online) We show the normal-to-the-axis macroscopic dielectric function ${ϵ}^M$ obtained through Eq. (17) for a 2D square array with lattice parameter $L$ of square dielectric prisms with response ${ϵ}_p=5.0$ placed in vacuum with filling fraction $f=0.3$ for a free-space wavelength ${\lambda }_0=10L$ as a function of the largest reciprocal-vector index vs $1/n_{\text{max}}$. We show a linear extrapolation of our results toward $1/n_{\text{max}}\to 0$ and we indicate the values predicted by Maxwell-Garnet formula and by some other authors mentioned in the text.

Figure 2

(Color online) Keller’s coefficient $K_x-1$ as a function of $1/n_{\text{max}}$ and its linear extrapolation toward $n_{\text{max}}\to \infty$ for the same system as in Fig. 1 in the cases of ${\lambda }_0/L=10$ and 100.

Figure 3

(Color online) ${ϵ}^M$ versus the filling fraction for the same system as shown in Fig. 1. Our result was obtained from Eqs. (17, 18) employing a $13120\times 13120$ matrix ${\left[{\mathcal{W}}_{\mathbf{0}}\left(\mathbf{G},{\mathbf{G}}^{\prime }\right)\right]}_{ij}$.

Figure 4

Unit cell of a 2D rectangular array of rectangular prisms with response ${ϵ}_p$ within a host with response ${ϵ}_h$. The aspect ratio of the rectangles is determined by the point $A=\left(L_x{\xi }_x,L_{y}f/{\xi }_x\right)$ that lies on the dashed line from $\left(L_{x}f,L_y\right)$ to $\left(L_x,L_{y}f\right)$.

Figure 5

(Color online) Reflectance $R_{\zeta }$ $\left(\zeta =x,y\right)$ vs photon energy for ${\xi }_x=1$, i.e., for a 1D multilayer of alternating slabs of gold $\left[{ϵ}_h=ϵ\left(\text{Au}\right)\right]$ and a dielectric $\left({ϵ}_p=4\right)$, with $L=40\text{nm}$ and $f=0.5$. The slabs are normal to the $y$ direction and the incoming light propagates along the $z$ direction. We compare numerical results obtained from Eq. (17) with the exact nonretarded results obtained through Eq. (27) for $R_x$ and Eq. (28) or Eq. (29) (Maxwell-Garnet in 1D) for $R_y$.

Figure 6

(Color online) Reflectance $R_x$ (thin lines) and $R_y$ (thick lines) vs the photon energy, for a gold host with inclusions of ${ϵ}_p=4$ and a fixed ${\xi }_y=0.5$, for three different values of ${\xi }_x=0.5$, 0.7, and 0.9 (see text for details). We also show the reflectivity of gold.

Figure 7

(Color online) Real (top panel) and imaginary (bottom panel) parts of ${ϵ}_{yy}^M$ vs photon energy for the same system as that presented in Fig. 6.

Figure 8

(Color online) Reflectance $R_x=R_y=R$ vs photon energy for a system made up of circular cylinders with ${ϵ}_p=4$ within an Au matrix ${ϵ}_h=ϵ\left(\text{Au}\right)$ for four values of the filling fraction $f=0.25$, 0.5, 0.6, and 0.7. We also show the reflectance of gold for comparison.