Lower limits for the homogenization of periodic metamaterials made from electric dipolar scatterers

Nonlocal constitutive relations promise to homogenize metamaterials even though the ratio of period over operational wavelength is not much smaller than unity. However, this ability has not yet been verified, as frequently only discrete structures were considered. This denies a systematic variation of the relevant ratio. Here, we explore, using the example of an electric dipolar lattice, the superiority of the nonlocal over local constitutive relation to homogenize metamaterials when the period tends to be comparable to the wavelength. Moreover, we observe a breakdown of the ability to homogenize the metamaterial at shorter lattice constants. This surprising failure occurs when energy is transported across the lattice thanks to a well-pronounced near-field interaction among the particles forming the lattice. Contrary to common wisdom, this suggests that the period should not just be much smaller than the operational wavelength to homogenize a metamaterial, but, for a given size of the inclusion, there is an optimal period.

Figure 1

Amplitude of the reflection coefficient, $|{\rho }^{\text{REF}}(k_0,k_x=0)|$, for a range of lattice constants generated using the general Mie method.

Figure 2

Amplitude value of the reflection coefficient $\left|\rho \right|$ retrieved using the WSD model for comparison with the data.

Figure 3

Effective material parameter, permittivity $\epsilon \left(k_0\right)$ and permeability $\mu \left(k_0\right)$, retrieved with the WSD model.

Figure 4

Amplitude of the reflection coefficient $|{\rho }^{\text{REF}}|$ as calculated with the general Mie method at different angles of incidence. (a) The value at normal incidence of light; here, only the symmetric oscillation of the dipole is visible. (b),(c) The value at angles ${30}^{\circ }$ and ${60}^{\circ }$, respectively, showing the growth of secondary resonance on increasing the angle of incidence.

Figure 5

Branching solution of the objective function $\delta \left(k_0\right)$ for the electric dipole array with a fixed lattice constant $a$ = 250 nm. The curve splits into two different branches at $k_0=5.1\mu {\mathrm{m}}^{-1}$ with the blue curve for $\mathrm{Im}\gamma \left(k_0\right)<0$ and the yellow curve for $\mathrm{Im}\gamma \left(k_0\right)>0$.

Figure 6

The objective function $\delta \left(k_0\right)$ in logarithmic scale as a function of frequency in the considered range of lattice constants. (a) The WSD model. (b) The SSD model. In comparison, the SSD model has a better fit compared to the WSD model.

Figure 7

Sum of objective function over the considered frequency range plotted against the respective lattice constant. The red squares denote the WSD model and the yellow squares denote the SSD model.

Figure 8

Poynting vector $\langle S_z\rangle$ for the electric dipole array with a fixed lattice constant $a=250$ nm. Here, the blue curve refers to $\mathrm{Im}\gamma \left(k_0\right)<0$ and the yellow curve refers to $\mathrm{Im}\gamma \left(k_0\right)>0$. The curve denoting $\mathrm{Im}\gamma \left(k_0\right)>0$ shows the energy flux to be in the direction away from the source $\left[\right(\langle S_z\rangle )>0]$, which is the right solution.