Spatial-dispersion-induced birefringence in metamaterials with cubic symmetry

We consider an array of isotropic particles possessing electric polarizability located in the sites of a cubic lattice. The dispersion properties of the structure and the polarization of eigenmodes are studied employing the discrete dipole model. We reveal the features beyond the effective medium model, namely, the anisotropy of the structure induced by spatial dispersion, and suggest a simple experiment allowing one to observe the effect. Additionally, we demonstrate that the eigenmodes of the structure are neither transverse nor longitudinal but have a “mixed” polarization state in the general case.

Figure 1

Metamaterial composed of isotropic particles arranged in the sites of a cubic lattice.

Figure 2

The dispersion diagram for the structure composed of isotropic particles with electric polarizability.

Figure 3

The typical isofrequency contours for the cubic lattice of isotropic particles with electric polarizability. $k_y=0$. $T_{xz}, T_y$, and $L$ mark isofrequency contours corresponding to the transverse waves with polarization in the plane $Oxz$, transverse waves polarized along the $y$ axis, and longitudinal waves, respectively.

Figure 4

(a) The dispersion diagram for the direction of propagation $(cos\phi ,0,sin\phi ), \phi ={30}^{\circ }$. Solid curve corresponds to the “transverse” wave polarized in the plane $Oxz$ (the angle between the electric field and the wave vector is greater than ${45}^{\circ }$); dashed curve corresponds to the transverse wave polarized along the $y$ axis, and the dot-dashed curve corresponds to the “longitudinal” mode (the angle between the electric field and the wave vector is smaller than ${45}^{\circ }$). (b) The angle ${\phi }_1$ between the local electric field and the wave vector for the “longitudinal” wave, that coincides with the angle ${\phi }_2-{90}^{\circ }$ for the “transverse” wave.

Figure 5

(a) A scheme of experiment allowing one to detect the anisotropy of metamaterial due to spatial dispersion by measuring the reflection coefficients for $x$ and $y$ polarizations of the incident wave. (b) Top view of the two upper layers of particles. Top particles are shown by solid line; bottom ones are shown by dashed contour. The choice of coordinate axes is different from that used in the previous sections.

Figure 6

Calculated reflection coefficients for different polarizations of the incident wave: solid curve for $x$ polarization; dashed curve for $y$ polarization in the geometry of Fig. 5; dot-dashed curve shows the result derived from the local effective medium model, the same for both polarizations. (a) Discrete dipole model. (b) cst microwave studio.

Figure 7

(a) Illustration of the experiment in Fig. 5 in terms of isofrequency contours. Structure supports propagation of $y$-polarized eigenmode whereas $x$-polarized mode is evanescent. (b) The boundary of the structure is cut along the (0,0,1) plane. Reflection at oblique incidence reveals the birefringence effect. $\vec{k}$ and $\vec{S}$ denote wave vector and Poynting vector, respectively.