High symmetry versus optical isotropy of a negative-index metamaterial

Optically isotropic metamaterials (MMs) are required for the implementation of subwavelength imaging systems. At first glance one would expect that their design should be based on unit cells exhibiting a cubic symmetry being the highest crystal symmetry. It is anticipated that this is a sufficient condition since it is usually assumed that light does not resolve the spatial details of MM but experiences the properties of an effective medium, which is then optically isotropic. In this work we challenge this assumption by analyzing the isofrequency surfaces of the dispersion relation of the split cube in carcass negative index MM. We show that this MM is basically optically isotropic but not in the spectral domain where it exhibits negative refraction. The primary goal of this contribution is to introduce a tool that allows to probe a MM against optical isotropy.

Figure 1

Schematic of the split cube in Carcass unit cell. The Carcass (a) serves to provide a plasmalike optical response resulting in a negative effective permittivity. The split-cube cut (cut is shown for clarity) (b) provides an artificial magnetic response.

Figure 2

(Color online) (a) Real and (b) imaginary parts of the propagation constants and effective refractive indices (c) and (d) for the effectively homogeneous finite and the infinite periodic structures at normal incidence. The results for the finite structure (solid lines) are given for an increasing number of layers [one layer—blue (upper), two layers—green (middle), and five layers—red (lower)]. The dotted black line corresponds to the values obtained from the Bloch mode with the smallest losses.

Figure 3

(Color online) (a) Real and (b) imaginary parts of the propagation constant of the lowest-order Bloch mode in the first Brillouin zone. (c) Real and (d) imaginary parts of the propagation constant of an isotropic medium with the same refractive index as the SCiC at normal incidence.

Figure 4

(Color online) (a) Real part of the propagation constant as a function of the frequency and the angle of incidence where the plane of incidence is parallel to the coordinate axes. (b) The same as (a) with plane of incidence rotated by $\pi /4$, i.e., the $\Gamma \text{M}$ direction. The green isoerror lines correspond to the relative deviations $\Delta ∊\left[0.02,0.1,0.5,2,5\right]\mathrm{\%}$ (from left to right).

Figure 5

(Color online) (a) Real part (solid lines) and imaginary part (dotted lines) of the propagation constant $k\left[\mu {\text{m}}^{-1}\right]$ of the two lowest damped Bloch modes at $k_{\text{t}}=k_0$ that are degenerated at normal incidence. (b) The difference between the real parts (blue, lower line) and the imaginary parts (green, upper line) of these two modes as function of the frequency. In the quasistatic limit, i.e., $\lambda \to \infty$ the difference is of course vanishing and both modes are degenerated.