Effects of discreteness in the Green's function of a hyperbolic medium

We present a theoretical study of the discrete Green's function for the two-dimensional metamaterial, realized as a transmission line array. We demonstrate that the Green's function possesses distinct ripples when the isofrequency contour is of hyperbolic shape. This effect is a characteristic nonlocal feature of the hyperbolic regime and is beyond the effective medium approximation.

Figure 1

Schematic illustration of the hyperbolic metamaterial based on a transmission line. The inset shows the unit cell with inductances and capacitances.

Figure 2

(a) The simulated spatial distribution of the absolute values of the Green's function in the two-dimensional elliptic medium with ${\eta }_x=28$, ${\eta }_y=31$. (b) Isofrequency curves obtained by numerical solution of the dispersion relation Eq. (5). The calculation has been performed for $x_0=y_0=26$, and the structure included $51\times 51$ unit cells.

Figure 3

The simulated spatial distribution of the absolute values of the Green's function in the two-dimensional hyperbolic medium with ${\eta }_x=-1.1$, ${\eta }_y=1.9$. (b) Isofrequency curves obtained by numerical solution of the dispersion relation Eq. (5).

Figure 4

(a) The simulated spatial distribution of the Green's function absolute values in the two-dimensional hyperbolic medium with ${\eta }_x=-1.5$, ${\eta }_y=1.5$. (b) Isofrequency curves obtained by numerical solution of the dispersion relation Eq. (5). (c) The absolute values of the Green's function at the fixed coordinate $y=10$.

Figure 5

(a) The spatial distribution of the Green function absolute values in the two-dimensional hyperbolic medium obtained after Eq. (7). (b) Isofrequency curves obtained by numerical solution of the dispersion relation Eq. (5). (c) Approximate spatial distribution of the Green function absolute values in the two-dimensional hyperbolic medium obtained after Eq. (8). (d) Approximate isofrequency curves obtained from the continuous-limit dispersion relation Eq. (2). Calculation parameters are the same as in Fig. 3.