Three-dimensional electromagnetic metamaterials that homogenize to uniform non-Maxwellian media

We consider three-dimensional structures consisting of multiple interlocking, disconnected metal networks. We show that, in the low-frequency limit, these systems support photonic bands with linear dispersion relation. The number of such photonic bands is directly controlled by the number of networks and can be arbitrarily large. We construct a non-Maxwellian effective medium for these systems. The effective medium supports multicomponent effective fields, with the numbers of field components designable by geometry.

Figure 1

(Color) Two-network structure. (a) A single cubic unit cell of the structure is depicted. Each network, represented by different colors, consists of wires along the edges of a cube. The two networks are displaced from one another by $\sqrt{3}a∕2$ along the [111] direction. (b) The band diagram is shown along high-symmetry $k$ points. The right inset shows the high-symmetry points in the first Brillouin zone. The left inset shows the electric field profile on the $z=0$ plane, for a mode at $\omega =0.033(2\pi c∕a)$, with its wave vector along the [001] direction. Here, $c$ is the speed of light in vacuum.

Figure 2

(Color) $N$-network cubic structure, where $N$ can be arbitrarily large. Each network has metal wires along the $x$, $y$, and $z$ directions. For the $i\text{th}$ network, these wires meet at $(\pm x_i,\pm x_i,\pm x_i)$ in the cubic unit cell. Shown here is a five network example with $x_i$’s chosen to be 0, $a∕12$, $3a∕16$, $a∕3$, and $a∕2$: (a) the unit cell and (b) the band diagram.

Figure 3

(Color) Four-network fcc structure. (a) Unit cell; (b) band diagram. Each network [inset of (b)] is identical in shape and is made by translating $x$-, $y$-, and $z$-directional wires on the edges of a cube by $\pm (a∕8)\widehat{y}$, $\pm (a∕8)\widehat{z}$, and $\pm (a∕8)\widehat{x}$, respectively. These wires are then connected by a small cube at each site of a face-centered-cubic (fcc) lattice. The basis element is topologically identical to Borromean rings (Ref. 29).

Figure 4

(Color) Detailed comparison of theory and simulations for the four-network fcc system shown in Fig. 3. (a) Isofrequency surfaces and contours of three low-frequency modes. The plot on the right shows the contour on the $k_x=0$ plane. Gray dashed lines represent a square tangential to some of the contours. (b) Frequencies of modes for $k$-vectors along a ${\mathbf{k}}_{\mathrm{a}}-{\mathbf{k}}_{\mathrm{b}}$ line segment, where ${\mathbf{k}}_{\mathrm{a}}=(2\pi ∕a)$ (0.0025, 0.016, 0.0175) and ${\mathbf{k}}_{\mathrm{b}}=(2\pi ∕a)$ (0.0225, 0, 0.0075). The lines are theory predictions and the circles are finite difference time domain results. These $k$ points are arbitrarily chosen. The agreement between theory and simulation is, in fact, excellent for all $k$ points.