Metamaterials with index ellipsoids at arbitrary k-points

Propagation behaviors of electromagnetic waves are governed by the equifrequency surface of the medium. Up to now, ordinary materials, including the medium exist in nature and the man-made metamaterials, always have an equifrequency surface (ellipsoid or hyperboloid) centered at zero k-point. Here we propose a new type of metamaterial possessing multiple index ellipsoids centered at arbitrary nonzero k-points. Their locations in momentum space are determined by the connectivity of a set of interpenetrating metallic scaffolds, whereas the group velocities of the modes are determined by the geometrical details. Such system is a new class of metamaterial whose properties arise from global connectivity and hence can have broadband functionality in applications such as negative refraction, orientation-dependent coupling effect, and cavity without walls, and they are fundamentally different from ordinary resonant metamaterials that are inherently bandwidth limited. We perform microwave experiments to confirm our findings.


Supplementary Figure 1 | 2x2 supercell of the double wire mesh metamaterials shown in
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In panel (a), all metals are shown in the same colour (grey). Panel (b) shows the same structure, but the red and blue colour illustrates the fact that the structure has two disjoint networks, with the connected parts rendered in the same colour. . We see that the 1D cone dispersion at quasistatic limit exists for arbitrary k x and k y . the horizontal and vertical lattice constants are a and d=0.6a. The helical wires are replaced by slanted cylinders and the in-plane connecting bars are changed to have a more tabular cross section. Here different colours highlight sets of unconnected meshes with independent potentials. We see that the structure in (a) still has two light cones at (0 0 2 / 3 ) − π d and (0 0 2 / 3 ) π d and that the structure in (b) still has three light cones at ( / / /2 ) π π π a a d , ( / / / 2 ) a a d π π −π and A point. . Therefore ( / / / ) a a a = π π π k which is consistent with the band structure in Fig. 1f.

Supplementary Note 2: Quasistatic potential analysis of the helical wire bundle
We can understand the shift of quasistatic modes in helical wire bundle by considering the quasistatic potential. Supplementary Figure 2 shows its unit cell. Suppose the quasistatic potentials of the green, violet and pink wires are 4 ϕ , 5 ϕ and 6 ϕ . Due to the C 6 symmetry, they are related by angular momentum, to be localized in the background. Note that these two bands can also be found in Fig. 2c for a single bundle. One is the cyan band when k z approaches to zero. The other is the circular polarized plane-wave propagating in the air background (the black dashed line in Fig. 2c).

Supplementary Note 5: The number of index ellipsoid and the connectivities of the wires
The number of index ellipsoid is determined by the number of independent potentials (disconnected wire meshes). We take the wire metamaterial in Fig. 3b for example. To see whether the wire meshes have independent potentials, one needs to plot the structure with many unit cells (see Supplementary Fig. 6a). Here different colours highlight different sets of wire meshes that have independent potentials. For clarity, only the interlayer connections (double lines) in the center unit cell are plotted. We found that the helical wires on opposite sides are connected and have the same potential. Thus this wire metamaterial is composed of three interpenetrating wire meshes (red, blue and green). Due to the uniqueness theorem, the system has two nontrivial and linearly independent solutions for the electric field at zero frequency. This is consistent with the computed band structure in Fig. 3f, which shows a light cone emerging from a nonsymmetric point between the Γ and the A points. The other zero frequency mode can be inferred by applying time-reversal symmetry. To see the connectivities, we also plot the other three wire metamaterials (those in Supplementary Fig. 7a, Figs. 4b & 4c) with many unit cells in Supplementary Fig. 6. Their numbers of index ellipsoids are also consistent with the numbers of disconnected wire meshes. Similarly, we can design wire metamaterials with index ellipsoid at other k-point with fractional number of reciprocal lattice and can also introduce more index ellipsoids by adding more wire meshes.

Supplementary Note 6: Excluding quasistatic modes by designing the in-plane connection
In Supplementary Fig. 7, we selectively exclude the quasistatic modes with 1, 2 m = ± ± by considering another configuration of in-plane connection in hexagonal array of wire bundle. We add six metallic bars (gray) in a way shown in Supplementary Fig. 7b and impose the condition that the connected metallic helixes to have equal potentials in the zero frequency limit. For example, the pink and yellow (violet and blue) helixes in Supplementary Fig. 7b are connected by metallic bars through the unit cell boundary along the y-direction. Then we have

Supplementary Note 7: Band structure description using a (2x2) unit cell
We note that band structure results are typically displayed using the reduced zone scheme. If we choose a bigger unit cell, the light cone at non-zero k-points can be "folded back" into the Γ point within the reduced zone scheme. But the physics cannot be changed by choosing a different cell. What is important is the Fourier component of the Bloch modes. If we perform a Fourier analysis of the Bloch modes, the "physically meaningful and dominant" k-components (measured in absolute units) will not change if we enlarge the unit cell. For example, if we purposely choose a bigger unit cell to fold the light cone back to zone center, the Bloch modes at low frequencies will still have predominately strong components at non-zero k-points.
To illustrate this point, we calculate the band structure of the square metasurface in Fig. 5a Supplementary Fig. 9a. We see that most of the fields localize inside the PCB board and their phases are almost opposite in the neighboring unit cells. Lower panel calculates the amplitudes of different k-components of this eigen mode. The colour of the centered patch in the 7x7 array represents the amplitude with (k x ,k y )=(π/20a, 0). It is seen that the Fourier field components concentrates on the four k-components with (k x ,k y )=(π/20a± π/a, ± π/a). This is consistent with the fact that it is an eigen mode near M point if we choose a primate unit cell. For comparison, Supplementary Figure 9c calculates the Fourier components of the propagating mode in air. Most of its Fourier field components concentrate in the zone center (the red patch at the center).

Supplementary Note 8: Engineering the equifrequency contour of the lowest band
The ability to control the number and position of the quasistatic modes gives us the freedom and flexibility to design the shape of the equifrequency surface at low frequency. To illustrate this idea, we consider the Brillouin zone of a 2D rhombic lattice (see for example, the black rhombus in Supplementary Fig. 12a). Such a system has one quasistatic mode locating at the zone corner.
The left, right and up (down) corners of the rhombus (Brillouin zone boundary) are equidistant and form a regular triangle. The quasistatic mode at these points can be viewed as the starting points of the low frequency contour and equifrequency contours are concentric circles emerging from these corners. As frequency increases from zero, equifrequency contours will become bigger and three contours (the left, right and up circles in Supplementary Fig. 12a) will come together to form a triangle. Likewise, another triangle forms in the lower half of the Brillouin zone. Supplementary Figure 12b shows the unit cell of metasurface with a rhombic lattice. This structure is similar to that in Fig. 5a, but its lattice is compressed along the y-direction in order to obtain the rhombic Brillouin zone in Supplementary Fig. 12a. The rhombic cell is 2a ( 2 / 3a )-long in the major (minor)-axis direction. Since it has the same connectivity as that in Fig. 5a, this metasurface has one quasistatic mode at zone corner. Supplementary Figure 12c calculates the equifrequency contour of the lowest band. The contour with frequency of 0.11 c/a forms two triangles although parts of them are shadowed by the projected light cone (gray solid circle) coming from free space above and below the meta-slab.
By introducing more quasistatic modes and controlling the slopes of the linear bands, different shapes of equifrequency contour/surface can in principal be designed to exhibit exotic refraction or wave transport behavior at low frequency. We note that complicated equifrequency surface can be obtained in the high frequency band of photonic crystals using band folding (Bragg scattering). Here we show that, with the new degree of freedom of quasistatic mode at nonzero k-point, exotic equifrequency surface (such as negative refraction medium) can also be obtained in low frequency and have broad bandwidth.

Supplementary Note 9: Cavity without wall
The quasistatic modes at nonzero k-points can be used to realize a cavity without wall. Consider a cube of the wire metamaterial shown in Fig. 4b, which has an index ellipsoid at Brillouin zone corner. When all of its interfaces are normal to the principle axis (x-, y-and z-direction), the EM wave propagating inside the medium can hardly couple out to the plane wave mode in air due to the mismatched k // component. This can serve as a cavity without wall at low frequency.
Supplementary Figure