Wave propagation in twisted metamaterials

Twisted metamaterials, or arrays of identical planar metasurfaces stacked with a sequential rotation, have been recently introduced to realize broadband circular dichroism. Here we develop a generalized Floquet analysis to obtain the exact modal solutions for eigenwaves supported by these structures. The dispersion relation and wave propagation in twisted metamaterials are discussed in detail. Our analysis shows how the modal dispersion in these metamaterials becomes inherently different from the one of conventional periodic structures and how the eigenmodes support specific circular polarization properties based on a lattice effect, even when achiral inclusions are considered. These wave properties are ideal to realize optical devices that manipulate the polarization state of light over broad bandwidths. By analyzing the physical nature of these modes, including complex modes, we also extend the application of twisted metamaterials to realize passband and stop-band nanophotonic structures with strong polarization manipulation properties.

Figure 1

Schematic geometry of a twisted metamaterial. Each layer is rotated by a constant angle compared to its immediate neighbor. The transfer matrix of each twisted unit cell can be obtained by suitably rotating the transfer matrix of the first unit cell. A twisted unit cell consists of a propagation length $d$ in free space and an ultrathin metasurface in the middle.

Figure 2

(a) The real and (b) the imaginary parts of the wave number of a twisted array composed of PEC dipoles. The angle of rotation between consecutive metasurfaces is π/3 rad, and the distance between consecutive metasurfaces is $d=85$ nm. Solid and dashed lines refer to real and imaginary parts of the wave number, respectively. Real and imaginary parts of each mode have the same color. Around $kd=0.9$ the metamaterial supports a pair of complex modes.

Figure 3

The dispersion diagram of a twisted metamaterial with supercell periodicity consisting of three twisted unit cells in the direction of propagation. (a) Dispersion diagram obtained from numerical simulations, and (b) analytical results obtained from (4). The real part of the complex modes is highlighted with purple.

Figure 4

(a) Real part of the Poynting vector in the direction of propagation $\left(\mathrm{Re}\right[S_z\left]\right)$ as a function of wave number $\left(k\right)$ for the same metamaterial as in previous figures. (b) Group velocity normalized to the speed of light in vacuum of the modes as a function of the wave number. Comparison between the figures reveals that both the group velocity and the Poynting vector show the same direction of energy propagation for different modes.

Figure 5

The evolution of polarization properties for the eigenmodes of Figs. 2, 3, 4 [defined as the logarithm of the ratio of left- to right-handed components $\left({\log }_{10}\right|E_{\mathrm{L}}|/|E_{\mathrm{R}}\left|\right)$] as a function of frequency. In some frequency bands, only one pair of propagating modes with approximately circular polarization exists.

Figure 6

(a) The excitation setup and scattering coefficients for scattering from a slab. In the case of scattering from the half-space the width of the twisted material slab approaches infinity. (b) Reflection coefficients from a half-space composed of the twisted array of Fig. 2 calculated using the analytical formulation developed in this paper.

Figure 7

Transmission coefficients for a slab of twisted metamaterial with a total thickness of 1 μm. (a) Full-wave numerical simulations and (b) analytical results.