Topological metasurfaces for symmetry-protected electromagnetic line waves

The discovery of topological condensed-matter systems has promoted extensive research on analogous classical photonic systems, motivated by the prospect of backscattering-immune wave propagation. So far, photonic topological insulators have mainly relied on engineering bulk modes in photonic crystals and waveguide arrays in two-dimensional systems. However, these realizations suffer from bulky structures, intricate design/material requirements, or limited operational bandwidth. Here, we present symmetry-protected topological states akin to quantum spin-Hall and valley-Hall effects by engineering surface modes over open-boundary metallic metasurfaces of infinitesimal thickness. As a result, the proposed structures support robust gapless edge states, which are confined and guided along a one-dimensional line rather than a surface interface. To emulate the spin degree of freedom, we exploit the electromagnetic-duality symmetry by stacking two complementary metasurfaces. Straightforwardly, the modal degeneracies are formed at high-symmetry K/K′ points due to the use of hexagonal unit cells, while the strong effective magneto-electric coupling inherent to the overlapped metasurfaces opens a wide non-trivial bandgap. To emulate the valley degree of freedom, on the other hand, we exploit the mirror symmetry of the structure by reducing the lattice symmetry of the hexagonal cell-based metasurface, which has either inductive or capacitive response, into C3υ point symmetry. Consequently, the degeneracy between the two valleys in reciprocal space is lifted. Owing to the simplicity, compactness, tunability, and openboundary nature of the proposed system, it constitutes an attractive tabletop platform for the study of classical topological phases, as well as for practical applications advancing the potential of photonic topological insulators.