Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides

Recent advances in designing time-reversal-invariant photonic topological insulators have been extended down to the deep subwavelength scale, by employing synthetic photonic matter made of dense periodic arrangements of subwavelength resonant scatterers. Interestingly, such topological metamaterial crystals support edge states that are localized in subwavelength volumes at topological boundaries, providing a unique way to design subwavelength waveguides based on engineering the topology of bulk metamaterial insulators. While the existence of these edge modes is guaranteed by topology, their robustness to backscattering is often incomplete, as time-reversed photonic modes can always be coupled to each other by virtue of reciprocity. Unlike electronic spins which are protected by Kramers theorem, photonic spins are mostly protected by weaker symmetries like crystal symmetries or valley conservation. In this paper, we quantitatively studied the robustness of subwavelength edge modes originating from two frequently used topological designs, namely metamaterial spin-Hall (SP) effect based on C6 symmetry, and metamaterial valley-Hall (VH) insulators based on valley preservation. For the first time, robustness is evaluated for position and frequency disorder and for all possible interface types, by performing ensemble average of the edge mode transmission through many random realizations of disorder. In contrast to our results in the previous study on the chiral metamaterial waveguide, the statistical study presented here demonstrates the importance of the specific interface on the robustness of these edge modes and the superior robustness of the VH edge stated in both position and frequency disorder, provided one works with a zigzag interface.