Quantifying topological protection in valley photonic crystals using triangular resonators

The realization of photonic crystal waveguides with topological protection enables robust light propagation against defect-induced scattering. It should allow the design of very compact devices by exploiting guiding through sharp bends with low losses and back-reflection. In this work, we use valley-topological triangular resonators coupled to an input waveguide to evaluate the quality of the topological protection. To that purpose, we analyze via numerical simulations the existence of backward scattering at cavity corners or transmission with pseudo-spin conversion at the splitter between the input waveguide and the cavity. We evidence that a breakdown of topological protection takes place, in particular at sharp corners, which results in transmission minima and split-resonances, otherwise non-existent. In order to evaluate the small coupling coefficients associated to this breakdown, a phenomenological model based on an ad hoc parameterization of scattering matrices at splitters and corners of the resonators is introduced. By comparison with the numerical simulations, we are able to quantify the loss of topological protection at sharp bends and splitters. Finally, varying the coupling rate between the input waveguide and the cavity by introducing a small gap allows reaching quality factors on the order of 10^4 to 10^6 . Our results suggest that even in a perfectly ordered system, topological protection is not complete at corners, sharp bends and splitters, which is crucial to design photonic devices which gather compactness and low losses through topological conduction of electromagnetic waves.