Scattering-matrix approach for a quantitative evaluation of the topological protection in valley photonic crystals

The realization of photonic crystal waveguides with high 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 backreflection. In this work we use valley topological triangular resonators coupled to an input waveguide to evaluate the conversion between helical topological edge modes with opposite helicity at sharp bends or routing elements like splitters. To that purpose, we first analyze via numerical simulations the existence of backward scattering at cavity corners or transmission with helicity conversion at the splitter between the input waveguide and the cavity. We show evidence that such processes take place, in particular at sharp corners, which results in transmission minima and split resonances, otherwise nonexistent. In order to evaluate the small coupling coefficients associated with this effect, a phenomenological model based on an exact parametrization of scattering matrices at splitters and corners of the resonators is then introduced. By comparison with the numerical simulations, we are able to quantify the helicity conversion at sharp bends and splitters. Finally, we use the obtained set of phenomenological parameters to compare the predictions of the model with full numerical simulations for fractal-inspired cavities based on the Sierpin ' ski triangle construction. We show that the agreement is overall good but shows more differences for the cavity composed of the smallest triangles. Our results suggest that even in a system exempt from geometrical and structural defects, helicity conversion is not negligible at corners, sharp bends, and splitters. However, simpler but predictive calculations can be realized with a phenomenological approach, allowing simulations of very large devices beyond the reach of standard numerical methods, which is crucial to the design of photonic devices which gather compactness and low losses through topological conduction of electromagnetic waves.