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In this work, we use valley-topological triangular resonators coupled to aninput waveguide to evaluate the quality of the topological protection. To thatpurpose, we first analyze via numerical simulations the existence of backwardscattering at cavity corners or transmission with pseudo-spin conversion at thesplitter between the input waveguide and the cavity. We evidence that abreakdown of topological protection takes place, in particular at sharpcorners, which results in transmission minima and split-resonances, otherwisenon-existent. In order to evaluate the small coupling coefficients associatedto this breakdown, a phenomenological model based on an exact parameterizationof scattering matrices at splitters and corners of the resonators is thenintroduced. By comparison with the numerical simulations, we are able toquantify the loss of topological protection at sharp bends and splitters.Finally, we use the obtained set of phenomenological parameters to compare thepredictions of the phenomenological model with full numerical simulations forfractal-inspired cavities based on the Sierpi\'nski triangle construction. Weshow that the agreement is overall good, but shows more differences for thecavity composed of the smallest triangles. Our results suggest that even in asystem exempt of geometrical and structural defects, topological protection isnot complete at corners, sharp bends and splitters. However, simpler butpredictive calculations can be realized with a phenomenological approach,allowing simulations of very large devices beyond the reach of standardsimulation methods, which is crucial to design photonic devices which gathercompactness and low losses through topological conduction of electromagneticwaves.

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