Supplyment to the article"Experimental observation of valley Hall topological waveguide based on interface engineering"

However, the scenario where topological edge states are localized and confined at specific interfaces in a frequency-dependent manner remains unexplored and unreported. In this work, by leveraging the coupling effects of different interface stacking types, we systematically investigate valley Hall edge states at distinct interfaces of valley photonic crystals (VPCs), including bearded and armchair interfaces.

Subsequently, we construct a U-shaped topological waveguide composed of these interfaces. Both numerically and experimentally, we demonstrate that valley Hall topological states in this U-shaped waveguide can be localized and confined at a designated interface in a frequency-dependent manner. Compared with conventional topological rainbow systems — where edge states of different frequencies are separated and trapped at distinct spatial positions — our proposed waveguide avoids complex techniques such as modulating external magnetic fields or designing structures with gradually varying parameters, thereby greatly facilitating photonic integration.