Dual-band rainbow trapping based on gradient topological edge states in a phononic crystal

While topological states offer a variety of wave manipulation functionalities, the multi-band wave engineering is usually hindered by the lack of a mechanism to produce multiple, well-isolated edge states simultaneously. In this work, we propose a valley topological phononic crystal that consists of two groups of rectangular cavities embedded in a hexagonal scatterer, enabling the simultaneous existence of multiple distinct edge states in different bandgaps along a single interface. By tuning the geometric parameters in the unit cell, the band structure is engineered to exhibit gradient edge states with clear bulk-edge separation. Such a gradient topological interface supports bidirectional frequency separation over two distinct edge states. Based on this principle, a bidirectional gradient “rainbow trapping” is further proposed, providing an efficient approach for multi-band acoustic separation and energy localization.