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In this work we propose a topological valley phononic crystal plate and weextensively investigate the refraction of valley modes into the surroundinghomogeneous medium. This phononic crystal includes two sublattices ofresonators (A and B) modeled by mass-spring systems. We show that two edgestates confined at the AB/BA and BA/AB type domain walls exhibit differentsymmetries in physical space and energy peaks in the Fourier space. As aresult, distinct refraction behaviors, especially through an armchair cut edge,are observed. On the other hand, the decay depth of these localized topologicalmodes, which is found to be solely determined by the relative resonant strengthbetween the scatterers, significantly affects the refraction patterns. Moreinterestingly, the outgoing traveling wave through a zigzag interface becomesevanescent when operating at deep sub-wavelength scale. This is realized bytuning the average resonant strength. We show that the evanescent modes onlyexist along a particular type of outlet edge, and that they can couple withboth topological interface states. We also present two designs of topologicalfunctional devices, including an elastic one-way transmission waveguide and anear-ideal monopole/dipole emitter, both based on our phononic structure.

Abnormal topological refraction into free medium at sub-wavelength scale in valley phononic crystal plates