  In this work we propose a topological valley phononic crystal plate and we
extensively investigate the refraction of valley modes into the surrounding
homogeneous medium. This phononic crystal includes two sublattices of
resonators (A and B) modeled by mass-spring systems. We show that two edge
states confined at the AB/BA and BA/AB type domain walls exhibit different
symmetries in physical space and energy peaks in the Fourier space. As a
result, distinct refraction behaviors, especially through an armchair cut edge,
are observed. On the other hand, the decay depth of these localized topological
modes, which is found to be solely determined by the relative resonant strength
between the scatterers, significantly affects the refraction patterns. More
interestingly, the outgoing traveling wave through a zigzag interface becomes
evanescent when operating at deep sub-wavelength scale. This is realized by
tuning the average resonant strength. We show that the evanescent modes only
exist along a particular type of outlet edge, and that they can couple with
both topological interface states. We also present two designs of topological
functional devices, including an elastic one-way transmission waveguide and a
near-ideal monopole/dipole emitter, both based on our phononic structure.
