Negative refraction of phonons and acoustic lensing effect of a crystalline slab

We study how good a flat slab of a bulk crystalline solid with a large elastic anisotropy exhibits a lensing action for phonons or sound waves. The slowness and group-velocity surfaces of an ideal elastic solid for a flat phonon lens are analyzed in the geometrical acoustic approximation. These surfaces are compared with the corresponding surfaces of an existing bulk crystal (a zinc crystal) with hexagonal symmetry. To demonstrate the lensing effect we calculate the intensity distribution of phonons emitted from a point source in an isotropic medium (on one side of the lens), propagating through the slab lens and then transmitted into the isotropic medium in the other side. A similar calculation for sound waves with a finite-difference-time-domain method is performed to see the effects neglected in the geometrical acoustic approximation, that is, the effects of finite wavelength, mode conversion, and finite transmission at the interfaces.