Klein-like tunneling in acoustic metamaterials

Quantum tunneling is the ability of relativistic particles to enter a classically forbidden region, and unimpededly cross an energy barrier of arbitrary height and width. Such tunneling is permitted by the uncertainty principle but becomes paradoxical when particles tunnel with certainty. This is known by the Klein paradox. A similar effect, known by the Klein tunneling of massless Dirac electrons, is associated with graphene. Based on pseudo-spin conservation, the tunneling occurs into a region augmented by lattice potential.
Here I present a realization of a completely analogous tunneling effect in a classical, effectively continuous acoustic metamaterial. I show that for a particular form of the constitutive parameters, the barrier crossing translates into the sign of the transmission angle. The resulting dispersion relation and the transmission characteristics follow exactly those of Klein tunneling in graphene, with unimpeded transmission at normal incidence.
I then show that the tunneling can be made omni-directional, supporting reflectionless Klein-like transmission at any incidence angle, by introducing anisotropy to the effective parameter distribution. I realize this distribution in real-time, by operating the metamaterial through an embedded feedback control mechanism.