Anomalous propagations of electromagnetic waves in anisotropic media with a unique dispersion relation

We investigate the reflection and refraction behaviors of electromagnetic waves at the interface between an isotropic material and the anisotropic medium with a unique dispersion relation. We show that the refraction angle of whether phase or energy flow for $E$-polarized waves is opposite to that for $H$-polarized waves, though the dispersion relations for $E$- and $H$-polarized waves are the same in such anisotropic media. For a certain polarized wave the refraction behaviors of wave vector and energy flow are also significantly different. It is found that waves exhibit different propagation behaviors in anisotropic media with different sign combinations of the permittivity and permeability tensors. Some interesting properties of propagation are also found in the special anisotropic media, such as amphoteric refraction, oblique total transmission for both $E$- and $H$-polarized waves, and the inversion of critical angle. The special anisotropic media can be realized by metamaterials and lead to potential applications, such as to fabricate polarization splitters with higher efficiencies than conventional counterparts.