In this paper we reveal the physics behind the formation of tri- and tetra-hyperbolic phases in anisotropic metamaterials without magnetoelectric coupling and describe the anti-crossing splitting phenomenon in the hyperbolic dispersion which arises due to the hybridization of the plasmonic and magnetic Bloch high-k polaritons. This considerably deepens the understanding of the high-k polaritons and the topology of the optical iso-frequency surfaces in k-space and will find applications in optical nano-resolution imaging and emission rate and directivity control. To accomplish this, we develop a range of new techniques of theoretical optics for bianisotropic materials, including the quadratic index of refraction operator method, suitable to study the high-k polaritons with finite indices of refraction and the explicit expression for the characteristic matrix in generic bianisotropic media. We introduce the spatial stratification approach for the electric and magnetic responses of anisotropic homogeneous media to analyze the underlying Bloch waves. We believe that the formalisms developed here can be useful for the researchers in the field of theoretical optics of anisotropic and bianisotropic media in the future.
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