We describe novel topological phases of iso-frequency k-space surfaces in bianisotropic optical materialstri-and tetra-hyperbolic materials, which are induced by introduction of chirality. This completes the classification of iso-frequency topologies for bianisotropic materials, since as we show all optical materials belong to one of the following topological classes: tetra-, tri-, bi-, mono-or non-hyperbolic. We show that phase transitions between these classes occur in the k-space directions with zero group velocity at high k-vectors [Eq. (19)]. This classification is based on the sets of high-k polaritons (HKPs), supported by materials. We obtain the equation describing these sets [Eqs. (14), (17)] and characterize the longitudinal polarization impedance of HKPs [Eqs. (16), (18)]Hyperbolic topologies spark imagination of the science fiction prosaists for almost a century [1]. In turn, the topology of iso-frequency k-surfaces in photonic materials fascinates the optics community workers. The known topologies include bounded k-surfaces such as spheres or ellipsoids, and unbounded k-surfacessingle-and double-leaf hyperboloids [2][3][4], and recently discovered bi-hyperboloids [5]. As can be seen from this list of k-surface topologies the main difference between them and the key to their classification are the high-k modes that propagate or not in these materials (see Fig. 1). The high-k modes are of primary interest in photonics and have already found applications for optical imaging with nanoscopic resolut ion using hyperlenses, emission control due to diverging optic al density of high-k states and emission directivity control [6].In this Letter we theoretically predict novel iso-frequency topology phasestri-and tetrahyperbolic materials and obtain an equation that describes the k-space directions in which the high-k modes exist in terms of the 36 material parameters of an arbitrary bi-anisotropic material [Eqs. (14), (17)]. We also employ a theorem due to Zeuthen (1873) [7], to show that our prediction of the tri-and tetra-hyperbolic topological phases completes the classification of bianisotropic materials, which means that an optical material belongs to one of the classes: tetra -, tri-, bi-, mono-or non-hyperbolic. The novel tetra-and tri-hyperbolic phases which we predict here are induced by introduction of chirality. Chirality induced modification of topology in the energy-momentum space was previously studied for mono-hyperbolic materials [8]; here we discuss the topology of iso-frequency surfaces in k-space.

We formulate and solve the inverse problem of quartic photonics, or photonics of metamaterials, whose k-surfaces (also known as iso-frequency surfaces or Fresnel surfaces) are quartic surfaces. This provides an approach to engineer the effective parameters of metamaterials starting from the desired plane waves. We apply our method to the design of the high-k limit of metamaterials, extreme non-reciprocity, and complex bi-anisotropic media.

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