  Hyperbolic materials offer a much wider freedom in designing optical
properties of nanostructures than ones with isotropic and elliptical
dispersion, both metallic or dielectric. Here, we present a detailed
theoretical and numerical study of the unique optical properties of spherical
nanoantennas composed of such materials. Hyperbolic nanospheres exhibit a rich
modal structure that, depending on the polarization and direction of incident
light, can exhibit either a full plasmonic-like response with multiple electric
resonances, a single, dominant electric dipole or one with mixed magnetic and
electric modes with an atypical reversed modal order. We derive resonance
conditions for observing these resonances in the dipolar approximation and
offer insight into how the modal response evolves with the size, material
composition, and illumination. Specifically, the origin of the magnetic dipole
mode lies in the hyperbolic dispersion and its existence is determined by two
diagonal permittivity components of different sign. Our analysis shows that the
origin of this unusual behavior stems from complex coupling between electric
and magnetic multipoles, which leads to very strongly scattering or absorbing
modes. These observations assert that hyperbolic nanoantennas offer a promising
route towards novel light-matter interaction regimes.
