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.
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