  The discovery of localized plasmon polariton resonances has been pivotal in
enabling tunability of the optical resonance. Recently, extensive research
efforts have aimed to expand these achievements to other polaritonic states
that exhibit less loss and in other spectral regions. However, these efforts
were limited to isotropic or uniaxial structures, and an eigenmode theory was
derived only for isotropic particles. Here, we present a breakthrough in
synthesizing biaxial nanostructures that exhibit localized hyperbolic phonon
resonances with high Q-factors in the mid-infrared. Furthermore, we develop a
theory that predicts high-order resonances in anisotropic particles with
coupling between the axial permittivites. Finally, we confirm the theoretical
predictions through near-field measurements, which demonstrate the existence of
both the first and higher-order resonant modes. Our findings provide the
foundation for designing a new generation of anisotropic resonators with
various applications in the mid-IR range. Our analysis applies to other fields,
such as quasi-magnetostatics and heat conduction.
