  We use scanning near-field optical microscopy to study the response of
hexagonal boron nitride nanocones at infrared frequencies, where this material
behaves as a hyperbolic medium. The obtained images are dominated by a series
of hot rings that occur on the sloped sidewalls of the nanocones. The ring
positions depend on the incident laser frequency and the nanocone shape. Both
dependences are consistent with directional propagation of hyperbolic phonon
polariton rays that are launched at the edges and zigzag through the interior
of the nanocones, sustaining multiple internal reflections off the sidewalls.
Additionally, we observe a strong overall enhancement of the near-field signal
at discrete resonance frequencies. These resonances attest to low dielectric
losses that permit coherent standing waves of the sub-diffractional polaritons
to form. We comment on potential applications of such shape-dependent
resonances and the field concentration at the hot rings.
