Why the hyperbolic polaritons are hyperbolic?

Polaritons travelling along a hyperbolic medium's surface have recently sparked significant interest in nanophotonics for the unprecedented manipulation ability on light at the nanoscale in a planar way, promising potential nano-optical applications, especially in two-dimensional circuitry. Despite of being named hyperbolic polaritons, the hyperbolic nature has not been thoroughly revealed since an analytical description of the Iso-frequency contour is still elusive. In this work, we proposed an analytical form for describing the iso-frequency contour of the hyperbolic polaritons, showcasing their strictly hyperbolic nature. Such an analytical form is obtained based on the focusing behavior of the hyperbolic polaritons and verified by both the published data from commonly used hyperbolic media systems of the hyperbolic polaritons and our own experimental characterizations on a hyperbolic metamaterial film. By presenting a concise and intuitive physical image, this work may provide a groundbreaking methodology in developing novel hyperbolic polaritons based optical devices.