Hyperbolic angular momentum of polaritons

We uncover a new type of symmetry in hyperbolic media. The conserved quantity corresponding to this symmetry introduces the notion of hyperbolic angular momentum. This concept allows the classification of optical modes in these media, revealing that the recently observed shear polaritons are manifestations of field profiles with nonzero hyperbolic angular momentum numbers. Using Huygens’ construction, we design an optical vortex to launch these shear modes. Additionally, we explore the hyperbolic angular momentum operator within a quantum framework, broadening our understanding of conjugate momentum in polaritonic systems.