Fig. 1: Leaky polaritons (LPs).

a Schematic of the excitation of ghost hyperbolic polaritons (g-HPs) in the left panel and LPs in the right panel, emerging at the interface between calcite and air and supporting directional waves that propagate on the surface and in the bulk of calcite. b, d Real space distribution of simulated field \({\mathfrak{R}}({{{{{{\rm{E}}}}}}}_{z})\) on the surface of calcite and c, e the corresponding Fourier Transformation (FT) in k-space, respectively. f 3D isofrequency contours (IFCs) for type-II hyperbolic regime at frequency (\(\omega=1470\,{{{{{{\rm{cm}}}}}}}^{-1}\)) supporting g-HPs and g 3D IFCs for type-I hyperbolic regime at frequency (\(\omega=887\,{{{{{{\rm{cm}}}}}}}^{-1}\)) supporting LPs, and their dependence of the in-plane momentum dispersion on the slanted optical axis of calcite. The green solid lines in f (green dashed lines in c) represent the light cones of extraordinary wave in type-II hyperbolic regime, blue and purple solid lines in g represent the light cones of ordinary wave and extraordinary wave in the type-I hyperbolic regime, respectively. The red solid lines in f (red dashed lines in c) are the dispersion of g-HPs, while the red solid lines in e, g show the dispersion of LPs, obtained from real- and complex-valued Eq. (1), respectively. The white sphere depicts the IFC of free space, and white circle is the free space light cone (FSLC).