Fig. 1: Schematics of refraction of polaritons between two hyperbolic media.

a Isofrequency curves of polaritons propagating in a hyperbolic slab (with ε_x = −5; ε_y = 1; ε_z = 5) placed on two different semi-infinite substrates with \({\varepsilon }_{{\mathrm{{sub}}}}\) = 1 (black curve) and \({\varepsilon }_{{\mathrm{{sub}}}}\) = 5 (gray curve) that define two different hyperbolic media (medium 1 and 2, respectively). The incident wave in medium 1 is characterized by collinear \({{\bf{k}}}_{{\rm{in}}}\) and \({{\bf{S}}}_{{\rm{in}}}\) (as in an isotropic medium, indicated by a dashed cyan circle). Upon refraction into medium 2, momentum conservation at the boundary (orange line), \({{\bf{k}}}_{{\boldsymbol{||}}}\) (\({k}_{\parallel }={k}_{{\rm{in}}}\,\bullet\, {\rm{sin }}\varphi\), where \(\varphi\) is the angle of the boundary), is fulfilled by non-collinear \({{\bf{k}}}_{{\rm{out}}}\) and \({{\bf{S}}}_{{\rm{out}}}\). The dashed orange lines represent the normal to the boundary. b The general case of refraction between two hyperbolic media is represented by an incident wave from medium 1 with non-collinear \({{\bf{k}}}_{{\rm{in}}}\) and \({{\bf{S}}}_{{\rm{in}}}\) (normal to the isofrequency curve). When the wave refracts into medium 2, momentum conservation at the boundary (orange line) is fulfilled by non-collinear \({{\bf{k}}}_{{\rm{out}}}\) and \({{\bf{S}}}_{{\rm{out}}}\). The dashed orange lines represent the normal to the boundary. c Real-space illustration of refraction between two hyperbolic media shown in a where the incident wave exhibits collinear \({{\bf{k}}}_{{\rm{in}}}\) and \({{\bf{S}}}_{{\rm{in}}}\), i.e. \({\theta }_{{\rm{in}}-k}={\theta }_{{\rm{in}}-S}\), giving rise to non-collinear \({{\bf{k}}}_{{\rm{out}}}\) and \({{\bf{S}}}_{{\rm{out}}}\), i.e. \({\theta }_{{\rm{out}}-S}\) ≠ \({\theta }_{{\rm{out}}-k}\). d Real-space illustration of the general case of refraction between two hyperbolic media shown in b where both the incident and the outgoing wave exhibits non-collinear k and S, i.e. \({\theta }_{{\rm{in}}-k}\ne {\theta }_{{\rm{in}}-{\rm{S}}}\) and \({\theta }_{{\rm{out}}-k}\) ≠ \({\theta }_{{\rm{out}}-S}\). The tangents parallel to both hyperbolas give rise to bending-free refraction, i.e. \({\theta }_{{\rm{in}}-S}\approx {\theta }_{{\rm{out}}-S}\). The orange dashed lines in c, d represent the normal to the boundary. The white and gray regions in c, d correspond to α-MoO₃/air and α-MoO₃/SiO₂, respectively.