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 ${\epsilon }_{\mathrm{sub}}$ = 1 (black curve) and ${\epsilon }_{\mathrm{sub}}$ = 5 (gray curve) that define two different hyperbolic media (medium 1 and 2, respectively). The incident wave in medium 1 is characterized by collinear ${\mathbf{k}}_{\mathrm{in}}$ and ${\mathbf{S}}_{\mathrm{in}}$ (as in an isotropic medium, indicated by a dashed cyan circle). Upon refraction into medium 2, momentum conservation at the boundary (orange line), ${\mathbf{k}}_{\mathit{\mid \mid }}$ ($k_{\parallel }=k_{\mathrm{in}}\bullet \sin \phi$, where $\phi$ is the angle of the boundary), is fulfilled by non-collinear ${\mathbf{k}}_{\mathrm{out}}$ and ${\mathbf{S}}_{\mathrm{out}}$. 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 ${\mathbf{k}}_{\mathrm{in}}$ and ${\mathbf{S}}_{\mathrm{in}}$ (normal to the isofrequency curve). When the wave refracts into medium 2, momentum conservation at the boundary (orange line) is fulfilled by non-collinear ${\mathbf{k}}_{\mathrm{out}}$ and ${\mathbf{S}}_{\mathrm{out}}$. 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 ${\mathbf{k}}_{\mathrm{in}}$ and ${\mathbf{S}}_{\mathrm{in}}$, i.e. ${\theta }_{\mathrm{in}-k}={\theta }_{\mathrm{in}-S}$, giving rise to non-collinear ${\mathbf{k}}_{\mathrm{out}}$ and ${\mathbf{S}}_{\mathrm{out}}$, i.e. ${\theta }_{\mathrm{out}-S}$ ≠ ${\theta }_{\mathrm{out}-k}$. 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. ${\theta }_{\mathrm{in}-k}\ne {\theta }_{\mathrm{in}-\mathrm{S}}$ and ${\theta }_{\mathrm{out}-k}$ ≠ ${\theta }_{\mathrm{out}-S}$. The tangents parallel to both hyperbolas give rise to bending-free refraction, i.e. ${\theta }_{\mathrm{in}-S}\approx {\theta }_{\mathrm{out}-S}$. 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.