Fig. 1. Field distribution in z direction for different dielectrics. (a) In-plane isotropic dielectric (ɛ_x=ɛ_y>0); (b) in-plane elliptic dielectric (ɛ_x>0, ɛ_y>0, ɛ_x≠ɛ_y); (c) in-plane hyperbolic dielectric (ɛ_x<0, ɛ_y>0); (d) in-plane hyperbolic dielectric (ɛ_x>0, ɛ_y<0)

Fig. 2. Natural hyperbolic metamaterials. (a) Infrared hyperbolic hypersurface of hBN nanostructure^[24]; (b) scanning electron microscope (SEM) image of hBN infrared hyperbolic hypersurface (nominal grating parameters are w≈75 nm and g≈25 nm); (c) tip launching phonon polaritons on hyperbolic hypersurface (indicated by simulated near fields); (d) schematic of unit cell of α-MoO₃ with lattice constants a=0.396 nm, b=1.385 nm and c=0.369 nm; (e) in-plane elliptical and hyperbolic phonon polaritons in α-MoO₃ disk; (f) thickness tunability of in-plane hyperbolic and elliptic phonon polaritons in α-MoO₃^[15]

Fig. 3. Controllable launching and focusing of hyperbolic phonon polariton. (a) Real-space imaging of edge-tailoring phonon polaritons at angle-dependent α-MoO₃; (b) angle-dependent k_e isofrequency contour of phonon polaritons in α-MoO₃ at ω=889.8 cm^-1; (c) edge modulated phonon polariton real-space image of isosceles triangles with different adjacent angles, squares and rectangles α-MoO₃ nano-cavity with different rotation angles^[23]; (d), (e) experimental near-field amplitude images in α-MoO₃ microstructures of varied shapes of square, regular pentagon and regular hexagon microstructures at frequencies of 915 cm^-1 and 986 cm^-1[25]

Fig. 4. High efficiency launching and focusing of phonon polariton. (a) Real-space imaging of hyperbolic phonon polaritons on surface of natural 220-nm-thick α-MoO₃ flake^[27]; (b) near-field images of hyperbolic phonon polaritons launched by an Au disk nanoantenna fabricated on top of a 165-nm-thick α-MoO₃ crystal^[28]; (c) radius-of-curvature-dependent hyperbolic phonon polaritons in-plane focusing^[29]

Fig. 5. Tunability of hyperbolic phonon polaritons. (a) Schematic showing nano-imaging of suspended α-MoO₃ flake using scattering-type scanning near-field optical microscope (s-SNOM); (b), (c) images taken at frequencies of 937 cm^-1 and 990 cm^-1[32]. (d) Optical micrographs and s-SNOM amplitude images (890 cm^-1) of MoO₃ flake on SiO₂/Si substrate before hydrogenation, 10 s after hydrogenation, and after dehydrogenation^[33]

Fig. 6. Optical topological transformation of hyperbolic phonon polaritons. (a)-(l) Experimental observation of topological polaritons launched by air, twisted angles are Δθ=-44°, Δθ=65°, and Δθ=-77°, respectively^[34]; (m)-(p) experimental observation of topological polaritons launched by air, twisted angles are Δθ=0°, Δθ=30°, Δθ=75°, and Δθ=90°, respectively^[35]

Fig. 7. Topological transition of hybrid polaritons in graphene/α-MoO₃ heterostructure. (a)-(l) Experimentally measured polariton near-field distributions with graphene doping Fermi energy of 0, 0.3, 0.7 eV^[42]; (m)-(p) thickness dependent hybrid polaritons in graphene/α-MoO₃ heterostructures with experimentally measured field distributions at 888 cm^-1 and 900 cm^-1[40]