Fig. 2: Real-space nano-imaging of THz hyperbolic phonon-polaritons (HPhPs) in PbI₂.

a Graphical illustration of the THz scattering scanning near-field optical microscopy (s-SNOM) scheme accessing HPhPs activity with a long metallic atomic force microscopy (AFM) tip, upper right: near-field amplitude image of a 655 nm-thick flake at 2.52 THz, lower right: corresponding calculated HPhP waves pattern (see Supplementary Information, Section 3.3). Scale bars in the insets represent 2 μm. b Fourier transform (FT) profiles of the near-field image of the 655 nm-thick flake, highlighting sharp peaks centered at q values inversely proportional to the HPhPs wavelength. Inset: Fourier transform image with indication (white dashed lines) of the directions from where the profiles were extracted (see Supplementary Information, Section 2.2). c Near-field images of flakes with various thicknesses (t) measured at 2.52 THz. Scale bars in (c) represent 5 μm. d Near-field amplitude profiles extracted from the nanoscopy images (dashed white line in c) displaying the HPhP wavelength variation with the flake thickness. The confinement factor (CF = λ₀/λ_p) is presented for each thickness. λ₀ (119 µm) and λ_p represent the excitation and polariton wavelengths, respectively. e Frequency–momentum (ω–q) dispersion relation of HPhPs between the transverse optical phonon frequency ν_Eu(TO) = 1.55 THz and the longitudinal phonon frequency ν_Eu(LO) = 3.18 THz calculated for various thicknesses (Supplementary Information, Section 3.1). f HPhP wavelength (λ_p) estimated from the FT analysis versus crystal thickness (open circles: experimental data, red line: calculation from the dispersion diagram).