Fig. 1. Real-space nano-spectroscopy of surface phonon polaritons in pristine SrTiO₃ at 15 K.

a Left axis: real (red line) and imaginary (blue line) parts of the STO permittivity ${\epsilon }_{\mathrm{STO}}\left(\omega \right)$ in the upper SPhP band (shaded area) corresponding to the highest-frequency phonon mode ν₄ (depicted in the inset). Right axis: the real part of the in plane SPhP momentum $q_p=q_0\sqrt{{\epsilon }_{\mathrm{STO}}/\left(1+{\epsilon }_{\mathrm{STO}}\right)}$ (green line) and the momentum of light in free space $q_0=\omega /c$ (purple line). The inset shows the unit cell of STO (Sr – green, Ti – gray, O – red), where the arrows denote the vibration eigen vector of the LO mode according to ref. ⁴⁰. b Schematics of the SPhP interferometry experiment. The incident infrared beam $E_{\mathrm{in}}$ (red arrows) illuminates both the metal edge and the s-SNOM tip. The edge launched SPhPs (orange arrows) propagate toward the tip. The back-scattered radiation $E_{\mathrm{sca}}$ contains contributions from two paths: the one mediated by the SPhPs (P1) and the one directly scattered by the tip (P2). As a result, an interference pattern is formed as a function of the edge-tip distance. α, β and φ represent the incident angle, the azimuthal angle and SPhP propagating angle, respectively. c Nano-FTIR distance-frequency map of the s-SNOM amplitude demodulated at the 2^nd tapping harmonics and normalized by the signal on metal. Light blue dashed line represents the surface optical phonon frequency ${\omega }_{\mathrm{SO}}$. d Symbols: line profile at 690 cm⁻¹ (dashed dark-blue line in c). Solid line: a damped-sine function fit of the data. e Symbols: the fringe spacing $d_{\exp }$ extracted from the fits, as a function of frequency. Red solid line: the best fit using formula (1) as described in the text. Green and purples lines: the SPhP wavelength ${\lambda }_p=2\pi /\mathrm{Re}\left(q_p\right)$ and the light wavelength ${\lambda }_0$ respectively. f Calculated propagation length $L_p=1/\mathrm{Im}\left(q_p\right)$ (top panel) and experimental SPhP decay length $L_{\exp }$ (bottom panel) as a function of frequency. All error bars correspond to the standard deviation.