Real-Space Double Crossing Reveals Symmetry-Driven Dispersion Back-Bending of Surface Phonon Polaritons on α-Quartz

Abstract

Surface phonon polaritons (SPhPs) in anisotropic crystals exhibit direction-dependent propagation and dispersion, yet how their dispersion back-bending manifests in real space has not been quantitatively resolved. Here, using nano-Fourier-transform infrared spectroscopy based on scattering-type scanning near-field optical microscopy, we image SPhP fringes on an $\alpha$-quartz surface for propagation parallel and perpendicular to the optic axis. Near 1160 cm${}^{-1}$, the fringe periods for the two directions cross twice, providing a real-space signature of back-bending. Spatial Fourier analysis attributes the dominant fringe to edge-launched SPhPs and enables reconstruction of the intrinsic dispersion. Inverse Fourier analysis further visualizes the spatial attenuation and continuous evolution of the fringe wavelength near the resonance. The reconstructed dispersions reveal that the polarization-selective E-mode phonon produces direction-dependent back-bending and, through its damping, reduces the visibility of the back-bending branch. These results establish how symmetry-controlled dispersion manifests in real-space surface polariton propagation on a bulk anisotropic crystal.

arXiv commentMain: 28 pages, 5 figures. SI: 17 pages, 8 figures. In this ver, we extend its analysis and interpretation