Fig. 4: Real-space SINS nanoimaging of SPhPs in SiO₂/Si-supported membranes.

a, b Experimental near-field amplitude (a) and phase (b) spectra obtained by a line scan perpendicular to the edge of the membrane. The trace of SINS linescan is shown in Fig. 1c. Dashed curves in (a) and (b) denote the peak and dip associated with the propagation of SPhSs. c, d Near-field SINS amplitude (c) and phase (d) spectra obtained at locations on the membrane with different distances to the edge. The locations are denoted by the red arrows in (a). Dashed curves in (c) and (d) denote the peak and dip associated with the propagation of SPhSs. e, f Simultaneously measured near-field amplitude (e) and phase (f) line profiles of the SrTiO₃ film at different frequencies (denoted by white arrows in (a)), with the scan direction perpendicular to the edge (denoted by green line) at x = 0. Experimental data is shown in blue. Black arrows in (e) and (f) show the shift of the peak in the amplitude profile and shift of the dip in the phase profiles. Red solid lines show the fitting of the experimental data using the complex-valued function \({s}_{2}{(x)e}^{i{\phi }_{2}(x)}=A\frac{{e}^{{iqx}}}{x}+B\). g Dispersion of SPhPs in bulk SrTiO₃ and membranes. Experimental data extracted from SINS imaging is shown as green circles. h The confinement factor of SPhPs in membranes and bulk SrTiO₃ as a function of frequency. i Propagation length L_p (purple circles) and quality factor Q (blue squares) of SPhPs in SrTiO₃ membrane versus frequency.