Fig. 4: Near-field imaging of directional propagation of LPs in real space.

a, c, e Nanoscale imaging of LPs at ω = 883 cm⁻¹, 887 cm⁻¹, 890 cm⁻¹, generated by recording the amplitude of the measured signal, including both type-I hyperbolic and transparent regimes. The indicated directions in c and e are closely aligned with the confined directional fields, where the black arrows are estimated from simulations. b, d, f Corrected FT (see Supplementary Information Fig. S11 for details) of near-field distribution in (a, c, e) (Note that the dispersions are flipped by 90° clockwise here for clearer presentation). The dispersions are corrected by considering the shift in transverse wave vector k_x caused by oblique incidence; see Supplementary Information for details. The dashed yellow lines indicate the FSLC, theory dispersions of LPs are marked by solid red lines. The directive LPs in (e) are expected in the flattened iso-frequency dispersion contour marked white arrows in (f). g The comparison of LPs and g-HPs damping extracted from experiments. The red balls show the s-SNOM optical signal(S₃) of LPs from (e), and the blue balls show the signal of g-HPs at ω = 1460 cm⁻¹ (shown in Fig. S17), which are both normalized on a smooth background signal. The black solid line indicates the fitting results by a modified damped wave function: S₃ = x^−0.5 · e^-γx· sin(kx), where x is the distance away from the source, the relative propagation length (Q) can be estimated from k/γ, L_p = 1/γ.