Fig. 3: Momentum dependence of HShPs computed analytically and probed by illuminating gold resonators of different sizes.

Experimental near-field images of HShP propagation, launched by a 6, b 4, and c 2 µm gold disc at a frequency of 720 cm⁻¹, with illumination incident from the lower left corner. d–f Fourier transform of the experimental images in (a–c), with a zero-filling factor = 5. g–i Finite element modeling of HShP propagation from gold discs to match experiment (a–c). j–l Fourier transforms of (g–i). m Fourier transform of an electromagnetic simulation of an HShP excited by a local dipole illustrating the strong anisotropy at high excitation momentum. n Analytical FOM dependence on the launcher radius—extracted by integrating the damping rate along the path of HShPs dispersion which is cut off in momentum determined by the launcher radius. o Dependence of FOM on the launcher radius, extracted from Fourier transforms of the experimental and simulated field profiles (a–c, g–i). White circles in (d–f) and green circles in (j–l) intersecting dispersion of HShP determine the upper bound of the integration in the k-space of FOM calculation in Eq. (1) for experimental data and simulated data, respectively. The green circles in the simulated Fourier spectra are shifted by a small distance from the origin in k-space, due to the oblique illumination. In contrast, the white circles in the experimental Fourier spectra are symmetric with respect to the coordinate frame because we obtained these spectra from real-valued experimental data.