## 3 Results
These two images (the area marked with the dotted box) show that the 2 DBSW is of transverse electronic (TE) polarization, meaning that the polarization direction of the light on the bright arc is perpendicular to the radial direction of the arc, similar to the azimuthal polarization of a vortex beam [35]. For the vertical bright line, the light spot on this line with K\({}_{\mathrm{x}}\)/K\({}_{0}\) = 0 is dark in the case of horizontal polarization (Figure 2f), and is bright in the case of perpendicular polarization (Figure 2e); this phenomenon reveals that the polarization direction of this light spot is along this bright line, and therefore this mode is the H1X mode [36]. Numerical simulations using the finite element method were also carried out to demonstrate the effect of top  
Figure 2: Controlling the effective refractive index of the surface plasmon polariton (SPP) waveguide mode with the top dielectric (SiO\({}_{2}\)) layer’s thickness. The thickness was selected as t = 160, 180, and 250 nm, respectively. The incident wavelength was selected as 590 nm. (**a–c**) the corresponding BFP images; (**d**) the curves of effective refractive index vs. the incident wavelength in the case of three top layer thicknesses; (**e,f**) BFP images in the case of t = 160 nm with a polarizer before the detector. The white double arrowhead lines represent the direction of the polarizer. On (**a–c**), the red rings with diameters at 1.49 and 1 are given by the numerical aperture (NA) of the objective and the critical angle, respectively. layer thickness and wavelengths on the effective refractive index. The permittivity of Ag NWs at different wavelengths were based on experimental values [37]. Due to the surface scattering and grain boundary effects in real thin films in the simulations, the refractive indices of SiO\({}_{2}\) and Si\({}_{3}\)N\({}_{4}\) were \(\text{n}_{\text{SiO2}}=1.46+\text{i}10^{-3}\) and \(\text{n}_{\text{Si3N4}}=2.14+\text{i}5\times 10^{-3}\), respectively. The refractive index of the glass substrate was \(\text{n}_{\text{glass}}=1.515\). The thickness of each dielectric layer was the same as that shown in Figure 1a. Figure 3a demonstrates the calculated effective refractive index of plasmonic H1X mode as a function of the top layer thicknesses with three incident wavelengths (570, 590, and 610 nm--identical to those used on the BFP images in Figure 2). The diameter of the nanowire is 70 nm. The simulation results show that the effective refractive index (\(\beta/\text{k}_{0}\)) of the H1X mode is decreased with decreasing top layer thickness, which is consistent with the results derived from the BFP images on Figure 2a-c. As the wavelength decreased, the cutoff thickness of H1X mode also decreased. For example, when the incident wavelength was 610 nm, this mode disappeared, as the top layer thickness was less than 160 nm. Whereas at a 570-nm wavelength, the same mode disappeared at a top layer thickness of 130 nm. To understand the wave guide behaviour of H1X mode placed on the dielectric multilayer, the electric field distribution of the H1X mode was calculated and shown in Figure 3c. Different from the case of the nanowire placed on the glass substrate, the predominantly electric field energy of H1X mode of the nanowire was localized near the top layer of dielectric multilayer. The changes of the top layer thickness can adjust the field localization of H1X mode, and greatly affect the wave guiding behaviour of this H1X mode. Figure 3b shows the effective refractive index of H1X mode at three different top layer thicknesses (t = 160, 180, and 250 nm, the same as those used in the experiment) as a function of the incident wavelength. As expected, the change of the effective refractive index with the incident wavelength is consistent with the experimental results shown in Figure 2d. At all three thicknesses, the index decreased with the increasing of the incident wavelength from 540 to 660 nm.