## 3 Results  
In this experiment, three dielectric-multilayer substrates were fabricated, with the top layer thickness (t) selected at about 160, 180, and 250 nm. The thicknesses of the other layers were kept the same as shown in Figure 1a. By using this dielectric multilayer substrate, SPPs can propagate for a long distance along the thin Ag NW (such as diameter of the Ag NW is less than 100 nm). However, if the same Ag NW was placed on a glass substrate, the propagation distance would be much shorter [34]. Figure 2a-c show the BFP images when the Ag NW of 70 nm diameter was placed on these three substrates. The incident wavelength was selected as 590 nm. In our LRM, an oil-immersed objective with numerical aperture (NA) at 1.49 was used to collect the leakage plasmonic signals. The outer red rings on Figure 2a-c represent the largest collection angle that can be derived as arcsin (1.49/1.515) = 79.58\({}^{\circ}\), where the refractive index of the oil is 1.515. The inner ring represents the critical angle where the total internal reflection occurs, hence the bright areas inside this inner ring are due to the direct transmission of the light through the dielectric multilayer. On the BFP images, the bright vertical line labelled with SPP is the signature of the SPP propagating along the Ag NW [32], and the line is parallel to the axis K\({}_{\rm{x}}\)/K\({}_{0}\), which is perpendicular to the NW long axis. From the distance (D) between this vertical line and the axis (K\({}_{\rm{x}}\)/K\({}_{0}\)), the diameter of the outer ring (R) and known NA (1.49) of the objective, the effective refractive index (or wavenumber, or the real part of the propagation constant) \(\beta\)/K\({}_{0}\) of this SPP waveguide mode can be calculated with the equation \(\beta\)/K\({}_{0}\) = (D/R)\({}^{\star}\)NA, where the K\({}_{0}\) is the wavenumber of the light in vacuum. The SPP propagates along the \(Z\)-axis, and the wavenumber along the \(Z\)-direction is K\({}_{\rm{z}}\)/K\({}_{0}\) = \(\beta\)/K\({}_{0}\) = 1.03 (a), 1.05 (b), and 1.21 (c), respectively. The BFP images at additional wavelengths ranging from 540 to 660 nm were also captured, such as the BFP images at 570 and 610 nm wavelength, which were used to derive the curves of the effective refractive index vs. incident wavelength, as shown in Figure 2d. With the increasing of incident wavelength, the effective index decreases, and at all these wavelengths, the effective index will be increased if the top layer thickness is increased from 160 to 250 nm. These results clearly demonstrate  
Figure 1: Schematic illustration of the sample. (**a**) The silver nanowire (Ag NW) was placed on a dielectric multilayer consisting of alternating layers of SiO\({}_{2}\) (105-nm-thick) and Si\({}_{3}\)N\({}_{4}\) (88-nm-thick). There were fourteen layers in total, with a top SiO\({}_{2}\) layer with a varied thickness of t nm. A fibre taper was used to couple the laser beam into the nanowire. The NW was oriented along the \(Z\)-axis and perpendicular to the \(X\)-axis; (**b**) SEM image of the Ag NW (diameter of 70 nm); (**c**) Schematic of the experimental set-up for imaging the plasmon’s propagation along the Ag NW. Two cameras are used for capturing the front focal plane (FFP) and back focal plane (BFP) images. A polarizer can be used to check the polarization state of this plasmonic mode. that the effective index of SPP waveguides mode can be tuned with the thickness (t) of the top SiO\({}_{2}\) layer (Figure 1a). There are bright arcs on the BFP images (labelled with 2 DBSW) appearing close to the bright lines, which are the two-dimensional Bloch Surface Waves (2 DBSW) generated by the light from fibre taper. The 2 DBSW is the two-dimensional surface wave spreading on the surface of the dielectric multilayer. To test the polarization of these modes (2 DBSW and of the NW), a linear polarizer was placed before the camera for BFP images, and the corresponding images with two orientations of the polarizer are shown in Figure 2e,f.