## 3 Results and Discussion  
Long-Range Dielectric-Loaded Surface Plasmon-Polariton Waveguides (LR-DLSPPW) Simulation and Analysis  
The considered plasmonic transmission line consists of a waveguide and light input/output diffraction grating. We considered two types of the waveguide cross-section with both silver and gold metal layers. However, in the paper we demonstrate only gold-based waveguides simulation results. For the first type, the lower Al\({}_{2}\)O\({}_{3}\) layer has the same width (Figure 2a) as the upper one, while for the second type, the lower layer has a much larger (infinite) width (Figure 2b). The two Al\({}_{2}\)O\({}_{3}\) layers' thicknesses were optimized in order to maximize the long-range plasmonic mode propagation length while keeping a sufficiently large long-range mode effective index compared with the substrate refractive index (to avoid too-large scattering losses into the substrate on waveguide roughness and bends). The Au layer thickness was taken to be 8-12 nm, which is a reliable technological minimum for continuous films. Then, for the optimized thicknesses of the three layers on the substrate, the waveguide width (the width of the three-layer Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) or the two-layer Au/Al\({}_{2}\)O\({}_{3}\) strip) was optimized in order to maximize the long-range waveguide plasmonic mode propagation length while keeping the waveguide to be single-mode (having only one long-range guided mode). Figure 2c shows the dependences of the effective index of the long-range mode in the infinitely wide three-layer system Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) on the upper Al\({}_{2}\)O\({}_{3}\) thickness for different lower Al\({}_{2}\)O\({}_{3}\) layer thicknesses. In Figure 2d, the corresponding dependences of the long-range plasmon propagation length are presented. All calculations are given for the wavelength of 785 nm. Based on the results presented in Figure 2, the thicknesses d1 and d3 were chosen so that the plasmon effective refractive index exceeded the substrate refractive index by more than 0.05, but the plasmon propagation length was maximal. This requirement approximately corresponds to d1 = 75 nm and the maximum of the corresponding curve in Figure 2d, i.e., d3 = 240 nm. In such structures, the calculated long-range plasmon propagation length is L\({}_{\text{prop}}\) = 0.27 mm for the wavelengths of 785 nm. As in/outcoupling systems for light, we considered tapers with periodic gratings of slits cut through the two upper layers of the three-layer Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) (see Figure 3). The tapers are made from three-layers Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) with the same layer thicknesses as in the LR-DLSPPW waveguides, which makes it possible to fabricate it in the same process steps. The tapers focus the long-range plasmonic waves into the LR-DLSPW aperture. In turn, to excite long-range plasmonic waves in the taper by an incident Gaussian beam, a periodic grating of slits in the taper is used. Figure 2: Waveguide cross-section optimization: (**a**) The lower and upper Al\({}_{2}\)O\({}_{3}\) layers have the same width, (**b**) The lower Al\({}_{2}\)O\({}_{3}\) width is much larger (infinite) than the width of the upper one, (**c**) Difference between the long-range plasmon effective index of the layered system (in the case of the infinite-width three-layer system) and the substrate refractive index as a function of the upper Al\({}_{2}\)O\({}_{3}\) layer thickness d3. Different curves correspond to different lower Al\({}_{2}\)O\({}_{3}\) layer thicknesses d1, (**d**) Dependence of the long-range plasmon propagation length in the infinite-width layered system on the upper Al\({}_{2}\)O\({}_{3}\) layer thickness d3.