## 3 Results and Discussion
To select the optimal parameters of taper with the grating of slits, we first optimized the parameters of a grating in the three-layer Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) of infinite width (two-dimensional (2D) grating), and then we optimized the taper opening angle, the taper length L\({}_{\text{tap}}\) and the distance from the grating to the taper-waveguide junction. As the exciting wave, we considered a Gaussian beam with a waist width of w\({}_{0}\) = 8.5 \(\upmu\)m, normally incident from the substrate (the focus is located at the substrate-Al\({}_{2}\)O\({}_{3}\)-layer interface). The grating period, D, and the slits width, W\({}_{\text{slit}}\), were varied in order to maximize the efficiency of the long-range wave excitation in a 2D layered system. Optimization was performed with respect according to the lattice period, D, and the width of the slots. The case of a 2D exciting Gaussian beam was considered (the waist is w\({}_{0}\) in the yz-plane perpendicular to the slits direction and is infinite in the \(x\)-direction along the slits). The number of slits was taken to be N = 38. For such N, the grating length is approximately equal to 20 \(\upmu\)m (or 2.3w\({}_{0}\)) for the grating period D = 531 nm. In Figure 3, the 2D grating efficiency (calculated as a ratio of the long-range plasmon power excited in the \(+z\) direction in the layered system of infinite width to the incident 2D Gaussian beam power) as a function of W\({}_{\text{slit}}\), and D are shown. As we can see from Figure 3b, the highest grating efficiency is observed at D = 531 nm and is about 13%. The taper width, W\({}_{\text{tap}}\), should be several times larger than the exciting Gaussian beam waist, w\({}_{0}\). For example, we take W\({}_{\text{tap}}\) = 30 \(\upmu\)m. The taper opening angle \(\theta\) (determined by the ratio \(W_{tap}/L_{tap}\)) should approximately meet the condition:  
\[W_{tap}/2L_{tap}=\text{tg}\frac{\theta}{2}=\frac{\sqrt{\left(\left(n^{\prime} _{effsurf}\right)^{2}-\left(n^{\prime}_{eff}\right)^{2}}\right.}{n^{\prime}_{ effsurf}} \tag{1}\]  
where, \(n^{\prime}_{effsurf}\) is the real part of the long-range plasmon effective refractive index in the three-layer Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) system on the substrate and \(n^{\prime}_{eff}\) is the real part of the long-range plasmonic mode of the LR-DLSPPW of finite width. This condition determines the maximal angle \(\theta\)/2 of a taper long-range plasmonic wave incidence for which this plasmonic wave could be coupled into the LR-DLSPPW. For  
Figure 3: (a) The geometry of the taper with a grating of slits for long-range mode excitation in the long-range dielectric-loaded surface plasmon-polariton waveguides (LR-DLSPPW). The taper is a triangular prism with a base in the xy substrate plane. L\({}_{\text{tap}}\) is the taper length along the z-axis (parallel to the LR-DLSPPW axis), W\({}_{\text{tap}}\) is the taper width. The red circle shows the exciting Gaussian beam incident from the substrate perpendicular to the substrate plane. (b) Numerically calculated efficiency of long-range plasmon excitation in a three-layer system Al\({}_{2}\)O\({}_{3}\)/Au/Al\({}_{2}\)O\({}_{3}\) by a Gaussian beam incident from the substrate on two-dimensional (2D) gratings of slits as functions of grating geometric parameters. W\({}_{\text{slit}}\) is a slit width. Different curves correspond to different grating periods D: the black curve is for D = 527 nm, the red curve for D = 529 nm, the blue curve for D = 531 nm, the green curve for D = 533 nm. The calculations were carried out for a grating of N = 38 slits and a Gaussian beam of width w\({}_{0}\) = 8.5 \(\upmu\)m. The three-layer parameters are d1 = 75 nm, d2 = 12 nm and d3 = 240 nm. The wavelength is 785 nm. a larger \(\theta\), however, the reflection of the long-range modes of the taper from the taper-waveguide junction is too large.