# Design of Power Splitters Based on Hybrid Plasmonic Waveguides
## 3 Results and Discussion  
The plasmonic mode was excited by a Gaussian beam, which is normally focused onto the end of the middle waveguide [17]. The arrangement of the five HPWsis shown in Figure 2a. The intensity distributions in the XZ plane are shown in Figure 2b, where the pattern was very complex due to multimode interference effects [18]. In ref. [18], the principles and properties of multimode interference devices are presented for conventional dielectric waveguides. To the device of five HPWs, supermodes were formed due to the coupling between adjacent waveguides [19; 20]. Customarily, supermodes are the modes of propagation of a periodic array of waveguides, which consists of \(N\) identical waveguides [20]. Figure 1: Schemes of five adjacent HPWs. (**a**) The top view of the HPWs. (**b**) The front view of the HPWs. The HPWs consist of GaAs nanowires embedded in SiO\({}_{2}\) dielectric. The diameter of the GaAs nanowire was 200 nm and the gap was h = 50 nm. The distance between adjacent waveguides was S = 500 nm. (**c**) The field distribution of the plasmonic mode for the single HPW. The wave functions of the supermodes were written as linear combinations of the unperturbed wave functions of the individual waveguides:  
\[H_{s}(x,z)=\sum_{n=1}^{N}C_{s,n}\psi_{n}(x,z) \tag{1}\]  
The total field in a periodic array of waveguides can be expressed as a sum of all supermodes, which can be written as:  
\[\varphi(x,z)=\sum_{s=1}^{N}a_{s}H_{s}(x)e^{-i\beta_{s}z} \tag{2}\]  
\(\beta_{s}\) is the propagation constant of a single supermode, \(a_{s}\) is the weight coefficient of excited supermodes [19]. Due to the incident field of SPPs being the input from the middle waveguide, even supermodes can be excited. These excited supermodes produced a multimode interference and formed a self-imaging phenomenon. The corresponding \(N\)-fold images were expected to form at distances [18]:  
\[L=\frac{P}{N}(\frac{3L_{\pi}}{4})\ (P\ =\ 1,\ 2,\ 3,\ \ldots) \tag{3}\]  
where \(P\) denotes the periodic nature of the imaging along the waveguides, \(L_{\pi}=\pi/(\beta_{1}-\beta_{2})\) is the beating length of the two lowest-order supermodes, and \(\beta_{1}\) and \(\beta_{2}\) are propagation constants of the two supermodes. In Figure 2b, the arrow A shows the position of a one-fold image (\(N=1\)), where \(L_{A}=11.2\)\(\upmu\)m. According to Equation (3), the positions of two-fold images and three-fold images were at \(L_{A}/2\) and \(L_{A}/3\), which are shown by the arrows B and C.  
The coupling transmission of the cross section between adjacent waveguides is shown in Figure 3, which corresponded to different z coordinates. It can be seen from Figure 3 that the energy was transferred from the middle waveguide to the waveguides on both sides due to the coupling between adjacent waveguides. Figure 2: (**a**) The arrangement of five HPWs in xz plane. (**b**) The propagation behavior of SPPs in five HPWs. As the distance increased, the coupling between waveguides became weaker. Figure 4 shows the propagation behavior of SPPs in three HPWs, in which the distance between waveguides was doubled. We removed the 2 nd and the 4 th waveguides on the basis of Figure 3. As can be seen from Figure 4, the length required for energy coupling from one waveguide to adjacent waveguides became longer. When the length of the HPWs was less than 10 um, the coupling between waveguides could be ignored. This also illustrates that the 5 th waveguide had little effect on the 3 rd waveguide in Figure 2.  
Figure 4: (**a**) The arrangement of three HPWs in xz plane. We removed the 2 nd and the 4 th waveguides on the basis of Figure 3 and the distance between adjacent waveguides was 1000 nm. (**b**) The propagation behavior of SPPs in three HPWs. Based on the simulation results of Figure 2b, we could design a \(1\times 3\) power splitter by the self-imaging phenomenon [18].