# O-Band Multimode Interference Coupler Power Combiner Using Slot-Waveguide Structures
## 3 Simulation Results  
The numerical investigations of the 2 \(\times\) 1 MMI power combiner based on using SiN slot-waveguide structures were performed using Rsoft-cad software. The data from FV-BPM and FDTD results were analyzed using MATLAB codes. Figure 2a shows the solution of the TE fundamental polarization mode and the mode profile at the x-y plane for two Gaussian laser sources that work at 1310 nm wavelength. As can be seen in the red color areas, a high intensity is obtained inside the SiN layers as expected from the light behavior under TE mode in the slot waveguide. Thus, a large range of high intensity is obtained over the \(y\)-axis with 794 nm. This large range can be useful for improving the light coupling adjustment between the two laser sources and the input waveguide taper over the \(y\)-axis. It is important to clarify that Figure 2a shows the profile of the TE fundamental polarization mode and not the physical location of the lasers. This means that the gap between the laser can be seen, as in Figure 1a. The gap between the two laser sources was set to 1.4 \(\upmu\)m, for the sake of diminishing the possibility of light coupling, as shown in Figure 2a. Usually, in classical waveguide techniques, the spacing is set to be over 2 \(\upmu\)m but, in our case, the slot waveguide allowed us to gain powerful light confinement, so it could be utilized to choose a narrower gap. According to the solutions of the TE fundamental polarization mode, we also optimized the grid convergence size in the \(x\)-axis and the \(y\)-axis to 20 nm and the \(z\)-axis to 15 nm using FV-BPM. These sizes were utilized for all other FV-BPM simulations. The normalized intensity level for the two sources over the \(x\)-axis at y = 250 nm, which is located at the SiN layer, as can be seen in Figure 2b. The n\({}_{\text{eff}}\) value was extracted by solving the fundamental mode inside the SiN slot waveguide and its value was 1.715. To obtain a compact combiner device the width of the MMI combiner coupler was set to 3.55 \(\upmu\)m. W\({}_{\text{e}}\) and L\({}_{\text{\pi}}\) were calculated using Equations (1) and (2) for the TE mode operation and their results were 2.94 \(\upmu\)m and 15.13 \(\upmu\)m, respectively. The optimal size of the MMI coupler length was found using FV-BPM simulations combined with MATLAB codes and the value was 9.81 \(\upmu\)m.  
Figure 3a,b shows the optimizations of the MMI combiner coupler and the tolerance range for the length and the width of the MMI combiner coupler. The width of the MMI coupler tolerance range was 3.48 to 3.62 \(\upmu\)m; this value was suitable to transmit more than 97% of the total power, as shown in Figure 3a. The length of the MMI coupler tolerance  
Figure 2: Solution of the TE fundamental mode for 2 \(\times\) 1 power combiner: (**a**) E\({}_{y}\) polarization mode profile; (**b**) horizontal view at y = 220 nm. range was 9.6 to 10.1 \(\upmu\)m, which was also ideal to transmit more than 97% of the total power, as shown in Figure 3b. From the fabrication overview, these tolerance parameter values were more than 200 nm for L\({}_{\text{MMI}}\) and 70 nm for W\({}_{\text{MMI}}\) from the optimal values. Thus, these results are considered great for a facility with a high accuracy of \(\pm\)40 nm from the optimal geometrical parameter. These tolerance ranges could also be utilized for the test structure measurements, which usually help to find the parameter's geometrical errors by comparing the experimental results to the theoretical results, as shown in Figure 3a,b. The input/output waveguide taper length was optimized to be suitable to the width changing size from 0.6 to 1.318 \(\upmu\)m, while keeping the mode solution stable, also called the adiabatic condition of the taper. From Figure 4, it can be seen that at more than 10 \(\upmu\)m, a normalized power level of 97% was obtained, however this length did not match the adiabatic condition, which is described in Equation (5).