## 3 Results  
Using BPM and FDTD in Rsoft photonic suite, the MMI coupler and the buried structure were simulated, and the results were processed using Python code to determine the best values. The TE fundamental mode profile inside the SiN strip for an operating wavelength of 1540 nm is shown in Figure 2a at the XY plane, and its horizontal view, where Y = 0.15 \(\upmu\)m, can be seen in Figure 2b. The red hue indicates high confinement, and an adiabatic taper can increase the mode 2.36 \(\times\) 2.22 \(\upmu\)m spot size to improve coupling between the demultiplexer and the laser source. For 1560 nm wavelength, a similar mode profile was obtained. Figure 2c shows the TM fundamental mode solution inside the SiN strip at the XY plane, and its vertical view, where X = 0 \(\upmu\)m, is shown in Figure 2d. As can be observed, the TM mode light in the center mode is not entirely confined at only 82%, which can increase the power light losses. Thus, it is better to use the TE mode in our design. In order to simulate the device, we used the mode solutions of both wavelengths as the initial launch conditions in the input waveguide of the demultiplexer. Figure 2: (**a**) The 1540 nm TE fundamental mode (SiN area in dotted rectangle) and (**b**) its horizontal view at Y = 0.15 \(\upmu\)m; (**c**) The 1540 nm TM fundamental mode (SiN area in dotted rectangle) and (**d**) its vertical view at X = 0 \(\upmu\)m.  
The fundamental mode for each operational wavelength was solved to determine the effective refractive indexes (n\({}_{\text{eff}}\)) values, which were 1.4823 and 1.4793 for 1540 nm and 1560 nm, respectively. Figure 3 illustrates the optimizations for the 320 nm SiN height, which was chosen as the ideal value, and its tolerance range of \(\pm\)20 nm (\(\pm\)6.25%) around the ideal value achieved normalized power of 75% to 82% (input power is considered to be 100%). Figure 3 illustrates the maximum power obtained for a height value of 310 nm. However, the maximum fabrication error that can currently be handled is not satisfied by this value from a fabrication standpoint. This restriction is a result of the fabrication process geometrical dimension inaccuracy, which is typically \(\pm\)20 nm away from the ideal value. As a result, with a SiN layer thickness error of \(\pm\)20 nm, the suggested device can function efficiently. Figure 4 illustrates the tolerance range of the MMI coupler width, which is \(\pm\)20 nm around the ideal value of 3.5 \(\upmu\)m (\(\pm\)0.6%), in order to achieve a normalized power (input power is considered to be 100%) of 77% to 81%. This restriction was established because a fabrication process with great accuracy typically deviates roughly \(\pm\)20 nm from the ideal value in terms of geometrical dimensions. These fabrication results can be used in the prototype test structure. Figure 4: Normalized power as a function of MMI width for both wavelengths. Figure 3: Normalized power as a function of SiN strip height. The beat length can be determined by using Equation (1) for each wavelength; the parameter p was found to be 65 using Python code, which takes into account the geometrical parameters and wavelengths and also calculates beat length, efficient MMI width, and the theoretical MMI length, combined with Equations (1)-(3). The parameter p was calculated using the beat lengths of the two selected wavelengths. Using Equation (3), we isolated the parameter p by dividing the smaller beat length by the absolute value of the difference between them. At last, the result was rounded to an integer. Using the calculated parameter p with Equation (3), the theoretical MMI length was determined to be 2072 \(\upmu\)m.  
The optimal MMI coupler length was determined using Equation (3) and BPM simulation. L\({}_{\text{mmi}}\) was chosen to be 2067 \(\upmu\)m to achieve optimum output power with low manufacturing error.