## 3 Design and Simulation
### Optimization of the Multimode Interference Region  
In this study, a uniform coupling region with a width of 3 \(\upmu\)m was designed. As stated in Equation (2), the beat length is in direct proportion to the width of the coupling region. As a result, decreasing the width of the coupler would result in a shortened length of the coupling region. This is beneficial for creating more compact optical devices. However, there is a trade-off, i.e., excessively reducing the width may lead to unintended coupling between the modes of the output ports. This phenomenon occurs because, as the width narrows, the spatial separation between modes decreases, increasing the likelihood of crosstalk and mode interference. To mitigate this issue, the coupling region is designed in a parabolic shape. The parabolic design helps to gradually taper the width, allowing for a smoother transition and better control over the coupling dynamics. The width change function of the coupling region is formulated to optimize performance while maintaining a balance between reducing the coupling region's length and avoiding unintended mode coupling. The width change function of the coupling region is formulated according to the following equation [22]:  
\[\text{y}=W_{0}+(W_{1}-W_{0})\ \frac{\text{x}^{2}}{L_{\text{MMI}}^{2}}, \tag{5}\]  
where x and y are the range of values for the length and width of the coupling region, respectively. \(W_{0}\) is the width of the coupling region at x = 0, whereas \(W_{1}\) is the width of the coupling region at x = \(L_{\text{MMI}}\). In order to determine the optimal imaging length and width of the coupling zone, this study used the EME function module of the Mode Solutions software (v.2020 r2) for simulation and analysis. The width of the narrow end of the coupling zone, \(W_{0}\), was first determined, and then the value of \(W_{0}\) was systematically varied to assess its effect on the normalized transmission as well as the value of \(L_{\text{MMI}}\). Through such simulations, the authors of this study were able to better understand the impact of width and length variations on coupling efficiency and transmission quality and, thus, optimize the design parameters to achieve optimal performance. Figure 4 presents the curves showing the variation in normalized transmission with \(L_{\text{MMI}}\) for different values of \(W_{0}\).  
From Figure 4, it can be clearly seen that when the value of \(W_{0}\) increases from 2.2 to 2.6 \(\upmu\)m, the normalized transmission shows an increasing and then decreasing trend with the change of \(L_{\text{MMI}}\). The normalized transmission is maximum when \(W_{0}\) is 2.4 \(\upmu\)m. When \(W_{0}\) is 2.4 \(\upmu\)m, the results are shown in Figure 4 when the \(L_{\text{MMI}}\) increases from 2.5 to 6 \(\upmu\)m. It can be seen that the total normalized transmission is highest at 0.992 when the length of the coupling region is kept at 4 \(\upmu\)m. Thus, in the subsequent analysis, the length of the coupling region was chosen as 4 \(\upmu\)m, and the width of the coupling region was chosen as 2.4 \(\upmu\)m. Figure 5 shows the optical field distribution of a 1 \(\times\) 3 power splitter based on  
Figure 4: Curves of normalized transmission with width and length in the coupling region.  
multimode interference in a parabolic Si-GaN slot-waveguide structure. The input light at the output port is evenly divided into 3 parts at the output port.