## 4 Results and Discussion  
Based on the simulation results, the parameters and indicators of the optimized power divider are as follows: The thicknesses of Si and GaN are 0.2 \(\upmu\)m and 0.1 \(\upmu\)m, respectively. The width of the single-mode waveguide is 0.4 \(\upmu\)m, and the length of the coupling area is 4 \(\upmu\)m. Furthermore, the two ends of the parabolic coupling region have widths of 2.4 \(\upmu\)m and 3 \(\upmu\)m. The length and width of the taper are 2 \(\upmu\)m and 0.65 \(\upmu\)m, respectively. Additionally, the total normalized transmission is up to 0.992, indicative of high efficiency. Specifically, the power divider exhibits excess loss and imbalance of 0.036 dB and 0.003 dB, respectively.  
A previous study [12] proposed a power splitter that utilizes gallium nitride and silicon dioxide. The dimensions of the coupling region are specified as 16.55 \(\upmu\)m in length and 4 \(\upmu\)m in width, resulting in a normalized transmission of 0.976. In contrast, the power splitter described in another study [13] employs silicon and gallium nitride and presents a coupling region with dimensions of 12.03 \(\upmu\)m in length and 5 \(\upmu\)m in width, resulting in a normalized transmission of 0.984. Our present study introduces a device with a coupling region that is approximately three and four times smaller in length and width, respectively, and displays a higher normalized transmission as compared to those present in the literature [12; 13].  
The power splitter presented in reference [17] employs Si and SiN\({}_{\text{x}}\). The refractive index of SiN\({}_{\text{x}}\) is 2.5. The coupling region features an isosceles trapezoidal structure with a length of 13.2 \(\upmu\)m and an upper base and lower base of 4 \(\upmu\)m and 5.38 \(\upmu\)m, respectively. The excess loss and imbalance are below 0.07 dB and 0.03 dB, respectively. Contrastingly, the power splitters developed in our work have coupling regions that are approximately three times smaller in length and width, exhibit roughly twice the amount of excess loss, and have nearly ten times less imbalance when compared to those in reference [17].  
To conclude, the size of the power splitter designed in this study is significantly smaller than the power splitters proposed in recent years. This compact design is crucial for integrated photonic circuits where space is at a premium. Additionally, the power splitter designed in this study incurs lower losses, which is a vital aspect for efficient signal transmission and overall device performance. The reduction in excess loss not only improves energy efficiency but also enhances the reliability of the device in practical applications.  
Figure 8: Effect of width of coupling region on excess loss and imbalance.  
Moreover, the splitting uniformity of the designed power splitter is notably more favorable. Uniformity in power splitting ensures that the signal is evenly distributed across all output ports, which is essential for maintaining signal integrity in complex photonic circuits. The proposed design provides an efficient solution for power splitting applications, offering a balance between compact size, low loss, and high splitting uniformity. These results highlight the potential of the proposed power splitter for application in integrated photonic optical circuits. Finally, it must be acknowledged that the designed devices may deviate from the expected results when actually prepared. This deviation mainly comes from the fact that the design of the multimode interference coupling region adopts a parabolic structure, which may have many difficulties in actual preparation. However, these preparation errors are relatively small and will not have a large impact on the device performance.