## 3 Design and Simulation
### Optimization of the Taper  
The tapered structure in a waveguide system is a key factor in optimizing performance. The tapered structure is a transition region that smooths the mode shift between the single-mode input waveguide and the multimode interference region. This transition is critical because abrupt changes in waveguide dimensions can cause scattering and coupling inefficiencies, leading to higher excess losses and imbalances. Therefore, a tapered structure was introduced in the connecting portion of the input/output waveguide and the multimode interference coupling region. This design feature effectively mitigates the excess loss problem while improving the device imbalance. In addition, the taper improves the coupling efficiency from the single-mode input to the coupling region, further improving the fabrication tolerance of the power divider [21]. According to Figure 6a, there is a clear trend of excess loss and imbalance when the taper length is varied in the range of 1.9 to 2.1 \(\upmu\)m. It can be concluded that the best performance is achieved when the taper length is 2 \(\upmu\)m. Therefore, the taper length was set to 2 \(\upmu\)m. The width of the tapered surface was scanned to assess any associated excess loss or imbalance.  
It is evident from Figure 6b that within the 0.5 to 0.7 \(\upmu\)m range for the width of the taper, there is a notable decrease and subsequent increase in both the excess loss and imbalance. This trend reveals the sensitivity of the device performance to the width of the cone. For smaller taper widths, specifically closer to 0.5 \(\upmu\)m, the excess loss and imbalance decrease. This reduction can be attributed to the decrease in waveguide crosstalk. Crosstalk occurs when there is interference between the signals in adjacent waveguides, leading to signal degradation. A narrower taper width reduces the overlap between the modes of  
Figure 5: Optical field distribution of a \(1\times 3\) power splitter with a parabolic slot-waveguide structure.  
Figure 6: (**a**) Effect of taper length on excess loss and imbalance. (**b**) Effect of taper width on excess loss and imbalance.  
adjacent waveguides, thereby minimizing crosstalk and improving performance. However, as the taper width increases beyond a certain threshold, particularly above 0.65 \(\upmu\)m, the crosstalk between the output waveguides becomes more pronounced. This increased crosstalk results in a deterioration in the performance of the power splitter, manifesting as higher excess loss and imbalance. Therefore, it is crucial to choose an appropriate value of taper width that balances the reduction in excess loss and imbalance against the increase in crosstalk. Through simulation analysis, it was found that a taper width of 0.65 \(\upmu\)m provides the optimal trade-off between these factors. At this width, the device achieves a minimum imbalance and an excess loss of less than 0.036 dB. This balance ensures that the device performs efficiently without significant signal degradation. Thus, based on these findings, the taper width for the device was set to 0.65 \(\upmu\)m. This value was chosen to optimize the performance of the power splitter, ensuring low excess loss and minimal imbalance while avoiding the adverse effects of increased crosstalk.