## 3 BPM Simulation Results and Discussion
In the whole C-band, the average splitting ratios of 50:50, 60:40, 70:30, 80:20, and 90:10 could be achieved by using BPM corresponding to the shifting length of 0, 0.063, 0.128, 0.2, and 0.297 mm, respectively. For C-band, regression analysis is employed to process these five points to derive regression polynomials:  
\[SR_{1}=-133.82\left(L_{s}\right)^{2}+175.76\ L_{s}+49.78, \tag{4}\]  
The R-squared value is 0.9996 representing a good fit to the actual data. The polynomial indicates that the average splitting ratio in C-band corresponding to the shifting length \(L_{s}\) can be predicted precisely. Further analysis on the wavelength dependent of splitting ratio with wavelengths between 1.53 and 1.57 um at the different shifting length \(L_{s}\) is shown in Figure 4. Five specific splitting ratios of output port guide 1 and guide 2 as the function of wavelength for each specific shifting lengths \(L_{s}\) are presented. When the splitting ratio is adjusted from 50:50 to 90:10, the deviation of the splitting ratio of the splitter with wavelengths between 1.53 and 1.57 um does not exceed 0.335 from a close examination of the data shown in Figure 4. It indicates that the proposed CVW splitter with variable splitting ratio is wavelength insensitivity. Figure 1.0 Five specific splitting ratios of output port guide 1 and guide 2 as the function of wavelength for each specific shifting lengths \(L_{s}\). Figure 1.1 The splitting ratio variations of output port guide 1, as the function of shifting length \(L_{s}\) at wavelength 1.55 μm. The second order polynomial curves of splitting ratio of guide 1 is demonstrated. In order to further demonstrate the wavelength-insensitive performance of our proposed CVW splitter, we compare the root mean square (RMS) values of the splitting ratio of CVW and MMI [11] splitters. For the 90:10 MMI splitter, the RMS value of the splitting ratio is approximately 0.36 with wavelengths between 1.53 and 1.57 \(\upmu\)m. For the proposed 90:10 CVW splitter, the RMS value of the splitting ratio is only 0.11, showing the superior performance of the CVW design. Figure 5a shows the variations of excess loss with splitting ratios of 50:50, 60:40, 70:30, 80:20, and 90:10 with wavelengths between 1.53 and 1.57 \(\upmu\)m. The excess loss of the device for is about 0.111 \(\sim\) 0.139 dB. For the MMI splitter [11], the excess loss is less than 0.312 dB while the splitting ratio is adjusted from 50:50 to 90:10 with wavelengths between 1.53 and 1.57 \(\upmu\)m. The excess loss of the CVW splitter is 0.173 dB lower than in the MMI splitter. Figure 5b shows that the undesired crosstalk \(P_{0}\) observed in guide 0 is below \(-\)22.75 dB. Those mean that the power transfer of the variable-ratio CVW splitter is stable with low crosstalk and low excess loss with wavelengths between 1.53 and 1.57 \(\upmu\)m.  
In order to investigate the polarization effect, the fundamental TM mode is launched in the guide 0. The polarization dependent loss can be written as:  
Figure 1.2  
Figure 1.3 Variations of (**a**) excess loss and (**b**) crosstalk with wavelengths between 1530 and 1570 nm. The symbols of splitting ratio: \(\Delta\) 50:50; + 60:40; \(\times\) 70:30; \(\square\) 80:20; \(\Circle\) 90:10.  
where \(EL_{TE}\) is the TE-polarized excess loss and \(EL_{TM}\) is the TM-polarized excess loss. Figure 6 shows the variations of the polarization dependent loss in C-band with splitting ratios of 50:50, 60:40,70:30, 80:20, and 90:10. One can see that all PDL simulation results are less than 0.006 dB. Figure 7 compares the two waveforms of output power guide 1, guide 0, and guide 2 for TE and TM modes with splitting ratio 90:10 at wavelength 1.55 \(\upmu\)m. The mean absolute percentage error (MAPE) function is used to evaluate the output waveforms for TE and TM modes. The MAPE value is 1.77%; in other words, the output waveforms for TE and TM modes coincide well. Figures 6 and 7 indicate that the CVW splitter is insensitive to polarization.