## 3 BPM Simulation Results and Discussion  
In the proposed splitter, the core index is 1.543 and the refractive index of cladding is 1.528 at wavelength 1.55 \(\upmu\)m. In order to keep single-mode [22] operation in C-band, the proposed waveguide height is less than 3.56 \(\upmu\)m and the width is less than 4.45 \(\upmu\)m. The 3D-BPM [23] is used to evaluate the design concept and the performances of the devices. The fundamental TE mode is launched in the guide 0 for 1.55 \(\upmu\)m wavelength. Here, an extra section waveguide must be added to the output port of guide 1, as shown in Figure 1, to avoid discontinuities on the output port in 3D-BPM simulation and obtain a meaningful result. The extra guide is considered to keep the cross section and the slope of the guide 1 at \(z=L_{\rm R}\). It can be readily achieved by directly extending the guide 1. Thus, the I/O ports of this extra guide and the output port of guide 1 exhibit the same cross section. The extended waveguide is described as a parallel hexahedron, in which the light energy from guide 1 can be maintained. Figure 1: Schematic of the variable splitting ratio splitter with coupling-weighted and velocity-tapered waveguides (CVW) structure. The slope of the extended guide can be defined by the derivative of the tapered gap [21] of guide 1 at \(z=L_{R}\). Its length in the \(z\) direction is the same as the shifting length \(L_{s}\). The seven parameters to be determined are shown in Figure 1. To fulfill the single mode condition, two out of the seven parameters of CVW, the width of the wider port \(W_{b}\) and the height of the waveguide \(H\), can be predetermined. Then, the shifting length \(L_{s}\) is set to zero for 50:50 splitting. Thus, only four parameters are used in the full factorial design method. It was found that a satisfactory result can be obtained to simultaneously satisfy \(-20\) dB crosstalk and equal power splitting in C-band. The results are as follows: \(G_{a}\) = 6.6, \(G_{b}\) = 5.6, \(W_{a}\) = 2.6, \(W_{b}\) = 4.4, \(H\) = 3.5 \(\upmu\)m, and \(L_{R}\) = 3.8 mm. The BPM simulation result of power evolution along the \(z\)-direction for CVW splitter with the shifting length \(L_{s}\) is shown in Figure 2. One can see that the input power from the center guide can be gradually divided into the two outer guides along the \(z\)-direction. In addition, the power output from guide 1 will be greater than that from guide 2. The output powers of guide 0, 1, and 2 are represented as \(P_{0}\), \(P_{1}\), and \(P_{2}\), respectively. The crosstalk is the residual power of guide 0. The definition of the excess loss (EL) is the ratio between the total three output powers and the input power. The splitting ratio of guide \(i\) (\(i\) = 1 or 2) is defined as:  
\[SR_{i}=\frac{P_{i}}{P_{1}+P_{2}}\cdot 100\%,\ i=1,2, \tag{2}\]  
Finally, the shifting length \(L_{s}\) is varied to achieve the specified splitting ratio. Eleven shifting lengths spaced equally in the interval 0.05 mm are generated and its corresponding splitting ratios can be obtained by using BPM as shown in Figure 3. The splitting ratio of guide 1 will increase as the shifting length \(L_{s}\) becomes longer. A second-order polynomial curve can be fitted from the eleven-point dataset by using least squares regression. For operating wavelength 1.55 \(\upmu\)m, the splitting ratios of output port guide 1 can be changed by the polynomial function of shifting length \(L_{s}\), which is expressed as  
\[SR_{1}=-172.87\left(L_{s}\right)^{2}+187.93L_{s}+49.32, \tag{3}\]  
In the regression, the R-squared value is found to be 0.9995 indicating that the curve is rather reliable. According to the polynomial function, the common splitting ratios of 50:50, 60:40, 70:30, 80:20, and 90:10 at operating wavelength 1.55 \(\upmu\)m could be obtained corresponding to the shifting length of 0, 0.06, 0.124, 0.2, and 0.298 mm, respectively. Figure 2: Power evolution for CVW splitter with the shifting length \(L_{s}\).