## 3 Component Design and Modeling
### Mach-Zehnder Optical Modulators  
To construct a Mach-Zehnder optical modulator [44], the subwavelength apodized bidirectional grating coupler is used as a power splitter at the input port, while a 1 \(\times\) 2 MMI coupler is utilized for optical power combining at the output port. A length difference \(\Delta L\) is introduced to form an asymmetric MZI. The schematic diagram of the optical modulator is shown in Figure 6a, and the cross-sectional view of the P-N phase shifter is shown in Figure 6b. As can be seen, a dual-drive push-pull operation scheme is utilized in the modulator design and on-chip terminator can be integrated for impedance matching. The cross-sectional view of the P-N phase shifter design is depicted in Figure 6b. The phase shifter is based on a ridge waveguide structure with width of 500 nm, height of 220 nm and slab thickness of 70 nm. Six different doping conditions are adopted to form the phase shifter. To improve the modulation efficiency, the P-N junction located in the waveguide rib is fully optimized with Electro-Optical simulations. As a result, the P-type doped region  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & \multirow{2}{*}{**CA \({}^{\star}\)**} & \multicolumn{2}{c}{**CE \({}^{\star}\) (dB)**} & \multicolumn{2}{c}{**BW \({}^{\star}\) (nm)**} & \multicolumn{2}{c}{**FS \({}^{\star}\)**} \\ \cline{3-8}  & & **Meas** & **Simu** & \(-\)**1 dB** & \(-\)**3 dB** & **CD \({}^{\star}\) (nm)** & **E \({}^{\star}\)** & **D \({}^{\star}\)** \\ \hline
[37] & 15\({}^{\circ}\) &  
in the rib is 300 nm in width and the N-type doping region is 100 nm in width. This means an intrinsic layer of 100 nm exists between P and N doping regions. This helps decrease the capacitance and improve the modulation speed. The P+ and N+ regions are medium-doped regions, which are located 250 nm away from the edge of the ridge waveguide to achieve a balanced performance between optical loss and modulation speed. P++ and N++ dopings are used for ohmic contact with the metal electrode. Table 3 listed all the design parameters of the PN phase shifter, which are used in the simulations of the optical modulator and the whole optical transmitter. To evaluate the performance of the phase shifter, we calculate the carrier density distribution change and PN junction capacitance change under different voltages using the commercially available software of Ansys Lumerical Charge. Then by importing the data to the optical waveguide and coplanar waveguide electrode simulations, the modulation efficiency, the optical loss and the characteristics of the travelling wave electrode can be obtained, which are depicted in Figure 7. The modulation efficiency and the optical loss are about 1.6 V-cm and 6.7 dB/cm with a bias voltage of 1 V. The microwave transmission loss and impedance of the travelling wave electrode (TWE) is shown in Figure 7b. The modulator speed is mainly decided by the 3-dB E-O bandwidth, which can be calculated by the following equation:  
\[EO_{S_{21}}=10 log\frac{|S_{21}|^{2}-2\cdot|S_{21}|\cdot\cos\left(\beta^{u}_{ opt}\cdot l\right)+1}{\left(\ln|S_{21}|\right)^{2}+\left(\beta^{u}_{opt}\cdot l \right)^{2}} \tag{8}\]  
\[\beta^{u}_{opt}=\frac{\omega_{m}}{c}\big{(}n_{\mu}-n_{opt}\big{)} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{3}{c}{**W\({}_{\text{rF}}\)** = 0.3 \(\mu\)m, \(\text{W}_{\text{rN}}\) = 0.1 \(\mu\)m, \(\text{W}_{\text{s}}\) = 0.125 \(\mu\)m} \\ \hline Doping region & P++ & N++ & P+ & N+ & P & N \\ Concentration cm\({}^{-3}\) & \(1\times 10^{20}\) & \(1\times 10^{20}\) & \(4\times 10^{18}\) & \(4\times 10^{18}\) & \(3.5\times 10^{17}\) & \(6\times 10^{17}\) \\ Span (\(\mu\)m) & 4.75 & 4.75 & 5.625 & 5.625 & 0.425 & 0.425 \\ \hline \end{tabular}
\end{table} 
Table 3: The design parameters of the PN phase shifter. Figure 6: (**a**) The schematic diagram of the silicon Mach–Zehnder optical modulator based on bidirectional grating coupler. (**b**) The schematic cross-sectional view of the P-N phase shifter.