## 3 Component Design and Modeling
### Mach-Zehnder Optical Modulators
In the formula, \(S_{21}\) represents the transmission coefficient of the traveling wave electrode scattering matrix, \(\beta_{opt}^{u}\) represents the velocity matching between the microwave and light wave, \(L\) is the length of the traveling wave electrode, \(\omega_{\text{m}}\) is the angular frequency of the modulator, \(c\) is the speed of light, \(n_{u}\) is the microwave refractive index, \(n_{opt}\) is the refractive index of light. It can be known from the formula that the electro-optical bandwidth is related to \(\beta_{opt}^{u}\), \(S_{21}\) and the length of the modulator. The refractive index of light wave \(n_{opt}\) is affected by waveguide shapes, doping profiles of the phase shifters. The refractive index of the microwave \(n_{u}\) is mainly related to the electrode geometry and capacitance of the PN junctions. Clearly, there are many factors affecting the 3 dB E-O bandwidth and performance tradeoffs are always faced when designing the modulators. For balanced performance between insertion loss, modulation speed and modulation depth, the phase shifter length is chosen as 2 mm. According to our simulation results, the corresponding 3-dB E-O bandwidth of the optical modulator is 35.83 GHz. To achieve impedance matching, on-chip TiN terminator of 30 \(\Omega\) is utilized in our simulations. The transmission spectra of the silicon Mach-Zehnder interferometer can be expressed as follows:  
\[T(\lambda)=\frac{C(\lambda)}{2}.\bigg{[}1+2\sqrt{S(\lambda)(1-S(\lambda)}. \cos(\varphi(\lambda))\bigg{]} \tag{10}\]  
\[\varphi(\lambda)=\frac{2\pi N_{effw}}{\lambda}.\Delta L+\frac{2\pi N_{effg}}{ \lambda}.2\Delta x \tag{11}\]  
where \(C(\lambda)\) represents the total coupling efficiency of the bidirectional grating coupler at a specific wavelength, \(\varphi(\lambda)\) represents the optical phase difference between the two optical arms of the MZI, \(N_{effw}\) and \(N_{effg}\) represent the effective refractive index of single mode waveguide and the grating structure, respectively. \(S(\lambda)\) represents the grating splitting ratio, which is defined as the optical power splitting into one waveguide arms divided by the total power coupling of the two waveguide arms. Specially, \(S(\lambda)\) equals to 0.5 when the fiber is perfectly aligned in the grating center. \(\Delta L\) and \(\Delta x\) represent the waveguide arm length difference and the lateral fiber misalignment with respect to the grating center, respectively. However, in order to obtain the optical transmission of the optical modulator, the waveguide loss and optical loss caused by doping cannot be ignored. Therefore, we use a circuit model to calculate the transmission spectra of the optical modulator. In the optical domain, the scattering parameter matrix of the bidirectional grating coupler and the MMI coupler are extracted from the FDTD simulations. Based on the compact model of those optical elements, the circuit model of the Mach-Zehnder interferometer is built up. In the electrical domain, the transmission line model of the travelling wave electrode is formed by incorporating the P-N junction parameters such as the series resistance and capacitance obtained from the commercially available software charge. Thus, both the static and dynamic response of the optical modulator to electrical signals can be calculated. Figure 7: The simulation results of PN phase shifter. (**a**) Modulation efficiency (dot and black line) and optical waveguide transmission loss (dot and red line) with different applied voltages. The inset picture shows the simulated doping profile of the PN phase shifter. (**b**) Microwave loss (dot and black line) and impedance of the travelling wave electrode (dot and red line). The inset shows the microwave mode profile. Figure 8a,b shows the normalized transmission spectrum of the MZ optical modulator and the spectra response with driving voltage of 2 V, respectively.