## 2 Notations and Methodology
### EO Performances of the Active Region of the MZM  
Silicon MZM electro-optical performances such as loss and modulation efficiency are more often referred to in terms of unit length: the loss \(\alpha\) in (dB/mm), and the phase modulation efficiency between 0 V and \(V_{\pi}\) noted as \(\Delta\varphi\) in (\({}^{\circ}\)/mm). The later parameter is linked to the product \(V_{\pi}L_{\pi}\) in (V\(\cdot\)cm) with the relation:  
\[\Delta\varphi=\frac{360\times\Delta n_{\rm eff}}{\lambda}=\frac{180\times V_{ \pi}}{V_{\pi}L_{\pi}} \tag{1}\]  
When the MZM electrode length \(l_{\rm MZM}\) is set, we can write IL\({}_{\rm MZM}=\alpha\times l_{\rm MZM}\). Since the voltage \(V_{\pi}\) represents the voltage needed for a given length \(l_{\rm MZM}\) to obtain a shift of \(\pi\), for an estimated efficiency of \(V_{\pi}L_{\pi}\) we have also the relation \(V_{\pi}=V_{\pi}L_{\pi}/l_{\rm MZM}\). Several phase shifting mechanisms exist [14] and give different compromises between these two parameters. In our case, we choose a doped p-n junction working in the depletion regime since it is currently the most common type used for high speed silicon modulators. The junction is embedded in a rib waveguide of 400 nm of width, 300 nm of height, and with a slab of 150 nm of height. In order to illustrate what may be achieved in terms of \(V_{\pi}L_{\pi}\) and \(\alpha\) for this junction, a parametric study was performed depending on the doping level of the N and P sides between (6.10\({}^{16}\)-5.10\({}^{18}\)) cm\({}^{-3}\), which is the order of magnitude of the doping levels using in the PN modulators. The doping levels impact the depletion width, and thus the capacitance of the p-n junction (see Section 2.4). The doping regions are assumed to have uniform doping densities and the junction is perfectly vertical. The performances of the modulator at 1310 nm and at an absolute bias voltage of 2.4 V are obtained by electro-optical simulations using Lumerical(r) software (2016a, Lumerical Solutions Inc., Vancouver, BC, Canada). The refractive indices of the materials were \(n_{\rm Si}=3.48\), and \(n_{\rm SiO2}=1.44\), and the Soref equations at 1310 nm were used [15]. Since the undoped propagation loss is mainly dominated by side wall roughness and thus is a foundry parameter, it is not included in the simulations in order to keep a more general approach. This variable is implicitly included in the tolerable path loss. The optimal position of the junction inside the waveguide, noted \(\delta_{\rm PN}\) and represented Figure 3, ensuring the best efficiency for each couple of doping \(N_{A}\) and \(N_{D}\) is implicitly included in the simulation. This offset can be obtained numerically by using a 1D model under the assumptions that: (1) the refractive index variations are induced by an abrupt p-n junction (equivalent of two correlated rectangular functions), and (2) at half height of the waveguide, the optical mode profile can be fitted by a center Gaussian profile with a beam's width \(\sigma\) of 87 nm for a waveguide of 400 nm of width at 1310 nm. The optimal position corresponds to the maximum of the convolution product between the two functions. Figure 3 displays the possible couples of efficiency and loss of the active region for all the doping levels simulated (purple region). The ideal MZM would have a small value of \(\alpha\) and large \(\Delta\varphi\), however simulations show that it is not possible to decrease one without decreasing the other. The unavoidable compromise between \(\alpha\) and \(V_{\pi}L_{\pi}\) expresses the intrinsic limitation of the junction itself. The best compromises are represented on the blue dash line on the graph. This curve is crucial for the design of optical modulator since it includes all the parameters of the p-n junction which allows us to obtain the best compromise between the efficiency and the losses.