## 3 MZM Parameter Optimization  
The optimization of the MZM parameters \(\left[ \begin{array}{cc}\mathrm{IL}_{\mathrm{MZM}}&V_{\pi}&\mathrm{BW}_{ \mathrm{MZM}}\end{array} \right]\) is equivalent to the optimization of \(\left[ \begin{array}{cc}\alpha_{\mathrm{MZM}}&\Delta\varphi&I_{\mathrm{MZM}} \end{array} \right]\) due to the relationship between these parameters as described in the section above. In order for a MZM with given parameters to be compliant with the transmission performance required, it must satisfy the condition \(P_{\mathrm{RX}}\geq P_{\mathrm{REQ}}\), where (in dB scale):  
\[P_{\mathrm{RX}}=P_{\mathrm{LAS}}-\alpha_{\mathrm{MZM}}\times I_{\mathrm{MZM}} -3-\mathrm{IL}_{\mathrm{PATH}} \tag{6}\]  
The 3 dB term accounts for the fact that the MZM is biased in quadrature. The target value of \(P_{\mathrm{REQ}}\) is determined form Figure 2 with \(V_{\pi}=\left(V_{\pi}L_{\pi}\right)/I_{\mathrm{MZM}}\), and the bandwidth value \(\mathrm{BW}_{\mathrm{MZM}}\) is taken from Figure 5.  
In order to illustrate the optimization process, we arbitrarily set the value of \(P_{\mathrm{LAS}}=+10\) dBm and the MZM length \(l_{\mathrm{MZM}}=1\) mm. Figure 7 shows the limiting curves in the \(\left[ \begin{array}{cc}\alpha_{\mathrm{MZM}}&V_{\pi}L_{\pi}\end{array} \right]\)  
Figure 5: MZM electrical modulation bandwidth vs. electrode length. Symbols are simulation results and the line is a polynomial fit to those results. Figure 6: MZM Transmission line equivalent circuit model. space. The upper limit (blue curve) is linked to the requirement described above i.e., the highest MZM insertion loss that satisfies \(P_{\text{RX}}\geq P_{\text{REQ}}\) when \(\text{IL}_{\text{PATH}}=0\):  
\[\alpha_{\text{MZM}}\leq\left(P_{\text{LAS}}-3-P_{\text{REQ}}\right)/l_{\text{ MZM}} \tag{7}\]  
where \(P_{\text{REQ}}\) is also a function of \(V_{\pi}\) and \(\text{BW}_{\text{MZM}}\) or, in other terms, of \(\alpha_{\text{MZM}}\) and \(V_{\pi}L_{\pi}\). The thick blue curve is extracted directly from the numerical results of Figure 2 while the dashed thin blue line is an analytical extrapolation of these results for large \(V_{\pi}\). When \(V_{\pi}\) increases, the eye diagram closes and a power penalty related to the degraded extinction ration is subtracted to the tolerable link budget, thus reducing the allowed MZM insertion loss. The thick black line shows the lower limit with the best performing MZM (Figure 3) based on our simulated waveguide and junction geometry. All the MZM within those two limits offer sufficient performances for a 25 Gbps link to operate with a BER \(<10^{-9}\). However, the MZM whose parameters are situated on the upper limit (blue dashed line) have the maximum allowable insertion loss and thus do not permit additional losses to be inserted between the TX and RX (i.e., \(P_{\text{RX}}=P_{\text{REQ}}\) for those parameters). On the contrary, the MZM whose parameters are situated on the black line offer the best performance in terms of efficiency and insertion loss and lead to the availability of path loss margin. Among all the possible parameters \(\left[ \begin{array}{cc}\alpha_{\text{MZM}}&V_{\pi}L_{\pi}\\ \end{array} \right]\) belonging to this lower limit, one particular point maximizes the optical loss \(L_{\text{PATH}}\) that can be tolerated between the transmitter (TX) and receiver (RX), this point is represented by a blue dot on Figure 7 at 1.7 dB/mm and 1.17 V\(\cdot\)cm. Indeed, if we draw a horizontal line from this optimal point to the upper limit curve, the intersect point is situated at 25.2 dB/mm and 1.17 V\(\cdot\)cm. The loss that can be tolerated is thus \((25.2-1.7)\times l_{\text{MZM}}=23.5\) dB with a MZM length of 1 mm. For clarity, we plot on Figure 8 the tolerable path loss as a function of \(V_{\pi}L_{\pi}\) directly computed from Figure 7 i.e., the difference along the \(x\)-axis between the black and dash-blue curves of Figure 7 multiplied by the MZM length.