## 2 Schematics
Assuming gray coding, the total BER equation is as follows:  
\[BER_{total}=\frac{Errorbits_{x}+Erorbits_{y}}{total\ bits}=\frac{SER_{x}+ SER_{y}}{N} \tag{3}\]  
For a typical MPAM, the SER equation is as follows [17]:  
\[SER_{MPAM}(\gamma_{s})=\frac{M-1}{M}erfc\left(\sqrt{\frac{3\gamma_{s}}{M^{2}-1 }}\right) \tag{4}\]  
where \(\gamma_{s}\) refers to \(\overline{E_{s}}/N_{0}\), the average symbol power-noise ratio. Here \(\overline{E_{s}}\) is defined as follows:  
\[b_{on}-b_{off}=2(M-1)\cdot\sqrt{\frac{3\overline{E_{s}}}{M^{2}-1}} \tag{5}\]  
where \(b_{on}\) refers to the maximum amplitude and \(b_{off}\) refers to the minimum amplitude that the modulator can generate. Therefore, for a given \(N_{0}\), if \(b_{off}\) and \(b_{on}\) are determined, SER can also be determined. Therefore, using Equations (4) and (5), we get:  
\[SER_{MPAM}(\gamma_{s})=SER_{MPAM}\left(\frac{(M+1)\left(b_{on}-b_{off}\right) ^{2}}{4(M-1)N_{0}}\right)=\frac{M-1}{M}erfc\left(\frac{b_{on}-b_{off}}{2(M-1) }\sqrt{\frac{1}{N_{0}}}\right) \tag{6}\]  
Then, if the MPAM is modulated along the \(X\)-axis and OOK is modulated along the \(Y\)-axis, Equation (3) can be modified as follows:  
Figure 3: Transfer curves and expected eye diagram for each polarization when the symbol mapping and polarization power rate is changed. (**a**) \(X\)-axis voltage vs. intensity transfer curve; (**b**) Expected \(X\)-axis eye diagram; (**c**) \(Y\)-axis voltage vs. intensity transfer curve; (**d**) Expected \(Y\)-axis eye diagram. \[E_{out,Y}=E_{in,Y}e^{j\pi\cdot V/6V_{\pi}}+E_{in,Y}e^{-j\pi\cdot V/6V_{\pi}}=E _{in,Y}\cos\frac{\pi V}{6V_{\pi}} \tag{2}\]0  
If \(N_{0}\) is a fixed value, the BER performance is only affected by \(\left(b_{on}-b_{off}\right)_{x}\) and \(\left(b_{on}-b_{off}\right)_{y}\). Each term must be normalized for optimization. From Equations (1) and (2), the intensity for each polarization is determined as follows:  
\[E_{out,Y}=E_{in,Y}e^{j\pi\cdot V/6V_{\pi}}+E_{in,Y}e^{-j\pi\cdot V/6V_{\pi}}=E _{in,Y}\cos\frac{\pi V}{6V_{\pi}} \tag{2}\]1  
\[E_{out,Y}=E_{in,Y}e^{j\pi\cdot V/6V_{\pi}}+E_{in,Y}e^{-j\pi\cdot V/6V_{\pi}}=E _{in,Y}\cos\frac{\pi V}{6V_{\pi}} \tag{2}\]2  
where \(\alpha\) refers to the polarization dividing ratio and \(\beta\) refers to the intensity. Therefore, if one voltage signal \(V\) (symbol mapping) and \(\alpha\) are determined, \(\left(b_{on}-b_{off}\right)_{x}\) and \(\left(b_{on}-b_{off}\right)_{y}\) are automatically determined by Equation (8). If the MZM has an ideal cosine curve, the optimal point is determined as shown in \[E_{out,Y}=E_{in,Y}e^{j\pi\cdot V/6V_{\pi}}+E_{in,Y}e^{-j\pi\cdot V/6V_{\pi}}=E _{in,Y}\cos\frac{\pi V}{6V_{\pi}} \tag{2}\]3  
The optimal point is the point where the eye per bit for each polarization is equal. This is because the power distance between the symbols determines the error rate. A smaller number of eyes per bit indicates that insufficient power is allocated to the polarization axis. Therefore, sufficient signal power should be allocated by adjusting the symbol mapping and polarization power division. In addition, to maximize efficiency, the symbol should be modulated in the \(3V_{\pi}\) range, as depicted in Figure 3, rather than in the \(2V_{\pi}\) range, as depicted in \[E_{out,Y}=E_{in,Y}e^{j\pi\cdot V/6V_{\pi}}+E_{in,Y}e^{-j\pi\cdot V/6V_{\pi}}=E _{in,Y}\cos\frac{\pi V}{6V_{\pi}} \tag{2}\]4 This is because the \(Y\)-axis distance decreases when it is modulated within the \(2V_{\pi}\) range.