## 2 Polarization Control Structure Furthermore, Principle
### Mach-Zehnder Interferometer-Based Polarization Control Structure  
An alternative polarization control structure is composed of a PSR and a tunable MZI, as shown in Figure 4. The two output ports of the PSR are connected to an MZI incorporating two phase shifters and two couplers [17; 18], which can be expressed as  
\[M_{2}=\frac{1}{2}\begin{bmatrix}1&\text{j}\\ j&1\end{bmatrix}\begin{bmatrix}e^{\beta_{2}}&0\\ 0&1\end{bmatrix}\begin{bmatrix}1&j\\ j&1\end{bmatrix}\begin{bmatrix}e^{\beta_{1}}&0\\ 0&1\end{bmatrix} \tag{7}\]  
where \(\theta_{1},\theta_{2}\) represents the phase of the first-stage and second-stage phase shifter, respectively. The first-stage \(\theta_{1}\) is used to rotate the input polarization around \(S_{1}\) axis on the Poincare ball to avoid \(\theta_{2}\) going beyond its boundary, which has only two choices \(0\) or \(\pi\).  
The second-stage \(\theta_{2}\) is used to rotate the polarization of light around the \(S_{3}\) axis. By changing the phases of them, the SOP of the output two paths of light can be changed accordingly, and achieve the purpose of controlling the output power of the two paths of light to be equal. Regardless of tracking speed, an MZI can rotate the input LO of any SOP to the target position.  
Figure 4: The control structure based on MZI.  
Figure 3: The trajectory controlled by HWP on Poincaré ball.  
Here, the photoelectric field of input LO can also be denoted as Equation (3). After being controlled by a two-stage MZI controller, the two output polarized lights can be represented as  
\[\begin{pmatrix}E_{xout}\\ E_{yout}\end{pmatrix}=M_{2}\cdot|E_{in}(\theta,\beta)\rangle\] \[=\frac{1}{2}\begin{pmatrix}a-bj+cje^{-j\theta_{1}}-de^{-j\theta_{ 1}}-ae^{-j\theta_{2}}\dots\\ +je^{-j\theta_{2}}+cje^{-j(\theta_{1}+\theta_{2})}-de^{-j(\theta_{1}+\theta_{ 2})}\\ aj+b-ce^{-j\theta_{1}}-dje^{-j\theta_{1}}+aje^{-j\theta_{2}}\dots\\ +be^{-j\theta_{2}}+ce^{-j(\theta_{1}+\theta_{2})}+dje^{-j(\theta_{1}+\theta_{ 2})}\end{pmatrix} \tag{8}\]  
where \(a=\cos\theta\cos\beta\), \(b=\sin\theta\sin\beta\), \(c=\sin\theta\cos\beta\), and \(d=\cos\theta\sin\beta\).  
The power difference between these two polarized lights can be obtained by using a BPD. The output of BPD under the prerequisite of \(\theta_{1}=0\) or \(\theta_{1}=\pi\) is:  
\[\varepsilon_{I}|_{\theta_{1}=0} =|\cos 2\beta\cos(2\theta+\theta_{2})| \tag{9}\] \[\varepsilon_{I}|_{\theta_{1}=\pi} =|\cos 2\beta\cos(2\theta-\theta_{2})| \tag{10}\]  
Similarly, a DSP module exports \(V_{1}\), \(V_{2}\) decided by \(\theta_{1}\), \(\theta_{2}\) through minimizing \(\varepsilon_{I}\) originating from BPD. After the \(\varepsilon_{I}\) minimum state is locked, \(\theta_{2}\) has:  
\[\theta_{2}|_{\theta_{1}=0} =\pm\frac{\pi}{2}-2\theta \tag{11}\] \[\theta_{2}|_{\theta_{1}=\pi} =\mp\frac{\pi}{2}+2\theta \tag{12}\]  
When stabilizing the polarization state of the input light, only the second-stage phase shifter needs to be tuned in most cases. However, when the second-stage phase shifter reaches its voltage bound, it needs to be reset with the help of the first-stage phase shifter. From Equations (11) and (12), we can conclude \(\theta_{2}|_{\theta_{1}=0}=-\theta_{2}|_{\theta_{1}=\pi}\), so we can flip the tunable direction of \(\theta_{2}\) by shifting \(\theta_{1}\) to 0 or \(\pi\). If \(\theta_{2}\) gradually increases to its limit value, shifting \(\theta_{1}\) can gradually decrease \(\theta_{2}\) so that \(\theta_{2}\) is always working within its working voltage range [19]. The change process of the polarization state through MZI from the perspective of the Poincare sphere is shown in Figure 5. It is supposed that the SOP of the input light of the PC can be represented by point \(A\). After passing through the polarization control loop, point \(A\) is rotated around the \(S_{3}\) axis to the target ring. Point \(A\) will make omissions to \(A^{\prime}\) when \(\theta_{1}\) is shifted between 0 and \(\pi\), which reverses the change direction of \(\theta_{2}\).  
Figure 5: The trajectory controlled by MZI on Poincaré ball.