## 2 Polarization Control Structure Furthermore, Principle
### Waveplate-Based Polarization Control Structure  
Figure 2 depicts the structure of the waveplate-based polarization controller. The received LO of any SOP can be seen as rotation \(\theta\) of elliptical polarized light \(|E\rangle\) in the laboratory coordinate system as  
\[|E\rangle^{\prime}{=} \begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}  \tag{1}\]  
Figure 1: The SHC transmission system. where \(|E\rangle\) can be expressed as  
\[|E\rangle= \begin{pmatrix}E_{x}\\ E_{y}\end{pmatrix} = \begin{pmatrix}A_{x}e^{i(wt+\varphi_{x})}\\ A_{y}e^{i(wt+\varphi_{y})}\end{pmatrix} = \begin{pmatrix}\cos\beta\\ j\sin\beta\end{pmatrix} \sqrt{{A_{x}}^{2}+{A_{y}}^{2}}e^{jwt} \tag{2}\]  
where \(A_{x}\), \(A_{y}\), \(\varphi_{x}\), \(\varphi_{y}\) correspond to amplitude and phase of X polarization and Y polarization, respectively. Assuming that the laboratory coordinate is its main axis coordinates, then \(\varphi_{y}-\varphi_{x}=\pi/2\), \(\beta\) is ellipticity, there are \(\cos\beta=A_{x}/\sqrt{{A_{x}}^{2}+{A_{y}}^{2}}\) and \(\sin\beta=A_{y}/\sqrt{{A_{x}}^{2}+{A_{y}}^{2}}\). For the convenience of analysis, we only consider variables that are affected by changes in SOP, so Equation (1) can be simplified as:  
\[|E_{in}(\theta,\beta)\rangle= \begin{pmatrix}\cos\theta\cos\beta-j\sin\theta \sin\beta\\ \sin\theta\cos\beta+j\cos\theta\sin\beta\end{pmatrix}  \tag{3}\]  
Since the control target is here to obtain equal power of LO light for two polarizations rather than control to a certain SOP as precise as the target of traditional APC, it can be realized by using only one stage of HWP in theory [5; 16]. It is known that HWP is a phase-fixed-angle-adjustable PC, which is expressed in Jones matrix as  
\[M_{1}= \begin{pmatrix}\cos\theta_{h}&-\sin\theta_{h}\\ \sin\theta_{h}&\cos\theta_{h}\end{pmatrix}\begin{pmatrix}e^{-j\frac{\pi}{2}}&0 \\ 0&e^{j\frac{\pi}{2}}\end{pmatrix}\begin{pmatrix}\cos\theta_{h}&\sin\theta_{h}\\ -\sin\theta_{h}&\cos\theta_{h}\end{pmatrix} , \tag{4}\]  
where \(\theta_{h}\) represents the tunable azimuth of HWP, which can be adjusted according to the input driving voltage \(V_{A}\), \(V_{C}\). The received LO controlled by rotatable HWP can then be expressed as:  
\[ \begin{pmatrix}E_{xout}\\ E_{yout}\end{pmatrix} =M_{1}\cdot|E_{in}(\theta,\beta)\rangle= \tag{5}\] \[ \begin{pmatrix}-j\cos 2\theta_{h}\cos\theta\cos\beta-\cos 2 \theta_{h}\sin\theta\sin\beta\cdots\\ -j\sin 2\theta_{h}\sin\theta\cos\beta+\sin 2\theta_{h}\cos\theta\sin\beta\\ -j\sin 2\theta_{h}\cos\theta\cos\beta-\sin 2\theta_{h}\sin\theta\sin\beta \cdots\\ +j\cos 2\theta_{h}\sin\theta\cos\beta-\cos 2\theta_{h}\cos\theta\sin\beta \end{pmatrix} \]  
After detected by BPD, the feedback error signal is obtained:  
\[\varepsilon_{I}=||E_{xout}|^{2}-|E_{yout}|^{2}|=|\cos 2\beta\cos(4\theta_{h}-2 \theta)| \tag{6}\]  
The DSP module is used to minimize the feedback signal \(\varepsilon_{I}\) by and adjust the azimuth \(\theta_{h}\) of HWP. It can be seen from Equation (6) that \(\theta_{h}\) should meet to be \(\theta_{h}=\pm\frac{\pi}{8}+\frac{\theta}{2}\) in order to keep it at the minimum value. Figure 3 illustrates the transition process of the SOP of the input LO through an HWP. \((S_{1},S_{2},S_{3})\) are stokes parameters that can visually describe the SOP of light on the Poincare sphere. \(S_{1}\) refers to the difference in intensity of the light after passing through the horizontal and vertical polarizers. \(S_{2}\) is the difference in intensity of the  
Figure 2: Waveplate-type polarization control structure. (polarization splitter–rotator (PSR), balanced photodetector (BPD), analog-to-digital conversion (ADC), and digital-to-analog conversion (DAC)). light after passing through \(\pm 45^{\circ}\) polarizers. \(S_{3}\) is the difference in intensity of the light after passing through a quarter-wave-plate (QWP) and \(\pm 45^{\circ}\) polarizers. From the perspective of Poincare sphere, the control target is the ring \(S_{1}=0\).