## 9 Stokes Space Polarization and Field Recovery Principle
### Polarization Recovery Techniques in the Direct Detection Stokes Vector Receiver
#### 9.1.3. Stoke Space Signal-Distribution-Based Polarization Demultiplexing  
Polarization rotation can also be ideally determined by neglecting the polarization-dependent loss and the polarization mode dispersion. In this context, two parameters between the two polarization modes, which are the phase difference, \(\Delta\phi\), and the rotation angle, \(\theta\), have to be well considered. Through the received Stokes vector distribution analysis, \(\Delta\phi\) and \(\theta\) can be obtained. Consequently, the \(3\times 3\) rotation matrix can be restored. For instance, for the 4-QAM modulation format shown in Figure 22a, a constant module is maintained by the signal. When the CSPR is set to 0 dB as illustrated in Figure 22b, the transmitted SVs can be restructured as: \(\mathbf{\mathcal{T}}=[0,2\Re(S\cdot C^{*}),2\Im(S\cdot C^{*})]^{T}\). With reference to the Poincare sphere, all the SVs are positioned at the plane \(X=0\) with the normal given by \((1,0,0)\)[11, 23]. Furthermore, the received signal polarization state can be characterized by the plane and the associated normal vector. Furthermore, it is worth noting that there is a combination of the received SVs and the noise. This hinders their ability to completely fall into a plane. However, the least-squares plane (LSP) can be estimated for all the SVs. The least-squares plane is the plane that minimizes the value \(\sum_{i}(d_{i})^{2}\), where \(d_{i}\) denotes the distance between the plane and the \(i\)th Stokes vector. Moreover, for the least-squares plane equation given  
Figure 22: The 4-QAM constellation in: (**a**) the Jones space; (**b**) the Stokes space. The 16-QAM constellation in: (**c**) the Jones space; (**d**) the Stokes space (adapted from [11]). as \(Ax+By+Cz=0\), the related least-squares plane normal vector would be \((A,B,C)\). \(d_{i}\) between the SVs (\(S_{i,1},S_{i,2},S_{i,3}\)) and the least-squares plane is given by [11]:  
\[\left[ \begin{array}{ccc}S_{1,1}&S_{1,2}&S_{1,3}\\ S_{2,1}&S_{2,2}&S_{2,3}\\ \vdots&\vdots&\vdots\\ S_{n,1}&S_{n,2}&S_{n,3}\end{array} \right]\cdot\left[ \begin{array}{c}A\\ B\\ C\end{array} \right]=\left[ \begin{array}{c}d_{1}\\ d_{2}\\ \vdots\\ d_{n}\end{array} \right]. \tag{52}\]  
In realistic scenarios, where the plane in the Stokes space suffers fast rotations induced by critical environmental conditions, affecting the fiber birefringence, the employment of alternative approaches for computing the normal vector may be advantageous [147]. In [148], the authors proposed an adaptive estimation of the inverse rotation matrix, without the calculation of the best fitting plane. This alternative method consists of an algorithm that uses the initial orientation of the normal, along with the polarization state of the upcoming signal sample, to adaptively compute the next normal. Moreover, the technique can integrate a stage for polarization-dependent loss compensation [149] without compromising its adaptive running [148]. It should be noted that one of the potential candidates for addressing the optimization problem is singular-value decomposition. Therefore, when the least-squares plane's normal is estimated as \(\mathbf{N}=(n_{1},n_{2},n_{3})\), afterwards, the rotation angle and the phase delay can be expressed respectively as [11]:  
\[\theta =a\tan\biggl{(}\sqrt{n_{2}^{2}+n_{3}^{2}},n_{1}\biggr{)} \tag{53a}\] \[\Delta\phi =a\tan(n_{3},n_{2}). \tag{53b}\]  
Furthermore, the 3 \(\times\) 3 rotation matrix can be obtained through multiplication of the polarization rotation matrix with the phase delay matrix as [11]:  
\[RM=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos(\Delta\phi)&-\sin(\Delta\phi)\\ 0&\sin(\Delta\phi)&\cos(\Delta\phi)\end{array} \right]\left[ \begin{array}{ccc} \cos(\theta)&-\sin(\theta)&0\\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{array} \right]. \tag{54}\]  
In addition, besides the 4-QAM, the aforementioned steps can be applied to other formats such as single-carrier modulations (i.e., 16-QAM) with discrete value modules and multicarrier modulation (i.e., OFDM) counterparts with arbitrary modules.