## 8 SCOH Transceiver Structures
### Optical Field Representation
#### 8.1.2 Stokes Space Representation
Figure 18: SV-DD transmitter structures with (**a**) dual-polarized IM, (**b**) dual-polarized IM-PM multiplexing, (**c**) single-polarization complex modulation, and (**d**) dual-polarized complex modulation. PBS/PBC: polarization beam splitter/ combiner; PM: phase modulator; \(IQM\): \(IQ\) modulator; IM: intensity modulator. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structures** & **Exploited DoFs (Highlighted)** & **Features** & **Reference** \\ \hline \hline
**Dual-Polarized IM** & \(\left[\begin{array}{c}|E_{x}|^{2}\\ E_{y}|^{2}\\ S_{3}\end{array}\right]=\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ 2|E_{x}|E_{y}|\cos(\phi_{x}-\phi_{y})\\ 2|E_{x}|E_{y}|\sin(\phi_{x}-\phi_{y})\end{array}\right]\) & \(\bullet\) & nonlinear mapping & [26, 63] \\
**Dual-Polarized IM-PM** & \(\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ S_{3}\end{array}\right]=\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ 2|E_{x}|E_{y}|\cos(\phi_{x}-\phi_{y})\\ 2|E_{x}|E_{y}|\sin(\phi_{x}-\phi_{y})\end{array}\right]\) & \(\bullet\) & nonlinear mapping & [26, 144] \\
**Single-Polarization Complex Modulation** & \(\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ |E_{y}|^{2}\\ S_{3}\end{array}\right]=\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ 2|E_{x}|\left|E_{y}|\cos(\phi_{x}-\phi_{y})\\ 2|E_{x}|\left|E_{y}|\sin(\phi_{x}-\phi_{y})\end{array}\right]\right]\) & \(\bullet\) & robust to CD & [26, 145] \\
**Dual-Polarized Complex Modulation** & \(\left[\begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ |E_{y}|^{2}\\ |E_{y}|^{2}\\ |E_{y}|^{2}\\ |E_{y}|\cos(\phi_{x}-\phi_{y})\\ 2|E_{x}|E_{y}|\sin(\phi_{x}-\phi_{y})\end{array}\right]\) & \(\bullet\) & depends on the CSPR value & [26, 41] \\ \hline \hline DoF: degree of freedom; CD: chromatic dispersion; IM: intensity modulation; PM: phase modulation; SE: spectral efficiency; CSPR: carrier-to-signal power ratio; SSBN: signal–signal beat noise. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: A summary of SV-DD transmitter structures based on the exploited DoFs. Therefore, the resulting received signal Stokes vector, \(\mathbf{\mathcal{R}}=(S_{0},S_{1},S_{2},S_{3})\) can be expressed as [11, 24, 140]:  
\[\mathbf{\mathcal{R}}=\left[ \begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{c}|X|^{2}+|Y|^{2}\\ |X|^{2}-|Y|^{2}\\ 2\Re(X\cdot Y^{*})\\ 2\Im(X\cdot Y^{*})\\ 2\Im(X\cdot Y^{*})\end{array} \right] \tag{44}\]  
Therefore, Equation (44) can be defined by linear transformation as [11, 24]:  
\[\mathbf{\mathcal{R}}=\left[ \begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{cccc}1&0&0&1\\ 1&0&0&-1\\ -1&1&0&-1\\ -1&0&1&-1\end{array} \right]\left[ \begin{array}{c}|X|^{2}\\ |X+Y|^{2}\\ |X+iY|^{2}\\ |Y|^{2}\end{array} \right] \tag{45}\]  
Based on the fact that the matrix in Equation (45) has a full rank of three, the four outputs are sufficient enough to comprise the entire information of the Stokes vector. Moreover, when the resulting received signal of the Stokes vector, \(\mathbf{\mathcal{R}}\), is linearly transformed, a matrix with the full rank of three will be obtained. Consequently, the four outputs are adequate enough to comprise the entire information of the Stokes vector. Furthermore, based on the configuration in Figure 19c, the \(S_{1}\) component of the Stokes vector can be retrieved as \(S_{1}=-S_{0}+2I_{x}\). Likewise, the \(S_{2}\) component can be recovered as  
Figure 19: Stokes vector receiver structures with: (**a**) three BPDs; (**b**) four PDs and an optical hybrid; and (**c**) four PDs coupled with polarizers. POL: polarizer; PBS: polarization beam splitter; \(\Phi\): \(90^{\circ}\) degrees polarization rotation; \(\lambda/4\): quarter-wave plane; X/Y: the X/Y polarization; PD, photodetector; B-PD: balanced photodetector (adapted from [11, 12, 23, 24, 33, 34, 39, 140]). \(S_{2}=-S_{0}+2I_{45^{\circ}}\), and the \(S_{3}\) component can be retrieved as \(S_{3}=-S_{0}+2I_{R}\). Therefore, the transformation can be expressed as [11]:  
\[\left[ \begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array}