## 8 SCOH Transceiver Structures
### Optical Field Representation
#### 8.1.2 Stokes Space Representation  
Unlike the Jones space representations, in the Stokes space, the entire Stokes parameters are real-valued second-order terms of the complex optical field (Figure 18). Based on this, the direct detection scheme can be employed to recover the second-order optical intensities [13; 20; 141]. Furthermore, the Jones space is a two-dimensional complex space in which each dimension denotes the optical field being propagated in the horizontal and vertical axis, while the Stokes space is a three-dimensional real-valued space. In physical terms, each dimension denotes the difference in the optical power after the field has passed through different orthogonal polarizer pairs. For instance, the first dimension represents the difference in the optical power after the linear polarizers with the principal axis orientation along the \(\hat{x}\)- and \(\hat{y}\)-directions. Furthermore, the second dimension denotes the outcome of the polarizer alignments at \(+45^{\circ}\) and \(-45^{\circ}\). Similarly, the third dimension represents the results of right-hand and left-hand circular polarizers [140]. Moreover, an optical field can be launched on two orthogonal polarizations and is then combined using a polarization beam combiner (Table 9). The resulting transmitted signal in the Jones vector (JV) can be represented by the Stokes vector. Moreover, an optical field can be launched on two orthogonal polarizations and then be combined using a polarization beam combiner. The resulting transmitted signal in the Jones vector can be represented by the Stokes vector. The Stokes space, \(\mathbf{\mathcal{T}}\), representation of an optical field can be defined as [26; 142]:  
\[\mathbf{\mathcal{T}}=\vec{S}=\left[ \begin{array}{c}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{c}E_{\delta}E_{x}^{*}+E_{\delta }E_{y}^{*}\\ E_{\delta}E_{x}^{*}-E_{\delta}E_{y}^{*}\\ 2\Re\{E_{x}E_{y}^{*}\}\\ -2\Im\{E_{\delta}E_{y}^{*}\}\end{array} \right]=\left[ \begin{array}{c}|E_{ \delta}|^{2}+|E_{\delta}|^{2}\\ |E_{\delta}|^{2}-|E_{\delta}|^{2}\\ 2\Re\{E_{x}E_{y}^{*}\}\\ 2\Im\{E_{\delta}E_{y}^{*}\}\end{array} \right] \tag{42}\]  
where \(S_{i}(i=0,\ldots,3)\) represent the Stokes vector components in which \(S_{0}\) denotes the total light power and \(\Re\) and \(\Im\) represent the real and imaginary part of a complex variable, respectively. In addition, Equation (42) can also be expressed based on a modified Stokes vector, \(\mathbf{\mathcal{I}}\), as [26; 143]:  
\[\mathbf{\mathcal{I}}=\left[ \begin{array}{c}\frac{1}{2}(S_{0}+S_{1})\\ \frac{1}{2}(S_{0}-S_{1})\\ S_{2}\\ S_{3}\end{array} \right]=\left[ \begin{array}{c}|E_{x}|^{2}\\ |E_{y}|^{2}\\ 2|E_{x}||E_{y}|\cos(\phi_{x}-\phi_{y})\\ 2|E_{x}||E_{y}|\sin(\phi_{x}-\phi_{y})\end{array} \right] \tag{43}\]Alternative versions of the SVRs in which four single-ended PDs are employed are depicted in Figure 19b,c. Based on the other structure illustrated in Figure 19b, four single-ended PDs and a \(2\times 4\) optical hybrid can be employed. After being split by the PBS, the output signals \(X\) and \(Y\) are the inputs to the couplers. Then, the output ports 1 and 4 of the couplers are individually launched directly into the PDs, while ports 2 and 3 are connected to the hybrid. Furthermore, only two outputs out of the two pairs of outputs from the hybrid are linked to the PDs. The assembly produces the outputs \(|X|^{2}\) and \(|Y|^{2}\) from the PDs that are connected to ports 1 and 4, respectively, while the outputs of \(|X+Y|^{2}\) and \(|X+iY|^{2}\) are obtained from the PDs that are connected to ports 2 and 3, respectively. Therefore, the \(S_{1}\) component of the Stokes vector can be recovered by \(S_{1}=|X|^{2}-|Y|^{2}\). Furthermore, the \(S_{2}\) component can be recovered by \(S_{2}=|X+Y|^{2}-|X|^{2}-|Y|^{2}=2\ \Re(X\cdot Y^{*})\) and the \(S_{3}\) by \(S_{3}=|X+iY|^{2}-|X|^{2}-|Y|^{2}=2\ \Im(X\cdot Y^{*})\)[8, 24].