# True Equalization of Polarization-Dependent Loss in Presence of Fast Rotation of SOP +
## 3 The True Equalization of PDL in the Presence of Fast RSOP
### The Power Equalization and RSOP Tracking  
After decoupling of the combined PDL and RSOP, we can then equalize them separately. We first treat the power imbalance equalization and RSOP tracking. We use the 3D Stokes space method [3] and Kalman filter to carry out this treatment. The 3D Stokes space provides a visual representation of the light SOP. By neglecting the common phase, the received optical signals can be represented with the Jones vector in the following form:  
\[\ket{s}=\left( \begin{array}{c}a_{x}\\ a_{y}exp(j\phi_{km})\end{array} \right) \tag{5}\]  
where \(a_{x}\), \(a_{y}\) are the amplitudes of two x- and y-polarization signals, and \(\phi_{km}\) is the phase difference. \(k,m=1,2,3,4\) denotes the four quadrature phase-shift keying (QPSK) constellation symbols in the x- and y-polarization constellation planes. The Stokes parameters of the signals represented in Stokes  
Figure 3: (**a**) The generalized polarization model in fiber; (**b**) the simplified polarization model. space can be obtained from the Jones vector \(|s\rangle\) by using the relation \(\overrightarrow{S}_{i}=\left\langle s|\overrightarrow{\sigma}|s\right\rangle\), \(i=0,1,2,3\), where \(\overrightarrow{\sigma}\) is the Pauli matrix. \[ \begin{array}{l}S_{0}=a_{x}^{2}+a_{y}^{2}\\ S_{1}=a_{x}^{2}-a_{y}^{2}\\ S_{2}=2a_{x}a_{y}cos\phi_{km}\\ S_{3}=2a_{x}a_{y}sin\phi_{km}\end{array}  \tag{6}\]  
As the amplitude in the x- and y-polarizations are the same and \(S_{1}\) equals zero, for the ideally compensated PDM-QPSK signals, the center of gravity of the constellations is at the origin of the Poincare sphere in Stokes space (denoted as \(C(0,0,0)\)), as shown in Figure 4a. When the signals suffer from PDL, their Jones vector will change to  
\[|E_{out}\rangle=J_{PDL}(\rho)|E_{in}\rangle=\left( \begin{array}{c}a_{x}\sqrt{ 1-\rho}\\ a_{y}\sqrt{1+\rho}e^{i\phi_{km}}\end{array} \right) \tag{7}\]  
whose corresponding Stokes vector components are  
\[ \begin{array}{c}S_{0,km}=2a_{xy}^{2}\\ S_{1,km}=-2\rho a_{xy}^{2}\\ S_{2,km}=2a_{xy}^{2}\sqrt{1-\rho}\sqrt{1+\rho}cos(\phi_{km})\\ S_{3,km}=2a_{xy}^{2}\sqrt{1-\rho}\sqrt{1+\rho}sin(\phi_{km})\end{array}  \tag{8}\]  
where \(a_{xy}=a_{x}=a_{y}\). We can obtain the mean value of each Stokes parameters as  
\[\begin{array}{c}\left\langle S_{1,km}\right\rangle=-2\rho a_{xy}^{2}=d\\ S_{2,km}\right\rangle=0\\ S_{3,km}\right\rangle=0 \tag{9}\]  
which means that when a power imbalance between the two polarization tributaries exists, the center of gravity of the constellation will shift from the origin (denoted as \(C(d_{1},0,0)\)), as shown in Figure 4b. When the signals suffer from both RSOP and PDL, the power inequality induced by PDL will cause a shift in the center of gravity of the constellation in Stokes space from the origin point of the Poincare sphere, and the RSOP will cause a decline in the constellation structure, as shown in Figure 4c. In this case, we can move \(C\) and then the entire constellation structure by the three steps of rotations and shifting until this \(C\) returns to the original position. This procedure of balancing the PDL-induced power inequality is shown in Figure 4d. Figure 4: PDL compensation in Stokes space for a polarization-division multiplexing polarization-division multiplexing (PDM) quadrature phase-shift keying (QPSK) (PDM)-QPSK signal: (**a**) Original position; (**b**) signals suffer from PDL; (**c**) signals suffer from PDL and rotation of state of polarization (RSOP); (**d**) power inequality compensation. After power imbalance equalization, the Kalman filter is applied to monitor and recover RSOP. We have investigated the Kalman filter structure thoroughly [23; 24; 25; 26] and found three key issues that we can design to let our Kalman filter structure be the most effective: (1) Select the right compensation matrices for compensating RSOP; (2) decide which state parameters should be correctly chosen; (3) design the appropriate measurement vector and innovation vector. Now, we will explain these issues in detail one by one. 1.