# True Equalization of Polarization-Dependent Loss in Presence of Fast Rotation of SOP +
## 2 The Effect of PDL with the Presence of RSOP  
PDL mainly comes from optical devices, which refers to the polarization-dependent attenuation of optical power to input signals. As shown in Figure 1, when a PDM signal passes through an optical component that has a pair of orthogonal high- and low-loss eigenmodes, the two polarizations \(\mathrm{E_{x}}\) and \(\mathrm{E_{y}}\) are transmitted along the high-loss eigen-mode and the low-loss eigen-mode, respectively; therefore, their losses are different, and the output power of the two orthogonal polarizations are also different. The PDL matrix can be expressed as  
\[\begin{array}{ll}\mathbf{U}_{PDL}&=\left(\begin{array}{cc}cos\gamma&-e^{-j \beta}sin\gamma\\ e^{j\delta}sin\gamma&cos\gamma\end{array}\right)\left(\begin{array}{cc}\sqrt {1-\rho}&0\\ 0&\sqrt{1+\rho}\end{array}\right)\left(\begin{array}{cc}cos\gamma&-e^{-j \delta}sin\gamma\\ e^{j\delta}sin\gamma&cos\gamma\end{array}\right)^{-1}\\ &=\mathrm{R}_{sig}\Lambda\mathrm{R}_{sig}^{-1}\end{array} \tag{1}\]  
where \(\gamma\) and \(\delta\) denote the amplitude ratio angle and phase difference, respectively. The two column vectors \(\left(cos\gamma,e^{j\delta}sin\gamma\right)^{T}\) and \(\left(-e^{-j\beta}sin\gamma,\;cos\gamma\right)^{T}\) in rotation matrix \(R_{sig}\) represent eigenmodes of PDL related to the high and low losses, which are actually the two orthogonal elliptical states of polarization (SOP). \(\rho\) represents the PDL ratio, and the diagonal matrix \(\Lambda\) represents the attenuation difference between the strongly and weakly attenuated SOP. The PDL value in dB is expressed by  
\[\Gamma_{dB}=10 log_{10}\frac{1+\rho}{1-\rho} \tag{2}\]  
PDL not only means the inequality of signal powers but also the OSNR imbalance between the two polarizations of a PDM signal. Figure 2a shows a PDL-induced constellation map of PDM signals. For each subfigure, the top row shows the strongly attenuated polarization branch and the bottom row represents the weakly attenuated polarization branch. Unfortunately, lots of PDL compensations only realize the power equalization; in this case, as shown in Figure 2b, the OSNR imbalance between the two polarizations is still unsolved. Hence, the true equalization of PDL should include the power equalization and the BER recovery caused by OSNR asymmetry, as shown in Figure 2c.  
In addition, RSOP and PDL may exist simultaneously in the optical fiber links. When we face the joint effect of RSOP and PDL, the generally adopted impairment model is RSOP1 + PDL + RSOP2, which is illustrated in Figure 3a and Equation (3). In this model, two matrices before and after the PDL matrix represent two arbitrary RSOP1 and RSOP2, which are time-varying in general.  
\[\mathbf{H}=R_{2}\mathbf{U}_{PDL}\mathbf{R}_{1} \tag{3}\]  
Figure 1: Schematic of polarization-dependent loss (PDL).  
Figure 2: Schematic representation of PDL equalization: (**a**) The signal suffers PDL; (**b**) after power equalization; (**c**) after bit-error-rate (BER) recovery coming from optical signal-to-noise ratio (OSNR) imbalance.  
where \(\mathbf{R}_{i}=\left(\begin{array}{cc}cos\gamma_{i}&-e^{-i\beta_{i}}\text{sint} \gamma_{i}\\ e^{i\beta_{i}}\text{sint}\gamma_{i}&cos\gamma_{i}\end{array}\right)\), \(i=1,2\). \(\mathbf{R}_{i}\) can take a similar form of matrix \(R_{eig}\).  
Based on the above analysis, we can conclude that the true PDL equalization scheme in the presence of RSOP should include the following three steps: (1) Obtaining decoupling model when PDL and RSOP coexist; (2) the power imbalance equalization and the RSOP tracking; (3) BER recovery caused by OSNR asymmetry. The three steps will be designed individually in Section 3.