# Clock Recovery Challenges in DSP-Based Coherent Single-Mode and Multi-Mode Optical Systems
## 4 Clock Recovery Performance in Single-Mode Fibers
### Polarization Mode Dispersion  
In order to analyze the clock recovery performance in single-mode fibers, we used the configuration as shown in Figure 5. We consider 32 GBd non-return-to-zero (NRZ)-QPSK modulated signals multiplexed in \(Z=2\) degrees of freedom (two polarizations). The transmission link was simulated numerically and consisted of a single-mode fiber with strong coupling between polarizations divided in 10,000 sections of fiber with 10 m each [27], resulting in a 100 km fiber link.  
To evaluate the behavior of timing recovery with different values of DGD, we considered random Givens rotation matrices and swept the group delay of each fiber segment, i.e., each segment had a random rotation with an angle uniformly distributed in the interval \(\theta\sim[0,2\pi)\). We swept the uncoupled DGD from 0.005 to 500 ps/km, i.e., the propagation delay difference between polarizations in each fiber section. Then, we measured the resulting DGD by the maximum difference of the two eigenvalues of the matrix, \(\mathbf{H}(\mathbf{\Omega})\), defined by [11]  
\[\mathbf{H}(\mathbf{\Omega})=j\frac{\partial\mathbf{M}(\Omega)}{\partial\Omega }\mathbf{M}^{*}(\Omega), \tag{23}\]  
with the differentiation being computed numerically. This resulting DGD is the DGD after all rotations between sections and discrete delays. The parameters used in this simulation are summarized in Table 2.  
We considered both the CTA extracted directly from the received signal at polarization X, \(\mathbf{E}_{out,X}\), and also the CTA extracted by a combination of the polarizations X and Y, \(\mathbf{E}_{out,XY}\), given by  
\[\mathbf{E}_{out,XY}=\mathbf{E}_{out,X}cos(\phi)+\mathbf{E}_{out,Y}\sin(\phi), \tag{24}\]  
where \(\phi\) is the angle that is tracked maximizing the CTA value [7]. The results are shown in Figure 6.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Modulation format & NRZ-QPSK \\ Symbol rate & 32 GBd \\ Number of fiber sections & 10,000 \\ Fiber length per section & 10 m \\ Total fiber length & 100 km \\ Rotation angle per section & uniform distribution \(\sim[0,360^{\circ})\) \\ Uncoupled DGD interval & \([0.005,500]\) ps/km \\ Uncoupled DGD step size & 25.89\% greater every iteration \\ Transmitter time skew & 0 \\ Receiver time skew & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for evaluation of CTA due to PMD in single-mode fiber transmission.  
Figure 5: General set-up for numerical simulation model of polarization multiplexed systems.  
We can see by Figure 6 that using the polarization rotation method [7], the CTA remain at the maximum level until a resulting DGD of 10 ps. Since a typical modern single-mode fiber has PMD parameter around 0.1 ps\(/\sqrt{\text{km}}\), the CTA would only be degraded after 10,000 km, needing at least ten times this transmission distance to drop the CTA to half of its maximum value. Therefore, for all practical applications the vanishing of clock tone is not a problem for polarization multiplexed transmission through single-mode fibers.