# Clock Recovery Challenges in DSP-Based Coherent Single-Mode and Multi-Mode Optical Systems
## 5 Clock Recovery Performance in Multi-Mode Fibers  
In order to analyze clock recovery performance in multi-mode systems we used the general configuration as shown in Figure 7. We considered 32 GBd NRZ-QPSK modulated signals spatially multiplexed in \(Z=6\) degrees of freedom. The transmission link was simulated numerically and consisted of a 3-mode FMF divided in sections of 10 km, with two degrees of freedom per mode. In the receiver DSP, the clock extraction and resampling was performed before dynamic MIMO equalization to enable the equalizer to keep up with slow timing drifts in steady state operation. The clock extraction was performed per mode, so we had one value of clock tone magnitude for each of the modes.  
To evaluate the behavior of timing synchronization in different coupling regimes, we swept the rotation distribution of the unitary matrices \(\mathbf{V}^{(k)}\) and \(\mathbf{U}^{(k)}\) in Equation (17), measuring the crosstalk into the fundamental mode group due to high order mode groups. The parameters used in this simulation  
Figure 6: Normalized clock tone amplitude as function of resulting DGD for a single-mode fiber.  
Figure 7: General set-up for numerical simulation model of multi-mode multiplexed systems.  
are summarized in Table 3. For simplicity we did not distinguish between inter- and intra-mode group coupling. We are not considering any losses, non-linearities and chromatic dispersion.  
We show in Figure 8 the results for a 1000 km transmission of 3-mode FMF. This fiber has two mode groups, two degrees of freedom (one mode) on fundamental group and four degrees of freedom (two modes) in the second group. Here, the crosstalk is the ratio of the contribution of the second mode group into the fundamental mode. We define a strong coupling regime when the crosstalk is approximately 3 dB, the weak coupling regime when the crosstalk is lower than \(-18\) dB and refer to intermediate coupling regime for the region in-between. The results shown are the average of 10 random simulations for each crosstalk value in a system with no losses or chromatic dispersion.  
Here, we parametrize the curves using the uncoupled group delay, \(\Delta\tau\), which is the time difference between the fastest and the slowest modes' group velocities. As expected for zero group delay, the clock tone remained unchanged at the maximum value irrespective of the crosstalk. However, increasing the uncoupled group delay, we saw a dramatic drop in the clock tone quality. In the strong coupling  
Figure 8: Normalized clock tone amplitude as a function of crosstalk between mode groups in a 1000 km transmission of a 3-mode FMF for different values of uncoupled group delay.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Modulation format & NRZ-QPSK \\ Symbol rate & 32 GBd \\ Number of degrees of freedom & 6 \\ Fiber length per section & 10 km \\ Total fiber length (Figure 8) & 1000 km \\ Total fiber length (Figure 9) & 100, 300, 500 and 1000 km \\ Uncoupled group delay (Figure 8) & 0, 0.03, 0.3, 3 and 300 ps/km \\ Uncoupled group delay interval (Figure 9) & \([0.001,1000]\) ps/km \\ Rotation angle per section & zero-mean normal distribution \\ Rotation angle variance interval & \([0,2000^{\circ}]\) \\ Transmitter time skew & 0 \\ Receiver time skew & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters for evaluation of CTA due to PMD in single-mode fiber transmission.  
case, an uncoupled group delay of less than 0.03 ps/km was required in order to still have a detectable clock tone higher than 50% of the maximum value.  
We also evaluated the behavior of clock tone quality for strong coupling regime as a function of uncoupled group delay for different transmission distances. Results are shown in Figure 9. The clock tone quality drops for smaller values of uncoupled group delay as the transmission distance increases, so for long-haul MDM transmission the delay spread requirements are even tighter.