## 4 Mode Scrambling for Few-Mode Fibers
### Deterministic Mode Scrambling  
In our scrambling approach, we propose to use a deterministic mode scrambler that permutes modes having more received power with modes having less received power. To have an insight into this strategy, we consider the previous 6-mode and 10-mode fibers with a misalignment std \(\sigma_{x,y}=4\%r_{c}\). At the transmitter we launch all modes with unit energy \(E_{s}=1\) and we compute the received energy per mode at the receiver side averaged over \(10^{5}\) channel realizations. From Figure 3a,b, we clearly notice that some modes arrive at the receiver with more power than the others. For the 6-mode fiber the less attenuated modes are the \(LP_{01}\) and \(LP_{11a,b}\) and the most attenuated modes are the \(LP_{02}\) and \(LP_{21a,b}\). Hence, we propose to permute the \(LP_{01}\) with the \(LP_{02}\) and the \(LP_{11a,b}\) with the \(LP_{21a,b}\). For the 10-mode fiber, modes arriving with more power are the \(LP_{01}\), \(LP_{11a,b}\) and \(LP_{21a,b}\), these modes will be permuted respectively with modes arriving with less power which are the \(LP_{02}\), \(LP_{31a,b}\) and \(LP_{12a,b}\). Our scrambling strategy, avoids permutations between modes arriving with close levels of power especially between degenerate modes that arrive with the same amount of power. However, this situations can occur if random mode scrambling is considered, which do not average power effectively between all modes. To compare the different scrambling strategies, we add 6 scramblers (\(K_{scr}=50\)) to the previous fibers (made of \(K=300\) sections). In Figure 4, we plot the PDF of MDL for \(10^{5}\) channel realizations and for a misalignment std \(\sigma_{x,y}=4\%r_{c}\). We can see that MS reduces the impact of MDL, and that the deterministic MS outperforms the random MS. For the 6-mode fiber, the average MDL without scramblers in the line is 25 dB, with random scrambling the MDL is reduced to 15 dB and for the proposed scrambling strategy is reduced to 9 dB. For the 10-mode fiber, the average MDL without scramblers in the line is 36 dB, with random scrambling the MDL is 15 dB and for the proposed scrambling strategy is reduced to 10 dB. To study the impact of MS, we simulate the system average and outage capacities based on \(10^{5}\) channel realizations of the same previous fibers made of \(K=300\) sections and using 6 scramblers. In Figure 5, we plot the average capacities for the 6-mode and 10-mode fibers for different scrambling strategies. We notice that MS increases the system capacity and that the deterministic MS outperforms the random MS. For an SNR = 15 dB and a std \(\sigma_{x,y}=4\%r_{c}\), the capacity gain brought by using deterministic MS to a strategy where no scrambler is used is 4.5 bits/s for the 6-mode fiber and 3.5 bits/s for the 10-mode fiber. Consequently, deterministic MS averages power more efficiently between all modes, which reduces significantly MDL and results in a better average capacity. Furthermore, we also notice that for low SNRs the capacity gain provided by MS is very low. This is due to the impact of noise that dominates the effect of MDL (given by the different eigenvalues \(\lambda_{i}\)'s) in the capacity expression of Equation (5). However, for high SNRs the gain provided by MS becomes more important. This behavior is different from the CSI gain that is more important for low SNR levels. In Figure 6a,b, we simulate the outage capacity as a function of the misalignment std \(\sigma_{x,y}\) for the different scrambling strategies and for an outage probability of \(10^{-3}\). We notice that as std \(\sigma_{x,y}\) increases which increases also MDL, the outage capacities are decreasing. However, the use of MS enhances these capacities significantly and the deterministic MS outperforms the random MS. For an std \(\sigma_{x,y}=4\%r_{c}\)  
Figure 4: PDF of of MDL for different scrambling strategies and a fiber misalignment std \(\sigma_{x,y}=4\%r_{c}\). Figure 3: Probability distribution function (PDF) of the average received energy per mode.