## 4 Conclusions  
We have numerically investigated the impact of the EEPN, in terms of the OSNR penalty, in a 25 Gbd 3-mode short-span FMF transmission system where the FMF is modeled in the weakly-coupled regime, for QPSK, 16-QAM, 32-QAM and 64-QAM modulation formats as a function of the link DMD spread and the transmitter and receiver lasers phase noise. The fundamental receiver subsystem was a time-domain equalizer and carrier phase recovery loop with optimized number of test phases \(B=32\) and filter length \(N=10\). In our study, we have considered the models A and B, corresponding to the case of ideal, full mode-selective couplers and ideal, non-mode-selective couplers, respectively. Furthermore, we have also investigated the effect of the distribution of the phase noise between the transmitter and receiver for QPSK and 16-QAM modulations and model B. The parameters of the BPS CPR algorithm were carefully optimized to avoid any biasing of the results, i.e., a sub-optimal operation of the CPR algorithm may be mistaken as a EEPN-caused penalty.  
The results reveal that the EEPN has a negligible penalty for model A--mode-selective couplers and weakly-coupled FMF. However, in model B, which assumes non-mode-selective couplers and weakly-coupled FMF, the EEPN penalty quickly grows as the DMD spread of the optical link increases for a given per-laser phase noise value, which may limit the maximum transmission distance of FMF transmission systems. Furthermore, we note that if the impulse response of the non-mode-selective MUX in model B, \(\mathbf{I}(\omega)\), is characterized a priori then a compensation by using \(\mathbf{I}^{-1}(\omega)\) in the digital domain may be performed at the transmitter to mitigate the effect of the EEPN. However, such a characterization is complex and time-consuming, requires the isolation of the non-mode-selective MUX during the characterization process [33], and on top of that, the frequency response may change over time due to environmental effects like temperature changes, which altogether renders this solution rather impractical in a real transmission system. For all M-QAM modulation formats analyzed, the onset of the penalty degradation is for a DMD of approximately 100 symbols. Furthermore, the results indicate that the EEPN penalty has a strong dependence on the amount of phase noise of the LO and hence noisier LOs have a larger impact on the system performance. The analysis of the combined effects of both the CD and the DMD in terms of the EEPN OSNR penalty, the modeling of the FMF in the strongly-coupled and intermediate-coupled regimes, and the impact of frequency-domain MIMO equalization--which is more computationally efficient but less able to track the phase noise compared with time-domain equalization--are left as future work.  
Conceptualization, J.M.D.M. and W.K.; Methodology, J.M.D.M.; Software, J.M.D.M.; Simulation and Data Preparation, J.M.D.M.; Writing--Original Draft Preparation, J.M.D.M.; Writing--Review & Editing, J.M.D.M., W.K., J.S., S.S., H.F., Y.A., N.W.; Supervision, J.S., S.S., H.F., Y.A., N.W.; Project Administration, S.S., H.F., Y.A., N.W.  
This research received no external funding.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l}
**AWGN** & additive white Gaussian noise \\
**BER** & bit-error rate \\
**BPS** & blind phase search \\
**CD** & chromatic dispersion \\
**CPR** & carrier phase recovery \\
**DEMUX** & demultiplexer \\
**DE/MUX** & demultiplexer and multiplexer \\
**DMD** & differential mode delay \\
**DMT** & discrete multi-tone \\
**DP-DPMZM** & dual-polarization dual-parallel Mach-Zehnder modulator \\
**DSP** & digital signal processing \\
**EEPN** & equalization enhanced phase noise \\ FMF & few-mode fiber \\
**LO** & local oscillator \\
**MDL** & mode-dependant loss \\ \end{tabular}