## 1 Introduction
Regarding the EEPN in FMF transmission systems, [PERSON] [16] analytically derived a simple expression for the signal to noise ratio (SNR) penalty, which grows linearly with the total amount of the DMD and the LO phase noise, for the worst case of total coupling between the two modes. [PERSON] and [PERSON] [17] extended the analysis of [16] to include all the possible ensemble averages of the unitary matrices modeling the mode coupling in the strongly-coupled regime. Note that both analytical analyses assume perfect and instantaneous estimation of the phase noise, yet practical phase estimation algorithms [18] must average over a number of samples to remove noise from the estimated phase value. [PERSON] et al. [19] experimentally showed that in a 28 GBd 3-mode QPSK system, duo-binary shaping combined with maximum likelihood sequence detection (MLSD) can relax the LO linewidth requirements due to EEPN. [PERSON] and [PERSON] [20] numerically studied the digital coherence enhancement technique to mitigate the effects of EEPN in 3-mode FMF transmission systems. In this work, we numerically assess the impact of the EEPN optical signal to noise ratio (OSNR) penalty on a typical 3-mode short-span FMF transmission link with M-quadrature amplitude modulation (QAM) formats at 25 GBd when using a MIMO time-domain equalizer Time-domain MIMO equalizers are less efficient in terms of complexity than their frequency-domain counterparts [21]. However, time-domain equalizers update the equalizer taps at a much higher rate than frequency-domain equalizers and hence are more robust against quick variations in the phase noise. We consider a short-span of FMF which can be modeled in the weakly-coupled regime and include two models, model A and model B, to account for the use of mode-selective demultiplexers and multiplexers (DE/MUXes) [22] or non-mode-selective DE/MUXes [2] at both ends of the FMF. In both models, coupling of modes within a mode-group--for example, the X and Y polarizations of the LP\({}_{01}\) mode--is modeled with random complex unitarys (RCUs) matrices. In our numerical simulations, we use the blind phase search (BPS) carrier phase recovery (CPR) algorithm, which has a high performance at a cost of an additional computational complexity compared to Viterbi and Viterbi-based CPR algorithms [23]. However, the BPS algorithm may be implemented in a hardware efficient way due to the absence of feedback loops [18]. The BPS parameters \(B\)--the number of test phases--and \(N\)--the number of averaged symbols--are carefully optimized to not bias the simulation results. Compared to our previous work [24], here we describe in detail the optimization procedure of the BPS CPR algorithm and extend the analysis to investigate the impact of the phase noise ratio between the transmitter and LO lasers for QPSK and 16-QAM modulation formats. The results show that the EEPN penalty is negligible for systems with mode-selective DE/MUXes with no crosstalk between mode-groups (model A). However, non-mode-selective DE/MUXes (model B), even for a short-span of FMF modeled in the weakly-coupled regime, induce a severe penalty due to the EEPN which makes the system phase noise requirements more stringent as the spread of the impulse response increases. The onset of the OSNR penalty in model B starts for a DMD impulse response spread of about 100 symbols for all modulation formats investigated. Furthermore, it is confirmed that to mitigate the EEPN penalty it is imperative that the LO has a very low phase noise compared with the transmitter laser phase noise. The remainder of this paper is organized as follows. Section 2 describes the EEPN in the context of FMF transmission systems and the numerical model used in this study. Then, the numerical results are presented and discussed in Section 3. Finally, Section 4 summarizes and concludes the paper.