## 2 Problem Description, Simulation Model and Methodology
### Few-Mode Fiber Transmission Systems Description
In the ideal weakly-coupled regime there is no exchange of energy between the \(\text{LP}_{\text{01}}\) and the \(\text{LP}_{\text{11}}\) mode-groups. The FMF strongly-coupled regime is defined as  
\[\mathbf{F}(\omega)=\mathbf{X}(\omega) \tag{6}\]  
where \(\mathbf{X}(\omega)\) is a RCU matrix with the energy of one input mode distributed equally among all the output modes and hence there is an energy exchange between the mode-groups. In an ideal FMF, with a perfectly circular core, no bending and no scattering, and whether is step-index or graded-index, there is no coupling between the mode-groups because of the different effective refractive indices of each mode-group. However, in a real FMF, coupling between the mode-groups may happen due to fiber bending [31], Rayleigh scattering, and core misalignments at splicing points [32; 33]. Both the weakly- and strongly-coupled regimes describe ideal situations not found in real fibers. Real FMF transmission systems can be attributed to the _intermediate-coupled_ regime [34], which depends on the total length \(L\) of the system and on the perturbations that introduce the mode coupling [31]. Real FMF systems with a relatively short \(L\) behave in the weakly-coupled regime but as the system length increases the behavior resembles the strongly-coupled regime. In spite of this, both the weakly- and strongly-coupled models are useful to investigate system behavioral boundaries as they are mathematically tractable and easy to simulate. For example, it has been shown that in the weakly-coupled regime, the DMD spread grows proportionally to the system distance \(L\) whereas in the strongly-coupled regime the DMD spread grows proportionally to \(\sqrt{L}\), whereas in real systems the DMD spread lies in between these two boundaries [32]. The concept of weakly- and strongly-coupled regimes can be applied to other few-mode optical components, such as the DE/MUXes. In the literature, spatial mode DE/MUXes are categorized into non-mode-selective and mode-selective. A MUX belonging to the former group is the non-mode selective photonic lantern [35], while the latter group is represented by mode-selective photonic lanterns [36], phase plates [37], planar lightwave circuit (PLC) based couplers [38], and tapered velocity couplers [39]. Mode-selective DE/MUXes can be ideally modeled with identity matrices and non-mode-selective DE/MUXes with RCU matrices. The EEPN originates when the MIMO equalizer at the receiver has to pick signal components from time-shifted copies of the signal that have uncorrelated phase noise, originating at the LO laser, which translates into different rotations of the constellation that cannot be compensated by the CPR algorithm. In other words, the time-varying phase of the LO will be distributed along the coefficients of the linear equalizer [8]. In this work, we consider the system impulse response, consisting of the combined MUX, fiber and DEMUX responses, as a black box where the coupling originates at discrete points at the edges of the fiber and the only impairment caused by the fiber is a differential mode group delay. With respect to the EEPN, the variables of interest are the total temporal length of the system impulse response and the amount of phase noise of the LO, regardless of whether the actual mode mixing originates in the fiber or in the DE/MUXes. Taking into account the weakly- and strongly-coupled regimes of the FMF and the nature of the DE/MUXes used, Figure 2a summarizes the most simple options to model a FMF transmission system. In this work, we focus on short-span systems where the FMF can be modeled in the weakly-coupled regime, as has been demonstrated experimentally many times in the literature [2; 22; 33], and investigate the OSNR penalty caused by the EEPN. In this simulation, it is assumed that the group velocity of the LP\({}_{11}\) mode-group is lower compared to the LP\({}_{01}\) mode-group and hence the LP\({}_{11}\) mode-group arrives later [40]. However, there are FMF designs where the LP\({}_{11}\) mode may arrive earlier [41].