## 2 Problem Description, Simulation Model and Methodology
### Few-Mode Fiber Transmission Systems Description
Taking into account this constraint, we can further factorize the response of the FMF into the unique principal modes,  
\[\mathbf{M}(\omega)=\mathbf{O}(\omega)\mathbf{V}(\omega)\boldsymbol{\Lambda}( \omega)\mathbf{U}^{*}(\omega)\mathbf{I}(\omega) \tag{2}\]  
where \(\mathbf{V}(\omega)\) and \(\mathbf{U}(\omega)\) are unitary matrices, \(*\) denotes conjugate transposition and \(\boldsymbol{\Lambda}(\omega)=\text{diag}[\mathbf{a}_{1}(\omega),\mathbf{a}_{2 }(\omega),\ldots,\mathbf{a}_{6}(\omega)]\) is a diagonal matrix with elements,  
\[\mathbf{a}_{n}(\omega)=\exp\Big{\{}\frac{g_{n}}{2}-j\omega\tau_{n}-j\omega^{2 }\frac{D_{n}\lambda^{2}L}{4\pi c}\Big{\}} \tag{3}\]  
where \(g_{n}\in\mathbb{R}\) represents the mode losses or gains, \(\tau_{n}\in\mathbb{R}\) the mode group delay and \(D_{n}\) the mode CD, all of them with respect to the principal modes of the FMF. For simplicity, in this work we consider an unamplified system with no mode-dependant loss (MDL) and CD and hence \(g_{n}=D_{n}=0\). The DMD is defined as the maximum temporal spread between the principal modes  
Figure 1: Schematic diagram of a typical optical transmission system consisting of a few-mode fiber supporting 6 spatial channels (2 mode-groups and 6 orthogonal modes). X polarization in-phase component (XI), X polarization quadrature component (XQ), Y polarization in-phase component (YI), Y polarization quadrature component (YQ), dual-parallel Mach-Zehnder modulator (DPMZM), transmitter (TX), multiplexer (MUX), demultiplexer (DEMUX), balanced photodiode (BPD), analog to digital converted (ADC), local oscillator (LO), digital signal processing (DSP), normalization (NORM), clock recovery (CR), multiple-input multiple-output (MIMO), time-domain equalizer (TDE), blind phase search (BPS), hard-detection (HD). \[\text{DMD}=|\max(\tau_{n})-\min(\tau_{n})| \tag{4}\]  
which corresponds to the temporal spread of the system impulse response. The complexity of the MIMO equalizer is directly proportional to this quantity. Step-index FMFs typically have large values of DMD, which produce clearly separated peaks in the impulse response [2]. Graded-index FMFs may be designed in such a way that the differential group delay of the principal modes is small [28; 29] and hence the impulse response for these fibers may have peaks that totally overlap and result in a lower DMD spread. A three mode FMF supports two mode-groups, \(\text{LP}_{\text{01}}\) and \(\text{LP}_{\text{11}}\), with effective refractive indices \(n_{eff}^{\text{01}}\) and \(n_{eff}^{\text{11}}\), respectively. The two modes belonging to the \(\text{LP}_{\text{01}}\) group and the four modes of the \(\text{LP}_{\text{11}}\) group are degenerated or quasi-degenerated and hence will always couple as the signal propagates through the FMF due to birefringence and other imperfections of the fiber [30], similarly to the polarization-mode dispersion (PMD) in single mode systems [27]. Depending on the properties of the matrix \(\mathbf{F}(\omega)\) with respect to the coupling between the two mode-groups, we can distinguish between the weakly-coupled FMF regime or the strongly-coupled FMF regime. The weakly-coupled regime, in its ideal case, is defined as,  
\[\mathbf{F}(\omega)= \begin{pmatrix}\mathbf{a}_{11}(\omega)&\mathbf{a}_{12}( \omega)&0&0&0&0\\ \mathbf{a}_{21}(\omega)&\mathbf{a}_{22}(\omega)&0&0&0&0\\ 0&0&\mathbf{b}_{11}(\omega)&\mathbf{b}_{12}(\omega)&\mathbf{b}_{13}(\omega)& \mathbf{b}_{14}(\omega)\\ 0&0&\mathbf{b}_{21}(\omega)&\mathbf{b}_{22}(\omega)&\mathbf{b}_{23}(\omega)& \mathbf{b}_{24}(\omega)\\ 0&0&\mathbf{b}_{31}(\omega)&\mathbf{b}_{32}(\omega)&\mathbf{b}_{33}(\omega)& \mathbf{b}_{34}(\omega)\\ 0&0&\mathbf{b}_{41}(\omega)&\mathbf{b}_{42}(\omega)&\mathbf{b}_{43}(\omega)& \mathbf{b}_{44}(\omega)\\ \end{pmatrix}  \tag{5}\]  
where the sub-matrices \(\mathbf{A}(\omega)\) and \(\mathbf{B}(\omega)\), with elements \(\mathbf{a}_{ij}(\omega)\) and \(\mathbf{b}_{ij}(\omega)\) and dimensions \(2\times 2\) and \(4\times 4\), respectively, are RCU matrices.