## 2 Few-Mode Fiber Channel Model  
Optical devices such as few-mode amplifiers and multiplexers represent the main sources of MDL [7; 8; 9; 10]. Nonetheless misaligned fiber splices also lead to an important end-to-end accumulated MDL [6]. In our work, we focus on distributed MDL arising from fiber splices and micro-bends. We consider graded-index FMFs (GI-FMFs) having parabolic profile with core radius \(r_{c}\) and a numerical aperture \(NA=\sqrt{n_{c}^{2}-n_{cl}^{2}}=0.205\), where \(n_{c}\) (resp. \(n_{cl}\)) is the refractive index of the core of the fiber (resp. of the cladding). GI-FMFs have a low DMGD and more modal coupling is obtained than step index fibers [15]. The wavelength is set to \(\lambda=1.55\)\(\upmu\)m, the number of propagating modes depends on the core radius \(r_{c}\), the wavelength \(\lambda\) and the numerical aperture \(NA\). In our work we fix \(\lambda\) and \(NA\) and we increase \(r_{c}\) to allow the propagation of higher order modes. In Table 1, we report the parameters of the GI-FMFs [6]. We can express the transverse field distribution of the linearly-polarized (LP) modes with a good approximation by Laguerre-Gauss polynomials as in [16]:  
\[E_{l,q}(r,\phi)=C_{l,q}\left(\frac{r}{w}\right)^{l}L_{q}^{(l)}\left(\frac{r^{2 }}{w^{2}}\right)\exp\left(\frac{-r^{2}}{2w^{2}}\right) \begin{cases}\sin\left( l\phi\right)\\ \cos\left(l\phi\right)\end{cases}  \tag{1}\]  
The field distribution of the last equation is written in polar coordinates given by \(r\) and \(\phi\). \(l\) denotes the circumferential order and \(q\) is the radial order of the \(LP_{lq}\) mode. \(L_{q}^{(l)}(x)\) is the generalized Laguerre polynomial of order \(l\) and degree \(q\) and \(w=\sqrt{r_{c}/\left(k_{0}NA\right)}\) is the spot size of the fundamental mode \(LP_{01}\) with \(k_{0}=2\pi/\lambda\) is the free space wavenumber. \(C_{l,q}\) is a normalization factor chosen to fulfill the orthogonal relation between spatial modes \(\iint_{A}E_{i}E_{j}dA=\delta_{i,j}\). In our capacity analysis, we consider a physical description of the propagation through GI-FMFs with realistic non-unitary modal coupling generated by fiber misalignments. This modelization is based on the electrical field distributions of spatial modes as described before rather than a mathematical modeling based on random matrices as in [12; 17]. Neglecting fiber non-linearities, the resulting linear MIMO transmission system is given by: \begin{table}
\begin{tabular}{c c c} \hline
**Number of Modes** & **Core Radius** & **Propagating Modes** \\ \hline
6 & 8.7 \(\upmu\)m & \(LP_{01}\), \(LP_{11a,b}\), \(LP_{21a,b}\), \(LP_{02}\) \\
10 & 11 \(\upmu\)m & \(LP_{01}\), \(LP_{11a,b}\), \(LP_{21a,b}\), \(LP_{02}\), \(LP_{12a,b}\), \(LP_{31a,b}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Graded-index-few-mode fibers (GI-FMF) parameters for 6- and 10-mode fibers with \(NA=0.205\). \[Y_{M\times 1}=H_{M\times M}X_{M\times 1}+N_{M\times 1}=\sqrt{L}\prod_{k=1}^{K} \left(T_{k}C_{k}\right)X_{M\times 1}+N_{M\times 1} \tag{2}\]  
where \(X_{M\times 1}\) (resp. \(Y_{M\times 1}\)) is the vector of transmitted (resp. received) symbols, \(N_{M\times 1}\) is an additive white Gaussian noise vector with zero mean and variance \(2\sigma^{2}\) per mode. The matrix \(H_{M\times M}\) represents the spliced FMF composed of \(K\) fiber sections. The factor \(L\) represents the mode averaged propagation loss. \(T_{k}\) is a diagonal matrix with random phase entries \(\exp(i\phi_{m})\) and \(\phi_{m}\in[0,2\pi]\). The dispersive effect of DMGD is not considered in our work since it does not impact the capacity of the system. The impact of DMGD only affects the complexity of the channel equalization. In the case of using OFDM modulation format, the size of the cyclic prefix should be longer enough to absorb the DMGD.