## 2 Inter-DCI Link Modeling and OP Estimation
### MCF Modeling and ICXT Generation  
In the following, the simulation model known as DP-DCM, used to characterize the ICXT induced by the cores of the MCF, is presented [13]. The DP-DCM characterizes the ICXT induced by the different cores of the MCF on the CUT. Several works have shown that the ICXT results mostly from the discrete contribution of points along the longitudinal propagation direction of the MCF, known as phase matching points (PMPs). The PMPs, where the difference between the effective refractive index of the interfering and interfered cores is zero, manifest randomly along the fiber [13]. The total ICXT induced by each interfering core can be approximated as the sum of the contributions associated with each PMP. Each contribution is weighted by an independent random phase shift (RPS) and the corresponding propagation delay [12]. The RPSs model random variations of the bending radius, twist rate, or other conditions in the MCF [11; 13]. The DP-DCM describes the ICXT generation in the two polarization directions \(\mathbf{x}\) and \(\mathbf{y}\). The transfer functions \(F_{a,b}(\omega)\) model the frequency response of the ICXT from polarization \(a\) (with \(a=x\) or \(y\)) at the input of one interfering core \(m\) (\(1\leq m\leq N_{i}\)) to polarization  
Figure 1: System equivalent model of an optically amplified inter-DCI link supported by a MCF and with CDC. (with \(b=x\) or \(y\)) at the output of the interfered core \(n\). The transfer function \(F_{a,b}(\omega)\) is given by [13]  
\[F_{a,b}(\omega)=-\frac{j}{\sqrt{2}}\overline{K}_{nm}\exp\bigl{(}- \vec{\beta}_{n}(\omega)L\bigr{)}\cdot\exp\Bigl{(}-\frac{\alpha}{2}L\Bigr{)}\cdot \\ \sum_{k=1}^{N_{p}}\exp\bigl{[}-j(\vec{\beta}_{m}(\omega)-\vec{ \beta}_{n}(\omega))z_{k}\bigr{]}\exp\Bigl{[}-j\phi_{mm,k}^{(a,b)}\Bigr{]} \tag{1}\]  
where \(\alpha\) is the power attenuation coefficient (assumed equal for all cores), \(\overline{K}_{nm}\) is the average inter-core coupling coefficient, which is given by the average of its contributions in the two polarization directions [13], \(\omega\) is the angular frequency, \(L\) is the MCF length, \(N_{p}\) is the number of PMPs [11], and \(\vec{\beta}_{i}(\omega)\) (with \(l=1...N_{i}\), or \(n\)) is the average of the propagation constants in the two polarization directions in core \(l\), given by [39]  
\[\vec{\beta}_{l}(\omega)=\beta_{0,l}+\beta_{1,l}\omega+\frac{\beta_{2,l}}{2} \omega^{2}+\frac{\beta_{3,l}}{6}\omega^{3} \tag{2}\]  
where \(\beta_{0,l}\) is the propagation constant at the operating wavelength \(\lambda_{0}\), \(\beta_{1,l}\) is the inverse of the group velocity, \(\beta_{2,l}\) is the group velocity dispersion and \(\beta_{3,l}\) is the higher order dispersion, for core \(l\). The skew between an interfering core \(m\), and the interfered core, \(n\), is given by \(S_{nm}=d_{nm}L\), where \(d_{mn}\) is the walkoff between cores \(m\) and \(n\) defined by \(d_{mn}=\beta_{1,m}-\beta_{1,n}\). In Equation (1), \(\phi_{mm,k}^{(a,b)}\) represents the RPS associated with the \(k\)-th PMP and the RPSs are independent and modeled by a uniform distribution between \(0\) and \(2\pi\). The longitudinal coordinate of the \(k\)-th PMP, \(z_{k}\), is randomly distributed between two consecutive PMPs and is given by \(z_{k}=L(r_{k}+k-1)/N_{p}\), where \(r_{k}\) (\(1\leq k\leq N_{p}\)) are independent random variables with uniform distribution in the interval \([0,1[\). Each PAM4 interfering signal is generated independently from the other interfering signals. The optical field at the input of each interfering core is split in the two polarization directions \(\mathbf{x}\) and \(\mathbf{y}\), given by  
\[c_{m,x}(t)=c_{m}(t)\times \sqrt{\xi_{m}}\] \[c_{m,y}(t)=c_{m}(t)\times \sqrt{1-\xi_{m}} \tag{3}\]  
where \(c_{m,x}(t)\) corresponds to the optical field transmitted in polarization \(\mathbf{x}\), \(c_{m,y}(t)\) corresponds to the optical field in polarization \(\mathbf{y}\) and \(\xi_{m}\) controls the power splitting between the two polarization directions.