# Multipath Interference Impact Due to Fiber Mode Coupling in C+L+S Multiband Transmission Reach
## 2 Characterization of Multipath Interference in G.654E SMF  
In G.654E SMFs, the coupling of optical power between the fundamental mode (LP\({}_{01}\)) and the high-order mode (LP\({}_{11}\)) originates in scattering events from fiber non-uniformities, from fiber microbending losses or from splices. A small fraction of the signal propagating in the LP\({}_{01}\) mode is coupled into the LP\({}_{11}\) mode and, after traveling some distance, is scattered back to the fundamental mode as indicated in Figure 1[14]. Therefore, many replicas of the signal co-propagated with the signal itself appear in the fundamental mode. As the refractive index is mode-dependent, the transmission speed in the two modes is different, giving rise to a time delay between the signal and its replicas. These signal replicas are called forward scattered MPI, and the time delay is described as follows [23]:  
\[\tau=\frac{L\Delta\overline{n_{i}}}{c} \tag{1}\]  
with \(L\) being the fiber length, \(c\) the speed of light in vacuum, and \(\Delta\overline{n_{i}}\) the difference between the refractive index of the fundamental mode and the higher-order mode. Note that the time delay, \(\tau\), between the signal and its replicas comes only from the refractive index difference in the two modes, since the path signal length and the path of replicas is the same. This interference is classified as coherent since the time delay between the signal and its replicas is typically shorter than the laser source coherence time, i.e.,  
\[\tau<\frac{1}{\Delta v} \tag{2}\]  
where \(\Delta v\) is the laser source linewidth. This condition is true for fibers with several hundred kilometers, e.g., considering a typical value for the refractive difference for the LP\({}_{11}\) mode of \(\Delta\overline{n_{i}}\approx 0.003\)[23] and \(\Delta v=100\) KHz. We can conclude that there is coherent interference for \(L<\frac{c}{\Delta v\cdot\Delta\overline{n_{i}}}=1000\) km. Figure 1: Mode-coupling scheme along G.654E SMF. Due to the presence of multiple delayed replicas at the SMF output, the MPI level can be defined as follows [20]:  
\[\text{MPI}=2\!\!\sum_{i=1}^{N}\sqrt{\varepsilon_{i}}cos\phi_{i}\, \tag{3}\]  
where \(\varepsilon_{i}\) is the ratio between the delayed replica power and the signal power, and the phase \(\phi_{i}\) is a random variable given by \(\phi_{i}=\frac{-2\pi\Lambda\overline{m_{i}}L}{\lambda_{i}}\) for the case of forward-scattered MPI from mode coupling [23]. This randomness comes from variations in the refractive index of the fundamental mode and higher modes. In this way, the MPI, given in Equation (3), is a random variable whose statistics must be characterized. In the presence of correlation, due to coherence nature of the MPI, the i.i.d. assumption no longer holds, and, as a consequence, the statistics of the MPI are difficult to predict and different from Gaussian statistics, as shown in [20], where it appears that they can be more rigorously described by fat-tailed statistics, like the extended skew-normal distribution [21], power-law, and accumulative law distributions [24]. Nevertheless, in this work, for the sake of simplicity, as was also performed in several works in the literature, e.g., [14], we consider that the MPI is described by Gaussian statistics, so that the QoT can be computed in a straightforward way with the GSNR formulation presented in the next section. Besides assuming that the MPI can be described by Gaussian statistics, the following analysis does not consider, for the sake of simplicity, the MPI wavelength dependency and the influence of the orthogonal modes \(\text{LP}_{11a}\) and \(\text{LP}_{11b}\). The MPI wavelength dependency is highlighted, for example, in [25], where it can be concluded that the wavelengths near the cut-off wavelength suffer much less from the MPI impact.