# Multipath Interference Impact Due to Fiber Mode Coupling in C+L+S Multiband Transmission Reach
## 3 Quality of Transmission Model  
The QoT of an optical transmission system in a coherent dispersion-uncompensated- and amplified-wavelength-division-multiplexed C+L+S MB system can be estimated through the GSNR [5]. Figure 2 depicts an optical transmission system with several identical spans, where each span is composed of a section of G.654E SMF with several splices and an optical amplifier stage composed of a band demultiplexer, an amplifier for each one of the bands--C, L, and S--and a band multiplexer. This MB C+L+S transmission system has several optical channels characterized by their bandwidths and central wavelengths, which can transport optical signals with a certain baud rate and modulation format. In such a scenario, the transmitted signal is impacted by the amplified spontaneous emission (ASE) noise originated from the optical amplifiers, by the non-linear interference (NLI) noise from the optical fiber transmission and by the MPI noise from the mode coupling in the G.654E SMF, so the GSNR of channel \(i\) after transmission along several identical spans can be given by [22]  
\[\text{GSNR}_{i}=\frac{P_{ch,i}}{P_{\text{ASE},i}+P_{\text{MPI},i}+P_{\text{ NLI},i}}, \tag{4}\]  
where \(P_{ch,i}\) is the launch power in channel \(i\), \(P_{\text{ASE},i}\) is the accumulated ASE noise power in channel \(i\), \(P_{\text{NLI},i}\) is the NLI noise power in channel \(i\) that also includes the ISRS effect, and \(P_{\text{MPI},i}\) is the MPI noise power in channel \(i\). The accumulated ASE noise power, \(P_{\text{ASE},i}\), in channel \(i\), after several identical spans, \(N_{span}\), can be modeled as [26]  
\[P_{\text{ASE},i}(f_{i})=N_{span}\cdot h\cdot\text{NF}_{i}\cdot f_{i}\cdot G_{i} \cdot B_{i}, \tag{5}\]  
where \(h\) is the Planck's constant, \(\text{NF}_{i}\) and \(G_{i}\) are, respectively, the amplifier noise figure and amplifier gain (in linear units) for channel \(i\), \(f_{i}\) is the central frequency of channel \(i\), and \(B_{i}\) is the channel's bandwidth considered to be the same as the symbol rate. Note that the amplifier gain, \(G_{i}\), is set to perfectly compensate for the span loss experienced by channel \(i\) and also for the loss or gain due to the ISRS effect in each span and can be given by [9; 26]  
\[G_{i}=\frac{G_{\alpha}}{G_{ISRS}}, \tag{6}\]  
where  
\[G_{\alpha}=e^{\alpha-L_{span}} \tag{7}\]  
is the gain that compensates for the span loss, with \(L_{span}\) being the span length in km and \(\alpha\) being the attenuation coefficient in Np/km, and  
\[G_{ISRS}=\frac{P_{ch,j}\big{(}L_{span}\big{)}}{P_{ch,i}\cdot e^{-\alpha-L_{span}}} \tag{8}\]  
is the gain or loss to compensate for the ISRS effect per span, with \(P_{ch,j}\big{(}L_{span}\big{)}\) being the optical power at the end of each span that can be given by Equation (8) of [26]. The NLI noise power, \(P_{\text{NLI},i}\), in channel \(i\) after several identical spans, \(N_{span}\), is calculated using an analytical approximation of the generalized Gaussian noise (GGN) model proposed in [9] and known in the literature as the ISRS-GN model, which allows for faster and still accurate GSNR estimations:  
\[P_{\text{NLI},i}(f_{i})=\ \eta_{NLI}(f_{i})\cdot P_{ch,j}^{3}(f_{i}), \tag{9}\]  
with the NLI coefficient, \(\eta_{NLI}(f_{i})\), given by  
\[\eta_{NLI}(f_{i})\approx N_{span}\cdot\big{[}\eta_{SPM,j}(f_{i})+\eta_{XPM,j}( f_{i})\big{]}, \tag{10}\]  
where \(\eta_{SPM,j}(f_{i})\) is the self-phase modulation (SPM) coefficient of channel \(i\) at frequency \(f_{i}\), defined by Equation (10) of [9], and \(\eta_{XPM,j}(f_{i})\) is the cross-phase modulation (XPM) coefficient of channel \(i\) at frequency \(f_{i}\), defined by Equation (11) of [9]. For simplicity, we assume incoherent accumulation of the NLI noise power, e.g., [26; 27]. Figure 2: Optical transmission system composed of a transmitter and a receiver connected by several spans for evaluating the QoT in a C+L+S MB system.