## 2 Modeling of Generalized Mathematical Formulas for Multi-Segment Transmission Link
The ASE noise has a flat spectral distribution across the operating bandwidth similar to a white noise and its impact is inherently dominant. The DRS noise creates cross-talk component that has nearly the same spectral shape as the signal itself [25] and leads to a large power penalty in multi-span long-haul transmission system [26]. RIN noise is a spectrum dependent quantity caused by power fluctuation of Raman pump laser source that will directly transfer to the signal's power [27] and can be reduced by improving the quality of the semiconductor laser source. Since this article focuses on single-span transmission systems, the initial analysis can be limited to ASE noise and the other two noise sources could be added in a future study. In the multi-segment hybrid fiber effective core area structure, the value of the fiber effective core area changes along the propagation distance. Therefore, the standard Raman amplification formulas should be rewritten to generalized formulas where the fiber effective core area is no longer a constant value but a function of propagation distance. To accommodate the variation in fiber core size along the propagation distance, the generalized form of Raman amplification formulas for multi-segment transmission link should be recalculated from the procedure discussed on the standard Raman amplification formulas [1; 4; 5]. Based on standard Raman coupled equations, we derived the generalized Raman formulas started from rewriting the fiber effective core area as a function of propagation distance \(A_{\textit{eff}}(z)\) as  
\[\frac{dP_{s}}{dz}=-\alpha_{s}P_{s}+\frac{g_{R}}{A_{eff}(z)}P_{p}P_{s}, \tag{2}\]  
\[\kappa\frac{dP_{p}}{dz}=-\alpha_{p}P_{p}-\frac{\lambda_{s}}{\lambda_{p}}\frac{ g_{R}}{A_{eff}(z)}P_{s}P_{p}, \tag{3}\]  
where \(z\) is the propagation distance, \(A_{\textit{eff}}(z)\) is the fiber effective core area as a function of propagation distance, \(P_{p}\) and \(P_{s}\) are pump and signal powers, \(\alpha_{s}\) and \(\alpha_{p}\) are signal and pump loss coefficients, and \(\lambda_{s}\) and \(\lambda_{p}\) are signal and pump operating wavelengths. The signal-pump coupling interaction of the Raman scattering process is determined by multiplying the signal and pump powers, and the intensity is determined by the fiber Raman gain coefficient \(g_{R}\), whose value depends on the signal and pump wavelengths. The value of parameter \(\kappa\) depends on the choice of Raman pump configuration, \(\kappa=1\) for the forward pumping and \(\kappa=-1\) for the backward pumping. In the bidirectional pumped Raman amplification model for fiber length \(z=L\), both pump lasers are located at opposite fiber ends. The contribution of pump power to the forward direction can be calculated from Equation (3) by setting \(\kappa=1\) and integrating along the span from 0 to \(z\). Similarly, the contribution of pump power to the backward direction can be calculated from Equation (3) by setting \(\kappa=-1\) and integrating along the span from \(z\) to \(L\). Considering a practical situation in which the pump powers are far larger than the signal power and pump power depletion can be ignored,the total pump power \(P_{p}\) at distance \(z\) can be obtained by adding the pump power from both directions and introducing the pump power ratio parameter \(r\) as the ratio of the forward pump power over the total pump power at launch \(P_{p0}\).