## 2 Modeling of Generalized Mathematical Formulas for Multi-Segment Transmission Link
Thus, the combined pump power can be written in the form  
\[P_{p}(z)=P_{p0}\big{\{}r\exp\bigl{(}-\alpha_{p}z\bigr{)}+(1-r)exp(-\alpha_{p}(L- z))\big{\}}. \tag{4}\]  
Signal power \(P_{s}\) along the propagation fiber can be calculated from Equation (2) by integrating along the \(z\) distance and can be written as  
\[P_{s}(z)=P_{s0}exp\left(\int_{0}^{z}\frac{g_{R}}{A_{eff}(z^{\prime})}P_{p}(z^{ \prime})dz^{\prime}-\alpha_{s}z\right). \tag{5}\]  
The profile for Raman net gain \(G_{net}\) along the propagation can be calculated by comparing the signal power at propagation distance \(P_{s}(z)\) to the signal power at initial point \(P_{s0}\) using Equation (5), which then yields  
\[G_{net}(z)=exp\left(\int_{0}^{z}\frac{g_{R}}{A_{eff}(z^{\prime})}P_{p}(z^{ \prime})dz^{\prime}-\alpha_{s}z\right). \tag{6}\]  
Since the fiber effective core area varies along transmission span, the standard Raman equation with constant fiber effective core area has to be modified to a generic formula that accommodates the position \(z\) dependency of fiber effective core area \(A_{eff}(z)\). To better understand the hybrid fiber core system, signal power density \(I_{s}(z)\) can be defined as the amount of signal power \(P_{s}(z)\) per unit effective core area \(A_{eff}(z)\) and can be written from Equation (5) as  
\[I_{s}(z)=\frac{P_{s0}}{A_{eff}(z)}exp\left(\int_{0}^{z}\frac{g_{R}}{A_{eff}(z^{ \prime})}P_{p}\bigl{(}z^{\prime}\bigr{)}dz^{\prime}-\alpha_{s}(z)\right). \tag{7}\]  
Based on the Raman analysis results, the Kerr nonlinear limit and Raman noise limit can be measured using the nonlinear phase shift and OSNR. In a single-channel system, the nonlinear phase shift parameter \(\Phi_{NL}\) indicates the degree of distortion caused by the nonlinear Kerr effect induced self-phase modulation is given by [23]  
\[\Phi_{NL}(z)=\int_{0}^{z}\gamma\cdot P_{s}(z)dz, \tag{8}\]  
where the nonlinear parameter \(\gamma\) used in our model varies along propagation distance \(z\) due to the variation of \(A_{eff}(z)\) in the span with multi-segmented structure. Therefore, the nonlinear parameter that changes along propagation \(\gamma(z)\) must be used and can then be modified from [23] in the form  
\[\gamma(z)=\frac{2\pi n_{2}}{\lambda_{s}A_{eff}(z)}, \tag{9}\]  
where \(n_{2}\) is the nonlinear Kerr parameter and \(\lambda_{s}\) is the signal wavelength. Finally, by inserting Equation (9) into Equation (8), accumulated nonlinear phase shift becomes  
\[\mathrm{V}=\frac{2\pi a}{\lambda}\left(n_{1}^{2}-n_{2}^{2}\right)^{1/2}<2.405, \tag{1}\]0  
The spontaneous Raman scattering (SRS) process generates random phases associated with all spontaneously generated photons and adds noise to the amplified signal that appears as a spontaneous Raman noise, which affects the performance of distributed Raman amplifiers and becomes the main sources of optical noise. The spectral density of amplified spontaneous emission (ASE) in Raman span \(S_{ASE}\) is defined as [8]  
\[\mathrm{V}=\frac{2\pi a}{\lambda}\left(n_{1}^{2}-n_{2}^{2}\right)^{1/2}<2.405, \tag{1}\]1where \(n_{sp}\) is the spontaneous scattering factor, \(\lambda_{p}\) is the Raman pump wavelength, and \(hc/\lambda_{p}\) is the photon energy of the Raman pump. The total ASE power can be calculated by integrating ASE spectral density \(S_{ASE}\) within the range of the filter's optical bandwidth \(B_{opt}\) after the amplifier. The OSNR shows the signal quality after it is affected by the substantial amount of spontaneous Raman noise generated by ASE. Since the OSNR of the amplified signal is defined by input signal power \(P_{s0}\) divided by total ASE power [8], the final calculation result is given by  
\[\mathrm{V}=\frac{2\pi a}{\lambda}\left(n_{1}^{2}-n_{2}^{2}\right)^{1/2}<2.405, \tag{1}\]2  
All these generalized formulas where the fiber effective core area is no longer a constant value but a function of link distance will be used for performance modeling on the following sections.