## 2 Principle  
The propagation of optical pulses in single-mode communication fibers is typically modeled with the generalized nonlinear Schrodinger equation by [14]:  
\[ \begin{array}{rl}\frac{\partial A}{\partial z}+\frac{\partial}{2}A&+i\frac{ \beta_{2}}{2}\frac{\partial^{2}A}{\partial T^{2}}-\frac{\beta_{3}}{6}\frac{ \partial^{2}A}{\partial T^{3}}\\ &=i\gamma\left(|A|^{2}A+iS\frac{\partial}{\partial T}\left(|A|^{2}A\right)-T_{ R}A\frac{\partial|A|^{2}}{\partial T}\right).\end{array}  \tag{1}\]  
In Equation (1), \(A(z,T)\) denotes the slowly varying field envelope of the optical signal, and \(z\) is the propagation distance. Equation (1) is valid in the frame of reference of the pulse traveling with signal group velocity \(v_{g}\). Using the relation \(T=t-z/v_{g}\) between the retarded time \(T\) and the present time \(t\), Equation (1) can be transformed into the laboratory frame of reference. The parameter \(\alpha\) models the linear signal loss, \(\beta_{2}\) and \(\beta_{3}\) are the coefficients of second- and third-order dispersion, and \(\gamma\) is the fiber's nonlinear parameter. Parameters \(S\) and \(T_{R}\) govern the effects of self-steepening and the Raman response. For pulses of width \(T_{0}>5 ps\), the last two terms in Equation (1) can be neglected, because the effects of self-steepening and the Raman response are not obvious [14]. As the contribution of the third-order dispersion term is also quite small for such pulses (as long as the carrier wavelength is not too close to the zero dispersion wavelength), we can employ the simplified equation [14]:\[\frac{\partial A}{\partial z}=-\frac{\alpha}{2}A-i\frac{\beta_{2}}{2}\frac{\partial^ {2}A}{\partial T^{2}}+i\gamma|A|^{2}A. \tag{2}\]  
In the optical fiber communication system with distributed FRAs, taking into account the Raman amplification effects, the simplified NLSE can be written as:  
\[\frac{\partial A}{\partial z}=\frac{-\alpha+g(z)}{2}A-i\frac{\beta_{2}}{2} \frac{\partial^{2}A}{\partial T^{2}}+i\gamma|A|^{2}A \tag{3}\]  
where \(g(z)\) is the Raman gain factor with the unit (1/km). However, the accuracy of the model based on Equation (3) is not high enough to allow for the simulation of the performance of the FRA system. To model the systems with FRAs more precisely, we should take into account not only the Raman amplification but also the distributed noise and the XPM between the signal and the pump lights when employing the NLSE. One of the limiting factors of the FRA system is the distributed noise, mainly induced by the SpRS and DRB. Figure 1 presents the schematic diagram of a bidirectional pumped distributed FRA system and the DRB noise generation process in the system. Two pump waves coupled at the beginning and the end of the fiber link led to distributed Raman gain in the fiber. The signal light was transmitted along the forward direction of the fiber and scattered randomly. A little part of the backscattered guided photons would generate the single Rayleigh backscattering (SRB) light. The SRB light was amplified by the Raman pump lights and backscattered randomly again to produce DRB light. The amplified DRB light was partially coherent with the signal light and could interfere with the signal light. The DRB noise was the summary of a mass of lights with different time delays when compared with the signal light, which can be regarded as the stochastic multi-path interference (MPI) noise [11]. The MPI noise can be negligible in the long-haul multi-span fiber system but can impact the performance of the long-reach unrepeated system and the distributed random feedback lasers. In addition, when the two pump waves were transmitted along the fiber, the SpRS photons were generated, and the forward-guided photons were also amplified [9; 10]. Besides, the intensities of the Raman pump lights varied with time, and the variety of pump intensities could change the Raman gain and the intensity of the signal light, referred to as the RIN transfer [13].