## 2 Principle
The RIN transfer and the XPM were dependent on the group delay between the signal and the pump lights [17]. Generally, the Raman pump sources are broadband continuous wave lasers. We can thus modify the loss parameter to model the amplification. Then, the SpRS and DRB noises can be regarded as bandlimited additive noises. The RIN transfer can be modeled by the variety of the Raman gain [13; 17]. Therefore, Equation (2) can be modified as:  
\[ \begin{array}{ll}\frac{\partial A_{s}(z,T)}{\partial z}&=\frac{(-\alpha_{s}+ x_{B}(z,T))}{2}A_{s}-i\frac{\beta_{2}}{2}\frac{\partial^{2}A_{s}(z,T)}{ \partial T^{2}}\\ &+i\gamma_{s}|A_{s}|^{2}A_{s}(z,T)+\frac{3}{2}i\gamma_{s}\big{(}2P_{p}(z,T^{ \prime})\big{)}A_{s}+N(z,T).\end{array}  \tag{4}\]  
Figure 1: Schematic diagram of a bidirectional pumped distributed fiber Raman amplifier (FRA) system and the double Rayleigh backscattering (DRB) noise generation in the system. In Equation (4), the first term at the right side refers to the effects of fiber attenuation and Raman gain. We recorded the Raman gain changes with time in order to model the RIN transfer. The second term describes the impact of the second-order dispersion, while the third and fourth terms refer to the SPM of the signal light and the XPM between the signal and the pump lights. We modeled the RIN of the pump laser as 1/(f\({}^{*}\)v) of the frequency noise [19]. Since the FRAs are independent from polarization, we should induce the factor 2/3 to describe the influence of the polarization effects on the XPM [14,17]. The last term, \(N(T,z)\), is used to model the bandlimited additive distributed noise in the FRA systems, such as the SpRS and DRB noises. To obtain the influence of the RIN transfer, we should solve the dynamic coupled intensity equations of the bidirectional Raman amplification regarding location and time, but the required computational cost is too high. As an alternative solution, we can solve the static coupled intensity equations in different time periods to acquire a series of \((-\alpha+\kappa_{R}(z))\). To reflect the influence of the group delay between the signal and the pump lights, the time variable \(T^{\prime}\) is induced. In the forward-pumped FRA system,  
\[T^{\prime}=T-\frac{z}{v_{s}-v_{p}} \tag{5}\]  
and in the backward-pumped FRA system,  
\[T^{\prime}=T-\frac{z}{v_{s}+v_{p}}. \tag{6}\]  
The static value of the parameter \((-\alpha+\kappa_{R}(z))\) can be calculated using the coupled intensity equations of Raman scattering [15,18]:  
\[ \begin{array}{l}\frac{dp_{p}^{+}}{dz}=-\alpha_{p}P_{p}^{+}-g_{R}\frac{v_{p} }{v_{s}}P_{p}^{+}\Big{(}P_{s}^{+}+P_{s}^{-}+4 hv_{s}\Delta v\Big{)}+\gamma_{p} P_{p}^{-}\\ \frac{dp_{p}^{-}}{dz}=-\alpha_{p}P_{p}^{-}-g_{R}\frac{v_{p}}{v_{s}}P_{p}^{-} \Big{(}P_{s}^{+}+P_{s}^{-}+4 hv_{s}\Delta v\Big{)}+\gamma_{p}P_{p}^{+}\\ \frac{dp_{s}^{+}}{dz}=-\alpha_{s}P_{s}^{+}+g_{R}\Big{(}P_{p}^{+}+P_{p}^{-} \Big{)}\Big{(}P_{s}^{+}+2 hv_{s}\Delta v\Big{)}+\gamma_{s}P_{s}^{-}\\ \frac{dp_{s}^{+}}{dz}=-\alpha_{s}P_{s}^{-}+g_{R}\Big{(}P_{p}^{+}+P_{p}^{-} \Big{)}(P_{s}^{-}+2 hv_{s}\Delta v)+\gamma_{s}P_{s}^{+}.\end{array}  \tag{7}\]  
In Equation (7), \(+\) and \(-\) denote the forward and backward propagation of lights in the fiber, respectively; \(P_{p}\) and \(P_{s}\) represent the pump power at frequency \(v_{p}\) and the power of the amplified signal light at frequency \(v_{S}\); \(\alpha_{p}\) and \(\alpha_{s}\) are the attenuation coefficients for the pump light and the signal light; \(g_{R}\) is the Raman gain factor from the pump light to the signal light; the term \(2 hv_{S}\Delta v\) represents the spontaneous Raman emission in the signal band; \(\Delta v\) is the bandwidth of the signal light or the generated spontaneous radiation; and \(\gamma_{p}\) and \(\gamma_{s}\) are the Rayleigh scattering coefficients for the pump light and the signal light. The fibers in our experiments were standard single-mode fibers (SSMF, SMF-28e).