## 2 Theoretical Model  
Consider an unrepeetered system with bi-directional Raman amplification and receiver-side DBP, as shown in Figure 1. Assuming that the nonlinear signal-to-signal (S-S) interaction is perfectly compensated by receiver-side DBP, the effective receiver signal-to-noise ratio (SNR) can be described as:  
\[SNR\approx\frac{P_{s}}{P_{ASE}+P_{S-N}}, \tag{1}\]  
where \(P_{s}\) is the average power of the signal; \(P_{ASE}\) is the average power of the ASE noise, including the ASE noise from EDFAs and spontaneous Raman estimation; and \(P_{S\text{-}N}\) is the power of NSNI. In the Raman amplified system, the spontaneous Raman scattering noise is generated along the fiber. Considering the spontaneous Raman scattering generated at position \(z\), denoted by \(E_{spRs}(z)\), it will propagate along the fiber with a length of \(L\)-\(z\) and interact with the signal, where \(L\) is the length of the whole fiber. After virtual transmission in DBP with a length of \(L\)-\(z\), the interaction of the noise and the signal in the practical fiber is eliminated. However, the noise continues to propagate and interact with the signal in the residual virtual fiber of length \(z\), causing the extra NSNI. The field of the extra NSNI can be described as a four-wave mixing (FWM) process [16]:  
\[ \begin{split} E_{NSNI}(z,\omega_{q})=\gamma\rho(2L-z,2L)E_{s}(z, \omega_{i})E_{s}(z,\omega_{j})E_{spRs}^{*}(z,\omega_{k})\\ +2\gamma\rho(2L-z,2L)E_{s}(z,\omega_{i})E_{s}^{*}(z,\omega_{k})E_{ spRs}(z,\omega_{j})\end{split} \, \tag{2}\]where the \(E_{NSNI}(z,\omega_{q})\) is the field of NSNI at angular frequency \(\omega_{q}=\omega_{i}+\omega_{j}-\omega_{k}\), \(\gamma\) is the nonlinear coefficient, \(E_{s}\) is the signal field, and \(\rho(2L-z,2L)\) is the FWM efficiency from position \(2L\)-\(z\) to \(2L\). The latter can be written as:  
\[\rho(2L-z,2L)=\int\limits_{2L-z}^{2L}P(\xi)\mathrm{e}^{-i\Delta\beta}d\xi, \tag{3}\]  
where \(P(\xi)\) is the normalized power profile in the fiber, \(\Delta\beta=\beta_{i}+\beta_{j}-\beta_{k}-\beta_{g}\) is the phase mismatch, and \(\beta_{i,j,k,g}\) represents the propagation constants for angular frequencies \(\omega_{i,j,k,g}\). Note that in Equation (2) the FWM of two noise frequencies and a signal frequency is neglected because the power of the signal is much greater than the noise. Integrating Equation (2) within the band of the signal, we can get the NSNI field at \(\omega_{g}\):  
\[E_{S-N}(z,\omega_{g})=\iint\limits_{\omega_{i,j,k}\in[-\frac{B}{2},\frac{B}{2 }]}E_{NSNI}(z,\omega_{g})d\omega_{i}d\omega_{j}, \tag{4}\]  
where \(B\) is the signal bandwidth. Assuming that the noise and the signal are independent random variables, the powers of the two terms in Equation (2) can be handled independently, and we can obtain the power of NSNI:  
\[ \begin{array}{ll}P_{S-N}(z)&=\int\limits_{\omega_{i,j,k}\in[-\frac{B}{2}, \frac{B}{2}]}3\gamma^{2}|\rho(2L-z,2L)|^{2}P_{s}(\omega_{i})P_{s}(\omega_{j})P _{spRs}(\omega_{i}+\omega_{j}-\omega_{g})d\omega_{i}d\omega_{j}d\omega_{g}\\ &\approx 3\gamma^{2}\eta(z)P_{s}^{2}P_{spRs}\end{array} , \tag{5}\]  
where \(P_{s,spRs}(\omega)\) is the power of the signal and spontaneous Raman scattering at angular frequency \(\omega\), \(P_{s,spRs}\) is the average power of the signal and spontaneous Raman scattering, and \(\eta(z)\) is the nonlinear interaction coefficient, which can be written as [15]:  
\[\eta(2L-z,2L)=\iint\limits_{\omega_{i,j,k}\in[-\frac{B}{2},\frac{B}{2}]}|\rho( 2L-z,2L)|^{2}d\omega_{i}d\omega_{j}, \tag{6}\]  
Integrating Equation (5) along the fiber, we can obtain the total NSNI power at the end of the DBP:  
\[P_{S-N}=\int\limits_{0}^{L}3\gamma^{2}\eta(2L-z,2L)P_{s}^{2}(z)P_{spRs}(z)dz, \tag{7}\]  
Note that in Equation (7), the transceiver noise can be added into \(P_{spRs}(z)\) at positions 0 and \(L\), for transmitter noise and receiver noise, respectively. Figure 1: Nonlinear signal-to-noise interaction (NSNI) in a bi-directional Raman system with receiver-side digital back-propagation (DBP).