## 2 Amplifier Configuration
\[\frac{dP_{R_{i}}}{dz}=-\sum_{j=1}^{i-1}g_{R_{g_{j}}}P_{R_{j}}P_{R_{k}}+\sum_{ k=i+1}^{N}\frac{\omega_{R_{i}}}{\omega_{R_{k}}}g_{R_{k}}P_{R_{k}}+\frac{ \omega_{R_{i}}}{\omega_{s}}g_{si}P_{s}P_{R_{k}}+\alpha_{R_{i}}P_{R_{i}} \tag{1}\] \[\frac{dP_{s}}{dz}=\sum_{i=1}^{N}g_{si}P_{s}P_{R_{i}}-\alpha_{s}P_ {s} \tag{2}\]  
where \(i=1\), \(2\), \(3\ldots N\) and total \(N\) represents the number of Raman pumps. \(j\) (\(=1\), \(2\), \(\ldots\), \(i-2\), \(i-1\)) and \(k\) (\(=i+1\), \(i+2\), \(\ldots\), \(N\)) represent the Raman pumps, such that \(\lambda_{Rj}<\lambda_{Ri}<\lambda_{Rk}\). The corresponding Raman gain coefficients are \(g_{Rij}\), \(g_{Rik}\) and \(g_{si}\), and \(\omega_{Ri}\), \(\omega_{Rk}\) and \(\omega_{s}\) are the angular frequencies of the Raman  
Figure 3: (**a**) Upstream HROPA with two Raman pumps; (**b**) measured DEMUX/MUX transfer function. Figure 2: (**a**) The modified Raman section of the HROPA that uses a single Raman pump and an optical circulator to couple the Raman pump in the HNLF; (**b**) optical spectrum at point A after the Raman amplification when the Raman pump is at 1270 nm, which highlights the problem of having the signal and leaked pump at the same wavelength. pumps and the signal, respectively. Equation (1) describes the optical power evolution of the \(i\)-th Raman pump. In particular, the first term in Equation (1) shows the power flow to the \(i\)-th Raman pump from the Raman pumps with a smaller wavelength, e.g., the \(j\)-th Raman pump. The second term describes the power transfer from the \(i\)-th Raman pump to the Raman pumps with a higher wavelength, e.g., the \(k\)-th Raman pump. Finally, the third term of Equation (1) describes the power transfer from the \(i\)-th Raman pump to the signal, and the last term represents the fiber attenuation. Equation (2) describes the evolution of the signal optical power as it propagates through the Raman amplifier section. It is known that the Raman gain efficiency increases as the pump wavelength decreases. Therefore, one must obtain the corresponding Raman gain coefficient when designing a broadband amplifier that includes Raman pumps covering a very wide range of wavelengths. As described in [12, 13, 14], we can scale the Raman gain coefficient from measurements made at one pump wavelength \(\lambda_{P}\) to a new pump wavelength \(\lambda_{P\text{-}new}\) as follows:  
\[g_{R}\left(\Delta\
u,\lambda_{P\text{-}new}\right)=g_{R}\left(\Delta\
u, \lambda_{P}\right)\left(\frac{\lambda_{s}}{\lambda_{s\text{-}new}}\right) \left[\frac{\text{A}_{eff}^{ps}\left(\Delta\
u,\lambda_{P\text{-}new}\right)} {\text{A}_{eff}^{ps}\left(\Delta\
u,\lambda_{P\text{-}new}\right)}\right] \tag{3}\]  
where,  
\[\text{A}_{eff}^{m}\left(\Delta\
u,\lambda_{p}\right)=\frac{\text{A}_{eff} \left(\lambda_{p}\right)+\text{A}_{eff}\left(\lambda_{s}\right)}{2} \tag{4}\]  
Here, \(g_{R}(\lambda\
u,\,\lambda_{P})\) is the measured Raman gain coefficient at a pump wavelength \(\lambda_{P}\), \(A_{eff}\) is the effective area of the fiber provided as a function of the wavelength and \(\lambda\
u\) is the frequency separation between the pump and the signal. Therefore, to model the HROPA accurately, we followed this analysis and generate the corresponding Raman gain efficiencies using Equations (3)-(4) and the numerical data shown in Figure 4a,c [15]. The calculated Raman gain efficiencies for the two Raman pumps at 1260 nm and 1280 nm are shown in Figure 5 along with the measured Raman gain efficiency at 1453 nm. Note that the Raman gain efficiency increases as the pump wavelength decreases [12, 13, 14], and it helps to overcome the high optical fiber attenuation at the bandwidth of interest (Figure 4b). We used these derived Raman gain efficiencies in our theoretical performance evaluation of the proposed HROPA. Figure 4: High non-linear fiber measured parameters [15]. (**a**) Raman gain efficiency; (**b**) fiber attenuation; (**c**) effective area as a function of wavelength. In HNLF2, we have the OPA process.