## 2 Optimized Distributed Raman Amplification Design over Single Fiber Span
### Distributed Raman Amplification with Bidirectional Pumping  
In our optimization of signal power profiles for mid-link OPC we consider three different distributed Raman amplification schemes, all of them bidirectionally pumped. Simulations are performed for each configuration, obtaining signal power excursion for different pump power ratios and span lengths using the tried and tested model fully described in [38] with the boundary conditions corresponding to each of the cases under consideration and assuming fully depolarized pumps as well as room temperature operation. Noise calculations are based on a 0.1 nm bandwidth. The coefficients for Raman gain and attenuation at the different wavelengths involved were extracted from measurements for SSMF [38], whereas the Rayleigh backscattering coefficients at 1366 and 1455 nm and the frequency of the signal are assumed to be 1.0 \(\times\) 10\({}^{-4}\), 6.5 \(\times\) 10\({}^{-5}\) and 4.5 \(\times\) 10\({}^{-5}\) km\({}^{-1}\), respectively. The first case considered (Figure 7a) corresponds to a conventional bidirectionally pumped first-order Raman amplifier, with pumps at 1455 nm that amplify the signal through the first Stokes shift. The second case corresponds to an ultra-long Raman fiber laser (URFL) amplifier (Figure 7b), that provides second-order pumping from single-wavelength pumps. In such an amplifier, the initial Raman fiber laser pumps operate at 1366 nm, that is, downshifted by two Stokes shift with respect to the frequency of the signal. Highly reflective (99%) fiber Bragg gratings (FBGs) centered at 1455 nm with a bandwidth of 200 GHz are located at both ends of the transmission line to back-reflect the first Stokes-shifted radiation at 1455 nm into the long cavity. Once a threshold of \(\sim\)0.8 W pump power is reached, the cavity forms a stable ultra-long laser that at that amplifies the signal around 1550 nm. This approach presents the advantage of having modifiable gain bandwidth and profiles by selecting appropriate FBGs, instead of requiring an active seed at the intermediate Stokes. In this case the reflectivity of the FBGs was chosen to be high to maximize pump-to-signal power conversion efficiency [38; 39; 40; 41]. Figure 6: Experimentally measured \(Q^{2}\) factors versus signal launch power in inline transmission systems without/with OPC using the optimized dual-order DRA scheme over 2 \(\times\) 50 km SSMF. The third and final approach uses a random distributed feedback Raman laser amplifier (Figure 7c). This is another fiber laser amplifier similar in part to the scheme shown in Figure 3. This is significantly different from a closed cavity with two FBGs. The scheme is essentially bidirectional Raman pumping using a half-open-cavity design with a single high-reflectivity FBG at 1455 nm located at the end of the span that reflects the 1455 nm Stokes in the backward direction. The second-order pump in front of the span, does not create a seed traveling in forward direction (either by inserting an FBG or an active seed) but rather amplifies the seed created at the end of the span. The lack of an FBG on the side of the forward pump reduces the RIN transfer to the 1455 nm Stokes in exchange for a reduction in conversion efficiency in comparison to the other two proposed setups [35]. This reduced interaction between the signal and the forward pumping is particularly important, as the RIN transfer from high-power forward pumps can be a limiting factor in data transmission [42; 43]. In order to perform a comparison of signal power asymmetry between the three proposed configurations, we simulated power profiles of a single channel at 1545 nm with a fixed launch power of 0 dBm. For each value of the forward pump power (FPP) and span lengths ranging from 40 to 100 km, the backward pump was adjusted to provide 0 dB net gain. Signal power asymmetry was calculated as in [12; 13].