## 4 Discussion
Realistic re-launch losses of 80\(\%\) still allow for a conversion efficiency of 40\(\%\), at optimum fiber length. Thus, with length optimization and modest improvement in the re-launch efficiency, we can considerably increase the experimentally obtained conversion efficiency, even without any improvements of fiber or pump. It is also interesting to investigate the effect of improvements in the pump power and fiber loss. For this, we introduce dimensionless variables as follows:  
\[ \begin{gathered} p=\frac{g_{R}I_{p}}{\alpha_{s}}\\ s=\frac{\frac{I_{s}}{h\
u_{s}}}{\frac{I_{p0}}{h\
u_{p}}}=\frac{I_{s} \lambda_{s}}{I_{p0}\lambda_{p}}\\ \zeta=\alpha_{s}z\end{gathered}  \tag{2}\]  
The normalized pump intensity \(p\) is given by the Raman gain it induces, relative to the signal propagation loss. The normalized Stokes intensity \(s\) is the Stokes photon flux density relative to the launched pump photon flux density. The normalized propagation coordinate \(\zeta\) is the physical propagation coordinate relative to the Stokes absorption length. With these substitutions, the Raman propagation equations (Equation (1)) can be rewritten as  
\[ \begin{gathered}\frac{dp}{d\zeta}=-(p_{0}s+\frac{\alpha_{p}}{ \alpha_{s}})p\\ \frac{ds}{d\zeta}=(p-1)s\end{gathered}  \tag{3}\]  
Here, \(p_{0}=g_{R}I_{p0}/\alpha_{s}\) is the normalized launched pump intensity, which is a free parameter that can be viewed as a figure of merit (FoM). For our DCRF with 65 W of pump power, \(p_{0}\) = 4.7. The loss  
Figure 10: **(a)** Simulations of conversion efficiency _vs._ DCRF length with optimized output couplings for different re-launch efficiencies (\(\eta_{1}\) and \(\eta_{2}\)); **(b)** optimum reflectivity of the out-coupling end R1 for different re-launch efficiency corresponding to Figure 10a. In the HR end, \(R_{2}\) = 1 in all cases. ratio \(\alpha_{p}/\alpha_{s}\) is the only other free parameter in the equations. This can often be approximated to unity. Equation (3) is for co-propagating pump and Stokes waves, but it is straightforward to add additional waves, at same or different wavelengths and propagating in the same or opposite direction. A solution to the normalized Raman propagation equations will be valid for any combination of pump and Stokes intensities (as well as powers) and losses that result in the same normalized parameters, if the boundary conditions and normalized fiber length are the same. This includes solutions that are optimized in terms of output coupling and normalized fiber length. The resulting quantum conversion efficiency is equal to the value of \(s\), as out-coupled from the cavity. Figure 11 shows how the maximum quantum conversion efficiency depends on the FoM, _i.e._, the normalized launched pump intensity \(p_{0}\), for different re-launch losses. We consider two loss ratios, \(\alpha_{p}/\alpha_{s}\) = 1.73 / 1.59 = 1.09 (as for our DCRF) and \(\alpha_{p}/\alpha_{s}\) = 1. This difference in loss ratio is small, as is the resulting difference in efficiency (compare the two uppermost curves in Figure 11). With a FoM of 4.7, which corresponds to our experiments with the DCRF fiber, and \(\alpha_{p}/\alpha_{s}\) = 1.09, a quantum conversion efficiency of 62\(\%\) is possible with 100\(\%\) re-launch efficiency at both ends, and 38\(\%\) with 50\(\%\) re-launch efficiency at the out-coupling end and 100\(\%\) re-launch at the high reflectivity end. In case of the GRIN fiber experiments, the FoM becomes 9.5 if we use an effective area of 1530 \(\upmu\)m\({}^{2}\). With this FoM, it is possible to reach 78\(\%\) conversion efficiency with 100\(\%\) re-launch efficiency at both ends, and 58\(\%\) with 50\(\%\) re-launch efficiency at the out-coupling end and 100\(\%\) re-launch at high reflectivity end. (This assumes \(\alpha_{p}/\alpha_{s}\) = 1.09 whereas actual value is 1.13.) Higher FoMs allow for conversion efficiencies of around 90\(\%\) for high far-end re-launch efficiencies, although this requires parameters considerably better than those we could reach.