## 2 Multimode Raman Gain Fiber
VBG: volume Bragg grating; DM: dichroic mirror; HR: high reflection mirror. Inset: refractive index profile (RIP) through the center of the core of the GRIN MM fiber. We measured the beam quality (\(M^{2}\)) of the pump incident on the fiber to 22.2, while it became 5.0 for the generated output beam. This means that the brightness (radiance) improved from 0.15 W\(\cdot\)sr\({}^{-1}\cdot\)um\({}^{-2}\) to 0.77 W\(\cdot\)sr\({}^{-1}\cdot\)um\({}^{-2}\), _i.e._, by a factor of 5.2 with respect to the incident pump beam. These measurements were at maximum power. We note the beam quality can be calculated from the refractive-index profile. In the highly-multimode regime, under the assumption that all modes are equally excited, we calculate the pump \(M^{2}\)-value to approximately 15.4 in this fiber, using the approach described in [16,17]. This corresponds to a brightness of the launched pump beam of 0.19 W\(\cdot\)sr\({}^{-1}\cdot\)um\({}^{-2}\) at full power. This slightly higher value than that of the incident pump beam is not surprising, since the launch can be adjusted to select the highest-brightness parts of the incident beam. The \(M^{2}\)-value of the laser output beam would be 14.7 if all modes were equally excited, which is considerably worse than the actual beam quality. We calculated the thresholds from the cavity losses, core area, effective lengths, and Raman gain coefficient. These become much higher than the measured thresholds, if we assume that the pump was uniformly distributed over the 3070 um\({}^{2}\) core area. However, the pump intensity is more likely to be higher in the center of the core, where the refractive index is higher. Therefore, the signal will build up there. This reduces the effective area. The precise value depends on which pump and signal modes are involved, but if we assume that all pump modes are uniformly excited then the effective area varies between 1930 um\({}^{2}\) (if lasing occurs on all modes, if they are strongly coupled) and 1307 um\({}^{2}\), if lasing occurs on the fundamental mode. The effective area is expected to vary with the power level, since the Stokes wave can selectively deplete pump modes as the Stokes power grows. Different loss of different modes can be a further complication, as is mode coupling occurring along the fiber and at the fiber ends. We adjusted the effective pump area to 1530 um\({}^{2}\) in order to achieve good agreement with measured data. The power characteristics with two fiber lengths were calculated using appropriate boundary conditions and standard power conversion equations [18]:  
\[ \begin{split}\pm\frac{1}{P_{p}^{\pm}}\frac{P_{p}^{\pm}}{dz}=- \alpha_{p}-\frac{g_{R}}{A_{eff}}\frac{\lambda_{s}}{\lambda_{p}}(P_{s}^{+}+P_{ s}^{-})\\ \pm\frac{1}{P_{s}^{\pm}}\frac{P_{s}^{\pm}}{dz}=-\alpha_{s}+\frac{g _{R}}{A_{eff}}(P_{p}^{+}+P_{p}^{-})\end{split}  \tag{1}\]  
Figure 3: Simulated and experimental CW FRL output power against the launched pump for GRIN-MM fiber lengths of 3 km (**a**) and 1.5 km (**b**). Here, subscript \(p\) refers to the pump and \(s\) to the Stokes wave. The plus and minus signs are used for forward and backward propagating waves, respectively. Furthermore, \(\alpha\) is the attenuation constant and \(\lambda\) is the wavelength. \(g_{R}\) refers to the Raman gain co-efficient and \(A_{eff}\) is the effective area. Figure 3 shows that with \(A_{eff}\) = 1530 \(\upmu\)m\({}^{2}\), the simulated characteristics agree with measurements, which confirms the high slope efficiency achievable in the 1.5 km fiber and the importance of the fiber length. Finally, Figure 4 shows the output spectrum from the 1.5 km fiber at full power. There is no sign of the second Stokes. The emission wavelength was 1019 nm for a Stokes shift of approximately 443 cm\({}^{-1}\) and a quantum defect of only 4.3\(\%\). The measured linewidth was resolution-limited (2 nm).