# Analysis and Reduction of Nonlinear Effects in Optical Fiber Frequency Transfer
## 3 Simulation and Results
### Raman Scattering  
Utilizing Equation (3), we calculate the threshold power \(P_{crit-R}\) for generating Raman scattering as a function of the length \(L\) of the optical fiber, considering an effective mode area \(A_{eff}\) of 1.0 \(\times\) 10\({}^{-10}\) m\({}^{2}\), an attenuation coefficient \(\alpha\) of 0.2 dB/km, and a gain coefficient \(\gamma_{R}\) for Raman scattering of 4.2 \(\times\) 10\({}^{-13}\) m/W. The results of this calculation are illustrated in Figure 6.  
As shown in Figure 6, when the fiber length \(L\) varies from 0.1 km to 1 km, the threshold powers, including \(P_{crit-R-B}\) for generating backward Raman scattering and \(P_{crit-R-F}\) for generating forward Raman scattering, are consistently in the vicinity of 10 W. This value is roughly two orders of magnitude greater than the threshold powers for Brillouin scattering at the same fiber length. This discrepancy can be attributed to the significantly higher gain coefficient of Brillouin scattering, which is approximately two orders of magnitude larger than the gain coefficient \(\gamma_{R}\) of Raman scattering. The threshold powers for generating Raman scattering are approximately 1 W for fiber lengths ranging from 1 km to 100 km. In cases where the fiber length \(L\) is fixed, \(P_{crit-R-B}\) for backward Raman scattering is approximately 25% higher than that for forward Raman scattering. Generally, the threshold powers for Brillouin scattering exhibit a decreasing trend as the fiber length \(L\) increases. For instance, for a 0.5 km fiber length, \(P_{crit-R-F}\) and \(P_{crit-R-B}\) are approximately 8 W and 10 W, respectively, and they decrease by approximately 3.8 W and 4.8 W, respectively, as the fiber length increases. However, for fiber lengths exceeding 10 km, the threshold powers for generating Raman scattering remain relatively constant, paralleling the behavior observed in Brillouin scattering. When the optical fiber's length is 10 km, Figure 7a displays the calculated threshold powers for generating Raman scattering as a function of the attenuation coefficient \(\alpha\) of the optical fiber. It is observed that as the attenuation coefficient \(\alpha\) of the optical fiber increases, the threshold powers for generating Raman scattering also increase. For example, when the attenuation coefficients are 1 dB/km and 2 dB/km, the threshold powers for backward Raman scattering are approximately 4.8 W and 9.5 W, respectively, while the threshold powers for forward Raman scattering are approximately 3.8 W and 7.6 W, respectively. Furthermore, when the attenuation coefficient \(\alpha\) of the optical fiber remains constant at 0.5 dB/km, the threshold power \(P_{crit-R-B}\) for backward Raman scattering surpasses that of forward Raman scattering, with values of about 2.4 W and 1.9 W, respectively. Figure 6: The threshold power for generating the Raman scattering as a function of the length \(L\) of the optical fiber. (**a**) The fiber with less than 1 km in length. (**b**) The fiber with the length from 1 km to 100 km. As shown in Figure 7b, the threshold powers of the Raman scattering as a function of the gain coefficient \(\gamma_{B}\) of the Raman scattering are presented, where the length \(L\) of the fiber is 10 km. In general, the threshold power \(P_{crit-R}\), including the backward Raman scattering and the forward Raman scattering, decreases as the gain coefficient \(\gamma_{R}\) of the Raman scattering increases. For instance, consider two scenarios with different gain coefficients for Raman scattering: 1 \(\times\) 10\({}^{-13}\) m/W and 1 \(\times\) 10\({}^{-12}\) m/W, resulting in average threshold powers of approximately 4.2 W and 0.4 W, respectively. When the gain coefficient \(\gamma_{R}\) for Raman scattering is established, the threshold power \(P_{crit-R-F}\) for forward Raman scattering is found to be lower than that for backward Raman scattering.