# Analysis and Reduction of Nonlinear Effects in Optical Fiber Frequency Transfer
## 3 Simulation and Results
### Stimulated Brillouin Scattering  
According to Equation (2), we assess the threshold power of generating the Brillouin scattering as a function of the optical fiber's length \(L\) with the various effective mode area \(A_{eff}\), including \(1\times 10^{-11}\) m\({}^{2}\), \(5\times 10^{-11}\) m\({}^{2}\), and \(1\times 10^{-10}\) m\({}^{2}\). The corresponding results for the threshold power \(P_{crit-B}\) are presented in Figure 3.  
In general, for fiber lengths of less than 1 km, the threshold power \(P_{crit-B}\) typically falls in the range of 100 mW. However, for fiber lengths ranging from 1 km to 100 km, the threshold powers for Brillouin scattering are typically around 10 mW. When the effective mode area \(A_{eff}\) of the fiber remains constant, the threshold powers for Brillouin scattering decrease as the fiber length increases. For instance, at fiber lengths of 100 m and 500 m, the threshold powers for Brillouin scattering with an effective mode area of 1 \(\times\) 10\({}^{-10}\) m\({}^{2}\) are approximately 424 mW and 88 mW, respectively. Conversely, when the fiber length exceeds 10 km, the threshold power \(P_{crit-B}\) remains relatively constant. Specifically, it hovers around 1 mW for an effective mode area \(A_{eff}\) of 1 \(\times\) 10\({}^{-11}\) m\({}^{2}\), approximately 4 mW for \(A_{eff}\) of 5 \(\times\) 10\({}^{-11}\) m\({}^{2}\), and about 8 mW for \(A_{eff}\) of 1 \(\times\) 10\({}^{-10}\) m\({}^{2}\). These calculations hold potential theoretical significance for guiding laser and optical fiber sensing applications that utilize Brillouin scattering characteristics. Additionally, when the fiber length \(L\) is held constant, the threshold powers for Brillouin scattering increase with a larger effective mode area \(A_{eff}\) of the fiber. For instance, at a fixed fiber length of 2 km, the threshold powers for Brillouin scattering with effective mode areas \(A_{eff}\) of 1 \(\times\) 10\({}^{-11}\) m\({}^{2}\), 5 \(\times\) 10\({}^{-11}\) m\({}^{2}\), and 1 \(\times\) 10\({}^{-10}\) m\({}^{2}\) are approximately 2.6 mW, 12.7 mW, and 25.5 mW, respectively. In Figure 4a, we present the threshold power \(P_{crit-B}\) for generating Brillouin scattering as a function of the attenuation coefficient \(\alpha\) of the optical fiber when the laser source linewidth \(\Delta v_{laser}\), Brillouin scattering linewidth \(\Delta v_{B}\), and Brillouin scattering gain coefficient \(\gamma_{B}\) are set at 1 kHz, 10 MHz, and 5 \(\times\) 10\({}^{-11}\) m/W, respectively. Generally, when the fiber length \(L\) is held constant, the threshold powers for Brillouin scattering increase with higher values of the attenuation coefficient \(\alpha\) of the optical fiber. For example, at \(L\) = 50 km, the threshold powers for \(\alpha\) = 0.1 dB/km, \(\alpha\) = 1 dB/km, and \(\alpha\) = 2 dB/km are approximately 4 mW, 42 mW, and 84 mW, respectively. However, with \(L\) = 1 km and \(\alpha\) = 0.5 dB/km, the threshold power for Brillouin scattering is approximately 53 mW, which is about 30 mW higher than that for \(L\) = 10 km. Interestingly, when \(\alpha\) is varied between 0.1 dB/km and 2 dB/km, the threshold power for Brillouin scattering at \(L\) = 50 km remains essentially unchanged compared to that at \(L\) = 10 km. The calculation results depicted in Figure 4b present the threshold powers of Brillouin scattering as a function of the Brillouin scattering gain coefficient \(\gamma_{B}\). Here, the parameters include the laser source linewidth \(\Delta v_{laser}\) at 1 kHz, Brillouin scattering linewidth \(\Delta v_{B}\) at 10 MHz, and optical fiber attenuation coefficient \(\alpha\) at 0.2 dB/km. Generally, when the fiber length \(L\) is constant, the threshold power \(P_{crit-B}\) of Brillouin scattering decreases with an increasing Brillouin scattering gain coefficient \(\gamma_{B}\). For instance, at \(L\) = 1 km, the threshold  
Figure 3: The threshold power \(P_{crit-B}\) for generating the Brillouin scattering as a function of the length \(L\) of the optical fiber.