## 3 The Evolution of MI in Long Optical Fiber  
The split-step Fourier method is adopted to simulate the evolution of MI in 50 km standard single-mode fiber. The ratio of the power at the central frequency versus the total transmission power (R) is studied by simulating the NLSE. The parameters are set as \(\gamma\approx 1.3\) W\({}^{-1}\)km\({}^{-1}\), \(\alpha=-1.9\) dB/km and \(\beta_{2}\approx-21\) ps\({}^{2}\)/km. The ASE noise is set as \(-130\) dB/Hz. The pulse duration is set as 200 ns, and the repetition rates range from tens to hundreds of kHz which presents typical parameters of the optical pulses in remote interferometric fiber sensing systems. The evolution of R along the transmission distance with different input power is shown in Figure 1. When the input power is lower than the MI threshold (the MI threshold is about 200 mW), almost all transmission power remains at the central frequency along the optical fiber. When the input power exceeds the MI threshold, MI occurs and transmission power gradually converts from the central frequency to MI sidebands (300 mW). It can be seen that the power remaining at the central frequency gradually decreases with the increase in input power. When input power further increases, an oscillation evolution with distance can be observed. We also simulate the evolution of R along the optical fiber with different ASE noise levels at an input power of 300 mW. The results are shown in Figure 2. We can see that higher ASE noise level leads to more serious pump power depletion. Notably, we notice that significant oscillation styles are shown in both Figures 1 and 2, which is related to the famous Fermi-Pasta-Ulam (FPU) recurrence in optical fiber [18]. It can be seen that the oscillation gets more and more drastic with the increase in the input power and ASE noise level. In addition, the oscillation is very drastic at the front end of the fiber. As the light further propagates, the oscillation becomes increasingly gentle. Apparently, MI seriously depletes the power remaining at the central frequency. Considering that the SBS threshold in the presence of MI is mainly determined by the power remaining at the central frequency, the pump depletion induced by MI decreases the interaction between the Brillouin pump and Stokes waves. Figure 1: The relationship of R versus the transmission distance with different input powers. **4. The Interaction Process between the Brillouin Pump and Stokes Waves for Optical Pulses**  
By solving Equation (4), the average Stokes power is obtained in the following equation:  
\[I_{S}(0,t)=I_{S}(z,t)\cdot\exp\{-\alpha z+\frac{1}{A_{eff}K}(\int_{L}^{0}I_{p1}(z,t)g_{B}(z,t)dz)dt\} \tag{6}\]  
in which \(T\) is the period of the optical pulses and \(L\) is the fiber length. The solution of Equation (6) presents the average Stokes power for optical pulse. The average Brillouin gain can be defined as:  
\[G=\int_{0}^{T}\int_{L}^{0}I_{p}(z,t)g_{B}(z,t)dzdt/(A_{eff}KT) \tag{7}\]  
For continuous light, the Brillouin gain is expressed as:  
\[G=[g(1-\exp(-\alpha L))/\alpha\cdot I_{p}]/(K\cdot A_{eff}) \tag{8}\]  
For periodic square optical pulses, which are widely used in a remote interferometric fiber sensing system:  
\[I_{p}(z,t)=I_{p0}\underset{m}{\sum}rect[z-t_{p}c/2-(mTc-tc)/n]\exp(-\alpha z) \tag{9}\]  
in which \(t_{p}\) is the pulse duration and n represents the refractive index. In the Brillouin scattering process, the interaction between the pump and backscattering Stokes waves determines the Brillouin gain. As shown in Figure 3, the backward Stokes light that initiates at the end of the optical fiber reaches the forepart of the fiber in \(t=Ln/c\). The Stokes light can be amplified when overlapping with the repeated optical pulses. So only the pump light within the range of \(2L\) can interact with the Stokes light that initiates at the end. When MI is not considered and the pump depletion induced by Brillouin scattering is neglected, the input light experiences linear loss in the optical fiber.