## 5 The SBS Threshold for Optical Pulses with MI Considered  
Taking the MI induced pump depletion into account, the Brillouin gain is updated as:  
\[G=\frac{1}{K\cdot A_{eff}T}\int\limits_{0}^{T}\int\limits_{L}^{0}\frac{1}{2}gI _{P0}\cdot R\cdot\sum\limits_{n}rect((z-t_{p}c/2-nTc-tc)/2) \tag{11}\] \[\cdot\exp(-\alpha(z-t_{p}c/2-nTc-tc)/2)dz\cdot dt\]  
Based on the updated Brillouin gain Equation (11), we simulate \(G\) versus the average power with different repetition rates. The result is shown in Figure 5. We define \(G=21\) (the dashed line in Figure 5) as the criterion of SBS threshold. Without considering MI (red dashed line), the Brillouin gain increases linearly with the average input power (assuming the Brillouin gain is linear), so the input power when \(G=21\) (SBS threshold) is unchanged with various repetition rates. However, the results change when MI is considered. It is found that the slope of the Brillouin gain versus the input power decreases as the input power increases, for the occurrence of MI. As a result, the input power at which \(G=21\) increases when the repetition rate is lower than a certain value (100 kHz), which agrees with the experimental results in reference [17]. Figure 4: The simulation result of the SBS threshold with different repetition rates. Figure 3: Schematic of the interaction between Brillouin pump and Stokes waves for optical pulse in optical fiber. To fully present the effect of MI on the SBS threshold, we calculate the average SBS threshold with different repetition rates and ASE noise levels. The peak SBS threshold can be calculated by the duty ratio of the optical pulses. The results are shown in Figure 6. When the repetition rates are higher than 200 kHz, the SBS threshold remains unchanged at about 4.2 mW, which is equal to the SBS threshold for continuous light. When the repetition rate decreases to 100 kHz, the SBS threshold remains constant when the ASE noise is lower than \(-125\) dB/Hz and increases with the ASE noise level when the ASE noise level is higher than \(-120\) dB/Hz. With the further decrease in the repetition rate, the ASE noise level at which SBS threshold starts increasing gradually decreases. The SBS threshold with the same ASE noise level increases. The mechanism by which MI affects the SBS threshold comes from the response mechanism difference between SBS and MI. The Brillouin scattering is a backward amplified process along the whole fiber, so the Brillouin gain mainly responds to the average input power within the optical fiber. While MI is a nonlinear effect that occurs in the forward transmission direction, the response time is at the order of \(\sim\)ps, so MI mainly responds to the peak power of the input light. In 50 km optical fiber, the SBS threshold is about 4.2 mW on average, while the MI threshold is about 150-200 mW at its peak. For continuous light, SBS occurs when the input average power is above 4.2 mW. The corresponding peak power is also 4.2 mW, which is much lower than the MI threshold. Therefore, the SBS threshold of continuous light is not affected by MI. For an optical pulse with the same average power, the lower repetition rates correspond to higher peak powers. When the repetition rate is lower than a certain value, the peak power of the input power reaches MI threshold before the average power reaches the SBS threshold of continuous light. Under these conditions, the occurrence of MI induces Brillouin pump depletion by converting the pump power to MI sidebands. So, the Brillouin gain accumulated along the optical fiber decreases correspondingly. As a result, the SBS threshold increases. In addition, Figure 6 shows that the SBS threshold increases with the ASE noise level. This is because that higher  
Figure 5: The Brillouin gain versus the average input power with different repetition rates. Figure 6: The evolution of the theoretical average SBS threshold versus ASE noise levels with different repetition rates. The pulse duration of the optical pulses is 200 ns.