## 2 Theoretical Noise Model  
We first model the intensity noise in MOPA with feedback-loop-based intensity noise suppression. Figure 1 shows a schematic of the intensity noise suppression feedback loop. The seed laser is amplified by an all-fiber amplifier, and a fraction (\(\varepsilon\)) of the amplified laser \(P_{a}\) is directed to a photodetector (PD\({}_{\text{in}}\)) to generate the error signal for the proportional-integral-derivative (PID) controller. The current of the amplifier pump LD changed in real time according to the control signal to suppress the temporal power fluctuations of the amplifier output. The components in the feedback loop, i.e., the pump LD, the fiber splitters, the PD\({}_{\text{in}}\), and the PID controller, induce excess noise in the laser output. The seed power can be described as  
\[P_{s}=\overline{P}_{s}+\delta P_{s}, \tag{1}\]  
where \(\overline{P}_{s}\) and \(\delta P_{s}\) stand for the average and fluctuations of the seed laser, respectively. The amplified laser can be written as  
\[P_{a}=GP_{s}+\delta P_{f}=G\cdot\overline{P}_{s}+G\cdot\delta P_{s}+\delta P_{ f}, \tag{2}\]  
where \(G\) and \(\delta P_{f}\) stand for the gain of the amplifier and the fluctuations induced by the feedback loop, respectively. The amplified laser is then divided into two beams by a fiber splitter with a coupling ratio \(\varepsilon\). The in-loop and out-of-loop power can be given by  
\[P_{in}=(\varepsilon-\delta\varepsilon)P_{a}=\varepsilon P_{a}-\delta \varepsilon P_{a}=\varepsilon P_{a}-\delta P_{\varepsilon}, \tag{3}\]  
\[P_{out}=(1-\varepsilon+\delta\varepsilon)P_{a}=(1-\varepsilon)P_{a}+\delta \varepsilon P_{a}=(1-\varepsilon)P_{a}+\delta P_{\varepsilon}, \tag{4}\]  
where \(\delta\varepsilon\) and \(\delta P_{\varepsilon}\) stand for the coupling ratio fluctuations and the power fluctuations induced by the fiber splitter, respectively. In the feedback loop, the photocurrent \(I_{in}\) can be expressed as  
\[I_{in}=(\Re-\delta\Re)P_{in}=\Re P_{in}-\delta\Re P_{in}=\Re P_{in}-\delta I_ {pd}, \tag{5}\]  
\[I_{in}=\Re\varepsilon G\cdot\overline{P}+\Re\varepsilon G\cdot\delta P_{s}+ \Re\varepsilon\cdot\delta P_{f}-\Re\cdot\delta P_{\varepsilon}-\delta I_{pd}, \tag{6}\]  
where \(\Re\) and \(\delta\Re\) represent the responsivity and responsivity fluctuations of the photodetector PD\({}_{\text{in}}\), respectively, and \(\delta I_{pd}\) represents the photocurrent fluctuations induced by the photodetector PD\({}_{\text{in}}\). Therefore, we can obtain  
\[\delta I_{in}=\Re\varepsilon G\cdot\delta P_{s}+\Re\varepsilon\cdot\delta P_{ f}-\Re\cdot\delta P_{\varepsilon}-\delta I_{pd}. \tag{7}\]  
Figure 1: Schematic for feedback-loop-based intensity noise suppression in the fiber amplifier. Although the fluctuations in the amplifier may depend on the transfer function of the feedback loop, we assume that the power fluctuations \(\delta P_{f}\) is frequency-independent [22], for simplicity. This means that \(\delta P_{f}=-K\delta I_{in}\). Consequently, we can get  
\[\delta P_{f}=\frac{-K\Re\epsilon G\cdot\delta\hat{P}_{s}+K\Re\cdot\delta P_{ \epsilon}+K\cdot\delta I_{pd}}{1+K\Re\epsilon}. \tag{8}\]  
Combining Equations (2) and (8), we obtain the expression for the total fluctuations of the amplified laser \(\delta P_{a}\) as  
\[\delta P_{a}=G\cdot\delta P_{s}+\delta P_{f}=\frac{G\cdot\delta P_{s}+K\Re \cdot\delta P_{\epsilon}+K\cdot\delta I_{pd}}{1+K\Re\epsilon}. \tag{9}\]  
Then, combining Equations (3), (4) and (9), we obtain the expression for the total fluctuations of the in-loop \(\delta P_{in}\) and out-of-loop \(\delta P_{out}\) as  
\[P_{a}=GP_{s}+\delta P_{f}=G\cdot\overline{P}_{s}+G\cdot\delta P_{s}+\delta P_{ f}, \tag{2}\]0  
\[P_{a}=GP_{s}+\delta P_{f}=G\cdot\overline{P}_{s}+G\cdot\delta P_{s}+\delta P_{ f}, \tag{2}\]1  
Equations (10) and (11) show that the total fluctuations of the in-loop and out-of-loop are quite different.