## 2 Materials and Methods
### Working Principle of Multi-Beam Position Detection  
QD is a photoelectric position detector which is composed of four identical photodiodes. When incident light strikes the photosensitive surface of the QD, the four quadrants will output corresponding photocurrents. The photocurrent is proportional to the energy of the beam. Suppose there are \(M\) beams irradiating the \(i\)-th quadrant of the QD at the same time, the energy density of the \(j\)-th Gaussian beam incident on the detector target surface is \(p_{ij}(x,y)\)[22], and the corresponding photocurrent is \(l_{ij}\). \[p_{ij}(x,y)=\frac{2p_{0}}{\pi\omega^{2}}\exp\left\{-\frac{2\left[\left(x-x_{j }\right)^{2}+\left(y-y_{j}\right)^{2}\right]}{\omega^{2}}\right\} \tag{14}\]  
where \(p_{0}\) is the total energy of the Gaussian beam, \(\omega\) is the beam waist radius of the Gaussian spot, and \(\left(x_{j},y_{j}\right)\) is the coordinate of the spot center. Then, the total photocurrent \(l_{i}\) received by the \(i\)-th quadrant of the detector can be recorded as [23]. \[l_{i}=\sum_{j=1}^{M}p_{ij}(x,y)dxdy=\sum_{j=1}^{M}\eta\iint\limits_{\Sigma^{i} }p_{ij}(x,y)dxdy=\sum_{j=1}^{M}l_{ij},\ i=A,B,C,D \tag{15}\]  
Here, \(\eta\) is the photoelectric conversion efficiency, and \(\sum_{i}\) is the integration area of the \(i\)-th quadrant. Equation (15) shows that the photocurrent output by QD in response to multiple light beams is linearly superposed. If single-frequency signals of different intensities are used to intensity modulate the incident beam separately, the photocurrent signal output by the detector is a superposition of single-frequency signals of different intensities. Assume that the photocurrent signal values generated in each quadrant after the \(j\)-th laser light which illuminates the detector are \(I_{A_{j}}\), \(I_{B_{j}}\), \(I_{C_{j}}\), \(I_{D_{j}}\), respectively. After FFT operation, the amplitude value are \(A_{A_{j}}\), \(A_{B_{j}}\), \(A_{C_{j}}\), \(A_{D_{j}}\), \(\hat{x}_{j}\) and \(\hat{y}_{j}\) represent the offset of the center position of the target spot measured by the \(j\)-th laser light relative to the center of the QD on the x-axis and y-axis, which can be obtained according to the addition and subtraction algorithm theory [24]:  
\[\left\{ \begin{array}{l}\hat{x}_{j}=\frac{\left(I_{A_{j}}+I_{D_{j}}\right)- \left(I_{B_{j}}+I_{C_{j}}\right)}{I_{A_{j}}+I_{B_{j}}+I_{C_{j}}+I_{D_{j}}}= \frac{\left(A_{A_{j}}+A_{D_{j}}\right)-\left(A_{B_{j}}+A_{C_{j}}\right)}{A_{A_ {j}}+A_{B_{j}}+A_{C_{j}}+A_{D_{j}}}\\ \hat{y}_{j}=\frac{\left(I_{A_{j}}+I_{B_{j}}\right)-\left(I_{D_{j}}+I_{C_{j}} \right)}{I_{A_{j}}+I_{B_{j}}+I_{C_{j}}+I_{D_{j}}}=\frac{\left(A_{A_{j}}+A_{B_{ j}}\right)-\left(A_{D_{j}}+A_{C_{j}}\right)}{A_{A_{j}}+A_{B_{j}}+A_{C_{j}}+A_{D_{j}}} \end{array} \right. \tag{16}\]  
\(\hat{x}_{j}\) and \(\hat{y}_{j}\) are the relative position information of the spot center solved by Equation (16) under the ideal situation without introducing noise. In the position detection system, the x-axis and the y-axis are independent and symmetrical. So, this paper only derives the mathematical model of position detection in the x-axis direction. In the actual detection process, the noise at the QD output signals includes thermal noise, shot noise of the detector, and the noise introduced by light, which are all Gaussian white noise. And the power spectral density can be regarded as evenly distributed over the entire frequency band, which is independent of each other and not affected [25; 26].