## 2 Materials and Methods
### Working Principle of Multi-Beam Position Detection
Taking the noise into account, the position solution is solved by Equation (17):  
\[\hat{x}_{j} =\frac{\left(I_{A_{j}}+I_{nA}+I_{D_{j}}+I_{D}\right)-\left(I_{B_ {j}}+I_{nB}+I_{C_{j}}+I_{nC}\right)}{I_{A_{j}}+I_{B_{j}}+I_{C_{j}}+I_{nA}+I_{ nB}+I_{nC}+I_{nD}} \tag{17}\] \[=\left(\frac{I_{A_{D_{j}}}-I_{BC_{j}}}{I_{AD_{j}}+I_{BC_{j}}}+ \frac{I_{nAD}-I_{nBC}}{I_{nAD}+I_{nBC}}\right)\cdot\left(\frac{1}{1+\frac{I_{ nAD}+I_{nBC}}{I_{AD_{j}}+I_{nB}}}\right)\] \[\approx\frac{I_{AD_{j}}-I_{BC_{j}}}{I_{AD_{j}}+I_{BC_{j}}}+2\cdot \frac{I_{BC_{j}}I_{nAC}-I_{AD_{j}}I_{nBC}}{\left(I_{AD_{j}}+I_{BC_{j}}\right)^ {2}}\]  
where \(I_{AD_{j}}=I_{A_{j}}+I_{D_{j}}\) represents the sum of useful photocurrents in the first and fourth quadrants, and \(I_{BC_{j}}=I_{B_{j}}+I_{C_{j}}\) represents the sum of useful photocurrents in the second and the third quadrants. \(I_{nAD}=I_{nA}+I_{nD}\) represents the sum of noise photocurrent in the first and fourth quadrants, and \(I_{nBC}=I_{nB}+I_{nC}\) represents the sum of noise photocurrent in the second and third quadrants. \(I_{nA}\), \(I_{nB}\), \(I_{nC}\), \(I_{nD}\) represent the noise currents in the four quadrants. The variance \(\sigma_{i_{mn}}^{2}\) of the spot relative to position \(\hat{x}_{j}\) is used to characterize the effect on the position detection accuracy:  
\[\sigma_{i_{mn}}^{2} =2\cdot\frac{\left(I_{AD_{j}}\right)^{2}+\left(I_{BC_{j}}\right)^ {2}}{I^{4}}\cdot\left(I_{nAD}^{2}+I_{nBC}^{2}\right) \tag{18}\] \[=k\cdot\frac{I_{n}^{2}}{I^{2}}\]  
where \(I^{2}\) represents the total power of useful photocurrent, \(I_{n}^{2}=I_{nAD}^{2}+I_{nBC}^{2}\) represents the total current noise power, and \(R_{SN}=I^{2}/I_{n}^{2}\) represents the system SNR. And the proportional coefficient \(k\) can be written as:  
\[k=2\cdot\frac{\left(I_{AD_{j}}\right)^{2}+\left(I_{BC_{j}}\right)^{2}}{I^{2}} \tag{19}\]  
The \(\hat{x}_{j}\) solved by Equation (17) is the relative position of the spot. The actual position \(\left(x_{j},y_{j}\right)\) of the spot needs to be solved according to the distribution model of the spot. Assume that the energy distribution model of the laser spot is a Gaussian spot and \(\omega\) is the spot radius of the Gaussian spot; when the spot position is near the center of the detector and \(x_{j},y_{j}\ll\omega\), the relative position value is substituted into the error function and the first-order approximation is made, then the actual position can be expressed as:  
\[x_{j}\approx\frac{\sqrt{\pi}\omega}{2\sqrt{2}}\hat{x}_{j} \tag{20}\]  
The standard deviation \(\sigma_{j_{xn}}\) and \(\sigma_{j_{pn}}\) of the actual value of the position detection is considered to be the standard to judge the accuracy of position detection in the process of position detection. The standard deviation of the position detection in the x-axis direction can be obtained by Equations (13), (18) and (20):  
\[\sigma_{j_{xn}}\approx\frac{\sqrt{\pi}\omega}{2\sqrt{2}}\cdot\frac{1}{\sqrt{ \frac{NK_{SN}}{2}}}\cdot\sqrt{1+erf^{2}\Bigg{(}\frac{\sqrt{2}x_{j}}{\omega} \Bigg{)}} \tag{21}\]  
A mathematical model of the position detection accuracy of the multi spot of the QD can be established. From Equation (21), it can be seen that the position detection accuracy is related to the number of FFT points \(N\), the Gaussian spot radius \(\omega\), the actual position value of the spot centroid position \(\left(x_{j},y_{j}\right)\), and the system SNR \(R_{SN}\).