## 3 Centroid Error Analysis  
### Signal-to-Noise Ratio  
The centroid error can be estimated based on the image size as well as SNR, which is defined as the ratio of the signal photoelectron counts to the noise fluctuation intensity during the integration time [27]. The centroid accuracy is proportional to SNR, but various noise sources during the integration time may lead to significant degradation of the centroid accuracy. In the quantitative signal analysis of beacon images on the beacon tracking detector, the centroid algorithm is used to compute the beacon location within the centroid window, which should be carefully selected so as not to deteriorate the centroid accuracy. Assuming the background noise is uniformly distributed over each pixel, the CMOS sensor's SNR corresponding to the exposure time is given by  
\[\gamma_{\text{CMOS}}(K_{s})=\frac{K_{s}}{\sqrt{K_{s}+N_{p}(N_{b}T_{s}+N_{d}T_{ s}+N_{r}^{2})}}, \tag{13}\]  
where \(N_{p}\) is the number of pixels involved in the centroid window; \(N_{d}\) is the dark current in electrons per second per pixel; \(N_{r}\) is the read noise in root-mean-square electrons per pixel; and \(N_{b}\) is the background noise in photoelectrons per second per pixel and can be calculated from \(\eta_{Q}\lambda P_{b}/\hbar cN_{FoV}\), where \(N_{FoV}\) is the number of pixels corresponding to the FPA detector's FoV, and \(P_{b}\) is the received power from background radiance. Note thatthe SNR equation for the CCD detector is exactly the same as for the CMOS detector. The background power is given by [4]  
\[P_{b}=L_{\lambda}\Omega_{FoV}\eta_{R}A_{R}\Delta\lambda_{filter}, \tag{14}\]  
where \(L_{\lambda}\) is the background spectral radiance; \(\Omega_{FoV}\) is the detector FoV in steradians; and \(\Delta\lambda_{filter}\) is the optical bandwidth.  
A quadrant position-sensitive detector consists of four individual yet identical photodiodes, and each quadrant generates an output photocurrent that is proportional to the amount of incident optical power, depending on the beam spot position and shape on the photodiode surface. Because of these properties, it provides continuous positioning data, higher positioning accuracy, and high-speed response. Unlike QPD detectors, QAPD detectors are used to detect the position of a faint beacon spot due to its high sensitivity. Assuming the responsivity and dark current are the same in each quadrant, the QAPD detector's SNR corresponding to the exposure time is derived as  
\[\gamma_{QAPD}(K_{s})=\frac{K_{s}}{\sqrt{FK_{s}+4(FG^{2}N_{b}T_{s}+N_{d}T_{s}+ \sigma_{n}^{2}T_{s}/q^{2})/G^{2}}}, \tag{15}\]  
where \(F\) is the excess noise factor, and \(G\) is the gain. \(N_{d}\) is the APD dark current consisting of two components (i.e., surface dark current \(I_{ds}\) and bulk dark current \(I_{db}\)), which can be calculated from the equation \(\left(FG^{2}I_{db}+I_{ds}\right)/q\), because the surface dark current does not trigger an avalanche process within the substrate but the bulk dark current does [38]. \(\sigma_{n}^{2}\) is the thermal noise, defined as \(2k_{B}T/R_{L}\), where \(k_{B}\) is the Boltzmann constant; \(T\) is the absolute temperature; and \(R_{L}\) is the load resistance of the detector circuit. Note that we can also get the SNR equation of the QPD detector from Equation (15) by taking \(F=1\) and \(G=1\).