## 3 Centroid Error Analysis
### Centroid Error under Atmospheric Turbulence  
Regardless of sensor type (i.e., image sensors or position-sensitive detectors), the centroid error is dominated by not only the angular size of the beacon image, but also SNR. Assuming there is no intensity fluctuation due to atmospheric turbulence, the instantaneous centroid error of a CMOS detector is expressed as [21; 22]  
\[\sigma_{iCMOS}^{noise}=\frac{1}{\gamma_{iCMOS}}\sqrt{\frac{N(N+1)}{3}}iFoV= \frac{\sqrt{K_{s}+K_{nC}}}{K_{s}}\sqrt{\frac{N(N+1)}{3}}iF_{0}V, \tag{16}\]  
where \(iFoV\) is the FoV of a single pixel; \(N\) is the truncated half width of a centroid window with the relation \(N_{p}=\left(2N+1\right)^{2}\); and \(K_{nC}\) is defined as \(N_{p}\big{(}N_{b}T_{s}+N_{d}T_{s}+N_{r}^{2}\big{)}\). Assuming diffraction-limited optics, the instantaneous centroid error of a QAPD detector is, in the absence of intensity fluctuation, given by [20; 39]  
\[\sigma_{iQAPD}^{noise}=\frac{3\pi}{16}\frac{1}{\gamma_{iQAPD}}\frac{\lambda} {D}=\frac{3\pi}{16}\frac{\lambda}{D}\frac{\sqrt{FK_{s}+K_{nQ}}}{K_{s}}, \tag{17}\]  
where \(D\) is the diameter of the receiving aperture, and \(K_{nQ}\) is defined as \(4\big{(}FG^{2}N_{b}T_{s}+N_{d}T_{s}+\sigma_{n}^{2}T_{s}/q^{2}\big{)}/G^{2}\). If the optics is not diffraction-limited, Equation (17) should be modified by replacing the factor \(\left(2.44\lambda/D\right)\) with the actual spot size in angular diameter [40]. However, the beacon beam does experience intensity fluctuation during atmospheric propagation, so the centroid error also fluctuates due to the varying SNR. Thus, the probability density function should take into account the intensity fluctuation. With some mathematical effort, the average centroid error of the CMOS detector can be written as  
\[ \begin{split}\sigma_{CMOS}^{noise}&=\int_{0}^{\infty} \sigma_{ICMOS}f(K_{s})dK_{s}\\ &=\sqrt{\frac{N(N+1)}{3}}\frac{iFoV}{\sqrt{2\pi c_{k}^{2}}}\int_{ 0}^{\infty}\frac{\sqrt{FeV_{s}+K_{m_{k}}}}{K_{s}^{2}}\exp\left(-\frac{\left( \ln K_{s}-m_{s}\right)^{2}}{2\sigma_{k}^{2}}\right)dK_{s},\\ &\approx\sqrt{\frac{N(N+1)}{3}}\frac{iFoV}{\sqrt{\pi}}\sum\limits _{i=1}^{n}w_{i}\frac{\sqrt{e^{\sqrt{2}}x_{i}+m_{k}+K_{m_{k}}}}{e^{\sqrt{2}}x_{ i}+m_{k}}\end{split}  \tag{18}\]  
where \(w_{i}\) and \(x_{i}\) denote the weight factors and the zero points of the nth-order Hermite polynomial, respectively, whose values are well tabulated in [37]. The order of Hermite polynomial should be chosen depending on the desired accuracy, which was taken as 20 in the numerical simulation. Following a procedure similar to Equation (18), the average centroid error of the QAPD detector can also be written as  
\[\sigma_{QAPD}^{noise}\approx\frac{3\lambda\sqrt{\pi}}{16D}\sum\limits_{i=1}^{ n}w_{i}\frac{\sqrt{Fe^{\sqrt{2}}x_{i}+m_{k}}+K_{nQ}}{e^{\sqrt{2}}x_{i}+m_{k}}. \tag{19}\]  
There is a finite delay time between the beacon sensing and the pointing of the optical beam toward the partner terminal. Due to the temporal evolution of atmospheric turbulence, the finite delay time also produces a centroid error, which involves the detector exposure time, the detector readout time, and the control-loop response time [41]. Taking only the exposure time latency into consideration in this study, the centroid error is given by [23; 24]  
\[\sigma^{time}=0.177\frac{T_{s}}{t_{0}}\left(\frac{r_{0}}{D}\right)^{1/6}\frac{ \lambda}{D}, \tag{20}\]  
where \(r_{0}\) and \(t_{0}\) are the atmospheric coherence length called the Fried parameter, and the coherence time, respectively. The coherence time is reported to be on the order of milliseconds. For a spherical wave, they are written by [12; 42]  
\[r_{0}=\left[0.423k^{2}\sec\theta_{z}\int_{h_{s}}^{h_{b}+H}C_{h}^{2}(h)\left( \frac{H+h_{s}-h}{H}\right)^{5/3}dh\right]^{-3/5}, \tag{21}\]  
\[t_{0}=0.314\frac{r_{0}}{\overline{v}}, \tag{22}\]  
where \(\overline{v}\) is the average wind speed over different atmospheric layers.