## 4 Numerical Results  
In this section, we present the numerical results of an analysis of the angular positioning performance of two beacon detectors (CMOS and QAPD) over log-normal tracking channels in terms of centroid error and fade probability. The numerical simulation was performed by considering two detectors onboard a small satellite placed at an altitude (\(H+h_{s}\)) of 500 km and a zenith angle \(\theta_{z}\) of 70\({}^{\circ}\), leading to a distance \(z\) of 1193 km from an optical ground station located at a height \(h_{s}\) of 100 m. The station transmits a beacon laser to the satellite with a wavelength \(\lambda\) of 850 nm, the transmitting power \(P_{T}\) of 10 W [19; 43], the transmitting optical efficiency \(\eta_{T}\) of 0.95, the divergence \(\theta_{D}\) of 2.5 mrad in half-angle, and the pointing error \(\theta_{e}\) of 1.5 mrad, leading to a received power loss of 3 dB. Taking a clear sky into account, the atmospheric attenuation coefficient and cirrus cloud thickness were \(\sigma(\lambda,V,0)=0.25\) and \(C_{T}=  
is larger than that of the CMOS detector due to the high quantum efficiency. However, the SNR of the QAPD detector was higher than that of the CMOS detector in the short exposure time regime, and vice versa in the long exposure time regime. The results come from the fact that the noise counts (i.e., background, dark current, and thermal noises) are significantly dominated by the exposure time for the QAPD detector, while the readout noise is independent of the exposure time for the CMOS detector excluding background and dark current noises. Given the optical configuration, the centroid error is influenced by not only noise sources, but also the finite delay time causing the temporal evolution of atmospheric turbulence. Note that the exposure time latency that causes the finite delay time is taken into consideration in this simulation. Figure 3 shows the average centroid errors due to noise sources over log-normal tracking channels, and Figure 4 shows the centroid errors due to the exposure time latency, for two atmospheric turbulence conditions (i.e., \(A_{0}=1.0\times 10^{-16}\,\mathrm{m}^{-2/3}\) and \(v=5\) m/s for the weak turbulence condition, \(A_{0}=1.7\times 10^{-14}\,\mathrm{m}^{-2/3}\) and \(v=21\) m/s for the moderate turbulence condition). Applying the average wind speed \(\overline{v}\) of 15 m/s, these two turbulence conditions provide values of the scintillation index, coherence length, and time (i.e., \(\sigma_{si}^{2}=0.28\), \(r_{0}=14.1\) cm and \(t_{0}=2.9\) ms for the weak turbulence condition, and \(\sigma_{si}^{2}=0.79\), \(r_{0}=7.4\) cm and \(t_{0}=1.6\) ms for the moderate turbulence condition). The average centroid error over log-normal tracking channels was obviously smaller under the weak turbulence condition than the moderate condition. Compared to the CMOS detector, in terms of the average centroid error, the QAPD detector showed better performance regardless of the exposure time, even though its SNR was lower than the CMOS detector in the long exposure time regime, as shown in Figure 2.  
As shown in Equation (20), the centroid error resulting from the exposure time latency is totally unrelated to the detector types, but depends on exposure time, coherence length, and diffraction-limited optics (\(\lambda/D\)). The centroid error resulting from the exposure time latency increases significantly as the exposure time is extended, and atmospheric turbulence becomes stronger because the coherence length is inversely proportional to the strength of atmospheric turbulence. The total centroid errors are shown in Figures 5 and 6 for weak and moderate atmospheric turbulence conditions, respectively. The average centroid error due to noise sources was inversely proportional to exposure time, but the centroid error resulting from exposure  
Figure 2: Received photoelectron counts and SNRs versus the exposure time under atmospheric turbulence with \(A_{0}=1.7\times 10^{-14}\,\mathrm{m}^{-2/3}\) and \(v=21\) m/s.