## 5 Numerical Results and Discussion
### Results of the Webb and Gaussian Modeled SER  
For performance analysis of APD-based optical communication systems, many papers have shown that the Webb model is more precise than the Gaussian model [33; 35; 49]. Of course, the Gaussian model is also widely used for performance analysis due to the simplicity of expressions [36; 37; 38; 39; 40; 41]. Here, we first investigate whether the Gaussian model is precise enough to characterize the considered deep space optical uplink system. In particular, we compare the SER estimated by Webb model and Gaussian model, with the Webb-estimated SER as the performance benchmark. SER performance is studied under various channel conditions. The applicable conditions where the Gaussian model can give a close fit to the Webb model, i.e., having a low approximation error, are discussed. Figure 5a shows comparisons of SER using Webb and Gaussian models under different average background noise and over the lossless link, where APD gain is \(40\). Parameters \(\bar{n}_{b}\) and \(\bar{n}_{s}\) denote the average number of background noise photons and received optical photons in a PPM symbol, respectively. Under both Webb and Gaussian models, SER  
Figure 4: Channel distribution under various: (**a**) atmospheric and coronal turbulence, (**b**) pointing errors, and (**c**) AOA fluctuations. decreases monotonically with increasing received photons, i.e., signal-to-noise ratio (SNR), aligning with expectations. In addition, SER exhibits an inverse dependence on the background noise level. In particular, under \(\bar{n}_{b}=1\), the Gaussian approximation achieves remarkable congruence with the Webb model, validating the efficacy of the Gaussian model in noise-dominated regimes as analyzed in Section 3.2. At \(\bar{n}_{b}=0.01\), the Gaussian model systematically underestimates SER relative to the Webb framework, highlighting its inherent limitations in high-sensitivity regimes. However, in this scenario, the Gaussian model can still be utilized to evaluate the system performance. This is because the communication performance estimated by the Gaussian model is worse than the actual system performance. Consequently, during system design, to achieve specific communication performance targets, the designed system redundancy (referring to the performance margin, such as additional power or error correction capability) becomes higher than the actual system redundancy. For instance, the actual received signal power might be greater than pre-research design assumptions. Consequently, over the lossless link, Webb with high accuracy and Gaussian with low complexity are both applicable in optical communications. In the fading channel, comparisons of SER using Webb and Gaussian models with APD gain being 40 are shown in Figure 5b. It can be seen that, when average received photons are given, the SER becomes worse as pointing errors, AOA fluctuations, and atmospheric and coronal turbulence intensity increase. Moreover, in Figure 5b, SER decreases with the increase in the average received photons under various \(\sigma_{ht}^{2}\), \(\sigma_{hs}^{2}\), and \(\sigma_{p}\), but in Figure 5c, the SER first decreases, and then remains constant, which means average received photons reaching saturation point. This is because \(\sigma_{\theta}\) can cause an optical receiving failure, which further results in SER error floors. In other words, the error floor level is decided primarily by \(\sigma_{\theta}\). In particular, the error floor increases with increasing \(\sigma_{\theta}\). Moreover, when the error floor is not present in Figure 5b,c, Gaussian model estimated SER is smaller than Webb model estimated SER under a given average received photons, which differs from Figure 5a. That is, communication redundancy estimated by the Gaussian model is higher than that by the Webb model. The actual redundancy of APD-based communication system can be evaluated by the Webb model, so the Gaussian model overestimates communication performance and no longer applies under the circumstances of Figure 5b,c.