## 2 Optical Uplink Channel Model
### Statistical Properties of Channel Fading Factors  
#### 2.1.1 Atmospheric and Coronal Turbulence-Induced Beam Scintillation  
Turbulence will cause optical intensity fluctuations, i.e., beam scintillation. Numerous statistical models for beam scintillation have been proposed, such as Log-normal, Gamma-Gamma, K-distribution, Exponentiated Weibull, Fisher-Snedecor, and Malaga distribution. We adopt the commonly used Gamma-Gamma model, whose PDF is defined as [14]  
\[f_{g}(x|\alpha,\beta)=\frac{2(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{\Gamma( \alpha)\Gamma(\beta)}x^{\frac{\alpha+\beta}{2}-1}\mathrm{K}_{\alpha-\beta} \left(2\sqrt{\alpha\beta x}\right), \tag{3}\]  
where \(\alpha\) and \(\beta\) represent the small-scale and large-scale turbulent eddies, respectively. \(\Gamma(\cdot)\), and \(\mathrm{K}_{\alpha-\beta}(\cdot)\) denote the standard gamma function and the modified Bessel function of the second kind of order \((\alpha-\beta)\), respectively. In addition, both \(\alpha\) and \(\beta\) are closely related to the Rytov variance \(\sigma_{g}^{2}\) of turbulence.  
For atmospheric turbulence-caused optical power fading coefficient, it is denoted as \(h_{t}\). The distribution of \(h_{t}\) is as (3), which is rewritten as \(f_{g}(h_{t}|\alpha_{t},\beta_{t})\). Rytov variance \(\sigma_{g}^{2}\) for atmospheric turbulence is rewritten as \(\sigma_{ht}^{2}\), which is given by [43]  
\[\sigma_{ht}^{2}\simeq 2.25k_{n}^{7/6}\mathrm{sec}^{11/6}\big{(}\zeta_{g}\big{)} \int_{H_{g}}^{H_{s}}\mathrm{C}_{n}^{2}(H)(H_{s}-H)^{\frac{5}{6}}dH, \tag{4}\]  
Figure 2: Block diagram of studied deep space optical communication system.  
where \(\zeta_{g}\) and \(k_{n}=2\pi/\lambda\) are the zenith angle and the optical wave number with \(\lambda\) being the wavelength, respectively. In addition, \(H,H_{s}\), and \(H_{g}\) denote the height parameter, satellite height, and ground station altitude, respectively. \(\mathcal{C}_{n}^{2}(H)\) is the structure parameter for refractive index fluctuation.  
Coronal turbulence-caused fading coefficient is expressed as \(h_{s}\). Similarly, the PDF of \(h_{s}\) is as (3) and rewritten as \(f_{g}(h_{s}|\alpha_{s},\beta_{s})\). Bytov variance \(\sigma_{g}^{2}\) for coronal turbulence is rewritten as \(\sigma_{hs}^{2}\), which takes the form [9]  
\[\sigma_{hs}^{2}\simeq\frac{1}{2}\tau_{e}^{2}(2\pi)^{\frac{p_{l}-3}{2}}\pi(3-p_{ l})\Gamma\Big{(}\frac{p_{l}}{2}\Big{)}\Gamma\bigg{(}\frac{p_{l}-1}{2}\Big{)} \Gamma\bigg{(}1+\frac{p_{l}}{2}\Big{)}\zeta_{s}^{2}N_{e}^{2}L_{o}^{3-p_{l}}(L _{l}+\lambda)^{1+\frac{p_{l}}{2}}\sec\Big{(}\frac{\pi p_{l}}{4}\Big{)}, \tag{5}\]  
where \(r_{e}\), \(p_{l}\), and \(\zeta_{s}\) represent the classical electron radius, spectral index, and relative solar wind density fluctuations ratio, respectively. In addition, \(N_{e}\), \(L_{o}\), and \(L_{l}\) represent the solar wind density, outer scale of coronal turbulence, and communication distance, respectively.  
The parameters of \(\sigma_{ht}^{2}\) and \(\sigma_{hs}^{2}\) are significant for the intensity of atmospheric and coronal turbulence, respectively. The typical values of \(\sigma_{ht}^{2}\) and \(\sigma_{hs}^{2}\) are analyzed in Appendix A.1.