## 2 Principle  
The communication performance of the system is analyzed in this section. Reference [22] calculates the outage probability in stratospheric FSOC system by Rician factor, inter-high-attitude platform distance and misalignment-induced fading. Since the UAV command and control does not need continuous communication, the connectivity probability within 1 s is calculated to characterize the communication quality. When the Gaussian beam transmitting in the turbulence, the irradiance strength at receiver is not a fixed value. The irradiance probability distribution function (PDF) is generally accepted to be lognormal, and takes the form of [23]:  
Figure 1: One-way communication link. \[\mathrm{P}(\mathrm{I})=\frac{1}{\mathrm{I}\sigma_{1}(\mathrm{r},\mathrm{L})\sqrt{2 \pi}}\exp\left\{-\frac{\left[\ln\left(\frac{\mathrm{I}}{\mathrm{I}\omega( \mathrm{I}(\mathrm{r},\mathrm{L}))}\right)+\frac{1}{2}\sigma_{1}^{2}(\mathrm{r },\mathrm{L})\right]^{2}}{2\sigma_{1}^{2}(\mathrm{r},\mathrm{L})}\right\}, \tag{1}\]  
where \(\sigma_{1}(\mathrm{r},\mathrm{L})\) is the scintillation index of Gaussian beam and \(\langle\mathrm{I}(\mathrm{r},\mathrm{L})\rangle\) is the irradiance of beam in normalized radiation medium, and the \(\mathrm{I}_{0}\langle\mathrm{I}(\mathrm{r},\mathrm{L})\rangle\) is the mean irradiance. The mean irradiance of Gaussian beam takes the form of:  
\[\langle\mathrm{I}(\mathrm{r},\mathrm{L})\rangle=\frac{\mathrm{W}_{0}{}^{2}}{ \mathrm{W}^{2}}\exp\left(-\frac{2\mathrm{r}^{2}}{\mathrm{W}^{2}}\right)\exp \left\{2.22\sigma_{1}{}^{2}\left[\frac{2\mathrm{L}}{\mathrm{k}\mathrm{W}^{2} }\right]^{\frac{5}{6}}\left[\frac{\mathrm{r}^{2}}{\mathrm{W}^{2}}\right]-1.3 3\sigma_{1}{}^{2}\left[\frac{2\mathrm{L}}{\mathrm{k}\mathrm{W}^{2}}\right]^ {\frac{5}{6}}\right\}, \tag{2}\]  
where the \(\mathrm{W}(\mathrm{L})\) is the radius of Gaussian spot and the \(\mathrm{W}_{0}\) is the waist radius. The \(\mathrm{k}\) is wave number, and the \(\sigma_{1}{}^{2}\) is the Rytov variance which has the following relationships with refractive index structure constant \(\mathrm{C_{n}}^{2}\)  
\[\sigma_{1}{}^{2}=1.23\mathrm{C_{n}}^{2}\mathrm{k}^{7/6}\mathrm{L}^{11/6} \tag{3}\]  
Based on zero inner scale model, the \(\sigma_{1}(\mathrm{r},\mathrm{L})\) has the following expression:  
\[\sigma_{1}{}^{2}(\mathrm{r},\mathrm{L})=4.42\sigma_{1}{}^{2}\Lambda_{\mathrm{ e}}{}^{5/6}\frac{\mathrm{r}^{2}}{\mathrm{W_{e}}{}^{2}}+\exp\left[\frac{0.49 \sigma_{\mathrm{B}}{}^{2}}{\left(1+0.56\sigma_{\mathrm{B}}{}^{12/5}\right)^{7 /6}}+\frac{0.51\sigma_{\mathrm{B}}{}^{2}}{\left(1+0.69\sigma_{\mathrm{B}}{}^ {12/5}\right)^{7/6}}\right]-1, \tag{4}\]  
where the \(\mathrm{W_{e}}=\mathrm{W}\left[1+1.63\sigma_{1}{}^{12/5}.\left(2\mathrm{L}/ \mathrm{k}\mathrm{W}^{2}\right)\right]\) is the effective radius of Gaussian spot under turbulence, and \(\sigma_{\mathrm{B}}{}^{2}\) is the Rytov variance for Gaussian beam wave, which can be approximately defined by:  
\[\sigma_{\mathrm{B}}{}^{2}=3.86\sigma_{1}{}^{2}\left\{0.40\left[\left(1+2\Theta \right)^{2}+4\cdot\left(\frac{2\mathrm{L}}{\mathrm{k}\mathrm{W}^{2}}\right)^{ 2}\right]^{5/12}\times\cos\left[\frac{5}{6}\tan^{-1}\left(\frac{1+2\Theta}{4 \mathrm{L}/\mathrm{k}\mathrm{W}^{2}}\right)\right]-\frac{11}{16}\left[\frac{2 \mathrm{L}}{\mathrm{k}\mathrm{W}^{2}}\right]^{5/6}\right\}. \tag{5}\]  
The parameter \(\Lambda_{\mathrm{e}}\) and \(\Theta\) shown in the equation can be written as:  
\[\Lambda_{\mathrm{e}}=\frac{2\mathrm{L}/\left(\mathrm{k}\mathrm{W}^{2}\right) }{1+1.63\sigma_{1}{}^{12/5}\cdot 2\mathrm{L}/\left(\mathrm{k}\mathrm{W}^{2} \right)}\, \tag{6}\]  
\[\Theta=1+\frac{\mathrm{L}}{\mathrm{R}(\mathrm{L})}. \tag{7}\]  
The \(\mathrm{R}(\mathrm{L})=\mathrm{L}\left[1+\left(\frac{\pi\mathrm{W}_{0}{}^{2}}{ \mathrm{L}}\right)\right]\) is the radius of curvature. Using the above formula, the PDF of irradiance is obtained. According to Reference [23], values of \(\mathrm{C_{n}}^{2}\) near the ground in warm climates generally vary between \(10^{-14}\) to \(10^{-12}\)\(\mathrm{m}^{-2/3}\).