# Atmospheric Effects on Satellite-Ground Free Space Uplink and Downlink Optical Transmissions
## 2 Uplink and Downlink Channels in Satellite-Ground FSO Communications
### Atmospheric Turbulence
For a plane wave(downlink), the coherence length is given as explained in [19]:  
\[r_{\rm o}=1.67\bigg{[}\sec(\theta)k^{2}\int_{h_{0}}^{H}C_{n}^{2}(h)\mathrm{dh} \bigg{]}^{-\frac{3}{5}} \tag{6}\]  
For a spherical wave(uplink), the coherence length is given as explained in [21]:  
\[r_{\rm o}=1.67\bigg{[}\sec(\theta)k^{2}\int_{h_{0}}^{H}C_{n}^{2}(h)\bigg{(} \left(1-\frac{h}{H}\right)^{5/3}\bigg{)}\mathrm{dh}\bigg{]}^{-\frac{3}{5}} \tag{7}\]  
where \(\theta\) is the zenith angle, k is the wave number, h\({}_{0}\) is the altitude of the ground station, and H is the satellite altitude. For FSO system designers, one of the main considerations when choosing an optical wavelength for transmission in an FSO system is atmospheric attenuation because the choice of wavelength determines the amount of optical signal attenuation due to the atmosphere. FSO systems employing a 1550-nanometer wavelength may transmit more than 10 times as much optical power than the systems employing 750-nanometer or 850-nanometer wavelengths [2]. From Equations (6) and (7), it is clear that coherence length depends on the wave number, which is inversely proportional to the operating wavelength. Hence, coherence length \(r_{0}\) varies with \(\lambda^{5/6}\). Therefore, a higher degree of coherence will be present in an FSO link operating at higher wavelengths than at lower wavelengths. Another important factor in the design of FSO channels is the turbulence model for scintillation. The severity of intensity fluctuations in transmission across FSO networks has been examined using a variety of models. Gamma-Gamma distribution is widely used to model fading approximation. For FSO channels with irradiance variations, the scintillation index using Gamma-Gamma distribution is given as follows [2]:  
\[\sigma_{\rm N}^{2}=\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\alpha\,\times\, \beta} \tag{8}\]  
For a plane wave (downlink channel), it is  
\[\alpha_{\rm downlink}=\left[\exp\left(\frac{0.49\,\times\,\sigma_{\rm I\,downlink }^{2}}{\left(1+1.11\,\times\,\sigma_{\rm I\,downlink}^{\frac{12}{5}}\right)^ {\frac{5}{6}}}\right)-1\right]^{-1} \tag{9}\]  
\[\beta_{\rm downlink}=\left[\exp\left(\frac{0.51\,\times\,\sigma_{\rm I\,downlink }^{2}}{\left(1+0.69\,\times\,\sigma_{\rm I\,downlink}^{\frac{12}{5}}\right)^ {\frac{5}{6}}}\right)-1\right]^{-1} \tag{10}\]  
\[\sigma_{\rm I\,downlink}^{2}=2.25k^{\frac{7}{6}}\sec(\theta)^{\frac{11}{6}} \int_{h_{0}}^{H}C_{n}^{2}(h)(h-h_{0})^{\frac{5}{6}}\mathrm{dh} \tag{11}\]For a spherical wave (uplink channel) [22], it is  
\[\alpha_{\text{uplink}}=\ \left[\exp\left(\frac{0.49\times\sigma_{1}^{2}\,\text{uplink }}{\left(1+0.56\times\sigma_{1}^{\frac{12}{5}}\right)^{\frac{2}{6}}}\right)-1 \right]^{-1} \tag{12}\]  
\[\beta_{\text{uplink}}=\ \left[\exp\left(\frac{0.51\times\sigma_{1}^{2}\,\text{uplink }}{\left(1+0.69\times\sigma_{1}^{\frac{12}{5}}\right)^{\frac{2}{6}}}\right)-1 \right]^{-1} \tag{13}\]  
\[\sigma_{1\,\text{uplink}}^{2}=2.25\text{k}^{\frac{7}{6}}\text{L}^{5/6}\int_{ \text{h}_{0}}^{\text{H}}\text{C}_{n}^{2}(\text{h})\left(1-\frac{\text{h}- \text{h}_{0}}{\text{H}-\text{h}_{0}}\right)^{5/6}\left(\frac{\text{h}-\text{ h}_{0}}{\text{H}-\text{h}_{0}}\right)^{5/6}\text{dh} \tag{14}\]  
where \(\theta\) is the zenith angle, k is the wave number, h\({}_{0}\) is the altitude of the ground station, L is the link distance, and \(\sigma_{1}^{2}\) is the Rytov variance. The numbers of small- and large-scale eddies in the scattering environment are denoted by the parameters \(\alpha\) and \(\beta\), respectively. Large-scale and small-scale eddies will behave as a prism or a lens and will interact with a laser beam traveling through the atmosphere, forming constructive or destructive interference. Equations (11) and (14) depict how the Rytov variance depends on the refractive index structure parameter and wave number. Rytov variance and the scintillation index parameter determine the strength of intensity fluctuations.