## 3 Optical Wireless Satellite Channel Model
### Atmospheric Turbulence: Beam Wander, Beam Spreading, and Beam Scintillation  
Atmospheric turbulence is a random phenomenon that is caused by the variation of the temperature and pressure on the atmosphere layers that are in the propagation path of the optical wireless signal. These temperature and pressure inhomogeneities form turbulent cells, known as _eddies_, which have different sizes and different diffractive indexes. The eddies act as if prisms/lenses were deployed in the propagation path, introducing constructive and destructive interference in the received optical signal. The perturbations that atmospheric turbulence introduces in the wave-front of the optical beam can be physically described by the Kolmogorov model [44]. Depending on the size of the turbulent eddies with respect to the transmitted beam size, three types of atmospheric turbulence-induced effects can be identified, namely: _beam wander, beam spreading_, and _beam scintillation_. **Turbulence-induced beam wander. ** This phenomenon takes place when the size of the turbulent eddies is _larger_ than the size of the optical beam. Beam wander results in a random deviation of the optical beam from its planned (rectilinear) propagating path and, in extreme displacement situations, may lead to the failure of the optical wireless link. Beam wander is a major concern in the uplink transmission of an optical feeder link, as the beam size in the ground-to-satellite transmission is often smaller than the size of the turbulent eddies, resulting in a beam displacement at the receiver side that can be as large as several hundred meters. In case of a collimated beam (plane wave model), the Root Mean Square (RMS) displacement due to beam wander for an uplink path with zenith angle \(\zeta\) can be written as:  
\[\sigma_{\text{BW}}^{2} = 7.25\left(H-h_{0}\right)^{2}\sec^{3}\left(\zeta\right)W_{0}^{-1/3} \,\int_{h_{0}}^{H}C_{n}^{2}(h)\left(1-\frac{h-h_{0}}{H-h_{0}}\right)^{2}dh \tag{14}\] \[\cong 0.54\left(H-h_{0}\right)^{2}\sec^{2}\left(\zeta\right)\left(\frac {\lambda}{2W_{0}}\right)^{2}\left(\frac{2W_{0}}{r_{0}}\right)^{5/3}, \tag{15}\]  
where \(H\) is the altitude of the GEO satellite (receiver), \(h_{0}\) is the altitude of ground station (transmitter), \(W_{0}\) is the initial beam size, and \(r_{0}\) is the atmospheric coherence width, which is also known now as _Fried's coherence length_, _Fried's parameter_, or simply coherence length [45]. The Fried's coherence length is a widely-used descriptor of the level of atmospheric turbulence at a particular site and, for a known structure constant profile \(C_{n}^{2}(h)\) and plane wave model (collimated beam), it is given by: [46]  
\[r_{0}\ =\ \left[0.423\,k^{2}\sec\left(\zeta\right)\,\int_{h_{0}}^{H}C_{n}^{2}(h) \,dh\right]^{-3/5}, \tag{16}\]  
where \(k\ =\ 2\pi/\lambda\) is the wavenumber of the optical beam. As expected, \(C_{n}^{2}(h)\) varies with the time of the day, the geographical location, and the altitude. Therefore, for vertical optical links (slant paths), the value of \(C_{n}^{2}(h)\) has to be integrated over the complete propagation path, starting from the height of the ground station above the sea level and ending at the altitude in which the Earth's atmosphere vanishes (i.e., at about 40 km). Various empirical models for \(C_{n}^{2}(h)\) have been proposed in the literature to estimate the turbulence profiles, using as reference the experimental measurements that were carried out at different geographical locations, time of the day, wind speed, terrain types, among others.