## 3 Optical Wireless Satellite Channel Model
### Atmospheric Turbulence: Beam Wander, Beam Spreading, and Beam Scintillation
The most widely-used model to characterize the refractive index structure of the atmosphere for vertical links (slant paths) is the so-called Hufnagel-Valley (H-V) model [45], i.e.,  
\[C_{n}^{2}(h)\ =\ A_{0}\exp\left(-\,\frac{h}{100}\right)+5.94\times 10^{-53} \left(\frac{v}{27}\right)^{2}h^{10}\,\exp\left(-\,\frac{h}{1000}\right)+2.7 \times 10^{-16}\,\exp\left(-\,\frac{h}{1500}\right), \tag{17}\]  
where \(h\) [m] is the altitude, \(v\) [m/s] is the RMS wind-speed, and parameter \(A_{0}\ =\ C_{n}^{2}(h_{0})\) [m\({}^{-2/3}\)] is the nominal value of the refractive index near the ground level. The RMS wind speed in (17) is determined from the Bufton wind model, and can take values that range from \(v\ =\ 10\) to 30 m/s for moderate and strong wind speeds, respectively. Similarly, the ground turbulence level can take values between \(A_{0}\ =\ 1.7\times 10^{-14}\) and \(1.7\times 10^{-13}\,\)m\({}^{-2/3}\), which depends on the location and day time, among other parameters. When using \(A_{0}\ =\ 1.7\times 10^{-14}\,\)m\({}^{-2/3}\) and \(v\ =\ 21\) m/s, this model is commonly referred to as the H-V\({}_{5/7}\) model because, for wavelength \(\lambda\ =\ 0.5\)\(\upmu\)m and a transmitter on the ground looking up (i.e., with \(\zeta\ =\ 0\) deg. ), it predicts a value of atmospheric coherence diameter \(r_{0}\ =\ 5\) cm according to (16) and a value of an isoplanatic angle:  
\[\theta_{0}\ =\ \frac{\cos^{8/5}(\zeta)}{\left[2.91\,k^{2}\,\int_{h_{0}}^{H}C_{n} ^{2}(h)\,(h-h_{0})^{5/3}\,dh\right]^{3/5}} \tag{18}\]  
of 7 \(\upmu\)rad in case of a spherical wave with output-plane beam parameters \(\Theta\ =\ \Lambda\ =\ 0\). The refractive index profile along the vertical/slant path is shown in Figure 5 for two nominal values of a refractive index at the ground level and three different RMS wind speeds. From this figure, it is possible to observe that the ground turbulence level \(A_{0}\) has little effect above 1 km, and that the wind speed governs the profile behavior primarily in the vicinity of altitudes in the 10 km range. Similarly, Figure 6 shows the RMS angular displacement due to beam wander (\(\sigma_{\rm BW}^{2}\)) as a function of the beam radius \(W_{0}\), when operating wavelength \(\lambda~{}=~{}1.55~{}\upmu\)m and refractive index structure \(C_{n}^{2}(h)\) follows the H-V\({}_{5/7}\) model. As expected, the RMS beam wander displacement is higher for the largest zenith angle, as the section of the atmosphere through which the optical beam needs passes through is thicker, the beam deviation with respect to the straight path grows. Finally, according to (16), the Fried's coherence length is \(r_{0}~{}=~{}19.25\) and \(12.70\,\)cm for zenith angle \(\zeta~{}=~{}0\) and \(60\,\)deg., respectively. Figure 5: Refractive index structure \(C_{n}^{2}(h)\) along the slant path for the H-V day model as a function of the altitude. Red lines: \(A_{0}~{}=~{}1.7\times 10^{-13}\,\)m\({}^{-2/3}\). Green lines: \(A_{0}~{}=~{}1.7\times 10^{-14}\,\)m\({}^{-2/3}\). Wind speed: \(v~{}=~{}10\) m/s (dotted lines with squares); \(v~{}=~{}21\) m/s (solid-lines with circles); and \(v~{}=~{}30\) m/s (dashed lines with diamonds). Figure 6: Root Mean Square (RMS) angular beam wander (\(\sigma_{\rm BW}^{2}\)) as a function of the beam radius (\(W_{0}\)) for a transmitter in the ground and a satellite in the space assuming \(\lambda~{}=~{}1.55\,\upmu\)m and a refractive index structure following the H-V\({}_{5/7}\) model (wind speed: \(v~{}=~{}21\) m/s). Zenith angle: \(\zeta~{}=~{}0\) deg. (solid orange line) and \(\zeta~{}=~{}60\) deg. (dashed purple line). **Turbulence-induced beam spreading:** This phenomenon takes place when the turbulent eddies are smaller than the size of the optical beam. Beam spreading generates a widening of the beam size, beyond the natural broadening due to diffraction that the non-turbulent atmosphere introduces. Beam spreading does not affect the direction of the optical beam but, in contrast, reduces the optical power at the receiver aperture due to the energy dispersion that takes place.