# Free Space Optical Communication: An Enabling Backhaul Technology for 6G Non-Terrestrial Networks
## 2 Fundamentals of FSO Signal Transmission
### Turbulence-Induced Fading for Airborne Nodes
In (9), \(\alpha\) and \(\beta\) are the effective number of large-scale and small-scale cells of the scattering process, respectively. It can be readily verified that the second moment of the fading coefficient, which represents the average change in the signal power, is given by \(E\big{[}I^{2}\big{]}=(1+1/\alpha)(1+1/\beta)\). Under plane wave approximation, these values are connected to the Rytov variance through  
\[\alpha=\Bigg{[}\exp\Bigg{(}\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{ 12/5}\right)^{7/6}}\Bigg{)}-1\Bigg{]}^{-1}. \tag{10}\]  
\[\beta=\Bigg{[}\exp\Bigg{(}\frac{0.51\sigma_{s}^{2}}{\left(1+0.69\sigma_{s}^{12 /5}\right)^{5/6}}\Bigg{)}-1\Bigg{]}^{-1}. \tag{11}\]The average variation in received power due to induced turbulence is then calculated as  
\[L_{\text{Tur,GG}} =\frac{10}{\ln 10}\left(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11 \sigma_{R}^{12/5}\right)^{7/6}}+\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R }^{12/5}\right)^{5/6}}\right) \tag{12}\] \[=10\log_{10}\left(1+\sigma_{s}^{2}\right).\]  
where \(\sigma_{s}^{2}\) is given by  
\[\sigma_{s}^{2}=\exp\left(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{12 /5}\right)^{7/6}}+\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R}^{12/5} \right)^{5/6}}\right)-1. \tag{13}\]  
The gamma-gamma model is particularly useful for moderate to strong turbulence conditions or long-range FSO links. Even in the presence of strong turbulence, it is interesting to note that the statistical distribution of the received optical power in FSO links tends to have a lognormal behavior [98; 99]. This is due to the effect of aperture averaging. Aperture averaging refers to the spatial averaging effect that occurs when the receiver's aperture size is much larger than the size of the turbulent eddies. In particular, the incident optical wavefront becomes distorted due to atmospheric fluctuations, causing intensity variations at different points in the receiver aperture. However, when these intensity fluctuations are averaged over the receiver aperture, the resulting distribution tends to converge to a lognormal distribution. To quantify the losses associated with turbulence, precise calculations of the Rytov variance are required. In such calculations, it is important to take into account that atmospheric turbulence affects the propagation of optical beams differently in uplink and downlink scenarios. In an uplink scenario, the optical signal is transmitted from the Earth's surface towards a satellite or distant target, and the beam encounters turbulence along its path. The strength of turbulence generally increases closer to the Earth's surface and decreases with altitude. Consequently, the beam begins to experience distortion nearer to the transmitter, causing the rays to deviate closer to the transmitter. In the downlink scenario, the optical beam is transmitted from a higher altitude, where the turbulence strength is typically lower or negligible, towards the Earth's surface. As the beam propagates downward, it starts to encounter turbulence, as it gets closer to the receiver. In other words, while the uplink is primarily affected by turbulence at the point of transmission when the signal is strong, the downlink signal is affected by turbulence when it is relatively weak and closer to the receiver. Assuming a downlink atmospheric channel and a Gaussian beam, the Rytov variance of the vertical link is approximated as [100]  
\[\sigma_{R}^{2}\approx 2.25\,k^{7/6}\int\limits_{0}^{h}C_{n}^{2}(\varsigma)(h- \varsigma)^{5/6}d\varsigma, \tag{14}\]  
where \(k=2\pi/\lambda\) is the wave number. The propagation direction cannot always be considered vertical but rather creates an angle \((90-\phi)\) with the vertical direction, i.e., the elevation angle is \(\phi\). Therefore, the transmission distance as a function of altitude can be written as \(z=h/\sin(\phi)\).