# _photonics_
## 3 FSO Channel Modeling
### Scintillation Loss  
Scintillation can heavily affect an FSO communication link by causing intensity fluctuations in the receiver [32]. These fluctuations are caused by thermal changes that lead to changes in the refractive indices in small air cells, resulting in beam diffraction and beam wander [23; 32]. In the case of satellite downlink, the beam front is usually much larger than the size of these air cells; therefore, the effect of scintillation in the receiver is small but should be taken into consideration. The intensity of the scintillation is often described as weak, moderate and strong, depending on the value of the refractive index structure parameter \(C_{n}^{2}\) (m\({}^{-2/3}\)), which strongly depends on the atmospheric conditions such as temperature and pressure. In this analysis, the modified expression of Hufnagel-Valley model was used [23; 32], which takes into account the altitude of the ground station and the elevation angle. The value of \(C_{n}^{2}\) can be calculated as follows [22; 23]:  
\[ \begin{array}{c}C_{n}^{2}(h)=A_{0}\exp\Bigl{(}-\frac{H_{\rm CS}}{700}\Bigr{)} \exp\Bigl{(}\frac{H_{\rm CS}-h}{100}\Bigr{)}+5.94\times 10^{-53}(\frac{u_{\rm max }}{27})^{2}h^{10}\exp\Bigl{(}-\frac{h}{1000}\Bigr{)}\\ +\ 2.7\times 10^{-16}\exp\Bigl{(}-\frac{h}{1500}\Bigr{)},\end{array}  \tag{11}\]where \(A_{0}\) (m\({}^{-2/3}\)) is the refractive index structure parameter at the ground level, \(u_{rms}\) (m/s) is the average wind speed along the slant path using the Bufton model, \(H_{GS}\) (m) is the OGS altitude height, and \(h\) (m) is the height above the ground station altitude. For a plane wave approximation and assuming the Kolmogorov model, the scintillation index for weak, mean, and strong turbulences can be given by the following expression [16; 32]:  
\[\sigma_{I,point}^{2}=\exp(\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{1 2/5}\right)^{7/6}}+\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R}^{12/5} \right)^{5/6}})-1 \tag{12}\]  
where \(\sigma_{R}\) is the Rytov index which when taking into account the OGS's height and can be calculated as [16; 32]:  
\[\sigma_{R}^{2}=2.25k\tilde{\xi}sec^{\frac{11}{6}}(\zeta)\int_{H_{GS}}^{H_{ Turb}}\ C_{n}^{2}(h)(h-H_{GS})^{\frac{5}{6}}dh, \tag{13}\]  
where \(k\) (rad/m) is the wavenumber, \(\zeta\) (rad) is the zenith angle, and \(H_{Turb}\) (m) is the turbulence altitude which is considered negligible for altitudes higher than 20 km. To continue, we also considered the aperture-averaging effect in order to take the receiver's aperture diameter into account. The aperture-averaging factor was expressed as [32; 48]:  
\[A(D_{r})=\sigma_{I}^{2}/\sigma_{I,point}^{2}, \tag{14}\]  
where \(\sigma_{I}^{2}\) is the scintillation index for a receiving telescope and is described as: \(\sigma_{I}^{2}=A(D_{r})\times\sigma_{I,point}^{2}\). The aperture-averaging factor is calculated according to the following expression [32; 49]:  
\[A(D_{r})=\left[1+1.062\bigg{(}\frac{D_{r}}{2\rho_{I}}\bigg{)}^{2}\right]^{- \frac{7}{6}}, \tag{15}\]  
where \(\rho_{I}\) is intensity structure size parameter [32; 49]:  
\[\rho_{I}=1.5\sqrt{\frac{\lambda}{2\pi}H_{d}\frac{\theta/90^{\circ}}{\left( \theta/90^{\circ}\right)^{2}+\left(10/90^{\circ}\right)^{2}}}, \tag{16}\]  
where \(\theta\) (deg) is the elevation angle of the link and \(H_{d}\) is 12,000 m.  
Finally, for modeling the signal fluctuation due to the scintillation effect, we used the log-normal distribution that suits for weak- and moderate-turbulence regimes. The log-normal normalized PDF is calculated as [32; 49]:  
\[p_{I}(I)=\frac{1}{I\sqrt{2\pi\sigma^{2}{}_{Int}}}\exp\left[-\frac{\left[ln(I)+ \frac{1}{2}\sigma^{2}{}_{Int}\right]^{2}}{2\sigma^{2}{}_{Int}}\right]\ \,I>0, \tag{17}\]  
where \(\sigma^{2}{}_{Int}=ln(\sigma^{2}{}_{I}+1)\). Given the PDF, we can now calculate the loss (\(dB\)) that is introduced by scintillation for a given probability (e.g., 1%) threshold as follows [49]:  
\[L_{sci}(dB)=4.343\times[\ erf^{-1}(2p_{0}-1)\cdot[2 ln(\sigma^{2}{}_{I}+1)]^{ \frac{1}{2}}-\frac{1}{2}ln(\sigma^{2}{}_{I}+1)]. \tag{18}\]