# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 3 Modified Rytov Theory for Gaussian Beam Waves
### Finite Inner and Finite Outer Scale Model (Modified Atmospheric Spectrum)  
In this model, \(l_{o}>0\) and \(L_{o}<\infty\). The finite inner and finite outer scale effects model is actually the modified atmospheric spectrum. The scintillation index for this model can be expressed as [44]:  
\[\sigma_{SI}^{2}(0,L)=\exp\Bigl{(}\sigma_{Inx}^{2}(l_{o},L_{o})+\sigma_{Iny}^{2} (l_{o})\Bigr{)}-1 \tag{15}\]  
where  
\[\sigma_{Inx}^{2}(l_{o},L_{o})=\sigma_{Inx}^{2}(l_{o})-\sigma_{Inx}^{2}(L_{o}) \tag{16}\]  
The large-scale log-irradiance variance component due to the inner scale of optical turbulence is given as [44]:  
\[\sigma_{Inx}^{2}(l_{o})=0.49\sigma_{1}^{2}\Bigl{(}0.333-0.5\overline{\Theta}_{ 1}+0.2\overline{\Theta}_{1}^{\ 2}\Bigr{)}\biggl{(}\frac{\eta_{x}Q_{l}}{\eta_{x}+Q_{l}} \biggr{)}^{1.167}\times\Biggl{(}1+1.75\biggl{(}\frac{\eta_{x}}{\eta_{x}+Q_{l} }\biggr{)}^{0.5}-0.25\biggl{(}\frac{\eta_{x}}{\eta_{x}+Q_{l}}\biggr{)}^{0.583} \Biggr{)} \tag{17}\]  
while the large-scale log-irradiance variance component due to outer scale of turbulence is expressed as [44]:  
\[\sigma_{Inx}^{2}(L_{o})=0.49\sigma_{1}^{2}\Bigl{(}0.333-0.5\overline{\Theta}_ {1}+0.2\overline{\Theta}_{1}^{\ 2}\Bigr{)}\biggl{(}\frac{\eta_{xo}Q_{l}}{\eta_{xo}+Q_{l}} \biggr{)}^{1.167}\times\Biggl{(}1+1.75\biggl{(}\frac{\eta_{xo}}{\eta_{xo}+Q_{l }}\biggr{)}^{0.5}-0.25\biggl{(}\frac{\eta_{xo}}{\eta_{xo}+Q_{l}}\biggr{)}^{0.583 }\Biggr{)} \tag{18}\]  
where  
\[\overline{\Theta}_{1}=1-\Theta_{1}=-\frac{L}{F} \tag{19}\]  
The nondimensional inner-scale parameter, \(Q_{l}\), is calculated as [44]:  
\[Q_{l}=\frac{10.89L}{kl_{o}^{2}} \tag{20}\]  
while the nondimensional low-pass cutoff frequency is expressed as [44]:  
\[\eta_{x}=\Biggl{(}\frac{0.38}{1-3.21\overline{\Theta}_{1}+5.29\overline{ \Theta}_{1}^{\ 2}}+\Biggl{(}0.47\sigma_{1}^{2}Q_{l}^{0.167}\Biggl{(}\frac{0.333-0.5 \overline{\Theta}_{1}+0.2\overline{\Theta}_{1}^{\ 2}}{1+2.20\overline{\Theta}_{1}}\Biggr{)}^{0.857} \Biggr{)}\Biggr{)}^{-1} \tag{21}\]  
The artificial quantity \(\eta_{xo}\), is defined as [44]:  
\[\eta_{xo}=\frac{\eta_{xo}Q_{o}}{\eta_{xo}+Q_{o}} \tag{22}\]  
and the nondimensional outer-scale parameter, \(Q_{o}\), is given as [44]:  
\[Q_{o}=\frac{64\pi^{2}L}{kl_{o}^{2}} \tag{23}\]  
The small-scale log-irradiance variance component due to the inner scale of optical turbulence is defined as [44]:  
\[\sigma_{Iny}^{2}(l_{o})=\frac{0.51\sigma_{G}^{2}}{\bigl{(}1+0.69\sigma_{G}^{2 4}\bigr{)}^{0.833}} \tag{24}\]  
where \(\sigma_{G}^{2}\) is the Rytov variance for a Gaussian-beam wave with inner scale and is derived in [38; 44]. Tables 2-5 show the RMS and ground wind speeds, \(C_{n}^{2}\), and scintillation indices for Gaussian beam waves based on the zero inner scale and infinite outer scale model andfinite inner and finite outer scale model for various cities in South Africa while taking into account periods not exceeding 50%, 99%, 99.9%, and 99.99% of the time, respectively. The values in these tables are generated based on RMS wind velocities during clear weather periods for each city and their corresponding C\({}_{n}^{2}\) calculated using the HAP model in Equations (1) and (2). It is important to reiterate that data spanning many years are necessary in order to accurately estimate the atmospheric turbulence losses that may be encountered by FSOC systems before their deployment to the desired locations. The scintillation index values based on the zero inner scale and infinite outer scale model in Tables 2-5 are calculated using Equations (3)-(14). These values range from \(\sim\)0.067 to \(\sim\)0.752. In the finite inner and finite outer scale model, the microscale and macroscale eddies are assigned to have the values of 0.005 m and 10 m, respectively. That is, \(l_{\rm o}=0.005\) m and \(L_{\rm o}=10\) m are used for all the computations of the scintillation indices based on the finite inner and finite outer scale model in Tables 2-5.