# Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa
## 3 Modified Rytov Theory for Gaussian Beam Waves
### Zero Inner Scale and Infinite Outer Scale Model (Infinite Kolmogorov Inertial Range)  
The microscale and macroscale effects are neglected in this model. That is, \(l_{o}\simeq 0\) and \(L_{o}\simeq\infty\). The scintillation index or normalized irradiance variance for the zero inner scale and infinite outer scale model is expressed as [44]:  
\[\sigma_{SI}^{2}(0,L)=\exp\Bigl{(}\sigma_{Inx}^{2}+\sigma_{Iny}^{2}\Bigr{)}-1 \tag{3}\]  
where \(\sigma_{Inx}^{2}\) is the large-scale log-irradiance variance and \(\sigma_{Iny}^{2}\) is the small-scale log-irradiance variance.  
The large-scale log-irradiance variance is given as [44]:  
\[\sigma_{Inx}^{2}=\frac{0.49\sigma_{B}^{2}}{\bigl{(}1+0.56\sigma_{B}^{2.4} \bigr{)}^{1.167}} \tag{4}\]  
Figure 3: RMS and ground wind speed against refractive index structure parameter during clear weather periods for various locations in South Africa.  
and the small-scale log-irradiance variance is expressed as [44]:  
\[\sigma_{Inv}^{2}=\frac{0.51\sigma_{B}^{2}}{\left(1+0.69\sigma_{B}^{2.4}\right)^{0.8 33}} \tag{5}\]  
where \(\sigma_{B}^{2}\) is the Rytov variance for a Gaussian-beam wave, and is expressed in [38] as:  
\[\sigma_{B}^{2}\cong 3.86\sigma_{1}^{2}\times\left\{0.40\Big{(}(1+2\Theta_{1})^{2 }+4\Lambda_{1}^{2}\Big{)}^{0.417}\times\cos\left(0.833\tan^{-1}\left(\frac{1+2 \Theta_{1}}{2\Lambda_{1}}\right)\right)-0.688\Lambda_{1}^{0.833}\right\} \tag{6}\]  
where \(\sigma_{1}^{2}\) is the Rytov variance for a plane wave, \(\Theta_{1}\) is the curvature parameter of the beam at the receiver, and \(\Lambda_{1}\) is the Fresnel ratio of the beam at the receiver. \(\sigma_{1}^{2}\) is given as [45]:  
\[\sigma_{1}^{2}=1.23C_{n}^{2}k^{1.167}L^{1.833} \tag{7}\]  
where \(k\) is the wave number of the plane wave. \(k\) is calculated in m\({}^{-1}\) as [45]:  
\[k=\frac{2\pi}{\lambda} \tag{8}\]  
where \(\lambda\) is the wavelength in metres. \(\Theta_{1}\) is expressed as [46]:  
\[\Theta_{1}=\frac{\Theta_{o}}{\Theta_{o}^{2}+\Lambda_{o}^{2}}=1+\frac{L}{F} \tag{9}\]  
where \(\Theta_{o}\) is the Beam curvature parameter at the transmitter, \(\Lambda_{o}\) is the Fresnel ratio of the beam at the transmitter, \(L\) is the propagation distance in metres, and \(F\) is the phase front radius of curvature of the beam at the receiver. \(\Lambda_{1}\) is given as [46]:  
\[\Lambda_{1}=\frac{\Lambda_{o}}{\Theta_{o}^{2}+\Lambda_{o}^{2}}=\frac{2L}{kW_{ 1}^{2}} \tag{10}\]  
where \(W_{1}\) is the beam radius in free space at the receiver. \(W_{1}\) can be calculated using [43; 47]:  
\[W_{1}=W_{o}\Big{(}\Theta_{o}^{2}+\Lambda_{o}^{2}\Big{)}^{0.5} \tag{11}\]  
where \(W_{o}\) is the beam radius at the transmitter. The beam curvature parameter at the transmitter can be expressed as [46; 47]:  
\[\Theta_{o}=1-\frac{L}{F} \tag{12}\]  
while the Fresnel ratio of the beam at the transmitter is given as [46; 47]:  
\[\Lambda_{o}=\frac{2L}{kW_{o}^{2}} \tag{13}\]  
The phase front radius of curvature of the beam at the receiver can be calculated using [38]:  
\[F=\frac{F_{o}\Big{(}\Theta_{o}^{2}+\Lambda_{o}^{2}\Big{)}(\Theta_{o}-1)}{ \Theta_{o}^{2}+\Lambda_{o}^{2}-\Theta_{o}} \tag{14}\]  
where \(F_{o}\) is the phase front radius of curvature of the beam at the transmitter.