## 3 Beam Wander  
The instantaneous center of the beam is randomly displaced in the receiver plane, producing what is commonly called the beam wander. Usually the phenomenon can be characterized statistically by the variance of the hot spot displacement along an axis or by the variance of the magnitude of the hot spot displacement [1]. Based on the Markov approximation, the beam wander or the variance of the instantaneous center of a Gaussian-beam wave in the receiver plane (\(z=L\)) is modeled by the following expression  
\[\langle\rho_{c}^{2}\rangle=4\pi^{2}k^{2}W^{2}\int_{0}^{L}\int_{0}^{\infty} \kappa\Phi_{n}(\kappa,z)H_{LS}(\kappa,z)\Big{(}1-e^{\Lambda Lx^{2}\xi^{2}/k} \Big{)}dxdz, \tag{12}\]  
where \(\xi\) is the normalized distance variable, \(W\) is the spot size radius at the receiver plane, \(\Lambda\) is called the Fresnel ratio at the output plane, \(\Phi_{n}(\kappa,z)\) represents the refractive-index fluctuation power spectrum, and \(H_{LS}(\kappa,z)\) denotes the large-scale filter function [27]  
\[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]  
where \(W_{0}\) denotes the radius of the beam size at the transmitter plane, \(\Theta_{0}\) is also called the curvature parameter and \(\Lambda_{0}\) is called the Fresnel ratio at the input plane, and \(\overline{\Theta}_{0}=1-\Theta_{0}\). Now, it is accepted that the influence of the atmospheric turbulence for the satellite-to-ground or ground-to-satellite link is a result of a combined action of the Kolmogorov turbulence in the boundary layer, non-Kolmogorov in the free troposphere, and non-Kolmogorov in the stratosphere. Based on Reference [1], beam wander for a Gaussian-beam wave propagating through Kolmogorov turbulence in the boundary layer takes the following form  
\[\langle\rho_{Bc}^{2}\rangle = 7.25L_{0}^{\,3}W_{0}^{-1/3}\int_{h_{0}}^{H_{1}}\,C_{n}^{2}(\xi) \xi^{2} \tag{14}\] \[\times \[PERSON]{\{}\frac{1}{|\[PERSON]_{0}+\overline{\[PERSON]}_{0}\xi|^{1/3}}- \[PERSON]{[}\frac{\kappa_{0}^{2}(h)W_{0}^{2}}{1+\kappa_{0}^{2}(h)W_{0}^{2}\left( \[PERSON]_{0}+\overline{\[PERSON]}_{0}\xi\right)^{2}}\Biggr{]}^{1/6}\Biggr{\}}d\xi,\]  
where \(h_{0}\) is is the height of an optical transmitter/receiver above ground and \(H_{1}\) is the altitude of the boundary layer. By following the approach in Reference [1], using Equation (5), beam wander a Gaussian-beam wave propagating through non-Kolmogorov turbulence in the free troposphere, is obtained (see Appendix A.1)  
\[\langle\rho_{Fc}^{2}\rangle = 1.59H_{1}^{3}W_{0}^{-2/3}\int_{H_{1}}^{H_{2}}\ C_{nF}^{2}(\xi)\xi^ {2} \tag{15}\] \[\times \left\{\frac{1}{|\Theta_{0}+\overline{\Theta}_{0}\xi|^{2/3}}- \left[\frac{\kappa_{0}^{2}(h)W_{0}^{2}}{1+\kappa_{0}^{2}(h)W_{0}^{2}(\Theta_{ 0}+\overline{\Theta}_{0}\xi)^{2}}\right]^{1/3}\right\}d\xi.\]  
In the above equation, \(H_{2}\) is the altitude of the free troposphere. In the same way, beam wander for a Gaussian-beam wave propagating through non-Kolmogorov turbulence in the stratosphere is also obtained (see Appendix A.2)  
\[\langle\rho_{Sc}^{2}\rangle = 0.34L_{2}^{3}W_{0}^{-1/3}\int_{H_{2}}^{H}\ C_{nS}^{2}(\xi)\xi^ {2} \tag{16}\] \[\times \left\{\left[\frac{\kappa_{0}^{2}(h)W_{0}^{2}}{1+\kappa_{0}^{2}( h)W_{0}^{2}\big{(}\Theta_{0}+\overline{\Theta}_{0}\xi\big{)}^{2}}\right]^{1/2}-| \Theta_{0}+\overline{\Theta}_{0}\xi|\right\}d\xi,\]  
where \(H\) is the altitude of the satellite. Here, it is noteworthy that the the geometrical optics approximation is used in the derivation of Equations (15) and (16), viz.,  
\[1-e^{\Delta Lx^{2}\xi^{2}/k}\simeq\frac{\Delta Lx^{2}\xi^{2}}{k},\qquad\ Lx^{ 2}/k\ll 1. \tag{17}\]  
In addition, to emphasize the refractive nature of beam wander, the last term in Equation (13) is also dropped.