## 3 Beam Wander
According to Equation (7), it can be concluded that the beam wander displacement variance over the whole satellite-to-ground or ground-to-satellite link for small zenith angles or under weak irradiance fluctuations is expressed as the sum of the beam wander displacement variance caused by the turbulence in the boundary layer, the free troposphere, and the stratosphere. \[\langle\rho_{c}^{2}\rangle = \langle\rho_{Sc}^{2}\rangle+\langle\rho_{Fc}^{2}\rangle+\langle \rho_{Sc}^{2}\rangle \tag{18}\] \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]0 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]1 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]2 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]3 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]4 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]5  
Here, it should be noted that beam wander is of little concern for a downlink path, but an uplink path can be large enough to cause the beam to move the several hundred meters. For a uplink path, the normalized variable \(\xi\) has this form, \(\xi=1-(h-h_{0})/(H-h_{0})\), and the beam wander displacement variance over the whole ground-to-satellite uplink for small zenith angles or under weak irradiance fluctuations is expressed as  
\[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]6 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]7 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]8 \[H_{LS}(\kappa,\xi)=\exp\Bigl{\{}-\kappa^{2}W_{0}^{2}\Bigl{[}\left(\Theta_{0}+ \overline{\Theta}_{0}\xi\right)^{2}+\Lambda_{0}^{2}(1-\xi)^{2}\Bigr{]}\,\Bigr{\}}, \tag{13}\]9 \[\langle\rho_{Bc}^{2}\rangle = 7.25L_{0}^{\,3}W_{0}^{-1/3}\int_{h_{0}}^{H_{1}}\,C_{n}^{2}(\xi) \xi^{2} \tag{14}\]0 \[\langle\rho_{Bc}^{2}\rangle = 7.25L_{0}^{\,3}W_{0}^{-1/3}\int_{h_{0}}^{H_{1}}\,C_{n}^{2}(\xi) \xi^{2} \tag{14}\]1 \[\langle\rho_{Bc}^{2}\rangle = 7.25L_{0}^{\,3}W_{0}^{-1/3}\int_{h_{0}}^{H_{1}}\,C_{n}^{2}(\xi) \xi^{2} \tag{14}\]2 \[\langle\rho_{Bc}^{2}\rangle = 7.25L_{0}^{\,3}W_{0}^{-1/3}\int_{h_{0}}^{H_{1}}\,C_{n}^{2}(\xi) \xi^{2} \tag{14}\]3  
where \(\zeta\) is the zenith angle. It can be seen from Equations (15) and (16) that beam wander of a Gaussian-beam wave propagating through non-Kolmogorov atmospheric turbulence is independent on optical wave length, similar to Kolmogorov atmospheric turbulence [1].