## 3 CV QKD over Satellite Channels
### Satellite-to-Ground Channel Model
The volume of air mass is related to the atmospheric extinction ratio \(\eta_{ext}\), which is a measure of Rayleigh scattering, scattering due to aerosols and molecular absorption experienced by the signal beam in terrestrial atmosphere [60]. The value of extinction ratio \(\eta_{ext,\xi=0}\) depends on the the signal wavelength, air constituents, temperature and weather conditions, and its change with increase of the path length through the atmosphere (in  
Figure 2: Geometrical representation of satellite-to-ground communication scheme. \(H\) is the circular orbit altitude; \(R_{\oplus}\) is Earth radius; \(\zeta\in[-Pi/2,Pi/2]\) is the angle between the ray in the meridian plane, pointing above the observer ([PERSON]), and the direct line connecting [PERSON] and satellite (Alice), in practice limited to \(\pm\zeta_{max}\); and \(L(\zeta)\) is the slant range. terms of zenith angle) can be approximated (for angles up to \(\zeta_{max}\) where refraction effects are small) as [61; 62]:  
\[\eta_{ext}(\zeta)=\eta_{ext,\zeta_{s}=0}^{sec(\zeta)}. \tag{7}\]  
The ratio at zenith can be obtained based on MODTRAN atmospheric transmittance and radiance model [63], which for rural or urban sea-level Mid-latitude location with clear sky visibility yields \(\eta_{ext}=0.908\). Aside from aforementioned effects, refraction and diffraction are also caused by wind shear and temperature fluctuations and consequently by spatial and temporal variations of refractive index in the channel. Such perturbations result in scintillation, deviation of beam-spot from the center of the receiving aperture and deformation of the beam-spot. For satellite links, the beam-spot radius is always significantly larger than the aperture, i.e., \(W>a\), which allows us to ignore the deformation of the Gaussian beam profile, hence making beam wandering the dominant effect governing the fluctuation statistic of the channel transmittance. The maximal transmission efficiency is reached when incoming signal is perfectly aligned with the receiving aperture (\(r=0\)) and is defined by the ratio \(a/W\) of the aperture and beam spot sizes as follows  
\[\eta_{0}=1-\exp\biggl{[}-2\Bigl{(}\frac{a}{W}\Bigr{)}^{2}\biggr{]}. \tag{8}\]  
Efficiency \(\eta\in[0,\eta_{0}]\) decreases with the increase of deflection distance \(r\), with approximate analytical solution [50] being  
\[\eta=\eta_{0}\exp\biggl{[}-\Bigl{(}\frac{r}{R}\Bigr{)}^{\lambda}\biggr{]}, \tag{9}\]  
where \(\lambda\) and \(R\) are shape and scale parameters respectively:  
\[\lambda=8\Bigl{(}\frac{a}{W}\Bigr{)}^{2}\frac{\exp\Bigl{[}-4\bigl{(}\frac{a}{W }\bigr{)}^{2}\Bigr{]}I_{1}\Bigl{[}4\bigl{(}\frac{a}{W}\bigr{)}^{2}\Bigr{]}}{1- \exp\Bigl{[}-4\bigl{(}\frac{a}{W}\bigr{)}^{2}\Bigr{]}I_{0}\Bigl{[}4\bigl{(} \frac{a}{W}\bigr{)}^{2}\Bigr{]}}\biggl{\{}\ln\biggl{[}\frac{2\eta_{0}}{1-\exp \Bigl{[}4\bigl{(}\frac{a}{W}\bigr{)}^{2}\Bigr{]}I_{0}\Bigl{[}4\bigl{(}\frac{a}{ W}\bigr{)}^{2}\Bigr{]}}\biggr{]}\biggr{\}}^{-1}, \tag{10}\]  
\[R=a\Biggl{\{}\ln\Biggl{[}\frac{2\eta_{0}}{1-\exp\Bigl{[}-4\bigl{(}\frac{a}{W} \bigr{)}^{2}\Bigr{]}I_{0}\Bigl{[}4\bigl{(}\frac{a}{W}\bigr{)}^{2}\Bigr{]}} \biggr{]}\Biggr{\}}^{-1/\lambda}, \tag{11}\]  
where \(I_{n}\) is \(n\)-th order Bessel function. The position of deflected beam center is assumed to be normally fluctuating around the aperture center [64] and described by random transverse vector \(\mathbf{r}_{0}\):  
\[L(\zeta)=\sqrt{H^{2}+2 HR_{\oplus}+R_{\oplus}^{2}\cos^{2}\zeta}-R_{\oplus}\cos\zeta, \tag{5}\]0  
where beam-wandering variance is limited by tracking accuracy and beam stabilization \(\sigma_{BW}=\theta_{p}L(\zeta)\), and \(r=|\mathbf{r}_{0}|^{2}\).