# Semi-Empirical Satellite-to-Ground Quantum Key Distribution Model for Realistic Receivers
## 3 Link Efficiency for 300-mm and 600-mm Aperture Ground Stations  
Channel losses between a satellite and a ground station dynamically change during quantum communication, which significantly distinguishes this method from fiber optic or terrestrial QKD. In general, diffraction losses vary with the change of the distance between a satellite and a ground station. Additionally, varying the elevation angle results in various effective thicknesses of the atmosphere during light transmission. The diffraction loss can be calculated using the laser source divergence \(\gamma\), communication channel length \(d\), and effective receiving area with telescope diameter \(D_{\mathrm{T}}\) and obstruction \(\varepsilon\) of the telescope secondary mirror, as follows \(\varepsilon D_{\mathrm{T}}^{2}/(\gamma d)^{2}\). However, the atmospheric extinction should be described in more detail. In astronomy, the magnitude of a star or another visible celestial object, such as a satellite, indicates its brightness class. The magnitudes \(m_{1}\) and \(m_{2}\) of two objects are related to the corresponding flux densities \(F_{1}\) and \(F_{2}\) by equation [35]:  
\[m_{1}-m_{2}=-2.5\log_{10}{(F_{1}/F_{2})}. \tag{4}\]  
Given that starlight travels a greater distance through the atmosphere near the horizon than it does at the zenith, it follows that a star near the horizon will be less luminous. Therefore, we can relate the magnitude of the observed object outside the atmosphere \(m_{0}\) (apparent magnitude) to the magnitude of the object at the Earth's surface \(m\), as follows:  
\[m-m_{0}=\varkappa\cdot f(\theta_{E_{I}}), \tag{5}\]  
Figure 2: Calculated characteristics of the satellite passages. Dashed curves: elevation angle for the actual passages using the satellite TLE data and for the limit zenith passage with a maximum elevation angle of \(90^{\circ}\). Solid curves: communication channel length for the actual passages and for the satellite passage through the zenith. The satellite passages are symmetrically aligned with respect to the highest elevation angle in each one that corresponds to zero time. where \(\varkappa\) is the atmospheric extinction coefficient and \(f(\theta_{El})\) is the air mass function depending on the elevation angle of the object. The amount of air at the zenith corresponds to one air mass. [PERSON] and [PERSON] (2007) proposed the relationship between the air mass function and elevation angles as:  
\[f(\theta_{El})=\csc\theta_{El}\cdot(1-0.0012\cot^{2}\theta_{El}). \tag{6}\]  
The atmospheric extinction coefficient was determined from the measurements of stars' fluxes with known apparent magnitudes outside the atmosphere in the spectral range 845 nm-855 nm ([PERSON], 1996) at several elevation angles by means of a 600-mm ground station. We assumed that, for the same ground station, the star fluxes were equally proportional to the photon count rate. Appendix B contains additional information and data. Figure 3 depicts the dependence of atmospheric extinction \(\varkappa f(\theta_{El})\) on air mass up to a constant, according to Equations (A13) and (6). The measurements of star fluxes were performed on 9 March 2022, with clear weather conditions (humidity \(<20\%\)) and on 24 June 2021, with hazy, foggy weather conditions (humidity \(<80\%\) near the Earth's surface). As a result, the slope of atmospheric extinction equals the atmospheric extinction coefficient and ranges from \(0.23\pm 0.08\) for clear nights to \(0.41\pm 0.09\) for foggy nights. Hence, link efficiency, or the overall transmission of photons at 850 nm between the Micius satellite and a ground station, including detection efficiency, is given by:  
\[\eta(t)=\frac{\varepsilon D_{T}^{2}}{(\gamma d)^{2}}\cdot 10^{-0.4\varkappa\csc \theta_{El}\cdot(1-0.0012\cot^{2}\theta_{El})}\eta_{opt}\eta_{det}, \tag{7}\]  
where \(\eta_{opt}\) is an optical efficiency of the respective ground station and \(\eta_{det}\) is a quantum efficiency of a single photon detector.