## 3 Satellite-to-Ground QKD Simulator
### Atmospheric Link Modeling
#### 3.2.6 Fiber Coupling Efficiency  
In addition to the factors affecting the channel efficiency that were discussed previously, this study also explored the influence of turbulence on the spatial coherence of the beam. The latter directly affects the beam's ability to be coupled with an SMF, a critical aspect when employing fiber-based detectors at the receiver's side for DV- and ENT-based QKD. AO systems present a viable approach to mitigate the perturbations caused by a free-space channel [61]. By compensating for phase fluctuations due to atmospheric turbulence in real time, AO systems are not only pivotal in ground-based astronomy [62], but they are also emerging as essential for free-space optical telecommunications [61; 63] and free-space QKD [63; 64; 65]. While existing studies have examined the uplink scenario to some extent [66; 67], most of the research on space-ground QKD has focused on the downlink channel. As mentioned before, our investigation also addresses the downlink configuration. We modeled the fiber coupling efficiency of a QKD downlink by expressing the coupling efficiency \(\eta_{FC}\) through Strehl ratio-like metrics, since the latter has been suggested as an estimator for SMF coupling efficiency [68]. We also took into account the optical coupling efficiency \(\eta_{0}\), which reflects the mismatch between an unperturbed received optical beam and the Mode Field Diameter (MFD) of the SMF. Although our model may not perfectly represent a true SMF coupling, it provides a solid approximation of fiber coupling efficiency. A more-detailed analytical model of SMF coupling, including detailed AO modeling, can be found in the literature [38; 69; 70]. In detail, the Strehl ratio (SR) in the presence of atmospheric turbulence is given as follows [53]:  
\[SR=\left[1+\gamma\left(\frac{D_{r}}{r_{0}}\right)^{5/3}\right]^{-\frac{6}{5}}, \tag{11}\]  
where factor \(\gamma\) accounts for the level of AO correction and can take the values 1, 0.28, and 0 for no correction, tip tilt correction, and full-AO correction, respectively; whereas \(r_{0}\) is the Fried parameter that characterizes he atmospheric turbulence across the slant path, which is given by the following equation:  
\[r_{0}=\left[0.42\;k^{2}sec(\zeta)\int_{H_{GS}}^{H_{turb}}C_{n}^{2}(h)dh\right]^ {-3/5}. \tag{12}\]  
It is important to note that only perfect correction was considered here, meaning that, for example, the tip-tilt correction (\(\gamma=0.28\)) completely removes any second-order distortion to the wavefront, while a full-AO correction (\(\gamma=0\)) removes all the aberrations, thus resulting in a diffraction limited spot. In accounting for the efficiency of the receiving telescope, we also modeled the optical coupling efficiency, which was determined by the design optics of the receiving telescope [35]. The ideal coupling efficiency can be parametrized by the following:  
\[\eta_{0}=2\left[\frac{\exp(-\beta^{2})-\exp(-\beta^{2}a^{2})}{\beta\sqrt{1- \alpha^{2}}}\right]^{2}, \tag{13}\]  
where \(\alpha\) and \(\beta\) are given by  
\[a=\frac{D_{obs}}{D_{r}},\;\beta=\pi\frac{\mathrm{D}_{r}}{4\lambda}\frac{MFD}{f}, \tag{14}\]  
where \(D_{obs}\) is the obscuration aperture diameter; \(MFD\) denotes the mode-field-diameter of an SMF fiber, which is equal to 10.4 \(\upmu\)m; and \(f\) is the focal length of the receiving telescope. Since the mode characteristic of an SMF can be represented by a Gaussian function, and as it is given that the aperture plane and the focal plane are interconnected through a Fourier transform [71], optimal coupling is possible only when a Gaussian beam matches the aperture within the coupling optics. Nevertheless, the average profile of incominglight tends to exhibit a flat-top distribution [72], which sets a theoretical upper limit to the coupling efficiency of approximately 81% when assuming a non-turbulent transmission, which can be extracted from Equation (13) for \(\alpha=0\) and \(\beta=1.12\). Finally, \(\eta_{sys}\) accounts for the AO/tip-tilt system optical components' transmittance.