## 4 Simulation Results
### Receiver Size and Design Characteristics  
Addressing the technical challenge of coupling an optical satellite signal into a ground receiver involves overcoming unique obstacles compared to traditional Radio Frequency (RF) communications. Optical satellite communications encounter several challenges due to the nature of light propagation and atmospheric conditions. Coupling from free space to SMF, particularly for large telescopes, presents significant challenges in the presence of turbulence as it requires matching between the incoming beam's intensity distribution and phase front [62]. To address this issue, AO systems were utilized, leading to a notable improvement in fiber coupling efficiency. These systems enable coupling efficiencies over 50%, even in the case of large telescopes. The upcoming section delves into how different system parameters, such as the telescope's focal length and aperture diameter, influence the overall efficiency of fiber coupling. Furthermore, it explores the viability of satellite QKD across a range of receiver telescope aperture sizes. It is important to highlight that, for each aperture size considered, a corresponding focal length was chosen to optimize fiber coupling. This focal length was determined to be ten times the value of the aperture diameter, thus ensuring that the fiber coupling capabilities are not compromised by a fixed focal length across different telescope sizes. Additionally, each telescope was assigned an obscuration ratio of 0.1, thus further tailoring the setup to maximize coupling efficiency. Incorporating the top-down approach outlined earlier, our focus here was solely on the LEO case. An in-depth discussion on how the characteristics of the receiver impact MEO and GEO cases will be addressed in Section 5. Figure 10 depicts a contour plot demonstrating the coupling efficiency \(\eta_{FC}\) of the SMF, computed using Equation (12), with respect to the receiver's telescope Fr (m) and aperture diameter Dr (m) under the conditions of full-AO and tip-tilt correction amidst moderate \(\left(Cn^{2}=10^{-14}\text{m}^{-2/3}\right)\) and strong turbulence \(\left(Cn^{2}=10^{-13}\text{m}^{-2/3}\right)\). The calculation assumed a satellite elevation angle of 90 degrees, with the receiver situated at an altitude of 500 m. To incorporate the direct impact of atmospheric turbulence, Equation (15) was employed. Specifically, the \(\gamma\) in Equation (11) was set to 0 for full-AO correction, while it took values of 0.28 and 1 for the tip-tilt correction and no correction, respectively. The telescope's efficiency was factored in using Equation (14), where \(D_{obs}\) remains constant at 0.1. Figure 9: SKRs over a single MEO satellite pass when varying the sky radiance values of (**a**) \(D_{r}=1\) m and SMF-coupled SNSPD detection and (**b**) \(D_{r}=2.3\) m and SMF-coupled SNSPD detection. Figure 11 depicts the calculated SKR over a single pass for (a) SNSPD detection with fiber coupling, (b) SPAD detection with fiber coupling, and (c) free-space SPAD detection. The calculation support in the case of SNSPD detection for telescopes with apertures as small as 0.28 m can support satellite QKD communications with a limited communication time window. Yet, SKRs reaching the order of hundreds of bits per second in average can be achieved. On the other hand, SPAD detection can support fiber coupling only with receiver telescope apertures larger than 0.8 m. On the contrary, for free-space SPAD detection, where the end-to-end loss remains limited, small-sized telescopes (e.g., a 0.28 m aperture) can support satellite QKD, thereby reaching maximum SKR values of up to 2 kbps. It was also evident that, as shown in Figure 11c, due to the dead time effect (note that \(t_{dead}=20\) us for our assumptions), if large apertures are employed, then the value of the calculated SKR is not drastically increased since the detectors come closer to the saturation limit as the receiver aperture increases.