## 4 Results  
The frequency-synchronization experiments conducted in the laboratory over a distance of 20 m commenced on 29 July 2024, and were concluded on 30 July 2024. The transfer end corresponds to the TX section in Figure 3, while the receiver end refers to the section after the \"Phase-shift Downlink\" in the same figure. The phase difference between these two ends represents the overall locking performance of the loop. The recovery clock refers to the clock reproduced at the transponder. The time domain frequency stabilization and waveforms of the residual phases of the closed-loop compensation experiment are respectively shown in Figure 11, panels (a) and (b). This experiment characterized the precision limits that the system could achieve. Its short-term  
Figure 10: Design of the Delay Line module. and long-term stability reached 2.00 \(\times\) 10\({}^{-13}\) at 1 s and 2.06 \(\times\) 10\({}^{-17}\) at 10,000 s. The phase values obtained in this time domain were statistically analyzed with the data recorded at 0.1 s intervals. The peak-to-peak values were approximately 1.49 ps, with a standard deviation of 0.13 ps, as determined by a statistical analysis. Regarding the transponder end, Figure 11, panels (c) and (d), respectively, show the frequency stability and time domain waveform of the remaining phase after proportional compensation. Although this method does not have a closed-loop structure for the uplink, as predicted by our theory, the time difference after compensation remained minimal. The peak-to-peak value was approximately 22.36 ps, and the standard deviation was 4.44 ps, as determined by statistical analysis. The system demonstrated short- and long-term stability, with 5.66 \(\times\) 10\({}^{-14}\) at 1 s and 3.56 \(\times\) 10\({}^{-16}\) at 10,000 s.  
For the dynamic experiment, the receiver end executed a reciprocal motion with a reciprocal distance of approximately 60 cm. Figure 12a shows the time domain waveforms of the remaining phases of the open- and closed-loop compensation experiments of 24 h for the ground-end. The standard deviation of the closed-loop and open-loop residual phase, converted to time, was calculated to be 1596.55 ps for the open-loop and 0.81 ps for the closed loop. These results demonstrate a considerable improvement in the closed-loop link performance. Figure 12b shows the time domain waveform and frequency stability of the remaining phase at the transponder end after proportional compensation. A comparison of the  
Figure 11: Results of the frequency-synchronization experiments. (**a**) Allan deviation of the phase difference between the ground station’s transfer and receiver ends. (**b**) Phase differences between the ground station’s transfer and receiver ends shown as time-difference changes. (**c**) Allan deviation of the recovery clock with the local frequency standard. (**d**) Phase difference in the recovery clock and local frequency standard shown as time-difference changes. standard deviation of the phase difference converted to time before and after compensation reveals a considerable improvement, with values of 796.02 ps for the open loop and 6.33 ps for the closed loop. Figure 13 illustrates the closed-loop frequency stability of the dynamic experiment over a 48 h period. The calculations demonstrate that the ground station end's 10,000 s stability reached 9.88 \(\times\) 10\({}^{-17}\), whereas the transponder end's 10,000 s stability reached 3.36 \(\times\) 10\({}^{-16}\). Notably, the ambient temperature in the vicinity of the RF device significantly correlated with the time-varying experimental results at the transponder end. Figure 14 illustrates the temperature fluctuations in the vicinity of the device throughout the experiment and depicts a comparison between the variations in temperature and the phase fluctuations of the recovered clock at the transponder side. The maximum Pearson product moment correlation coefficient (r) was \(-\)0.87, indicative of a strong correlation between the two variables.