## 5 Conclusions  
This paper investigated the RIS-assisted satellite-HAP-terrestrial downlink laser communication system. The atmospheric channel was modeled using the M distribution characterization. The effect of light intensity scintillation in the satellite-to-HAP link was considered. Furthermore, the influence of both light intensity scintillation and pointing error was considered in the HAP-to-terrestrial link. Closed expressions for the BER of the RIS-SHTLC system were derived. The BER performance of the system was simulated under weak turbulence and compared with that of the SHTLC system. The effects of different modulation methods, the zenith angle, receiving aperture and divergence angle on the BER performance of the system were also investigated.  
The simulation results showed that better system performance could be obtained with the RIS-SHTLC system. The BER difference between the RIS-SHTLC and SHTLC systems gradually increased with the increase in the number of RIS elements at the same average SNR. In order to meet the requirement of a minimum bit error rate of \(10^{-5}\) for the communication system, the transmission power of the SHTLC system with an RIS element number of 60 is 12 dB and 8 dB lower than that of system with an RIS element number of 20 and 40 respectively. For the same communication performance, the RIS-SHTLC system could obtain better communication performance with CBPSK modulation compared to CBPSK, NBPSK, and DBPSK modulations. At the same time, when the zenith angle is \(30^{\circ}\), the bit error rate of the SHTLC system with an RIS element number of 60 is 4 and 2 orders of magnitude lower than that of the system with an RIS element number of 20 and 40, respectively. In addition, the RIS-SHTLC system can reduce the aperture requirement of the receiver by increasing the number of RIS elements. When the system bit error rate is \(10^{-5}\), the receiving aperture of SHTLC systems with 60 RIS elements is 0.15 m and 0.1 m smaller than that of systems with 20 and 40 RIS elements, respectively. When the required divergence angle was greater than \(20\)\(\times\)\(10^{-5}\) rad, an optimal divergence angle existed that minimized the BER of the system for the RIS-SHTLC system. The optimal divergence angle became larger as the number of RIS elements increased. The optimal divergence angle of the SHTLC system with an RIS element number of 60 increases by \(20\)\(\times\)\(10^{-5}\) rad and \(5\)\(\times\)\(10^{-5}\) rad compared to that of the system with an RIS element number of 20 and 40, respectively. In summary, for SHTLC systems, RIS technology could effectively optimize system performance and improve the stability of the communication link. This work can provide a theoretical reference for the engineering implementation of RIS technology for satellite-terrestrial laser communication.  
Figure 7: Effect of divergence angle on the performance of RIS-SHTLC system with different numbers of RIS elements.  
**Author Contributions:** Conceptualization, [PERSON] and H.W.; methodology, H.W.; software, H.W.; validation, Y.W., H.W. and X.J.; formal analysis, H.W.; investigation, X.J.; resources, Y.W.; data curation, H.W.; writing--original draft preparation, H.W.; writing--review and editing, H.W.; visualization, H.W.; supervision, Y.W.; project administration, Y.W.; funding acquisition, Y.W. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by Open Fund (PLN2022-06) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) and the National Natural Science Foundation of China (Grant No. 51704267).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.