## 5 Conclusions  
This paper focuses on optimizing satellite trajectory and designing orbits in a common-view model for networks of multiple SLR stations. The goal is to improve laser time transfer and signal transmission from ground laser stations by maximizing the longest feasible common-view time intervals. Addressing this, we introduce an improved simulated annealing algorithm designed to optimize satellite orbit trajectories. Anchored in the ground-to-ground laser time transfer principle and satellite visibility to design system model, the algorithm aims to determine optimal satellite orbits among multiple SLR station networks with a common-view perspective. Initial analysis establishes minimum satellite orbit altitudes for varying SLR station distances, based on this, conducted two sets of simulations. Comparative simulations against conventional iterative traverse and particle swarm methods reveal a substantial 10 to 20-fold reduction in computation time with an rms value of 0.45486 over 30 runs, ensuring convergence, global optimality, and stability. Selecting five satellites, including HY-2C, Sentinel-6A, LARES, Ajisai, and LAGEOS-2, their common-view times were compared with the optimized satellite orbits. Post-optimization, the common-view times at the same altitude exhibited enhancements ranging from 2.97 to 12.815 times. Further simulations optimize satellite orbits among 10 stations in the Asia-Pacific and Europe-to-Asia regions, resulting in a common-view model between all 45 station pairs in each region. Evaluation primarily focuses on common-view times among any two stations within this network, revealing intervals of 6.183 to 60.623 min in the Asia-Pacific and 5.583 to 61.75 min in the Europe-to-Asia regions. Ultimately, the refined satellite orbits foster mutual common viewing among multi-SLR station networks, significantly maximizing overall common-view durations.  
To achieve longer common-view durations between stations, an ideal satellite orbit model algorithm can be created from the perspective of multiple stations. This will provide a wider range of options for selecting satellite orbit configurations during launches, simplifying time and signal transmission and improving the precision of inter-station time and signal transfer. The findings of this paper can serve as reference factors in the selection of SLR station locations, ultimately contributing to the development of a global laser ranging station network. By integrating these insights, laser time transmission and space laser communications can be achieved with higher accuracy and faster transmission rates, allowing for a more detailed design of satellite constellations.  
Conceptualization, H.X. and Z.Z.; methodology, H.X., Z.Z. and S.D.; validation, H.X. and Z.Z.; formal analysis, H.X.; investigation, H.X.; resources, Z.Z.; data curation, H.X.; writing--original draft preparation, H.X.; writing--review and editing, H.X., Z.Z. and S.D.; visualization, H.X., L.M. and L.D.; supervision, Z.Z.; project administration, J.F.; funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 12173059, 11673082, 11103047; The Jilin Provincial Natural Science Foundation, China, grant number 20220101157 JC.  
All data that support the findings of this study are available from the corresponding authors upon reasonable request. The data are not publicly available due to privacy restrictions.  
The authors would like to thank the International Laser Ranging Service (ILRS), International Earth Rotation Service (IERS) for providing relevant data and products, all of which enable this study.  
The authors declare no conflicts of interest.