## 5 Conclusions  
In this paper, we proposed an exponential power spectra of refractive-index fluctuations for non-Kolmogorov turbulence in the free troposphere and the stratosphere, and on this basis, modified a three-layer altitude-dependent power spectrum of refractive-index fluctuations for satellite-to-ground and ground-to-satellite links, which is composed of the exponential Kolmogorov turbulence power spectrum of the boundary layer, the exponential non-Kolmogorov power spectrum turbulence of the free troposphere, and the exponential non-Kolmogorov power spectrum of the stratosphere. Based on the Markov approximation, beam wander displacement variances of a Gaussian-beam wave for non-Kolmogorov turbulence in the free troposphere and the stratosphere were derived, and the propagation theory of optical wave through atmospheric turbulence was further enriched. Further, the theoretical model of beam wander displacement variances of a Gaussian-beam wave propagating through Kolmogorov and non-Kolmogorov turbulence along laser-satellite communication uplink was developed and applied to characterize the displacement of the instantaneous center of the beam for the small zenith angle. Finally, the expression was used to analyze the variations of beam wander with beam radius, zenith angles, and a nominal value of refractive-index structure parameter on the ground.  
Through our work in the paper, wander theoretical model of a Gaussian-beam wave propagating through Kolmogorov and non-Kolmogorov turbulence along laser-satellite communication uplink was established. According to the actual situation of the atmospheric turbulence along the satellite-ground link, the combined influence of Kolmogorov and non-Kolmogorov turbulence on beam wander was studied. It provides a theoretical basis for the design and practical application of a satellite-ground laser communication system. The research results not only further enrich the theory of atmospheric turbulent optical wave propagation, but also lay a theoretical foundation for the realization of high-speed satellite-ground laser communication, which has important guiding significance for the system design, in-orbit test, and practical application of satellite laser communication, and will further promote the development of satellite-ground laser communication technology.  
Figure 3: Beam wander displacement variance of a Gaussian-beam wave for the three-layer model plotted as a function of nominal value of refractive-index structure parameter on the ground \(C_{n}^{2}(0)\) for a transmitter on the ground and a satellite in space.  
**Author Contributions:** Conceptualization, F.W. and W.D.; methodology, D.L., S.F. and W.D.; formalism, dyslexia, Q.Y.; investigation, F.W. and Q.Y.; data curation, Q.Y. and F.W., writing--original draft preparation, F.W. and W.D.; writing--review and editing, D.L., S.F., F.W. and W.D.; supervision, Q.Y. and W.D.; project administration, Q.Y. and W.D.; funding acquisition, Q.Y. and W.D. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Foundation of Heilongjiang Educational Committee (FHEC) grant 12521603, the Natural Science Foundation of Heilongjiang Province of China (NSFHPC) grant LH2020F050, and the Scientific Research Project of Basic Scientific Research Funding for Universities in Heilongjiang Province (SRPBSRFUHP), grant 135309453 and 145109402.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data set available on request to corresponding authors.  
**Acknowledgments:** We thank [PERSON], [PERSON], and [PERSON], who helped with the Matlab numerical simulation.  
**Conflicts of Interest:** The authors declare no conflict of interest.