## 5 Conclusions  
In this work, the WSF of Gaussian beam-waves propagation through maritime atmospheric turbulence has been derived. Using this new expression, the long-exposure MTF for Gaussian-beam waves can be obtained. These new expressions can be used under the terrestrial atmospheric turbulence as well as the maritime atmospheric turbulence. The beam curvature parameter \(\Theta_{0}\) and Fresnel ratio \(\Lambda_{0}\) at the receiver, the power law exponent \(\alpha\), the anisotropic parameter \(\zeta\), the turbulence strength \(\sigma_{R}^{2}\), and the outer scale of the turbulence cells \(L_{0}\) for the WSF and long-exposure MTF were analyzed. The degrading effects for the plane and spherical waves are also evaluated as the special case of Gaussian-beam waves. These parameters have significant influence on the curves of the WSFs and long-exposure MTFs. The curves of the WSFs decrease with respect to increasing values of \(\Lambda_{0}\), \(\alpha\), and \(\zeta\); thus, these values obviously increase as \(\sigma_{R}^{2}\) increases. Furthermore, we have also studied the properties of optical waves with respect to those parameters in order to evaluate their effects on MTFs of three optical waves working in the imaging systems. The MTF curves observed from numerical simulations show that smaller degradation is obtained with the increasing beam curvature parameter \(\Theta_{0}\). Additionally, as the power law exponent \(\alpha\) and the anisotropic parameter \(\zeta\) increase, the imaging systems degradation gets smaller. Moreover, the increasing outer scale of the turbulence \(L_{0}\) has the bigger effect on the imaging system than those of small values. In addition, the degrading effects under maritime atmospheric turbulence are bigger than that of terrestrial one, which is related to higher humidity and temperature fluctuations in the maritime atmospheric turbulence.  
The simulation results that are shown in this work can be useful for analyzing the degrading effects on the imaging systems for all of the optical waves. In addition, the expressions of WSF are also can be used to derive spatial coherence radius, root-mean-square angle of arrival, and image jitter. Our results are useful for theoretical and practical study on FSO communication and remote sensing systems involving turbulent maritime atmospheric channels. We are also interested in the study of degradation behavior of optical waves propagation through oceanic turbulence. Furthermore, the effects of the short-exposure MTFs for those optical waves working in the imaging systems will be also considered in the future work.  
B.G. (Writing--Original Draft Preparation); H.Y. (Software and Validation); J.C. (Supervision and Writing--Review); W.S. and B.G. (Writing--Review & Editing). All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
This work was supported by The Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 19 KJ8510058, Brain Korea 21 Program for Leading Universities & Students, and The Professorial and Doctoral Scientific Research Foundation of Huizhou University(2020 JB014 and 2019 JB019).  
The authors declare no conflict of interest.  
Figure 6: Long-exposure MTF as a function of normalized spatial frequency of outer scale of the turbulence for **(a)** Gaussian beam wave **(b)** plane and spherical wave