## 2 Anisotropic Maritime Atmospheric Spectrum  
Based on the Hill spectrum, [PERSON] [27] has developed a modified atmospheric spectrum, which follows the form of von [PERSON] spectrum and includes an outer scale factor. The Andrews' power spectrum is given, as follows:  
\[\Phi_{n}(\kappa)=0.033C_{n}^{2}\exp(-\kappa^{2}/\kappa_{l}^{2})(\kappa^{2}+ \kappa_{0}^{2})^{-11/6}\left[1+a_{1}(\kappa/\kappa_{l})+a_{2}(\kappa/\kappa_{l })^{7/6}\right], \tag{1}\]  
where \(\kappa\) is the scalar spatial wave number; \(C_{n}^{2}\) is refractive-index structure parameter in \(m^{-2/3}\); \(\kappa_{l}=3.3/l_{0}\); \(\kappa_{0}=2\pi/L_{0}\); \(l_{0}\) and \(L_{0}\) are inner scale and outer scale parameters; \(a_{1}=1.802\) and \(a_{2}=-0.254\). To study the effects of optical wave propagation through the marine environment, a new maritime atmospheric power spectrum is proposed for the weak optical turbulence for small inner scale values by changing the coefficient's value as \(a_{1}=-0.061\) and \(a_{2}=2.836\)[15]. [PERSON] et al. [18] have extended this spectrum considering a non-Kolmogorov turbulence and the power spectrum is defined, as follows:  
\[\Phi_{n}(\kappa,\alpha)=A(\alpha)\tilde{C}_{n}^{2}[1+a_{1}\frac{\kappa}{ \kappa_{H}}+a_{2}(\frac{\kappa}{\kappa_{H}})^{3-\frac{\alpha}{2}}][\kappa^{2} +\kappa_{0}^{2}]^{-\frac{\alpha}{2}}\exp(-\frac{\kappa^{2}}{\kappa_{H}^{2}}), \tag{2}\]where \(\alpha\) is the power law exponent; \(\tilde{C}_{n}^{2}\) is the generalized structure parameter with \(m^{3-\alpha}\) as its unit; \(\kappa_{H}=c(\alpha)/l_{0}\). The function \(A(\alpha)\) maintains the consistency between the index structure function and its power spectrum. \[A(\alpha)=\frac{1}{4\pi^{2}}\Gamma(\alpha-1)\cos\left[\frac{\alpha\pi}{2} \right], \tag{3}\]  
\[C(\alpha)=\left\{\pi A(\alpha)\left[\Gamma(\frac{3-\alpha}{2})(\frac{3-\alpha }{3})\ +a_{1}\Gamma(\frac{4-\alpha}{2})(\frac{4-\alpha}{3})\ +a_{2}\Gamma(3-\frac{3\alpha}{4})(\frac{4- \alpha}{2})\right]\right\}^{\frac{1}{\alpha-5}}. \tag{4}\]  
In this paper, we add the anisotropic parameter \(\zeta\) in the power spectrum that s proposed in [18]. Subsequently, the anisotropic maritime refractive-index power spectrum can be written, as follows:  
\[\Phi_{n}(\kappa,\alpha,\zeta)=A(\alpha)\tilde{C}_{n}^{2}\zeta^{2-\alpha}[1+a_ {1}\frac{\kappa}{\kappa_{H}}+a_{2}(\frac{\kappa}{\kappa_{H}})^{3-\frac{\alpha }{2}}][\kappa^{2}+\tilde{\kappa}_{0}^{2}]^{-\frac{\alpha}{2}}\exp(-\frac{ \kappa^{2}}{\tilde{\kappa}_{H}^{2}}),(\kappa>0,3<\alpha<4), \tag{5}\]  
where \(\tilde{\kappa}_{0}^{2}=\kappa_{0}^{2}/\zeta^{2}\) and \(\tilde{\kappa}_{H}^{2}=\kappa_{H}^{2}/\zeta^{2}\). **3. Gaussian-Beam Wave Structure Function**  
The WSF that is associated with a Gaussian-beam waves is defined as the sum of the log-amplitude function and the phase structure function, which depends on the position of two observation points \(r_{1}\) and \(r_{2}\). Regarding the special case \(r_{1}=r_{2}\), the WSF of the Gaussian-beam wave can be written, as follows [1]:  
\[D(\rho,L)=8\pi^{2}k^{2}L\int_{0}^{1}\int_{0}^{\infty}\kappa\Phi_{n}(\kappa, \alpha,\zeta)\exp(-\frac{\Lambda L\kappa^{2}\xi^{2}}{k})\left\{I_{0}(\Lambda \rho\xi\kappa)-J_{0}[(1-\bar{\Theta}\xi)\kappa\rho]\right\}dxd\xi. \tag{6}\]  
where \(k=2\pi/\lambda\) is wave number of laser beam; \(\rho\) is scalar separation between two observation points; L is the propagation path length; \(\Lambda=2L/kW^{2}\) is the Fresnel ratio of beam at receiver; \(\Theta=1+L/R\) is the beam curvature parameter at receiver; R is the phase front radius of curvature; \(\overline{\Theta}=1-\Theta\), \(\Theta=\frac{\Theta_{0}}{\Theta_{0}^{2}+\Lambda_{0}^{2}}\), \(\Lambda=\frac{\Lambda_{0}}{\Theta_{0}^{2}+\Lambda_{0}^{2}}\). Moreover, \(\Theta_{0}=1-L/R_{0}\) and \(\Lambda_{0}=2L/kW_{0}^{2}\) are beam curvature parameter and Fresnel ratio at transmitter, respectively. \(J_{0}(\cdot)\) is \(zero^{th}\) order of the Bessel function of the first kind and \(I_{0}(\cdot)\) is \(zero^{th}\) order of the modified Bessel function of the first kind [28].