## 2 Spectrum of Anisotropic Oceanic Turbulence  
Plenty of turbulence models, such as Kolmogorov spectrum, Tatarskii spectrum and von Karman spectrum, have been used to characterize the turbulent environment. Considering the effects of both the inner and outer scales of oceanic turbulence, the modified Kolmogorov spectrum of refractive-index fluctuations is employed in this study and is expressed as [33].  
\[\Phi_{n}(\kappa)=0.033C_{n}^{2}\bigg{[}1+a_{1}\bigg{(}\frac{\kappa}{\kappa_{i }}\bigg{)}+a_{2}\bigg{(}\frac{\kappa}{\kappa_{i}}\bigg{)}^{7/6}\bigg{]}\frac{ \exp\left(-\kappa^{2}/\kappa_{i}^{2}\right)}{\left(\kappa^{2}+\kappa_{o}^{2} \right)^{11/6}}. \tag{1}\]Note that we mainly consider the temperature-induced fluctuation in this study. Therefore, only the seawater temperature-induced spectrum model is employed here. Where \(C_{n}^{2}\) represents the refractive-index structure parameter for Kolmogorov turbulence and has units of m\({}^{-2/3}\), \(\kappa_{i}=3.3/L_{i}\) and \(\kappa_{o}=2\pi/L_{o}\), where \(L_{i}\) and \(L_{o}\) are the inner and outer scales, respectively. Note that \(a_{1}=-0.061\) and \(a_{2}=2.836\) in a maritime turbulent environment according to [34]. \(\kappa=\sqrt{\kappa_{x}^{2}+\kappa_{y}^{2}+\kappa_{z}^{2}}\) is the spatial frequency of the oceanic turbulence fluctuations. This spectral model is extensively used for theoretical calculations over both atmosphere and ocean environment [36]. The accuracy of this model has been well confirmed with the experimental data under oceanic turbulence [37].  
According to the anisotropic property of oceanic turbulence, as discussed in Section 1, we use the concept of anisotropy in atmospheric turbulence [38] together with the spectral density index of refraction fluctuations under isotropic oceanic turbulence in this study [39]. Therefore, the spectral density index of refraction fluctuations under the anisotropic oceanic turbulence can be written as  
\[\Phi_{n}(\kappa)= 0.033C_{n}^{2}\mu_{x}\mu_{y}\left[1+a_{1}\left(\frac{\kappa^{ \prime}}{\kappa_{i}}\right)+a_{2}\left(\frac{\kappa^{\prime}}{\kappa_{i}} \right)^{7/6}\right]\frac{\exp\left(-\kappa^{2}/\kappa_{i}^{2}\right)}{\left( \kappa^{\prime 2}+\kappa_{o}^{2}\right)^{11/6}}\quad. \tag{2}\]  
In this paper, only the anisotropy turbulence along the propagation direction is considered for simplicity, i.e., the \(z\) direction of the beam, and is accounted for by the influential anisotropic factors \(\mu_{x}\) and \(\mu_{y}\), which are in the \(x\) and \(y\) directions, respectively. Note that this model can be recast to an isotropic case when \(\mu_{x}=\mu_{y}=1\). We also have \(\kappa_{x}^{\prime}=\mu_{x}\kappa_{x}\), \(\kappa_{y}^{\prime}=\mu_{y}\kappa_{y}\), \(\kappa_{z}^{\prime}=\kappa_{z}\), and \(\kappa^{\prime}=\sqrt{\kappa_{x}^{\prime 2}+\kappa_{y}^{\prime 2}+\kappa_{z}^{ \prime 2}}=\sqrt{\mu_{x}^{2}\kappa_{x}^{2}+\mu_{y}^{2}\kappa_{y}^{2}+\kappa_{ z}^{2}}\), where \(\kappa_{x}\), \(\kappa_{y}\) and \(\kappa_{z}\) are the \(x\), \(y\) and \(z\) components of the spatial frequency, respectively.  
For simplification, we use the Markov approximation [40], which is usually used in the theory of wave propagation in random media, with the assumption that the refractive index is delta-correlated at any pair of points located along the direction of propagation. By invoking the Markov approximation, we ignore \(\kappa_{z}\), i.e., the spatial wavenumber component along the direction of propagation. Therefore, Equation (2) can be further rewritten as  
\[\begin{split}\Phi_{n}\left(\mu_{x},\mu_{y},\kappa_{x},\kappa_{y} \right)&=0.033C_{n}^{2}\mu_{x}\mu_{y}\left[1+a_{1}\left(\frac{ \sqrt{\mu_{x}^{2}\kappa_{x}^{2}+\mu_{y}^{2}\kappa_{y}^{2}}}{\kappa_{i}}\right) \right.\\ &\left.+a_{2}\left(\frac{\sqrt{\mu_{x}^{2}\kappa_{x}^{2}+\mu_{y}^ {2}\kappa_{y}^{2}}}{\kappa_{i}}\right)^{7/6}\right]\frac{\exp\left[-\left(\mu _{x}^{2}\kappa_{x}^{2}+\mu_{y}^{2}\kappa_{y}^{2}\right)/\kappa_{i}^{2}\right] }{\left(\mu_{x}^{2}\kappa_{x}^{2}+\mu_{y}^{2}\kappa_{y}^{2}+\kappa_{o}^{2} \right)^{11/6}}\end{split} \tag{3}\]