## 2 Power Spectrum Density in Anisotropic Turbulence  
In this section, we briefly review the derivation of the expression for the anisotropic power spectrum density (PSD) as an extension from the PSD in isotropic turbulence. Atmospheric (optical) turbulence is generated by a temperature differential between the earth's surface and the atmosphere and is the cause of the random fluctuations of the refractive index of air. Due to its random nature, the statistical theory is the most appropriate method for the description of this phenomenon. Let us start with the refractive index structure function in anisotropic turbulence obeying Kolmogorov statistics  
\[D_{n}(\mathbf{R})=C_{n}^{2}\Big{(}x^{2}/\mu_{x}^{2}+y^{2}/\mu_{y}^{2}+z^{2}/ \mu_{z}^{2}\Big{)}^{1/3}, \tag{1}\]  
where \(\mathbf{R}\) is a vector spatial variable; \(C_{n}^{2}\) is the structure parameter with unit m\({}^{-2/3}\). \(\mu_{x}\), \(\mu_{y}\) and \(\mu_{z}\) are the anisotropy factors in \(x\), \(y\), and \(z\) directions, respectively. Please note that Equation (1) is only valid in the so-called inertial subrange which is \(l_{0}<x^{2}/\mu_{x}^{2}+y^{2}/\mu_{y}^{2}+z^{2}/\mu_{z}^{2}<L_{0}\), where \(l_{0}\) and \(L_{0}\) denote the inner and outer scale of the turbulence, respectively. To evaluate the PSD from the structure function, we make the following change of the variables  
\[x^{\prime}=x/\mu_{x},y^{\prime}=y/\mu_{y},z^{\prime}=z/\mu_{z}, \tag{2}\]  
\[\kappa_{x}^{\prime}=\mu_{x}\kappa_{x},\kappa_{y}^{\prime}=\mu_{y}\kappa_{y}, \kappa_{z}^{\prime}=\mu_{z}\kappa_{z}, \tag{3}\]  
where \(\kappa\equiv\big{(}\kappa_{x},\kappa_{y},\kappa_{z}\big{)}\) is the spatial frequency. On substituting Equation (2) into Equation (1), the calculation of the PSD becomes the same as in the case of isotropic and homogeneous turbulence in the (R\({}^{\prime}\),\(\kappa^{\prime}\)) domain. Thus, the PSD is directly related to the structure function by the following integral formula [1]  
\[\Phi_{n}(\kappa^{\prime})=\frac{\mu_{x}\mu_{y}\mu_{z}}{4\pi^{2}\kappa^{\prime 2 }}\int_{0}^{\infty}\frac{\sin\kappa^{\prime}R^{\prime}}{\kappa^{\prime}R^{ \prime}}\frac{d}{dR^{\prime}}\bigg{[}{R^{\prime}}^{2}\frac{d}{dR^{\prime}}D_{ n}(R^{\prime})\bigg{]}dR^{\prime}. \tag{4}\]  
By inserting Equation (1) into Equation (4) and after integrating over R\({}^{\prime}\) we obtain the analytical expression for the anisotropic PSD  
\[ \begin{array}{l}\Phi_{n}(\kappa)=0.033\mu_{x}\mu_{y}\mu_{z}C_{n}^{2}\kappa^ {\prime-11/3}\\ =\frac{0.033\mu_{x}\mu_{y}\mu_{z}C_{n}^{2}}{\big{[}\mu_{x}^{2}\kappa_{z}^{2}+ \mu_{y}^{2}\kappa_{y}^{2}+\mu_{z}^{2}\kappa_{z}^{2}\big{]}^{11/6}},\end{array}  \tag{5}\]  
Equation (5) is the Kolmogorov power spectrum for anisotropic turbulence with anisotropy along \(x\), \(y\), and \(z\) direction. When the anisotropy factors \(\mu_{x}\), \(\mu_{y}\) and \(\mu_{z}\) equal unity, Equation (5) reduces to the expression for the well-known Kolmogorov power-law spectrum. However, this model is only valid in the inertial subrange \(1/\text{L}_{0}\ll\kappa^{\prime}\ll 1/\text{l}_{0}\) due to the range of validity for R\({}^{\prime}\) in the structure function. Following the method for extending the Kolmogorov power spectrum into the dissipation range \(\kappa^{\prime}>1/\text{l}_{0}\) and the input range \(\kappa^{\prime}\gg 1/\text{L}_{0}\) used by [PERSON] and [PERSON] [1], we may modify the PSD in Equation (5) including the effect of the inner and outer scale of turbulence into it, which is \[\Phi_{n}(\kappa)=\frac{0.033u_{x}u_{y}u_{z}C_{z}^{2}}{\left(\kappa^{ \prime 2}+\kappa_{eff}^{2}\right)^{11/6}}\exp\left(-\frac{\kappa^{\prime 2}}{ \kappa_{m}^{2}}\right) \tag{6}\] \[=\frac{0.033u_{x}u_{y}u_{y}u_{z}C_{z}^{2}}{\left(\kappa_{x}^{2}+ \kappa_{y}^{2}\kappa_{y}^{2}+\kappa_{z}^{2}\kappa_{z}^{2}+\kappa_{eff}^{2} \right)^{11/6}}\exp\left(-\frac{\kappa_{x}^{2}\kappa_{x}^{2}+\kappa_{y}^{2} \kappa_{y}^{2}+\mu_{x}^{2}\kappa_{z}^{2}}{\kappa_{m}^{2}}\right),\]  
where \(\kappa_{\rm eff}=2\pi/L_{0}\) and \(\kappa_{\rm m}=5.92/l_{0}\).