## 3 Propagation Characteristics of the NUC Beams in Anisotropic Turbulence Along the Horizontal Links  
### Spectral Density of the NUC Beams Propagation through Anisotropic Turbulence  
According to [35], the cross-spectral density (CSD) function of the NUC beams in the source plane is represented by the following integral formula:  
\[W_{x}(x_{1},x_{2})=\int p_{x}(v)H_{x}^{*}(x_{1},v)H_{x}(x_{2},v)dv. \tag{7}\]  
The functions \(p_{x}(v)\) and \(H_{x}(x,v)\) entering in Equation (7) take the form:  
\[p_{x}(v)=\left(\pi a_{x}^{2}\right)^{-1/2}\exp(-v^{2}/a_{x}^{2}), \tag{8}\]  
\[H_{x}(x,v)=\exp\left(-\frac{x^{2}}{2w_{0}^{2}}\right)\exp[-ik(x-x_{0})^{2}v]. \tag{9}\]  
where k = 2\(\pi/\lambda\) is the wavenumber with \(\lambda\) being the wavelength of a light beam; \(w_{0}\) and \(x_{0}\) are real constants. On substituting Equations (8) and (9) into Equation (7) and integrating over \(v\), the expression for the CSD function of the NUC beam becomes:  
\[W_{x}(x_{1},x_{2})=\exp\left(-\frac{x_{1}^{2}+x_{2}^{2}}{2w_{0}^{2}}\right)\exp \left[-\frac{\left(x_{2}-x_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}}{ \delta_{x}^{4}}\right]. \tag{10}\]  
where \(\delta_{x}=\sqrt{2/\rm{ka}_{x}}\), is the r.m.s correlation width. Please note that Equation (10) only represents the one-dimensional case for the NUC beams. The extension of the NUC beams to the two-dimensional case is straightforward, i.e., W\({}_{0}\)(r\({}_{1}\),r\({}_{2}\)) = W\({}_{x}\)(x\({}_{1}\),x\({}_{2}\))W\({}_{y}\)(y\({}_{1}\),y\({}_{2}\)). The function W\({}_{y}\)(y\({}_{1}\),y\({}_{2}\)) is immediately derived from W\({}_{x}\)(x\({}_{1}\),x\({}_{2}\)) replacing \(x\) with \(y\) in Equation (10). r\({}_{1}\equiv\) (x\({}_{1}\),y\({}_{1}\)) and r\({}_{2}\equiv\) (x\({}_{2}\),y\({}_{2}\)) are two position vectors in the source plane. It can be seen that the two-dimensional CSD function is a separable function with respect to the variables (x\({}_{1}\),x\({}_{2}\)) and (y\({}_{1}\),y\({}_{2}\)). In the presence of anisotropic turbulence and within the validity of the paraxial approximation, the spectral density of the NUC beams in a plane at distance \(z\) from the source plane (\(z=0\)) can be treated by the following eHF integral  
\[S(\rho,z)=\left(\frac{k}{2\pi z}\right)^{2}\int\int W_{0}({\bf r }_{1},{\bf r}_{2})\exp\left[\frac{ik}{2z}\left(\left|{\bf r}_{2}-\rho\right|^{2 }-\left|{\bf r}_{1}-\rho\right|^{2}\right)\right] \tag{11}\] \[\times\langle\exp(\psi^{*}({\bf r}_{1},\rho,z)+\psi({\bf r}_{2}, \rho,z))\rangle_{m}d^{2}{\bf r}_{1}d{\bf r}_{2},\]  
with  
\[ \begin{array}{l}T=\int_{0}^{\infty}\kappa^{\prime 3}\Phi_{n}(\kappa^{\prime})dx^{ \prime}/\mu_{x}\mu_{y}\\ =0.0033\mu_{x}C_{n}^{2}\biggl{[}\Gamma\biggl{(}\frac{1}{\delta},\frac{x_{eff}^ {2}}{\kappa_{n}^{2}}\biggr{)}\frac{(6x_{eff}^{2}+5x_{m}^{2})}{\kappa_{n}^{2}/ 6}-6\kappa_{eff}^{1/3}\biggr{]},\end{array}  \tag{13}\]  
where \(\mathrm{x_{d}}=\mathrm{x_{2}}-\mathrm{x_{1}}\) and \(\mathrm{y_{d}}=\mathrm{y_{2}}-\mathrm{y_{1}}\). \(\Gamma\) is the Gamma function. As shown in Equation (13), \(\mathrm{F_{2}}\) is also a separable function with respect to (\(\mathrm{x_{1}}\),\(\mathrm{x_{2}}\)) and (\(\mathrm{y_{1}}\),\(\mathrm{y_{2}}\)). Therefore, we can rewrite Equation (11) as a product of the \(x\) and \(y\) parts of the spectral density in anisotropic turbulence, i.e.,  
\[S(\rho,z)=S_{x}(\zeta,z)S_{y}(\eta,z), \tag{14}\]  
with  
\[ \begin{array}{l}S_{\alpha}(\beta,z)=\frac{k}{2\pi z}\int W_{\alpha}(\alpha_ {1},\alpha_{2})\exp\biggl{[}\frac{ik}{2z}(\alpha_{2}^{2}-\alpha_{1}^{2})- \frac{ik\beta}{z}(\alpha_{2}-\alpha_{1})\biggr{]}\\ \hskip 14.226378 pt\times\exp\biggl{(}-\frac{\pi^{2}k^{2}Tz(\alpha_{2}-\alpha_ {1})^{2}}{3\mu_{x}^{2}}\biggr{)}dx_{1}da_{2},(\alpha=x,y;\beta=\zeta,\eta). \end{array}  \tag{15}\]  
In general, it is impossible to acquire the analytical expression for the spectral density from Equation (15).