## 3 Propagation Characteristics of the NUC Beams in Anisotropic Turbulence Along the Horizontal Links
### Spectral Density of the NUC Beams Propagation through Anisotropic Turbulence
By inserting Equation (17) into (19) and after integrating over \(\xi\) and \(\eta\), we can obtain the analytical expression for the _PIB_:  
\[PIB=\frac{p_{x}(D)p_{y}(D)}{\pi w_{0}^{2}}, \tag{20}\]  
with  
\[ \begin{array}{c}p_{\alpha}=\sum\limits_{n=1}^{N}\frac{kw_{0}\delta_{\alpha}^ {2}\Delta v}{4}\exp\Bigl{(}-\frac{k^{2}\delta_{\alpha}^{2}v_{\alpha}^{2}}{4} \Bigr{)}\Bigl{[}Erf\Bigl{(}\frac{22v_{\alpha}a_{0}+D/2}{\sqrt{M_{\alpha}}} \Bigr{)}\\ -Erf\Bigl{(}\frac{22v_{\alpha}a_{0}-D/2}{\sqrt{M_{\alpha}}}\Bigr{)}\Bigr{]},( \alpha=x,y)\end{array}  \tag{21}\]  
where _Erf_ denotes the error function. Figure 6: 3D-plots (black lines) of the evolution of the position of the intensity maxima during propagation in free space and in the presence of atmospheric turbulence. Red, green, and blue lines are the projections in the \(\xi\)-\(\eta\), \(\xi\)-\(z\) and \(\eta\)-\(z\) planes, respectively. **(a)** free space; **(b)**\(\mu_{x}=1\), \(\mu_{y}=1\); **(c)**\(\mu_{x}=2\), \(\mu_{y}=1\); **(d)**\(\mu_{x}=3\), \(\mu_{y}=1\). The parameters \(x_{0}\) and \(y_{0}\), used in the calculation, are \(x_{0}=y_{0}=0.7\omega_{0}\). The dependence of the _PIB_ of the NUC beam (\(x_{0}=y_{0}=0\)) on the propagation distance in free space, in the presence of isotropic turbulence or anisotropic turbulence, is shown in Figure 7a-c. The size of the aperture \(D\) used in the calculation is 0.04 m. For comparison, the corresponding results of the _PIB_ of the Gaussian beam which have the same initial beam width as the NUC beam (\(\omega_{0}=30\) mm), are also plotted in Figure 7a-c (dashed lines). It is shown that the _PIB_ of the NUC beam is larger than that of the Gaussian beam under the same condition on free-space propagation from the source z = 0 to a distance of 3 km away from it. This means that more power will be received by the detector if illuminated by the NUC beam. However, this advantage is weakened or disappears in the presence of turbulence, be it isotropic or anisotropic turbulence. As shown in Figure 7b, the advantage of the NUC beam is only in the region 0-2.5 km. The _PIB_ of the NUC beam drops rather quickly when the propagation distance is larger than 1.5 km. In the case of anisotropic turbulence, the situation becomes worse. It only keeps its superior behavior for propagation distances less than 2.0 km [see in Figure 7c]. Our other numerical results (not shown here) indicate that when the strength of turbulence is larger than \(10^{-15}\) m\({}^{-2/3}\), the NUC beam will lose its advantages. To quantitatively assess the superiority of the NUC beam, compared to the Gaussian beam, the following parameter is introduced to characterize it, i.e.,  
\[P=\frac{PIB_{NUC}-PIB_{Gau}}{PIB_{Gau}}. \tag{22}\]  
In Equation (22), the _PIB_ with the subscript \"\(NUC\)\" and \"\(Gau\)\" denote the power-in-bucket of NUC beams and Gaussian beams, respectively. The positive of \(P\) means the NUC beams are superior to the Gaussian beams. Figure 7d shows the variation of \(P\) against the propagation distance both in free space and in the presence of atmospheric turbulence. One can see that the maximum value of \(P\) is about 25% or 20% near the propagation distance z = 1.5 km in the case of isotropic or anisotropic turbulence, respectively. Figure 8 illustrates the dependence of P on the width of the receiver aperture, the anisotropic factor \(\mu_{x}\) with \(\mu_{y}=1\) and \(D=0.02\) m, and the strength of turbulence with \(D=0.02\) m at \(z=1.5\) km. It is shown in Figure 8a that for \(C_{n}^{2}=10^{-17}\) m\({}^{-2/3}\) more power can be received using a NUC beam than using a Gaussian beam when the aperture is relatively small (\(D<0.05\) m), irrespectively of whether isotropic or anisotropic turbulence is present.