## 3 Propagation Characteristics of the NUC Beams in Anisotropic Turbulence Along the Horizontal Links
### Spectral Density of the NUC Beams Propagation through Anisotropic Turbulence
On substituting Equation (16) into Equation (15) and after some manipulations and integration, we obtain the approximate analytical expression for the \(x(y)\) part of the spectral density  
\[\begin{array}{l}S_{\alpha}(\beta,zLet us consider the NUC beams propagating along the horizontal path near the ground with the \(z\) axis being the main propagation direction. It is reasonable to assume that the anisotropic factor \(\mu_{x}\) equals to \(\mu_{z}\) and is larger than \(\mu_{y}\). The initial parameters used in the calculation are \(\omega_{0}=30\) mm, \(\delta_{x}=\delta_{y}=30\) mm, \(\lambda=632.8\) nm, \(\mu_{x}=\mu_{z}=2\), \(\mu_{y}=1\), \(l_{0}=0.01\) m, \(L_{0}=1.0\) m and fixed throughout the text unless other values are specified.  
Figure 2 illustrates the spectral density of the NUC beam with \(x_{0}=0\) at different propagation distances both in free space (\(C_{n}^{2}=0\)) and in the presence of anisotropic turbulence with two different values of the constant structures \(C_{n}^{2}=10^{-17}\) m\({}^{-2/3}\) and \(C_{n}^{2}=10^{-15}\) m\({}^{-2/3}\). In free space, because of the initial state of partial coherence, the beam gradually transits from a Gaussian profile to a diamond shape as the propagation distance increases. The position of the intensity maxima keeps unchanged on propagation since the initial value of \(x_{0}=y_{0}=0\) equals zero. Please note that the maximum intensity at the transverse plane is enhanced on propagation, i.e., the values of the intensity maxima exceed one which is the maxima value in the source plane [see the color bar in Figure 2a-d]. In the presence of anisotropic turbulence, the beam profile loses its diamond shape, and changes into an elliptical shape in the far field [see in Figure 2h,l]. When the strength of turbulence increases, the beam profile turns into an elliptical shape at a relative short distance [seen in Figure 2i-l] and keeps its elliptical shape on further propagation. The maximum intensity (on-axis point) is greatly suppressed owing to the effects of turbulence. As shown Equation (12), (i.e., the second-order statistics of phase correlation), the effect of turbulence on the beam propagation is that it looks like imposing an elliptical Gaussian-Schell-model term on the beam. However, when the propagation distance is short, or the turbulence is weak, the parameter \(T\) in Equation (12) is very small. Hence the turbulence effect can then be neglected. In this case, the beam profile keeps its free-space features. When the propagation distance is large enough, or the turbulence is strong, the situation is reversed. The effect of turbulence dominates the central role to determine the beam profile and turns the beam profile into an elliptical shape. In our numerical examples, the anisotropy factor in \(x\) direction is larger than that in the \(y\) direction, implying that the strength of the turbulence in the \(y\) direction is much stronger. Thus, the spectral density of the NUC beams are elongated along the \(y\)-axis. Here, we should emphasize that the turbulence we consider belongs to weak turbulence. According to [1], a widely used parameter to judge whether the turbulence is weak is the Rytov variance, defined as \(\sigma_{R}^{2}~{}=~{}1.23C_{n}^{2}K^{2/6}z^{11/6}\). If \(\sigma_{R}^{2}<1\), the turbulence is weak. In our numerical examples, when the propagation distance is  
Figure 1: Variation of spectral density \(S_{x}(\xi,z)\) with \(\xi\) at three different propagation distances in free space (**a–c**) and in the presence of turbulence (**d–f**). The solid lines are calculated from the approximate analytical formula shown in Equation (17). The circular dots are the numerical integration directly from Equation (15). It shows that irrespective of propagation in free space or in turbulence, the spectral density obtained from the approximate analytical formula Equation (17) is consistent with that obtained from direct numerical integration.  
z = 3.0 km, the calculated values of the parameter is about 0.004 and 0.4 for \(C_{n}^{2}=10^{-17}\) m\({}^{-2/3}\) and \(C_{n}^{2}=10^{-15}\) m\({}^{-2/3}\), respectively.  
Figure 3 shows the spectral density of the NUC beam with \(x_{0}=y_{0}=0.7\omega_{0}\) at four different propagation distances in free space and in the presence of anisotropic turbulence. In the case of free space, the beam profile gradually turns into a sector-like shape during free-space propagation and lateral shifting of the intensity maxima occurs, which is much different from those with \(x_{0}=y_{0}=0\) [see Figures 2a-d and 3a-d]. It is shown in Figure 3e-l that the spectral density is strongly affected by anisotropic turbulence, especially for \(C_{n}^{2}=10^{-15}\) m\({}^{-2/3}\), i.e., the evolution of the spectral density for \(x_{0}=y_{0}=0.7\omega_{0}\) is quite similar to that of the NUC beam for \(x_{0}=y_{0}=0\). This phenomenon can be explained by the fact that there are two factors determining the evolution of the spectral density with the propagation distance. One is the initial correlation properties of the field, and the other one is the effect of turbulence. When the turbulence is weak, or the propagation distance is short, the former factor determines the feature of the spectral density. When the turbulence is relative strong, or the propagation distance is long enough, the latter factor plays a central role in determining the evolution properties of the spectral density. Therefore, the evolution properties of the NUC beams between \(x_{0}=y_{0}=0\) and \(x_{0}=y_{0}=0.7\omega_{0}\) are almost the same when \(C_{n}^{2}=10^{-15}\) m\({}^{-2/3}\).  
Figure 4 illustrates the evolution of the on-axis intensity (intensity maxima) of the NUC beams (\(x_{0}=y_{0}=0\)) with different values of coherence widths as a function of the propagation distance \(z\) in free space and in the presence of anisotropic turbulence. For comparison, the variation of the on-axis intensity of the fundamental Gaussian beam (dash-dot lines) with the propagation distance is also plotted in Figure 4. In the calculation, the initial beam width of the Gaussian beam is the same as that of the NUC beams. One finds that the value of the on-axis intensity of the NUC beams exceeds that in the source plane during free-space propagation, implying that a sharpened intensity distribution occurs. For NUC beams with large values of the coherence width the sharpened intensity distribution can last for long propagation distances, compared to those with smaller values of the coherence width. However, the maximum value decreases as the coherence width increases. Please note that after reaching the maximum value, the on-axis intensity decays much faster than that of the Gaussian beam when the propagation distance increases further. In the presence of turbulence, the sharpened intensity distribution of the NUC beams is strongly affected by the strength of the turbulence. The propagation  
Figure 2: Density plots of the spectral density of the NUC beam with \(x_{0}=y_{0}=0\) propagation through free space and in the presence of anisotropic turbulence at several different propagation distances.  
distance, which retains the sharpened intensity, is greatly shortened, compared to the case in free space. When the strength of the turbulence is \(C_{n}^{2}=10^{-15}\) m\({}^{-2/3}\), the NUC beams with three values of coherence width almost have the same values for the on-axis intensity. These values of the on-axis intensity are much lower than those for a Gaussian beam with a propagation distance larger than 600 m [see in Figure 4d].  
Figure 5 gives the variation of the on-axis intensity of the NUC beams (\(x_{0}=y_{0}=0\)) with \(\omega_{0}=0.01\) m, 0.02 m, and 0.03 m as a function of the propagation distance in free space and in the presence of anisotropic turbulence with different anisotropy factors. One can see in Figure 5a that for different initial beam sizes the maximum values of the on-axis intensity are the same. However,  
Figure 3: Density plots of spectral density of the NUC beam with \(x_{0}=y_{0}=0.7\omega_{0}\) in free space and in anisotropic turbulence at different propagation distances.  
the propagation distance for which the maximum intensity occurs is closely related to the initial beam size. The larger the values of the initial beam size, the longer the propagation distances needed for obtaining the maximum intensity. In turbulence with different anisotropy factors [see in Figure 5b-d], it can be seen that as the anisotropy factor \(\mu_{x}\) increases, the values of the on-axis intensity of the NUC beams with larger beam size decrease more rapidly. From Figures 4 and 5, one may also conclude that the NUC beams still have the ability to retain the sharpened intensity distribution in atmospheric turbulence when the structure constant \(C_{n}^{2}\) is smaller than \(10^{-16}\) m\({}^{-2/3}\). One can modulate the initial beam size or coherence width to acquire the maximum intensity at the receiver plane. This maximum intensity may even be higher than that of the Gaussian beam under the same condition. This peculiar property of the maximum intensity is quite useful for an information carrier in free space. The anisotropy factors also have the additional effects to affect the evolution of the on-axis intensity.  
If the parameters \(x_{0}\) and \(y_{0}\) of the NUC beams are not zero, the position of the intensity maxima will shift laterally from the on-axis point upon free-space propagation due to the non-uniform source correlations. To examine the behavior of the lateral shifting in the presence of anisotropic turbulence, we plot in Figure 6 the 3D-graphs of the position of the intensity maxima (black lines) both in free space and in the presence of turbulences with different anisotropy factors. The red, green, and blue lines in Figure 6 denote the projection of the black lines in the \(\xi\)-\(\eta\), \(\xi\)-\(z\) and \(\eta\)-\(z\) planes, respectively. In free space, the lateral shift of the intensity maxima first increases with the increase of the propagation distance, and then keeps a fixed value almost unchanged upon propagation. The shifts along the \(x\) and \(y\) direction are the same because we set the initial parameter \(x_{0}\) equals to \(y_{0}\) in the calculation. In the case of isotropic turbulence [see in Figure 6b], the effect of the turbulence is to suppress the shift of the intensity maxima, which is consistent with the analysis in [36]. Please note that the shifts along the \(x\) and \(y\) direction are the same, as expected. In the case of anisotropic turbulence [see in Figure 6c,d], the projection of the trace in the \(\xi\)-\(\eta\) plane becomes a curved line instead of a line, as in the case of isotropic turbulence. This indicates that the traces of the lateral shifts of the intensity maxima are not lying in an arbitrary plane. In our numerical case (\(\mu_{x}>\mu_{y}\)), the strength of the turbulence in the \(x(\xi)\) direction is weaker than that in the \(y(\eta)\) direction. Therefore, the lateral shift in the \(x(\xi)\) direction is always larger than that in the corresponding \(y(\eta)\) direction.  
Figure 5: Variation of the intensity maxima of the NUC beams (\(x_{0}=y_{0}=0\)) with the propagation distance in free space in (**a**) and in anisotropic turbulence with different anisotropic factors in (**b–d**).  
In a practical situation one is concerned about the amount of energy carried by the beam received by a detector in the receiver plane. The more energy the detector receives, the higher the efficiency of the optical system. However, the receiver's aperture has finite dimensions in a practical system. Consequently, a beam whose intensity is concentrated on a small region during propagation has a certain ability to improve the energy receiving rate. Based on the previous analysis, the NUC beams may have a higher receiving energy rate, even compared to the Gaussian beam under the same conditions. Let us consider such an optical system for free-space optical communication. A laser beam, which is acting as an information carrier, is sent to the receiver via an atmospheric channel. The receiver lens collects a portion of the transmitted optical field, which is focused onto a photodetector. This portion is used to convert the optical information into electrical signals for processing.  
Assume that the aperture of the receiver lens has a square shape for simplicity. Therefore, the ratio of the beam power captured by the photodetector and the total power carried by the beam, known as power-in-the-bucket (_PIB_), can be expressed as  
\[PIB=\frac{\int_{-D/2}^{D/2}\int_{-D/2}^{D/2}S(\xi,\eta,z)d\xi d\eta}{\int\int S (\xi,\eta,z)d\xi d\eta}, \tag{19}\]  
where \(D\) is the width of the square aperture of the lens.