Considerable amount of data has been collected in the past asserting that atmospheric turbulence has regions where it exhibits anisotropic statistics. For instance, it is known that the fluctuations in the refractive index within the first meter above the ground are typically stronger in the vertical direction compared with those in the horizontal directions. We have investigated the second-order statistical properties of a Gaussian Schell-model (GSM) beam traversing anisotropic atmospheric turbulence along a horizontal path. Analytical expression is rigorously derived for the cross-spectral density function of a GSM beam. It is shown that the spread of the beam and its coherence properties become different in two transverse directions due to anisotropy. In the limiting case when the source coherence width becomes infinite our results reduce to those for Gaussian beam propagation. Source partial coherence is shown to mitigate anisotropy at sub-kilometer distances.
© 2016 Optical Society of America
Since the seminal experimental work of Consortini et al.  (see also [2,3]) it became apparent that in some regions of the Earth’s atmosphere the optical turbulence may exhibit anisotropic features. Namely, while in two horizontal directions it has essentially the same statistics, in the vertical direction it may substantially differ (from horizontal) being typically somewhat stronger [4–7]. Hence the atmospheric anisotropic turbulence can be viewed as an ensemble of “eddy ellipsoids” or “eddy pancakes” whose smallest semi-axis coincides with the vertical direction. In the isotropic limit the ellipsoidal eddies reduce to spheres. Hence two types of propagation problems are of interest: along vertical and along horizontal paths (slant path propagation is being a much more involved subject at the moment). In the former case the beam experiences equal perturbations along two directions transverse to its propagation path and, hence, turbulent anisotropy acts only as a modifier to the turbulence’s strength. Theoretical results on the optical beam propagation in such scenario belong to [8–11]. On the other hand, beam propagation along the horizontal paths is more challenging since anisotropy affects the wave differently in two transverse directions. This leads to interesting phenomena, for example, after propagation at a sufficiently large distance an initially circularly symmetric Gaussian beam acquires elliptical shape, with the ellipse being elongated in the vertical direction [12,13].
Atmospheric anisotropic turbulence can be also simulated with the help of the Spatial Light Modulators (SLM)  that produce spatially and temporally resolved phase perturbations with the given power spectra. Both vertical and horizontal propagation scenarios have been recently investigated by this method [15,16] for the Gaussian beam and agreed with the theoretical predictions.
In this paper we will extend the work in Refs [12,13,16]. by considering propagation of scalar Gaussian Schell-Model (GSM) beams  in anisotropic turbulence along the horizontal paths, theoretically. In the coherent limit our results reduce to those for the Gaussian beam. Moreover, in contrast with a heuristic approaches taken in [13,16] to model beam spreading and coherence state in two transverse directions, we derive these statistics directly from the anisotropic power spectrum of the refractive index fluctuations. For our derivation we employ the extended Huygens-Fresnel integral given in  for anisotropic case, and the non-Kolmogorov, anisotropic power spectrum of . Moreover, we show that for beams with Cartesian symmetry the propagating spectral density can be expressed as a product of two terms, each being the convolution function of Fourier transforms of the scaled source spectral density, source degree of coherence and complex phase correlation function of turbulence, where each of the six terms is calculated at a direction vector to the point of interest. One of the important results of our study is the fact that for random beams the effects of turbulent anisotropy are reduced, at least at sub-kilometer distances, as compared to the deterministic beams, in the sense that the ellipse-like transverse cross-section of the beam reduces to a circular one with the decrease of source degree of coherence.
The paper is organized as follows: in section 2 the theory of the scalar GSM beam propagation in anisotropic turbulence is given; in section 3 the examples relating to the spectral density and to the degree of coherence are given; section 4 offers a qualitative analysis of the beam propagation and section 5 summarizes the results.
2. Scalar GSM beam propagation along horizontal paths in anisotropic turbulence
Let a beam generated in the source plane z = 0 propagate through an anisotropic turbulence along a horizontal path coinciding with axis z (see Fig. 1). Suppose that the incident beam is generated by a scalar GSM source whose spectral density (SD) and the Degree of Coherence (DOC) are Gaussian functions . The second-order statistical properties of such a source are characterized by the following cross-spectral density (CSD) function
Within the validity of paraxial approximation, the CSD function of a partially coherent beam after propagating through atmospheric turbulence at distance z can be related with that in the source plane with the help of the extended Huygens-Fresnel integral formula , i.e.,Eq. (3), the Markov approximation is applied, implying that the fluctuations in the refractive index are delta-correlated at any pair of points in the direction of propagation. For isotropic turbulence, Eq. (3) reduces to expressionEq. (4) is widely applied in exploring the second-order statistical properties of the laser beams propagation through isotropic turbulence [20–23].
In this paper we will employ the non-Kolmogorov anisotropic power spectrum having the van Karman formEq. (5) reduces to that introduced in Ref . Further, when and , the power spectrum reduces to the conventional isotropic Kolmogorov spectrum, i.e., .Eq. (8) takes formEq. (8) turns out to be21], the Bessel function in Eq. (10) can be expressed as the first two terms of the Tylor expansion, to a good approximation: . By applying the approximation and integrating over t,, we find that Eq. (10) becomesEq. (11) that the second-order correlation function of the complex phase perturbation induced by the anisotropic turbulence is a separable function with respect to the variables and . This result is consistent with that of a wave optical simulation and a spatial light modular-based experiment presented in Ref .Eq. (14) areEq. (14), respectively. Eq. (13) provides a convenient analytical expression to investigate the statistical properties of the GSM beam in anisotropic turbulence such as the SD and the DOC 
3. Numerical examples
We will now illustrate the evolution of the SD and the DOC of the GSM beam propagating in the anisotropic turbulence along horizontal paths based on the analytical expressions derived in section 2 by a number of typical examples. The initial parameters in the numerical calculation are chosen to be , ,,,, and unless different values are specified.
Fig. 2 shows density plots (false-color renderings) of the SD of the GSM beam at several propagation distances for three different values of initial coherence widths and . One finds that in the case of a fairly coherent source, with the beam profile gradually transits from a circular shape in the source plane [see Fig. 1(a)] to an elliptical shape [see Figs. 1(c)-1(d)] on propagation. However, the transition from the circular shape to the elliptical shape becomes much slower as the initial coherence width decreases. When , the SD shape remains the nearly circular from the source plane to the propagation distance z = 5km. Therefore, the GSM beam with a small coherence width can reduce the anisotropic effect of turbulence compared to that with large coherence width.
The density plots (false-color renderings) of the modulus of the DOC at two points and corresponding to Fig. 1 are shown in Fig. 3. It is interesting to find that the evolution of the DOC distribution is very similar for three different values of the initial coherence width. The DOC profile exhibits the obviously elliptical shape at the selected propagation distances. We also notice that the long axis of the DOC profile is along the x (horizontal) direction, while the long axis of the SD is along y (vertical) direction (refer to Fig. 1 for the orientations).
4. Analysis and discussion
To investigate the difference between the behavior of the SD and DOC of the GSM beam in detail, we formulate the beam width and the coherence width of the GSM beam along the x(y) direction in the plane of z by comparing Eq. (13) to Eq. (1), i.e.,Eq. (19) that the first term in the right side of the square root is the initial SD width; the second and third terms stand for diffraction of the beam on free-space propagation induced by σ0 and δ0, respectively. Diffraction in the second and third terms is isotropic and proportional to the square of the propagation distance z. The fourth term stands for the extra diffraction induced by turbulence and this diffraction factor is anisotropic, proportional to the anisotropic factor . When the propagation distance is short, the contribution from the fourth term is negligible because parameter T is a small number on the order of from weak to strong turbulence. As a result, the beam shape still displays the circular profile [see Figs. 2(b), 2(f) and 2(j)] for short propagation distances. As the distance increases, the diffraction induced by turbulence starts to gradually play a major role since the value of the fourth term increases with the cube of z, much faster than those of the second or the third terms. Consequently, the SD acquires an elliptical shape at larger propagation distances, say z = 5km. However, at the intermediate propagation distance, z = 2.5km, the coherence-induced diffraction increases dramatically with the decrease of the coherence width, and may play a role comparable to the turbulence-induced diffraction. That is the reason why the beam shape of the GSM beam withδ0 = 1mm still keeps circular profile at z = 2.5km, while the beam shape with δ0 = 1mm is elliptical. As an example, we plot in Fig. 4(a) the evolution of three terms in Eq. (19) with the propagation distance. One finds that the turbulence-induced diffraction in the y direction gradually plays an important role in determining the beam width in the y direction when the propagation distance is larger than 3km. While the diffraction induced by turbulence in the x direction is still smaller than the diffraction induced by initial coherence width in the propagation distance range from 0 to 5km, which leads to the elliptical beam shape at the certain propagation distance.
In Fig. 5 we present the evolution of the beam width as a function of propagation distance z and the Rytov variance of a plane wave for different values of turbulence strength withδ0 = 1mm. For comparison, the corresponding evolution of the beam width of the coherent beam () is also shown in Fig. 5. We want to mention that under the coherence limiting case, the third term in Eq. (19) disappears and the fourth term is proportional to , which is consistent with the third term in heuristic Eq. (5) in Ref . One finds from Fig. 5 the GSM beam with low coherence has the ability to maintain the circular shape for a longer distance compared to the coherent beam regardless of the strength of turbulence. For sufficiently large propagation distances, the turbulence-induced diffraction determines the beam width, despite of the initial beam parameters. Therefore, the beam widths of the GSM beam along the x and y directions are almost the same as those of the coherent beam, respectively, indicating that the GSM and the coherent beams exhibit the same beam profile.
Now, let us examine evolution of the DOC in anisotropic turbulence. From Eq. (20), it is seen that three factors affect the coherence width of the beam. The first term in the square brackets characterizes diffraction induced by the initial coherence width (can be considered as a diffraction factor), having similar effect with that for diffraction of the beam width, leading to the increase of the coherence width on propagation. The second term stands for the de-coherence effects due to turbulence, deteriorating the beam coherence with increasing propagation distance. In the third term, the diffraction and the de-coherence effects are coupled and lead to complex beam behavior at the intermediate distances.
In addition, in the presence of anisotropic turbulence along the horizontal path (μx > μy) the diffraction term [the first term in Eq. (20)] results in a larger coherence width in y (vertical) direction than that in x (horizontal) direction at sufficiently large distances from the source. On the other hand, the de-coherence term [the second term in Eq. (20)] causes the coherence width in x direction to become larger than that in the y direction immediately after the beam leaves the source plane. In other words, the effects of anisotropy on the DOC induced by the diffraction and de-coherence terms are the opposite. To show the contribution of each term in Eq. (20), Fig. 4(b) and 4(c) presents the evolution of each term in the x and y direction with the propagation distance, respectively. It can be seen that the first term drops rapidly as the propagation distance increases. The second term (de-coherence effects) becomes the determinant role when the propagation distance are about 1km for x component and about 500m for y component, which leads to an elliptical DOC profile elongated along x direction.
Fig. 6 illustrates the evolution of the coherence width of the GSM beam () and coherent beam () with the propagation distance and the Rytov variance for three different values of the turbulence strength . In the case of the GSM beam, the coherence width first increases with the propagation distance due to the diffraction induced by , reaches the maximum value, and then decreases when the turbulence-induced de-coherence effects become the dominant factor in determining the coherence width. Note that the coherence widths in x and y directions have little discrepancy within the several hundred meters of propagation, implying that the GSM beam has a certain ability to maintain the circular DOC profile. In case of the coherent beam, the coherence width decreases monotonically with distance because the diffraction induced by the initial coherence width (first term) disappears. We stress again that for the coherent beam, the difference of coherence widths in x and y directions already holds when the beam leaves the source plane. In practical application, it may provide an efficient method for detection of the anisotropy of turbulence through measuring the discrepancy of the coherence widths in x and y direction at a short propagation distance, i.e., 100-300 m. When the propagation distance is sufficiently large, the values of the coherence width in x(y) direction of the GSM beam and the coherent beam saturate to the same values due to the fact that the de-coherence effects become major factors in determining the coherent widths, regardless of source coherence.
In order to quantitatively assess the anisotropy of the SD and the DOC of beams propagating in the anisotropic turbulence, we introduce a parameter which is similar to the ellipticity of an ellipse, defined as
Fig. 7 illustrates the dependence of the ellipticities of the SD of the GSM beam () and the coherent Gaussian beam () on the propagation distance, the anisotropic factor , the power law and the turbulence structure parameter. In Fig. 7(a), it is shown that the SD of the GSM beam maintains the circular profile for a longer distance than that of the coherent beam. However, for sufficiently large propagation distance, the ellipticity of the GSM beam and the coherent beam tend to the same constant 0.667, i.e., . At a fixed propagation distance z = 2km [see in Fig. 7(b)-7(d)], one finds that the ellipticity of the partially coherent beam is always smaller than that of the coherent beam with any variation of the anisotropic factor, power law and structure constant, implying that the partially coherent source can reduce the effect of turbulent anisotropy on the propagating SD profile.
The dependence of the ellipticities of the DOC distribution on the propagation distance, the anisotropic factor μx, the power law and the turbulence structure parameter is shown in Fig. 8. As seen from Fig. 8(a), the ellipticity of the GSM beam keeps the value 0 over about 100m of propagation, and then increases with further increase of distance, finally tending to a constant value 0.667, as expected. For the coherent beam, the ellipticity remains constant 0.667 for the short propagation distance (z < 200m), and then have a dip, finally tending to the constant 0.667 again. This phenomenon can be explained by the fact that the diffraction induced by the initial coherence state disappears for the coherent beam. In this situation, the de-coherence effects dominate for the short propagation distance (z < 200m), while in the intermediate distances (200m < z < 5km) the diffraction of the turbulence-induced de-coherence [the third term in Eq. (20)] has certain contribution to the coherence width of the beam. As stated above, the anisotropy induced by de-coherence and diffraction on coherence widths are the opposite. Thus, the ellipticity slightly decreases. When the propagation distance is large (z > 5km), the diffraction factor in Eq. (20) decreases dramatically with the distance. The de-coherence effects play the major role again and the ellipticity tends to 0.667. At the fixed propagation distance z = 200 m, it is found that the relation between the ellipticity of the DOC for the coherent beam and the anisotropic factors of turbulence is approximately and is almost independent on the power law and structure parameter [see in Fig. 7(c) and 7(d)]. This is in contrast with the ellipticity of the GSM beam which is closely related to the power law and structure constant, due to the diffraction induced by the initial coherence state.
Anisotropy in turbulence is pertinent to a variety of atmospheric links and meteorological conditions and can affect optical system operation, for instance imaging MTF  or pulsed LIDAR outputs [26–28]. Accurate predictions of the statistics of optical beams propagating in such a medium are hence a necessity.
Our major contribution to this research area is the rigorous derivation of the spatial, second-order correlation function of a beam generated by a scalar Gaussian Schell-model source, i.e. characterizing waves with Gaussian intensity spread and Gaussian coherence state, and propagating in anisotropic turbulence in horizontal direction (perpendicular to the anisotropy ellipsoid’s axis of symmetry). Our results imply that for the initially highly coherent beams the coherence widths in vertical and horizontal directions are affected already at very small distances from the source according to the corresponding anisotropic factors. Hence, the coherence widths form an ellipse, elongated horizontally with semi-axes directly proportional to the anisotropic factors. On the other hand, the r.m.s. beam widths in vertical and horizontal directions are gradually changing with distance and form an ellipse elongated vertically, with semi-axes inversely proportional to the anisotropic factors in the strong turbulent regime. As the source coherence decreases the beam become less susceptible to the anisotropy, at least for sub-kilometer propagation distances, maintaining circular shape in both intensity profile and coherence state. However, for sufficiently large propagation distances (on the order of several kilometers) its behavior is undistinguishable from that of the coherent beam. Thus, source partial coherence may be considered as an efficient tool for mitigation of the anisotropic effects for optical systems operating over short horizontal links.
Among the practical applications that could benefit from our results are the wireless optical communication channel analysis and the direct energy delivery budgeting in the presence of anisotropic turbulence statistics.
Air Force Office of Scientific Research (AFOSR) (FA9550-121-0449); National Natural Science Foundation of China (NSFC) (11474213).
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