The effects of anisotropic, non-Kolmogorov turbulence on propagating stochastic electromagnetic beam-like fields are discussed for the first time. The atmosphere of interest can be found above the boundary layer, at high (more than 2 km above the ground) altitudes where the energy distribution among the turbulent eddies might not satisfy the classic assumption represented by the famous 11/3 Kolmogorov’s power law, and the anisotropy in the direction orthogonal to the Earth surface is possibly present. Our analysis focuses on the classic electromagnetic Gaussian Schell-model beams but can either be readily reduced to scalar and/or coherent beams or generalized to other beam classes. In particular, we explore the effects of the anisotropic parameter on the spectral density, the spectral degree of coherence and on the spectral degree of polarization of the beam.
© 2014 Optical Society of America
The atmosphere can be divided into two parts: the atmospheric boundary layer (up to 2 km in altitude), where heating of the surface leads to convective instability, and the free atmosphere (above the atmospheric boundary layer) where the effect of the Earth’s surface friction on the air motion is negligible. Inside the atmospheric boundary layer optical turbulence can be considered homogeneous and isotropic therefore the Kolmogorov model for power spectrum of the fluctuating refractive index is generally valid within the inertial sub-range. However, in the free atmosphere, in particular inside the stably layered stratosphere, optical turbulence can be anisotropic (mostly at large scales) and the Kolmogorov power spectrum does not properly describe the real turbulence behavior. Although some evidence of anisotropy was measured more than forty years ago by Consortini  near the ground (about 1 m above), the evidence of anisotropy of optical turbulence is more likely in zones of the atmosphere with inversion layers, i.e. in the presence of the positive temperature gradients, where the vertical component of turbulence is strongly reduced. Dalaudeier et al.  showed experimentally the presence in the atmosphere, at least up to 25 km, of very strong positive temperature gradients within very thin layers, termed “sheets”. Such sheets are anisotropic having vertical extension up to 10 m and horizontal extensions larger than 100 m. It is well known that temperature fluctuations yield to optical turbulence, therefore if the temperature field is layered or anisotropic the turbulence field should also assume the same behavior. Grechko et al.  reported a strong anisotropy of temperature and density in the middle atmosphere from experimental observations of star scintillation. Biferale et al.  used probes with two different geometries (horizontally and vertically) to detect information about anisotropic turbulence in the boundary layer and concluded that the atmospheric boundary layer exhibits three-dimensional statistical turbulence intermingled with flow patterns whose statistics have a quasi-two-dimensional nature. Belenkii et al. [5,6] experimentally observed anisotropy of the wavefront tilt’s statistics. They observed that the horizontal outer scale is bigger than the vertical one, on-axis tilt variances are unequal and the horizontal tilt variance is consistently greater than the vertical one. Also, the evidence of anisotropy in the stratosphere based on the balloon-borne measurements has been reported in , where the authors employed a power spectrum model with two components: anisotropic and isotropic. Results showed a major contribution to scintillation of the anisotropic component relative to the isotropic one. Experimental measurements [8,9] have implied that the outer scale of turbulence in the horizontal direction can be many times larger than that in the vertical direction. The horizontal size of these eddies is typically tens of meters across or, in some cases, kilometers across . In vertical direction the size of the outer scale cells is usually confined to a few meters. In addition to anisotropy, turbulence does not always follow the Kolmogorov power spectrum density model but sometimes it follows different power laws [11–13]. Kyrazis et al. measured non-Kolmogorov turbulence in the upper troposphere and lower stratosphere [14–17].
On the other hand, the interaction of deterministic and random optical beams [18–20] with turbulent atmosphere has been the active research area during several decades (see [21–23] and references wherein). In all these studies the atmospheric fluctuations in the refractive index have been assumed to be homogeneous and isotropic which allowed the treatment of beam propagation with the help of the power spectrum depending on the scalar wave number. In particular the details of beam evolution in non-Kolmogorov (isotropic) turbulence were discussed in [24–30]. So far the modeling of anisotropic turbulence  (see also ) has only resulted in analysis of scalar deterministic beams propagating in it ,  and . The degree of anisotropy has been shown to play an important part in predicting the evolution of such beams.
The objective of this paper is to extend the analysis of anisotropic turbulence from scalar deterministic to electromagnetic stochastic beams and to explore the influence of the anisotropy on all the major second-order statistics along the propagation path. We will restrict our considerations only to the case when the beam travels along the anisotropy axis, i.e. only along up or down link. Therefore our results will be of interest for optimization of the earth-satellite communication and imaging links as well as in astronomical studies. In such cases light must travel for large (tens of kilometers) distances through the upper atmospheric layers where the effects of anisotropy become substantial. Since the sunlight and beams used in communications are in general partially coherent and partially polarized our results are of importance, being able to cover a large variety of possible coherence and polarization states. All the previous studies that included atmospheric anisotropy only treated coherent and fully polarized radiation.
Recently a variety of stochastic sources and beams have been introduced [34–44] all differing by the shape of either the intensity distribution or the degree of coherence. Most of these source and beam models have been derived on the basis of the sufficient condition developed in [45–47]. In this paper however we will restrict ourselves to the classic electromagnetic Gaussian Schell-model beam [18, 23] in order to focus on the effects stemming from atmospheric anisotropy rather than on different source properties. After establishing the propagation laws for the cross-spectral density matrix of the beam we investigate in details the changes in its derivatives: the spectral density, the degree of coherence and the polarization features. The examples pertaining to the limiting case of a coherent beam are presented first followed by those for general electromagnetic random beam.
2. Anisotropic power spectrum with inner and outer scale
For our presentation we will employ the anisotropic non-Kolmogorov power spectrum reported in  also including the inner and outer scale effects by using a generalized von Karman model . We assume that the anisotropy exists only along the direction of propagation, say z, of the beam and accounted via an effective anisotropic factor as discussed in [32,33]. Hence the power spectrum of refractive-index fluctuations is given by the expression:48]24]
Let us now also ignore the spatial wave number component along the direction of propagation, by invoking the Markov approximation, which is usually used in the theory of wave propagation in random media (c.f .). The Markov approximation implies that the index of refraction is delta-correlated at any pair of points located along the direction of propagation. Under the Markov approximation turbulence is essentially supposed to be layered along the direction of propagation, i.e. the energy transfer process in the inertial sub-range develops only over planes orthogonal to the propagation direction.
3. Propagation of the cross-spectral density matrix of the beam in anisotropic turbulence
Let us now recall that the propagation law for the components of the cross-spectral density matrix that characterizes any stochastic electromagnetic beam at a pair of points, say r1 and r2 on propagation in any random linear medium has form :21]:Eq. (5) results in the expression
We will now select the electromagnetic Gaussian-Schell model (EMGSM)  as the model source:23]. On substituting from Eqs. (5)-(7) into Eq. (4), the elements of the EMGSM beam at distance z from the source can be shown to be20]20]49]:
3. Examples for fully coherent case
We will first examine the limiting case of a coherent, unpolarized beam with parameters,(unless other values of parameters are specified). This case is equivalent to a deterministic Gaussian beam with radius We set the following parameters: , , , , ,. The average refractive index structure constant turn out to be .
Figure 1 shows the on-axis spectral density of the propagating coherent beam for the chosen combinations of the power law and anisotropy parameter. Generally, for higher values of the effect of the atmosphere reduces. Regardless of the effect on the beam is the greatest for about 3.1, similarly to the case of isotropic non-Kolmogorov turbulence.
Figure 2 presents the similar analysis of the r.m.s. beam width calculated by the expression32,33].
Figure 3 illustrates the changes in the spectral degree of coherence of the propagating beam. In Fig. 3(a) the separation between two points is fixed at . The degree of coherence decreases with propagation distance with higher rate corresponding to lower values of anisotropy parameter. Figure 3(b) shows the changes continuously for broad range of both and at fixed (large) propagation distance. As for the spectral density and the r.m.s. beam width the degree of coherence is minimal for between 3.0 and 3.2 for all values of . Figure 3(c) demonstrates the evolution of the degree of coherence as a function and separation between two points, for the fixed value of .
4. Examples for partially coherent case
We will now turn to the more general case of a partially coherent beam radiated by source (9). Let the transverse coherence widths be ,, while that other parameters are the same as for fully coherent beam, unless other values of parameters are specified.
Figure 4 shows the changes in the on-axis spectral density for the chosen values of anisotropy and power law parameters. The trends are similar to those in coherent case [see Fig. 1], however Fig. 4(a) implies that in partially coherent case the discrepancy between different beams is much smaller, since the beam behavior is more affected by the source rather than by turbulence. Such small discrepancies in the spectral density of random beams are in line with their propagation in isotropic turbulence investigated in detail previously , . It is indeed well known that the behavior of random beams in random media is greatly affected by source correlations, even at large propagation distances, hence the changes in the turbulent statistics can only induce relatively small effects. This is in striking difference with deterministic beams for which a slight change in random media statistics (here the change in anisotropy parameter ζ) lead to substantial changes in beam behavior, in particular, in its spectral density.
Figure 5 shows the evolution of the r.m.s. beam width Fig. (a), Fig. (b) and Fig. (c), Fig. (d) of the EMGSM beam. The trends are similar to that in coherent case [see Fig. 3]. However, due to generally different values of r.m.s. correlations and the r.m.s. beam widths are evolving at different rates. As is seen from the figure, anisotropy plays a crucial part in predicting the r.m.s. beam width especially at low values of.
Figure 6 illustrates the changes in the spectral degree of coherence. In the case of a partially coherent beam it generally grows first due to source correlations and then may start decreasing for sufficiently small anisotropy parameter values [see Fig. 6(a)] due to turbulence effects. Further, the comparison of Figs. 3 and 6, parts Fig. (b) and Fig. (c), implies that the variation of the degree of coherence at fixed large distances and with separation distance are the same in trend for coherent and stochastic beams, while the actual values are lower for the later cases.
Figure 7 shows the evolution of the degree of polarization of the beam on the optical axis. Just like other parameters, it is modulated by turbulence at most for lower values of anisotropy parameter, and for power law parameter close to 3.1. For very high values of anisotropy such as ζ = 10 [see Fig. 7(a)] the degree of polarization is mostly influenced by the source (but not turbulence) and grows monotonically, similarly to its free space behavior. However, for lower anisotropy parameter values (ζ = 1;1.5;2) the atmosphere becomes dominant, making the degree of polarization decrease after reaching a certain maximum.
Finally, Fig. 8 shows the typical behavior of the polarization ellipse and its parameters for several selected values of anisotropy. Just like the degree of polarization the rest of the polarimetric features are affected less by turbulence with high anisotropy parameter values.
Thus, anisotropic atmosphere does not influence the polarization properties of propagating beams as much as the isotropic one. This conclusion may be of importance for polarization communications and imaging, in which the polarization state is chosen to carry the information instead of (in addition with) conventional average intensity-based transfer.
5. Concluding remarks
In this paper we investigated the effects of anisotropic turbulence on propagation of the electromagnetic Gaussian Schell-model beam. In particular, we have examined how anisotropy affects the spectral density, the states of coherence and polarization of the beam. Our theoretical research was based on a generalized von Karman power spectrum of the index of refraction with a non-Kolmogorov power law and an effective anisotropic parameter, to describe anisotropy along the vertical direction. Anisotropy introduces a rescaling of turbulence, which for the case of the zero inner scale and infinite outer scale is given by the factor . The major conclusion that comes out of our analysis is that the high values of the anisotropy parameter lead to suppression of the turbulence effects and enhancement of the source effects on both coherent and random beams. Thus due to anisotropy pertinent to uplink or downlink propagation the turbulence may be effectively weaker compared to conventional cases associated with horizontal links.
M. Yao’s research is sponsored by the National Natural Science Foundation of China under Grant No. 11304287; O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449) and US ONR (N0018913P1226).
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