5654 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 10, OCTOBER 2017 Communication Electromagnetic Wave Propagation in the Turbulent Atmosphere With an Anisotropic Exponent of the Spectrum Elad Dakar, Ephim Golbraikh, Natan Kopeika, and Arkadi Zilberman Abstract—In this communication, we propose a new model for describ- ing the anisotropy of turbulence relative to the vertical and horizontal directions, through the anisotropy of the structure function exponent. Using the obtained anisotropy spectrum, we calculate various parameters such as log-amplitude variance for various atmospheric communication channels with comparison to the classical Kolmogorov model and newer models developed in previous works. Index Terms—Atmospheric propagation, electromagnetic wave prop- agation, optical propagation in anisotropic media. I. I NTRODUCTION As is well known, the atmosphere can be an anisotropic turbulent medium [1]–[5], which complicates the description of its character- istics. Recently, there have many theoretical works on the study of electromagnetic wave propagation through the anisotropic atmosphere (see [6]–[8]). However, all of these models, one way or another, are based on the model proposed in [9]. In the context of index of refraction fluctuations, one of the first articles which suggested an anisotropic model was [10]. The basic assumption of the model is that the refractivity fluctuations are flattened with respect to the isotropic case in the vertical direction by a factor, which is constant for all scales, but is allowed to vary with altitude. Due to the fact that the factor is constant, the mathematical way in which the model is expressed is by taking a spectrum which is elongated in the vertical wavenumber k z direction in comparison to the isotropic spectrum. The consequence of this assumption is that the surfaces on which the spectrum has a constant value, known as isosurfaces, are ellipsoids with their main axis of the vertical wavenumber. The equation describing the coordinates of the points belonging to a specific isosurface is (see [1], [11]) q 2 = η 2 k 2 ρ + k 2 z (1) where k ρ ≡  k 2 x + k 2 y is the horizontal wavenumber, q is the length of half of the ellipsoid major axis, and η is the main axis length divided by a minor axis (there are actually two minor axes) length so η> 1. The value of the spectrum  on each isosurface is taken as  n (q ) = η 2 · C n 4π · q μ (2) where C n is the constant amplitude of the spectrum and μ is the parameter that determines the power law of the spectrum. It is usually Manuscript received July 22, 2016; revised July 28, 2017; accepted August 13, 2017. Date of publication August 24, 2017; date of current version October 5, 2017. (Corresponding author: Ephim Golbraikh.) E. Dakar and N. Kopeika are with the Electrooptical Engineering Depart- ment, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail: dakare@post.bgu.ac.il; kopeika@bgu.ac.il). E. Golbraikh is with the Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail: golbref@bgu.ac.il). A. Zilberman is with the Electrical and Computer Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail: arkadiz@gmail.com). Digital Object Identifier 10.1109/TAP.2017.2743740 the same value of the exponent of k ≡  k 2 x + k 2 y + k 2 z in the isotropic spectrum. In order for the model to be useful, the spectrum it gives must be determined for each point in Fourier space. However, since each point in this space is defined by k x and k y which yield k ρ and, together with k z determine q , each point has a determined value of the spectrum. The values for η were calculated in [9] to be between 2.5 and 25, where each value was probably calculated for measurements targeting a specific range of altitudes. This constant anisotropy model can be developed into a more accurate [7], [9], [12] one by allowing η to be a function of the scale, i.e., of the magnitude of the wave vector. The result is a spectrum which has ellipsoids as isosurfaces as in the constant anisotropic case, but those ellipsoids become more and more like a sphere as q becomes larger than some characteristic scale. Thus, the models based on the parametric anisotropy become much more complex, but the control parameters of this model cannot be measured directly. Usually, as a result of the experiments in the laboratory and atmosphere where turbulence is studied, we obtain the turbulent spectrum along some directions (see [13]–[15]). In this case, the anisotropy of the turbulence can be manifested by changing the spectral index and/or the amplitude C n of the structure function depending on the direction in the atmosphere. Therefore, to describe the anisotropy of the atmospheric turbulence, the most physical approach is to introduce the dependence of the structure function on the zenith angle. For example, such an approach is used for the description of the turbulence anisotropy in plasma [16]. In this communication, a new model of anisotropic 3-D turbulent structure function for the stratified atmosphere is proposed. For the description of the turbulence anisotropy, it is suggested to take into account the dependence of the spectral index on the zenith angle. II. MODEL The 3-D turbulent structure function in the stratified atmosphere is D n (ρ, z) = ˜ C 2 n · (  ρ 2 + z 2 ) α(γ ) (3) where ˜ C 2 n is the generalized structure constant, ρ is the distance from the vertical axis, z is the location on the vertical axis, and α is the power of the structure function which changes according to γ —the angle of the propagation of the electromagnetic wave above or below the horizontal direction (see Fig. 1). For γ = 0, the direction is horizontal, and for γ = 90°, the direction is vertical and α(-γ) = α(γ ). We assume that when an electromagnetic wave is penetrating in a specific direction through the atmosphere, then the correlations along this direction and the directions close to it affect its behavior significantly more than the correlations along any other direction. The solid angle defined by the propagation direction includes all the points whose line to the origin has an angle with the horizon, which equals approximately the angle γ of the propagation direction 0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.