Abstract

Analytical expressions for the variance of angle of arrival (AOA) fluctuations based on the Rytov approximation theory are derived for plane and spherical waves’ propagation through weak anisotropic non-Kolmogorov turbulence atmosphere. The anisotropic spectrum model based on the assumption of circular symmetry in the orthogonal plane throughout the path is adopted and it includes the same degree of anisotropy along the direction of propagation for all the turbulence cells size in the inertial sub-range. The derived expressions consider a single anisotropic coefficient describing the turbulence anisotropic property and a general spectral power law value in the range 3 to 4. They reduce correctly to the previously published analytic expressions for the cases of plane and spherical waves’ propagation through weak isotropic non-Kolmogorov turbulence for the special case of anisotropic factor equaling one. To reduce the complexity of the analytical results, the asymptotic-fit expressions are also derived and they fit well with the close-form ones. These results are useful for understanding the potential impact of deviations from the standard isotropic non-Kolmogorov turbulence atmosphere.

© 2015 Optical Society of America

1. Introduction

Atmospheric turbulence has a significant degrading impact on the optical imaging and laser systems due to the random inhomogeneity of the refractive index of air. That is because a series of turbulence effects are produced when optical wave propagation through atmosphere turbulence, including irradiance scintillation, AOA fluctuations, beam wander, beam spread and so on. In which, the AOA fluctuations are related to the signal distortion of the long-range infrared imaging system or laser communication systems and they reduce greatly the performance of these systems. Investigations of this turbulence effect become very important.

In recent years, study on AOA fluctuations has been focused on the isotropic non-Kolmogorov atmospheric turbulence [1–6]. For weak isotropic non-Kolmogorov turbulence, with the Rytov theory, the non-Kolmogorov spectrum [7], the generalized von Karman spectrum [8] and the generalized exponential spectrum [9] have been used to investigate the variance of AOA fluctuations [8–10]. When the turbulence strength increases up to moderate-to-strong (or strong) turbulence regime, with the extended Rytov theory, the effective non-Kolmogorov spectrum [11] and the effective generalized exponential spectrum [12] which are the modifications to the conventional non-Kolmogorov spectrum and the generalized exponential spectrum, have been adopted to investigate the variance of AOA fluctuations [13,14], and the close-form solutions to AOA variances have been obtained for plane and spherical waves by approximating the phase structure function in certain asymptotic cases.

However, experimental and theoretical results have shown that the atmospheric turbulence can also be anisotropic [15–28]. Grechko and et al. [17] reported a strong anisotropy in the middle atmosphere from experimental observations of star scintillation. Biferale and et al. [18] detected the information about anisotropic turbulence in the boundary layer by using two probes with two different geometries (horizontally and vertically). Benlenky and et al. [19,20] experimentally observed anisotropy of the statistics of wavefront tilt. They observed a horizontal outer scale bigger than the vertical one and the horizontal tilt variance is consistently greater than the vertical one. Also, evidence of anisotropy in the stratosphere has been reported in [21], where authors used a power spectrum with two components: anisotropic and isotropic, and the validity of the spectrum was verified by balloon-borne experiments. Experimental measurements [16] have shown that the outer scale of turbulence in the horizontal direction can be many times larger than in the vertical direction. The horizontal size of these eddies is typically tens of meters across or, in some cases, kilometers across [27]. Anisotropy is usually present at high altitude, above the atmospheric boundary layer, which extends to about 2 km in altitude and it is more evident for large turbulence cells or eddies [27]. Anisotropy can be present also at few meters above the ground [15]. These experiments promote the study of optical waves’ propagation through anisotropic non-Kolmogorov atmosphere turbulence [27–30].

In this study, the variance of AOA fluctuations in weak anisotropic non-Kolmogorov turbulence will be investigated. Firstly, the anisotropic non-Kolmogorov spectrum will be introduced. For simplicity, the assumption of circular symmetry around the direction of propagation is adopted, which makes the spectrum takes a much simpler form. Secondly, based on the Rytov approximation theory and the simplified anisotropic non-Kolmogorov spectrum, the analytic expressions for the variance of AOA fluctuations will be obtained. To Lastly, the calculations will be performed to analyze the impacts of anisotropic factor and general spectral power law values on the variance of AOA fluctuations.

2. Anisotropic non-Kolmogorov spectrum for anisotropic non-Kolmogorov turbulence

The conventional isotropic non-Kolmogorov power spectrum over the inertial sub-range takes the form as [7]:

Φn_isotropic(κ,α)=A(α)C^n2κα,(0κ<,3<α<4).
A(α)=14π2Γ(α1)cos[απ2].
where, A(α) is a constant which maintains consistency between the refractive index structure function and its power spectrum, α is the general spectral power law value, C^n2 is the generalized structure parameter with units m3α, Γ() is the gamma function. κ is the wavenumber related to the turbulence cell size, κ=κx2+κy2+κz2. κx, κy, and κz are the components of κ in the x, y, and z directions.

For anisotropic non-Kolmogorov turbulence, the structure function takes the form as [25,27]:

Dn(R,α,ς)=A(α)C^n2(x2+y2ς2+z2)α/2,l0RL0,3<α<4.

It is noted that Eq. (3) was established to represent the horizontal symmetry employed to analyze anisotropic turbulence for simplicity just as [25,27] did. In [29,30], they take more general forms. R is a vector spatial variable, ς is the anisotropic factor and it carries the burden of representing the anisotropy in this format by assuming different values. ς parameterizes the asymmetry of turbulence cells or eddies in both horizontal and vertical directions. When ς equals one, the isotropic turbulence is shown. If the turbulence cells are anisotropic, horizontal extension is higher than vertical one, the value of ς is always bigger than one.

By making the changes of variables x=ςx', y=ςy', the resulting structure function becomes isotropic in the new spatial variable R'=(x',y',z). In view of the relationship between the structure function and the turbulence spectrum Φn() [7], the resulting Φn() will be isotropic in the stretched wave number space κx'=ςκx, κy'=ςκy, κz'=κz. Φn() for the anisotropic turbulence becomes [25,27]:

Φn(κ,α,ς)=A(α)C^n2ς2(κ')α=A(α)C^n2ς2[κz2+ς2(κx2+κy2)]α2.
where κ'=κz2+ς2(κx2+κy2). When ς=1, the anisotropic non-Kolmogorov spectrum reduces to the conventional isotropic non-Kolmogorov turbulence spectrum.

For the analysis directly below, it is assumed that the propagation is in the z direction (κz=0) and the circular symmetry is maintained in the orthogonal xy-plane throughout the path just like [25,27]. κz is ignored by invoking the Markov approximation, which is usually used in the theory of wave propagation in random media. 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 orthogonal to the propagation direction. Hence, the spectrum model (Eq. (4)) in this case can be reduced to a much simple form [25,27]:

Φn(κ,α,ς)=A(α)C^n2ς2ακα,κ=κx2+κy2.

3. Variance of AOA fluctuations for anisotropic non-Kolmogorov turbulence

Considering the relation between the variance and covariance, the variance of AOA fluctuations can be derived from the covariance of AOA fluctuations. According to the Wiener-Kinchin theorem, the spatial covariance function of the AOA fluctuations Cθ() can be expressed as [5,6]:

Cθ(ρ,β)=πk20κ3Wϕ(κ)GD(κ)[J0(ρκ)cos(2β)J2(ρκ)]dκ
where ρ represents the geometrical separation between points in the plane transverse to the direction of propagation, β is the angle between the baseline (z-axis) and the AOA observation axis, k=2π/λ and λdenotes the optical wavelength. J0(ρκ) denotes the zero order Bessel function. Wϕ(κ) is the wave-front phase power spectrum. GD(κ) represents the point spread function of the receiver aperture [31]:

GD(κ)exp(b2D2κ24),b=0.52.

Using the relation between the variance and covariance, when ρ and β in Eq. (9) equal to zero, the variance of the AOA fluctuations can be obtained as follows:

σ2=Cθ(ρ=0,β=0)=πk20κ3Wφ(κ)GD(κ)dκ,

The anisotropic power spectrum used in this paper can be useful only for vertical paths where anisotropy is present along the direction of propagation. In the following work, we analyzed only horizontal path for simplicity (C^n2 will be a constant), but for real applications a turbulence profile with altitude, such as Hafnagen-Valley (HV) model, should be considered (C^n2 varies with atmosphere altitude).

3.1 Variance of AOA fluctuations for plane wave under anisotropic non-Kolmogorov turbulence

In view of the Rytov approximation theory, the wave-front phase power spectrum of the plane wave is given by

Wϕ(pl)(κ)=2πk20LΦn(κ,α,ς)cos2(κ2z2k)dz,

Substituting the wave-front phase power spectrum of the plane wave into Eq. (8), the variance of the AOA fluctuations of the plane wave is obtained,

σ(pl)2=2π20κ3Φn(κ,α,ς)GD(κ)dκ0Lcos2(κ2z2k)dz.

It is worth noting that we do not need an explicit form of the refractive-index power spectrum in deriving our expression. When the conventional isotropic non-Kolmogorov spectrum is substituted into Eq. (10), the conventional result of the AOA fluctuation variance for the isotropic non-Kolmogorov turbulence is obtained. Here, with the anisotropic non-Kolmogorov spectrum, Eq. (5), the analysis of the AOA fluctuations will be carried out for the plane wave propagating through weak anisotropic non-Kolmogorov turbulence.

Substituting Eqs. (5) and (7) into Eq. (10), yields

σ(pl)2=π20dκ01dzκ3Φn(κ,α,ς)[1+kLκ2sin(Lκ2k)]exp[b2D2κ24],

Now the problem is reduced to a double integration over the turbulence frequency variable κ and the path z. Using the gamma function [32]:

Γ(x)=0κx1eκdκ,
and its property [33]:
0κμ1exp(aκ)sin(bκ)dκ=Γ(μ)(a2+b2)μ/2sin[μtan1(ba)],μ>1,a>0
the analytic expression of the variance of AOA fluctuations for plane wave propagating though weak anisotropic non-Kolmogorov turbulence is obtained

σ(pl)2(α,ζ)=π2A(α)C^n2Lς2α2{Γ(2-α2)(b2D24)α2-2+kLΓ(1α2)(b4D416+L2k2)2α4sin[2α2tan1(4Lkb2D2)]}.

From Eq. (14), it can be seen that the variance of AOA fluctuations of the plane wave is a concise close-form expression, and it contains both general spectral power law and anisotropic factor.

Defining q=D/λL, q is the Fresnel number which is the ratio between the aperture diameter and the Fresnel length λL, and Eq. (14) becomes

σ(pl)2(α,ζ)=rp(q,α,ζ)C^n2LDα4,
rp(q,α,ζ)=π2A(α)ς2α2Γ(2α2)(b2)α-4×{1+2α2(π2)2α/2b4αq4α(1+b4q4π24)α24sin[α22tan1(2b2q2π)]}.

Next, we discuss the cases of small-aperture (q1) and large-aperture (q1). When q1, the second term in the curly brackets of Eq. (16) is much smaller than 1 and negligible. Then, we have the asymptotic value,

σ(pl)2(q1,α,ζ)=rp(q1,α,ζ)C^n2LDα4,
rp(q1,α,ζ)=π2A(α)ς2α2Γ(2α2)(b2)α-4.

For large-aperture (q1), we can approximate several terms of (1+b4q4π24)α24(b4q4π24)α24, sin[α22tan1(2b2q2π)]α22tan1(2b2q2π), and tan1(2b2q2π)2b2q2π. Then, the second term in the curly brackets of Eq. (16) approaches 1, and we obtain the asymptotic values,

σ(pl)2(q1,α,ζ)=rp(q1,α,ζ)C^n2LDα4,
rp(q1,α,ζ)=π2A(α)ς2αΓ(2α2)(b2)α-4.

Equation (20) is consistent with the results derived for plane wave propagating through isotropic non-Kolmogorov turbulence [34] where the geometrical optic assumption (DλL, that is q1) was adopted.

For intermediate range (q<1), (1+b4q4π24)α241 and tan1(2b2q2π)π2. Thus, Eq. (15) can be approximated expressed as

σ(pl)2(q<1,α,ζ)=rp(q<1,α,ζ)C^n2LDα4,
rp(q<1,α,ζ)=π2A(α)ς2α2Γ(2α2)(b2)α-4×{1+2α2(π2)2α/2b4αq4αsin[π(α2)4]}.

3.2 Variance of AOA fluctuations for spherical wave under anisotropic non-Kolmogorov turbulence

In view of the Rytov theory, the wave-front phase power spectrum of spherical wave is given by

Wϕ(sp)(κ)=2πk20LΦn(κ,α,ς)(zL)2cos2[κ2z(Lz)2kL]dz.

Substituting Eqs. (5) and (23) into Eq. (8), the variance of AOA fluctuations for spherical can be expressed as

σ(sp)2=π2L0dκ01dξκ3Φn(κ,α,ς)[1+cos(κ2ξ(1ξ)Lk)]ξ2exp[b2D2κ2ξ24],

Using Eq. (12) and the gauss hypergeometric function F21(A,B;C;Z) [32]:

F21(A,B;C;Z)=Γ(C)Γ(B)Γ(CB)01tB1(1t)CB1(1tZ)Adt.

the analytic expression of the variance of AOA fluctuations for spherical wave propagating through weak anisotropic non-Kolmogorov turbulence is derived and takes the form as

σsp2(α,ζ)=π2A(α)C^n2Lς2α2Γ(2α2){1α1(b2D24)α/22+Re[(iLk)4α222+αF21(4α2,2+α2;4+α2;1+ib2D2k4L)]}.

From Eq. (26), it can be seen that the variance of AOA fluctuations of the spherical wave is also a concise closed-form expression, and it contains both general spectral power law and anisotropic factor.

Defining q=D/λL, Eq. (26) becomes

σsp2(α,ζ)=rs(q,α,ς)C^n2LDα4,
rs(q,α,ζ)=π2A(α)ς2α2(α1)Γ(2α2)(b24)α/22{1+2(α1)2+α×Re[(2iπb2q2)4α2F21(4α2,2+α2;4+α2;1+iπb2q22)]}.

Here, we discuss two special cases of small-aperture (q1) and large-aperture (q1). When q1, we can simply Eq. (28) by means of 2iπb2D20. Thus, we have the asymptotic value,

σsp2(q1,α,ζ)=rp(q1,α,ζ)C^n2LDα4.
rp(q1,α,ζ)=π2A(α)ς2α2(α1)Γ(2α2)(b24)α/22.

For very large aperture (q1), we can approximate the fourth argument of the hypergeometric function in Eq. (28) as 1+iπb2q22iπb2q22, and adopting the property of F21(a,b;c;z) [32]:

F21(a,b;c;z)=Γ(c)Γ(ba)Γ(b)Γ(ca)(z)aF21(a,1c+a;1b+a;1/z)+Γ(c)Γ(ab)Γ(a)Γ(cb)(z)bF21(b,1c+b;1a+b;1/z)
Equation (27) becomes

σsp2(q1,α,ζ)=rs(q1,α,ζ)C^n2LDα4,
rs(q1,α,ζ)=π2A(α)ς2α(α1)Γ(2α2)(b24)α/22.

Equation (33) is consistent with the results derived for spherical wave propagating through isotropic non-Kolmogorov turbulence [34] where the geometrical optic assumption (DλL, that is q1) was adopted.

For intermediate range (q<1), the hypergeometric function in Eq. (28) can be approximated as F21(4α2,2+α2;4+α2;1+iπb2q22)F21(4α2,2+α2;4+α2;1). Using the property of [33]:

F21(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)Γ(cb).

Equation (27) becomesF21(a,b;c;z)

σsp2(q<1,α,ζ)=rs(q<1,α,ζ)C^n2LDα4,
rs(q<1,α,ζ)=π2A(α)ς2α2(α1)Γ(2α2)(b24)α/22×{1+2(α1)2+αRe[(2iπb2q2)4α2Γ(2+α/2)Γ(α/21)Γ(α)]}.

4. Calculations and analysis

In Section 3, the close-form expressions of variance of AOA fluctuations under anisotropic non-Kolmogorov turbulence (Eqs. (15) and (27)) have been derived for both plane and spherical waves. Also, special cases of small-aperture, large-aperture and intermediate-aperture are discussed. At this time, the asymptotic fits for the complex closed-form expressions of the variance of AOA fluctuations can be approximately expressed as

σ(pl)2(q,α,ζ)rp(q,α,ζ)C^n2LDα4,
σsp2(q,α,ζ)rs(q,α,ζ)C^n2LDα4,
rp(q,α,ζ)={π2A(α)ς2α2Γ(2α2)(b2)α-4×{1+2α2(π2)2α/2b4αq4αsin[π(α2)4]},q1π2A(α)ς2αΓ(2α2)(b2)α-4.q>1
rs(q,α,ζ)={π2A(α)ς2α2(α1)Γ(2α2)(b24)α/22×{1+2(α1)2+αRe[(2iπb2q2)4α2Γ(2+α/2)Γ(α/21)Γ(α)]},q1π2A(α)ς2α(α1)Γ(2α2)(b24)α/22.q>1

Compared with the closed-form expressions (Eqs. (15), (16), (17) and (18)), the asymptotic fit results (Eqs. (37)-(40)) take much simpler forms. In this section, comparisons for the complex closed-form and the asymptotic fits results will be made, and they are plotted as a function of q and shown in Figs. 1 and 2. Different general spectral power law and anisotropic factor values are chosen for both plane and spherical waves. By using Matlab (version: R2012a) on Windows pc with Intel Core i3-2370M CPU and 6GB memory, the calculation time for the closed-form expressions is 1.034 milliseconds (plane wave) and 97.85 milliseconds (spherical wave), and it is 1.31 milliseconds (plane wave) and 0.45 milliseconds (spherical wave) for the asymptotic fit expressions. The calculation time is comparable for plane wave case and it takes much longer time for the closed-form expressions for spherical wave case. That is because it spends more time to calculate F21(4α2,2+α2;4+α2;1+iπb2q22) function.

 

Fig. 1 Aperture-averaged AOA variance, normalized by C^n2LDα4, versus q for plane and spherical waves with different anisotropic factor ς values. (a): Plane wave; (b): Spherical wave.

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Fig. 2 Aperture-averaged AOA variance, normalized by C^n2LDα4, versus q for plane and spherical waves with different anisotropic factor ς and general spectral power law values. (a): Plane wave; (b): Spherical wave.

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From Figs. 1 and 2, it can be seen that the asymptotic fit expressions is very accurate far away from q=1 and is still almost useful in the intermediate range. The maximum differences in Fig. 1 are the same for various ς values, and they are 1.40% and 2.51% for plane and spherical waves, respectively. In Fig. 2, the maximum differences for plane wave are 7.56% (α=10/3), 4.30% (α=11/3), and 1.4% (α=3.9). For spherical waves, they are 15.32% (α=10/3), 8.12% (α=11/3), 2.51% (α=3.9). To further analyze the discrepancy between the closed-form and asymptotic fit results, the percentage differences are calculated and plotted as a function of ς and α in Fig. 3. It can be seen that the percentage differences are the same for different ς values, and decrease obviously with the increased α values. That is because the variance of AOA fluctuations is proportional to ς2α.

 

Fig. 3 Difference (%) between the close-form expression with the asymptotic fit results with different anisotropic factor ς and α values. (a): Differences as a function of ς; (b): Differences as a function of α.

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Next, the influences of anisotropic factor and general spectral power law values on the variance of AOA fluctuations will be discussed for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. Figure 4 analyzes the anisotropic factor’s influence on the variance of AOA fluctuations. The variance of AOA fluctuations as a function of anisotropic factor values is plotted. The other parameters are set as: D=0.05m, C^n2=1015m3α, L=1000m, λ=1.55μm. They are set as an example and other values can also be chosen. As shown, the anisotropic non-Kolmogorov turbulence produces different effects on the final results compared with the isotropic non-Kolmogorov turbulence (anisotropic factor equals one). As ς increases, the anisotropic non-Kolmogorov turbulence influences less obviously on the AOA fluctuations for both plane and spherical waves. This can be physically explained by mentioning the change of curvature of the anisotropic turbulence cells with respect to the isotropic case. Anisotropic cells change the focusing properties of the turbulence; in particular, a beam propagating along the short axis of the anisotropic cells (z direction) will be less deviated from the direction of propagation because these cells act as lenses with a higher radius of curvature. Accordingly, the turbulence effect will be reduced.

 

Fig. 4 The influence of anisotropic factor on the variance of AOA fluctuations with different spectral power law values. (a): Plane wave; (b): Spherical wave.

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The influence of general spectral power law value on the variance of AOA fluctuations for optical waves propagating through weak anisotropic non-Kolmogorov turbulence is analyzed and shown in Fig. 5. It can be seen that the general spectral power law’s variation trend on the final expressions for weak anisotropic non-Kolmogorov turbulence is similar to that for the weak isotropic non-Kolmogorov turbulence. Specifically, with the increase of general spectral power law values, the variance of AOA fluctuations firstly increases until arrives at a maximum value and then decreases. The maximum differences between different spectral power law values for plane wave are 98.3% (ζ=1), 87.0% (ζ=10), and 86.3% (ζ=50). For spherical wave, they are 97.6% (ζ=1), 85.9% (ζ=10), and 89.5% (ζ=50).

 

Fig. 5 The influence of general spectral power law values on the variance of AOA fluctuations with different anisotropic factor values. (a): Plane wave; (b): Spherical wave.

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5. Discussions and conclusions

In this work, with the Rytov approximation theory and the anisotropic non-Kolmogorov turbulence spectrum, the close-form expressions for the variance of AOA fluctuations have been derived for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. Both the anisotropic factor which increases with the atmospheric layer altitude and the general spectral power law values have been considered. They describe the influence of anisotropic turbulence appearing in the free atmosphere layer above the ground on the AOA fluctuations. Compared with the isotropic turbulence which appears near the ground, the anisotropic factor which parameterizes the asymmetry of turbulence cells in both horizontal and vertical directions is included and it varies with the atmospheric layer altitude. Also, the asymptotic fit expressions have been derived, which take much simpler form than the close-form results. It was found that the asymptotic fit results have good consistency with those from the close-form expressions, and the discrepancy decreases with the increased general spectral power law values. Compared with the isotropic non-Kolmogorov turbulence (ζ=1), increasing anisotropic factor to values greater than unity will greatly enlarge atmospheric turbulence’s influence on the AOA fluctuations.

In the investigation, the simplified anisotropic non-Kolmogorov turbulence (circular symmetry is maintained in the orthogonal xy-plane throughout the path) is adopted. In [29], the more general anisotropic non-Kolmogorov power spectrum was proposed to analyze the optical waves’ propagation through anisotropic non-Kolmogorov turbulence. In the propagation, the orthogonal xy-plane will no longer be circularly symmetric (i.e., isotropic). In general this more general spectrum may lead to different statistical values in the horizontal and vertical transverse directions. In addition, it is also supposed that the anisotropic factor has the same effect on all the turbulence scales just like [25,27] for mathematical derivation convenience. This assumption is not suitable in the stable atmospheric boundary layers, where isotropy probably prevails at small scales [21]. In [30], the concept of anisotropy at different scales has been introduced and an effective anisotropic parameter has been defined for two specific cases of anisotropic laws of linear and parabolic. The effective anisotropic parameter has been used to introduce anisotropy inside a generalized von Karman spectrum and finite turbulence inner and outer scales are included.

In future, based on the more general anisotropic non-Kolmogorov spectrum [29] and the anisotropic non-Kolmogorov power spectrum [30] considering degree of anisotropy at different turbulence eddy scales, the improved AOA fluctuations models of an optical wave under anisotropic non-Kolmogorov turbulence will be obtained. Not only will be the turbulence inner and outer scales considered, but the variable anisotropic factors for different turbulence eddy scales will also be included. Though still more experiments and theoretical investigations about anisotropic turbulence need to be performed, this work should be considered as a first step of a more complete analysis.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (61405004), the Fundamental Research Fund for the Central University (YWF-14-RSC-033), and the Open Research Fund of Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences.

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15. R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986). [CrossRef]  

16. F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998). [CrossRef]  

17. G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992). [CrossRef]  

18. L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005). [CrossRef]  

19. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999). [CrossRef]  

20. M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999). [CrossRef]  

21. C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008). [CrossRef]   [PubMed]  

22. L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

23. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999). [CrossRef]  

24. A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).

25. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011). [CrossRef]   [PubMed]  

26. V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29(12), 2622–2627 (2012). [CrossRef]   [PubMed]  

27. L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013). [CrossRef]  

28. V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere,” J. Opt. Soc. Am. A 31(1), 148–154 (2014). [CrossRef]   [PubMed]  

29. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014). [CrossRef]  

30. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014). [CrossRef]   [PubMed]  

31. Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007). [CrossRef]   [PubMed]  

32. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).

33. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering, 2005).

34. W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009). [CrossRef]  

References

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  1. D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
    [Crossref]
  2. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
    [Crossref]
  3. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
    [Crossref]
  4. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
    [Crossref]
  5. A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
  6. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
    [Crossref] [PubMed]
  7. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [Crossref]
  8. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]
  9. B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011).
    [Crossref] [PubMed]
  10. L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
    [Crossref]
  11. L. Y. Cui, B. D. Xue, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012).
    [Crossref] [PubMed]
  12. X. Yi, Z. J. Liu, and P. Yue, “Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
    [Crossref] [PubMed]
  13. L. Y. Cui, B. D. Xue, and F. G. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30(11), 2188–2195 (2013).
    [Crossref] [PubMed]
  14. L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 31(4), 829–835 (2014).
    [Crossref] [PubMed]
  15. R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
    [Crossref]
  16. F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
    [Crossref]
  17. G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
    [Crossref]
  18. L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
    [Crossref]
  19. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
    [Crossref]
  20. M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
    [Crossref]
  21. C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
    [Crossref] [PubMed]
  22. L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).
  23. M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
    [Crossref]
  24. A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).
  25. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
    [Crossref] [PubMed]
  26. V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29(12), 2622–2627 (2012).
    [Crossref] [PubMed]
  27. L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
    [Crossref]
  28. V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere,” J. Opt. Soc. Am. A 31(1), 148–154 (2014).
    [Crossref] [PubMed]
  29. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
    [Crossref]
  30. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
    [Crossref] [PubMed]
  31. Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007).
    [Crossref] [PubMed]
  32. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).
  33. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering, 2005).
  34. W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
    [Crossref]

2014 (4)

2013 (2)

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

L. Y. Cui, B. D. Xue, and F. G. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30(11), 2188–2195 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (2)

2009 (2)

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

2008 (2)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
[Crossref] [PubMed]

2007 (2)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

2006 (1)

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

2005 (1)

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

1999 (3)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

1998 (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

1997 (1)

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

1995 (4)

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[Crossref] [PubMed]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

1994 (1)

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

1992 (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

1986 (1)

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

Agrawal, B.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Antoshkin, L. V.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Bai, X. Z.

Barchers, J. D.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Belen’kii, M. S.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[Crossref] [PubMed]

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Bishop, K. P.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Botygina, N. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Brown, J. M.

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Cao, X. G.

Cheon, Y.

Conan, J. M.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Cuellar, E.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

Cui, L. Y.

Dalaudier, F.

Du, W. H.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

Eaton, F. D.

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Emaleev, O. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Fortes, B. V.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Golbraikh, E.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Grechko, G. M.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Gudimetla, V. S.

Gurvich, A. S.

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Holmes, R. B.

Hughes, K. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

Jiang, Y.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Kan, V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Karis, S. J.

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Keating, D. B.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Kireev, S. V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Kyrazis, D. T.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Lavrinova, L. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Leclerc, T.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Liu, Z. J.

Lukin, V. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Ma, J.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

Manning, R. M.

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

Michau, V.

Muschinski, A.

Nastrom, G. D.

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Osmon, C. L.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Preble, A. J.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Renard, J. B.

Restaino, S.

Riker, J. F.

Robert, C.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Rostov, A. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Rye, V. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

Savchenko, S. A.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Tan, L. Y.

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Toselli, I.

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Wissler, J. B.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Xie, W.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Xue, B. D.

Xue, W. F.

Yankov, A. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Yi, X.

Yu, S.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Yue, P.

Zheng, S. L.

Zhou, F. G.

Zilberman, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Adv. Space Res. (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Atmos. Oceanic Opt. (1)

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

J. Opt. Soc. Am. A (10)

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29(12), 2622–2627 (2012).
[Crossref] [PubMed]

V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere,” J. Opt. Soc. Am. A 31(1), 148–154 (2014).
[Crossref] [PubMed]

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007).
[Crossref] [PubMed]

C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
[Crossref] [PubMed]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

L. Y. Cui, B. D. Xue, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012).
[Crossref] [PubMed]

L. Y. Cui, B. D. Xue, and F. G. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30(11), 2188–2195 (2013).
[Crossref] [PubMed]

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 31(4), 829–835 (2014).
[Crossref] [PubMed]

J. Russ. Laser Res. (1)

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

Opt. Commun. (1)

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rep. (1)

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Proc. SPIE (10)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Radio Sci. (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Other (3)

A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering, 2005).

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Figures (5)

Fig. 1
Fig. 1 Aperture-averaged AOA variance, normalized by C ^ n 2 L D α4 , versus q for plane and spherical waves with different anisotropic factor ς values. (a): Plane wave; (b): Spherical wave.
Fig. 2
Fig. 2 Aperture-averaged AOA variance, normalized by C ^ n 2 L D α4 , versus q for plane and spherical waves with different anisotropic factor ς and general spectral power law values. (a): Plane wave; (b): Spherical wave.
Fig. 3
Fig. 3 Difference (%) between the close-form expression with the asymptotic fit results with different anisotropic factor ς and α values. (a): Differences as a function of ς ; (b): Differences as a function of α .
Fig. 4
Fig. 4 The influence of anisotropic factor on the variance of AOA fluctuations with different spectral power law values. (a): Plane wave; (b): Spherical wave.
Fig. 5
Fig. 5 The influence of general spectral power law values on the variance of AOA fluctuations with different anisotropic factor values. (a): Plane wave; (b): Spherical wave.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

Φ n_isotropic ( κ,α )=A( α ) C ^ n 2 κ α , ( 0κ<, 3<α<4 ).
A( α )= 1 4 π 2 Γ( α1 )cos[ απ 2 ].
D n ( R,α,ς )=A( α ) C ^ n 2 ( x 2 + y 2 ς 2 + z 2 ) α/2 , l 0 R L 0 ,3<α<4.
Φ n ( κ,α,ς )=A( α ) C ^ n 2 ς 2 ( κ ' ) α =A( α ) C ^ n 2 ς 2 [ κ z 2 + ς 2 ( κ x 2 + κ y 2 ) ] α 2 .
Φ n ( κ,α,ς )=A( α ) C ^ n 2 ς 2α κ α ,κ= κ x 2 + κ y 2 .
C θ ( ρ,β )=π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( ρκ )cos( 2β ) J 2 ( ρκ ) ]dκ
G D ( κ )exp( b 2 D 2 κ 2 4 ),b=0.52.
σ 2 = C θ ( ρ=0,β=0 )=π k 2 0 κ 3 W φ ( κ ) G D ( κ )dκ ,
W ϕ( pl ) ( κ )=2π k 2 0 L Φ n ( κ,α,ς ) cos 2 ( κ 2 z 2k )dz ,
σ ( pl ) 2 =2 π 2 0 κ 3 Φ n ( κ,α,ς ) G D ( κ )dκ 0 L cos 2 ( κ 2 z 2k )dz .
σ ( pl ) 2 = π 2 0 dκ 0 1 dz κ 3 Φ n ( κ,α,ς )[ 1+ k L κ 2 sin( L κ 2 k ) ]exp[ b 2 D 2 κ 2 4 ] ,
Γ( x )= 0 κ x1 e κ dκ ,
0 κ μ1 exp( aκ )sin( bκ )dκ = Γ( μ ) ( a 2 + b 2 ) μ/2 sin[ μ tan 1 ( b a ) ],μ>1,a>0
σ ( pl ) 2 ( α,ζ )= π 2 A( α ) C ^ n 2 L ς 2α 2 { Γ( 2- α 2 ) ( b 2 D 2 4 ) α 2 -2 + k L Γ( 1 α 2 ) ( b 4 D 4 16 + L 2 k 2 ) 2α 4 sin[ 2α 2 tan 1 ( 4L k b 2 D 2 ) ] }.
σ ( pl ) 2 ( α,ζ )= r p ( q,α,ζ ) C ^ n 2 L D α4 ,
r p ( q,α,ζ )= π 2 A( α ) ς 2α 2 Γ( 2 α 2 ) ( b 2 ) α-4 ×{ 1+ 2 α2 ( π 2 ) 2α/2 b 4α q 4α ( 1+ b 4 q 4 π 2 4 ) α2 4 sin[ α2 2 tan 1 ( 2 b 2 q 2 π ) ] }.
σ ( pl ) 2 ( q1,α,ζ )= r p ( q1,α,ζ ) C ^ n 2 L D α4 ,
r p ( q1,α,ζ )= π 2 A( α ) ς 2α 2 Γ( 2 α 2 ) ( b 2 ) α-4 .
σ ( pl ) 2 ( q1,α,ζ )= r p ( q1,α,ζ ) C ^ n 2 L D α4 ,
r p ( q1,α,ζ )= π 2 A( α ) ς 2α Γ( 2 α 2 ) ( b 2 ) α-4 .
σ ( pl ) 2 ( q<1,α,ζ )= r p ( q<1,α,ζ ) C ^ n 2 L D α4 ,
r p ( q<1,α,ζ )= π 2 A( α ) ς 2α 2 Γ( 2 α 2 ) ( b 2 ) α-4 ×{ 1+ 2 α2 ( π 2 ) 2α/2 b 4α q 4α sin[ π( α2 ) 4 ] }.
W ϕ( sp ) ( κ )=2π k 2 0 L Φ n ( κ,α,ς ) ( z L ) 2 cos 2 [ κ 2 z( Lz ) 2kL ]dz .
σ ( sp ) 2 = π 2 L 0 dκ 0 1 dξ κ 3 Φ n ( κ,α,ς )[ 1+cos( κ 2 ξ( 1ξ )L k ) ] ξ 2 exp[ b 2 D 2 κ 2 ξ 2 4 ] ,
F 2 1 ( A,B;C;Z )= Γ( C ) Γ( B )Γ( CB ) 0 1 t B1 ( 1t ) CB1 ( 1tZ ) A dt.
σ sp 2 ( α,ζ )= π 2 A( α ) C ^ n 2 L ς 2α 2 Γ( 2 α 2 ){ 1 α1 ( b 2 D 2 4 ) α/22 + Re[ ( i L k ) 4α 2 2 2+α F 2 1 ( 4α 2 , 2+α 2 ; 4+α 2 ;1+i b 2 D 2 k 4L ) ] }.
σ sp 2 ( α,ζ )= r s ( q,α,ς ) C ^ n 2 L D α4 ,
r s ( q,α,ζ )= π 2 A( α ) ς 2α 2( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 { 1+ 2( α1 ) 2+α × Re[ ( 2i π b 2 q 2 ) 4α 2 F 2 1 ( 4α 2 , 2+α 2 ; 4+α 2 ;1+i π b 2 q 2 2 ) ] }.
σ sp 2 ( q1,α,ζ )= r p ( q1,α,ζ ) C ^ n 2 L D α4 .
r p ( q1,α,ζ )= π 2 A( α ) ς 2α 2( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 .
F 2 1 ( a,b;c;z )= Γ( c )Γ( ba ) Γ( b )Γ( ca ) ( z ) a F 2 1 ( a,1c+a;1b+a;1/z ) + Γ( c )Γ( ab ) Γ( a )Γ( cb ) ( z ) b F 2 1 ( b,1c+b;1a+b;1/z )
σ sp 2 ( q1,α,ζ )= r s ( q1,α,ζ ) C ^ n 2 L D α4 ,
r s ( q1,α,ζ )= π 2 A( α ) ς 2α ( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 .
F 2 1 ( a,b;c;1 )= Γ( c )Γ( cab ) Γ( ca )Γ( cb ) .
σ sp 2 ( q<1,α,ζ )= r s ( q<1,α,ζ ) C ^ n 2 L D α4 ,
r s ( q<1,α,ζ )= π 2 A( α ) ς 2α 2( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 ×{ 1+ 2( α1 ) 2+α Re[ ( 2i π b 2 q 2 ) 4α 2 Γ( 2+α/2 )Γ( α/21 ) Γ( α ) ] }.
σ ( pl ) 2 ( q,α,ζ ) r p ( q,α,ζ ) C ^ n 2 L D α4 ,
σ sp 2 ( q,α,ζ ) r s ( q,α,ζ ) C ^ n 2 L D α4 ,
r p ( q,α,ζ )={ π 2 A( α ) ς 2α 2 Γ( 2 α 2 ) ( b 2 ) α-4 ×{ 1+ 2 α2 ( π 2 ) 2α/2 b 4α q 4α sin[ π( α2 ) 4 ] }, q1 π 2 A( α ) ς 2α Γ( 2 α 2 ) ( b 2 ) α-4 . q>1
r s ( q,α,ζ )={ π 2 A( α ) ς 2α 2( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 ×{ 1+ 2( α1 ) 2+α Re[ ( 2i π b 2 q 2 ) 4α 2 Γ( 2+α/2 )Γ( α/21 ) Γ( α ) ] }, q1 π 2 A( α ) ς 2α ( α1 ) Γ( 2 α 2 ) ( b 2 4 ) α/22 . q>1

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