Generalized anisotropic turbulence spectra and applications in the optical waves&#x2019; propagation through anisotropic turbulence

Linyan Cui; Bindang Xue; Fugen Zhou

doi:10.1364/OE.23.030088

1. Introduction

The atmospheric turbulence refractive-index fluctuations spectra represent the statistical property of atmospheric refractive-index random fluctuations, and are the fundamental models to investigate optical waves’ propagation through atmospheric turbulence media [1]. In recent years, the researches have mainly focused on the isotropic non-Kolmogorov atmospheric turbulence. The generalized non-Kolmogorov spectrum [2], the generalized von Karman spectrum [3], the generalized exponential spectrum [4], and the generalized modified atmospheric spectrum [5] have been derived theoretically and applied to investigate the for optical plane and spherical waves’ propagation through isotropic non-Kolmogorov atmospheric turbulence [6-10]. In which, the generalized non-Kolmogorov spectrum is only valid in the inertial sub-range. When it is used to estimate the optical wave’s propagation through atmospheric turublence, it is ordinarily extended to all ranges by assuming the turbulence inner scale is zero and the turbulence outer scale is infinite. In comparison, the generalized von Karman and the generalized exponential spectra consider mathematically the turbulence inner and outer scales, and are widely used to study theoretically the optical wave’s propagation through isotropic turbulence. The generalized modified atmospheric spectrum can represent physically the bump property introduced by the turbulence inner scale. But, this spectral model is derived with the experimental data measured for the isotropic Kolmogorov turbulence, so it needs to be improved with further experiments [5].

Experimental and theoretical results have shown that the atmospheric turbulence can also be anisotropic [11-28]. Grechko and et al. [13] reported a strong anisotropy in the middle atmosphere from experimental observations of star scintillation. Biferale and et al. [14] detected the information about anisotropic turbulence in the boundary layer by using two probes with two different geometries (horizontally and vertically). Belen’kii and et al. [15,16] experimentally observed anisotropy of the statistics of wavefront tilt. They observed a horizontal turbulence 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 [17], where authors used a power spectrum with two components: anisotropic and isotropic, and the validity of the spectrum was verified by balloon-borne experiments. 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 eddies [23]. Anisotropy can be present also at few meters above the ground [11].

For anisotropic Kolmogorov atmospheric turbulence, the anisotropic Kolmogorov three dimensional atmospheric turbulence spectral model was derived [21] to describe the anisotropy property of the Kolmogorov turbulence. For the anisotropic non-Kolmogorov turbulence, the anisotropic generalized non-Kolmogorov spectrum with circular symmetric assumption of turbulence eddies in the orthogonal xy-plane throughout the path was derived [21] and applied to investigate the optical waves’ propagation through the anisotropic non-Kolmogorov turbulence [21,23,25]. Then, the more anisotropic non-Kolmogorov turbulence spectrum was proposed by Gudimetla [22] and Andrews [26] to analyze the optical waves’ propagation through the anisotropic non-Kolmogorov turbulence [22,24]. In the investigation, the turbulence eddies in the orthogonal xy-plane are no longer circularly symmetric (i.e., isotropic). This more general spectrum may lead to different statistical values in the horizontal and vertical transverse directions. In [27], the concept of anisotropy at different turbulence eddy scales was introduced to the general anisotropic non-Kolmogorov turbulence spectrum by an effective anisotropic parameter which was defined for two specific cases of linear and parabolic anisotropic laws. But, these three anisotropic turbulence spectral models are only valid in the inertial sub-range and the finite turbulence inner and outer scales are not considered. Toselli [28] introduced the finite turbulence inner and outer scales into the turbulence refractive-index fluctuations spectrum by combining the generalized von Karman spectrum [3] and the conventional anisotropic non-Kolmogorov spectrum reported in [21]. Also, the concept of anisotropy at different turbulence eddy scales was considered. However, the circular symmetric assumption of turbulence eddies in the orthogonal xy-plane throughout the path was still adopted in [28].

In this study, to consider simultaneously the asymmetric property of turbulence eddies in the orthogonal xy-plane throughout the path and the finite turbulence inner and outer scales, two anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectral models will be derived and applied to investigate the variance of AOA fluctuations for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. First, the atmospheric turbulence refractive-index fluctuations spectra (the generalized von Karman spectrum and the generalized exponential spectrum) for isotropic non-Kolmogorov turbulence will be introduced. Second, taking the asymmetric property of turbulence eddies in the orthogonal xy-plane into consideration, the generalized von Karman spectrum and the generalized exponential spectrum will be further generalized, and two anisotropic non-Kolmogorov turbulence spectra (the anisotropic generalized von Karman spectrum and the anisotropic generalized exponential spectrum) will be derived. The parameters of finite turbulence inner and outer scales, general spectral power law value, and two anisotropic factors whch parameterize the asymmetry of turbulence eddies in both horizontal and vertical directions will be included. Third, based on the derived anisoytropic turbulence spectrum and the Rytov approximation theory, the variances of AOA fluctuations will be derived for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. Last, calculations will be performed to analyze the derived anisotropic non-Kolmogorov turbulence spectra and variance of AOA fluctuations.

2. Generalized refractive-index fluctuations spectra for isotropic non-Kolmogorov turbulence

The generalized von Karman spectrum and the generalized exponential spectrum for the isotropic non-Kolmogorov turbulence take the forms as [3,4]:

Φ_{n_v o n} (κ, α) = \frac{A (α) {\hat{C}}_{n}^{2}}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} \begin{matrix} \exp (- \frac{κ^{2}}{κ_{l}^{2}}), & (\begin{matrix} 0 \leq κ < \infty, & 3 < α < 4 \end{matrix}) \end{matrix} .

Φ_{n_\exp} (κ, α) = A (α) {\hat{C}}_{n}^{2} κ^{- α} [1 - \exp (- \frac{κ^{2}}{κ_{0}^{2}})] \exp (- \frac{κ^{2}}{κ_{l}^{2}}) \begin{matrix} , & (\begin{matrix} 0 \leq κ < \infty, & 3 < α < 4 \end{matrix}) \end{matrix} .

where

Φ_{n_v o n} (κ, α)

and

Φ_{n_\exp} (κ, α)

are the generalized von Karman spectrum and the generalized exponential spectrum, respectively.

α

is the general spectral power law and varies from 3 to 4.

{\hat{C}}_{n}^{2} = γ C_{n}^{2}

is the generalized structure parameter with unit [

m^{3 - α}

] and

γ

is a dimensional constant with unit [

m^{11 / 3 - α}

]. For Kolmogorov power law (

α = 11 / 3

), the generalized structure parameter reduces to the structure parameter

C_{n}^{2}

with unit [

m^{- 2 / 3}

].

κ

denotes the magnitude of the spatial-frequency vector with units of

r a d / m

and it is related to the size of turbulence eddies.

κ_{l} = c (α) / l_{0}

,

κ_{0} = C_{0} / L_{0}

,

C_{0} = 2 π

for the generalized von Karman spectrum and

C_{0} = 4 π

for the generalized exponential spectrum.

l_{0}

and

L_{0}

are separately the turbulence inner and outer scales.

A (α)

is a constant which maintains consistency between the refractive-index structure function and its refractive-index fluctuations spectrum.

A (α)

and

c (α)

were given in [3,4].

For anisotropic non-Kolmogorov turbulence, the anisotropic factors which parameterize the asymmetry of turbulence eddies in different propagation paths should be considered. At this time, the generalized von Karman spectrum and the generalized exponential spectrum are not applicable. In the next section, these two spectral models will be further generalized and two turbulence refractive-index fluctuations spectra will be derived for anisotropic non-Kolmogorov turbulence.

3. Generalized refractive-index fluctuations spectra for anisotropic non-Kolmogorov turbulence

Before deriving the anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectra, it needs to introduce the refractive-index fluctuations structure function $D_{n} (\cdot)$ , which describes the behavior of the correlations of turbulence refractive-index field fluctuations between two given points separated by a distance of $\vec{R}$ . $R$ is a vector spatial variable, $R = (R_{x}, R_{y}, R_{z})$ , $R_{x}$ , $R_{y}$ , and $R_{z}$ are the components of $R$ in the x, y, and z directions.

If the turbulence is isotropic, the refractive-index fluctuations structure function $D_{n} (\cdot)$ has been well known. The relationship between the refractive-index fluctuations structure function $D_{n} (\cdot)$ and refractive-index fluctuations spectrum $Φ_{n} (\cdot)$ satisfies [3,4]:

Φ_{n} (κ, α) = \frac{1}{4 π^{2} κ^{2}} \int_{0}^{\infty} \frac{\sin (κ R)}{κ R} \frac{\partial}{\partial R} [R^{2} \frac{\partial D_{n} (R, α)}{\partial R}] d R .

For isotropic turublence,

R = | \vec{R} | = \sqrt{R_{x}^{2} + R_{y}^{2} + R_{z}^{2}}

. If the turbulence eddies are anisotropic, the horizontal extension of turbulence eddies is higher than the vertical one, so the anisotropic factors of

μ_{x}

and

μ_{y}

need to be considered.

μ_{x}

and

μ_{y}

are the ratios between the turbulence eddies in horizontal direction and those in vertical direction. As the horizontal turbulence eddy is always larger than the vertical one for the anisotropic turbulence,

μ_{x}

and

μ_{y}

are always bigger than one. When both

μ_{x}

and

μ_{y}

equal one, the isotropic turbulence is shown. When they increase, the anisotropic property exhibits more obviously. In the following part, we will give the detailed derivations about how to transfer the investigations for anisotropic turbulence to the cases for isotropic turbulence.

If the turbulence is anisotropic, by making the changes of variables $x = μ_{x} x^{'}$ and $y = μ_{y} y^{'}$ [26], the refractive-index structure function for the anisotropic turbulence will become isotropic in this new spatial variable $(x^{'}, y^{'}, z)$ . The rectangular coordinate differentials in Eq. (3) are related by $d x^{'} d y^{'} d z = \frac{d x d y d z}{μ_{x} μ_{y}}$ and $d R^{'} = \frac{d R}{μ_{x} μ_{y}}$ . And the resulting wavenumber $\vec{κ}$ will be isotropic in the stretched wavenumber space $\vec{κ^{'}} = (κ_{x}^{'}, κ_{y}^{'}, κ_{z}^{'})$ , $κ_{x}^{'} = μ_{x} κ_{x}$ , $κ_{y}^{'} = μ_{y} κ_{y}$ , $κ_{z}^{'} = κ_{z}$ , and $κ^{'} = | \vec{κ^{'}} | = \sqrt{μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2}}$ . With the above variable substitutions and in view of the relationship between the refractive-index fluctuations structure function $D_{n} (\cdot)$ and refractive-index fluctuations spectrum $Φ_{n} (\cdot)$ under isotropic turbulence, the anisotropic turbulence refractive-index fluctuations spectrum and the refractive-index fluctuations structure function will satisfy the following relationship:

Φ_{n_a n i s o} (κ, α, μ_{x}, μ_{y}) = μ_{x} μ_{y} \frac{1}{4 π^{2} {(κ')}^{2}} \int_{0}^{\infty} \frac{\sin (κ^{'} R^{'})}{κ^{'} R^{'}} \frac{\partial}{\partial R^{'}} [{(R')}^{2} \frac{\partial D_{n} (R^{'}, α)}{\partial R^{'}}] d R^{'} .

In which,

Φ_{n} (κ^{'}, α) = \frac{1}{4 π^{2} {(κ')}^{2}} \int_{0}^{\infty} \frac{\sin (κ^{'} R^{'})}{κ^{'} R^{'}} \frac{\partial}{\partial R^{'}} [{(R')}^{2} \frac{\partial D_{n} (R^{'}, α)}{\partial R^{'}}] d R^{'} .

Therefore, the anisotropic generalized von Karman spectrum and the anisotropic generalized exponential spectrum can be expressed as

Φ_{n_a n i s o_\exp} (κ, α, μ_{x}, μ_{y}) = μ_{x} μ_{y} Φ_{n_\exp} (κ^{'}, α),

Φ_{n_a n i s o_v o n} (κ, α, μ_{x}, μ_{y}) = μ_{x} μ_{y} Φ_{n_v o n} (κ^{'}, α) .

Considering the variable substitutions mentioned above, the wavenumber $κ^{'}$ associated with turbulence inner and outer scales become $κ_{l}^{'} = c^{'} (α) / l_{0}$ and $κ_{0}^{'} = C_{0} / L_{0}$ . $C_{0}$ is set to the same value as for the generalized von Karman spectrum and the generalized exponential spectrum [28]. In the following derivations, $c^{'} (α)$ and the constant of $\hat{A} (α)$ which maintains consistency between the anisotropic turbulence refractive-index structure function and its refractive-index fluctuations spectrum will be derived.

Here, the detailed derivations for the anisotropic generalized exponential spectrum will be given. Substituting Eq. (6) into Eq. (4), it yields

D_{n} (R^{'}, α) = 8 π \int_{0}^{\infty} {(κ^{'})}^{2} \cdot Φ_{n_\exp} (κ^{'}, α) \cdot (1 - \frac{\sin κ^{'} R^{'}}{κ^{'} R^{'}}) d κ^{'} .

Following the procedures shown in Appendix A, $\hat{A} (α)$ and $c^{'} (α)$ are finally derived and take the expressions as

\hat{A} (α) = \frac{Γ (α - 1)}{4 π^{2}} \sin [(α - 3) \frac{π}{2}] .

c^{'} (α) = {[π \hat{A} (α) Γ (- \frac{α}{2} + \frac{3}{2}) (\frac{3 - α}{3})]}^{\frac{1}{α - 5}} .

At this time, the anisotropic generalized exponential spectrum is obtained. Following the same procedures as described above, the anisotropic generalized von Karman spectrum is also obtained. They consider simultaneously the finite turbulence inner and outer scales, the general spectral power law value, and two anisotropic factors which parameterize the asymmetry of turbulence eddies in different propagation paths.

Φ_{n_a n i s o_v o n} (κ, α, μ_{x}, μ_{y}) = \frac{\hat{A} (α) {\hat{C}}_{n}^{2} μ_{x} μ_{y}}{{(μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2} + κ_{0}^{' 2})}^{α / 2}} \exp (- \frac{μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2}}{κ_{l}^{' 2}}),

\begin{matrix} Φ_{n_a n i s o_e x p} (κ, α, μ_{x}, μ_{y}) = μ_{x} μ_{y} \hat{A} (α) {\hat{C}}_{n}^{2} {(μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2})}^{- α / 2} \\ [1 - \exp (- \frac{μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2}}{κ_{0}^{' 2}})] \exp (- \frac{μ_{x}^{2} κ_{x}^{2} + μ_{y}^{2} κ_{y}^{2} + κ_{z}^{2}}{κ_{l}^{' 2}}) . \end{matrix}

4. Variance of AOA fluctuations based on the derived spectrum under anisotropic non-Kolmogorov turbulence

In this section, the derived anisotropic generalized exponential spectrum will be adopted to investigate the variance of AOA for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. The AOA fluctuations are related to the signal distortion of the long-range imaging system or laser communication systems and they reduce greatly the performance of these systems. Considering the relation between the variance and covariance, the variance of AOA fluctuations can be derived from the spatial covariance of AOA fluctuations. According to the Wiener-Khinchin theorem, the spatial covariance function of the AOA fluctuations $C_{θ} (\cdot)$ can be expressed as [29]:

C_{θ} (ρ, β) = π k^{- 2} \int_{0}^{\infty} κ^{3} W_{ϕ} (κ) G_{D} (κ) [J_{0} (ρ κ) - \cos (2 β) J_{2} (ρ κ)] d κ,

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 = 2 π / λ

and

λ

denotes the optical wavelength.

J_{0} (ρ κ)

denotes the zero order Bessel function.

W_{ϕ} (κ)

is the wave-front phase power spectrum.

G_{D} (κ)

represents the point spread function of the receiver aperture [30]:

G_{D} (κ) \approx \exp (- \frac{b^{2} D^{2} κ^{2}}{4}), b = 0.52.

In view of the Rytov approximation theory, the wave-front phase power spectra for plane and spherical waves are given by

W_{ϕ (p l)} (κ) = 2 π k^{2} \int_{0}^{L} Φ_{n_a n i s o_e x p} (κ, α, u_{x}, u_{y}) \cos^{2} (\frac{κ^{2} z}{2 k}) d z,

W_{ϕ (s p)} (κ) = 2 π k^{2} \int_{0}^{L} Φ_{n_a n i s o_e x p} (κ, α, u_{x}, u_{y}) {(\frac{z}{L})}^{2} \cos^{2} [\frac{κ^{2} z (L - z)}{2 k L}] d z .

W_{ϕ (p l)} (κ)

and

W_{ϕ (s p)} (κ)

are the wave-front phase power spectra for plane and spherical waves, respectively. Using the relation between the variance and covariance, when

ρ

and

β

in Eq. (13) equal to zero, the variance of the AOA fluctuations can be obtained as follows:

σ_{}^{2} = C_{θ} (ρ = 0, β = 0) = π k^{- 2} \int_{0}^{\infty} κ^{3} W_{φ} (κ) G_{D} (κ) d κ .

In the following analysis, by invoking the Markov approximation, which assumes that the index of refraction is delta-correlated at any pair of points located along the direction of propagation, the component of $κ_{z}$ in Eq. (12) can be ignored. At this time, the turbulence is essentially supposed to be layered along the direction of propagation.

Substituting Eqs. (14), (15) and (16) into Eq. (17), the variances of the AOA fluctuations for plane and spherical waves become

σ_{(p l)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \int_{0}^{\infty} \int_{0}^{\infty} κ^{2} Φ_{n_a n i s o_e x p} (κ, α, μ_{x}, μ_{y}) [1 + \frac{k}{L κ^{2}} \sin (\frac{L κ^{2}}{k})] \exp [- \frac{b^{2} D^{2} κ^{2}}{4}] d κ_{x} d κ_{y},

\begin{matrix} σ_{(s p)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{1} d ξ κ^{2} Φ_{n_a n i s o_e x p} (κ, α, μ_{x}, μ_{y}) \\ [1 + \cos (\frac{κ^{2} ξ (1 - ξ) L}{k})] ξ^{2} \exp [- \frac{b^{2} D^{2} κ^{2} ξ^{2}}{4}] d κ_{x} d κ_{y} . \end{matrix}

where,

κ^{2} = κ_{x}^{2} + κ_{y}^{2}

,

ξ = z / L

. Then we change from a stretched coordinate system for the spectrum to an isotropic one through the substitutions

\begin{array}{l} κ_{x} = \frac{q_{x}}{μ_{x}} = \frac{q \cos θ}{μ_{x}}, κ_{y} = \frac{q_{y}}{μ_{y}} = \frac{q \sin θ}{μ_{y}}, q = \sqrt{q_{x}^{2} + q_{y}^{2}} . \\ d κ_{x} d κ_{y} = \frac{d q_{x} d q_{y}}{μ_{x} μ_{y}} = \frac{q d q d θ}{μ_{x} μ_{y}}, \\ Φ_{n_a n i s o_e x p} (κ, α, μ_{x}, μ_{y}) = μ_{x} μ_{y} \hat{A} (α) {\hat{C}}_{n}^{2} q^{- α} [1 - \exp (- \frac{q^{2}}{κ_{0}^{' 2}})] \exp (- \frac{q^{2}}{κ_{l}^{' 2}}) . \end{array}

Substituting Eq. (20) into Eqs. (18) and (19), it arrives at

\begin{matrix} σ_{(p l)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \hat{A} (α) {\hat{C}}_{n}^{2} \int_{0}^{2 π} \int_{0}^{\infty} q^{3 - α} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}) [1 - \exp (- \frac{q^{2}}{κ_{0}^{' 2}})] \exp (- \frac{q^{2}}{κ_{l}^{' 2}}) \\ [1 + \frac{k}{L q^{2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1} \sin (\frac{L q^{2}}{k} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}))] \\ \exp [- \frac{b^{2} D^{2} q^{2}}{4} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})] d q d θ, \end{matrix}

\begin{matrix} σ_{(s p)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \hat{A} (α) {\hat{C}}_{n}^{2} \int_{0}^{2 π} \int_{0}^{\infty} \int_{0}^{1} q^{3 - α} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}) [1 - \exp (- \frac{q^{2}}{κ_{0}^{' 2}})] \\ \exp (- \frac{q^{2}}{κ_{l}^{' 2}}) [1 + \cos (\frac{q^{2} ξ (1 - ξ) L}{k} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}))] ξ^{2} \\ \exp [- \frac{b^{2} D^{2} q^{2} ξ^{2}}{4} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})] d q d θ d ξ . \end{matrix}

Then, making the variable substitution of $q_{1} = q \sqrt{\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}}$ , it yields

\begin{matrix} σ_{(p l)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \hat{A} (α) {\hat{C}}_{n}^{2} \int_{0}^{2 π} \int_{0}^{\infty} q_{1}^{3 - α} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- \frac{2 - α}{2}} \\ [1 - \exp (- \frac{q_{1}^{2}}{κ_{0}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1})] e x p [- \frac{q_{1}^{2}}{κ_{l}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1}] \\ [1 + \frac{k}{L q_{1}^{2}} \sin (\frac{L q_{1}^{2}}{k})] \exp [- \frac{b^{2} D^{2} q_{1}^{2}}{4}] d q_{1} d θ, \end{matrix}

\begin{matrix} σ_{(s p)}^{2} (α, μ_{x}, μ_{y}) = \frac{π L}{2} \hat{A} (α) {\hat{C}}_{n}^{2} \int_{0}^{2 π} \int_{0}^{\infty} \int_{0}^{1} q_{1}^{3 - α} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\cos^{2} θ}{μ_{y}^{2}})}^{- \frac{2 - α}{2}} \\ [1 - \exp (- \frac{q_{1}^{2}}{κ_{0}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\cos^{2} θ}{μ_{y}^{2}})}^{- 1})] \exp [- \frac{q_{1}^{2}}{κ_{l}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\cos^{2} θ}{μ_{y}^{2}})}^{- 1}] \\ [1 + \cos (\frac{q_{1}^{2} ξ (1 - ξ) L}{k})] ξ^{2} \exp [- \frac{b^{2} D^{2} q_{1}^{2} ξ^{2}}{4}] d q_{1} d θ d ξ . \end{matrix}

In view of the geometrical optics assumption (the Fresnel scale ${(L / k)}^{1 / 2}$ is much smaller than the receiver aperture diameter), $\frac{k}{L q_{1}^{2}} \sin (\frac{L q_{1}^{2}}{k})$ and $\cos (\frac{q_{1}^{2} ξ (1 - ξ) L}{k})$ in Eqs. (23) and (24) can be expressed approximately as $\frac{k}{L q_{1}^{2}} \sin (\frac{L q_{1}^{2}}{k}) \approx 1$ and $\cos (\frac{q_{1}^{2} ξ (1 - ξ) L}{k}) \approx 1$ . Based on this, considering the definition of gamma function $Γ (x)$ and the gauss hypergeometric function ${}_{2}F_{1} (a, b; c; z)$ [31]:

{}_{2}F_{1} (a, b; c; z) = \frac{Γ (c)}{Γ (b) \cdot Γ (c - b)} \int_{0}^{1} t^{b - 1} \cdot {(1 - t)}^{c - b - 1} \cdot {(1 - t z)}^{- a} d t,

the variances of AOA fluctuations for plane and spherical waves under anisotropic non-Kolmogorov turbulence become

\begin{matrix} σ_{(p l)}^{2} (α, μ_{x}, μ_{y}) = \frac{π}{2} \hat{A} (α) {\hat{C}}_{n}^{2} L Γ (2 - \frac{α}{2}) \int_{0}^{2 π} d θ {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{\frac{α - 2}{2}} \\ {{[\frac{b^{2} D^{2}}{4} + \frac{1}{κ_{l}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1}]}^{- 2 + \frac{α}{2}} - \\ {[\frac{b^{2} D^{2}}{4} + (\frac{1}{κ_{0}^{' 2}} + \frac{1}{κ_{l}^{' 2}}) {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1}]}^{- 2 + \frac{α}{2}}}, \end{matrix}

\begin{matrix} σ_{s p}^{2} (α, μ_{x}, μ_{y}) = \frac{π}{2} \hat{A} (α) {\hat{C}}_{n}^{2} L \frac{1}{3} Γ (2 - \frac{α}{2}) \int_{0}^{2 π} d θ (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}) {{(\frac{1}{κ_{l}^{' 2}})}^{- 2 + \frac{α}{2}} \\ {}_{2}F_{1} (2 - \frac{α}{2}, \frac{3}{2}; \frac{5}{2}; - \frac{b^{2} D^{2} κ_{l}^{2}}{4} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})) - {(\frac{1}{κ_{l}^{' 2}} + \frac{1}{κ_{0}^{' 2}})}^{- 2 + \frac{α}{2}} \\ {}_{2}F_{1} (2 - \frac{α}{2}, \frac{3}{2}; \frac{5}{2}; - \frac{b^{2} D^{2}}{4} {(\frac{1}{κ_{l}^{' 2}} + \frac{1}{κ_{0}^{' 2}})}^{- 1} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}}))} . \end{matrix}

For the AOA fluctuations, the turbulence outer scale plays a dominant role in the analyses and the turbulence inner scale’s influence can be ignored [8-10]. In addition, the diameter of the receiver aperture is on the order of millimeter and the turbulence outer scale is on the order of meters, and $μ_{x}$ and $μ_{y}$ are always bigger than one. Therefore, $\frac{b^{2} D^{2}}{4} {(\frac{1}{κ_{l}^{' 2}} + \frac{1}{κ_{0}^{' 2}})}^{- 1} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})$ is much smaller than one. ${}_{2}F_{1} (a, b; c; z)$ in Eq. (27) can be expressed approximately as [31]:

\begin{array}{l} {}_{2}F_{1} (a, b; c; z) = \frac{Γ (c) Γ (b - a)}{Γ (b) Γ (c - a)} {(- z)}^{- a} (1 + \frac{a (1 - c + a)}{1 - b + a} \frac{1}{z}) \\ + \frac{Γ (c) Γ (a - b)}{Γ (a) Γ (c - b)} {(- z)}^{- b} (1 + \frac{b (1 - c + b)}{1 - a + b} \frac{1}{z}), | z | ≫ 1 \end{array}

{}_{2}F_{1} (a, b; c; z) = 1 + \frac{a b}{c} z, | z | ≪ 1

At this time, the expressions of the variance of AOA fluctuations can be further simplified and take the forms as

\begin{matrix} σ_{(p l)}^{2} (α, μ_{x}, μ_{y}) = \frac{π}{2} \hat{A} (α) {\hat{C}}_{n}^{2} L Γ (2 - \frac{α}{2}) \int_{0}^{2 π} d θ {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{\frac{α}{2} - 1} \\ {{(\frac{b^{2} D^{2}}{4})}^{- 2 + \frac{α}{2}} - {[\frac{b^{2} D^{2}}{4} + \frac{1}{κ_{0}^{' 2}} {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{- 1}]}^{- 2 + \frac{α}{2}}}, \end{matrix}

\begin{matrix} σ_{s p}^{2} (α, μ_{x}, μ_{y}) = \frac{π}{2} \hat{A} (α) {\hat{C}}_{n}^{2} L \frac{1}{3} Γ (2 - \frac{α}{2}) \int_{0}^{2 π} d θ {(\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})}^{\frac{α}{2} - 1} {\frac{3}{α - 1} {(\frac{b^{2} D^{2}}{4})}^{\frac{α}{2} - 2} - \\ {[κ_{0}^{' 2} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})]}^{2 - \frac{α}{2}} \cdot [1 - \frac{3}{5} (2 - \frac{α}{2}) \frac{b^{2} D^{2} κ_{0}^{' 2}}{4} (\frac{\cos^{2} θ}{μ_{x}^{2}} + \frac{\sin^{2} θ}{μ_{y}^{2}})]} . \end{matrix}

When $μ_{x} = μ_{y} = 1$ (corresponding to the isotropic non-Kolmogorov atmospheric turbulence) and the finite turbulence inner and outer scales are ignored, Eqs. (30) and (31) have good consistency with the previously published results reported in [7] which focused on the isotropic non-Kolmogorov turbulence.

5. Calculations and analyses

In this section, first, the derived anisotropic generalized exponential spectrum and anisotropic generalized von Karman spectrum will be analyzed. To compare them with the spectra derived for the isotropic non-Kolmogorov turbulence, the ratios between the anisotropic and isotropic non-Kolmogorov spectra will be calculated:

r a t i o_{v o n} = \frac{Φ_{n_a n i s o_v o n} (κ, α, μ_{x}, μ_{y})}{Φ_{n_v o n} (κ, α)},

r a t i o_{\exp} = \frac{Φ_{n_a n i s o_\exp} (κ, α, μ_{x}, μ_{y})}{Φ_{n_\exp} (κ, α)} .

Figure 1 plots the ratios of $r a t i o_{v o n}$ and $r a t i o_{\exp}$ as a function of wavenumber $κ$ with different anisotropic factors and general spectral power law values. In the calculation, $μ_{x}$ is fixed and $μ_{y}$ takes different values. Three different general spectral power law values (10/3, 11/3 and 3.9) are chosen. Note that the calculations performed in this work should be considered as an arbitrary example to show only some trend of results.

Fig. 1 The ratios between the anisotropic non-Kolmogorov turbulence spectra and the isotropic non-Kolmogorov turbulence spectra ( $l_{0} = 1 m m$ , $L_{0} = 10 m$ ). (a): $α = 10 / 3$ ; (b): $α = 11 / 3$ ; (c): $α = 3.9$ .

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It can be deduced from Fig. 1 that the atmospheric turbulence refractive-index fluctuations spectra are affected by a turbulence rescaling due to both anisotropy and non-Kolmogorov general spectral power law value. In the inertial sub-range of refractive-index spectrum, the ratios of $r a t i o_{v o n}$ and $r a t i o_{\exp}$ keep constant. They decrease sharply both in the input and dissipation sub-ranges. When the anisotropic factors increase, the values for the ratios of $r a t i o_{v o n}$ and $r a t i o_{\exp}$ decrease obviously. It can be explained from the definitions of the anisotropic generalized spectra, which are inversely proportional to the anisotropic factors. The physical explanation for this phenomenon has been already stated in [21], that is the curvature of the anisotropic turbulence eddies will change with respect to the isotropic turbulence case. These anisotropic turbulence eddies act as lenses with a longer radius of curvature compared with the isotropic ones, and they will change the focusing properties of the turbulence [21]. When the anisotropic factor increases (the anisotropy property exhibits more obviously), the focusing properties of the turbulence will become more and more weak. Eventually, the anisotropic turbulence eddies will make the optical wave less deviated from its propagation path. That is, the anisotropic turbulence eddies’ influence on the optical waves’ propagation becomes weaker and weaker with the increased anisotropic factors. Because the atmospheric turbulence spectrum represents the statistical property of atmospheric refractive-index fluctuations, it will take smaller value with the increased anisotropic factors.

Second, the variance of AOA fluctuations (with units of $r a d^{\land} 2$ ) will be analyzed for plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. The influences of anisotropic factors and general spectral power law value on the final results will be analyzed. Figure 2 shows the variances of AOA fluctuations for the chosen combinations of the general spectral power law $α$ and anisotropic factors $μ_{x}$ and $μ_{y}$ . In which, ${\hat{C}}_{n}^{2} = 3.7 \times 10^{- 16} m^{3 - α}$ , the values of $α$ and $μ_{x}$ are fixed and $μ_{y}$ varies from 1 to 100 like Andrews did in [26]. $α$ is in the range 3 to 4, in Fig. 2(a), the spectral power law index is $α = 10 / 3$ , in Fig. 2(b) the spectral power law value is the conventional Kolmogorov value $α = 11 / 3$ , and in Fig. 2(c) the value is $α = 3.9$ . For real atmosphere, the turbulence inner scale is usually in the order of millimeter and the turbulence outer scale is in the order of meter. In this calculation, they are set to $l_{0} = 1 m m$ and $L_{0} = 10 m$ as examples for theoretical analysis purpose and other values can also be chosen according to the specific propagation scenario.

Fig. 2 Variance of AOA fluctuations as a function of $μ_{y}$ with different $α$ and $μ_{x}$ values for both plane and spherical waves. (a): $α = 10 / 3$ ; (b): $α = 11 / 3$ ; (c): $α = 3.9$ .

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From Fig. 2, it can be seen that when $μ_{y}$ increases, the effect of the anisotropic atmosphere turbulence on the AOA fluctuations reduces greatly firstly when $μ_{x}$ is comparable with $μ_{y}$ and then reduces very slowly especially when $μ_{y}$ is much bigger than $μ_{x}$ . When $μ_{x}$ increases from 1 to 2, the anisotropic turbulence produces more effects on the variance of AOA fluctuations. In addition, when $μ_{x} = μ_{y}$ , the effects of anisotropic atmosphere turbulence on the final results fall rapidly for both optical plane and spherical waves. This is because the influences of $μ_{y}$ and $μ_{x}$ on the final results are connected together. When $μ_{y}$ is much bigger than $μ_{x}$ , $\frac{\sin^{2} θ}{μ_{y}^{2}}$ is much smaller than $\frac{\cos^{2} θ}{μ_{x}^{2}}$ . At this time, the part of $\frac{\cos^{2} θ}{μ_{x}^{2}}$ plays a dominant role in the derived AOA fluctuations models. Therefore, the anisotropic turbulence’s influence on the AOA fluctuations will almost keep unchanged when $μ_{x}$ is fixed and $μ_{y}$ is much bigger than $μ_{x}$ . The phenomenon shown in Fig. 2 can also be explained physically by the change of curvature of the anisotropic turbulence eddies just like it is explained for Fig. 1. When $μ_{x}$ is fixed and $μ_{y}$ is much bigger than $μ_{x}$ , the variation of the curvature of turbulence eddies will be almost ignorable. Therefore, the anisotropic turbulence’s influence on the AOA fluctuations will almost keep unchanged.

In Fig. 3, the variance of AOA fluctuations as a function of turbulence outer scale is plotted with different anisotropic factors. The turbulence inner scale is fixed to 10mm and the general spectral power law of 10/3 is chosen. As shown, with the increase of turbulence outer scale, the anisotropic non-Kolmogorov turbulence produces more effects on the AOA fluctuations. This conclusion is consistent with [8-10] which focus on the isotropic non-Kolmogorov turbulence. The phenomenon shown in Fig. 2 can be explained directly from the definition of the anisotropic generalized exponential spectrum, which takes the expression proportional to the turbulence outer scale. When turbulence outer scale increases, this spectrum takes a higher value, so that the variance of AOA fluctuations of plane and spherical waves increases. It can also be explained from the physical point of view: the phase fluctuations are contributed mostly by turbulence eddies with size of Fresnel scale or larger, when the turbulence outer scale is assumed with high value, the wave meets a major number of large-scale turbulent eddies along its propagation length and they lead to higher variance of AOA fluctuations with respect to the case of lower outer scale value, where more large turbulence scales are cut out.

Fig. 3 Variance of AOA fluctuations as a function of $L_{0}$ with different $μ_{x}$ and $μ_{y}$ values for both plane and spherical waves ( $α = 10 / 3$ ). (a): $μ_{x} = 1, μ_{y} = 2, 5, 10$ ; (b): $μ_{x} = 2, μ_{y} = 2, 5, 10$ ; (c): $μ_{x} = μ_{y} = 1, 2, 10$ .

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From Figs. 2 and 3, it can also be seen that the anisotropic turbulence produces more effects on the AOA fluctuations for plane wave case than that for the spherical wave case. This conclusion is consistent with that for the isotropic turbulence. Note that due to the lack of sufficient experiments about the anisotropic non-Kolmogorov turbulence, the atmospheric turbulence parameters in the calculations cannot be well related to the specific anisotropic atmospheric turbulence. The parameters adopted in this work are set as examples for theoretical analyses purpose and other values can also be adopted according to the specific atmospheric propagation scenarios if the atmospheric turbulence parameters can be measured in this specific atmospheric propagation scenario.

6. Discussions and conclusions

In this work, two anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectral models, that is the anisotropic generalized von Karman spectrum and the anisotropic generalized exponential spectrum have been developed to analyze theoretically the optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence. Compared with the isotropic turbulence, two anisotropic factors which parameterize the asymmetry of turbulence eddies in both horizontal and vertical directions are introduced. These two anisotropy factors are introduced to the orthogonal xy-plane so that the circular symmetry assumption of turbulence eddies in the xy-plane is no longer required. Also, the finite turbulence inner and outer scales and the general spectral power are included. Calculations show that with the increase of anisotropic factors, the ratios between the anisotropic turbulence spectra derived in this work and the turbulence spectra derived for isotropic turbulence decrease obviously due to the degraded focusing properties of the anisotropic turbulence.

In addition, based the derived anisotropic generalized exponential spectrum and the Rytov approximation theory, the expressions of the variances of AOA fluctuations have been obtained for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. Compared with the previously derived results, both the asymmetry of turbulence eddies in the orthogonal xy-plane and the finite turbulence inner and outer scales are taken into consideration. Results show that increasing the anisotropic parameters to values greater than unity reduces greatly the influences of anisotropic turbulence on the variance of AOA fluctuations due to the degraded focusing properties of the anisotropic turbulence. When one of these two anisotropic factors is much bigger than the other one, the smaller anisotropic factor plays a dominant role in the optical waves’ propagation through anisotropic turbulence. In addition, with the increase of turbulence outer scale, the anisotropic turbulence produces more effects on the final results. This conclusion is consistent with former researches [8-10] which focused on the isotropic turbulence.

In the investigation, it is supposed that the anisotropic factors have the same effects on all the turbulence eddy scales just like [21,26] for mathematical derivation convenience. This assumption is not suitable in the stable atmospheric boundary layers, where isotropy probably prevails at small scales [12]. In [27], the author introduced the concept of anisotropy at different scales through two effective anisotropic parameters which was defined mathematically for two specific cases of linear and parabolic anisotropic laws. In the future, with the further development of experiments about the anisotropic turbulence, the anisotropy at different scales will be introduced to the anisotropic turbulence spectrum based on the experimental data. Based on this, the improved expressions for the variance of AOA fluctuations will also be obtained for optical waves propagating through weak anisotropic non-Kolmogorov turbulence. Though still more experiments and theoretical investigations about anisotropic turbulence need to be performed, this work should be considered as a first step to a more complete analysis of optical waves’ propagation through anisotropic non-Kolmogorov atmosphere turbulence.

Appendix A

Here the detailed derivations for $\hat{A} (α)$ and $c^{'} (α)$ will be given as follows.

First, expanding $1 - \frac{\sin κ^{'} R^{'}}{κ^{'} R^{'}}$ in Eq. (8) in Maclaurin series [31]:

1 - \frac{\sin κ^{'} R^{'}}{κ^{'} R^{'}} = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{(2 n + 1)!} {(κ^{'})}^{2 n} {(R^{'})}^{2 n},

and then using the definitions of gamma function

Γ (x)

and hypergeometric function

{}_{1}F_{1} (a; b; z)

[31]:

Γ (x) = \int_{0}^{\infty} κ^{x - 1} \cdot e^{- κ} d κ \begin{matrix} (κ > 0, x > 0) \end{matrix},

{}_{1}F_{1} (a; b; z) = \sum_{n = 0}^{\infty} \frac{{(a)}_{n} \cdot z^{n}}{{(b)}_{n} \cdot n!},

where

{(a)}_{n}

is the Pochhammer symbol and has the form

{(a)}_{n} = \frac{Γ (a + n)}{Γ (a)} = a (a + 1) \cdot \cdot \cdot (a + n - 1),

the refractive-index structure function for the anisotropic turbulence becomes

D_{n} (R^{'}, α) = 4 π \hat{A} (α) {\hat{C}}_{n}^{2} {(κ_{l}^{'})}^{3 - α} Γ (- \frac{α}{2} + \frac{3}{2}) [1 - {}_{1}F_{1} (- \frac{α}{2} + \frac{3}{2}; \frac{3}{2}; - \frac{{(R^{'})}^{2} {(κ_{l}^{'})}^{2}}{4})] .

For statistically homogeneous and isotropic atmospheric turbulence, the related refractive-index structure function is given by [32]:

D_{n} (R, α) = {\begin{matrix} {\hat{C}}_{n}^{2} l_{0}^{α - 5} R^{2} & 0 \leq R ≪ l_{0} \\ {\hat{C}}_{n}^{2} R^{α - 3} & l_{0} ≪ R ≪ L_{0} \end{matrix} .

Because the refractive-index structure function becomes isotropic in the new spatial variable $\vec{R^{'}}$ just as we mentioned above, the related refractive-index structure function under anisotropic turbulence can be expressed as [28]:

D_{n} (R^{'}, α) = {\begin{matrix} {\hat{C}}_{n}^{2} l_{0}^{α - 5} {(R^{'})}^{2} & 0 \leq R^{'} ≪ l_{0} \\ {\hat{C}}_{n}^{2} {(R^{'})}^{α - 3} & l_{0} ≪ R^{'} ≪ L_{0} \end{matrix} .

Note that the structure function in Eq. (40) focuses on the theoretical analyses, and has not been well tested in long range propagation environments either via measurements and/or simulations.

Using Eq. (38) and Eq. (40), the unknown $\hat{A} (α)$ and $c^{'} (α)$ will be derived. When $l_{0} ≪ R^{'} ≪ L_{0}$ , then $\frac{{(R^{'})}^{2} {(κ_{l}^{'})}^{2}}{4} = \frac{{(R^{'})}^{2} {(c^{'} (α))}^{2}}{4 l_{0}^{2}} ≫ 1$ . At this time, ${}_{1}F_{1} (a; b; - x)$ in Eq. (38) can be expanded approximately with big argument given by [31]:

{}_{1}F_{1} (a; b; - x) \approx \frac{Γ (b)}{Γ (b - a)} x^{- a} \begin{matrix} (x ≫ 1) \end{matrix}

Substituting Eq. (41) into Eq. (38), $D_{n} (R^{'}, α)$ becomes

D_{n} (R^{'}, α) \approx - 4 π \hat{A} (α) {\hat{C}}_{n}^{2} Γ (- \frac{α}{2} + \frac{3}{2}) \frac{Γ (3 / 2)}{Γ (α / 2)} {(\frac{1}{2})}^{α - 3} {(R^{'})}^{α - 3} \begin{matrix} (l_{0} ≪ R^{'} ≪ L_{0}) \end{matrix}

Comparing Eq. (42) with Eq. (40), and considering the properties of gamma function [31]:

\begin{matrix} Γ (α + 1) = α Γ (α), Γ (1 - α) Γ (α) = \frac{π}{\sin (π α)}, \\ Γ (α) Γ (α + \frac{1}{2}) = 2^{1 - 2 α} \sqrt{π} Γ (2 α) . \end{matrix}

as a result,

\hat{A} (α)

can be expressed with the form as

\hat{A} (α) = \frac{Γ (α - 1)}{4 π^{2}} \sin [(α - 3) \frac{π}{2}] .

When $0 \leq R^{'} ≪ l_{0}$ , then $\frac{{(R^{'})}^{2} {(κ_{l}^{'})}^{2}}{4} = \frac{{(R^{'})}^{2} {(c^{'} (α))}^{2}}{4 l_{0}^{2}} ≪ 1$ . ${}_{1}F_{1} (a; b; x)$ in Eq. (38) can be approximately expanded with small argument given by [31]:

{}_{1}F_{1} (a; b; x) \approx \sum_{n = 0}^{1} \frac{{(a)}_{n} \cdot z^{n}}{{(b)}_{n} \cdot n!} = 1 + \frac{a}{b} \begin{matrix} x & (x ≪ 1) \end{matrix}

Substituting it into Eq. (38), $D_{n} (R^{'}, α)$ becomes

D_{n} (R^{'}, α) \approx π \hat{A} (α) {\hat{C}}_{n}^{2} {(κ_{l}^{'})}^{5 - α} {(R^{'})}^{2} [Γ (- \frac{α}{2} + \frac{3}{2}) (\frac{3 - α}{3})] \begin{matrix} (0 \leq R^{'} ≪ l_{0}) \end{matrix} .

Comparing Eq. (46) with Eq. (40), the expression of $c^{'} (α)$ is finally derived:

c^{'} (α) = {[π \hat{A} (α) Γ (- \frac{α}{2} + \frac{3}{2}) (\frac{3 - α}{3})]}^{\frac{1}{α - 5}} .

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (NSFC) (61405004) and the China Scholarship Council (201506025046).

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Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence

Abstract

1. Introduction

2. Generalized refractive-index fluctuations spectra for isotropic non-Kolmogorov turbulence

3. Generalized refractive-index fluctuations spectra for anisotropic non-Kolmogorov turbulence

4. Variance of AOA fluctuations based on the derived spectrum under anisotropic non-Kolmogorov turbulence

5. Calculations and analyses

6. Discussions and conclusions

Appendix A

Acknowledgments

References and links

Cited By

Figures (3)

Equations (47)

Optics Express