Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence

Bindang Xue; Linyan Cui; Wenfang Xue; Xiangzhi Bai; Fugen Zhou

doi:10.1364/OE.19.008433

1. Introduction

AOA fluctuations play an important role in a diverse range of fields including atmospheric turbulence [1,2], free space optical communication [3], ground-based astronomical observations [4], etc. It involves estimating the variance of AOA fluctuations through integrals of atmospheric turbulence strength along the propagation path. For a long time, study on AOA fluctuations is based on Kolmogorov atmospheric turbulence model, and many methods have been developed for analyzing AOA fluctuations. These methods can generally be divided into two classes: geometrical optics based method and covariance of AOA based method. The former is conditioned by $\sqrt{λ L} ≪ D$ ( $\sqrt{λ L}$ is the Fresnel zone, L is the wave propagation distance, λ is the wavelength, andD is the diameter of the receiver aperture). In this method, the variance of AOA fluctuations is expressed as the derivative of the phase structure function with aperture average, and the result is independent of optical wavelength. Tatarskii [5], Wheelon [6], and Andrews [7] presented close-form solutions to AOA variances for plane and spherical waves by approximating the phase structure function in certain asymptotic cases, and they contained finite turbulence inner and outer scales. Tofsted [8] and Eyyuboglu [9] derived the expression of the variance of AOA fluctuations for the Gaussian beam waves. The latter adopted an original approach to derive the variance of AOA fluctuations from its covariance, and it is applicable in more complete cases than the first method. Conan et al. [10] expressed the AOA variance as convergent series using the Mellin transforms, and the close-form solution was obtained in certain cases by means of a simple and analytical approximation. Cheon [11] developed closed form for the AOA variances without geometrical optics assumption, but the influences of finite inner and outer scales were not considered.

However, over the past decades, both experimental evidences [12–15] and theoretical investigations [16,17] have shown that atmospheric turbulence derivates from the Kolmogorov case in certain portions of the atmosphere. So the non-Kolmogorov spectral model and the generalized von Karman spectral model were proposed and used to investigate the AOA fluctuations for wave propagating through non-Kolmogorov turbulence [18–20]. In these studies, the geometrical optics approximation assumption is adopted to simplify the calculations, and that makes the results independent of optical wavelength.

In this study, without particular assumption, the generalized exponential spectral model [21] is used to research the AOA fluctuations for plane/spherical optical wave propagating through non-Kolmogorov turbulence. And then, the impacts of the optical wavelength, turbulence inner scales, outer scales and receiver aperture diameter on the variance of AOA fluctuations have been analyzed.

2. Generalized Exponential Spectrum

The generalized exponential spectral model [21] can be applied in non-Kolmogorov atmospheric turbulence, which considers finite turbulence inner and outer scales and has a general spectral power law value in the range of 3 to 5 instead of standard power law value 11/3. Specifically, this spectral model has the following form

Φ_{n} (κ, α, l_{0}, L_{0}) = A (α) \cdot C_{n}^{2} \cdot \begin{matrix} κ^{- α} \cdot f (k, l_{0}, L_{0}, α) & (\begin{matrix} 0 \leq κ < \infty, & 3 < α < 5 \end{matrix}) \end{matrix},

f (κ, l_{0}, L_{0}, α) = [1 - \exp (- κ^{2} / κ_{0}^{2})] \cdot \exp (- κ^{2} / κ_{l}^{2}) .

where

C_{n}^{2}

is the generalized refractive-index structure parameter with units

m^{3 - α}

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

r a d / m

and is related to the size of turbulence cells,

f (k, l_{0}, L_{0}, α)

describes the influence of finite turbulence inner and outer scales,

κ_{l} = c (α) / l_{0}

,

κ_{0} = C_{0} / L_{0}

,

l_{0}

and

L_{0}

are the turbulence inner and outer scales, respectively. The choice of

C_{0}

depends on the specific application, in this study, it is set to

4 π

just as [7].

A (α)

and

c (α)

are given by [21]

A (α) = \frac{Γ (α - 1)}{4 π^{2}} \sin [(α - 3) \frac{π}{2}] \begin{matrix} , & c (α) = {π A (α) [Γ (- \frac{α}{2} + \frac{3}{2}) (\frac{3 - α}{3})}^{\frac{1}{α - 5}} \end{matrix} .

When

α = 11 / 3

,

A (11 / 3) = 0.033

and

c (11 / 3) \approx 5.92

, Eq. (1) is reduced to the Kolmogorov turbulent exponential spectral model. And when

l_{0} \to 0

,

L_{0} \to \infty

, Eq. (1) becomes the general non-Kolmogorov spectrum

Φ_{n} (κ, α) = A (α) \cdot C_{n}^{2} \cdot \begin{matrix} κ^{- α} & (\begin{matrix} 0 \leq κ < \infty, & 3 < α < 5 \end{matrix}) \end{matrix} .

3. Variance of AOA fluctuations

The variance of AOA fluctuations can be expressed with different forms [10,11], but they are equivalent to each other after simple mathematical transforms. In this study, we adopt the form [11]

σ^{2} = π^{2} L \int_{0}^{\infty} d κ \int_{0}^{1} d ξ κ^{3} Φ_{n} (κ) h (κ, ξ),

where ξ is the normalized path coordinate and

h (κ, ξ)

is a weighting function. For plane and spherical waves,

h (κ, ξ)

has different form as

h_{p l} (κ) = [1 + \frac{k}{κ^{2} L} \sin (\frac{κ^{2} L}{k})] A (a κ),

h_{s p} (κ) = [1 + \cos (\frac{κ^{2} ξ (1 - ξ) L}{k})] ξ^{2} A (a κ ξ),

where

k = 2 π / λ

, and

a = D / 2

is the radius of the receiver aperture.

A (x)

is the wavenumber-weighting function which can suppress the influence of turbulence on the variance of AOA fluctuations when the scale of turbulent eddy is smaller than the radius of receiver [6].

A (x) = {[\frac{2 J_{1} (x)}{x}]}^{2},

where

J_{1} (x)

is the first-order Bessel function and

A (x)

can be approximated by a Gaussian pattern [10,11]

A (x) \approx \exp [- {(β x)}^{2}] .

where β can be assigned through theoretically calculating (

β = 0.5216

) [11] or curve fitting (

β = 0.4832

) [22] for Kolmogorov turbulence cases. For non-Kolmogorov turbulence cases, we assume that β is the function of α.

In the next section, the expression forms of $β (α)$ and the variance of AOA fluctuations for plane and spherical waves propagating through weak non-Kolmogorov turbulence will be derived. It should be mentioned that in the following calculation, α is restricted to the range of 3 to 4 just for comparison with [18–20,23]. The expressions of the variance of AOA fluctuations themselves are effective for α in the range of 3 to 5.

3.1 Expression of $β (α)$

Following the same procedure as [11], $β (α)$ for the non-Kolmogorov turbulence is obtained

β (α) = \frac{1}{2} \cdot {\frac{Γ (α - 1)}{{[Γ (α / 2)]}^{2} + Γ (1 + α / 2)}}^{1 / (α - 4)} \begin{matrix} , & (3 < α < 4) . \end{matrix}

Figure 1 shows $β (α)$ as the function of α. When $α = 11 / 3$ , $β (α) = 0.5216$ . As shown, $β (α)$ hardly varies with α and this is physically correct. Because $β (α)$ is the parameter of $A (x)$ , while $A (x)$ describes the inherence property (aperture averaging) of receiver and is only related to the receiver itself, it does not vary with α. Here, we verify it again from the theoretical analysis point of view. In this study, for calculation purposes, $β (α)$ is set to $0.52$ .

Fig. 1 $β (α)$ as a function of α.

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3.2 Variance of AOA fluctuations for plane wave

Substituting Eq. (6) and Eq. (9) into Eq. (5), the variance of AOA fluctuations for plane wave propagating through atmospheric turbulence becomes

σ_{(p l)}^{2} = π^{2} L \int_{0}^{\infty} d κ \int_{0}^{1} d ξ κ^{3} Φ_{n} (κ) [1 + \frac{k}{L κ^{2}} \sin (\frac{L κ^{2}}{k})] \exp [- \frac{β^{2} D^{2} κ^{2}}{4}],

Using the generalized exponential spectral model [21], Eq. (11) becomes

σ_{(p l)}^{2} (α, λ, D, l_{0}, L_{0}) = π^{2} L \int_{0}^{\infty} d κ \int_{0}^{1} d ξ κ^{3} Φ_{n} (κ, α, l_{0}, L_{0}) [1 + \frac{k}{L κ^{2}} \sin (\frac{L κ^{2}}{k})] \exp [- \frac{β^{2} D^{2} κ^{2}}{4}],

For analysis purposes, $f (κ, l_{0}, L_{0}, α)$ is divided into two parts

\begin{matrix} f (κ, l_{0}, L_{0}, α) = f_{1} (κ, l_{0}, L_{0}, α) + f_{2} (κ, l_{0}, L_{0}, α), \\ f_{1} (κ, l_{0}, L_{0}, α) = \exp (- κ^{2} / k_{l}^{2}) \begin{matrix} , & f_{2} (κ, l_{0}, L_{0}, α) = - \exp [- κ^{2} (1 / k_{0}^{2} + 1 / k_{l}^{2})] \end{matrix} . \end{matrix}

Using the gamma function [24]

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

and its property

\int_{0}^{\infty} κ^{μ - 1} \exp (- a κ) \sin (b κ) d κ = \frac{Γ (μ)}{{(a^{2} + b^{2})}^{μ / 2}} \sin [μ \tan^{- 1} (\frac{b}{a})],

The variance of AOA fluctuations for plane wave propagating though weak non-Kolmogorov becomes

σ_{(p l)}^{2} (α, λ, D, l_{0}, L_{0}) = π^{2} A (α) C_{n}^{2} L \cdot [g_{1} (A_{11}, B_{11}, C_{11}) - g_{1} (A_{21}, B_{21}, C_{21})],

where

g_{1} (A_{11}, B_{11}, C_{11})

and

- g_{1} (A_{21}, B_{21}, C_{21})

are obtained from

f_{1} (κ, l_{0}, L_{0}, α)

and

f_{2} (κ, l_{0}, L_{0}, α)

separately, and can be expressed with the forms

g_{1} (A_{i j}, B_{i j}, C_{i j}) = \frac{1}{2} Γ (\frac{A_{i j} + 1}{2}) B_{i j}^{- \frac{A_{i j} + 1}{2}} + \frac{1}{2 C_{i j}} \frac{Γ (\frac{A_{i j} - 1}{2})}{{(B_{i j}^{2} + C_{i j}^{2})}^{\frac{A_{i j} - 1}{4}}} \cdot \sin [\frac{A_{i j} - 1}{2} \tan^{- 1} (\frac{C_{i j}}{B_{i j}})],

where

A_{11} = A_{21} = 3 - α, B_{11} = \frac{β^{2} D^{2}}{4} + \frac{1}{k_{l}^{2}}, C_{11} = C_{21} = \frac{L}{k}, B_{21} = \frac{β^{2} D^{2}}{4} + \frac{1}{k_{l}^{2}} + \frac{1}{k_{0}^{2}} .

The analytical expression of variance of AOA fluctuations for plane wave propagating through weak non-Kolmogorov turbulence with horizontal path has been obtained, and it contains variable wavelength, the receiver aperture diameter, and the finite inner and outer scales.

3.3 Variance of AOA fluctuations for spherical wave

For a spherical wave propagating through atmospheric turbulence, the variance of the AOA fluctuations is given by

σ_{(s p)}^{2} = π^{2} L \int_{0}^{\infty} d κ \int_{0}^{1} d ξ κ^{3} Φ_{n} (κ) [1 + \cos (\frac{κ^{2} ξ (1 - ξ) L}{k})] ξ^{2} \exp [- \frac{β^{2} D^{2} κ^{2} ξ^{2}}{4}],

Using the generalized exponential spectral model [21], Eq. (18) becomes

\begin{matrix} σ_{(s p)}^{2} (α, λ, D, l_{0}, L_{0}) = π^{2} L \int_{0}^{\infty} \int_{0}^{1} κ^{3} Φ_{n} (κ, α, l_{0}, L_{0}) \\ [1 + \cos (\frac{κ^{2} ξ (1 - ξ) L}{k})] ξ^{2} \exp [- \frac{{(β D κ ξ)}^{2}}{4}] d κ d ξ . \end{matrix}

Similarly, $f (κ, l_{0}, L_{0}, α)$ is expressed with Eq. (13). Using Eq. (14) and the gauss hypergeometric function ${}_{2}F_{1} (A, B; C; Z)$ [24]

{}_{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 variance of AOA fluctuations for spherical wave propagating through weak non-Kolmogorov turbulence can be expressed as

σ_{s p}^{2} (α, λ, D, l_{0}, L_{0}) = π^{2} A (α) C_{n}^{2} L \cdot [g_{2} (A_{11}, B_{11}, C_{11}) - g_{2} (A_{21}, B_{21}, C_{21})] .

where

g_{2} (A_{11}, B_{11}, C_{11})

and

- g_{2} (A_{21}, B_{21}, C_{21})

are obtained from

f_{1} (κ, l_{0}, L_{0}, α)

and

f_{2} (κ, l_{0}, L_{0}, α)

separately

\begin{matrix} g_{2} (A_{i j}, B_{i j}, C_{i j}) = \frac{1}{2} Γ (\frac{A_{i j} + 1}{2}) {\frac{1}{3} B_{i j}^{- \frac{A_{i j} + 1}{2}} \cdot {}_{2}F_{1} (\frac{A_{i j} + 1}{2}, \frac{3}{2}; \frac{5}{2}; - \frac{b_{2}}{B_{i j}}) + \\ Re {\int_{0}^{1} ξ^{2} {[B_{i j} + b_{2} ξ^{2} + i C_{i j} ξ (1 - ξ)]}^{- \frac{A_{i j} + 1}{2}} d ξ}} . \end{matrix}

where

A_{i j}

and

C_{i j}

have the same forms as in section 3.2, while

B_{i j}

has the form as

B_{11} = \frac{1}{k_{l}^{2}}, B_{21} = \frac{1}{k_{l}^{2}} + \frac{1}{k_{0}^{2}}, b_{2} = \frac{β^{2} D^{2}}{4} .

For the atmospheric turbulence, usually $l_{0}$ is in the order of magnitude of millimeter and $L_{0}$ is in the order of magnitude of meter [5-7], so the condition of $l_{0} ≪ \sqrt{λ L}$ and $L_{0} ≫ \sqrt{λ L}$ is basically satisfied, then Eq. (22) can be expressed with closed form just as follows. When $l_{0} ≪ \sqrt{λ L}$ and $L_{0} ≫ \sqrt{λ L}$ , then $B_{11} = \frac{1}{k_{l}^{2}} = \frac{l_{0}^{2}}{c^{2} (α)} ≪ C_{11}$ and $B_{21} \approx \frac{1}{k_{l}^{2}} = \frac{L_{0}^{2}}{C_{0}^{2}} ≫ C_{21}$ , the integration part in $g_{2} (A_{11}, B_{11}, C_{11})$ and $g_{2} (A_{21}, B_{21}, C_{21})$ are approximately expressed as

\begin{array}{l} Re {\int_{0}^{1} ξ^{2} {[B_{11} + b_{2} ξ^{2} + i C_{11} ξ (1 - ξ)]}^{- \frac{A_{11} + 1}{2}} d ξ} \\ \approx Re {\int_{0}^{1} ξ^{2} {[b_{2} ξ^{2} + i C_{11} ξ (1 - ξ)]}^{- \frac{A_{11} + 1}{2}} d ξ} \\ = Re [{(i C_{11})}^{- \frac{A_{11} + 1}{2}} \cdot \frac{2}{5 - A_{11}} \cdot {}_{2}F_{1} (\frac{A_{11} + 1}{2}, \frac{5 - A_{11}}{2}; \frac{7 - A_{11}}{2}; 1 + i \frac{b_{2}}{C_{11}})], \end{array}

\begin{array}{l} Re {\int_{0}^{1} ξ^{2} {[B_{21} + b_{2} ξ^{2} + i C_{21} ξ (1 - ξ)]}^{- \frac{A_{21} + 1}{2}} d ξ} \\ \approx \int_{0}^{1} ξ^{2} {[B_{21} + b_{2} ξ^{2}]}^{- \frac{A_{21} + 1}{2}} d ξ \\ = \frac{1}{3} B_{21}^{- \frac{A_{i j} + 1}{2}} \cdot {}_{2}F_{1} (\frac{A_{21} + 1}{2}, \frac{3}{2}; \frac{5}{2}; - \frac{b_{2}}{B_{21}}) . \end{array}

As a result, $σ_{s p}^{2} (α, D, l_{0}, L_{0})$ becomes

σ_{s p}^{2} (α, λ, D, l_{0}, L_{0}) = π^{2} A (α) C_{n}^{2} L \cdot [g_{2} (A_{11}, B_{11}, C_{11}) - g_{2} (A_{21}, B_{21}, C_{21})],

\begin{matrix} g_{2} (A_{11}, B_{11}, C_{11}) = \frac{1}{2} Γ (\frac{A_{11} + 1}{2}) {\frac{1}{3} B_{11}^{- \frac{A_{11} + 1}{2}} \cdot {}_{2}F_{1} (\frac{A_{11} + 1}{2}, \frac{1}{2}; \frac{3}{2}; - \frac{b_{2}}{B_{11}}) - \\ Re [{(i C_{11})}^{- \frac{A_{11} + 1}{2}} \cdot \frac{2}{5 - A_{11}} \cdot {}_{2}F_{1} (\frac{A_{11} + 1}{2}, \frac{5 - A_{11}}{2}; \frac{7 - A_{11}}{2}; 1 + i \frac{b_{2}}{C_{11}})]}, \end{matrix}

g_{2} (A_{21}, B_{21}, C_{21}) = \frac{1}{3} Γ (\frac{A_{21} + 1}{2}) B_{21}^{- \frac{A_{21} + 1}{2}} \cdot {}_{2}F_{1} (\frac{A_{21} + 1}{2}, \frac{1}{2}; \frac{3}{2}; - \frac{b_{2}}{B_{21}}) .

The analytical expression of variance of AOA fluctuations for spherical wave propagating through weak non-Kolmogorov turbulence with horizontal path has been obtained, and it contains variable wavelength, the receiver aperture diameter, and the finite inner and outer scales.

5. Numerical results

In this section, simulations are conducted to analyze the influences of changeable parameters (D, $l_{0}$ ,λ and $L_{0}$ ) on the variance of AOA fluctuations. To avoid the mutual interferences between parameters, in the following simulations, we fix three of the four parameters at a time and analyze only one parameter’s influence on the variance of AOA fluctuations. All the experiments are conducted for plane/spherical optical wave propagating in horizontal path with the settings

C_{n}^{2} = 1 \times 10^{- 14} m^{3 - α}, L = 1000 m

5.1 Effect of wavelength’s variation on the variance of AOA fluctuations

In the first simulation experiment, we fix $D = 0.05 m$ , $l_{0} = 1 m m$ , $L_{0} = 10 m$ and different wavelengths, including visible light ( $λ = 0.55 μ m$ ), near infrared light ( $λ = 1.55 μ m$ ), intermediate infrared light ( $λ = 4 μ m$ ) and far infrared light ( $λ = 10 μ m$ ) are chosen to analyze their influences on the variance of AOA fluctuations. Simulation experimental results are shown in Fig. 2 .

Fig. 2 Variance of AOA fluctuations as a function of α with different wavelength values. (a): plane wave. (b): spherical wave.

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As shown, variable wavelengths bring different effects on the variance of AOA fluctuations. With the increase of wavelength, the turbulence produces less effect on the wave propagation, and this is the supplement to the previous results [18–20] where the influence of various wavelength is ignored under the geometrical optics approximation assumption.

5.2 Effect of inner scale’s variation on the variance of AOA fluctuations

To analyze turbulence inner scale’s influence on the variance of AOA fluctuation, D, $L_{0}$ and λ are fixed to constant values of $D = 0.05 m$ , $L_{0} = 10 m$ and $λ = 0.55 μ m$ . Different inner scale sizes are chosen (for the real atmospheric turbulence condition, $l_{0}$ is in the order of magnitude of millimeter, here it is set to 1 $m m$ , 2 $m m$ , 3 $m m$ and 5 $m m$ , respectively, and it satisfy $l_{0} ≪ \sqrt{λ L}$ ). Figure 3 shows the simulation experimental results. As shown, when turbulence inner scale is much smaller than the Fresnel zone, effects of inner scale on the variance of AOA fluctuations can be ignored, and this result is consonant with [18].

Fig. 3 Variance of AOA fluctuations as a function of α with different inner scale values.(a): plane wave. (b): spherical wave.

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This phenomenon can also be explained from the point of view: the phase fluctuations are contributed mostly by turbulence cells with size of $\sqrt{λ L}$ or larger [5,6]. When $l_{0} < < \sqrt{λ L}$ , the number of turbulent cells with size of $\sqrt{λ L}$ almost keeps unchanged. Therefore, the phase fluctuations will not be changed, which makes the variance of AOA fluctuations unchanged.

5.3 Effect of outer scale’s variation on the variance of AOA fluctuations

To analyze turbulence outer scale’s influence on the variance of AOA fluctuations, D, $l_{0}$ and λ are set to constant values ( $D = 0.05 m$ , $l_{0} = 1 m m$ , $λ = 0.55 μ m$ ). Different outer scales are chosen, and the simulation results are shown in Fig. 4 .

Fig. 4 Variance of AOA fluctuations as a function of α with different outer scale values. (a): plane wave. (b): spherical wave.

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As shown, with the increase of the turbulent outer scale, the variance of AOA fluctuations increases. This trend is consistent with [18,20]. The results can be explained directly from the function $f (κ, α, l_{0}, L_{0})$ in Eq. (2). When $L_{0}$ increases, $f (κ, α, l_{0}, L_{0})$ raises, so that the variance of AOA fluctuations of plane/spherical waves also increase.

It can also be interpreted from another point of view: the phase fluctuations are contributed mostly by turbulence cells with size of $\sqrt{λ L}$ or larger [5,6], when the $L_{0}$ is assumed with high value, the wave meets a major number of large-scale turbulent cells along its propagation length and these cells lead to higher variance of AOA fluctuations with respect to the case of lower outer scale value, where more large scales are cut out [18].

5.4 Effect of D’s variation on the variance of AOA fluctuations

To analyze D ’s influence on the variance of AOA fluctuations, λ, $l_{0}$ and $L_{0}$ are set to constant values ( $λ = 0.55 μ m$ , $l_{0} = 1 m m$ , $L_{0} = 10 m$ ), and different Dis chosen. Figure 5 shows that when Dincreases, the variance of AOA fluctuations decreases obviously, and this is the aperture averaging effects.

Fig. 5 Variance of AOA fluctuations as a function of α with different Dvalues. (a): plane wave. (b): spherical wave.

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6. Conclusion

In this study, new theoretical expressions of the variance of AOA fluctuations with variable wavelength, finite inner and outer scales and general power law are derived under the assumption of weak fluctuation theory for plane and spherical optical wave propagating through non-Kolmogorov atmospheric turbulence with horizontal path.

Simulation results show that the variance of AOA fluctuations as the function of power law α for plane wave is similar to that for spherical wave. The turbulence outer scale’s variation produces obvious effects on the variance of AOA fluctuations, while the inner scale’s influence is ignorable when it is much smaller than the Fresnel zone. This is consistent with [18,20]. As the receiver aperture diameter D increases, strong averaging effects are produced, and it will alleviate the value of the variance of AOA fluctuations. Also, without geometrical optics assumption, the influence of variable wavelength is included in the expressions, and this is the obvious difference between the expressions derived in this study and the existed expressions where the influence of wavelength is ignored for calculation purpose [18–20]. The results in this study will help to better investigate the effects of turbulence on the plane and spherical optical waves propagating through non-Kolmogorov atmospheric turbulence with horizontal path.

Acknowledgments

The authors would like to thank the anonymous reviewers for their very constructive comments and suggestions. This work is partly supported by the National Natural Science Foundation of China (No.60832011) and the Aeronautical Science Foundation of China (20080151009).

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Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence

Abstract

1. Introduction

2. Generalized Exponential Spectrum

3. Variance of AOA fluctuations

3.1 Expression of $β (α)$

3.2 Variance of AOA fluctuations for plane wave

3.3 Variance of AOA fluctuations for spherical wave

5. Numerical results

5.1 Effect of wavelength’s variation on the variance of AOA fluctuations

5.2 Effect of inner scale’s variation on the variance of AOA fluctuations

5.3 Effect of outer scale’s variation on the variance of AOA fluctuations

5.4 Effect of D’s variation on the variance of AOA fluctuations

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (29)

Optics Express

Abstract

1. Introduction

2. Generalized Exponential Spectrum

3. Variance of AOA fluctuations

3.1 Expression of β(α)

3.2 Variance of AOA fluctuations for plane wave

3.3 Variance of AOA fluctuations for spherical wave

5. Numerical results

5.1 Effect of wavelength’s variation on the variance of AOA fluctuations

5.2 Effect of inner scale’s variation on the variance of AOA fluctuations

5.3 Effect of outer scale’s variation on the variance of AOA fluctuations

5.4 Effect of D’s variation on the variance of AOA fluctuations

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (29)

Optics Express

3.1 Expression of $β (α)$