Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence

Liying Tan; Wenhe Du; Jing Ma; Siyuan Y u; Qiqi Han

doi:10.1364/OE.18.000451

1. Introduction

At present, it has been accepted that the Kolmogorov model is not the only possible turbulent one in the atmosphere, which is supported by numerous experimental evidences [1–6] and some results of theoretical investigation [7–10]. This has prompted the investigation of optical wave propagation through the atmospheric turbulence exhibiting non-Kolmogorov statistics.

Beland analyzed a representative amplitude effect, the log-amplitude variance, and a phase effect, the coherent length, when the refractive-index fluctuations deviate from Kolmogorov statistics [11]. Stribling et al defined the turbulence, in which the structure function for the index of refraction and the corresponding power spectrum obeyed an arbitrary power law, as non-Kolmogorov turbulence and presented an analysis of optical propagation in non-Kolmogorov atmospheric turbulence, mainly including the wave structure function, the Strehl ratio, etc [12]. Boreman and Dainty studied the expressions of non-Kolmogorov turbulence in the light of Zernike polynomials [13]. Rao et al analyzed the spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence [14]. Gurvich and Belen’kii introduced a model for the power spectrum of stratospheric non-Kolmogorov turbulence and investigated the stratospheric turbulence on the scintillation and the coherence of starlight and on the degradation of star image [15]. Belen’kii studied the influence of the stratosphere on star image motion again based on the model for the power spectrum of stratosphere [16]. Recently, Tosellia et al presented a non-Kolmogorov theoretical power spectrum model and estimated free space optical system performance for laser beam propagating horizontally through non-Kolmogorov atmospheric turbulence [17]. And then they analyzed the angle-of-arrival fluctuations for free space laser beam again [18]. So far all of theoretical investigations for optical propagation in non-Kolmogorov atmospheric turbulence have focused on the unbounded plane or spherical-wave models. However, in many applications the plane and spherical-wave approximations do not suffice to describe the propagation properties of optical wave. Therefore,it is very necessary to extend the investigation of optical propagation in non-Kolmogorov atmospheric turbulence to Gaussian-beam wave model.

In this paper, a non-Kolmogorov theoretical power spectrum is considered [17], which has a generalized power law that takes all the values ranging from 3 to 4. As the power law α is set to the standard Kolmogorov value 11/3, the spectrum reduces to the conventional Kolmogorov one. Based on this spectrum and following the same procedure already used from Miller et al in the Kolmogorov case [19], the log-amplitude variances for a Gaussian-beam wave have been derived in weak turbulence for a horizonal path and the influence of spectral power-law variations on the log-amplitude fluctuations is analyzed.

2. Non-Kolmogorov spectrum

For the purpose of this paper, a theoretical power spectrum model that describes non-Kolmogorov optical turbulence is considered [17], which obeys a more general power law and in which the power-law exponents can take all the values ranging from 3 to 4,

Φ_{n} (κ, α) = A (α) \tilde{C_{n}^{2}} κ^{- α}, 2 π / L_{0} ≪ κ ≪ 2 π / l_{0}, 3 < α < 4,

where κ is the magnitude of three dimensional wave number vector (in units of rad/m), α is the spectral power-law exponent, C̃_n ² is a generalized refractive-index structure parameter (in units of m ^3-α) that describes the strength of the turbulence along the path, l ₀ and L ₀ denote the inner and outer scales of turbulence, respectively, and A(α) is a function defined by

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}),

where the symbol Γ(x) represents the gamma function. When the power law α is equal to 11/3, A(11/3)= 0.033,C̃_n ² = C _n ², and the spectrum reducesto the conventional Kolmogorov spectrum [20],

Φ_{n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3},

where C ² represents the conventional refractive-index structure parameter and has units of m ^-2/3. In addition, as α → 3, A (α) → 0. As a result, the power spectrum for refractive-index fluctuations disappears in the limiting case α = 3. Finally, it can be seen from Eq. (1) that all of the analyses performed in this paper are only related to the inertial interval of turbulent spectrum, i.e., 2π/L ₀ ≪ κ ≪ 2π/l ₀.

3. Log-amplitude variance

Although our study is concerned with non-Kolmogorov turbulence, we still start our work from the conventional Kolmogorov results. In the weak-fluctuation regime the log-amplitude variance for a Gaussian-beam wave propagating through the conventional Kolmogorov turbulence is given by [20]

\begin{matrix} σ_{χ}^{2} (ρ) = 2 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) {I_{0} (2 Λ ρξκ) - \cos [\frac{L κ^{2}}{k} (1 - \tilde{Θ} ξ) ξ]} \\ \times [\exp (- \frac{Λ L ξ^{2} κ^{2}}{k})] dκdξ, \end{matrix}

where ρ is distance from the beam center line in the plane transverse to the propagation direction (z axis), k = 2π/λ and λ is the optical wavelength, L is the propagation distance of the beam in the turbulent atmosphere, Φ_n(κ) is the power spectrum for refractive-index fluctuations, I ₀(x) is a modified Bessel function of the first kind, ξ is related to z by z = 1 - ξ/L, and the complementary parameter Θ̃ = 1 - Θ. Θ and Λ are the output plane (or receiver) beam parameters that are related to the input plane (or transmitter) beam parameters Θ₀ and Λ₀ by

Θ = \frac{Θ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}}, Λ = \frac{Λ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}},

where Θ₀ and Λ₀ are defined by Θ₀ = 1-L/R ₀ and Λ₀ = λL/πW ₀ ², respectively. Here, W ₀ and R ₀ denote the radius of the beam size and the radius of curvature of the phase front at the transmitter. The parameter Θ₀ is also called the curvature parameter and Λ₀ is called the Fresnel ratio at the input plane, while the quantity Θ is called the curvature parameter and Λ is called the Fresnel ratio at the output plane. Either the transmitter beam parameters or the receiver beam parameters can describe the diffractive characteristics of Gaussian-beam wave.

For interpretation purposes, the log-amplitude variance is usually expressed as the sum of the radial component, σ_{χ, r} ²(ρ), and the longitudinal component,σ_χ,r ²,

σ_{χ}^{2} (ρ) = σ_{χ, r}^{2} (ρ) + σ_{χ, l}^{2},

where

\begin{matrix} σ_{χ, r}^{2} (ρ) = 2 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) [I_{0} (2 Λ ρξκ) - 1] \\ \times [\exp (- \frac{Λ L ξ^{2} κ^{2}}{k})] dκdξ, \end{matrix}

\begin{matrix} σ_{χ, l}^{2} = 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) \sin^{2} [\frac{L κ^{2}}{2 k} (1 - \tilde{Θ} ξ) ξ] \\ \times [\exp (- \frac{Λ L ξ^{2} κ^{2}}{k})] dκdξ . \end{matrix}

The radial component physically denotes the off-axis contribution to the log-amplitude variance and vanishes at the beam center line (ρ = 0) or as Λ = 0 (corresponding to an infinite wave such as a plane or spherical wave), whereas the longitudinal component is constant throughout the beam cross section in any transverse plane.

It is noteworthy that the explicit form of the refractive-index power spectrum is not taken in the development of the above equations. Therefore, when the Kolmogorov power spectrum is substituted into Eq. (4), the conventional results will be obtained. Here, using a non-Kolmogorov power spectrum and following the same procedure as discussed in Ref. [17] for the standard Kolmogorov spectrum, the log-amplitude fluctuations for a Gaussian-beam wave propagating in non-Kolmogorov atmospheric turbulence are analyzed.

For the radial component, our analysis results in

\begin{matrix} σ_{χ, r}^{2} (ρ, α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - α / 2} L^{α / 2} Γ (\frac{2 - α}{2}) Λ^{\frac{α - 2}{2}} \\ \times [_{1} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}}) - 1], \end{matrix}

where ₁ F ₁(a;c;z) is the confluent hypergeometric function of the first kind and W is the beam radius at the receiver. When the relation ρ ≤ W or ρ ≫ W is satisfied, based on the first few terms of the small-argument series representation or the large-argument asymptotic form of the ₁ F ₁ function, the approximate results are also obtained,

\begin{matrix} σ_{χ, r}^{2} (ρ, α) = \frac{2}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - α / 2} L^{α / 2} Λ^{\frac{α - 2}{2}} \frac{ρ^{2}}{W^{2}} \\ \times [Γ (\frac{4 - α}{2}) + Γ (\frac{6 - α}{2}) \frac{ρ^{2}}{{2 W}^{2}}], ρ \leq W, \end{matrix}

\begin{matrix} σ_{χ, r}^{2} (ρ, α) = \frac{1}{α - 1} 2^{- α / 2} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Λ^{\frac{α - 2}{2}} \\ \times {(\frac{W}{ρ})}^{α} \exp (\frac{2 ρ^{2}}{W^{2}}), ρ ≫ W . \end{matrix}

For the longitudinal component, the corresponding result is expressed as

\begin{matrix} σ_{χ, l}^{2} (α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{Λ}^{\frac{α - 2}{2}} \\ - Re [\frac{2 α - 2}{α} i^{\frac{α - 2}{2}}_{2} F_{1} (\frac{2 - α}{2}, \frac{α}{2}; \frac{2 + α}{2}; (\tilde{Θ} + i Λ))]}, \end{matrix}

where ₂ F ₁(a,b;c;z) is the hypergeometric function and Re means the real part of the expression in the square brackets.

Finally, the log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence is given by

\begin{matrix} σ_{χ}^{2} (ρ, α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{Λ}^{\frac{α - 2}{2}}_{1} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}}) \\ - Re [\frac{2 α - 2}{α} i^{\frac{α - 2}{2}}_{2} F_{1} (\frac{2 - α}{2}, \frac{α}{2}; \frac{2 + α}{2}; (\tilde{Θ} + i Λ))]} . \end{matrix}

Here it is noted that the assumptions that non-Kolmogorov turbulence is homogeneous along the propagation path and the inner scale is much smaller than the size of the Fresnel zone are involved in the above evaluations.

As for the Kolmogorov case, for the special beam form for the non-Kolmogorov case, the hypergeometric function ₂ F ₁ in Eqs. (12) and (13) can also be simplified to more tractable analytic functions. For example, for |Θ̃ + iΛ| ≤ 1 (as Θ₀ ≥ 0.5), which corresponds to all divergent and collimated beam and some convergent beam models, the series denotation of the hypergeometric function leads to

\begin{matrix} Re [\frac{2 α - 2}{α} i^{\frac{α - 2}{2}}_{2} F_{1} (\frac{2 - α}{2}, \frac{α}{2}; \frac{2 + α}{2}; (\tilde{Θ} + i Λ))] = \\ \frac{2 α - 2}{2} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(\frac{α}{2})}_{n}}{{(\frac{2 + α}{2})}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{n / 2} \cos [n \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(α - 2) π}{4}], \end{matrix}

where (a)_n = Γ(a + n)/Γ(a), n = 0,1,2,⋯. Moreover, for |Θ̃ + iΛ| = 1 (as Θ₀ < 0.5), using the analytic continuation formula of the hypergeometric function [21],

\begin{matrix} _{2} F_{1} (a, b; c; y) = \frac{Γ (c) Γ (b - a)}{Γ (b) Γ (c - a)} {(- y)}^{- a}_{2} F_{1} (a, 1 - c + a; 1 - b + a; \frac{1}{y}) \\ + \frac{Γ (c) Γ (a - b)}{Γ (a) Γ (c - b)} {(- y)}^{- b}_{2} F_{1} (b, 1 - c + b; 1 - a + b; \frac{1}{y}), \end{matrix}

yields

\begin{matrix} Re [\frac{2 α - 2}{α} i^{\frac{α - 2}{2}}_{2} F_{1} (\frac{2 - α}{2}, \frac{α}{2}; \frac{2 + α}{2}; (\tilde{Θ} + i Λ))] \\ {= (\tilde{Θ^{2}} + Λ^{2})}^{\frac{α - 2}{4}} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(1 - α)}_{n}}{{(2 - α)}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{- n / 2} \cos [(n - \frac{α - 2}{2}) \\ \times \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(α - 2) π}{4}] + \frac{2 α - 2}{α} \frac{Γ (1 + α / 2) Γ (1 - α)}{Γ (1 - α / 2)} {(\tilde{Θ^{2}} + Λ^{2})}^{- \frac{α}{4}} \\ \times \cos [\frac{α}{2} \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(2 - 3 α) π}{4}] . \end{matrix}

Table 1. Expressions of Log-amplitude Variance for Various Beam Types

View Table

3.1. Collimated beam

For a collimated beam (Θ₀ = 1), based on Eqs. (5) and (14), the longitudinal component is expressed as

\begin{matrix} σ_{χ, l}^{2} (α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{(\frac{Λ_{0}}{1 + Λ_{0}^{2}})}^{\frac{α - 2}{2}} - \frac{2 α - 2}{α} \\ \times \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(\frac{α}{2})}_{n}}{{(\frac{2 + α}{2})}_{n} n!} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2})})}^{n / 2} \cos [n \tan^{- 1} (\frac{1}{Λ_{0}}) + \frac{(α - 2) π}{4}]} . \end{matrix}

Because the radial component disappears in the limiting case of a plane wave or a spherical wave, the longitudinal component, Eq. (17), reduces to the plane-wave log-amplitude variance in the limiting case Λ₀ = 0,

σ_{χ, p}^{2} (α) = - \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) \cos [\frac{(α - 2) π}{4}],

while reducing to the spherical-wave one in the limiting case Λ₀ = ∞,

σ_{χ, s}^{2} (α) = - \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} \frac{Γ (\frac{α}{2}) Γ (\frac{2 - α}{2}) Γ (\frac{2 + α}{2})}{Γ (α)} \cos [\frac{(α - 2) π}{4}] .

Although Eqs. (18) and (19) are formally different from the results obtained by Beland [11], they are equivalent each other. In addition, when α is set to 11/3, Eqs. (18) and (19) match the conventional Kolmogorov results for the plane and spherical waves perfectly [20].

The sum of longitudinal and radial components for a collimated beam is listed in Table 1. In order to analyze the influence of spectral power-law variations on the log-amplitude fluctuations of a collimated beam, the log-amplitude variance is plotted in Fig. 1 as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several values of the ratio ρ/W for a special case, taking L = 1000m; C̃_n ² = 1.0×10^-14m^3-α; λ = 1.55×10^-6 m. Figure 1 (a) depicts the on-axis log-amplitude fluctuations (ρ/W = 0), while Figure 1 (b) and 1(c) represent the off-axis log-amplitude fluctuations (ρ/W = 0.5,1.0). Since the path length L and optical wavelength λ are fixed, all changes in the Fresnel ratio at the transmitter Λ₀ = 2L/kW ₀ ² correspond to variations in the transmitter beam radius W ₀.

As it is shown in Fig. 1(a), for some fixed Fresnel ratio for lower alpha values than 3.5 the on-axis log-amplitude variance increases up to a maximum value that occurs near α = 3.2. At the maximum point the curve changes its slopes and the log-amplitude variance begins to decrease down to zero. In addition, for larger alpha values than 3.5 the log-amplitude variance slightly decreases. These comments are similar to those on the scintillations for a plane or spherical wave in Ref. [15]. Figure 1(b) and 1(c) show that the above comments about the on-axis log-amplitude variance also adapt to the off-axis log-amplitude variance for small or large Λ₀, while the off-axis log-amplitude variance in the vicinity of Λ₀ = 1 firstly increases up to a peak value, then decreases down to a trough, lastly begins to increase again rather that tends to zero as Fig. 1(a). This is due to the variations in the ratio ρ/W. In order to further analyze the influence of the ratio ρ/W on the log-amplitude fluctuations, the log-amplitude variance for several values of the Fresnel ratio at the transmitter Λ₀ = 0.01,1.00,100 is plotted in Fig. 2 as a function of α and the ratio ρ/W for the same case as Fig. 1. It is deduced from Fig. 2 that the log-amplitude variance increases monotonically with the ratio ρ/W for some fixed value of alpha and increases slightly for small or large value of Λ₀ as shown in Fig. 2(a) and 2(c), while increasing sharply near Λ₀ = 1 as shown in Fig. 2(b). Furthermore, Figure 2(b) also shows that for different values of alpha the log-amplitude variance in the vicinity of Λ₀ = 1 increases with the ratio ρ/W with different speeds. Finally, it is emphasized that the radial component for a collimated beam disappears for all ρ and α between the range 3 to 4 only as Λ₀ = 0 and Λ₀ = ∞.

Fig. 1. The log-amplitude variance for a collimated beam (Θ₀ = 1) as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

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Fig. 2. The log-amplitude variance for a collimated beam (Θ₀ = 1) as a function of power law α and the ratio ρ/W for several values of the Fresnel ratio at the transmitter Λ₀, with (a) for Λ₀ = 0.05, (b) for Λ₀ = 1.00, and (c) for Λ₀ = 100.

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3.2. Divergent beam

For a divergent beam (Θ₀ = 1), using Eq. (14), the longitudinal component is given by

\begin{matrix} σ_{χ, l}^{2} (ρ, α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {\times {Λ}^{\frac{α - 2}{2}} \\ - \frac{2 α - 2}{α} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(\frac{α}{2})}_{n}}{{(\frac{2 + α}{2})}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{n / 2} \cos [n \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(α - 2) π}{4}]} . \end{matrix}

The sum of longitudinal and radial components for a divergent beam is also listed in Table 1 and are plotted in Figs. 3 and 4, respectively, in a similar manner to that for the collimated beam in Figs. 1 and 2 for the curvature parameter at the transmitter Θ₀ = 2 for a special case. We take L = 1000m; C̃_n ² = 1.0×10^-14 m ^3-α; λ = 1.55×10^-6 m. Figs. 3 and 4 are much like Figs. 1 and 2, respectively, thus the same comments as for the collimated beam are also deduced from Figs. 3 and 4. In addition, the radial component for the divergent beam, like that for the collimated beam, diminishes for all ρ and α when Λ₀ = 0 and Λ₀ = ∞. The longitudinal component for the divergent beam approaches the log-amplitude variance of the spherical wave as Λ₀ → ∞, while not tending to the log-amplitude variance of the plane wave as Λ₀ → 0 because the curvature parameter at the receiver Θ for a divergent beam tends to the value Θ = 1/Θ₀ rather than unity as for a collimated beam. For this limiting case Λ₀ → 0, the longitudinal component, like the Kolmogorov case, moves from the log-amplitude variance of the plane wave to that of the spherical wave as Θ₀ increases from unity. Here it is noted that the log-amplitude variances for the plane and spherical waves mentioned above are all the function of the power law α.

Fig. 3. The log-amplitude variance for a divergent beam (Θ₀ = 2) as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

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Fig. 4. The log-amplitude variance for a divergent beam (Θ₀ = 2) as a function of power law α and the ratio ρ/W for several values of the Fresnel ratio at the transmitter Λ₀, with (a) for Λ₀ = 0.05, (b) for Λ₀ = 1.00, and (c) for Λ₀ = 100.

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3.3. Convergent beam

For a convergent beam (Θ₀ < 1), when Θ₀ ≥ 0.5, the longitudinal component has the same expression as that of the divergent beam given by Eq. (20) above. But when Θ₀ < 0.5, using Eq. (16) leads to

\begin{matrix} σ_{χ, l}^{2} (ρ, α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{Λ}^{\frac{α - 2}{2}} - {(\tilde{Θ^{2}} + Λ^{2})}^{\frac{α - 2}{4}} \\ \times \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(1 - α)}_{n}}{{(2 - α)}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{- n / 2} \cos [(n - \frac{α - 2}{2}) \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) \\ + \frac{(α - 2) π}{4}] - \frac{2 α - α}{α} \frac{Γ (1 + α / 2) Γ (1 - α)}{Γ (1 - α / 2)} {(\tilde{Θ^{2}} + Λ^{2})}^{- \frac{α}{4}} \\ \times \cos [\frac{α}{2} \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(2 - 3 α) π}{4}] . \end{matrix}

Furthermore, for the traditional case of a perfectly focused beam characterized by Θ₀ = 0, the longitudinal component is given by

\begin{matrix} σ_{χ, l}^{2} (ρ, α) = \frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {Λ_{0}^{\frac{2 - α}{2}} - {(\frac{1 + Λ_{0}^{2}}{Λ_{0}^{2}})}^{\frac{α - 2}{4}} \\ \times \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(1 - α)}_{n}}{{(2 - α)}_{n} n!} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2}})}^{n / 2} \cos [(n - \frac{α - 2}{2}) \tan^{- 1} (\frac{1}{Λ_{0}}) \\ + \frac{(α - 2) π}{4}] - \frac{2 α - 2}{α} \frac{Γ (1 + α / 2) Γ (1 - α)}{Γ (1 - α / 2)} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2}})}^{\frac{α}{4}} \\ \times \cos [\frac{α}{2} \tan^{- 1} (\frac{1}{Λ_{0}}) + \frac{(2 - 3 α) π}{4}]} . \end{matrix}

Fig. 5. The log-amplitude variance for a perfectly focus beam (Θ₀ = 0) as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

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The expressions of the log-amplitude variance for the convergent beam and the perfectly focused beam appear in Table 1. The log-amplitude variance for the perfectly focused beam is plotted in Fig. 5 as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several values of the ratio ρ/W for a special case, taking L = 1000/m; C̃_n ² = 1.0 × 10^-14 m ^3-α; λ = 1.55 × 10^-6/w. Figure 5 is completely different from Figs. 1 and 3. The on-axis log-amplitude variance transmits smoothly from zero (Λ₀ → 0) to the log-amplitude variance of the spherical wave (Λ₀ → ∞). The off-axis log-amplitude variance for small Λ₀ becomes very large and increases rapidly with α, while that for large Λ₀ also increases to a peak value and then decreases up to zero, which are similar to the log-amplitude variance of the spherical wave. This is because the longitudinal component for a perfectly focused beam tends to the log-amplitude variance of the spherical wave and the radial component disappears as Λ₀ → ∞, whereas as Λ₀ → 0 the longitudinal component disappears and the radial component becomes unbounded for ρ ≠ 0, while vanishing for 𐏁 = 0.

Fig. 6. The on-axis log-amplitude variance as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several curvature parameters at the transmitter Θ₀, with (a) for Θ₀ = 0.01, (b) for Θ₀ = 0.10, and (c) for Θ₀ = 0.50.

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Fig. 7. The diffractive edge (ρ/W = 1) log-amplitude variance as a function of power law α and the Fresnel ratio at the transmitter Λ₀ for several curvature parameters at the transmitter Θ₀, with (a) for Θ₀ = 0.01, (b) for Θ₀ = 0.10, and (c) for Θ₀ = 0.50.

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In order to analyze the log-amplitude fluctuations for small but not zero Θ₀, the on-axis and diffractive edge (ρ/W = 1) log-amplitude variances are plotted in Figs. 6 and 7 as a function of power law α and the Fresnel ratio at the transmitter Λ₀, respectively, for a special case, taking L = 1000/m; C̃_n ² = 1.0×10^-14 m ^3-α; λ = 1.55×10^-6 m. Figure 6 shows that, when Θ₀ is small but not zero, the on-axis log-amplitude variance has a minimum value near Λ₀ = 1 and increases rapidly as Λ₀ approaches zero (Θ₀ is the smaller, the rapider is the increase), eventually tending to a nonzero limiting value. And it approaches the log-amplitude variance of the spherical wave as Λ₀ tends to the unbounded. Furthermore, the on-axis log-amplitude variance for small but not zero Θ₀ increases up to a peak value, and at the maximum point the curve changes slope and decreases down to zero with the increase of the power law α for some fixed Fresnel ratio, which are similar to that for the collimated or divergent waves (Figure 1(a) or 3(a)). Figure 7(a) shows that the diffractive edge log-amplitude variance for Θ₀ = 0.01, similar to that for the perfectly focused beam (Figure 5(c)), also becomes very large for small Λ₀ and increases rapidly with α, while that for large Λ₀ increases to a peak value and then decreases up to zero. Figure 7(b) and 7(c) show that the diffractive edge log-amplitude variances for Θ₀ = 0.1 and Θ₀ = 0.5, similar to that for the collimated or divergent waves (Figure 1(c) or 3(c)), increase to a peak value and then decrease up to zero with increase of alpha for small or large Λ₀, while they in the vicinity of Λ₀ = 1 firstly increase up to a peak value, then arrive at a trough, lastly begin to increase again. Finally, the radial component for a convergent beam, like the Kolmogorov case, tends to zero as Λ₀ → 0 and Λ₀ → ∞, while the longitudinal component approaches the log-amplitude variance of the spherical wave for Λ₀ → ∞ while tending to a limiting value for Λ₀ → 0.

Finally, the series in main equations in Table 1 converge quite rapidly so that the errors are smaller than 1 % when the first five terms are calculated. Therefore, the first five terms of the series are used to calculated the log-amplitude variance in all of Figures in the paper.

4. Conclusion

In this paper, the log-amplitude variances for a Gaussian-beam wave propagating in non-Kolmogorov turbulence are derived in the weak-fluctuation regime for a horizonal path following the procedure already used from Miller and Ricklin [19] and using a non-Kolmogorov theoretical power spectrum, which has a generalized power law and a generalized amplitude factor instead of the standard Kolmogorov power law 11/3 and a constant amplitude factor 0.033. The analytical expressions are obtained and also summarized in Table 1, like the Kol-mogorov case, for convenient reference and comparison with the conventional Kolmogorov results. Here it is especially noted that the expressions developed match perfectly with the conventional Kolmogorov results [19], respectively, when the spectral power law is equal to the standard Kolmogorov value 11/3.

Based on the expressions developed, the effect of spectral power-law variations on the log-amplitude fluctuations for Gaussian-beam wave is analyzed for a particular case. It can be concluded that the on-axis log-amplitude variance and the off-axis log-amplitude variance for small or large Λ₀ for a collimated or divergent beam, similar to the scintillations for the plane or spherical waves, increase up to a peak value, and at the maximum point the curve change slope and decrease down to zero with the increase of the power law α for some fixed Fresnel ratio, but the off-axis log-amplitude variance in the vicinity of Λ₀ = 1 is completely different from the on-axis one owing to the variations of distance from the beam center line, it firstly increases up to a peak value, then arrives at a trough, lastly begins to increase again. However, for a perfectly focused beam, the on-axis log-amplitude variance transmits smoothly from zero to the log-amplitude variance of the spherical wave. The off-axis log-amplitude variance for small Λ₀ becomes very large and increases rapidly with α, while that for large Λ₀ also increases to a peak value and then decreases up to zero, which are similar to the log-amplitude variance of the spherical wave. Finally, for some fixed alpha value the influence of the Fresnel ratio Λ₀ and the ratio ρ/W on the log-amplitude variance is also analyzed, it is shown that the influence of the Fresnel ratio Λ₀ and the ratio ρ/W on the log-amplitude variance is similar to that for the Kolmogorov case.

Acknowledgement

This research was financially supported by the National Natural Science Foundation of China (NSFC)(No.10374023 and 60432040). The authors are grateful for a grant from NSFC.

References and links

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Collimated Beam: Θ₀ = 1
Λ₀ = 0	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - / α 2} L^{α / 2} Γ (\frac{2 - α}{2}) \cos [\frac{(α - 2) π}{4}]$
0 ≤ Λ₀ < ∞	$\frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{(\frac{Λ_{0}}{1 + Λ_{0}^{2}})}^{\frac{α - 2}{2}}_{1} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}})$
	$- \frac{2 α - 2}{2} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(\frac{α}{2})}_{n}}{{(\frac{2 + α}{2})}_{n} n!} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2}})}^{n / 2} \cos [n \tan^{- 1} (\frac{1}{Λ_{0}}) + \frac{(α - 2) π}{4}]}$
Λ₀ = ∞	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{Γ (\frac{α}{2}) Γ (\frac{2 - α}{2}) Γ (\frac{2 + α}{2})}{Γ (α)}) \cos [\frac{(α - 2) π}{4}]$
Divergent Beam and Convergent Beam: 0.5 ≤ Θ₀
Λ₀ = 0	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2})_{2} F_{1} (\frac{2 - α}{2}, \frac{α}{2}; \frac{2 + α}{2}; \tilde{Θ}) \cos [\frac{(α - 2) π}{4}]$
0 ≤ Λ₀ < ∞	$\frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{Λ^{\frac{α - 2}{2}}}_{1} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}})$
	$- \frac{2 α - 2}{2} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(\frac{α}{2})}_{n}}{{(\frac{2 + α}{2})}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{n / 2} \cos [n \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(α - 2) π}{4}]}$
Λ₀ = ∞	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} \frac{Γ (\frac{α}{2}) Γ (\frac{2 - α}{2}) Γ (\frac{2 + α}{2})}{Γ (α)} \cos [\frac{(α - 2) π}{4}]$
Convergent Beam: Θ₀ < 0.5, Θ₀ ≠ 0
Λ₀ = 0	${\begin{matrix} - \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) Θ^{\frac{α - 2}{2}} \\ \times_{2} F_{1} (\frac{2 - α}{2}, 1; \frac{2 + α}{2}; 1 - \frac{1}{Θ}) \cos [\frac{(α - 2) π}{4}], & Θ_{0} > 0 \\ - A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} \\ \times {\frac{2}{α} Γ (1 + α / 2) Γ (1 - α) {∣ \tilde{Θ} ∣}^{- \frac{α}{2}} \cos [\frac{(2 - 3 α) π}{4}] + Γ (\frac{2 - α}{2}) \\ \times \frac{1}{α - 1} {∣ \tilde{Θ} ∣}^{\frac{α - 2}{2}}_{2} F_{1} (\frac{2 - α}{2}, 1 - α; 2 - α; \frac{1}{∣ \tilde{Θ} ∣}) \cos [\frac{(α - 2) π}{4}]}, & Θ_{0} > 0 \end{matrix}$
0 ≤ Λ₀ < ∞	$\frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {{Λ}^{\frac{α - 2}{2}}_{1} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}})$
	${- (\tilde{Θ^{2}} + Λ^{2})}^{\frac{α - 2}{4}} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(1 - α)}_{n}}{{(2 - α)}_{n} n!} {(\tilde{Θ^{2}} + Λ^{2})}^{- n / 2} \cos [(n - \frac{α - 2}{2})$
	$\times \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(α - 2) π}{4}] - \frac{2 α - α}{α} \frac{Γ (1 + α / 2) Γ (1 - α)}{Γ (1 - α / 2)} {(\tilde{Θ^{2}} + Λ^{2})}^{- \frac{α}{4}}$
	$\times \cos [\frac{α}{2} \tan^{- 1} (\frac{Λ}{\tilde{Θ}}) + \frac{(2 - 3 α) π}{4}]}$
Λ₀ = ∞	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} \frac{Γ (\frac{α}{2}) Γ (\frac{2 - α}{2}) Γ (\frac{2 + α}{2})}{Γ (α)} \cos [\frac{(α - 2) π}{4}]$
Perfect Focused Beam: Θ₀ = 0
Λ₀ = 0	${\begin{matrix} 0, & ρ = 0 \\ \infty, & ρ > 0 \end{matrix}$
0 ≤ Λ₀ < ∞	$\frac{1}{α - 1} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} Γ (\frac{2 - α}{2}) {Λ_{0}^{\frac{2 - α}{2}} F_{1} (\frac{2 - α}{2}; 1; \frac{2 ρ^{2}}{W^{2}})$
	$- {(\frac{1 + Λ_{0}^{2}}{Λ_{0}^{2}})}^{\frac{α - 2}{4}} \sum_{n = 0}^{\infty} \frac{{(\frac{2 - α}{2})}_{n} {(1 - α)}_{n}}{{(2 - α)}_{n} n!} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2}})}^{n / 2} \cos [(n - \frac{α - 2}{2})$
	$\times \tan^{- 1} (\frac{1}{Λ_{0}}) + \frac{(α - 2) π}{4}] - \frac{2 α - 2}{α} \frac{Γ (1 + α / 2) Γ (1 - α)}{Γ (1 - α / 2)} {(\frac{Λ_{0}^{2}}{1 + Λ_{0}^{2}})}^{\frac{α}{4}}$
	$\times \cos [\frac{α}{2} \tan^{- 1} (\frac{1}{Λ_{0}}) + \frac{(2 - 3 α) π}{4}]}$
Λ₀ = ∞	$- \frac{2}{α} A (α) {\tilde{C}}_{n}^{2} π^{2} k^{3 - a / 2} L^{a / 2} \frac{Γ (\frac{α}{2}) Γ (\frac{2 - α}{2}) Γ (\frac{2 + α}{2})}{Γ (α)} \cos [\frac{(α - 2) π}{4}]$

Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence

Abstract

1. Introduction

2. Non-Kolmogorov spectrum

3. Log-amplitude variance

3.1. Collimated beam

3.2. Divergent beam

3.3. Convergent beam

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (7)

Tables (1)

Equations (22)

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