Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence

Xiang Yi; Zengji Liu; Peng Yue

doi:10.1364/OE.20.004232

1. Introduction

An optical wave propagating through a turbulent atmosphere will experience random fluctuations in irradiance commonly known as scintillation. Turbulence-induced scintillation is a serious issue for optical wireless communications for both terrestrial as well as ground-to-space or space-to-ground data links, as it can produce deep random fades in the received signal [1–3]. Past studies concerning the scintillation of optical waves have led to many analytical results for Kolmogorov turbulence [4–8]. However, recent experimental data indicate turbulence in certain portions of the atmosphere does not exhibit Kolmogorov statistics [9–11]. This prompts the need for reinvestigations of optical scintillation in the context of atmospheric turbulence representing non-Kolmogorov properties.

In order to facilitate the theoretical researches on optical wave propagation through non-Kolmogorov turbulence, a generalized Kolmogorov spectrum model that owns a generalized power law ranging from 3 to 4 instead of standard Kolmogorov power law value 11/3 and a generalized amplitude factor instead of constant Kolmogorov value 0.033 has been proposed [12]. Partly because of its relatively simple mathematical form, it has been widely used in previous studies of optical scintillation [12–17]. Nonetheless, this spectrum model is theoretically valid only over inertial subrange where both inner scale and outer scale effects are negligible. To account for the behavior of the power spectrum outside the inertial subrange, some other spectrum models have recently been suggested, such as the generalized von Kármán spectrum [18], the generalized exponent spectrum [19], and the generalized modified atmospheric spectrum [20]. These models include the effects from the inner scale and outer scale, and also include the generalized Kolmogorov spectrum model as special case. In attempting to develop the more general scintillation models for non-Kolmogorov turbulence, it is necessary to use these newly proposed spectrum models. Recently, the irradiance scintillations for horizontally propagating optical waves have been studied in [21] and [22], respectively, based on the generalized von Kármán spectrum and the generalized modified atmospheric spectrum. Although their scintillation results include inner- and outer-scale effects, they are limited to weak irradiance fluctuations. For a propagating optical wave that experiences stronger conditions of non-Kolmogorov turbulence, to the best of the authors’ knowledge, there is no publishing work analyzing the impacts of the inner- and outer-scale on scintillation. This has motivated the work herein.

In this paper, by use of the generalized von Kármán spectrum model and the extended Rytov method, we first develop the theoretical expressions for the scintillation index (SI) of a plane wave and a spherical wave both propagating through moderate-to-strong non-Kolmogorov turbulence along a horizontal path. Then, we present some numerical results to illustrate the influences of the inner- and outer-scale on scintillation under various fluctuation conditions.

2. Generalized von Kármán spectrum

The generalized von Kármán spectrum model is given by [18]

Φ_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} \frac{exp (- κ^{2} / κ_{m}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, 0 ⩽ κ < \infty, 3 < α < 4,

where κ denotes the magnitude of the spatial-frequency with units of rad/m, α is the generalized power law that takes all values ranging from 3 to 4; A(α) is the generalized amplitude factor that has the form A(α) = Γ (α – 1) · cos(απ/2)/(4π²) with Γ(·) being the Gamma function [24]. The factor

{\tilde{C}}_{n}^{2}

is the generalized refractive index structure parameter with the units m³⁻^α. The remaining parameters κ_m and κ₀ are basically reciprocals, respectively, of the inner scale l₀ and outer scale L₀ of refractive index fluctuations, that is, κ_m = c (α)/l₀ with c(α) = {Γ[(5 – α)/2] · A(α) · 2π/3}^1/(^α⁻⁵⁾ and κ₀ = 2π/L₀. The inner and outer scale parameters l₀ and L₀ are characteristic length scales associated with the smallest and largest scale sizes, respectively, in the turbulent atmosphere that causes random fluctuations in refractive index. The inner scale is generally on the order of millimeters while the outer scale is on the order of meters. It is noted that the spectrum model Eq. (1) reduces to the generalized Kolmogorov spectrum [12, Eq. (2)] when l₀ tends to zero and L₀ tends to infinite, and reduces to the conventional von Kármán spectrum [6. Eq. (1)] when α = 11/3, A(α) = 0.033, and

{\tilde{C}}_{n}^{2} = C_{n}^{2}

. Here,

C_{n}^{2}

represents the conventional structure parameter that has units of m^−2/3.

It is known that the conventional von Kármán spectrum cannot feature the high wave number rise (or bump) prior to the dissipation range that is revealed in experimental data [25], but, because of its relatively tractable mathematical form, the conventional von Kármán spectrum is widely used in theoretical calculations [4, 6, 8]. Furthermore, although the von Kármán model does not agree with all scintillation values predicted by the modified atmospheric spectrum (a well-known analytically tractable model that can characterize the high wave number bump) [4, Eq. (3.31)], the von Kármán spectrum captures most of the behavior of the optical wave [4, Chap. (9.6.4), 8]. The modified atmospheric spectrum has also been generalized to apply in non-Kolmogorov turbulence [20]. But due to lack of experimental data for non-Kolmogorov turbulence in the upper troposphere and stratosphere, some coefficients in this spectrum model have not yet been determined. Therefore, our analysis here is based on the generalized von Kármán spectrum like [21].

3. The plane-wave spatial coherence radius and phase structure function

3.1. Spatial coherence radius

It is known that weak-fluctuation and strong-fluctuation conditions for an unbounded plane wave or spherical wave based on the generalized Kolmogorov spectrum are characterized by ${\tilde{σ}}_{R}^{2} < 1$ and ${\tilde{σ}}_{R}^{2} ≫ 1$ , respectively [15, 16]. Here

{\tilde{σ}}_{R}^{2} = - 8 π^{2} A (α) Γ (1 - α / 2) α^{- 1} sin (α π / 4) {\tilde{C}}_{n}^{2} k^{3 - α / 2} L^{α / 2}

is the non-Kolmogorov Rytov variance for a plane wave [15], k = 2π/λ is the wave number at the wavelength λ, and L is the path length. However, for an arbitrary spectrum model, we should rely on the more general parameter

L / k ρ_{p l}^{2} (α)

as a measure of the strength of irradiance fluctuations [4, 6]. Here ρ_pl (α) is the spatial coherence radius of a plane wave [15]. The corresponding criterion for distinguishing between weak-fluctuation and strong-fluctuation conditions in terms of the parameter

L / k ρ_{p l}^{2} (α)

is defined by the following sets of inequalities [4, Chap.7]:

\begin{array}{l} L / k ρ_{p l}^{2} (α) & ≪ 1, & weak - fluctuation conditions, \\ L / k ρ_{p l}^{2} (α) & ≫ 1, & strong - fluctuation conditions . \end{array}

Obviously if we want to develop a general scintillation model that can cover a wide range of fluctuation conditions (weak to strong) on the basis of the generalized von Kármán spectrum, we need an in-depth understanding of the plane-wave spatial coherence radius. Specifically, to include the effects from inner scale, we need to know the form of the plane-wave spatial coherence radius which is much less than the inner scale of turbulence.

The spatial coherence radius is determined by the wave structure function (WSF) [4, 23]. For non-Kolmogorov turbulence, the WSF associated with a plane wave is given by [19, 23]

D_{p l} (ρ, α) = 8 π^{2} k^{2} L \int_{0}^{\infty} κ Φ_{n} (κ, α) [1 - J_{0} (κ ρ)] d κ,

where ρ is the separation distance between two observation points at the receiver plane; J₀ (·) is a Bessel function of the first kind [24]. By substituting Eq. (1) into Eq. (4) and expanding the Bessel function in the integrand in a Maclaurin series, we find

D_{p l} (ρ, α) = 8 π^{2} A (α) {\tilde{C}}_{n}^{2} k^{2} L \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1} {(ρ)}^{2 n}}{2^{2 n} {(n!)}^{2}} \int_{0}^{\infty} \frac{κ^{2 n + 1} exp (- κ^{2} / κ_{m}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} d κ .

Then, by making an appropriate change of variable, we deduce that

D_{p l} (ρ, α) = 4 π^{2} A (α) {\tilde{C}}_{n}^{2} k^{2} L κ_{0}^{2 - α} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n!} {(κ_{0} ρ / 2)}^{2 n} U (n + 1; n + 2 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}}),

where we have used [24, Eq. (10.28)] to evaluate the integral in terms of the confluent hyper-geometric function of the second kind U (a;c;x) [24]. In the present study, it suffices to know the form of the WSF only in the asymptotic regime ρ ≪ l₀. For small ρ, we can approximate Eq. (6) by the first term of the series to obtain

D_{p l} (ρ, α) ≅ π^{2} A (α) {\tilde{C}}_{n}^{2} k^{2} L κ_{0}^{4 - α} U (2; 3 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}}) ρ^{2}, ρ ≪ l_{0} .

Under most conditions of atmospheric turbulence we find that

κ_{0}^{2} / κ_{m}^{2} ≪ 1

, so by making the further approximation U (a;c;x) ≅ Γ(c – 1)x^1−c/Γ(a),0 < x ≪ 1 [24, Eq. (10.30a)], we obtain

D_{p l} (ρ, α) ≅ π^{2} A (α) Γ (2 - α / 2) {\tilde{C}}_{n}^{2} k^{2} L κ_{m}^{4 - α} ρ^{2}, ρ ≪ l_{0} .

When α = 11/3, Eq. (8) reduces to

D_{p l} (ρ, 11 / 3) = 3.28 C_{n}^{2} k^{2} L l_{0}^{- 1 / 3} ρ^{2}

which is the correct result for conventional von Kármán spectrum [4, Eq. (8.88)].

By setting Eq. (8) equal to 2 and solving for ρ, we deduce the plane-wave spatial coherence radius as

ρ_{p l} (α) = {[0.5 π^{2} A (α) Γ (2 - α / 2) {\tilde{C}}_{n}^{2} k^{2} L κ_{m}^{4 - α}]}^{- 1 / 2}, ρ_{p l} (α) ≪ l_{0} .

It is noted that the spatial coherence radius given by Eq. (9) is consistent with conventional Kolmogorov result [4, Eq. (6.64)] when α is set to 11/3.

3.2. Phase structure function

Knowledge of the plane-wave phase structure function (PSF) is crucial for developing scintillation results in the strong fluctuation regimes [4, 7, 15]. To include inner-scale effects, we need to know the form of the plane-wave PSF in the asymptotic regime ρ ≪ l₀.

The plane-wave PSF can be determined by [23]

D_{S, p l} (ρ, α) = 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) [1 - J_{0} (κ ρ)] [1 + cos (\frac{L κ^{2} ξ}{k})] d κ d ξ,

where ξ is the normalized path length parameter. For small separation distances in which ρ ≪ l₀, we can use the small-argument approximation for the Bessel function (i.e., J₀ (κρ) ≅ 1 – (κρ)²/4) to obtain

D_{S, p l} (ρ, α) = π^{2} k^{2} L ρ^{2} \int_{0}^{1} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, α) [1 + cos (\frac{L κ^{2} ξ}{k})] d κ d ξ .

If we write the cosine function in Eq. (11) as cosx = Re(e^−ix) through use of Euler’s formula, the substitution of Eq. (1) into Eq. (11) leads to

\begin{array}{l} D_{S, p l} (ρ, α) & = & A (α) π^{2} {\tilde{C}}_{n}^{2} k^{2} L ρ^{2} Re \int_{0}^{1} \int_{0}^{\infty} \frac{κ^{3}}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} \\ \times {exp (- \frac{κ^{2}}{κ_{m}^{2}}) + exp [- \frac{κ^{2} (1 + i Q_{m} (α) ξ)}{κ_{m}^{2}}]} d κ d ξ, \end{array}

where

Q_{m} (α) = {[c (α)]}^{2} L / (k l_{0}^{2})

is a nondimensional inner-scale parameter. Next, the inside integration yields ([24, Eq. (10.28)])

\begin{array}{l} D_{S, p l} (ρ, α) & = & 0.5 Γ (2) A (α) π^{2} {\tilde{C}}_{n}^{2} k^{2} L ρ^{2} Re \int_{0}^{1} {κ_{0}^{4 - α} U (2; 3 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}}) \\ + κ_{0}^{4 - α} U (2; 3 - \frac{α}{2}; \frac{κ_{0}^{2} (1 + i Q_{m} (α) ξ)}{κ_{m}^{2}})} d ξ . \end{array}

Because

κ_{0}^{2} / κ_{m}^{2} \sim l_{0}^{2} / L_{0}^{2} ≪ 1

, we can use the small-argument approximation of the confluent hypergeometric function given by [24, Eq. (10.30a)] to reduce Eq. (13) to

\begin{array}{l} D_{S, p l} (ρ, α) & = & 0.5 A (α) π^{2} Γ (2 - \frac{α}{2}) {\tilde{C}}_{n}^{2} k^{2} L κ_{m}^{4 - α} ρ^{2} Re \int_{0}^{1} {1 + {(1 + i Q_{m} (α) ξ)}^{\frac{α}{2} - 2}} d ξ \\ = & A (α) π^{2} Γ (2 - \frac{α}{2}) {\tilde{C}}_{n}^{2} k^{2} L κ_{m}^{4 - α} ρ^{2} Re {\frac{1}{2} + \frac{{[1 + i Q_{m} (α)]}^{α / 2 - 1} - 1}{i Q_{m} (α) (α - 2)}} . \end{array}

By simplifying the last expression above, we are led to our general result

D_{S, p l} (ρ, α) = 0.5 Ξ A (α) π^{2} Γ (2 - α / 2) {\tilde{C}}_{n}^{2} k^{2} L κ_{m}^{4 - α} ρ^{2}, ρ ≪ l_{0},

where

Ξ = 1 + \frac{2 sin [\frac{α - 2}{2} {tan}^{- 1} Q_{m} (α)]}{(α - 2) {[Q_{m} (α)]}^{2 - α / 2}} {1 + \frac{1}{{[Q_{m} (α)]}^{2}}}^{\frac{α - 2}{4}} .

When α = 11/3, Eq.(15) reduces to

D_{S, p l} (ρ, 11 / 3) = 1.64 C_{n}^{2} k^{2} L l_{0}^{- 1 / 3} [1 + 0.64 {(k l_{0}^{2} / L)}^{1 / 6}] ρ^{2}

which matches well with conventional Kolmogorov result [4, Eq. (6.66)].

4. Scintillation index in the strong fluctuation regime

The extended Rytov method presented in [4, 7] makes use of known expressions for both the weak and strong fluctuation regimes. Fortunately, Cui et al. [21] have developed analytical expressions for the weak fluctuation regime based on the generalized von Kármán spectrum. Here we develop expressions for the strong fluctuation regime.

In the strong fluctuation regime, the asymptotic theory predicts that the SI for an unbounded plane wave or spherical wave can be expressed in the form [4, 15]

\begin{array}{l} σ_{I}^{2} (α) & = & 1 + 32 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) {sin}^{2} [\frac{L κ^{2} ξ}{2 k}] \\ \times exp {- \int_{0}^{1} D_{S, p l} [\frac{L κ}{k} w (τ, ξ), α] d τ} d κ d ξ, L / k ρ_{p l}^{2} (α) ≫ 1, \end{array}

where τ is a normalized distance variable and the exponential function acts like a low-pass spatial filter defined by the plane-wave PSF D_S,pl (ρ,α). The function w(τ,ξ) is defined by [4]

w (τ, ξ) = {\begin{matrix} τ (1 - \bar{Θ} ξ), τ < ξ, \\ ξ (1 - \bar{Θ} τ), τ > ξ, \end{matrix}

where the parameter Θ̅ = 0 for a plane wave and Θ̅ = 1 for a spherical wave.

When the spatial coherence radius of the optical wave ρ_pl (α) is much less than the inner scale of turbulence l₀, the PSF for a plane wave based on the generalized von Kármán spectrum can be calculated by Eq. (15). Hence, for the plane wave case, we find that

\begin{array}{l} \int_{0}^{1} D_{S, p l} [\frac{L κ}{k} w (τ, ξ), α] d τ & = & - \frac{α Γ (2 - α / 2) Ξ}{16 Γ (1 - α / 2) sin (α π / 4)} \\ \times {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2} (\frac{L κ^{2}}{k}) ξ^{2} (1 - \frac{2}{3} ξ) . \end{array}

Also, the sine function in Eq. (17) may be approximated by its leading term, which yields

{sin}^{2} [\frac{L κ^{2} ξ}{2 k}] ≅ \frac{L^{2} κ^{4} ξ^{2}}{4 k^{2}} .

By substituting Eq. (1), Eq. (19), and Eq. (20) into Eq. (17), we deduce that

\begin{array}{l} σ_{I, p l}^{2} (α) & = & 1 - \frac{α}{Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(\frac{L}{k})}^{3 - α / 2} \int_{0}^{1} ξ^{2} \int_{0}^{\infty} \frac{κ^{5}}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} \\ \times exp {- \frac{κ^{2}}{κ_{m}^{2}} {1 - \frac{α Γ (2 - α / 2) Ξ}{16 Γ (1 - α / 2) sin (α π / 4)} \\ \times {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - \frac{α}{2}} ξ^{2} (1 - \frac{2}{3} ξ)}} d κ d ξ . \end{array}

Next, we evaluate the inside integration in Eq. (21) by [24, Eq. (10.28)], which leads to

\begin{array}{l} σ_{I, p l}^{2} (α) & = & 1 - \frac{α}{2 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(\frac{L}{k})}^{3 - α / 2} \int_{0}^{1} ξ^{2} κ_{0}^{6 - α} Γ (3) \\ \times U (3; 4 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}} {1 - \frac{α Γ (2 - α / 2) Ξ}{16 Γ (1 - α / 2) sin (α π / 4)} \\ \times {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - \frac{α}{2}} ξ^{2} (1 - \frac{2}{3} ξ)}) d ξ . \end{array}

As discussed in [8, Appendix H], we can use the small-argument approximation for U (a;c;x) [24, Eq. (10.30a)] and then obtain

\begin{array}{l} σ_{I, p l}^{2} (α) & = & 1 - \frac{α Γ (3 - α / 2)}{2 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2} \\ \times \int_{0}^{1} \frac{ξ^{2}}{{1 - \frac{α Γ (2 - α / 2) Ξ}{16 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - \frac{α}{2}} ξ^{2} (1 - \frac{2}{3} ξ)}^{3 - α / 2}} d ξ . \end{array}

The evaluation of the last integral in closed form is not known. But it can be integrated easily by the numerical integration method. For brevity, we rewrite Eq. (23) as

σ_{I, p l}^{2} (α) = 1 + \frac{C_{p l} (α)}{{{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2}}^{2 - α / 2}}, - \frac{α Γ (2 - α / 2) {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2}}{16 Γ (1 - α / 2) sin (α π / 4)} ≫ 1,

where we have used −αΓ(2 − α/2)

{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2} / [16 Γ (1 - α / 2) sin (α π / 4)] ≫ 1

to represent the strong-fluctuation conditions when ρ_pl (α) ≪ l₀ [based on Eq. (9)], and defined

\begin{array}{l} C_{p l} (α) & = & \frac{- α Γ (3 - α / 2)}{2 Γ (1 - α / 2) sin (α π / 4)} \\ \times \int_{0}^{1} \frac{ξ^{2}}{{{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{α / 2 - 3} - \frac{α Γ (2 - α / 2) Ξ}{16 Γ (1 - α / 2) sin (α π / 4)} ξ^{2} (1 - \frac{2}{3} ξ)}^{3 - α / 2}} d ξ . \end{array}

In the spherical-wave case, we first note that

\begin{array}{l} \int_{0}^{1} D_{S, p l} [\frac{L κ}{k} w (τ, ξ)] d τ & = & - \frac{α Γ (2 - α / 2) Ξ}{48 Γ (1 - α / 2) sin (α π / 4)} \\ \times {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2} (\frac{L κ^{2}}{k}) ξ^{2} (1 - ξ^{2}), \end{array}

which leads to

\begin{array}{l} σ_{I, s p}^{2} (α) & = & 1 - \frac{α Γ (3 - α / 2)}{2 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2} \\ \times \int_{0}^{1} \frac{ξ^{2} {(1 - ξ)}^{2}}{{1 - \frac{α Γ (2 - α / 2) Ξ}{48 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - \frac{α}{2}} ξ^{2} (1 - ξ^{2})}^{3 - α / 2}} d ξ . \end{array}

For brevity, we rewrite Eq. (27) as

σ_{I, s p}^{2} (α) = 1 + \frac{C_{s p} (α)}{{{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2}}^{2 - α / 2}}, - \frac{α Γ (2 - α / 2) {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2}}{16 Γ (1 - α / 2) sin (α π / 4)} ≫ 1,

where

\begin{array}{l} C_{s p} (α) & = & \frac{- α Γ (3 - α / 2)}{2 Γ (1 - α / 2) sin (α π / 4)} \\ \times \int_{0}^{1} \frac{ξ^{2} {(1 - ξ)}^{2}}{{{\tilde{σ}}_{R}^{- 2} {[Q_{m} (α)]}^{α / 2 - 3} - \frac{α Γ (2 - α / 2) Ξ}{48 Γ (1 - α / 2) sin (α π / 4)} ξ^{2} {(1 - ξ)}^{2}}^{3 - α / 2}} d ξ . \end{array}

5. Scintillation index for plane wave

In this and the following section, the extended Rytov method suggested in [4, 7, 15] is used to derive expressions for the SI of an optical wave that are valid under moderate-to-strong irradiance fluctuations. In applying the extended Rytov method, we here rely on a minor modification of the “effective non-Kolmogorov spectrum”

{\tilde{Φ}}_{n, e} (κ, α) = {\tilde{Φ}}_{n} (κ, α) G (κ, l_{0}, L_{0}, α) = {\tilde{Φ}}_{n} (κ, α) [G_{X} (κ, l_{0}, L_{0}, α) + G_{Y} (κ, α)],

where

{\tilde{Φ}}_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} κ^{- α}

is the generalized Kolmogorov spectrum, G(κ,l₀, L₀,α) is an amplitude (irradiance) spatial filter that consists of a large-scale spatial filter G_X (κ,l₀, L₀,α) and a small-scale spatial filter G_Y (κ,α). To include the effects from the inner scale and outer scale, we represent the large-scale and small-scale spatial filters by

G_{X} (κ, l_{0}, L_{0}, α) = f (κ, l_{0}, α) g (κ L_{0}) exp (- \frac{κ^{2}}{κ_{X}^{2}}),

G_{Y} (κ, α) = \frac{κ^{α}}{{(κ^{2} + κ_{Y}^{2})}^{α / 2}},

where f (κ,l₀,α) is a factor that describes inner scale modifications of the basic non-Kolmogorov power law, and g(κL₀) describes outer scale effects. The parameter κ_X in the large-scale filter function is a cutoff spatial frequency for the large-scale turbulent cell effects, and κ_Y in the small-scale filter function is a cutoff spatial frequency for the small-scale turbulent cell effects. In this fashion, G(κ,l₀, L₀,α) acts like a linear spatial filter that only permits low-pass spatial frequencies κ < κ_X and high-pass spatial frequencies κ > κ_Y at a given propagation distance L. The effects of mid-range scale sizes which contribute little to scintillation under strong fluctuations can therefore be filtered out by using G(κ,l₀, L₀,α). Note that outer-scale effects and the inner-scale factor f (κ,l₀,α) is assumed to act only on the large-scale fluctuations, i.e., the small-scale filter has exactly the same form as assumed for zero inner scale [15]. However, we do assume κ_Y depends on inner scale. In our modeling of inner scale and outer scale effects, we choose

f (κ, l_{0}, α) = exp (- κ^{2} / κ_{m}^{2})

and

g (κ L_{0}) = 1 - exp (- κ^{2} / κ_{0}^{2})

with κ₀ = 8π/L₀ for the purpose of capturing the underlying physics, but also for mathematical tractability. In this context, the large scale filter function takes the form

G_{X} (κ, l_{0}, L_{0}, α) = exp (- \frac{κ^{2}}{κ_{m}^{2}}) [exp (- \frac{κ^{2}}{κ_{X}^{2}}) - exp (- \frac{κ^{2}}{κ_{X 0}^{2}})],

where

κ_{X 0}^{2} = κ_{X}^{2} κ_{0}^{2} / (κ_{X}^{2} + κ_{0}^{2})

. Hence, it follows that the large-scale log-irradiance scintillation in this case can be expressed as a difference, that is,

σ_{ln X}^{2} (l_{0}, L_{0}, α) = σ_{ln X}^{2} (l_{0}, α) - σ_{ln X}^{2} (L_{0}, α) .

If we invoke the extended Rytov method, then the total SI can be expressed in the form

σ_{I}^{2} (α) = exp [σ_{ln X}^{2} (l_{0}, α) - σ_{ln X}^{2} (L_{0}, α) + σ_{ln Y}^{2} (l_{0}, α)] - 1,

where

σ_{ln Y}^{2} (l_{0}, α)

represent the small-scale log-irradiance scintillation in the presence of a finite inner scale.

For the plane-wave case, the first term on the right-side in Eq. (34) is given by

\begin{array}{l} σ_{ln X, p l}^{2} (l_{0}, α) & = & 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ {\tilde{Φ}}_{n} (κ, α) exp (- \frac{κ^{2}}{κ_{m}^{2}}) exp (- \frac{κ^{2}}{κ_{X}^{2}}) \\ \times {1 - cos [\frac{L κ^{2}}{k} ξ]} d κ d ξ \\ ≅ & - \frac{α}{4 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} \int_{0}^{1} ξ^{2} d ξ \\ \times \int_{0}^{\infty} η^{2 - α / 2} exp (- \frac{η}{Q_{m} (α)} - \frac{η}{η_{X}}) d η, \end{array}

where we have used a geometrical optics approximation

1 - cos [\frac{L κ^{2}}{k} ξ] ≅ \frac{1}{2} {[\frac{L κ^{2}}{k} ξ]}^{2}, κ ≪ κ_{X},

and the parameter changes η = Lκ²/k in the second step, and defined

η_{X} = L κ_{X}^{2} / k

. The geometrical optics approximation is valid here because the large scale filter excludes high spatial frequency contributions in the moderate-to-strong fluctuation regimes. Upon evaluation of the last expression in Eq. (36), we obtain

σ_{ln X, p l}^{2} (l_{0}, α) = - \frac{α Γ (3 - α / 2)}{12 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(\frac{η_{X} Q_{m} (α)}{η_{X} + Q_{m} (α)})}^{3 - α / 2} .

Note that when l₀ → 0 (i.e., Q_m (α) → ∞), Eq. (38) reduces to

σ_{ln X, p l}^{2} (α) = - \frac{α Γ (3 - α / 2)}{12 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(η_{X})}^{3 - α / 2},

which is the large-scale log-irradiance variance result for the zero scale case [15, Eq. (28)].

To determine the low-pass nondimensional cutoff frequency η_X, we use the asymptotic result [7, Eq. (26)]

η_{X} = \frac{1}{c_{1} (α) + c_{2} (α) L / k ρ_{p l}^{2} (α)} \sim {\begin{array}{l} 1 / c_{1} (α), & L / k ρ_{p l}^{2} (α) ≪ 1, \\ k ρ_{p l}^{2} (α) / c_{2} (α) L & L / k ρ_{p l}^{2} (α) ≫ 1, \end{array}

as well as established behavior of the SI in these asymptotic regimes provided by [21, Eq. (35)]

{\tilde{σ}}_{P L}^{2} = \frac{{\tilde{σ}}_{R}^{2}}{sin (α π / 4)} {{[1 + \frac{1}{Q_{m}^{2} (α)}]}^{α / 4} sin [\frac{α}{2} {tan}^{- 1} Q_{m} (α)] - \frac{α}{2} {[Q_{m} (α)]}^{1 - \frac{α}{2}}}, {\tilde{σ}}_{R}^{2} < 1,

and Eq. (24). In the weak fluctuation regimes where inner-scale effects can be ignored, we approximate

σ_{ln X, p l}^{2} (l_{0}, α)

by

σ_{ln X, p l}^{2} (α)

, and assume that

{\tilde{σ}}_{P L}^{2}

reduces to

{\tilde{σ}}_{R}^{2}

. In this fashion, the scale constant c₁ (α) can be deduced by imposing

σ_{ln X, p l}^{2} (α) \sim 0.49 {\tilde{σ}}_{R}^{2}

similar to the zero scale case. The result is

c_{1} (α) = {[- \frac{5.88 Γ (1 - α / 2) sin (α π / 4)}{α Γ (3 - α / 2)}]}^{\frac{2}{α - 6}} .

Since inner-scale effects tend to diminish under strong irradiance fluctuations, we approximate

σ_{ln X, p l}^{2} (l_{0}, α)

by

σ_{ln X, p l}^{2} (α)

once again. Based on Eq. (24) and the identity

σ_{I}^{2} (α) ≅ 1 + 2 σ_{ln X}^{2} (α)

,

L / k ρ_{p l}^{2} (α) ≫ 1

[4, Eq. (9.35)], we impose

σ_{ln X, p l}^{2} (α) \sim \frac{C_{p l} (α)}{2 {{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2}}^{2 - α / 2}}, - \frac{α Γ (2 - α / 2) {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2}}{16 Γ (1 - α / 2) sin (α π / 4)} ≫ 1 .

Thus, we find

\frac{c_{2} (α) L}{k ρ_{p l}^{2} (α)} = {- \frac{C_{p l} (α)}{2 {{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2}}^{2 - α / 2}} \cdot \frac{12 Γ (1 - α / 2) sin (α π / 4)}{α Γ (3 - α / 2) {\tilde{σ}}_{R}^{2}}}^{\frac{2}{α - 6}}

By substituting Eq. (42) and Eq. (44) into Eq. (40), we obtain

η_{X} = \frac{1 / c_{1} (α)}{1 + {[1.02 C_{p l} (α)]}^{\frac{2}{α - 6}} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - \frac{α}{2}}} .

Thus, by inserting Eq. (45) into Eq. (38), we find that the large-scale log-irradiance scintillation including inner scale parameter can be written in the form

\begin{array}{l} σ_{ln X, p l}^{2} (l_{0}, α) & = & - \frac{α Γ (3 - α / 2)}{12 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} \\ \times {\frac{Q_{m} (α)}{1 + c_{1} (α) Q_{m} (α) + c_{1} (α) {[1.02 C_{p l} (α)]}^{\frac{2}{α - 6}} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - \frac{α}{2}}}}^{3 - α / 2} \end{array}

By an entirely analogous evaluation, it follows that the large-scale log-irradiance scintillation with outer scale parameter becomes

σ_{ln X, p l}^{2} (L_{0}, α) = - \frac{α Γ (3 - α / 2)}{12 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(\frac{η_{X 0} Q_{m} (α)}{η_{X 0} + Q_{m} (α)})}^{3 - α / 2},

where

η_{X 0} = \frac{η_{X} Q_{0}}{η_{X} + Q_{0}} = \frac{Q_{0}}{1 + c_{1} (α) Q_{0} + c_{1} (α) {[1.02 C_{p l} (α)]}^{\frac{2}{α - 6}} {\tilde{σ}}_{R}^{2} Q_{0} {[Q_{m} (α)]}^{2 - \frac{α}{2}}},

and where

Q_{0} = L κ_{0}^{2} / k = 64 π^{2} L / k L_{0}^{2}

.

Because the form of the filter function is the same for the small-scale log-irradiance scintillations in this case as that for the zero inner scale case, these scintillations are again described by [15, Eq. (37)]

σ_{ln Y, p l}^{2} (l_{0}, α) = \frac{α}{(α - 2) Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} η_{Y}^{1 - α / 2},

where

η_{Y} = L κ_{Y}^{2} / k

. To determine the high-pass nondimensional cutoff frequency η_Y, we impose

σ_{ln Y}^{2} (l_{0}, α) \sim 0.51 {\tilde{σ}}_{P L}^{2}

in weak fluctuations and

σ_{ln Y}^{2} (l_{0}, α) \sim ln 2

in strong fluctuations. By using the asymptotic results [7, Eq. (27)]

η_{Y} = c_{3} (α) + c_{4} (α) L / k ρ_{p l}^{2} (α) \sim {\begin{array}{l} c_{3} (α), & L / k ρ_{p l}^{2} (α) ≪ 1, \\ c_{4} (α) L / k ρ_{p l}^{2} (α), & L / k ρ_{p l}^{2} (α) ≫ 1, \end{array}

we find

η_{Y} = {[- \frac{α}{0.51 (α - 2) Γ (1 - α / 2) sin (α π / 4)} \frac{{\tilde{σ}}_{R}^{2}}{{\tilde{σ}}_{P L}^{2}}]}^{\frac{2}{α - 2}} (1 + {0.73}^{\frac{2}{α - 2}} {\tilde{σ}}_{P L}^{\frac{4}{α - 2}}) .

And therefore Eq. (49) becomes

σ_{ln Y, p l}^{2} (l_{0}, α) = 0.51 {\tilde{σ}}_{P L}^{2} {(1 + {0.73}^{\frac{2}{α - 2}} {\tilde{σ}}_{P L}^{\frac{4}{α - 2}})}^{1 - α / 2} .

We note here that although the small-scale filter Eq. (32) does not explicitly contain a inner scale factor like the large-scale filter Eq. (33), the cutoff spatial frequency κ_Y for the small-scale filter does depend on the inner scale [see (51)] and, hence, small-scale scintillation described by Eq. (52) also depends on the inner scale, particularly, in the weak fluctuation regime. Outer-scale effects are negligible here. Finally, by combining Eq. (46), Eq. (47), and Eq. (52), the SI for an infinite plane wave in the presence of a nonzero inner scale and a finite outer scale is given by

σ_{I, p l}^{2} (α) = exp [σ_{ln X, p l}^{2} (l_{0}, α) - σ_{ln X, p l}^{2} (L_{0}, α) + σ_{ln Y, p l}^{2} (l_{0}, α)] - 1, 0 ⩽ {\tilde{σ}}_{R}^{2} < \infty .

6. Scintillation index for spherical wave

Under weak irradiance fluctuations, the SI based on the generalized von Kármán spectrum leads to [21, Eq. (35)]

\begin{array}{l} {\tilde{σ}}_{S P}^{2} & = & \frac{{\tilde{σ}}_{R}^{2}}{sin (α π / 4)} {3^{1 - \frac{α}{2}} {[1 + \frac{9}{Q_{m}^{2} (α)}]}^{α / 4} sin [\frac{α}{2} {tan}^{- 1} (\frac{Q_{m} (α)}{3})] \\ - \frac{α}{2} {[Q_{m} (α)]}^{1 - \frac{α}{2}}}, {\tilde{σ}}_{R}^{2} < 1 . \end{array}

In the strong fluctuation regime, the corresponding approximation to the SI is given by Eq. (28). As in the plane-wave case, we assume the large-scale filter function is Eq. (33), which contains both inner scale and outer scale parameters. Under the geometrical optics approximation, the large-scale scintillation due to inner scale alone is described by the log-irradiance variance

\begin{array}{l} σ_{ln X, s p}^{2} (l_{0}, α) & = & 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ {\tilde{Φ}}_{n} (κ, α) exp (- \frac{κ^{2}}{κ_{m}^{2}}) exp (- \frac{κ^{2}}{κ_{X}^{2}}) \\ \times {1 - cos [\frac{L κ^{2}}{k} ξ (1 - ξ)]} d κ d ξ \\ ≅ & - \frac{α}{4 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} \int_{0}^{1} ξ^{2} (1 - ξ^{2}) d ξ \\ \times \int_{0}^{\infty} η^{2 - α / 2} exp (- \frac{η}{Q_{m} (α)} - \frac{η}{η_{X}}) d η \\ = & - \frac{α Γ (3 - α / 2)}{120 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(\frac{η_{X} Q_{m} (α)}{η_{X} + Q_{m} (α)})}^{3 - α / 2} . \end{array}

When l₀ → 0 (i.e., Q_m (α) → ∞), Eq. (55) reduces to

σ_{ln X, s p}^{2} (α) = - \frac{α Γ (3 - α / 2)}{120 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} {(η_{X})}^{3 - α / 2} .

Similar to the plane-wave case, we rely on Eq. (56) rather than Eq. (55) to determine η_X. Thus, we impose the conditions

σ_{ln X, s p}^{2} (α) \sim {\begin{array}{l} 0.49 {\tilde{β}}_{0}^{2}, & {\tilde{σ}}_{R}^{2} < 1, \\ \frac{C_{s p} (α)}{2 {{\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{3 - α / 2}}^{2 - α / 2}}, & - \frac{α Γ (2 - α / 2) {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - α / 2}}{16 Γ (1 - α / 2) sin (α π / 4)} ≫ 1, \end{array}

where

{\tilde{β}}_{0}^{2} = Γ (α / 2) Γ (1 + α / 2) {\tilde{σ}}_{R}^{2} / Γ (α)

is the non-Kolmogorov spherical wave Rytov variance [14, Eq. (9)]. Then, by using the asymptotic results for η_X [Eq. (40)], we derive

η_{X} = \frac{1 / d_{1} (α)}{1 + Δ (α)},

where

d_{1} (α) = {[- \frac{58.8 Γ (1 - α / 2) sin (α π / 4) Γ (α / 2) Γ (1 + α / 2)}{α Γ (3 - α / 2) Γ (α)}]}^{\frac{2}{α - 6}},

and

Δ (α) = {[\frac{1.02 Γ (α) C_{s p} (α)}{Γ (α / 2) Γ (1 + α / 2)}]}^{\frac{2}{α - 6}} {\tilde{σ}}_{R}^{2} {[Q_{m} (α)]}^{2 - \frac{α}{2}} .

By substituting Eq. (58) into Eq. (55), we obtain

\begin{array}{l} σ_{ln X, s p}^{2} (l_{0}, α) & = & - \frac{α Γ (3 - α / 2)}{120 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} \\ \times {(\frac{Q_{m} (α)}{1 + d_{1} (α) Q_{m} (α) + d_{1} (α) Δ (α) Q_{m} (α)})}^{3 - α / 2} . \end{array}

Similarly, the large-scale scintillation due to outer scale effects is derived by

\begin{array}{l} σ_{ln X, s p}^{2} (L_{0}, α) & = & - \frac{α Γ (3 - α / 2)}{120 Γ (1 - α / 2) sin (α π / 4)} {\tilde{σ}}_{R}^{2} \\ \times {\frac{Q_{0} Q_{m} (α)}{Q_{0} + Q_{m} (α) + d_{1} (α) Q_{0} Q_{m} (α) [1 + Δ (α)]}}^{3 - α / 2} . \end{array}

The small-scale log-irradiance variance is once again given by Eq. (49), but in this case η_Y is described by

η_{Y} = {[- \frac{α}{0.51 (α - 2) Γ (1 - α / 2) sin (α π / 4)} \frac{{\tilde{σ}}_{R}^{2}}{{\tilde{σ}}_{S P}^{2}}]}^{\frac{2}{α - 2}} (1 + {0.73}^{\frac{2}{α - 2}} {\tilde{σ}}_{S P}^{\frac{4}{α - 2}}) .

Hence, the small-scale log-irradiance variance becomes

σ_{ln Y, s p}^{2} (l_{0}, α) ≅ 0.51 {\tilde{σ}}_{S P}^{2} {(1 + {0.73}^{\frac{2}{α - 2}} {\tilde{σ}}_{S P}^{\frac{4}{α - 2}})}^{1 - α / 2} .

If we combine Eq. (61), Eq. (62), and Eq. (64), we obtain the SI for a spherical wave in the presence of a finite inner scale and outer scale described by

σ_{I, s p}^{2} (α) = exp [σ_{ln X, s p}^{2} (l_{0}, α) - σ_{ln X, s p}^{2} (L_{0}, α) + σ_{ln Y, s p}^{2} (l_{0}, α)] - 1, 0 \leq {\tilde{σ}}_{R}^{2} < \infty .

7. Numerical results

In this section, we show numerical results of the analytical SI for the plane wave and the spherical wave model. For numerical calculations we set the wavelength λ = 1.55μm, the generalized structure parameter ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , and allow the path length L to vary. In this fashion, the increment of the path length corresponds to increases in turbulence strength.

In Fig. 1(a) and 1(b) we illustrate the SI of a plane wave Eq. (53) and a spherical wave Eq. (65) as a function of the path length and several values of power law α (solid line). We take the atmospheric conditions described by the generalized von Kármán spectrum with inner scale size of l₀ = 3mm and outer scale L₀ = ∞. For comparison purposes, we also plot the SI based on the asymptotic behavior Eq. (41) and Eq. (54) under weak irradiance fluctuations (dotted line) and the asymptotic theory approximation Eq. (24) and Eq. (28) for strong irradiance fluctuations (dashed line). As expected, our general scintillation models Eq. (53) and Eq. (65) track the corresponding weak and strong turbulence curves fairly closely, providing visual indications that calculations and assumptions used to develop them are correct. In addition, we note that when power law is lower than 11/3 (≈ 3.67), the SI increases initially within the regime of weak irradiance fluctuations with increasing values of the path length. It then increases beyond unity and reaches its maximum value. And after that, the SI gradually decreases, saturating at a level on the order of unity as the path length increases without bound. This general behavior is basically the same as that for conventional von Kármán spectrum (also see [8. Fig. 4 and 5]). However, as the value of the power law decreases, peak scintillation occurs at shorter and shorter path lengths. We also note that for power law values higher than 11/3, the peak scintillation phenomenon vanishes and the SI increases monotonically toward a limiting value above unity with the increment of the path length. This behavior is similar to what is observed for the zero inner scale case [15, Fig. 3].

Fig. 1 Scintillation index vs. path length, (a) plane wave model, (b) spherical wave model.

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Fig. 3 Spherical wave scintillation vs.L.

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As presented in Section 4, in arriving at Eq. (24) we have used the small-argument approximation for the confluent hypergeometric function U (a;c;x) that appears in Eq. (22). Such approximation has been successfully applied to develop expression for the SI in the saturation regime based on the conventional von Kármán spectrum with α = 11/3 [8, Appendix H]. To justify the validity of such approximation for other values of α, we evaluate the integral in Eq. (22) numerically. All of the parameters selected in numerical integration of Eq. (22) are the same as that used in the evaluation of Eq. (24) except that L₀ = 1000m (a finite outer scale parameter is required in calculating Eq. (22), so, we use a large outer scale value to approximate an infinite outer scale.) The results of the numerical integration of Eq. (22) are also plotted in Fig. 1(a) with several values of α (open circles). It can be seen that the approximation results based on Eq. (24) agree well with the numerical results of Eq. (22) for all α values, indicating that the approximation that we have made is valid. From Fig. 1(b) it is clear that the approximation results based on Eq. (28) are also valid in the spherical-wave case. In addition, note in Fig. 1(a) and 1(b) that the total SI Eq. (53) and Eq. (65) match with the SI in the saturation regime Eq. (24) and Eq. (28) at different path lengths for different α values. This is because the validity of Eq. (24) and Eq. (28) are determined by the conditional statement in Eq. (24) that is a function of α. For example, the conditional statement in Eq. (24) is satisfied at L = 2000m for α = 3.07 (in this case, the conditional statement is 113≫1), but for α = 3.87, L need to increase to 6000m to make the conditional statement established (in this case, the conditional statement is 125≫1). The studies based on the generalized Kolmogorov spectrum [15, 16] have used ${\tilde{σ}}_{R}^{2} ≫ 1$ to define the strong-fluctuation conditions. However, this conditional statement is not applicable in the present case. For example, ${\tilde{σ}}_{R}^{2}$ for α = 3.07 at L = 2000m is equal to 9.8 which is not satisfied ${\tilde{σ}}_{R}^{2} ≫ 1$ , but as revealed in Fig. 1(a), the SI for α = 3.07 does indeed enter into the saturation regime at L = 2000m.

The influence of the inner scale and outer scale on plane wave scintillation is shown in Fig. 2. Two groups of curves are plotted against power law. The first group of curves are based on Eq. (53) with L₀ = ∞ and inner scale values l₀ = 1mm and l₀ = 10mm, illustrating the effect of the inner scale alone on scintillation. The second group of curves are for the same inner-scale values but with finite outer scale value L₀ = 1m. In Fig. 2(a) we set L = 200m which corresponds to weak fluctuation conditions. In this figure, we note that with a given inner scale value, the SI curves of the finite outer scale case and the infinite outer scale case coincide regardless of the value of α. This implies that the outer scale has negligible effect on scintillation under weak fluctuations, consistent with previous results in this regime [21]. As for the inner-scale effects, we note that when 3< α <3.2, the SI predicted for the smaller inner scale value (l₀ = 1mm) is less than that predicted for the larger inner scale value (l₀ = 10mm). However, the situation is opposite for power law values in the range of 3.2< α <4. This is a consequence of the behavior in the generalized von Kármán spectrum as a function of inner scale and power law. In Fig. 2(b), L is set to 1000m, corresponding to stronger conditions of non-Kolmogorov turbulence. Here we note that for power law values lower than 3.7 but not close to 3, the SI predicted for l₀ = 10mm is significantly larger than that predicted for l₀ = 1mm. We also note that the presence of a finite outer scale causes a slight drop in the SI within the range 3< α <3.7 except for α close to 3. In Fig. 2(c), L increases to 2000m. Here we note that for either the finite outer scale or the infinite outer scale cases, the gap between two inner scale SI curves becomes narrow when compared with that depicted in Fig. 2(b). Such narrowing gap implies that the effect of inner scale on scintillation begins to diminish as turbulence strength increases. We also note that the presence of a finite outer scale leads to increased scintillation reduction in this figure than in Fig. 2(c), particularly for larger values of inner scale and power law. Such increased reduction of scintillation implies that the outer-scale effect on the SI becomes stronger as the strength of turbulence increases. In Fig. 2(d), we set L = 6000m, representing an extremely strong fluctuation condition. From this figure it is clear that the inner scale has no appreciable effect on scintillation when outer scale is finite. Although there exists a gap between two inner scale SI curves in the infinite outer scale case, it becomes narrower in comparison with Fig. 2(c), also implying that the effects of inner scale on scintillation weaken with increasing strength of turbulence. On the other hand, for a given inner scale, the gap between the finite outer-scale curve and the infinite outer-scale curve increases apparently, especially for larger values of α. This indicates that the outer-scale effects cannot be ignored for larger power law values under extremely strong fluctuation conditions.

Fig. 2 Plane wave scintillation vs.α.

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In Fig. 3 we plot the SI of spherical wave deduced from Eq. (65) as a function of L, with inner scale values of l₀ = 3 and 8mm and outer scale L₀ = ∞ (solid lines). The dashed lines are comparable scintillation values that arise with all parameters the same except L₀ = 1m. We set α = 3.07, 3.37, 3.67, and 3.87 in the Fig. 3(a), 3(b), 3(c), and 3(d), respectively. It is observed that for an arbitrary value of α, the outer scale has a negligible effect on the SI of a spherical wave under weak fluctuations associated with short path lengths. However, with an increase in path lengths, the presence of a finite outer scale in the scintillation model is quite clear. Namely, the outer-scale effect initially reduces scintillation at a steeper rate toward its limiting value of unity than would occur with an infinite outer scale. Also, for an arbitrary value of α, the influence of inner scale on scintillation is not significant under weak fluctuation conditions. Under moderate-to-strong fluctuations, the SI predicted for the smaller inner scale value (l₀ = 3mm) is always less than that predicted for the larger inner scale value (l₀ = 8mm). Specifically, for power law values lower than 11/3 (viz., α = 3.07, 3.37), the increment of inner scale significantly increases the SI near its peak value. As the path length increases, the gap between the larger and smaller inner scale SI curves becomes narrower and narrower, suggesting that inner-scale effect on scintillation tends to diminish with the increment of turbulence strength. For power law values higher than 11/3 (viz., α = 3.87), the predicted SI over the whole moderate-to-strong fluctuation regimes is only slightly higher for a large inner-scale value than for a small inner-scale value.

8. Conclusion

In this paper, by using the generalized von Kármán spectrum and the extended Rytov method, we have derived theoretical expressions for the SI of a horizontally propagating plane wave and spherical wave that are valid under moderate-to-strong irradiance fluctuations. Numerical results show that our general models also match well with previous results in weak-fluctuation regime. Based on the developed models, we have analyzed the inner- and outer-scale effects on the SI with different values of power law and under different turbulence conditions.

In summary, the outer scale has a negligible effect on scintillation under weak turbulence conditions for all values of power law. However, the presence of a finite outer scale causes a more rapid drop in scintillation at larger values of the path length. The decreasing rate of the SI from its peak value to its saturation value is dependent on power law. On the other hand, the inner scale plays a noticeable role in the SI under weak fluctuations. In this regime, whether the predicted SI for the larger inner scale values is greater than that predicted for the smaller inner scale values depends on the value of power law. However, under moderate-to-strong fluctuations, the predicted SI for the smaller inner scale values is always less than that predicted for the larger inner scale values. Furthermore, when power law is lower than 11/3, the increment of inner scale significantly increases the SI near its peak value. As turbulence strength increases, the inner-scale effect on scintillation tends to diminish. When power law is higher than 11/3, the increment of inner scale leads to a slight increase in scintillation over all strong fluctuation regime.

Acknowledgments

This work is supported by the project of State Key Laboratory on Integrated Services Networks, and partly by the National Nature Science Foundation of China grant (no. 60902038).

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Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence

Abstract

1. Introduction

2. Generalized von Kármán spectrum

3. The plane-wave spatial coherence radius and phase structure function

3.1. Spatial coherence radius

3.2. Phase structure function

4. Scintillation index in the strong fluctuation regime

5. Scintillation index for plane wave

6. Scintillation index for spherical wave

7. Numerical results

8. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (65)

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