Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels

Mingjian Cheng; Lixin Guo; Yixin Zhang

doi:10.1364/OE.23.032606

1. Introduction

Free-space optical (FSO) communication has attracted much attention with increasing activities in the maritime environment, as it can offer a high-speed, large-capacity wireless network [1]. Nevertheless, FSO link performance is severely degraded by optical turbulence along the beam propagation path between the transmitter and receiver because of turbulence-induced scintillation [2]. The irradiance scintillation index is an important parameter used to determine the effects of turbulence-induced scintillation on the performance of FSO systems. Moreover, the irradiance scintillation index of Gaussian-beam waves have been extensively studied in Kolmogorov [2–4 ] or non-Kolmogorov [5–8 ] terrestrial atmospheric turbulence environment, by utilizing several atmospheric spectrum models, such as Kolmgorov, Tatarskii, von Karman, exponential and modified spatial power spectrum.

Maritime atmospheric turbulence differs from terrestrial atmospheric turbulence, because of their different environments [9,10 ]. Grayshan et al. [10] introduced a new power spectrum for the fluctuations of refractive index in the maritime atmospheric turbulence; since then, the influence of maritime atmospheric turbulence on laser beam propagation has been extensively studied theoretically and experimentally. The developed spectrum has been usd to derive analytical expression for the irradiance scintillation index of a spherical wave in all turbulent maritime atmosphere [11]. Toselli et al. [12] used the new maritime atmospheric spectrum to evaluate beam spread and beam wander of Gaussian-beam waves propagating in weak turbulent maritime atmosphere. The probability density function of a fluctuating intensity can be theoretically reconstructed using a number of measurements conducted in the maritime environment [13]. Modulation transfer functions [14], temporal power spectra of irradiance scintillation [15], and angle of arrival fluctuations [16] for infrared imaging system over maritime atmospheric turbulent channels have been developed. Khannous [17] studied the effect of maritime atmospheric turbulence on Li’s flattened Gaussian beams. Moreover, the Bit error rate (BER) performance of a coherent FSO system with quadrature array phase-shift keying (QPSK) modulation over maritime atmospheric turbulence was investigated in Ref [18]. However, this new maritime atmospheric spectrum was originally developed for Kolmogorov turbulence with a specific power-law exponent value of 11/3, which is theoretically valid only in the inertial sub-range and cannot be directly applied in non-Kolmogorov turbulence cases [2]. Based on non-Kolmogorov anisotropic spectrum, the receiver-aperture-averaged irradiance scintillation index for a plane wave, spherical wave and Gaussian-beam wave are presented in [9]. The current study was performed to determine the effects of non-Kolmogorov maritime atmospheric turbulence on FSO communication links. Results could be a basis for ship-to-ship/shore optical wireless communication system design and mission planning in the maritime environment. To the best of our knowledge, the effects of non-Kolmogorov turbulence of maritime atmosphere environment and aperture averaging on the irradiance scintillation index of Gaussian-beam waves and the average BER of the FSO system with Gaussian-beam waves have not been reported.

In this study, Kolmogorov maritime atmospheric spectrum [10] was generalized to apply in non-Kolmogorov or generalized maritime atmospheric turbulence. The non-Kolmogorov maritime atmospheric spectrum has a general spectral power-law exponent, which all values ranging from 3 to 5, instead of the standard Kolmogorov value of 11/3. Rytov theory was employed first to derive the closed-form expression for the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves in weak non-Kolmogorov turbulent maritime atmosphere. The associated average BER of FSO links was then investigated versus normalized average signal-to-noise ratio (SNR) in non-Kolmogorov turbulent maritime atmospheric environment.

2. Atmospheric spatial power spectrum model

The Kolmogorov modified atmospheric spatial power spectrum model of refractive index fluctuation, considering the effects of turbulence inner and outer scales, is defined as [2]:

Φ_{n} (κ) = 0.033 C_{n}^{2} \exp (- κ^{2} / κ_{l}^{2}) {(κ^{2} + κ_{0}^{2})}^{- 11 / 6} [1 + a_{1} (κ / κ_{l}) + a_{2} {(κ / κ_{l})}^{7 / 6}],

where

C_{n}^{2}

represents the refractive-index structure parameter for Kolmogorov turbulence and has the unit of m^−2/3;

κ_{l} = 3.41 / l_{0}

,

κ_{0} = 4 π / L_{0}

; l ₀ is inner scale and L ₀ is outer scale; in maritime turbulent atmospheric environment [10],

a_{1} = - 0.061

,

a_{2} = 2.836

; and in terrestrial turbulent atmospheric environment [2],

a_{1} = 1.802

,

a_{2} = - 0.254

.

In this paper, we examine a more general spatial power spectrum model to describe non-Kolmogorov atmospheric turbulence, in which the power-law exponent of 11/3 is allowed to deviate somewhat from this value. To extend the Kolmogorov atmospheric spatial power spectrum model to non-Kolmogorov case, the corresponding generalized or non-Kolmogorov atmospheric spectrum model should takes the form as [2],

Φ_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} \frac{\exp (- κ^{2} / κ_{H}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} [1 + a_{1} (\frac{κ}{κ_{H}}) + a_{2} {(\frac{κ}{κ_{H}})}^{3 - α / 2}], 3 < α < 5,

where

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

is the generalized refractive-index structure parameter with units m^3- ^α,

κ_{H} = c_{0} (α) / l_{0}

, and

A (α)

is a constant that maintains consistency between the refractive-index structure function and its power spectrum [19],

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

c_{0} (α) = {π A (α) [Γ (\frac{3}{2} - \frac{α}{2}) \frac{3 - α}{3} + a_{1} Γ (2 - \frac{α}{2}) \frac{4 - α}{3} + a_{2} Γ (3 - \frac{3 α}{4}) \frac{4 - α}{2}]}^{\frac{1}{α - 5}} .

In the Appendix, the expressions of $A (α)$ and $c_{0} (α)$ are derived.

Equation (4) can be split into 3 intervals, and we can solve each interval separately by determining a general form $μ [{(κ / κ_{H})}^{γ}]$ with

μ [{(κ / κ_{l})}^{γ}] = μ_{1} [{(κ / κ_{l})}^{γ_{1}}] + μ_{2} [{(κ / κ_{l})}^{γ_{2}}] + μ_{3} [{(κ / κ_{l})}^{γ_{3}}] .

Hence, the non-Kolmogorov turbulent atmospheric spatial power spectrum model for maritime or terrestrial atmospheric environments becomes,

Φ_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} μ \exp (- κ^{2} / κ_{H}^{2}) {(κ^{2} + κ_{0}^{2})}^{- α / 2} [{(κ / κ_{l})}^{γ}], 3 < α < 5.

3. Aperture-averaged scintillation index model

In the weak-fluctuation regime, based on Rytov theory, the receiver-aperture-averaged irradiance scintillation index for Gaussian-beam waves propagating through non-Kolmogorov turbulent atmosphere can be expressed as [9]

\begin{matrix} δ_{IG}^{2} (D) = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) \exp (- \frac{Λ L κ^{2} ξ^{2}}{k}) \exp (- \frac{D^{2} κ^{2} ξ^{2}}{16}) \\ \times [1 - \cos (\frac{L κ^{2}}{k} ξ (1 - \bar{Θ} ξ))] d κ d ξ, \end{matrix}

where L is the wave propagation distance,

ξ

is the normalized path coordinate and related to z by

ξ = z / L

,

k = 2 π / λ

, λ is the wavelength, D is the aperture diameter of the receiver, and the complementary parameter

\bar{Θ} = 1 - Θ

. Θ and Λ are the output plane (or receiver) beam parameters, which are related to the curvature parameter Θ₀ and Fresnel ratio Λ₀ of the Gaussian-beam waves at the input (or transmitter) plane by

Θ = Θ_{0} / (Θ_{0}^{2} + Λ_{0}^{2})

,

Λ = Λ_{0} / (Θ_{0}^{2} + Λ_{0}^{2})

. Beam parameters Θ₀ and Λ₀ are defined by Θ₀ = 1 + L/F and

Λ_{0} = 2 L / (k w_{0}^{2})

, respectively. Here, F denotes the phase front radius of curvature and w ₀ is the spot size of the Gaussian-beam waves at the input plane [3].

By substituting Eq. (6) into Eq. (7), the receiver-aperture-averaged irradiance scintillation index for Gaussian-beam waves can be written as:

\begin{matrix} δ_{IG}^{2} (D) = 8 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} μ \int_{0}^{1} \int_{0}^{\infty} κ \frac{\exp (- κ^{2} / κ_{H}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} {(\frac{κ}{κ_{H}})}^{γ} \exp (- \frac{Λ L κ^{2} ξ^{2}}{k} - \frac{D^{2} κ^{2} ξ^{2}}{16}) \\ \times [1 - \cos (\frac{L κ^{2}}{k} ξ (1 - \bar{Θ} ξ))] d κ d ξ . \end{matrix}

Using Euler’s formula, $\cos (x) = Re [\exp (- i x)]$ , Eq. (8) becomes:

\begin{matrix} δ_{IG}^{2} (D) = \frac{8 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} μ}{κ_{H}^{γ}} Re \int_{0}^{1} \int_{0}^{\infty} \frac{κ^{γ + 1}}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} {\exp [- (\frac{1}{κ_{H}^{2}} + \frac{Λ L ξ^{2}}{k} + \frac{D^{2} ξ^{2}}{16}) κ^{2}] \\ - \exp [- (\frac{1}{κ_{H}^{2}} + \frac{Λ L ξ^{2}}{k} + \frac{D^{2} ξ^{2}}{16} + i \frac{L}{k} ξ (1 - \bar{Θ} ξ)) κ^{2}]} d κ d ξ . \end{matrix}

For mathematical analysis convenience, we define two parameters $Q_{L} = (Λ L / k + D^{2} / 16) κ_{H}^{2}$ , $Q_{H} = L κ_{H}^{2} / k$ and use the confluent hypergeometric function of the second kind $U (a; c; x)$ [20],

U (a; c; x) = \frac{1}{Γ (a)} \int_{0}^{\infty} \exp (- x t) t^{a - 1} {(1 + t)}^{c - a - 1} d t,

we can obtain,

\begin{matrix} δ_{IG}^{2} (D) = \frac{8 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} μ}{κ_{H}^{γ}} κ_{0}^{2 + γ - α} Re \int_{0}^{1} Γ (\frac{γ + 2}{2}) {U (\frac{γ + 2}{2}, 2 + (γ - α) / 2; (1 + Q_{L} ξ^{2}) \frac{κ_{0}^{2}}{κ_{H}^{2}}) \\ - U (\frac{γ + 2}{2}, 2 + (γ - α) / 2; [1 + Q_{L} ξ^{2} + i Q_{l} ξ (1 - \bar{Θ} ξ)] \frac{κ_{0}^{2}}{κ_{H}^{2}})} d ξ . \end{matrix}

As l ₀ << L ₀, and $| κ_{0}^{2} / κ_{l}^{2} |$ << 1, $U (a; c; x)$ has an approximate result [20],

U (a; c; x) \approx \frac{Γ (1 - c)}{Γ (1 + a - c)} + \frac{Γ (c - 1)}{Γ (a)} x^{1 - c}, | x | < < 1.

Thus, Eq. (9) can be reduced to,

\begin{matrix} δ_{IG}^{2} (D) = 8 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} μ κ_{H}^{2 - α} Γ (1 - \frac{(α - γ)}{2}) Re \int_{0}^{1} {{(1 + Q_{L} ξ^{2})}^{(α - γ) / 2 - 1} \\ - {[1 + Q_{L} ξ^{2} + i Q_{H} ξ (1 - \bar{Θ} ξ)]}^{(α - γ) / 2 - 1}} d ξ . \end{matrix}

Evaluate $\int_{0}^{1} {(1 + Q_{L} ξ^{2})}^{(α - γ) / 2 - 1} d ξ$ with the generalized hypergeometric function [21], $_{2} F_{1} (a, b; c; z) = \frac{Γ (c)}{Γ (b) Γ (c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} {(1 - z t)}^{- a} d t, c > b > 0$ , results in,

\int_{0}^{1} {(1 + Q_{L} ξ^{2})}^{(α - γ) / 2 - 1} d ξ =_{2} F_{1} (1 - \frac{(α - γ)}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) .

And evaluate $Re \int_{0}^{1} {[1 + Q_{L} ξ^{2} + i Q_{H} ξ (1 - \bar{Θ} ξ)]}^{(α - γ) / 2 - 1} d ξ$ with binomial expansion ${(1 + x)}^{k} = \sum_{n = 0}^{\infty} (\begin{array}{l} k \\ n \end{array}) x^{n}, | x | < 1$ , $(\begin{array}{l} k \\ n \end{array}) = \frac{{(- 1)}^{n} {(- k)}_{n}}{n!}$ , we have,

\begin{array}{l} \int_{0}^{1} {[1 + Q_{L} ξ^{2} + i Q_{H} ξ (1 - \bar{Θ} ξ)]}^{(α - γ) / 2 - 1} d ξ \\ = \sum_{n = 0}^{\infty} (\begin{array}{l} (α - γ) / 2 - 1 \\ n \end{array}) {(i Q_{H})}^{n} \int_{0}^{1} ξ^{n} {(1 - (\bar{Θ} - \frac{Q_{L}}{i Q_{H}}) ξ)}^{n} d ξ . \end{array}

Following the same derivation procedure as Eq. (14), we can obtain,

\int_{0}^{1} ξ^{n} {(1 - (\bar{Θ} - \frac{Q_{L}}{i Q_{H}}) ξ)}^{n} d ξ = \frac{1}{n + 1} {}_{2}F_{1} (- n, n + 1; n + 2; \bar{Θ} - \frac{Q_{L}}{i Q_{H}}) .

As $| \bar{Θ} | < 1$ , and $| \bar{Θ} - \frac{Q_{L}}{i Q_{H}} | < 1$ , we can use the approximation [20], $_{2} F_{1} (- n, n + 1; n + 2; x)$ $\approx {(1 - 2 x / 3)}^{n}, | x | < 1$ , and $_{2} F_{1} (a, b; c; z) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(b)}_{k} z^{k}}{{(c)}_{k} k!},$ Eq. (15) can be reduced to

\int_{0}^{1} {[1 + Q_{L} ξ^{2} + i Q_{H} ξ (1 - \bar{Θ} ξ)]}^{(α - γ) / 2 - 1} d ξ = {}_{2}F_{1} (1 + \frac{γ - α}{2}, 1; 2; i Q_{H} - \frac{2}{3} (i Q_{H} \bar{Θ} - Q_{L})) .

Using the property of generalized hypergeometric function [20], ${}_{2}F_{1} (1 - a, 1; 2; - x)$ $= \frac{{(1 + x)}^{a} - 1}{a x}$ , we can obtain,

\begin{array}{l} Re {}_{2}F_{1} (1 + \frac{γ - α}{2}, 1; 2; i Q_{H} - \frac{2}{3} (i Q_{H} \bar{Θ} - Q_{L})) \\ = Re \frac{{[(3 + 2 Q_{L}) / 3 + i (1 + 2 Θ) Q_{H} / 3]}^{(α - γ) / 2} - 1}{[2 Q_{L} / 3 + i (1 + 2 Θ) Q_{H} / 3] (α - γ) / 2} . \end{array}

By polar coordinates transformation with $x + i y = \sqrt{x^{2} + y^{2}} \exp [i \tan^{- 1} (y / x)]$ and Re[exp(ix)] = cos(x),

\begin{array}{l} Re {\frac{{[(3 + 2 Q_{L}) / 3 + i (1 + 2 Θ) Q_{H} / 3]}^{(α - γ) / 2} - 1}{[2 Q_{L} / 3 + i (1 + 2 Θ) Q_{H} / 3] (α - γ) / 2}} \\ = {[(α - γ) Τ]}^{- 1} {Ζ^{(α - γ) / 2} \cos [\frac{(α - γ)}{2} φ_{1} - φ_{2}] - \cos (φ_{2})}, \end{array}

where

Τ = \frac{1}{6} \sqrt{4 Q_{L}^{2} + {(1 + 2 Θ)}^{2} Q_{H}^{2}}

;

Ζ = (\frac{1}{3} \sqrt{{(3 + 2 Q_{L})}^{2} + {(1 + 2 Θ)}^{2} Q_{H}^{2}})

;

φ_{1} = \tan^{- 1} (\frac{(1 + 2 Θ) Q_{H}}{3 + 2 Q_{L}})

and

φ_{2} = \tan^{- 1} (\frac{(1 + 2 Θ) Q_{H}}{2 Q_{L}})

.

Thus, By substituting Eq. (14) and Eq. (19) into Eq. (13), yields,

\begin{matrix} δ_{IG}^{2} (D) = 8 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} μ κ_{H}^{2 - α} Γ (1 + \frac{γ - α}{2}) {_{2} F_{1} (1 + \frac{γ - α}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) \\ + {[(α - γ) Τ]}^{- 1} [Ζ^{(α - γ) / 2} \cos [\frac{(α - γ)}{2} φ_{1} - φ_{2}] - \cos (φ_{2})]} . \end{matrix}

As a result, the final solution to the irradiance scintillation index for Gaussian-beam waves propagating through weak non-Kolmogorov maritime atmospheric turbulence considering finite aperture receiver is obtained from the above, by solving $μ {(κ / κ_{H})}^{γ} = 1 - 0.061 (κ / κ_{H}) + 2.836 {(κ / κ_{H})}^{3 - α / 2}$ for each interval independently, then adding them together,

\begin{matrix} δ_{IG,Mar}^{2} (D) = \frac{- α Q_{H}^{1 - α / 2} δ_{R}^{2}}{2 Γ (1 - α / 2) \sin (α π / 4)} {Γ (1 - α / 2) [_{2} F_{1} (1 - \frac{α}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) \\ - {(α Τ)}^{- 1} [Ζ^{α / 2} \cos (\frac{α φ_{1}}{2} - φ_{2}) - \cos (φ_{2})]] - 0.061 Γ (3 / 2 - α / 2) \\ \times [_{2} F_{1} (\frac{3 - α}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) + {[(α - 1) Τ]}^{- 1} \times [Ζ^{(α - 1) / 2} \cos [(α - 1) φ_{1} / 2 - φ_{2}] - \cos (φ_{2})]] \\ + 5.672 Γ (5 / 2 - 3 α / 4) [_{2} F_{1} (\frac{10 - 3 α}{4}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) - {[(3 α - 6) Τ / 2]}^{- 1} \\ \times [Ζ^{(3 α - 6) / 4} \cos [(3 α - 6) φ_{1} / 4 - φ_{2}] - \cos (φ_{2})]]}, \end{matrix}

where

δ_{R}^{2}

is the irradiance scintillation index for a plane wave propagating through weak non-Kolmogorov turbulence [5],

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

As Gaussian-beam waves propagating through weak non-Kolmogorov terrestrial atmospheric environment, $μ {(κ / κ_{H})}^{γ} = 1 + 1.802 (κ / κ_{H}) - 0.254 {(κ / κ_{H})}^{3 - α / 2}$ , we can obtain,

\begin{matrix} δ_{IG,Ter}^{2} (D) = \frac{- α Q_{H}^{1 - α / 2} δ_{R}^{2}}{2 Γ (1 - α / 2) \sin (α π / 4)} {Γ (1 - α / 2) [_{2} F_{1} (1 - \frac{α}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) \\ - {(α Τ)}^{- 1} (Ζ^{α / 2} \cos (\frac{α φ_{1}}{2} - φ_{2}) - \cos (φ_{2}))] + 1.802 Γ (3 / 2 - α / 2) \\ \times [_{2} F_{1} (\frac{3 - α}{2}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) + {((α - 1) Τ)}^{- 1} \times (Ζ^{(α - 1) / 2} \cos [(α - 1) φ_{1} / 2 - φ_{2}] - \cos (φ_{2}))] \\ - 0.508 Γ (5 / 2 - 3 α / 4) [_{2} F_{1} (\frac{10 - 3 α}{4}, \frac{1}{2}; \frac{3}{2}; - Q_{L}) - {[(3 α - 6) Τ / 2]}^{- 1} \\ \times [Ζ^{(3 α - 6) / 4} \cos [(3 α - 6) φ_{1} / 4 - φ_{2}] - \cos (φ_{2})]]} . \end{matrix}

For the case of Θ = 1 and Λ = 0, we have the receiver-aperture-averaged irradiance scintillation index of a plane wave propagating through weak atmospheric turbulence, and for the case of Θ = Λ = 0, we have the receiver-aperture-averaged irradiance scintillation index of a spherical wave.

4. Analysis of average BER performance

Here, we consider an intensity-modulated and direct-detection (IM/DD) FSO system using On-Off Keying (OOK) modulation with Gaussian-beam wave propagating through weak non-Kolmogorov maritime atmospheric turbulence channel with additive white Gaussian noise (AWGN). And the received electrical signal is given by

y = I x + n,

where y is the received signal, x is the modulated signal, I is the instantaneous fading coefficient because of atmospheric turbulence and n is the AWGN with zero mean with variance

σ_{n}^{2}

.

Under the weak irradiance fluctuations, we usually model instantaneous fading coefficient I by log-normal model with the following probability density function (PDF) [2],

f_{I} (I) = {[\sqrt{2 π δ_{I}^{2} (α, L)} I]}^{- 1} \exp {- {[\ln (I) + δ_{IG,Mar}^{2} (D) / 2]}^{2} / [2 δ_{IG,Mar}^{2} (D)]} .

For an OOK FSO communication system, average BER can be obtained by [22]:

BER = \int_{0}^{\infty} f_{I} (I) Q (\sqrt{S N R_{0}} I) d I,

where SNR ₀ is the average SNR with

R^{2} P_{t}^{2} / σ_{n}^{2}

, R is the receiver responsivity and P_t is the average transmitted optical power, Q(x) is the Gaussian Q function,

Q (x) = \frac{1}{π} \int_{0}^{2 π} \exp (- \frac{x^{2}}{2 \sin^{2} θ}) d θ, x > 0,

Let $x = [\ln (I) + δ_{I}^{2} (α, L) / 2] / {[2 δ_{I}^{2} (α, L)]}^{1 / 2}$ and use the Gauss-Hermite quadrature integration approximation, $\int_{0}^{\infty} \exp (- x^{2}) f (x) d x \approx \sum_{i = 1}^{n} w_{i} f (x_{i})$ [23] average BER can be efficiently and accurately approximated as:

\begin{matrix} BER = \frac{1}{π} \int_{0}^{2 π} \frac{1}{\sqrt{π}} w_{i} \sum_{i = 1}^{n} \exp {\frac{- S N R_{0}^{2} \exp [2 (\sqrt{2} δ_{I} (α, L) x_{i} - δ_{I}^{2} (α, L) / 2)]}{2 \sin^{2} θ}} d θ \\ = \frac{1}{\sqrt{π}} \sum_{i = 1}^{n} Q {S N R_{0} \exp [\sqrt{2} δ_{I} (α, L) x_{i} - δ_{I}^{2} (α, L) / 2]} \end{matrix}

where x_i are the roots of the Hermite polynomials

L_{n} (x)

and the associated weight w_i is given by

w_{i} = \frac{2^{n - 1} n! x_{i}}{n^{2} {[H_{n - 1} (x_{i})]}^{2}}

.

5. Numerical results

In this section, previously derived formulas were used to analyze differences in the irradiance scintillation index of Gaussian-beam waves propagating through weak non-Kolmogorov maritime and terrestrial atmospheric environments and considering the effect of aperture averaging. The influences of non-Kolmogorov maritime atmospheric turbulence and aperture averaging on the error performance of the FSO communication system with Gaussian-beam waves over horizontal non-Kolmogorov maritime atmospheric turbulence channels were also analyzed. The following parameters are assumed unless ortherwise specified: non-Kolmogorov spectral power-law exponent, α = 3.3; atmospheric turbulence inner scale, l ₀ = 7 mm; aperture diameter of the receiver, D = 4 cm; turbulent structure parameter, ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ ; wavelength, λ = 1550 nm; and propagation distance, L = 2 km.

The receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves propagating through the weak non-Kolmogorov maritime and terrestrial atmospheric turbulence channels was plotted in Fig. 1 , as a function of propagation distance with different values of non-Kolmogorov spectral power-law exponent, α = 3.2, 3.4, 3.6, and 3.8. Evidently, the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves is affected by propagation distance and non-Kolmogorov spectral power-law exponent. Low values of α, result in satisfactory performance of the irradiance scintillation index in non-Kolmogorov maritime and terrestrial atmospheric environments. The receiver-aperture-averaged irradiance scintillation index is least affected by atmospheric turbulence when α approaches 4 and most affected when α is approximately 3.2. When α approaches 4, the wavefront pure tile plays a dominant role, resulting in relatively low irradiance scintillation index value.

Fig. 1 Scintillation index of Gaussian-beam waves as a function of L with different α values, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.

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Figure 2 reveals the effect of turbulence inner scale (l ₀ = 1, 3, 5, 7, and 9 mm) on the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves in non-Kolmogorov maritime and terrestrial atmospheric environments. Atmospheric turbulence exerts more effects on the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves with the increasing turbulence inner scale. Moreover, the influence of turbulence inner scale on the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves in maritime atmospheric turbulence channels is more obvious than that in terrestrial atmospheric turbulence channels.

Fig. 2 Scintillation index of Gaussian-beam waves as a function of L with different l ₀ value, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.

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In Fig. 3 , we evaluate the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves propagating through weak non-Kolmogorov maritime and terrestrial atmospheric turbulence channels with different receiver aperture diameter as a function of propagation distance. Here we take receiver aperture diameter D to be 0 (point receiver), 2, 4 and 6 cm. Expectedly, increasing the receiver aperture diameter results in strong aperture averaging effects, and it will obviously results in the value reduction of the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves.

Fig. 3 Scintillation index of Gaussian-beam waves as a function of L with different D, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.

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Atmospheric turbulent structure parameter, ${\tilde{C}}_{n}^{2}$ is another important parameter that affects atmospheric turbulence strength. The influence of atmospheric turbulent structure parameter on the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves in non-Kolmogorov maritime and terrestrial atmospheric environments was analyzed in Fig. 4 , by changing ${\tilde{C}}_{n}^{2}$ = 10⁻¹⁵, 5 × 10⁻¹⁵ and 10⁻¹⁴ m^3- ^α. It is shown that the high ${\tilde{C}}_{n}^{2}$ values result in strong turbulence. Thus, the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves increases with increasing ${\tilde{C}}_{n}^{2}$ values.

Fig. 4 Scintillation index of Gaussian-beam waves as a function of L with different ${\tilde{C}}_{n}^{2}$ , in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.

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To analyze wavelength’s influence on receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves, Fig. 5 demonstrates the evolution of the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves in non-Kolmogorov maritime and terrestrial atmospheric environments as a function of propagation distance with different wavelength, λ = 1550, 1064, 850, and 632.8 nm. It is obvious that receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves is sensitive to wavelength, λ, and decreases with increasing λ values. This phenomenon is reasonable, because atmospheric coherence length is directly proportional to wavelength, and laser beams with large wavelength can more effectively mitigate the effects of atmospheric turbulence.

Fig. 5 Scintillation index of Gaussian-beam waves as a function of L with different λ, in weak non-Kolmogorov maritime (a) and terrestrial (b) atmospheric environment.

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Fig. 1-5 show the comparison of the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves for non-Kolmogorov maritime and terrestrial atmospheric environments. The receiver-aperture-averaged irradiance scintillation index in maritime atmospheric environment predicts higher than that in the terrestrial atmospheric environment. Furthermore, these values of the receiver-aperture-averaged irradiance scintillation index present the some change trends in the two atmospheric environments.

In Fig. 6 , we present the average BER of the FSO communication system with Gaussian-beam wave propagation over horizontal non-Kolmogorov maritime atmospheric turbulence channels, with different values of non-Kolmogorov spectral power-law exponent, α = 3.37, 2.67, and 3.97, and altered normalized average SNR. It is observed that average BER decreases with the increase of non-Kolmogorov spectral power-law exponent α. For high values of α, e.g., 3.97, the irradiance scintillation index of Gaussian-beam waves is low, and the FSO system performance is less degraded because of atmospheric turbulence.

Fig. 6 BER of FSO links against normalized SNR ₀ for different α values in weak non-Kolmogorov maritime atmospheric environment.

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Figure 7 explores the influence of turbulence inner scale, l ₀ on the average BER of the FSO link in maritime atmospheric environment as a function of normalized average SNR by changing l ₀ as l ₀ = 1, 3, 5, 7, and 9 mm. It is clearly that average BER is affected by turbulence inner scale. The normalized average SNR for reaching the benchmark BER increases with the increasing turbulence inner scale. For instance, a normalized average SNR of approximately 27 dB is required to achieve a BER of 10⁻⁵ in maritime atmospheric with l ₀ = 1 mm, and it penalty rises to 34 dB for the case of l ₀ = 9 mm.

Fig. 7 BER of FSO links against normalized SNR ₀ for different l ₀ in weak non-Kolmogorov maritime atmospheric environment.

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The effect of aperture averaging on the average BER of the FSO system with Gaussian-beam wave propagation in weak non-Kolmogorov maritime atmospheric turbulence channels is discussed in Fig. 8 . Here we take receiver aperture diameter, D to be 0, 2, 4 and 6 cm, respectively. As shown, when receiver aperture diameter increases, the average BER of the FSO link decreases obviously, this phenomenon is known as the aperture averaging effects. Large receiver aperture diameter significantly alleviates the effect of atmospheric turbulence. The performance gain of the FSO link is 10 dB for D = 6 cm relative to that for D = 2 cm to achieve specific average BER of 10⁻⁵ in terms of the normalized average SNR. Aperture averaging can be used to mitigate the detrimental effect of turbulence-induced irradiance fading.

Fig. 8 BER of FSO links against normalized SNR ₀ for different D in weak non-Kolmogorov maritime atmospheric environment.

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In Fig. 9 , we illustrate the impact of wavelength on the average BER of the FSO links over weak non-Kolmogorov maritime atmospheric turbulence channels. To do this, we change wavelength λ = 632.8, 850, 1064, and 1550 nm. Increasing wavelength significantly improves on the FSO links performance in the maritime atmospheric environment, which is consistent with the observation presented in terrestrial atmospheric turbulence. Hence, larger wavelength signal source is a good alternative to improve the performance of ship-to-ship/shore optical wireless communication systems in weak maritime environments.

Fig. 9 BER of FSO links against SNR ₀ for different λ in weak non-Kolmogorov maritime atmospheric environment

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6. Conclusions

In summary, we developed closed-form expressions for the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves propagating through weak non-Kolmogorov maritime and terrestrial atmospheric environments, by establishing a non-Kolmogorov or generalized maritime atmospheric turbulent refractive index spectrum. We investigated the average BER of the FSO links in weak non-Kolmogorov maritime atmospheric turbulence channels by employing log-normal PDF model under weak turbulence conditions. The influences of propagation distance, non-Kolmogorov spectral power-law exponent, maritime atmospheric turbulence inner scale, receiver aperture diameter, atmospheric turbulent structure parameter, wavelength, and normalized average SNR, were discussed. Numerical results show that the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves exhibits the some change tendencies in non-Kolmogorov maritime and terrestrial atmospheric environments. The receiver-aperture-averaged irradiance scintillation index in maritime atmospheric environment is higher than that in the terrestrial atmospheric environment. The influence of maritime atmospheric turbulence on the receiver-aperture-averaged irradiance scintillation index of Gaussian-beam waves increases with the increase of turbulence inner scale, propagation distance, and atmospheric turbulent structure parameter, and the decrease of non-Kolmogorov spectral power-law exponent, receiver aperture diameter, and wavelength. Increasing receiver aperture diameter and wavelength is a good alternative to improve the performance of FSO communication system in weak maritime turbulent atmosphere channels. This phenomenon is consistent with the observation presented in terrestrial atmospheric turbulence. The results presented in this paper can be beneficial in the design of ship-to-ship/shore optical wireless communication systems in the maritime environment.

Appendix

In this appendix, we derive the expressions of $c_{0} (α)$ and $A (α)$ for non-Kolmogorov modified atmospheric turbulence.

The relationship between $D_{n} (R, α)$ and $Φ_{n} (κ, α)$ can be described as [19],

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

Substituting Eq. (2) into Eq. (29), and L ₀ is set to infinity for calculation purposes, Eq. (32) becomes:

\begin{matrix} D_{n} (R, α) = 8 π \int_{0}^{\infty} κ^{2 - α} A (α) {\tilde{C}}_{n}^{2} \exp (- κ^{2} / κ_{H}^{2}) \\ \times [1 + a_{2} (\frac{κ}{κ_{H}}) + a_{2} {(\frac{κ}{κ_{H}})}^{3 - α / 2}] (1 - \frac{\sin (κ R)}{κ R}) d κ . \end{matrix}

By expanding $(1 - \frac{\sin (κ R)}{κ R})$ within a Maclaurin series,

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

we can obtain,

\begin{matrix} D_{n} (R, α) = 8 π A (α) {\tilde{C}}_{n}^{2} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{(2 n + 1)!} R^{2 n} \int_{0}^{\infty} κ^{2 + 2 n - α} \exp (- κ^{2} / κ_{H}^{2}) \\ \times [1 + a_{1} (\frac{κ}{κ_{H}}) + a_{2} {(\frac{κ}{κ_{H}})}^{3 - α / 2}] d κ . \end{matrix}

Making use of the gamma function $Γ (x)$ ,

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

and hypergeometric function

{}_{1}F_{1} (a, b; x)

,

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

the expression of

D_{n} (R, α)

can be derived:

D_{n} (R, α) = 4 π A (α) {\tilde{C}}_{n}^{2} κ_{H}^{3 - α} {\begin{cases} Γ (\frac{3}{2} - \frac{α}{2}) [1 - {}_{1}F_{1} (\frac{3}{2} - \frac{α}{2}; \frac{3}{2}; - \frac{R^{2} κ_{H}^{2}}{4})] \\ + a_{1} Γ (2 - \frac{α}{2}) [1 - {}_{1}F_{1} (2 - \frac{α}{2}; \frac{3}{2}; - \frac{R^{2} κ_{H}^{2}}{4})] \\ + a_{2} Γ (3 - \frac{3 α}{4}) [1 - {}_{1}F_{1} (3 - \frac{3 α}{4}; \frac{3}{2}; - \frac{R^{2} κ_{H}^{2}}{4})] \end{cases}} .

For statistically homogeneous, isotropic, non-Kolmogorov atmospheric turbulence, the related refractive-index structure function, $D_{n} (R, α)$ is given by [2],

D (R, α) = {\begin{cases} {\tilde{C}}_{n}^{2} R^{α - 3}, l_{0} < < R < < L_{0} \\ {\tilde{C}}_{n}^{2} l_{0}^{α - 5} R^{2}, 0 \leq R < < l_{0} . \end{cases}

When $l_{0} < < R < < L_{0}$ , $R^{2} κ_{H}^{2} / 4 > > 1$ , ${}_{1}F_{1} (a; b; x)$ approximately expanded as [20],

{}_{1}F_{1} (a; b; - x) \approx \frac{Γ (b)}{Γ (b - a)} x^{- α}, (x > > 1) .

Substituting Eq. (37) into Eq. (35), we can obtain

D_{n} (R, α) = - 4 π A (α) {\tilde{C}}_{n}^{2} R^{α - 3} Γ (\frac{3}{2} - \frac{α}{2}) \frac{Γ (3 / 2)}{Γ (α / 2)} {(\frac{1}{2})}^{α - 3} .

Using Eqs. (36) and (38) , and considering the properties of the gamma function [21], $Γ (a + 1) = a Γ (a)$ , $Γ (1 - a) Γ (a) = π / \sin (π a)$ , $Γ (a + 1 / 2) Γ (a) = 2^{1 - 2 α} Γ (2 a)$ , we can obtain the expression of $A (α)$ ,

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

When $0 \leq R < < l_{0}$ , $R^{2} κ_{H}^{2} / 4 < < 1$ , ${}_{1}F_{1} (a; b; x)$ approximately expanded as [20],

{}_{1}F_{1} (a; b; x) \approx 1 + a x / b (x < < 1) .

Substituting Eq. (40) into Eq. (35), and Eq. (35) can be expressed as,

\begin{matrix} D_{n} (R, α) = 4 π A (α) {\tilde{C}}_{n}^{2} κ_{H}^{5 - α} R^{2} [Γ (\frac{3}{2} - \frac{α}{2}) \frac{3 - α}{12} \\ + a_{1} Γ (2 - \frac{α}{2}) \frac{4 - α}{12} + a_{2} Γ (3 - \frac{3 α}{4}) \frac{4 - α}{8}] . \end{matrix}

Using Eqs. (36) and (41) , we can obtain the expression of $c_{0} (α)$ ,

c_{0} (α) = {π A (α) [Γ (\frac{3}{2} - \frac{α}{2}) \frac{3 - α}{3} + a_{1} Γ (2 - \frac{α}{2}) \frac{4 - α}{3} + a_{2} Γ (3 - \frac{3 α}{4}) \frac{4 - α}{2}]}^{\frac{1}{α - 5}} .

Acknowledgments

This work was supported by China National Funds for Distinguished Young Scientists (Grant No.61225002), and the Fundamental Research Funds for the Central Universities.

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Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels

Abstract

1. Introduction

2. Atmospheric spatial power spectrum model

3. Aperture-averaged scintillation index model

4. Analysis of average BER performance

5. Numerical results

6. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (42)

Optics Express