Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model

doi:10.1364/OE.19.019067

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Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model

Ji Cang and Xu Liu »View Author Affiliations

Optics Express, Vol. 19, Issue 20, pp. 19067-19077 (2011)
http://dx.doi.org/10.1364/OE.19.019067

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Abstract

Based on the generalized spectral model for non-Kolmogorov atmospheric turbulence, analytic expressions of the scintillation index (SI) are derived for plane, spherical optical waves and a partially coherent Gaussian beam propagating through non-Kolmogorov turbulence horizontally in the weak fluctuation regime. The new expressions relate the SI to the finite turbulence inner and outer scales, spatial coherence of the source and spectral power-law and then used to analyze the effects of atmospheric condition and link length on the performance of wireless optical communication links.

1. Introduction

The propagation of optical wave through the turbulent atmosphere has recently generated more and more interest owing to the possibility of high-data-rate free-space optical (FSO) communication systems [1

A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Res. 2(4), 345–396 (2005). [CrossRef]

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]

]. Their performance depends largely on the laser source property and the atmospheric conditions between the transmitter and the receiver [3

H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average capacity of optical wireless communication systems over atmospheric turbulence channels,” J. Lightwave Technol. 27(8), 974–979 (2009). [CrossRef]

–5

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

]. The variance of irradiance fluctuation at the receiver is a paramount parameter that determines the performance of FSO links, which can be lowered through the use of a spatially partially coherent beam (PCB) [6

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43(2), 330–341 (2004). [CrossRef]

–9

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

]. Traditionally, the performance of FSO communication links are estimated with the assumption that atmospheric turbulence belongs to the Kolmogorov type [10

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Trans. NOAA by Israel Program for Scientific Translations, 1971).

]. However, recent experimental data and theoretical investigations have indicated significant deviations from the Kolmogorov model in some portions of the atmosphere, which has prompted research of optical waves propagation through the atmospheric turbulence exhibiting non-Kolmogorov statistics [11

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]

–14

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]

]. Several non-Kolmogorov spectral models have been proposed, which possess variable spectral index ranging from 3 to 5 and a more general amplitude factor rather than constant value 0.033 [15

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]

–17

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]

]. Toselli et al investigated the scintillation of plane wave in non-Kolmogorov moderate-strong turbulence and estimated the FSO system performance for laser beam propagating horizontally through non-Kolmogorov weak turbulence [18

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6457, 64570T (2007). [CrossRef]

–20

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009). [CrossRef]

]. Tan et. al examined the log-amplitude variance for a Gaussian-beam propagating through non-Kolmogorov weak turbulence [21

L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010). [CrossRef] [PubMed]

]. Zilberman et al investigated the influence of non-Kolmogorov turbulence statistics on laser communication links for different propagation scenarios and presented the effects of different turbulence spectral models on optical communication links [22

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010). [CrossRef]

]. Xue et. al studied the angle of arrival variance based on generalized atmospheric turbulence model for wave propagating through non-Kolmogorov turbulence [23

B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011). [CrossRef] [PubMed]

]. However, above-mentioned work on irradiance fluctuations didn’t take turbulence-scale effects and spatial coherence of the source into account.

In this study, the new proposed generalized modified exponential spectrum model including finite turbulence inner and outer scales [24

B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011). [CrossRef] [PubMed]

] is applied to beam propagation through non-Kolmogorov atmospheric turbulence. Based on the new generalized exponential spectrum model, an approximate expression of scintillation index (SI) is derived for a partially coherent Gaussian beam propagating horizontally through non-Kolmogorov turbulence in weak fluctuation regime. And then the influences of variations of turbulence inner scale, outer scale, link length and spectral index on SI, outage probability, mean bit error rate (BER) and average channel capacity for horizontal links are analyzed.

2. Generalized atmospheric spectral model

The generalized modified spectral model is applicable to non-Kolmogorov atmospheric turbulence, which includes finite turbulence inner and outer scales and has a general spectral index ranging from 3 to 5 rather than standard power law value 11/3. Specifically, this spectrum has the following form [24

\begin{array}{l} Φ_{n} (κ, α, l_{0}, L_{0}) = A (α) {\tilde{C}}_{n}^{2} κ^{- α} f (κ, l_{0}, L_{0}, α) \\ (0 \leq κ < \infty, 3< α <5), \end{array}

(1)

f (κ, l_{0}, L_{0}, α) = [1 - \exp (- \frac{κ^{2}}{κ_{0}^{2}})] [1 + a_{1} (\frac{κ}{κ_{l}}) - b_{1} {(\frac{κ}{κ_{l}})}^{β}] \exp (- \frac{κ^{2}}{κ_{l}^{2}})

(2)

where

κ_{l} = c (α) / l_{0}

κ_{0} = C_{0} / L_{0}

, and

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

is the generalized refractive-index structure parameter with units m ^3-α, κ is the magnitude of the spatial-frequency and is related to the size of eddies,

f (κ, l_{0}, L_{0}, α)

describes the influence of finite turbulence inner and outer scales, l ₀ and L ₀ are the inner and outer scales, respectively. The choice of C ₀ depends on the specific application, and it is set to 4π in this study just as [5

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

]. A(α) and c(α) are derived by [24

]

\begin{array}{l} A (α) = \frac{Γ (α - 1)}{4 π^{2}} \sin [(α - 3) \frac{π}{2}] \\ c (α) = {π A (α) [Γ (- \frac{α}{2} + \frac{3}{2}) (1 - \frac{α}{3}) + a_{1} Γ (- \frac{α}{2} + 2) (\frac{4 - α}{3}) - b_{1} Γ (\frac{- α + 3 + β}{2}) (\frac{3 + β - α}{3})]}^{\frac{1}{α - 5}} \end{array}

(3)

when α=11/3, Eq. (1) is reduced to the Kolmogorov modified spectral model, and when

l_{0} \to 0

L_{0} \to \infty

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

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

(4)

3. Variance of irradiance fluctuations

In the weak fluctuation regime, where the log-amplitude variance satisfies

σ_{χ}^{2} ≪ 1

, the normalized irradiance variance has the relation

σ_{I}^{2} \approx σ_{\ln I}^{2} = 4 σ_{χ}^{2}

Following the approach in Ref [5

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

], the irradiance variance at the receiver plane can be expressed as the sum:

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

(5)

\begin{array}{l} σ_{I, l}^{2} (L) = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α, l_{0}, L_{0}) \exp (- Λ_{e d} L κ^{2} ξ^{2} / k) \\ \times {1 - \cos [\frac{L κ^{2}}{k} ξ (1 - {\bar{Θ}}_{e d} ξ)]} d κ d ξ \end{array}

(6)

\begin{array}{l} σ_{I, r}^{2} (ρ, L) = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α, l_{0}, L_{0}) \exp (- Λ_{e d} L κ^{2} ξ^{2} / k) \\ \times [I_{0} (2 Λ_{e d} ρ ξ κ) - 1] d κ d ξ \end{array}

(7)

In the above formulae,

Θ_{e d} = \frac{Θ_{1}}{1 + 4 q_{c} Λ_{1}}

and

Λ_{e d} = \frac{Λ_{1} N_{s}}{1 + 4 q_{c} Λ_{1}}

are the effective beam parameters at the receiver plane.

{\bar{Θ}}_{e d} = 1 - Θ_{e d}

and

q_{c} = \frac{L}{k l_{c}^{2}}

is a nondimensional coherence parameter, where l _c is the spatial coherence length of the source. The values Θ₁ and Λ₁ are the curvature parameter and Fresnel ratio at the receiver plane for vacuum propagation, which given in terms of their respective values at the source plane are

Θ_{1} = \frac{Θ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}}

and

Λ_{1} = \frac{Λ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}}

. The source plane parameters are

Θ_{0} = 1 - \frac{L}{F_{0}}

and

Λ_{0} = \frac{2 L}{k W_{0}^{2}}

, where L is the propagation distance, F ₀ is the initial radius of curvature, W ₀ is the initial beam waist, and k = 2π/λ is the wavenumber.

N_{s} = 1 + \frac{4 q_{c}}{Λ_{0}}

is the number of speckle cells; for fully coherent beam,

Θ_{e d} = Θ_{1}

and

Λ_{e d} = Λ_{1}

In the above formulae, the radial component

σ_{I, r}^{2} (ρ, L)

represents the off-axis contribution to the total irradiance variance and disappears at the beam centerline (ρ=0) or when Λ=0, while the longitudinal component

σ_{I, l}^{2} (L)

is invariant throughout the beam cross section in any transverse plane.

For horizontal path,

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

is constant and substituting Eq. (1) into Eq. (6), we can obtain

\begin{array}{l} σ_{I, l}^{2} (L) = g (E_{1}, H_{1}) + \frac{a_{1}}{κ_{l}} g (E_{2}, H_{1}) - \frac{b_{1}}{κ_{l}^{β}} g (E_{3}, H_{1}) \\ - g (E_{1}, H_{2}) - \frac{a_{1}}{κ_{l}} g (E_{2}, H_{2}) + \frac{b_{1}}{κ_{l}^{β}} g (E_{3}, H_{2}) \end{array}

(8)

where

\begin{array}{l} g (E_{i}, H_{j}) = 4 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} Γ (- \frac{E_{i}}{2} + 1) {(\frac{L}{k})}^{\frac{E_{i}}{2} - 1} H_{j}^{- \frac{E_{i}}{2} + 1} {{}_{2}F_{1} (- \frac{E_{i}}{2} + 1, \frac{1}{2}; \frac{3}{2}; - Λ_{e d} H_{j}) \\ - Re [\sum_{n = 0}^{\infty} \frac{{(- E_{i} / 2 + 1)}_{n} {(1)}_{n}}{n! {(2)}_{n}} {(- i H_{j})}^{n} {}_{2}F_{1} (- n, n + 1; n + 2; {\bar{Θ}}_{e d} + i Λ_{e d})]}, H_{j} < 1 \end{array}

(9)

With the application of the following two formulae [25

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]

{}_{2}F_{1} (- n, n + 1; n + 2; x) \approx {(1 - 2 x / 3)}^{n}, | x | < 1

;

{}_{2}F_{1} (1 - b, 1; 2; - x) = \frac{{(1 + x)}^{b} - 1}{b x}

the approximate analytic expression can be derived by [25

]

\begin{array}{l} g (E_{i}, H_{j}) ≅ 4 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} Γ (- \frac{E_{i}}{2} + 1) {(\frac{L}{k})}^{\frac{E_{i}}{2} - 1} H_{j}^{- \frac{E_{i}}{2} + 1} \\ \times {{}_{2}F_{1} (- \frac{E_{i}}{2} + 1, \frac{1}{2}; \frac{3}{2}; - Λ_{e d} H_{j}) - Re 〚 \frac{{[1 + H_{j} (2 Λ_{e d} + i+2 Θ_{e d} i) / 3]}^{\frac{E_{i}}{2}} - 1}{\frac{E_{i}}{2} H_{j} [2 Λ_{e d} + i(1+2 Θ_{e d})] / 3} 〛}, \\ | {\bar{Θ}}_{e d} + i Λ_{e d} | < 1 \end{array}

(10)

The above expression is valid for all H _j. where

E_{1} = α, E_{2} = α - 1, E_{3} = α - β,

H_{1} = Q_{l}, H_{2} = \frac{Q_{l} Q_{0}}{Q_{0} + Q_{l}}

; and

Q_{l} = \frac{L}{k} κ_{l}^{2}

Q_{0} = \frac{L}{k} κ_{0}^{2}

;

{(a)}_{n} = Γ (a + n) / Γ (a)

is Pochhammer symbol,

Γ (\cdot)

is the Gamma function and

{}_{2}F_{1} (a, b; c; x)

denotes the hypergeometric function [26

L. C. Andrews, Special Functions of Mathematics for Engineers , 2nd ed. (SPIE Optical Engineering Press, 1998).

Substituting Eq. (1) into Eq. (7), and expanding I ₀ in the form of Maclaurin series, we can obtain

\begin{array}{l} σ_{I, r}^{2} (L) = f (E_{1}, H_{1}) + \frac{a_{1}}{κ_{l}} f (E_{2}, H_{1}) - \frac{b_{1}}{κ_{l}^{β}} f (E_{3}, H_{1}) \\ - f (E_{1}, H_{2}) - \frac{a_{1}}{κ_{l}} f (E_{2}, H_{2}) + \frac{b_{1}}{κ_{l}^{β}} f (E_{3}, H_{2}) \end{array}

(11)

where

\begin{array}{l} f (E_{i}, H_{j}) = 4 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} \cdot Γ (- \frac{E_{i}}{2} + 1) {(\frac{L}{k})}^{\frac{E_{i}}{2} - 1} H_{j}^{1 - \frac{E_{i}}{2}} \sum_{n = 1}^{\infty} \frac{{(- E_{i} / 2 + 1)}_{n} {(1 / 2)}_{n}}{n! {(1)}_{n} {(3 / 2)}_{n}} \\ \times {(\frac{2 ρ^{2}}{W_{e d}^{2}})}^{n} {(Λ_{e d} H_{j})}^{n} {}_{2}F_{1} (n + 1 - \frac{E_{i}}{2}, n + \frac{1}{2}; n + \frac{3}{2}; - Λ_{e d} H_{j}) \\ ≅ 4 π^{2} k^{2} L A (α) {\tilde{C}}_{n}^{2} \cdot Γ (- \frac{E_{i}}{2} + 1) {(\frac{L}{k})}^{\frac{E_{i}}{2} - 1} H_{j}^{1 - \frac{E_{i}}{2}} \frac{2 - E_{i}}{3} \\ \times (\frac{ρ^{2}}{W_{e d}^{2}}) (Λ_{e d} H_{j}) {}_{2}F_{1} (2 - \frac{E_{i}}{2}, \frac{3}{2}; \frac{5}{2}; - Λ_{e d} H_{j}), ρ / W_{e d} < 1. \end{array}

(12)

It should be mentioned that there is generally good agreement between approximate analytical expressions (10) and (12) and exact results for collimated beams [25

For the case of plane wave, Θ=1 and Λ=0, we have

\begin{array}{l} σ_{I, p l}^{2} (L, α, l_{0}, L_{0}) = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α, l_{0}, L_{0}) [1 - \cos (\frac{L κ^{2}}{k} ξ)] d κ d ξ \\ = h_{p l} (E_{1}, H_{1}) + \frac{a_{1}}{κ_{l}} h_{p l} (E_{2}, H_{1}) - \frac{b_{1}}{κ_{l}^{β}} h_{p l} (E_{3}, H_{1}) \\ - h_{p l} (E_{1}, H_{2}) - \frac{a_{1}}{κ_{l}} h_{p l} (E_{2}, H_{2}) + \frac{b_{1}}{κ_{l}^{β}} h_{p l} (E_{3}, H_{2}) \end{array}

(13)

h_{p l} (E_{i}, H_{j}) = \frac{- 8 π^{2} A (α)}{1.23 E_{i}} Γ (1 - \frac{E_{i}}{2}) {\tilde{σ}}_{R}^{2} (E_{i}) [{(1 + \frac{1}{H_{j}^{2}})}^{E_{i} / 4} \sin (\frac{E_{i}}{2} \tan^{- 1} H_{j}) - \frac{E_{i}}{2} H_{j}^{- \frac{E_{i}}{2} + 1}]

(14)

where

{\tilde{σ}}_{R}^{2} (E_{i}) = 1.23 {\tilde{C}}_{n}^{2} k^{3 - \frac{E_{i}}{2}} L^{\frac{E_{i}}{2}}

For the case of spherical wave, Θ=Λ=0, we have

\begin{array}{l} σ_{I, s p}^{2} (L, α, l_{0}, L_{0}) = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α, l_{0}, L_{0}) [1 - \cos (\frac{L κ^{2}}{k} ξ (1 - ξ))] d κ d ξ \\ = h_{s p} (E_{1}, H_{1}) + \frac{a_{1}}{κ_{l}} h_{s p} (E_{2}, H_{1}) - \frac{b_{1}}{κ_{l}^{β}} h_{s p} (E_{3}, H_{1}) \\ - h_{s p} (E_{1}, H_{2}) - \frac{a_{1}}{κ_{l}} h_{s p} (E_{2}, H_{2}) + \frac{b_{1}}{κ_{l}^{β}} h_{s p} (E_{3}, H_{2}) \end{array}

(15)

\begin{array}{l} h_{s p} (E_{i}, H_{j}) = \frac{- 4 π^{2} A (α)}{1.23} {\tilde{σ}}_{R}^{2} (E_{i}) Γ (- \frac{E_{i}}{2} + 1) H_{j}^{- \frac{E_{i}}{2} + 1} \\ \times Re [{}_{2}F_{1} (- \frac{E_{i}}{2} + 1, 1; \frac{3}{2}; - i \frac{H_{j}}{4}) - 1] \\ = \frac{- 4 π^{2} A (α)}{1.23} {\tilde{σ}}_{R}^{2} (E_{i}) Γ (- \frac{E_{i}}{2} + 1) H_{j}^{- \frac{E_{i}}{2} + 1} \\ \times [{}_{3}F_{2} (1, \frac{2 - E_{i}}{4}, \frac{4 - E_{i}}{4}; \frac{3}{4}, \frac{5}{4}; - \frac{H_{j}^{2}}{16}) - 1] \end{array}

(16)

4. Optical communication link performance statistics

A point-to-point FSO communication system using intensity modulation/direct detection (IM/DD) scheme is considered [2

]. The partially coherent laser beam propagates horizontally through a non-Kolmogorov turbulence channel with additive white Gaussian noise (AWGN). The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fading statistics. We also consider that the channel state information (CSI) is available at both the transmitter and the receiver. In this case, the statistical channel model is given by

y=ηIx+n where ηI is the instantaneous intensity gain, η is the effective photo-current conversion ratio at the receiver, I is the irradiance, x is the modulation signal taking values 0 or 1 and n is the AWGN with zero mean and variance N ₀/2. For weak-to-moderate atmospheric fluctuation conditions, the turbulence-induced fading is assumed to be a random process following the log-normal distribution.

4.1 Outage probability

The probability density function (PDF) of the log-normal model is given by [5

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

]

p_{I} (I) = \frac{1}{I σ_{I} \sqrt{2 π}} \exp {- \frac{{[\ln (I) + σ_{I}^{2} / 2]}^{2}}{2 σ_{I}^{2}}}, I > 0

(17a)

By defining the instantaneous electrical signal-to-noise ratio (SNR) as μ=(ηI)²/N ₀, the average electrical SNR will be given by

\bar{μ} = η {(E [I])}^{2} / N_{0}

, where

E [\cdot]

denotes the expectation. Then, considering that

E [I] = 1

since I is normalized to unity, and after a power transformation of the RV I in above model, the electrical SNR PDF can be rewritten as:

p_{μ} (μ) = \frac{1}{2 μ σ_{I} \sqrt{2 π}} \exp {- \frac{{[\ln (μ / \bar{μ}) + σ_{I}^{2}]}^{2}}{8 σ_{I}^{2}}}, I > 0

(17b)

The outage probability represents the probability that the instantaneous SNR falls below a critical threshold μ _th, which corresponds to sensitivity limit of the receiver. Thus, the outage probability for weak-to-moderate fluctuation regime is obtained from the log-normal distribution model, given by [2

]

P_{o u t} = \Pr (μ \leq μ_{t h}) = \int_{0}^{μ_{t h}} p_{μ} (μ) d μ = \frac{1}{2} erfc [\frac{\ln (\bar{μ} / μ_{t h}) - σ_{I}^{2}}{2 \sqrt{2} σ_{I}}]

(18)

where erfc(x) is the complementary error function.

4.2 Mean Bit-Error-Rate (BER) performance

In the presence of optical turbulence, the probability of error is regarded as a conditional probability averaged over the PDF of the random signal to determine the mean BER, i.e [5

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).

BER = \frac{1}{2} \int_{0}^{\infty} p_{I} (I) e r f c (\frac{I 〈 SNR 〉}{2 \sqrt{2} 〈 I 〉}) d I = \frac{1}{2} \int_{0}^{\infty} p_{μ} (μ) e r f c (\frac{u 〈 SNR 〉}{2 \sqrt{2}}) d u

(19)

4.3 Average channel capacity

The average (ergodic) capacity denotes the practically achievable capacity of an FSO channel with atmospheric turbulence-induced fading and is a paramount metric for evaluating the linkperformance [2

]. The average achievable capacity can be defined as

〈 C 〉 = \int_{0}^{\infty} B \log_{2} (1 + \frac{{(η I)}^{2}}{N_{0}}) p_{I} (I) d I

(20)

where B is the signal transmission bandwidth.

Substituting Eq. (13) into Eq. (16), it yields [2

]

\begin{array}{l} 〈 C 〉 = \frac{B}{2 σ_{I} \sqrt{2 π} \ln (2)} \int_{0}^{\infty} \frac{\ln (1 + μ)}{μ} \exp (- \frac{{(\ln μ - A)}^{2}}{8 σ_{I}^{2}}) d μ \\ = \frac{B \exp (- A^{2} / 8 σ_{I}^{2})}{2 \ln (2)} \sum_{m = 1}^{\infty} \frac{{(- 1)}^{m + 1}}{m} [erfcx (\sqrt{2} σ_{I} m + \frac{A}{2 \sqrt{2} σ_{I}}) + erfcx (\sqrt{2} σ_{I} m - \frac{A}{2 \sqrt{2} σ_{I}})] \\ + \frac{B \exp (- A^{2} / 8 σ_{I}^{2})}{2 \ln (2)} [\frac{4 σ_{I}}{\sqrt{2 π}} + A \exp (\frac{A^{2}}{8 σ_{I}^{2}}) erfc (- \frac{A}{2 \sqrt{2} σ_{I}})] \end{array}

(21)

where

A = \ln (\bar{μ}) - σ_{I}^{2}

and

e r f c x (x) = \exp (x^{2}) e r f c (x)

5. Numerical results and discussion

Using the above mathematical expressions for scintillation index [Eqs. (5), (8) and (11)], the outage probability [Eq. (18)], the mean BER [Eq. (19)] and the average capacity [Eq. (21)], we now investigate the irradiance fluctuation, the reliability and performance of FSO links over non-Kolmogorov weak fluctuation channel by employing practical link parameters. The parameters under consideration that affects the system performance are the link length L, spectral index alpha, turbulence inner and outer scales. The transmitted beam is taken to be collimated partially coherent Gaussian beams with parameters of λ=1.55μm, Θ₀=1, W ₀=2.5cm, l _c=0.02m; the coefficients of the spectrum are set to be a ₁=1.802, b ₁=0.254, and β=7/6; and the turbulence strength is

{\tilde{C}}_{n}^{2} = 1 \times 10^{- 15} m^{3 - α}

. In the results presented below, only the on-axis scintillation is considered, however, it is noticeable that employing Eqs. (5), (8) and (11), the off-axis scintillation can be easily evaluated under weak fluctuation regime.

In Fig. 1 , the SI is plotted against turbulence inner scale for different spectral indices and propagation distance. The propagation distance corresponding to Fig. 1(a)–1(d) is 0.2, 1, 2, 4km, respectively. The turbulence outer scale is chosen to be 1m. For L=4km,

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

and λ=1.55μm, Rytov variance

σ_{R}^{2} < 1

and it satisfies the condition of weak fluctuations. It can be seen from Fig. 1(a) that with the increment of inner scale, the SI increases first, and then decreases slowly for shorter propagation distance. This phenomenon is obvious when the spectral index is equal to 10/3. The current case corresponds to 200m. For longer propagation distance (longer than 1 km and inner scale is smaller than Fresnel size), with the increment of inner scale, the SI increases slowly first, and then decreases slightly. When the width of a Fresnel zone is much less than l ₀ (

\sqrt{λ L} < < l_{0}

), geometrical optics holds and the scintillations are dominated by the inner scale of turbulence l ₀. As the path length increases,

\sqrt{λ L}

increases (

\sqrt{λ L} > l_{0}

), diffraction effects are important and the scintillations are dominated by the scale size of a Fresnel zone. When α = 10/3, the corresponding SI is maximal, and when α=3.9, the SI is minimal. Scintillation is caused primarily by small-scale inhomogeneities. The interpretation for this phenomenon is that when alpha tends to 4, the wavefront tilt aberration plays a dominant role, thus the SI is relatively small. Inner scale has weak influence on scintillations in weak fluctuation regime.

Fig. 1 Scintillation index as a function of inner scale for different link lengths and spectral indices.

The SI is depicted against spectral index α for different values of link length in Fig. 2 . In Fig. 2(a), path-length is L=200m, the size of Fresnel zone

\sqrt{λ L}

is 1.761cm; while in Fig. 2(b), path-length is L=4km, the size of Fresnel zone

\sqrt{λ L}

is 7.874cm. We deduce from Fig. 2 that for alpha values larger than Kolmogorov value α=11/3, there is a decrease of scintillation. There is a maximum value of scintillation for each specific link for a partially coherent Gaussian beam. It can be also deduced that when alpha value is fixed, the SI increases with the increment of inner scale. The explanation is that the sizes of Fresnel zone are larger than the inner scale and larger inner scale leads to higher scintillations in weak fluctuation regime (

l_{0} < \sqrt{λ L}

). For shorter propagation distance (L=200m), SI obtains maximum value when α=3.2; while for longer propagation distance (L=4km), SI has maximum value when α=3.27. The alpha value corresponding to maximum value of SI is unchanged for different inner scale values. For alpha values higher than α=11/3, or close to 3, the scintillation index decreases, and therefore it will lead to a gain in the performance of FSO communication systems. The physical interpretation of alpha approaching 3 is that turbulence tends to vanish and the explanation for alpha approaching 4 is that the power spectrum contains fewer eddies of high wave numbers, i.e. the wavefront tilt is the primary aberration.

Fig. 2 Scintillation index as a function of alpha for different link lengths and inner scales.

Figures 3 and 4 illustrate the impact of the variations of spectral index and inner scale on the outage probability, respectively. It can be seen from Fig. 3 that, when α=10/3, the outage probability is maximal; while the outage probability is minimal when α=3.9. From Fig. 4, we can deduce that the outage probability increases slightly with the increment of inner scale and the effects of inner scale become a little evident for larger normalized average SNR. The effects of outer scale on scintillations within the beam cross-section are negligible.

Fig. 3 Outage probability as a function of normalized average SNR for different spectral indices.

Fig. 4 Outage probability as a function of normalized average SNR for different spectral indices and inner scales.

Figures 5 and 6 illustrate the influence of the variations of spectral index and inner scale on the mean BER, respectively. It can be seen from Fig. 5 that, when α=10/3, the BER is maximal; while the BER is minimal when α=3.9. From Fig. 6, we can deduce that the BER increases slightly with the increment of inner scale and the effects of inner scale become a little evident for larger SNR. The effect of outer scale on the BER is negligible. Following Fig. 2, larger inner scale leads to higher scintillation in weak fluctuation regime under the condition of

l_{0} < \sqrt{λ L}

, therefore, larger inner scale leads to higher outage probability and BER.

Fig. 5 Mean BER as a function of SNR for different alpha values.

Fig. 6 Mean BER as a function of SNR for different spectral indices and inner scale values.

In addition, different spectral indices have slight influence on average channel capacity, especially for larger SNR. The effect of inner scale on the average channel capacity is negligible in weak fluctuation regime. The explanation is that inner scale has weak influence on scintillations in weak fluctuation regime and average channel capacity is insensitive to little variation of scintillations in weak fluctuation regime.

6. Conclusions

In this paper, by employing the generalized atmospheric spectral model, we have derived analytical expressions of the scintillation index for plane wave, spherical wave and a partially coherent Gaussian beam propagating horizontally through non-Kolmogorov turbulence in weak fluctuation regime. Using the log-normal distribution model for weak-to-moderate fluctuation conditions, we studied the dependence of the reliability of FSO links on the atmospheric condition and link length. For horizontal links, it is shown that with the increment of inner scale, the SI increases slightly for relatively longer links in weak fluctuation regime. The alpha value corresponding to maximum value of SI is invariant for different values of inner scale.

References and links

1.	A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Res. 2(4), 345–396 (2005). [CrossRef]
2.	H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009). [CrossRef]
3.	H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average capacity of optical wireless communication systems over atmospheric turbulence channels,” J. Lightwave Technol. 27(8), 974–979 (2009). [CrossRef]
4.	L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002). [CrossRef]
5.	L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).
6.	O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43(2), 330–341 (2004). [CrossRef]
7.	C. Y. Chen, H. M. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009). [CrossRef] [PubMed]
8.	D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express 18(20), 20746–20758 (2010). [CrossRef] [PubMed]
9.	G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]
10.	V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Trans. NOAA by Israel Program for Scientific Translations, 1971).
11.	D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]
12.	B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]
13.	M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]
14.	A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]
15.	N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]
16.	I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]
17.	L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]
18.	I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6457, 64570T (2007). [CrossRef]
19.	I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007). [CrossRef]
20.	I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009). [CrossRef]
21.	L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010). [CrossRef] [PubMed]
22.	A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010). [CrossRef]
23.	B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011). [CrossRef] [PubMed]
24.	B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011). [CrossRef] [PubMed]
25.	W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]
26.	L. C. Andrews, Special Functions of Mathematics for Engineers , 2nd ed. (SPIE Optical Engineering Press, 1998).

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 30, 2011
Revised Manuscript: August 12, 2011
Manuscript Accepted: September 2, 2011
Published: September 15, 2011

Citation
Ji Cang and Xu Liu, "Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model," Opt. Express 19, 19067-19077 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19067

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References

A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Res.2(4), 345–396 (2005). [CrossRef]
H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun.3(8), 1402–1409 (2009). [CrossRef]
H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average capacity of optical wireless communication systems over atmospheric turbulence channels,” J. Lightwave Technol.27(8), 974–979 (2009). [CrossRef]
L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE4489, 23–34 (2002). [CrossRef]
L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005).
O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng.43(2), 330–341 (2004). [CrossRef]
C. Y. Chen, H. M. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett.34(4), 419–421 (2009). [CrossRef] [PubMed]
D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express18(20), 20746–20758 (2010). [CrossRef] [PubMed]
G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett.35(5), 715–717 (2010). [CrossRef] [PubMed]
V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Trans. NOAA by Israel Program for Scientific Translations, 1971).
D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE2120, 43–55 (1994). [CrossRef]
B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE2471, 181–196 (1995). [CrossRef]
M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE3126, 113–123 (1997). [CrossRef]
A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt.47(34), 6385–6391 (2008). [CrossRef] [PubMed]
N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE7588, 758808 (2010). [CrossRef]
I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6551, 65510E (2007). [CrossRef]
L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express18(20), 21269–21283 (2010). [CrossRef] [PubMed]
I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6457, 64570T (2007). [CrossRef]
I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE6747, 67470B (2007). [CrossRef]
I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag.57(6), 1783–1788 (2009). [CrossRef]
L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express18(2), 451–462 (2010). [CrossRef] [PubMed]
A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun.283(7), 1229–1235 (2010). [CrossRef]
B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express19(9), 8433–8443 (2011). [CrossRef] [PubMed]
B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A28(5), 912–916 (2011). [CrossRef] [PubMed]
W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A11(10), 2719–2726 (1994). [CrossRef]
L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

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