Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence

Chunyi Chen; Huamin Yang; Zhou Zhou; Weizhi Zhang; Mohsen Kavehrad; Shoufeng Tong; Tianshu Wang

doi:10.1364/OE.21.029731

1. Introduction

Recently, free-space optical (FSO) communications have been considered as a promising candidate for low-cost broadband access applications. However, the turbulence-induced beam scintillations can severely degrade the performance of FSO links. It has been shown that the use of a spatially partially coherent source can reduce the scintillations in FSO links through atmospheric turbulence [1,2]; on the other hand, it has also been revealed that spatial partial coherence of the source causes additional beam spreading [3], which results in an additional decrease in the average irradiance of the beam at the receiver plane. As a result, optimization for the transverse correlation length of the source is needed and has been addressed by many authors [4–6]. The temporal fade statistics of the received irradiance, e.g., the average fade frequency and average fade duration, are critical to analyze a packet FSO communication system, using forward error correction (FEC), impaired by atmospheric turbulence [7]. Thus it is interesting to understand how spatial partial coherence of the source impacts the temporal characteristics of irradiance fading of the produced beam passing through atmospheric turbulence.

Up to now, although both the average irradiance and scintillation index of beams, radiated from spatially partially coherent sources, propagating in atmospheric turbulence have been investigated in detail [1,3,8–10], the temporal characteristics of irradiance fading of these spatially partially coherent beams are rather unexplored. The Taylor frozen-turbulence hypothesis [10] has been widely used to study the temporal behavior of irradiance fluctuations induced by atmospheric turbulence. Yura and McKinley [11] developed the temporal statistics of the turbulence-induced irradiance fluctuations in a ground-to-space FSO link. The temporal spectra of irradiance fluctuations for plane, spherical and Gaussian-beam waves in weak turbulence and for plane waves in weak-to-strong turbulence, respectively, were derived by Andrews and Phillips [10]. By taking a spatially incoherent optical field into account, Holmes et al. [12] studied the temporal spectrum of irradiance fluctuations for incoherent speckle propagation through atmospheric turbulence. Nevertheless, all these authors did not consider the case of beam waves radiated from spatially partially coherent sources. On the other hand, Xiao and Voelz [13] and Korotkova et al. [14] treated the probability density function (PDF) of irradiance fluctuations in FSO links transmitting a partially coherent beam though atmospheric turbulence and analyzed the probability of fade, described by the integral of the PDF from 0 to the threshold value, whereas their work did not involve the temporal fade statistics such as the average fade frequency and average fade duration, which depend on the root-mean-square (RMS) bandwidth of the irradiance temporal spectrum. Furthermore, practical FSO communication systems always employ receivers with a finite aperture size, and aperture averaging occurs if the aperture size in the presence of atmospheric turbulence is larger than the irradiance correlation width; in this case, it is the irradiance flux that should be taken into account rather than the irradiance. By considering a plane wave propagating in atmospheric turbulence, Andrews and Phillips [10] have shown that aperture averaging has an impact on the irradiance-flux temporal spectrum. Hence, the receiver aperture size should be taken into account when one examines the temporal fade statistics of the received light signal in a FSO link, employing a spatially partially coherent source, through atmospheric turbulence.

The purpose of this paper is to gain a deep insight into the effects of source spatial partial coherence on the temporal characteristics of irradiance-flux fading in FSO links through atmospheric turbulence. By considering a beam radiated from a spatially partially coherent source, propagating in atmospheric turbulence, and received by a finite-size aperture, we first develop the temporal covariance function of irradiance-flux fluctuations, and then derive the expressions for the RMS bandwidth of the irradiance-flux temporal spectrum. With the help of these expressions, the temporal fade statistics of the irradiance flux are calculated and analyzed for beams, radiated from sources with different transverse correlation lengths, propagating in atmospheric turbulence with various strengths.

2. Theoretical formulations

Let us consider a FSO link consisting of a transmitter, employing a spatially partially coherent source from which a beam is radiated and propagated horizontally into atmospheric turbulence, and a receiver situated at a distance L from the transmitter, which uses a Gaussian lens to collect the light and focus it onto a photodetector. For mathematical tractability, here we model the beam radiated from the spatially partially coherent source as a Gaussian Schell-model (GSM) beam ([15], Sec. 5.6.4).

2.1. Temporal covariance function of irradiance-flux fluctuations

Here we develop the temporal covariance function of irradiance-flux fluctuations for GSM beams propagating in atmospheric turbulence based on the method of effective beam parameters ([16], Sec. 5.2.1). In what follows, we begin by deriving the temporal covariance function of irradiance-flux fluctuations for spatially fully coherent Gaussian beams propagating in atmospheric turbulence, and then extend it to the case of GSM beams.

Based on the derived statistical moments for beam waves propagating through complex paraxial ABCD optical systems in the presence of atmospheric turbulence ([10], p. 401, Eqs. (20) and (21)), recalling the modified Rytov theory ([16], Sec. 2.3.1) and the Taylor frozen-turbulence hypothesis for Gaussian-beam waves ([10], Sec. 8.5.3), the temporal covariance function of on-axis irradiance-flux fluctuations at the photodetector plane for a spatially fully coherent Gaussian beam can be written as

B_{I} (τ) = exp [B_{ln X} (τ) + B_{ln Y} (τ)] - 1

with the large- and small-scale log-irradiance-flux temporal covariance given by

\begin{array}{l} B_{ln X} (τ) & = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) G_{X} (κ) J_{0} (κ v τ) \\ \times exp {- \frac{L κ^{2}}{k (Λ_{1} + Ω_{G})} \cdot [{(1 - {\bar{Θ}}_{1} ξ)}^{2} + Λ_{1} Ω_{G} ξ^{2}]} \\ \times {1 - cos [\frac{L κ^{2}}{k} \cdot \frac{Ω_{G} - Λ_{1}}{Ω_{G} + Λ_{1}} \cdot ξ (1 - {\bar{Θ}}_{1} ξ)]} d κ d ξ \end{array}

and

\begin{array}{l} B_{ln Y} (τ) & = 8 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) G_{Y} (κ) J_{0} (κ v τ) \\ \times exp {- \frac{L κ^{2}}{k (Λ_{1} + Ω_{G})} \cdot [{(1 - {\bar{Θ}}_{1} ξ)}^{2} + Λ_{1} Ω_{G} ξ^{2}]} \\ \times {1 - cos [\frac{L κ^{2}}{k} \cdot \frac{Ω_{G} - Λ_{1}}{Ω_{G} + Λ_{1}} \cdot ξ (1 - {\bar{Θ}}_{1} ξ)]} d κ d ξ, \end{array}

respectively, where k = 2π/λ denotes the wavenumber, λ is the wavelength; L is the separation distance between the transmitter and receiver; J₀ (·) is a Bessel function of the first kind; v is the average transverse wind velocity;

Λ_{1} = Λ_{0} / (Λ_{0}^{2} + Θ_{0}^{2})

, Θ̄₁ = 1 − Θ₁,

Θ_{1} = Θ_{0} / (Λ_{0}^{2} + Θ_{0}^{2})

,

Λ_{0} = 2 L / (k W_{0}^{2})

, Θ₀ = 1 − L/F₀; W₀ and F₀ are the beam radius and phase front radius of curvature at the transmitter plane, respectively;

Ω_{G} = 2 L / (k W_{G}^{2})

is the lens Fresnel parameter, W_G is the effective transmission radius of the Gaussian lens which relates to the hard aperture diameter D_G by

D_{G}^{2} = 8 W_{G}^{2}

([10], p. 408); Φ_n(κ) is the spatial power spectrum of refractive-index fluctuations;

G_{X} (κ) = exp (- κ^{2} / κ_{X}^{2})

and

G_{Y} (κ) = κ^{11 / 3} exp (Λ_{1} L κ^{2} ξ^{2} / k) / {(κ^{2} + κ_{Y}^{2})}^{11 / 6}

are the large- and small-scale filter function, respectively ([10], pp. 333 and 352); κ_X and κ_Y are the large- and small-scale cutoff spatial frequency, respectively.

By introducing the Kolmogorov spectrum $Φ_{n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3}$ with $C_{n}^{2}$ being the refractive-index structure constant into Eq. (2), and employing the geometrical optics approximation, one obtains

\begin{array}{l} B_{ln X} (τ) & ≅ 0.53 σ_{R}^{2} {(\frac{Ω_{G} - Λ_{1}}{Ω_{G} + Λ_{1}})}^{2} \int_{0}^{1} ξ^{2} {(1 - {\bar{Θ}}_{1} ξ)}^{2} \int_{0}^{\infty} η^{1 / 6} J_{0} (ω_{t} τ \sqrt{η}) \\ \times exp (- \frac{η}{η_{X}}) exp {- \frac{η}{Λ_{1} + Ω_{G}} \cdot [{(1 - {\bar{Θ}}_{1} ξ)}^{2} + Λ_{1} Ω_{G} ξ^{2}]} d η d ξ, \end{array}

where

σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6}

denotes the Rytov variance for a plane wave, ω_t = v(k/L)^1/2 is the Fresnel frequency, η = Lκ²/k, and

η_{X} = L κ_{X}^{2} / k

. Note that, Eq. (4) reduces to Eq. (60) in Sec. 6.5.1 of [16] by letting τ = 0. By first performing the variable change η = x², and then making use of the identity ([10], p. 764)

\int_{0}^{\infty} x^{u} exp (- a^{2} x^{2}) J_{p} (b x) d x = \frac{b^{p} Γ (\frac{p + u + 1}{2})}{2^{p + 1} a^{p + u + 1} Γ (p + 1)} \cdot {}_{1}{F_{1}} (\frac{p + u + 1}{2}; p + 1; - \frac{b^{2}}{4 a^{2}}),

where Re(u + p) > −1, a > 0, b > 0, and ₁F₁ (·) is a confluent hypergeometric function of the first kind, to evaluate the inside integration in Eq. (4), it follows that

B_{ln X} (τ) = 0.49 σ_{R}^{2} {(\frac{Ω_{G} - Λ_{1}}{Ω_{G} + Λ_{1}})}^{2} \int_{0}^{1} ξ^{2} {(1 - {\bar{Θ}}_{1} ξ)}^{2} R^{7 / 6} (ξ) \cdot {}_{1}{F_{1}} [\frac{7}{6}; 1; - \frac{ω_{t}^{2} τ^{2} R (ξ)}{4}] d ξ

with

R (ξ) = η_{X} {1 + \frac{η_{X}}{Λ_{1} + Ω_{G}} [{(1 - {\bar{Θ}}_{1} ξ)}^{2} + Λ_{1} Ω_{G} ξ^{2}]}^{- 1} .

Along a similar line presented in Sec. 6.5.1 of [16], substituting the Kolmogorov spectrum into Eq. (3) results in

B_{ln Y} (τ) ≅ 1.06 σ_{R}^{2} \int_{0}^{1} \int_{0}^{\infty} \frac{J_{0} (ω_{t} τ \sqrt{η})}{{(η + η_{Y})}^{11 / 6}} exp {- \frac{{(1 - {\bar{Θ}}_{1} ξ)}^{2} η}{Λ_{1} + Ω_{G}}} d η d ξ,

where

η_{Y} = L κ_{Y}^{2} / k

. Notice that, Eq. (8) reduces to the first line of Eq. (63) in Sec. 6.5.1 of [16] if τ = 0. By evaluating the integration over ξ in Eq. (8) and performing the variable change η = x², one finds

\begin{array}{l} B_{ln Y} (τ) & = 1.06 π^{1 / 2} σ_{R}^{2} \frac{\sqrt{Λ_{1} + Ω_{G}}}{{\bar{Θ}}_{1}} \int_{0}^{\infty} \frac{J_{0} (ω_{t} τ x)}{{(x^{2} + η_{Y})}^{11 / 6}} \\ \times [erf (\frac{x}{\sqrt{Λ_{1} + Ω_{G}}}) + erf (\frac{x ({\bar{Θ}}_{1} - 1)}{\sqrt{Λ_{1} + Ω_{G}}})] d x, \end{array}

where erf(·) is the error function.

Equations (6) and (9) are our first theoretical contribution. By employing the aforementioned method of effective beam parameters, the large- and small-scale log-irradiance-flux temporal covariance for GSM beams can be obtained by formally replacing the conventional beam parameters Λ₁ and Θ̄₁ in Eqs. (6) and (9) with the effective beam parameters Λ_e and Θ̄_e, which are given by [1]

Λ_{e} = \frac{Λ_{1} N_{S}}{1 + 4 Λ_{1} q_{c}}, {\bar{Θ}}_{e} = \frac{{\bar{Θ}}_{1} + 4 Λ_{1} q_{c}}{1 + 4 Λ_{1} q_{c}},

where N_S = 1 + 4q_c/Λ₀,

q_{c} = L / (2 k σ_{c}^{2})

with σ_c being the transverse correlation length of the source. Although Λ_e and Θ̄_e have been developed based on a random phase screen model, it has been proven that the random phase screen model is equivalent to the GSM ([10], p. 675). For GSM beams, η_X and η_Y introduced in Eqs. (4) and (8) are given by ([10], pp. 351, 352, 682)

η_{X} = {(\frac{1}{3} - \frac{{\bar{Θ}}_{e}}{2} + \frac{{\bar{Θ}}_{e}^{2}}{5})}^{- 6 / 7} {(\frac{σ_{B}}{σ_{R}})}^{12 / 7} {[1 + 0.56 (2 - {\bar{Θ}}_{e}) σ_{B}^{12 / 5}]}^{- 1}

and

η_{Y} = 3 {(\frac{σ_{R}}{σ_{B}})}^{12 / 5} + 2.07 σ_{R}^{12 / 5},

respectively, with

σ_{B}^{2} = 3.86 σ_{R}^{2} Re [i^{5 / 6} {}_{2}{F_{1}} (- \frac{5}{6}, \frac{11}{6}, \frac{17}{6}; {\bar{Θ}}_{e} + i Λ_{e}) - \frac{11}{16} Λ_{e}^{5 / 6}],

where ₂F₁(·) is a hypergeometric function.

2.2. RMS bandwidth of irradiance-flux temporal spectrum

Although the dynamics of irradiance-flux fluctuations are completely described by their temporal spectra, generally it is the RMS bandwidth of these temporal spectra that attracts the most attention in FSO communication applications due to its importance in determining the quantities such as the average fade frequency and average fade duration of the received light signal [7,10,11].

The RMS bandwidth of the irradiance-flux temporal spectrum is defined by [7]

B_{RMS} = \frac{1}{2 π} {[\frac{\int_{0}^{\infty} ω^{2} S_{I} (ω) d ω}{\int_{0}^{\infty} S_{I} (ω) d ω}]}^{1 / 2},

where ω denotes the temporal angular frequency, and S_I(·) represents the irradiance-flux temporal spectrum. Theoretically, S_I(·) can be obtained by taking the Fourier transform of B_I(·). Nevertheless, this approach is not very tractable in our case. Thus, an alternative definition of the RMS bandwidth of the irradiance-flux temporal spectrum is employed to develop an expression for B_RMS, viz., [7]

B_{RMS} = \frac{1}{2 π} {[- \frac{B_{I}^{″} (0)}{B_{I} (0)}]}^{1 / 2}

with B″_I(·) being the second derivative of B_I(·), which takes the form

B_{I}^{″} (τ) = exp [B_{ln X} (τ) + B_{ln Y} (τ)] {{[B_{ln X}^{'} (τ) + B_{ln Y}^{'} (τ)]}^{2} + B_{ln X}^{″} (τ) + B_{ln Y}^{″} (τ)},

where B′_ln_X (·) and B′_ln_Y (·) represent the first derivatives of B_ln_X (·) and B_ln_Y (·), respectively; B″_ln_X (·) and B″_ln_Y (·) denote the second derivatives of B_ln_X (·) and B_ln_Y (·), respectively. It needs to be pointed out that the RMS bandwidth of the irradiance-flux temporal spectrum is also referred to in the literature [10,11] as the quasi-frequency. By making use of the Leibniz rule [17,18], one finds B′_ln_X (0) = 0, B′_ln_Y (0) = 0,

B_{ln X}^{″} (0) = C_{1} ω_{t}^{2}

, and

B_{ln Y}^{″} (0) = C_{2} ω_{t}^{2}

with

C_{1} = - 0.29 σ_{R}^{2} {(\frac{Ω_{G} - Λ_{e}}{Ω_{G} + Λ_{e}})}^{2} \int_{0}^{1} ξ^{2} {(1 - {\bar{Θ}}_{e} ξ)}^{2} R_{e}^{13 / 6} (ξ) d ξ,

\begin{array}{l} C_{2} = - 0.53 π^{1 / 2} σ_{R}^{2} \frac{\sqrt{Λ_{e} + Ω_{G}}}{{\bar{Θ}}_{e}} \int_{0}^{\infty} \frac{x^{2}}{{(x^{2} + η_{Y})}^{11 / 6}} \\ \times [erf (\frac{x}{\sqrt{Λ_{e} + Ω_{G}}}) + erf (\frac{x ({\bar{Θ}}_{e} - 1)}{\sqrt{Λ_{e} + Ω_{G}}})] d x, \end{array}

where R_e (·) is obtained by replacing Λ₁ and Θ̄₁ in Eq. (7) with Λ_e and Θ̄_e. Based on the results above, Eq. (15) can be written as

B_{RMS} = \frac{C_{3} ω_{t}}{2 π},

where

C_{3} = {- (C_{1} + C_{2}) [1 + B_{I}^{- 1} (0)]}^{1 / 2}

. It is noted that B_I(0) is actually the irradiance-flux variance for GSM beams, of which a good approximation has been given by [1]

B_{I} (0) = σ_{I}^{2} = exp (σ_{ln X}^{2} + σ_{ln Y}^{2}) - 1

with

σ_{ln X}^{2} ≅ 0.49 σ_{R}^{2} {(\frac{Ω_{G} - Λ_{e}}{Ω_{G} + Λ_{e}})}^{2} (\frac{1}{3} - \frac{{\bar{Θ}}_{e}}{2} + \frac{{\bar{Θ}}_{e}^{2}}{5}) {[\frac{η_{X}}{1 + 0.4 η_{X} (2 - {\bar{Θ}}_{e}) / (Λ_{e} + Ω_{G})}]}^{7 / 6},

σ_{ln Y}^{2} ≅ \frac{1.27 σ_{R}^{2} η_{Y}^{- 5 / 6}}{1 + 0.4 η_{Y} / (Λ_{e} + Ω_{G})} .

Equation (19) as our second theoretical contribution indicates that the RMS bandwidth of the irradiance-flux temporal spectrum is proportional to the Fresnel frequency ω_t with a scaling coefficient which depends on the effective beam parameters, Rytov variance and lens Fresnel parameter. Note that B_RMS is in hertz. It is really difficult to develop an accurate analytical solution for the integral in Eq. (17). Hence, numerical integration is needed to evaluate it. On the contrary, upon employing the Mellin-transform-based method suggested by Sasiela ([19], Chaps. 5 and 6), the integral in Eq. (18) can be evaluated to give

\begin{array}{l} C_{2} & = - 0.53 σ_{R}^{2} {\frac{3 π {(Λ_{e} + Ω_{G})}^{1 / 6}}{{\bar{Θ}}_{e} Γ (5 / 6)} [{}_{2}{F_{2}} (\frac{11}{6}, \frac{1}{3}, \frac{4}{3}, \frac{5}{6}; \frac{η_{Y}}{Λ_{e} + Ω_{G}}) \mp {| 1 - {\bar{Θ}}_{e} |}^{2 / 3} \\ \times {}_{2}{F_{2}} (\frac{11}{6}, \frac{1}{3}; \frac{4}{3}, \frac{5}{6}; \frac{η_{Y} {(1 - {\bar{Θ}}_{e})}^{2}}{Λ_{e} + Ω_{G}})] - \frac{36 η_{Y}^{1 / 6}}{5 {\bar{Θ}}_{e}} [{}_{2}{F_{2}} (2, \frac{1}{2}; \frac{7}{6}, \frac{3}{2}; \frac{η_{Y}}{Λ_{e} + Ω_{G}}) \\ - (1 - {\bar{Θ}}_{e}) \cdot {}_{2}{F_{2}} (2, \frac{1}{2}; \frac{7}{6}, \frac{3}{2}; \frac{η_{Y} {(1 - {\bar{Θ}}_{e})}^{2}}{Λ_{e} + Ω_{G}})]}, \end{array}

where the upper sign corresponds to Θ̄_e ≤ 1 and the lower one to Θ̄_e > 1; ₂F₂ (·) is a generalized hypergeometric function.

2.3. Temporal fade statistics

The temporal fade statistics of the received light signal, such as the fractional fade time, average fade frequency and average fade duration, are of great interest in FSO communication systems using on-off-keying (OOK) modulation combined with direct detection in the presence of atmospheric turbulence. These statistics are used, in this paper, as measures to characterize the temporal characteristics of irradiance-flux fading in FSO links through atmospheric turbulence. Note that, although the turbulence-induced fading of the received light signal is sensed with the help of a photodetector, to avoid the mathematical complications associated with the detection noise ([20], Sec. 7.2.3), here we only concentrate on the irradiance-flux fading caused by atmospheric turbulence.

By assuming that the irradiance-flux fluctuations are an ergodic process, the fractional fade time, defined as the percentage of time when the irradiance flux is below a given threshold I_th, equals the probability of fade given by [21]

p_{fade} (μ_{th}) = \int_{0}^{μ_{th}} p_{I} (μ) d μ,

where p_I(μ) = 2(αβ)⁽^α⁺^β^)/2μ⁽^α⁺^β^)/2−1K_α₋_β [2(αβμ)^1/2]/[Γ(α)Γ(β)] is the PDF of irradiance-flux fluctuations under weak-to-strong turbulence conditions [10,16,21], Γ(·) denotes the gamma function, K_α₋_β(·) represents a modified Bessel function of the second kind,

α = 1 / [exp (σ_{ln X}^{2}) - 1]

,

β = 1 / [exp (σ_{ln Y}^{2}) - 1]

, the normalized irradiance flux μ = I/〈I〉 with I being the irradiance flux and 〈I〉 the average irradiance flux, and the normalized threshold μ_th = I_th/〈I〉. Equation (24) can be written as an analytical expression involving two generalized hypergeometric functions ([10], p. 452). However, numerical errors may occur under certain conditions when this analytical expression is evaluated by current software programs. Hence, in this paper, the integral appearing in Eq. (24) is numerically evaluated. It should be noted that the fractional fade time does not depend on the Fresnel frequency.

The average fade frequency, defined as the expected number per unit time of negative crossings of the irradiance flux below a given threshold I_th, is written by ([10], p. 456)

〈 f_{fade} (μ_{th}) 〉 = \frac{2 \sqrt{2 π α β} σ_{I} B_{RMS}}{Γ (α) Γ (β)} {(α β μ_{th})}^{(α + β - 1) / 2} K_{α - β} (2 \sqrt{α β μ_{th}}) .

The average fade duration characterizes the average time at which the irradiance flux stays below a given threshold I_th and is defined by ([10], p. 456)

〈 t_{fade} (μ_{th}) 〉 = \frac{p_{fade} (μ_{th})}{〈 f_{fade} (μ_{th}) 〉} .

3. Numerical calculations and analysis

In this section, we examine the effects of source spatial partial coherence on the temporal characteristics of irradiance-flux fading in FSO links using a receiver with a finite-size aperture through atmospheric turbulence. The basic parameters used in the calculations are as follows: λ = 1550 nm, W₀ = 2.5 cm, F₀ = ∞, and L = 2 km. To account for weak-to-strong atmospheric turbulence, the Rytov variance $σ_{R}^{2}$ , used as a measure of turbulence strength, is specified as 0.3, 1 and 10, corresponding to weak, moderate and strong atmospheric turbulence, respectively. The parameter W_G is assumed to be 10⁻³, 0.1 and 1 cm. W_G = 10⁻³ cm is used to consider the case of a receiver with a very small aperture size, which nearly behaves like a “point receiver.”

The RMS bandwidth of the irradiance-flux temporal spectrum, i.e., the quasi-frequency, is very important for accurately determining the average fade frequency and average fade duration. Hence, it is worthwhile to first provide some insights into the effects of source spatial partial coherence on the RMS bandwidth of the irradiance-flux temporal spectrum. Figure 1 illustrates the scaled RMS bandwidth B_RMS/ω_t of the irradiance-flux temporal spectrum as a function of the relative transverse correlation length σ_c/W₀ with various combinations of $σ_{R}^{2}$ and W_G. Notice that, the Fresnel frequency ω_t depends on neither $C_{n}^{2}$ nor W_G. As a result, the differences between the curves in Fig. 1 are only caused by the distinctions in their associated parameters concerning the turbulence strength and effective transmission radius of the Gaus-sian lens. It is seen from Fig. 1 that with the same values of $σ_{R}^{2}$ and σ_c/W₀, a greater W_G results in a smaller B_RMS/ω_t. This fact manifests that aperture averaging can reduce the RMS band-width of the irradiance-flux temporal spectrum. It is observed from Fig. 1 that for all curves, although the scaled RMS bandwidth only exhibits very slight variations when σ_c/W₀ < 10⁻¹ or σ_c/W₀ > 10¹, there exists an observable increase in its value as σ_c/W₀ reduces from 10¹ to 10⁻¹. This phenomenon suggests that decreasing the transverse correlation length of the source can lead to an increase in the RMS bandwidth of the irradiance-flux temporal spectrum. Furthermore, it is found from Fig. 1 that in the case of W_G = 10⁻³ or 0.1 cm, with the same value of σ_c/W₀, stronger atmospheric turbulence leads to a larger B_RMS/ω_t regardless of what value of σ_c/W₀ is considered, and in the case of W_G = 1 cm, however, B_RMS/ω_t with a relatively large value of σ_c/W₀, e.g., σ_c/W₀ = 10², decreases as the turbulence strength increases. Hence, the effects of turbulence strength on the RMS bandwidth of the irradiance-flux temporal spectrum depend on both the receiver aperture size and the transverse correlation length of the source.

Fig. 1 RMS bandwidth of the irradiance-flux temporal spectrum, scaled by the Fresnel frequency ω_t, as a function of σ_c/W₀.

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Now we investigate the effects of source spatial partial coherence on the temporal fade statistics of the irradiance flux. According to the analysis above, we choose four typical values of the transverse correlation length as σ_c = 0.05W₀, 0.5W₀, 10W₀ and ∞; the first three values of σ_c correspond to spatially partially coherent sources, and the last one denotes a spatially fully coherent source. Figure 2 shows the fractional fade time in terms of the threshold parameter in decibels, $F_{T} = 10 {log}_{10} (μ_{th}^{- 1})$ , with different combinations of W_G and σ_c. It is found from Fig. 2 that both source spatial partial coherence and aperture averaging can affect the fractional fade time under all conditions of turbulence strength, i.e., a larger receiver aperture size or smaller transverse correlation length leads to a lower fractional fade time. The reason is qualitatively explained as follows. Physically, on the one hand, spatial partial coherence of the source can smear the scintillation pattern at the receiver plane and hence fill many of the fade regions; on the other hand, aperture averaging can average multiple independent intensity patches over the aperture area and thus reduce the probability of fade. Moreover, it is observed from Figs. 2(a)–2(c) that the curves associated with the same W_G and various σ_c become closer to each other as atmospheric turbulence grows stronger. This fact means that the reduction in the fractional fade time due to a decrease in the transverse correlation length of the source becomes less signifi-cant in stronger atmospheric turbulence. Nevertheless, it is illustrated by Fig. 2(c) that even in strong atmospheric turbulence with $σ_{R}^{2} = 10$ , aperture averaging can still reduce the fractional fade time substantially. Finally, it is noted that with the same W_G, an indiscernible difference between the curves for σ_c = 10W₀ and those for σ_c = ∞ is shown by Fig. 2. This implies that as far as our particular numerical example is concerned, for a fixed W_G, a partially coherent source with σ_c = 10W₀ nearly leads to the same fractional fade time as a fully coherent one with σ_c = ∞. Physically, if σ_c/W₀ ≫ 1, the global coherence parameter [22] of the nominal partially coherent source approaches 1, and hence it almost behaves like a fully coherent source.

Fig. 2 Fractional fade time in terms of the threshold parameter in decibels. (a) $σ_{R}^{2} = 0.3$ ; (b) $σ_{R}^{2} = 1$ ; (c) $σ_{R}^{2} = 10$ .

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Figure 3 illustrates the scaled average fade frequency as a function of the threshold parameter F_T with different combinations of W_G and σ_c. Notice that, as in Fig. 1, the average fade frequency in Fig. 3 is scaled by the Fresnel frequency, which is independent of the transverse correlation length, turbulence strength, threshold parameter and effective transmission radius of the Gaussian lens. It is observed from Figs. 3(a) and 3(b) that any two curves associated with the same W_G intersect with each other within the range 0 < F_T < 10. For description convenience, here we refer to the specific threshold level in decibels associated with the aforementioned intersection as the critical threshold level F_c for a pair of curves associated with the same W_G. By comparing each pair of curves in Figs. 3(a) and 3(b) associated with the same W_G and different σ_c, one finds that a smaller σ_c leads to a larger average fade frequency when F_T < F_c and a lower one when F_T > F_c. This fact results in a statement that for two spatially partially coherent sources with different σ_c, if the receiver aperture size is fixed, the source with the smaller σ_c can lead to a lower average fade frequency only when the normalized threshold μ_th is specified as a value smaller than 10^−0.1F_c. It is also found from Fig. 3(c) that any two curves associated with W_G = 10⁻³ cm also intersect with each other within the range 0 < F_T < 10, but those associated with W_G = 1 cm do not. At a first glance, it would appear that the curves associated with W_G = 1 cm in Fig. 3(c) contradict the said statement. However, this seeming contradiction is resolved if we extend the range of the F_T -axis from 10 to a much larger value; namely, when F_T grows further more beyond 10, in the case of W_G = 1 cm, the aforementioned intersection actually occurs too, although this is not displayed in Fig. 3(c). By comparing the curves in Figs. 3(a) – 3(c) associated with different $σ_{R}^{2}$ , it is found that with a fixed receiver aperture size, the critical threshold level F_c associated with a given pair of curves with different σ_c increases as the turbulence strength enhances. Unlike the source spatial partial coherence, one finds from Figs. 3(a) – 3(c) that aperture averaging can always reduce the average fade frequency significantly under all conditions of turbulence strength no matter what value of the threshold parameter F_T is specified. Furthermore, it is interesting to be noted that in the case of strong atmospheric turbulence with $σ_{R}^{2} = 10$ , the dB threshold level at which the average fade frequency reaches its maximum is obviously greater than 0, i.e., the level below which the received irradiance flux most often crosses is smaller than the average irradiance flux. Once again, we note that for a fixed W_G, the curves associated with σ_c = 10W₀ and ∞ almost merge together.

Fig. 3 Average fade frequency, scaled by the Fresnel frequency ω_t, as a function of the threshold parameter in decibels. (a) $σ_{R}^{2} = 0.3$ ; (b) $σ_{R}^{2} = 1$ ; (c) $σ_{R}^{2} = 10$ .

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The scaled average fade duration in terms of the threshold parameter F_T with different combinations of W_G and σ_c is shown in Fig. 4. By comparing the curves in Figs. 4(a) – 4(c), it is found that although the average fade duration reduces monotonously as F_T rises, i.e., as μ_th decreases, under all conditions of turbulence strength, the reduction in the average fade duration with increasing F_T becomes slightly less significant as atmospheric turbulence grows stronger. In addition, it needs to see from Fig. 4 that with the same W_G, a smaller transverse correlation length of the source always leads to a shorter average fade duration. This behavior is different from that of the average fade frequency in Fig. 3, i.e., source spatial partial coherence can reduce the average fade duration no matter what value of the threshold parameter is considered. However, one also finds from Fig. 4(c) that the reduction in the average fade duration caused by decreasing the transverse correlation length becomes very inconsiderable in strong atmospheric turbulence for the relatively small receiver aperture with W_G = 10⁻³ cm. Thus, a statement can be made that source spatial partial coherence becomes nearly ineffective in reducing the average fade duration in strong atmospheric turbulence for a FSO communication system using a relatively small receiver aperture. It needs to be noted from Fig. 4 that a larger receiver aperture size results in a longer average fade duration. This is due to the reason that enlarging the receiver aperture size leads to a more remarkable decrease in the average fade frequency than that in the fractional fade time. Finally, as before, we find from Fig. 4 that with a fixed W_G, the average fade duration for σ_c = 10W₀ can not be clearly differentiated from that for σ_c = ∞.

Fig. 4 Average fade duration, multiplied by the Fresnel frequency ω_t, as a function of the threshold parameter in decibels. (a) $σ_{R}^{2} = 0.3$ ; (b) $σ_{R}^{2} = 1$ ; (c) $σ_{R}^{2} = 10$ .

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4. Conclusions

In this paper, novel expressions for the temporal covariance function of irradiance-flux fluctua-tions of GSM beams propagating in weak-to-strong atmospheric turbulence have been derived by making use of the method of effective beam parameters. According to the obtained expressions, by employing the Leibniz rule, formulae for the RMS bandwidth of the irradiance-flux temporal spectrum due to GSM beams passing through weak-to-strong atmospheric turbulence have been further developed. Based on the achieved theoretical results, the temporal fade statistics of the received light signal in FSO links, using spatially partially coherent sources, through atmospheric turbulence can be easily determined, and hence the effects of source spatial partial coherence on the temporal fade statistics of the received light signal can be examined and analyzed.

It has been found that reducing the transverse correlation length of the source can induce an increase in the RMS bandwidth of the irradiance-flux temporal spectrum. With a fixed receiver aperture size, a smaller transverse correlation length of the source results in a lower fractional fade time and shorter average fade duration; however, when atmospheric turbulence grows strong, source spatial partial coherence becomes less effective in decreasing the fractional fade time for the case of both large and small receiver apertures, and in reducing the average fade duration for the case of small receiver apertures. On the other hand, for a FSO link with a given receiver aperture size, a reduction in the average fade frequency, caused by changing the transverse correlation length of the source from a larger value to a smaller one, can occur only when the threshold parameter in decibels is specified as a value greater than the critical threshold level.

The research work in this paper provides a better understanding of how source spatial partial coherence impacts the temporal fade statistics of the received light signal in FSO communication systems, employing spatially partially coherent sources, in the presence of atmospheric turbulence. Our results are useful in analyzing and designing a packet FSO communication system impaired by atmospheric turbulence.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61007046 and 61275080), the Jilin Provincial Development Programs of Science & Technology of China (No. 201101096) and the China Scholarship Council (No. 201208220008).

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Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence

Abstract

1. Introduction

2. Theoretical formulations

2.1. Temporal covariance function of irradiance-flux fluctuations

2.2. RMS bandwidth of irradiance-flux temporal spectrum

2.3. Temporal fade statistics

3. Numerical calculations and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (26)

Optics Express