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 [1,2]. Their performance depends largely on the laser source property and the atmospheric conditions between the transmitter and the receiver [3–5]. 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) [6–9]. Traditionally, the performance of FSO communication links are estimated with the assumption that atmospheric turbulence belongs to the Kolmogorov type [10]. 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 [11–14]. 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 [15–17]. 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 [18–20]. Tan et. al examined the log-amplitude variance for a Gaussian-beam propagating through non-Kolmogorov weak turbulence [21]. 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]. Xue et. al studied the angle of arrival variance based on generalized atmospheric turbulence model for wave propagating through non-Kolmogorov turbulence [23]. 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] 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]:

3. Variance of irradiance fluctuations

In the weak fluctuation regime, where the log-amplitude variance satisfies${\sigma }_{\chi }^2\ll 1$, the normalized irradiance variance has the relation ${\sigma }_I^2\approx {\sigma }_{\ln I}^2=4{\sigma }_{\chi }^2$.

Following the approach in Ref [5], the irradiance variance at the receiver plane can be expressed as the sum:

In the above formulae, ${\Theta }_{ed}=\frac{{\Theta }_1}{1+4q_c{\Lambda }_1}$ and ${\Lambda }_{ed}=\frac{{\Lambda }_1{N}_s}{1+4q_c{\Lambda }_1}$ are the effective beam parameters at the receiver plane. ${\overline{\Theta }}_{ed}=1-{\Theta }_{ed}$ 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 ${\Theta }_1=\frac{{\Theta }_0}{{\Theta }_0^2+{\Lambda }_0^2}$ and ${\Lambda }_1=\frac{{\Lambda }_0}{{\Theta }_0^2+{\Lambda }_0^2}$. The source plane parameters are ${\Theta }_0=1-\frac{L}{F_0}$ and ${\Lambda }_0=\frac{2L}{k{W}_0^2}$, 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. $N_s=1+\frac{4q_c}{{\Lambda }_0}$ is the number of speckle cells; for fully coherent beam, ${\Theta }_{ed}={\Theta }_1$ and ${\Lambda }_{ed}={\Lambda }_1$.

In the above formulae, the radial component${\sigma }_{I,r}^2(\rho ,L)$ 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 ${\sigma }_{I,l}^2(L)$ 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

With the application of the following two formulae [25], ${}_{2}F{}_1\left(-n,n+1;n+2;x\right)\approx {(1-2x/3)}^n,\left|x\right|<1$; ${}_{2}F{}_1\left(1-b,1;2;-x\right)=\frac{{(1+x)}^b-1}{bx}$ the approximate analytic expression can be derived by [25]

The above expression is valid for all H _j. where $E_1=\alpha ,E_2=\alpha -1,E_3=\alpha -\beta ,$ $H_1=Q_l,H_2=\frac{Q_l{Q}_0}{Q_0+Q_l}$; and $Q_l=\frac{L}{k}{\kappa }_l^2$, $Q_0=\frac{L}{k}{\kappa }_0^2$; ${(a)}_n=\Gamma (a+n)/\Gamma (a)$ is Pochhammer symbol, $\Gamma (\cdot )$ is the Gamma function and ${}_{2}F{}_1\left(a,b;c;x\right)$ denotes the hypergeometric function [26].

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

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

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

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]

By defining the instantaneous electrical signal-to-noise ratio (SNR) as μ=(ηI)²/N ₀, the average electrical SNR will be given by $\overline{\mu }=\eta {\left(E[I]\right)}^2/N_0$, 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:

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]

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].

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

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

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 ${\tilde{C}}_n^2=1\times {10}^{-15}{\text{m}}^{3-\alpha }$. 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, ${\tilde{C}}_n^2={10}^{-15}{\text{m}}^{3-\alpha }$and λ=1.55μm, Rytov variance ${\sigma }_R^2<1$ 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 ($\sqrt{\lambda L}<<l_0$), geometrical optics holds and the scintillations are dominated by the inner scale of turbulence l ₀. As the path length increases, $\sqrt{\lambda L}$ increases ($\sqrt{\lambda L}>l_0$), 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.

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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 $\sqrt{\lambda L}$ is 1.761cm; while in Fig. 2(b), path-length is L=4km, the size of Fresnel zone $\sqrt{\lambda L}$ 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 ($l_0<\sqrt{\lambda L}$). 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.

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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.

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

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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 $l_0<\sqrt{\lambda L}$, therefore, larger inner scale leads to higher outage probability and BER.

 

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

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

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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.