# Performance Evaluation of FSO Communications under Sand-Dust Conditions

Authors: Minghua Cao; Huiqin Wang; Yu Yao; Shanglin Hou
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

Sand-dust weather conditions are considered the primary challenge to free-space optical (FSO) communications. It may cause severe attenuation that is malignant to FSO link performance. This study investigates the impact of sand-dust particles on a laser signal using the radiation propagation method and the small-angle approximation method. Numerical simulation shows that in sand-dust weather conditions, the multiple scattering effect is dominant and results in signal pulse delay and pulse broadening. Furthermore, the signal attenuation follows a negative exponential distribution to the laser wavelength. Superior performance can be achieved by employing a longer wavelength laser to reduce pulse delay and mutual interference.


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## Body

## 1. Introduction

Free-space optical communication (FSO) is considered one of the most promising alternatives in metropolitan and local area networks as a viable solution to the access network last mile problem, in which a compromise between the available data rates and the cost is desirable [1, 2]. However, the performance of an FSO system is dependent on atmospheric conditions, particularly in the presence of aerosols (e.g., fog, smoke, sand-dust, and water vapour) which result in scattering and absorption effects [3–5]. Combined with aerosols, sand-dust particles may cause serious transmission interruption. Therefore, we focus on FSO link performance under sand-dust weather conditions to accurately predict the impact of sand-dust particles on FSO transmission.Early researches on laser scattering primarily utilized Lambert–Beer’s law to analyse the single scattering of spherical particles [6–8]. However, it is insufficient to analyse only the single scattering scenario because the corresponding optical thickness increases with increasing transmission distance and atmospheric visibility. To resolve this issue, Islam proposed a simple model for the calculation of signal attenuation induced by sand-dust particles [9]. Wang investigated the influence of atmospheric visibility on pulse delay and pulse broadening through Mie theory and Monte Carlo method [10]. The spherical shape of the sand-dust particle is assumed in all these researches. However, the geometries of sand-dust particles cannot be classified as spheres, ellipsoids, cubes, or otherwise. Thus, the results may not be sufficiently accurate to describe the practical situation in FSO links.On the problem of nonspherical particle scattering, Mackowski proposed a computational method to calculate the T-matrix of arbitrary-shaped particles by introducing moment solution to the discrete dipole approximation interaction equations [11]. Wang employed the finite element method to evaluate three types of boundary conditions and modelled the light propagation process in turbid media [12]. These researches provide novel perspectives to the problem of laser signal transmission under sand-dust weather conditions. Drawing inspiration from these researches, the radiation propagation method and the small-angle approximation method were utilized in this study to macroscopically analyse the characteristics of laser transmission in a sand-dust channel.


## 2. Theoretical Analysis

### 2.1. Laser Intensity

When a laser transmits in the scattering medium, its intensity can be expressed as(1)dLe,esds=−ρCeLe,es+ρCe∫4πpes,es′Le,es′dα′+εe,es,where Le,es is the light intensity, e is the particle position, es is the incident laser direction, es′ is the scattering light direction from other particles, ρ is the particle density, and pes,es′ is the scattering phase function. Ce=Ca+Cs is the extinction coefficient, where Ca and Cs represent the absorption coefficient and the scattering coefficient, respectively. Assume ɑ as the solid angle of the scattering light, a′ as the solid angle of the es′ direction, and εe,es as the radiation power of a unit volume along the es direction. Consequently, equation (1) can be illustrated by Figure 1.Figure 1
Schematic geometry of radiative transmission.It is evident from the plot that the received signal consists of three main parts: (1) the attenuated signal in thees direction which went through the particle scattering and absorption effects; (2) the signal from other directions, e.g., es′, which are superimposed on the es direction; and (3) the signal emitted by the internal radiation source in the es direction.Assume the medium is completely absorptive and the photons follow the elastic scattering mode, and equation (1) can be simplified as(2)dLe,esdτ=−Le,es+∫4πpes,es′Le,es′dα′,where τ=∫ρCeds denotes the optical thickness. pes,es′ can be unfolded by the Legendre function as(3)pθ,φ;θ′,φ′=∑j=0∞∑i=j∞dijPijcosθPijcosθ′cosjφ−φ′,where θ and φ denote the zenith angle and azimuth angle of the incident light, respectively; θ′ and φ′ denote the zenith angle and azimuth angle of the emitting light, respectively; Pijcosθ denotes the Legendre function; and dij denotes the Legendre constant.In contrast, equation (2) can be expressed in Cartesian coordinates as(4)cosθdLz,θdθ=−CeLz,θ+Cs2∫0πpθ,θ′Lz,θ′sinθ′dθ′.When it is assumed thatμ = cos θ, μ′=cosθ′, and j ≠ 0, ∫02πcosjφ−φ′dφ′=0. Furthermore, when the scattering phase function is assumed as pμ,μ′=∑i=0∞diPiμPiμ′ and the scattering albedo as ω=Cs/Ce, equation (4) can be simplified as(5)μdLτ,μdτ=−Lτ,μ+ω2∫−11pμ,μ′Lτ,μ′dα′.However, the laser beam will gradually be deviated away from the optical axis due to the sand-dust particles’ multiple scattering effect. Therefore, the light intensity can be expanded by the sum of the scattered lights [13]:(6)Lτ,μ=∑m=0∞Lmτ,μ.If the scattered laser from other directions can be neglected, the light intensity can be derived by a recursive relation as(7)Lmτ,μ=∫0τJmτ′,μexp−τ−τ′μdτ′μ,where(8)Jmτ,μ=ω2∫−11pμ,μ′Lm−1τ,μ′dμ′,where m is the scattering number which satisfies m ≥ 1.Considering that the laser wavelength is much smaller than the average radius of the sand-dust particles, the sand-dust particles show a significant forward scattering selectivity in the light. Consequently, the laser power is concentrated mainly in a very small range along the propagation direction.Substituting equations (7) and (8) into equation (6), it can then be expressed in the form of Lmτ,μ=Amτ,μ0Bmμ,μ0, where μ0 denotes the initial value of cos θ before scattering:(9)Bmμ,μ0=ω2∫−11pμ,μ′Bm−1μ′,μ0dμ′,(10)Amτ,μ0=∫0τAm−1τ′,μ0exp−τ−τ′μdτ′μ.According to the orthogonality of the Legendre polynomials,∫−11PiμPjμdμ=δi,j/i+1/2, where δi,j=1,i=j,0,i≠j,. Therefore, equation (9) can be rewritten as(11)Bmμ,μ0=ωm∑i=0∞i+12dimPiμPiμ0.According to small-angle approximation (whereμ = μ0 and μ′≈μ), equation (10) can be rewritten as(12)A1τ,μ0=∫0τA0τ′,μ0exp−τ−τ′μdτ′μ=E0exp−τμ∫0τexpμ0−μμ0μτ′dτ′μ≈E0exp−τμ0τμ0,(13)Amτ,μ0=∫0τAm−1τ′,μ0exp−τ−τ′μdτ′μ≈E01m!τμ0mexp−τμ0,where E0 denotes the initial laser intensity.Substituting equations (11) and (13) into equation (6), the light intensity of multiple scattering can be expressed as(14)Lτ,μ=∑m=0∞Amτ,μ0Bmμ,μ0≈∑m=0∞E0Bmμ,μ0τ/μ0mm!exp−τμ0.In contrast, the laser intensity without scattering can be written as(15)L0τ,μ=E0exp−τμ0δμ−μ0,where δ is the Dirac function.


### 2.2. Laser Pulse Characteristics

The scattering of sand-dust particles leads to light intensity attenuation. It also results in laser pulse delay and pulse broadening. Suppose a laser beam is transmitting along thez-axis, and a sand-dust particle lies at the original point O. The laser propagation direction will be deflected to point P due to the scattering effect, as described in Figure 2. The average photon path length per unit time (denoted as dR/dt) can be expressed as(16)dRdt=dxdt2+dydt2+dzdt21/2.Figure 2
Geometrical schematic of laser scattering.According to the classical electron scattering theory [14], the scattering projection angle α can be written as(17)α=ωτθ021/2,where θ is the scattering angle and θ02=∫0πθ2pθdθ. pθ is the scattering phase function, which describes the laser power spatial distribution after scattering.A revised two-term Henyey–Greenstein (TTHG) scattering phase function is proposed in [15]. The authors use two asymmetric factors to consider both forward scattering and backward scattering. Consequently, the pθ function can be expressed as(18)pμ;λ,g1,g2=λpμ;g1+1−λpμ;g2=λ1−g121−2μg1+g123/2+1−λ1−g221−2μg2+g223/2,where g1 and g2 denote the asymmetric factor parameters, and the asymmetric factor g=λg1+1−λg2, 0 ≤ λ ≤ 1, g1⋅g2 ≤ 0.According to Koschmieder’s law [16], the relationship between atmospheric visibility (V) and the extinction coefficient (Ce) can be written as(19)Ce=3.91Vλ0.55−q,where λ is the laser wavelength, and q is a parameter related to atmospheric visibility, which is described by the following equations:(20)q=0.585V1/3,V<6km,1.3,6km≤V<50km,1.6,V≥50km.According to the particle forward scattering selectivity and the small-angle approximation theory,tanθ≈θ. Therefore, the x-direction component can be expressed as(21)x=z⋅tanωτθ021/2≈z⋅ωτθ021/2,(22)dxdz=32ωτθ021/2.Since they-direction component of the XOZ plane is 0, the average laser scattering path length is(23)dRdz=94ωτθ02+11/2.Integral to both sides of equation (23), the photon scattering path length R is(24)R=0.3zωτθ0294ωτθ02+13/2−1.As a result, the pulse delay Δt can be expressed as(25)Δt=R−zc=z⋅nc⋅0.3ωτθ0294ωτθ02+13/2−1−1,where n denotes the complex refractive index of sand-dust particles, and c = 3 × 108 m/s is the speed of light in a vacuum. It is evident from equation (25) that the laser pulse delay in sand-dust conditions is related to the parameters of transmission distance, optical thickness, scattering angle, scattering coefficient, and attenuation coefficient. Furthermore, the laser pulse delay is in direct proportion to the transmission distance and the sand-dust complex refractive index.However, the pulse delay is in nanosecond order of magnitude, and the scattering process can be considered as a laser pulse throughout a linear time-invariant system. Consequently, the signal at the receiver end can be expressed as(26)yt=ft∗Pst,where ft denotes the laser pulse, which follows the Gaussian distribution and can be expressed as(27)ft=exp−ln16T02t2,where T0 is the pulse width. Pst denotes the normalized pulse power, which can be expressed as [17](28)Pst=ERtΔt2exp−tΔt,t≥0,where ER is the unit pulse power of the received laser signal.Substituting equations (27) and (28) into equation (26), the received signal is(29)yt=ft∗Pst=∫0∞ERξΔt2exp−ξΔt−ln16T02t−ξ2dξ.The received signal is primarily related to the parameters of normalized pulse power, pulse delay, and pulse width.Definingtm as a 3 dB pulse time extension width of Pst,(30)∫02tm1ERPstdt=95.6%.This indicates that 95.6% of the pulse power is concentrated in2tm. Therefore, the amount of pulse broadening can be written as(31)tm=2.45Δt=z⋅nc⋅0.735ωτθ0294ωτθ02+13/2−1−2.45.It is evident from equation (31) that the amount of laser pulse broadening is affected primarily by the parameter of time delay and is related to the parameters of transmission distance, sand-dust particle refractive index, scattering albedo, optical thickness, and scattering angle.


### 2.3. Light Attenuation

Sand-dust concentration describes the quantity of sand-dust particles within a unit volume, which directly reflects the severity of the sand-dust weather condition [18]. The distribution of the sand-dust concentration with a certain particle size can be expressed as [19](32)W=π6Gρ10−9∫0∞r3prdr=0.001388ρ∫0∞r3prdr,where G = 2.65 × 106 g/m3 denotes the sand-dust proportion; r denotes the radius of sand-dust particles; and pr denotes the sand-dust particle radius distribution, which normally follows the log-normal distribution [20].Therefore, the probability density distribution function of the sand-dust particle radius can be written as(33)Nr=prρ.According to Koschmieder’s law, the relationship between atmospheric visibilityV and the attenuation coefficient Ce is(34)V=15Ce,(35)Ce=8.686×10−3πρ∫0∞r2Nrdr.In terms of atmospheric visibility [21], ρ can be expressed as(36)ρ=158.686×103πV∫0∞r2Nrdr.Additionally, the relationship between atmospheric visibility and antenna transmission height can be demonstrated as [22](37)V=V0×expbh,where h is the vertical height, b = 1.25, and V0 denotes horizontal visibility near the ground.Substituting equations (36) and (37) into equation (32), the distribution of sand-dust particle concentration can be expressed as(38)W=5.085×10−8∫0∞r3prdrV0exp1.25h∫0∞r2Nrdr.However, sand-dust particles are randomly distributed in the air and can be considered as composite media composed of dry sand-dust particles and moisture. Its complex permittivity is determined by the dielectric constant of sand and moisture and varies with frequency. According to the Maxwell-Garnett equations, the complex permittivity of sand-dust particles takes the form [23](39)εe=εr+iεi=εs1+3Pεw−εs/εw+2εs1−Pεw−εs/εw+2εs,where εr and εi are the real and imaginary parts of εe, P denotes the volume fraction occupied by sand-dust particles in a unit volume, and εs and εw denote the complex permittivity of dry sand-dust particles and the relative dielectric constant of moisture, respectively. According to Wang et al. [24], the real parts εs′ and imaginary parts εs″ of εs are(40)εs′=3,εs″=1.8×102lgf−2.8f,0.8GHz≤f<80GHz,18.256f,f≥80GHz..According to the empirical formula proposed by Ellison [25], the real parts εw′ and imaginary parts εw″ of εw are(41)εw′=ε∞+ε1−ε∞1−λs/λ1−αsinαπ/21+2λs/λ1−αsinαπ/2+λs/λ21−α,εw″=ε∞+ε1−ε∞λs/λ1−αcosαπ/21+2λs/λ1−αsinαπ/2+λs/λ21−α+σwλ18.8496×1010,where σw=12.5664×108S⋅m−1; and α, λs, ε1, and ε∞ are four atmospheric temperature-dependent quantities. Consequently, the light attenuation in sand-dust weather conditions can be written as [26]:(42)A=92.7⋅εiεr+22+εi2⋅Wλ,where, εr and εi are the real and imaginary parts of εe and λ is the laser wavelength.Substituting equations (39)–(41) into equation (42), the light attenuation in sand-dust weather conditions can be derived as(43)A=4.714×10−6⋅εi∫0∞r3prdrλ⋅V0⋅εr+22+εi2⋅exp1.25h∫0∞r2Nrdr.


## 2.1. Laser Intensity

When a laser transmits in the scattering medium, its intensity can be expressed as(1)dLe,esds=−ρCeLe,es+ρCe∫4πpes,es′Le,es′dα′+εe,es,where Le,es is the light intensity, e is the particle position, es is the incident laser direction, es′ is the scattering light direction from other particles, ρ is the particle density, and pes,es′ is the scattering phase function. Ce=Ca+Cs is the extinction coefficient, where Ca and Cs represent the absorption coefficient and the scattering coefficient, respectively. Assume ɑ as the solid angle of the scattering light, a′ as the solid angle of the es′ direction, and εe,es as the radiation power of a unit volume along the es direction. Consequently, equation (1) can be illustrated by Figure 1.Figure 1
Schematic geometry of radiative transmission.It is evident from the plot that the received signal consists of three main parts: (1) the attenuated signal in thees direction which went through the particle scattering and absorption effects; (2) the signal from other directions, e.g., es′, which are superimposed on the es direction; and (3) the signal emitted by the internal radiation source in the es direction.Assume the medium is completely absorptive and the photons follow the elastic scattering mode, and equation (1) can be simplified as(2)dLe,esdτ=−Le,es+∫4πpes,es′Le,es′dα′,where τ=∫ρCeds denotes the optical thickness. pes,es′ can be unfolded by the Legendre function as(3)pθ,φ;θ′,φ′=∑j=0∞∑i=j∞dijPijcosθPijcosθ′cosjφ−φ′,where θ and φ denote the zenith angle and azimuth angle of the incident light, respectively; θ′ and φ′ denote the zenith angle and azimuth angle of the emitting light, respectively; Pijcosθ denotes the Legendre function; and dij denotes the Legendre constant.In contrast, equation (2) can be expressed in Cartesian coordinates as(4)cosθdLz,θdθ=−CeLz,θ+Cs2∫0πpθ,θ′Lz,θ′sinθ′dθ′.When it is assumed thatμ = cos θ, μ′=cosθ′, and j ≠ 0, ∫02πcosjφ−φ′dφ′=0. Furthermore, when the scattering phase function is assumed as pμ,μ′=∑i=0∞diPiμPiμ′ and the scattering albedo as ω=Cs/Ce, equation (4) can be simplified as(5)μdLτ,μdτ=−Lτ,μ+ω2∫−11pμ,μ′Lτ,μ′dα′.However, the laser beam will gradually be deviated away from the optical axis due to the sand-dust particles’ multiple scattering effect. Therefore, the light intensity can be expanded by the sum of the scattered lights [13]:(6)Lτ,μ=∑m=0∞Lmτ,μ.If the scattered laser from other directions can be neglected, the light intensity can be derived by a recursive relation as(7)Lmτ,μ=∫0τJmτ′,μexp−τ−τ′μdτ′μ,where(8)Jmτ,μ=ω2∫−11pμ,μ′Lm−1τ,μ′dμ′,where m is the scattering number which satisfies m ≥ 1.Considering that the laser wavelength is much smaller than the average radius of the sand-dust particles, the sand-dust particles show a significant forward scattering selectivity in the light. Consequently, the laser power is concentrated mainly in a very small range along the propagation direction.Substituting equations (7) and (8) into equation (6), it can then be expressed in the form of Lmτ,μ=Amτ,μ0Bmμ,μ0, where μ0 denotes the initial value of cos θ before scattering:(9)Bmμ,μ0=ω2∫−11pμ,μ′Bm−1μ′,μ0dμ′,(10)Amτ,μ0=∫0τAm−1τ′,μ0exp−τ−τ′μdτ′μ.According to the orthogonality of the Legendre polynomials,∫−11PiμPjμdμ=δi,j/i+1/2, where δi,j=1,i=j,0,i≠j,. Therefore, equation (9) can be rewritten as(11)Bmμ,μ0=ωm∑i=0∞i+12dimPiμPiμ0.According to small-angle approximation (whereμ = μ0 and μ′≈μ), equation (10) can be rewritten as(12)A1τ,μ0=∫0τA0τ′,μ0exp−τ−τ′μdτ′μ=E0exp−τμ∫0τexpμ0−μμ0μτ′dτ′μ≈E0exp−τμ0τμ0,(13)Amτ,μ0=∫0τAm−1τ′,μ0exp−τ−τ′μdτ′μ≈E01m!τμ0mexp−τμ0,where E0 denotes the initial laser intensity.Substituting equations (11) and (13) into equation (6), the light intensity of multiple scattering can be expressed as(14)Lτ,μ=∑m=0∞Amτ,μ0Bmμ,μ0≈∑m=0∞E0Bmμ,μ0τ/μ0mm!exp−τμ0.In contrast, the laser intensity without scattering can be written as(15)L0τ,μ=E0exp−τμ0δμ−μ0,where δ is the Dirac function.


## 2.2. Laser Pulse Characteristics

The scattering of sand-dust particles leads to light intensity attenuation. It also results in laser pulse delay and pulse broadening. Suppose a laser beam is transmitting along thez-axis, and a sand-dust particle lies at the original point O. The laser propagation direction will be deflected to point P due to the scattering effect, as described in Figure 2. The average photon path length per unit time (denoted as dR/dt) can be expressed as(16)dRdt=dxdt2+dydt2+dzdt21/2.Figure 2
Geometrical schematic of laser scattering.According to the classical electron scattering theory [14], the scattering projection angle α can be written as(17)α=ωτθ021/2,where θ is the scattering angle and θ02=∫0πθ2pθdθ. pθ is the scattering phase function, which describes the laser power spatial distribution after scattering.A revised two-term Henyey–Greenstein (TTHG) scattering phase function is proposed in [15]. The authors use two asymmetric factors to consider both forward scattering and backward scattering. Consequently, the pθ function can be expressed as(18)pμ;λ,g1,g2=λpμ;g1+1−λpμ;g2=λ1−g121−2μg1+g123/2+1−λ1−g221−2μg2+g223/2,where g1 and g2 denote the asymmetric factor parameters, and the asymmetric factor g=λg1+1−λg2, 0 ≤ λ ≤ 1, g1⋅g2 ≤ 0.According to Koschmieder’s law [16], the relationship between atmospheric visibility (V) and the extinction coefficient (Ce) can be written as(19)Ce=3.91Vλ0.55−q,where λ is the laser wavelength, and q is a parameter related to atmospheric visibility, which is described by the following equations:(20)q=0.585V1/3,V<6km,1.3,6km≤V<50km,1.6,V≥50km.According to the particle forward scattering selectivity and the small-angle approximation theory,tanθ≈θ. Therefore, the x-direction component can be expressed as(21)x=z⋅tanωτθ021/2≈z⋅ωτθ021/2,(22)dxdz=32ωτθ021/2.Since they-direction component of the XOZ plane is 0, the average laser scattering path length is(23)dRdz=94ωτθ02+11/2.Integral to both sides of equation (23), the photon scattering path length R is(24)R=0.3zωτθ0294ωτθ02+13/2−1.As a result, the pulse delay Δt can be expressed as(25)Δt=R−zc=z⋅nc⋅0.3ωτθ0294ωτθ02+13/2−1−1,where n denotes the complex refractive index of sand-dust particles, and c = 3 × 108 m/s is the speed of light in a vacuum. It is evident from equation (25) that the laser pulse delay in sand-dust conditions is related to the parameters of transmission distance, optical thickness, scattering angle, scattering coefficient, and attenuation coefficient. Furthermore, the laser pulse delay is in direct proportion to the transmission distance and the sand-dust complex refractive index.However, the pulse delay is in nanosecond order of magnitude, and the scattering process can be considered as a laser pulse throughout a linear time-invariant system. Consequently, the signal at the receiver end can be expressed as(26)yt=ft∗Pst,where ft denotes the laser pulse, which follows the Gaussian distribution and can be expressed as(27)ft=exp−ln16T02t2,where T0 is the pulse width. Pst denotes the normalized pulse power, which can be expressed as [17](28)Pst=ERtΔt2exp−tΔt,t≥0,where ER is the unit pulse power of the received laser signal.Substituting equations (27) and (28) into equation (26), the received signal is(29)yt=ft∗Pst=∫0∞ERξΔt2exp−ξΔt−ln16T02t−ξ2dξ.The received signal is primarily related to the parameters of normalized pulse power, pulse delay, and pulse width.Definingtm as a 3 dB pulse time extension width of Pst,(30)∫02tm1ERPstdt=95.6%.This indicates that 95.6% of the pulse power is concentrated in2tm. Therefore, the amount of pulse broadening can be written as(31)tm=2.45Δt=z⋅nc⋅0.735ωτθ0294ωτθ02+13/2−1−2.45.It is evident from equation (31) that the amount of laser pulse broadening is affected primarily by the parameter of time delay and is related to the parameters of transmission distance, sand-dust particle refractive index, scattering albedo, optical thickness, and scattering angle.


## 2.3. Light Attenuation

Sand-dust concentration describes the quantity of sand-dust particles within a unit volume, which directly reflects the severity of the sand-dust weather condition [18]. The distribution of the sand-dust concentration with a certain particle size can be expressed as [19](32)W=π6Gρ10−9∫0∞r3prdr=0.001388ρ∫0∞r3prdr,where G = 2.65 × 106 g/m3 denotes the sand-dust proportion; r denotes the radius of sand-dust particles; and pr denotes the sand-dust particle radius distribution, which normally follows the log-normal distribution [20].Therefore, the probability density distribution function of the sand-dust particle radius can be written as(33)Nr=prρ.According to Koschmieder’s law, the relationship between atmospheric visibilityV and the attenuation coefficient Ce is(34)V=15Ce,(35)Ce=8.686×10−3πρ∫0∞r2Nrdr.In terms of atmospheric visibility [21], ρ can be expressed as(36)ρ=158.686×103πV∫0∞r2Nrdr.Additionally, the relationship between atmospheric visibility and antenna transmission height can be demonstrated as [22](37)V=V0×expbh,where h is the vertical height, b = 1.25, and V0 denotes horizontal visibility near the ground.Substituting equations (36) and (37) into equation (32), the distribution of sand-dust particle concentration can be expressed as(38)W=5.085×10−8∫0∞r3prdrV0exp1.25h∫0∞r2Nrdr.However, sand-dust particles are randomly distributed in the air and can be considered as composite media composed of dry sand-dust particles and moisture. Its complex permittivity is determined by the dielectric constant of sand and moisture and varies with frequency. According to the Maxwell-Garnett equations, the complex permittivity of sand-dust particles takes the form [23](39)εe=εr+iεi=εs1+3Pεw−εs/εw+2εs1−Pεw−εs/εw+2εs,where εr and εi are the real and imaginary parts of εe, P denotes the volume fraction occupied by sand-dust particles in a unit volume, and εs and εw denote the complex permittivity of dry sand-dust particles and the relative dielectric constant of moisture, respectively. According to Wang et al. [24], the real parts εs′ and imaginary parts εs″ of εs are(40)εs′=3,εs″=1.8×102lgf−2.8f,0.8GHz≤f<80GHz,18.256f,f≥80GHz..According to the empirical formula proposed by Ellison [25], the real parts εw′ and imaginary parts εw″ of εw are(41)εw′=ε∞+ε1−ε∞1−λs/λ1−αsinαπ/21+2λs/λ1−αsinαπ/2+λs/λ21−α,εw″=ε∞+ε1−ε∞λs/λ1−αcosαπ/21+2λs/λ1−αsinαπ/2+λs/λ21−α+σwλ18.8496×1010,where σw=12.5664×108S⋅m−1; and α, λs, ε1, and ε∞ are four atmospheric temperature-dependent quantities. Consequently, the light attenuation in sand-dust weather conditions can be written as [26]:(42)A=92.7⋅εiεr+22+εi2⋅Wλ,where, εr and εi are the real and imaginary parts of εe and λ is the laser wavelength.Substituting equations (39)–(41) into equation (42), the light attenuation in sand-dust weather conditions can be derived as(43)A=4.714×10−6⋅εi∫0∞r3prdrλ⋅V0⋅εr+22+εi2⋅exp1.25h∫0∞r2Nrdr.


## 3. Numerical Evaluation

Numerical simulations were carried out based on the preceding derivation. The simulation parameters were set as follows: the initial light intensityE0 = 1, the signal pulse width T0 = 60 ns, the temperature t = 20°C, and n = 1.55−0.005i, where the real part of n indicates the scattering capacity and the imaginary part of n indicates the absorption capacity.


### 3.1. Laser Intensity

According to Mie theory and equation (14), the laser intensity under single and multiple scattering scenarios are shown in Figures 3 and 4.Figure 3
Laser intensity under (a) single scattering scenario and (b) multiple scattering scenarios.
(a)
(b)Figure 4
Laser intensity versus the optical thickness for (a) laser intensity of different scattering times and (b) total received laser intensity of different scattering times.
(a)
(b)It is evident from Figure3(a) that the laser intensity is directly proportional to the optical thickness and inversely proportional to the transmission distance in the single scattering scenario. Figure 3(b) shows that the laser intensity is inversely proportional to the scattering number and the optical thickness. The laser intensity follows exponential decay to the scattering number and the optical thickness.Figure4(a) illustrates the laser intensity versus the atmospheric optical thickness for different scattering number values. It can be observed that the maximum value is between 1 and 3. Furthermore, the laser intensity decreases rapidly with increasing scattering number values. This indicates that the multiple scattering will strongly attenuate the laser intensity in sand-dust weather conditions. In contrast, the laser intensity of multiple scattering is slightly larger than that of single scattering when the optical thickness is more than 5. This indicates that multiple scattering is dominant in the sand-dust channel when the optical thickness is high. The total received laser intensity versus the atmospheric optical thickness for different scattering number values is depicted in Figure 4(b). It is evident from the figure that the total received laser intensity increases with increasing scattering number values. The reason for this phenomenon is that the total received laser intensity is comprised of the single scattering light and the corresponding multiple scattering lights. Moreover, the total received laser intensity changes a little when the scattering number is more than three. Therefore, it can be inferred that the laser intensity of multiple scattering of more than three can be ignored.


### 3.2. Laser Pulse Characteristics

The received laser pulse waveform and the laser pulse characteristics under sand-dust weather conditions are shown in Figures5–9 based on equations (15) and (29).Figure 5
The normalized laser pulse waveform at different transmission distances.Figure 6
The pulse time delay versus the transmission distance and atmospheric visibility under different laser wavelengths.Figure 7
Pulse time delay of different laser wavelengths at (a) various transmission distances and (b) various atmospheric visibilities.
(a)
(b)Figure 8
Pulse time delay as a function of transmission distance under (a) different scattering albedos and (b) different scattering coefficients.
(a)
(b)Figure 9
Pulse time delay as a function of the asymmetry factor.Figure5 demonstrates the normalized laser pulse waveform at different transmission distances, where the scattering albedo is set as ω = 0.5, and the scattering coefficient is set as Cs = 7 × 10−4. The figure indicates that pulse width will broaden with increasing transmission distance. Table 1 shows the extended received pulse width with the different transmission distances, where the initial pulse width is set as 100 ns.Table 1
Received pulse width at different transmission distances.
Transmission distances (m)
Received pulse width (ns)
Initial pulse width
100
100
121
150
136
200
149
250
166
300
182Figure6 illustrates the effect of transmission distance and atmospheric visibility on laser pulse time delay under different laser wavelengths, where the laser wavelength becomes shorter from the bottom to the top. The pulse time delay is an increasing trend when atmospheric visibility decreases. Moreover, the short wavelength leads to greater pulse time delay.The pulse time delay of different laser wavelengths at various transmission distances is shown in Figure7(a), where the atmospheric visibility is set as 1 km. The pulse time delay grows incrementally over the transmission distance under different light wavelengths.Table2 shows the pulse time delay of different laser wavelengths, where transmission distance is 300 m. This illustrates that the pulse time delay gradually increases when using short-wavelength lasers as the carrier. Consequently, longer wavelength is conducive to reducing the pulse time delay and the mutual interference when transmitting signals over a short distance in sand-dust weather conditions. The influence of atmospheric visibility on pulse time delay is described in Figure 7(b), where the transmission distance is set as 1 km. It is evident from the plot that the decline of pulse time delay follows a negative exponential trend relative to the atmospheric visibility. Furthermore, the pulse time delay tends to be a wavelength-dependent constant when the atmospheric visibility is greater than 3.5 km.Table 2
Pulse time delay of different laser wavelengths.
Laser wavelength (μm)
Pulse time delay (ns)
10.60
93.2
8.50
103.1
6.32
118.3
3.25
162.8
1.55
234.6
0.55
392.1Figure8(a) demonstrates the curves of pulse time delay versus the transmission distance at different scattering albedos, where the scattering coefficient is set as 7 × 10−4. It is evident from the plot that the pulse time delay shows an exponential growth with increasing transmission distance. Table 3 shows the pulse time delay with the transmission distances, where ω = 0.6. Moreover, the pulse time delay increases with increasing scattering albedo. As a result, the multipath effect caused by scattering is the main factor behind the pulse time delay when the scattering effect is stronger than the absorption effect. Figure 8(b) shows the influence of the scattering coefficient on pulse time delay when ω = 0.7. It is evident from the figure that larger scattering coefficients result in longer pulse time delay. The reason for this phenomenon is that the larger scattering coefficient indicates higher sand-dust particle concentration in the atmosphere. Therefore, the probability of scattering is increased and leads to more serious multipath effect.Table 3
Pulse time delay at different transmission distances (ω = 0.6).
Transmission distances (m)
Pulse time delay (ns)
50
5.1
100
13.7
150
25.9
200
41.7
250
60.9
300
83.6Figure9 illustrates the relationship between pulse time delay and the asymmetry factor at different transmission distances when λ = 10.6 μm, V = 1 km, and Cs = 7 × 10−4. As depicted in the plot, the pulse time delay decreases in a negative exponential trend with an increasing asymmetry factor. This is because forward scattering is predominant when the asymmetry factor gets close to 1.


### 3.3. Light Attenuation

The light signal attenuation characteristics under sand-dust weather conditions are shown in Figures10–12 based on the result of equation (43).Figure 10
Laser signal attenuation as a function of atmospheric visibility and humidity under different laser wavelengths.Figure 11
Laser attenuation of different laser wavelengths in (a) different atmospheric visibilities and (b) different atmospheric humidities.
(a)
(b)Figure 12
Sand-dust concentration and signal attenuation as a function of antenna transmission height under (a) different atmospheric visibilities and (b) different laser wavelengths.
(a)
(b)Figure10 depicts laser signal attenuation as a function of atmospheric visibility and humidity under three different laser wavelengths, where the laser wavelength becomes shorter from the bottom to the top. It is evident from the figure that the signal attenuation is inversely proportional to atmospheric visibility and directly proportional to atmospheric humidity. This phenomenon is particularly evident in the short laser wavelength scenario.Figure11(a) shows signal attenuation versus atmospheric visibility with different laser wavelengths in linear-log coordinates, where the atmospheric humidity is set as 10%. It is evident from the figure that the attenuation follows a negative exponential decay trend with increasing atmospheric visibility. The attenuation eventually tends to be a constant when the atmospheric visibility increases to 5 km. Furthermore, it is clear that the attenuation decreases with increasing laser wavelength. Figure 11(b) illustrates signal attenuation versus atmospheric humidity under different laser wavelengths. The attenuation increases with increasing atmospheric humidity.Table4 shows the signal attenuation of different laser wavelengths, where the atmospheric humidity is 5%, 15%, and 25%, respectively. This indicates that the signal attenuation decreases with increasing laser wavelength. As a result, we can conclude that longer laser wavelengths can help reduce signal attenuation when operating a laser communication system in sand-dust weather conditions.Table 4
Attenuation of different laser wavelengths.
Laser wavelength (μm)
Attenuation (dB/km)
Atmospheric humidity (5%)
Atmospheric humidity (15%)
Atmospheric humidity (25%)
10.6
17.50
20.45
23.10
8.5
21.83
25.51
28.82
6.328
29.32
34.29
38.75Figure12(a) demonstrates the relationship between sand-dust concentration and antenna transmission height under different levels of atmospheric visibility. It can be concluded from the figure that the sand-dust concentration shows a negative exponential decay trend to the antenna transmission height. Particularly, when the antenna transmission height increases beyond 2.5 km, the atmospheric sand-dust concentration reduces below 2.8 μg/m3 and gradually tends to be a constant. Figure 12(b) shows the influence of the antenna transmission height on signal attenuation with different laser wavelengths. It can be observed from the figure that the attenuation decreases with increasing antenna transmission height. Furthermore, it tends to be a constant value when the antenna transmission height exceeds 2.5 km. This is because the increased height leads to smaller atmospheric sand-dust concentration. Consequently, the effect of multiple scattering becomes less, which results in less attenuation.


## 3.1. Laser Intensity

According to Mie theory and equation (14), the laser intensity under single and multiple scattering scenarios are shown in Figures 3 and 4.Figure 3
Laser intensity under (a) single scattering scenario and (b) multiple scattering scenarios.
(a)
(b)Figure 4
Laser intensity versus the optical thickness for (a) laser intensity of different scattering times and (b) total received laser intensity of different scattering times.
(a)
(b)It is evident from Figure3(a) that the laser intensity is directly proportional to the optical thickness and inversely proportional to the transmission distance in the single scattering scenario. Figure 3(b) shows that the laser intensity is inversely proportional to the scattering number and the optical thickness. The laser intensity follows exponential decay to the scattering number and the optical thickness.Figure4(a) illustrates the laser intensity versus the atmospheric optical thickness for different scattering number values. It can be observed that the maximum value is between 1 and 3. Furthermore, the laser intensity decreases rapidly with increasing scattering number values. This indicates that the multiple scattering will strongly attenuate the laser intensity in sand-dust weather conditions. In contrast, the laser intensity of multiple scattering is slightly larger than that of single scattering when the optical thickness is more than 5. This indicates that multiple scattering is dominant in the sand-dust channel when the optical thickness is high. The total received laser intensity versus the atmospheric optical thickness for different scattering number values is depicted in Figure 4(b). It is evident from the figure that the total received laser intensity increases with increasing scattering number values. The reason for this phenomenon is that the total received laser intensity is comprised of the single scattering light and the corresponding multiple scattering lights. Moreover, the total received laser intensity changes a little when the scattering number is more than three. Therefore, it can be inferred that the laser intensity of multiple scattering of more than three can be ignored.


## 3.2. Laser Pulse Characteristics

The received laser pulse waveform and the laser pulse characteristics under sand-dust weather conditions are shown in Figures5–9 based on equations (15) and (29).Figure 5
The normalized laser pulse waveform at different transmission distances.Figure 6
The pulse time delay versus the transmission distance and atmospheric visibility under different laser wavelengths.Figure 7
Pulse time delay of different laser wavelengths at (a) various transmission distances and (b) various atmospheric visibilities.
(a)
(b)Figure 8
Pulse time delay as a function of transmission distance under (a) different scattering albedos and (b) different scattering coefficients.
(a)
(b)Figure 9
Pulse time delay as a function of the asymmetry factor.Figure5 demonstrates the normalized laser pulse waveform at different transmission distances, where the scattering albedo is set as ω = 0.5, and the scattering coefficient is set as Cs = 7 × 10−4. The figure indicates that pulse width will broaden with increasing transmission distance. Table 1 shows the extended received pulse width with the different transmission distances, where the initial pulse width is set as 100 ns.Table 1
Received pulse width at different transmission distances.
Transmission distances (m)
Received pulse width (ns)
Initial pulse width
100
100
121
150
136
200
149
250
166
300
182Figure6 illustrates the effect of transmission distance and atmospheric visibility on laser pulse time delay under different laser wavelengths, where the laser wavelength becomes shorter from the bottom to the top. The pulse time delay is an increasing trend when atmospheric visibility decreases. Moreover, the short wavelength leads to greater pulse time delay.The pulse time delay of different laser wavelengths at various transmission distances is shown in Figure7(a), where the atmospheric visibility is set as 1 km. The pulse time delay grows incrementally over the transmission distance under different light wavelengths.Table2 shows the pulse time delay of different laser wavelengths, where transmission distance is 300 m. This illustrates that the pulse time delay gradually increases when using short-wavelength lasers as the carrier. Consequently, longer wavelength is conducive to reducing the pulse time delay and the mutual interference when transmitting signals over a short distance in sand-dust weather conditions. The influence of atmospheric visibility on pulse time delay is described in Figure 7(b), where the transmission distance is set as 1 km. It is evident from the plot that the decline of pulse time delay follows a negative exponential trend relative to the atmospheric visibility. Furthermore, the pulse time delay tends to be a wavelength-dependent constant when the atmospheric visibility is greater than 3.5 km.Table 2
Pulse time delay of different laser wavelengths.
Laser wavelength (μm)
Pulse time delay (ns)
10.60
93.2
8.50
103.1
6.32
118.3
3.25
162.8
1.55
234.6
0.55
392.1Figure8(a) demonstrates the curves of pulse time delay versus the transmission distance at different scattering albedos, where the scattering coefficient is set as 7 × 10−4. It is evident from the plot that the pulse time delay shows an exponential growth with increasing transmission distance. Table 3 shows the pulse time delay with the transmission distances, where ω = 0.6. Moreover, the pulse time delay increases with increasing scattering albedo. As a result, the multipath effect caused by scattering is the main factor behind the pulse time delay when the scattering effect is stronger than the absorption effect. Figure 8(b) shows the influence of the scattering coefficient on pulse time delay when ω = 0.7. It is evident from the figure that larger scattering coefficients result in longer pulse time delay. The reason for this phenomenon is that the larger scattering coefficient indicates higher sand-dust particle concentration in the atmosphere. Therefore, the probability of scattering is increased and leads to more serious multipath effect.Table 3
Pulse time delay at different transmission distances (ω = 0.6).
Transmission distances (m)
Pulse time delay (ns)
50
5.1
100
13.7
150
25.9
200
41.7
250
60.9
300
83.6Figure9 illustrates the relationship between pulse time delay and the asymmetry factor at different transmission distances when λ = 10.6 μm, V = 1 km, and Cs = 7 × 10−4. As depicted in the plot, the pulse time delay decreases in a negative exponential trend with an increasing asymmetry factor. This is because forward scattering is predominant when the asymmetry factor gets close to 1.


## 3.3. Light Attenuation

The light signal attenuation characteristics under sand-dust weather conditions are shown in Figures10–12 based on the result of equation (43).Figure 10
Laser signal attenuation as a function of atmospheric visibility and humidity under different laser wavelengths.Figure 11
Laser attenuation of different laser wavelengths in (a) different atmospheric visibilities and (b) different atmospheric humidities.
(a)
(b)Figure 12
Sand-dust concentration and signal attenuation as a function of antenna transmission height under (a) different atmospheric visibilities and (b) different laser wavelengths.
(a)
(b)Figure10 depicts laser signal attenuation as a function of atmospheric visibility and humidity under three different laser wavelengths, where the laser wavelength becomes shorter from the bottom to the top. It is evident from the figure that the signal attenuation is inversely proportional to atmospheric visibility and directly proportional to atmospheric humidity. This phenomenon is particularly evident in the short laser wavelength scenario.Figure11(a) shows signal attenuation versus atmospheric visibility with different laser wavelengths in linear-log coordinates, where the atmospheric humidity is set as 10%. It is evident from the figure that the attenuation follows a negative exponential decay trend with increasing atmospheric visibility. The attenuation eventually tends to be a constant when the atmospheric visibility increases to 5 km. Furthermore, it is clear that the attenuation decreases with increasing laser wavelength. Figure 11(b) illustrates signal attenuation versus atmospheric humidity under different laser wavelengths. The attenuation increases with increasing atmospheric humidity.Table4 shows the signal attenuation of different laser wavelengths, where the atmospheric humidity is 5%, 15%, and 25%, respectively. This indicates that the signal attenuation decreases with increasing laser wavelength. As a result, we can conclude that longer laser wavelengths can help reduce signal attenuation when operating a laser communication system in sand-dust weather conditions.Table 4
Attenuation of different laser wavelengths.
Laser wavelength (μm)
Attenuation (dB/km)
Atmospheric humidity (5%)
Atmospheric humidity (15%)
Atmospheric humidity (25%)
10.6
17.50
20.45
23.10
8.5
21.83
25.51
28.82
6.328
29.32
34.29
38.75Figure12(a) demonstrates the relationship between sand-dust concentration and antenna transmission height under different levels of atmospheric visibility. It can be concluded from the figure that the sand-dust concentration shows a negative exponential decay trend to the antenna transmission height. Particularly, when the antenna transmission height increases beyond 2.5 km, the atmospheric sand-dust concentration reduces below 2.8 μg/m3 and gradually tends to be a constant. Figure 12(b) shows the influence of the antenna transmission height on signal attenuation with different laser wavelengths. It can be observed from the figure that the attenuation decreases with increasing antenna transmission height. Furthermore, it tends to be a constant value when the antenna transmission height exceeds 2.5 km. This is because the increased height leads to smaller atmospheric sand-dust concentration. Consequently, the effect of multiple scattering becomes less, which results in less attenuation.


## 4. Conclusion

We analysed the influential factors relevant to laser signal transmission in sand-dust weather conditions by utilizing the radiation propagation method and the small-angle approximation method. Simulation results were presented to evaluate the performance of laser signals transmitting through a sand-dust channel with scattering and absorption effects. Results indicate that sand-dust weather conditions cause significant light attenuation and pulse time delay. However, the results also indicate that longer laser wavelengths could potentially be utilized to improve the communication reliability of FSO systems in sand-dust weather conditions.



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*Source: 2046896-2019-10-31.xml*