Performance Evaluation of FSO Communications under Sand-Dust Conditions

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. )is 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.


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][4][5]. Combined with aerosols, sand-dust particles may cause serious transmission interruption. erefore, 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][7][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]. e spherical shape of the sanddust particle is assumed in all these researches. However, the geometries of sand-dust particles cannot be classified as spheres, ellipsoids, cubes, or otherwise. us, 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].
ese 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.

Theoretical Analysis
where L(e, e s ) is the light intensity, e is the particle position, e s is the incident laser direction, e s ′ is the scattering light direction from other particles, ρ is the particle density, and p(e s , e s ′ ) is the scattering phase function. C e C a + C s is the extinction coe cient, where C a and C s represent the absorption coe cient and the scattering coe cient, respectively. Assume ɑ as the solid angle of the scattering light, a ′ as the solid angle of the e s ′ direction, and ε(e, e s ) as the radiation power of a unit volume along the e s direction. Consequently, equation (1) can be illustrated by Figure 1. It is evident from the plot that the received signal consists of three main parts: (1) the attenuated signal in the e s direction which went through the particle scattering and absorption e ects; (2) the signal from other directions, e.g., e s ′ , which are superimposed on the e s direction; and (3) the signal emitted by the internal radiation source in the e s direction.
Assume the medium is completely absorptive and the photons follow the elastic scattering mode, and equation (1) where τ ρC e ds denotes the optical thickness. p(e s , e s ′ ) can be unfolded by the Legendre function as 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; P j i (cos θ) denotes the Legendre function; and d j i denotes the Legendre constant.
In contrast, equation (2) can be expressed in Cartesian coordinates as When it is assumed that μ cos θ, μ ′ cos θ ′ , and j ≠ 0, 2π 0 cos[j(φ − φ ′ )]dφ ′ 0. Furthermore, when the scattering phase function is assumed as p(μ, μ ′ ) ∞ i 0 d i P i (μ)P i (μ ′ ) and the scattering albedo as ω C s /C e , equation (4) can be simpli ed as However, the laser beam will gradually be deviated away from the optical axis due to the sand-dust particles' multiple scattering e ect. erefore, the light intensity can be expanded by the sum of the scattered lights [13]: If the scattered laser from other directions can be neglected, the light intensity can be derived by a recursive relation as where where m is the scattering number which satis es m ≥ 1.
Considering that the laser wavelength is much smaller than the average radius of the sand-dust particles, the sanddust particles show a signi cant 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 L m (τ, μ) A m (τ, μ 0 )B m (μ, μ 0 ), where μ 0 denotes the initial value of cos θ before scattering: According to the orthogonality of the Legendre polynomials, . erefore, equation (9) can be rewritten as According to small-angle approximation (where μ μ 0 and μ ′ ≈ μ), equation (10) can be rewritten as  International Journal of Antennas and Propagation where E 0 denotes the initial laser intensity. Substituting equations (11) and (13) into equation (6), the light intensity of multiple scattering can be expressed as In contrast, the laser intensity without scattering can be written as where δ is the Dirac function.

Laser Pulse
Characteristics. e 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 the z-axis, and a sand-dust particle lies at the original point O. e laser propagation direction will be de ected to point P due to the scattering e ect, as described in Figure 2. e average photon path length per unit time (denoted as dR/dt) can be expressed as According to the classical electron scattering theory [14], the scattering projection angle α can be written as where θ is the scattering angle and θ 2 0 π 0 θ 2 p(θ)dθ. p(θ) 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]. e authors use two asymmetric factors to consider both forward scattering and backward scattering. Consequently, the p(θ) function can be expressed as where g 1 and g 2 denote the asymmetric factor parameters, and the asymmetric factor g λg 1 According to Koschmieder's law [16], the relationship between atmospheric visibility (V) and the extinction coe cient (C e ) can be written as where λ is the laser wavelength, and q is a parameter related to atmospheric visibility, which is described by the following equations: According to the particle forward scattering selectivity and the small-angle approximation theory, tan θ ≈ θ. erefore, the x-direction component can be expressed as dx dz Since the y-direction component of the XOZ plane is 0, the average laser scattering path length is dR dz Integral to both sides of equation (23), the photon scattering path length R is Receiving end

International Journal of Antennas and Propagation
As a result, the pulse delay Δt can be expressed as where n denotes the complex refractive index of sand-dust particles, and c � 3 × 10 8 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 where f(t) denotes the laser pulse, which follows the Gaussian distribution and can be expressed as where T 0 is the pulse width. P s (t) denotes the normalized pulse power, which can be expressed as [17] where E R is the unit pulse power of the received laser signal. Substituting equations (27) and (28) into equation (26), the received signal is e received signal is primarily related to the parameters of normalized pulse power, pulse delay, and pulse width.
Defining t m as a 3 dB pulse time extension width of P s (t), is indicates that 95.6% of the pulse power is concentrated in 2t m . erefore, the amount of pulse broadening can be written as 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.

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]. e distribution of the sand-dust concentration with a certain particle size can be expressed as [19] where G � 2.65 × 10 6 g/m 3 denotes the sand-dust proportion; r denotes the radius of sand-dust particles; and p(r) denotes the sand-dust particle radius distribution, which normally follows the log-normal distribution [20]. erefore, the probability density distribution function of the sand-dust particle radius can be written as According to Koschmieder's law, the relationship between atmospheric visibility V and the attenuation coefficient C e is In terms of atmospheric visibility [21], ρ can be expressed as Additionally, the relationship between atmospheric visibility and antenna transmission height can be demonstrated as [22] where h is the vertical height, b � 1.25, and V 0 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 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] 4 International Journal of Antennas and Propagation International Journal of Antennas and Propagation where ε r and ε i are the real and imaginary parts of ε e , P denotes the volume fraction occupied by sand-dust particles in a unit volume, and ε s and ε w 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 ε s ′ and imaginary parts ε s ′′ of ε s are ε s ′ � 3, According to the empirical formula proposed by Ellison [25], the real parts ε w ′ and imaginary parts ε w ′′ of ε w are where σ w � 12.5664 × 108 S·m − 1 ; and α, λ s , ε 1, and ε ∞ are four atmospheric temperature-dependent quantities.
Consequently, the light attenuation in sand-dust weather conditions can be written as [26]: where, ε r and ε i are the real and imaginary parts of ε e 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

Numerical Evaluation
Numerical simulations were carried out based on the preceding derivation. e simulation parameters were set as follows: the initial light intensity E 0 � 1, the signal pulse width T 0 � 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.

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. It is evident from Figure 3(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. e laser intensity follows exponential decay to the scattering number and the optical thickness. Figure 4(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. is 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. is indicates that multiple scattering is dominant in the sand-dust channel when the optical thickness is high. e 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. e 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. erefore, it can be inferred that the laser intensity of multiple scattering of more than three can be ignored.

Laser Pulse
Characteristics. e received laser pulse waveform and the laser pulse characteristics under sanddust weather conditions are shown in Figures 5-9 based on equations (15) and (29). Figure 5 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 C s � 7 × 10 − 4 . e 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. Figure 6 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. e pulse time delay is an increasing trend when atmospheric visibility decreases. Moreover, the short wavelength leads to greater pulse time delay. e pulse time delay of different laser wavelengths at various transmission distances is shown in Figure 7(a), where the atmospheric visibility is set as 1 km. e pulse time delay grows incrementally over the transmission distance under different light wavelengths. Table 2 shows the pulse time delay of different laser wavelengths, where transmission distance is 300 m. is 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. e 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. Figure 8(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. e reason for this phenomenon is that the larger scattering coefficient indicates higher sand-dust particle concentration in the atmosphere. erefore, the probability of scattering is increased and leads to more serious multipath effect. Figure 9 illustrates the relationship between pulse time delay and the asymmetry factor at different transmission distances when λ � 10.6 μm, V � 1 km, and C s � 7 × 10 − 4 . As depicted in the plot, the pulse time delay decreases in a negative exponential trend with an  International Journal of Antennas and Propagation 7 increasing asymmetry factor.
is is because forward scattering is predominant when the asymmetry factor gets close to 1.

Light Attenuation.
e light signal attenuation characteristics under sand-dust weather conditions are shown in Figures 10-12 based on the result of equation (43). Figure 10 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. is phenomenon is particularly evident in the short laser wavelength scenario. Figure 11(a) shows signal attenuation versus atmospheric visibility with different laser wavelengths in linearlog 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. e attenuation eventually tends to be a constant when the atmospheric visibility increases to 5 km.    International Journal of Antennas and Propagation 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. e attenuation increases with increasing atmospheric humidity. Table 4 shows the signal attenuation of different laser wavelengths, where the atmospheric humidity is 5%, 15%, and 25%, respectively. is 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. Figure 12(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/m 3 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. is 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.

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.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.