The Multiple Scattering of Laser Beam Propagation in Advection Fog and Radiation Fog

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Introduction
Fogs contains small water droplets, these droplets suspend in the air, and they attenuate laser beam; fogs have adverse efects on optical communication, lidar, remote sensing, etc. Many scientists have conducted researches in this feld. Nebuloni derived relations between an extinction coefcient and visibility of fog for near, mid, and far-IR (infrared) light transmission of free-space optic communications; the extinction coefcients in the visible 0.55 μm, the near IR 1.2 μm, and the mid IR 3.7 μm are comparable to and roughly twice as much as that in the far-IR10.6 μm when visibility is less than a few hundred meters [1]. Grabner and Kvicera measured the receiving power after beam scattering in free-space optical communication system at wavelengths of 0.83 μm and 1.55 μm; several fog attenuation events were observed on experimental paths. Important relations between fog microphysical parameters, atmospheric visibility, optical signal attenuation and bit error ratio based on measured data are presented in this case study [2]. Nebuloni and Capsoni used liquid water content and visibility to identify laser attenuation characteristics in fog; results show that visibility is almost proportional to fog attenuation at short wavelengths but not in the far-IR region [3]. Ijaz et al. proposed a new wavelength dependent empirical model to predict the spectrum attenuation of free-space optical communication systems operating at visible and near infrared wavelengths under fog and smoke in a laboratory condition, for the same fog and smoke conditions, it was found that the attenuation of light is almost linearly decreasing when laser wavelength increase from visible to NIR range [4]. Grabner and Kvicera analyzed the multiple scattering of light and the impulse response of optical channel in rain and fog for free-space optical links. Simulations show that optical attenuation of the multiple scattering due to rain is about two times lower than predicted by single scattering approach [5]. Mori and Marzano proposed an atmospheric water particles scattering model (a unifed microphysically oriented atmospheric particle scattering model) and simulated the scattering efects on free-space optics links, scattering, absorption, and extinction coefcients and the asymmetry factor are numerically simulated for each particle class, and then parametrized with respect to particle water content, fall rate, and visibility, spanning from visible to infrared wavelengths [6]. Wu et al. analyzed the changes of extinction coefcients and visibilities based on short and long time lidar data, and evaluated the free-space optical communication link for diferent wavelengths under the sea surface environment. Te horizontal extinction coefcients and visibilities are obtained by inverting the collected data. On this basis, the free-space optical communication link availabilities of the 850 and 1550 nm links are evaluated during the observation period [7]. Al Naboulsi and Sizun presented results on fog attenuation at wavelengths 650, 850, and 950 nm; they have compared the experimental data with our fog attenuation models; the advection and the radiation one deduced from FASCODE (Fast Atmospheric Signature Codes). Tis comparison clearly shows that, for the three studied wavelengths, their models lead to the best fts for the measured data [8]. Shapiro used ultrashort (femtosecond duration) light pulses for line-of-sightfree-space optical (FSO) communication through fog; their study show that scattering-induced multipath spread is less than the reciprocal of the scattering-induced doppler spread [9]. Su et al. investigates scintillation and fog attenuation efects for THz and IR signals by measuring bit error rates (BER), signal power, and phase front distortions [10]. Shah et al. established an engineering model of laser transmission in fog, and compared with other models, selected nine models including theoretical and empirical models, which were simulated in MATLAB, and the comparison results were discussed [11]. Wang et al utilized a large quantity of high-quality water vapor lidar data collected by a multifunction Raman lidar system, investigated the air mass trajectories, variations of water vapor over Xi'an. Te results indicate that local airfow trajectories mainly afect water vapor transport below the boundary layer, and that these fows are closely related to the formation of fog and haze events in the Xi'an area [12]. Di et al. proposed a method to measure the aerosol and fog particle size distribution by optimizing scattering angle and wavelengths; seven wavelengths for extinction coefcients and fve wavelengths for forward scattering coefcients were chosen for the retrieval of particle size distribution in the measurement [13]. Zeng et al. studied the propagation of linear and circular polarization in fog, and comparing their transmission characteristic with diferent condition, investigated the transmission performance of circular and linear polarization in variable foggy environments, exploring the impact of the detection range in particular [14]. Zhang et al. calculated on the refection and transmission of light propagating through sea fog by dividing fog into diferent layers with diferent refractive indices. Te scattering processes of the radiation in the polydisperse sea fog layer are traced in their improved Monte Carlo (MC) simulation. Tey provide more accurate calculations on the refection and transmission when radiation propagates through poly-disperse sea fog media of two diferent refractive indices [15]. Vasseur and Gibbins developed an experimental method to infer the physical characteristics of fog from concurrent attenuation measurements at millimeter, infrared, and visible wavelengths; a gamma function comprised of three parameters is assumed for the drop size distribution. Simulations of fog attenuation based on the inferred drop size distribution are found in excellent agreement with the measurements [16]. Al Naboulsi et al. investigated laser system performance in the advection and convection fog in the 0.4 μm to 15 μm spectral zone, the author also proposes a fast transmission relations based on an exact Mie theory calculation valid in the 0.69 μm to 1.55 μm spectral bands [17].
Te visibility is an important parameter of fog, which can be measured directly and quickly; however, the correlation between fog droplet parameter and the visibility and the scattering parameters of beams changing with fog's visibility for diferent beams' wavelengths have not been reported by now. Te current laser transmission model is simple, for laser with diferent wavelength, it is urgent to study scattering character of fog particles and the multiple scattering of low visibility fog.
In this paper, we propose on the correlation between visibility and fog droplet parameter, such as distribution, droplet extinction coefcient, and asymmetry factor of diferent types of fogs. Based on this, using Monte Carl method, for infrared laser beams with diferent wavelength, the multi scattering of laser beam in fogs is studied. Te coefcients versus visibility and scattering intensity distribution versus scattering angle are studied in detail.

The Particle Size Distribution of Fog
Te fog particles are small water droplets. Te particle size distribution function is given as follows [8,14]: where n is the number of droplets per unit volume and unit radius and r is the droplet radius. Other parameters are determined by the size distribution of fog. Visibility of fog is related to the optical attenuation c, which is expressed as follows [18]: Te optical attenuation c is as follows [16]: where Q e (r) is the optical extinction efciency. Te dimensions of fog droplets are much greater the visible optical wavelength, so Q e (r) of fog droplets can be consider as 2 [17]. c can be written as follows [16]: Liquid water content is expressed as follows [5]: where ρ∼10 6 g/m 3 . With equation (1), W is rewritten as follows: 2 International Journal of Optics W � 10 6 4π 3 ∞ 0 ar 5 e − br dr � 4π5!a 3b 6 × 10 6 .

(6)
By equations (2)-(6), a and b is expressed by V and W as follows: Te droplet concentration can be obtained as follows: Te model radius corresponds to the peak value of droplet size distribution curve as follows: Te average radius of droplet size distribution is as follows: According to diferent regions, fog can be divided into two types, advection fog and radiation fog. For advection fog, relations between the liquid water content W and the V is shown as follows [19]: For radiation fog, it is shown as follows: For advection fog, droplet size distribution is calculated from equations (1), (7), (11), and (12), as follows: For radiation fog, droplet size distribution is as follows:

Scattering Coefficient of Fog Droplets
Using Mie scattering theory, the extinction coefcient Q e , scattering coefcient Q s , absorption coefcient Q a , asymmetry factor g, and single albedo w 0 of a single particle can be calculated as follows [20]: (2n + 1) Re a n + b n , where α λ is particle size parameter and a n and b n are Lorentz-Mie scattering coefcients.
With the above formula, we obtain coefcients of single fog droplet. However, in practical application, we need to consider the size distribution of fog droplets; the average coefcients are to be calculated [21].
Te average extinction coefcient is as follows: Te average scattering coefcient is as follows: Te average absorbtion coefcient is as follows: Te droplet concentration is as follows: Te average single albedo is as follows: International Journal of Optics Te average asymmetry factor is as follows: With the size distribution of the fog, we can calculate the average coefcients of fogs.

Monte Carlo Method for Laser Beam Propagation in Fog
Te Monte Carlo method can be employed for the complex case of light scattering [15,22,23], the incident photon and the particles of the medium are regarded as the dispersive particles interacting with each other. Tere are scattering and absorption during the transmission of photons in the random medium. In the multiple scattering processes, it can be described by Markov process. Te transmission history of photons is expressed as a phase space point s � (τ, μ, φ), and the particle transport equation can be expressed as follows [22]: Obviously, Equation (22) has the Neumann series solution as follows: I 0 (s) � (s), Te integral operator shows that photons have experienced a transmission and collision. I 0 (s) means these photons propagating in the medium without collision, and I m (s) means the photons arriving at the phase space points after m collisions.
Te probability model can well describe equation (25). Here, the event is that photon transmissions collides with particles of the scattering medium and reaches the point S. It can be described by the law of total probability as follows: Te probability of photons arriving at point S of phase space after m times collisions is P m (s). It corresponds to I m (s) in equation (25).
Te latter term in equation (28) indicates that the random transmission of a photon in the medium are Markov processes. Comparing equations (25) with (28), the integral operator K(s l−1 ⟶ s l )ds l−1 corresponds to the conditional probability P(s l |s l−1 ).
Using Monte Carlo simulation, the initial position of the incident photon is in the plane of Z � 0, as shown in Figure 1. For Lambert source, the probability sampling in the initial direction is μ 0 � cos θ 0 � �� ξ 1 . ξ i (i � 1,2,3,4) is a random number uniformly distributed in the interval [0, 1].
Te free path length L is the distance of a photon propagation until an interaction event, scattering, or absorption occurs, ξ 2 � exp(−Q e L) means existence probability of photons after colliding with particles, Q e is the extinction coefcient. Te free path length L is as follow: When scattering occurs, a new propagation direction must be specifed, the H-G function is an appropriate phase function for fog particle scattering, the probability density function corresponding to normalized phase function is as follows [23]: Te scattering angle in the particle local coordinate system is v and g is the asymmetry factor; the photon scattering direction is as follows: By using the propagation distance L and scattering angle v, the scattering direction and position of photons can be determined.

Calculation Results
Based on previous theories, the following calculations are proposed and advection fog and radiation fog are considered. Te fog spectrum distribution is gamma distribution, as shown in equations (1)- (8). Te value of visibility is between 0.01 km and 10 km. Te laser wavelengths are 0.86 μm, 0.91 μm, 1.06 μm, 1.3015, and 10.6 μm, respectively. Firstly, we calculate the extinction coefcient, albedo, asymmetry factor of fog droplets versus visibility, then we use these parameters in the calculation of Monte Carlo method, the forward scattering and backscattering of laser light are calculated. Figure 2 shows the average extinction coefcient of advection fog and radiation fog versus visibility at the light wavelength of 0.86 μm, 0.91 μm, 1.06 μm, 1.315 μm, and 10.6 μm. As shown in Figure 2, the average extinction coefcient decreases as the visibility increases, and the average extinction coefcient of advection fog is bigger than that of radiation fog under the same visibility. Te average extinction coefcient fog is basically the same under the wavelength of 0.86 μm, 0.91 μm, 1.06 μm, and 1.315 μm. For wavelength of 10.6 μm, the average extinction coefcient is smaller than that of other wavelengths, and decreases quickly as visibility increases. According to equations (16) and (19), 〈σ e 〉 mainly depends on Q e and r. Te droplet particle radius is much bigger than the wavelength of the incident light wave, so the change of Q e is small, Q e is close to 2. Terefore, the change trend of 〈σ e 〉 is mainly determined by r 2 in equation (16). From equations (9)- (11), it is shown that with the increase of V, r 0 decreases, r decreases, correspondingly r 2 decreases, so 〈σ e 〉 decreases. According to equation (15), Q e contains parameter 2/a λ 2 , a λ � 2πr/λ, λ is wavelength, the wavelength of 10.6 μm is about 10 times bigger than other wavelengths (0.86 μm, 0.91 μm, 1.06 μm,  and 1.315 μm), and its Q e is calculated to be less than 2, which is smaller than that of other wavelengths, and with the same visibility, 〈σ e 〉 of 10.6 μm is smaller than that of other wavelengths. Figure 3 shows the average albedo of advection fog and radiation fog versus visibility. It can be seen that the average albedo of wavelengths of 0.86 μm, 0.91 μm, 1.06 μm, and 1.315 μm in the two kinds of fog is close to 1. Te average albedo of wavelength of 10.6 μm decreases as the visibility increases, it is obviously diferent from other wavelengths. Study on the refractive index of droplet of fog is as follows: where n′ is the real part of the refractive index that represent the scattering capacity of the particle (Q s ) and n″ is the complex part that represents the absorption by the particle (Q a ). Te extinction of the particle includes scattering and absorption (Q e � Q s + Q a ), as shown in Table 1, the refractive index is related to the wavelength of the incident wave. For the wavelength of incident wave (0.86 μm, 0.91 μm, 1.06 μm, and 1.315 μm), the complex part of its refractive index is small, Q a is small. According to equations (16)- (18), 〈σ a 〉 can be ignored, 〈σ e 〉 is close to 〈σ s 〉, and by equation (20), 〈w 0 〉 is approximately equal to 1. For the incident wave with wavelength of 10.6 μm. Te complex part of refractive index is relatively big, 〈σ e 〉 is relatively big. Te particle average size decreases with the increase of visibility, and particle size parameter α λ � 2πr/λ is close to 1 and 〈σ s 〉 decreases rapidly, but 〈σ a 〉 changes slowly, so 〈w 0 〉 decrease as visibility increase. Figure 4 shows the average asymmetry factor 〈g〉 of advection fog and radiation fog versus visibility. According to equation (21), average asymmetry factor 〈g〉 describes the relative proportion of forward and backward scattering energy of particles. When the incident wavelengths are 0.86 μm,0.91 μm, 1.06 μm, and 1.315 μm, the droplet particle radius is much bigger than the wavelength of the incident light wave. According to equation (21), the forward scattering is bigger, 〈g〉 is between 0.8 and 1. When the wavelength of incident light is 10.6 μm, and its real part of refractive index is smaller, 〈g〉 of this wavelength is slightly bigger than that of other wavelengths for lower visibility, with increasing of visibility. According to equation (10), r decreases, the particle size parameter α λ � 2πr/λ is close to 1, forward scattering decrease, backward scattering increase, and 〈g〉 decreases rapidly.
Asymmetry factor and albedo are important parameters for Monte Carlo calculation. Te scattering intensity versus scattering angle is calculated according to these parameters. Combined with the numerical results of Mie theory, the foundation for calculating the attenuation and scattering of laser in fog is laid. Figure 5 shows the change of forward scattering intensity and back scattering intensity with scattering angle, the laser wavelength is 1.06 μm, visibility is 0.1 km, the particle asymmetry factor g � 0.75, and w 0 is albedo. As shown in Figure 5(a), the forward scattering intensity increases when albedo increases, when the scattering angle is greater than 20°, the scattering intensity is relatively small, indicating that the forward scattering beam is relatively concentrated. As can be seen from Figure 5(b), the backward scattering intensity increases when single albedo increases, however, compared with the forward scattering intensity, the backward scatter intensity is much smaller. Te backscattering intensity decreases when the scattering angle is larger than 60°, which indicates that the backscattering beams are more dispersed.
In Figure 6, the relationship between scattering intensity and scattering angle is calculated, particle albedo w 0 � 0.9. Figure 6(a) show that, the forward scattering increases when the asymmetry factor increases. Te scattering intensity decreases when the scattering angle increases. It is shown in Figure 6(b), the backscattering intensity decreases when the asymmetry factor increases. At the same time, when the scattering angle is less than 50°, the scattering intensity changes little, and decreases with the increase of the scattering angle when the scattering angle is bigger than 50°. And the backward scatter intensity is much smaller than the forward scattering intensity. Figure 7 shows the backscattering intensity versus scattering angle at several typical wavelengths when the visibility is 0.6 km. It can be seen from the Figure 7(a), for radiation fog, when the scattering angle is less than 40°, the scattering intensity oscillates rapidly, and there is little diference in the scattering intensity curves for wavelengths of 0.86 μm, 0.91 μm, and 1.06 μm. Figure 7(b) show that, for advection fog, the scattering intensity curves of 0.86 μm, 0.91 μm, and 1.06 μm of radiation fog are almost same. Comparing the two fgures, it can be seen that the backscattering intensity of radiation fog is bigger than that of advection fog, and the backward scattering intensity of 10.6 μm is much smaller than that of other wavelengths, that means the forward transmittance of 10.6 μm is much bigger than that of other wavelengths.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  International Journal of Optics 9