1,

Introduction

Lasers play an important role in the fields of optical communications, lidar, directional damage, aerospace and other fields, and have great scientific and practical value in environmental media measurement and target detection ^[1]. However, due to the scattering and absorption of the droplets, the laser energy will continue to decay, obviously decay and result in the effectiveness weakness of the laser system. Therefore, it is of great value to study the laser transmission in the fog.

Wei Hailiang used the fog droplet spectral index distribution model to study the attenuation characteristics of laser in fog^[2]. Li Wei compared the applicability of the fog drop error model in sea fog of different intensities ^[3], and concluded that the greater the fog density, the closer the model value is to the measured value. Leng Kun studied the influence of propagation distance, visibility and initial RMS radius of the laser on the spatial distribution of photons reaching the receiving plane^[4]. Rao Ruizhong mainly introduced the optical properties of the atmosphere, atmospheric refraction, molecular absorption and scattering, light scattering of aerosol particles, etc. In the second chapter of the book, he gives the composition, geometric structure and spatial distribution of droplet particles ^[5]. Dong Feibiao based on the Mie scattering theory, applied the Monte Carlo multiple scattering model to numerically simulate the transmission characteristics of laser pulses in the fog ^[6]. Yuan Hui calculated the extinction coefficient of 1.064 μm laser radiation under different visibility conditions using experimental data and empirical formulas ^[7]. Han Yong calculated the atmospheric extinction coefficient at a wavelength of 0.55 μm through visibility ^{[7] [8]}, and then obtained the atmospheric extinction coefficient at a wavelength of 1.064 μm. Research on the impact of the above documents on laser transmission includes theoretical research, experimental research and numerical simulation research. Theoretical research is based on the theory of laser propagation attenuation in the atmosphere. In many cases, analytical expressions cannot be obtained; experimental research requires real laser systems and field test conditions, and the work is complicated and costly ^[9]. Therefore, as a numerical simulation method, Monte Carlo method can simulate all different situations, avoid a lot of experiments, and reflect the real situation of laser transmission to a certain extent. Based on the Monte Carlo method, this paper quantitatively analyzes the spatial distribution of photons on the receiving plane and the temporal distribution of photons on the receiving plane, and compares the calculation results with the MODTRAN online calculation results.

2,

Theoretical Analysis

2.1,

Analysis of laser attenuation characteristics in fog

Generally, the scattering generated when the size of the droplet particles is similar to the laser wavelength, which AOPC 2020: Advanced Laser Technology and Application, edited by Zhiyi Wei, Jing Ma, Wei Shi, Xuechun Lin, can be solved by Mie theory. Due to the arbitrariness and randomness of the orientation of the droplet particles in space and the small particle size, it is statistically regarded as an isotropic spherical particle for analysis ^[2].

According to the Mie scattering theory, the attenuation cross section σ_t and scattering cross section of a single droplet particle σ_s are defined as follows ^[5]:

Where a_n and b_n are Mie coefficients, k = 2π/λ,n is the eigenvalue.

At present, the commonly used droplet Gamma particle size distribution model is ^[6]:

Where N is the number density of droplet particles and r is the radius of droplet particles. r is in meters.

Where V is the visibility of fog (km) and W is the water content (g/m³).

To calculate the attenuation coefficient of fog droplets for laser transmission, the relationship between visibility and attenuation coefficient can be approximated ^[8].

Where μ_t0.55 is the attenuation coefficient of the most sensitive wavelength of the human eye. The attenuation coefficient and scattering coefficient of fog droplets to laser transmission can also be obtained by using the attenuation cross section and scattering cross section:

Where n(r) is the probability density function of the droplet particle size distribution ^[5], N_i is the total number of particles of component i per unit volume, and the two characteristic parameters r_modN,i and σ_i are the average radius and standard deviation, respectively.

The asymmetry factor g of the particles indicates the degree of asymmetry of the front and back scattering. The calculation formula is as follows ^[11]:

Where, Q_s is the scattering efficiency factor, and ω = μ_s/μ_t is the single scattering efficiency.

Table 1 shows the calculation results of the extinction parameters of 1.064 μm laser with different droplet particles using the above formula, where the visibility is V = 100m and the complex refractive index of the droplet on the infrared laser is n=1.327-2.89*10-6i ^[6], and the scale parameter of different droplet particles can be found in^[10].

Table 1

the comparison of the calculation results of the extinction parameters in different fogs

Fog	Radiation fog	Advection fog	Misty rain
Attenuation coefficient	0.0187	0.0191	0.0200
Single scattering rate	0.9999	0.9996	0.9999
Asymmetry factor	0.7761	0.8554	0.7832

Through theoretical analysis of droplet particles, it is found that the attenuation of fog on a laser with a wavelength of 1.064 μm is closely related to the type of fog. Different types of droplets have different particle radius and different attenuation coefficients for light. At the same time, when the visibility is V = 1000m, the attenuation coefficient of advection fog is 0.002, the single scattering rate is 0.9999, and the asymmetry factor is 0.7214.

2.2,

Monte Carlo calculation model

2.2.1,

Initial beam analysis

According to the theory of laser resonator cavity, the self-reproducing mode generated by the stabilized cavity follows the gaussian distribution, and the output of the optical cavity is the gaussian beam. In the plane perpendicular to the transmission direction of the laser beam, its light intensity distribution is ^[12].

Where, γ_S is the RMS radius of the laser beam, which is used to describe the radial distribution of the beam, x and y are the abscissa and ordinate on the plane perpendicular to the transmission direction of the laser beam, respectively.

In the source sampling of the Monte Carlo method, a Gaussian distribution sampling model is introduced to make the statistical characteristics of the emitted photons satisfy the characteristics of the Gaussian beam.

2.2.2

Analysis of photon transport in fog

The basic idea of the Monte Carlo method to simulate photon transmission is: photons enter the random medium with a certain direction and weight and initial time, then the next transmission step is given after being scattered by the droplet particles, and the scattering phase function is sampled to obtain new direction of motion after scattering. This process is repeated until the photon is received by the target screen or the weight is less than the threshold (10^-6 in this paper)^[6]. The propagation process is shown in Figure 1 ^[4].

Figure 1

is a schematic diagram of laser transmission in fog

The motion state of the photon is represented by the spatial position (x,y,z), the moving direction (θ, φ), and the photon weight W. θ and φ are the scattering angle and azimuth angle, respectively. The geometric path length of each step of the photon is as follows.

Where ε_r1 is a random number evenly distributed between [0,1]. After the step l is determined, the next collision position (x’, y’, z’) of the photon is determined by the current photon position (x,y,z) and the direction cosine (μ_x,μ_y,μ_z).

Then determine whether the photon is scattered. Taking a random number ε_r2 evenly distributed between [0,1]. If ε_r2 < ω, the photon is scattered; otherwise, the photon is absorbed and the tracking of the photon is terminated ^[13]. Assuming that the initial weight of the photon is W = 1, after scattering, the weight of the photon becomes W = W·ω.

After the photon is scattered, the new propagation direction is determined by the scattering phase function, and the Henyey-Greenstein function can be used to approximate its scattering phase probability function; sometimes in order to obtain a higher accuracy of the scattering phase probability function, the weighted sampling method can also be used find the phase function ^[14]. In this paper, the Henyey-Greenstein function is selected in article ^[15], which has the following form.

Where g is the asymmetry factor. Assuming that the scattering azimuth φ is evenly distributed in [0, 2π], the sampled value can be expressed as:

Where, ε_r3 is a random number evenly distributed between [0,1]. According to the scattering direction of the photon, the new direction cosine in the global coordinate system after the photon collision can be obtained by coordinate transformation as

3.

Simulation results and analysis

As shown in Figure 1, after the laser been transmitted for a distance L, a receiving plane is set perpendicular to the propagation direction to study the spatial distribution of photons. The total number of simulated photons is 2000000, The divergence angle β is 0.001rad. The attenuation coefficient μ_t, the single scattering rate ω), and the asymmetry factor g are given in ^[6]. The influence of the propagation distance on the spatial distribution of photons is firstly considered in the receiving plane. The Monte Carlo simulation results are shown in Figure 2, where the visibility V = 100m and the initial beam RMS radius γ_s =0.5m. Then, the influence of visibility on the spatial distribution of photons in the receiving plane is considered. The Monte Carlo simulation results are shown in Figure 3, where the propagation distance L = 10m and the initial beam root-mean-square radius γ_s =0.5m. Finally, we consider the effect of the initial RMS radius on the spatial distribution of the photons in the receiving plane is considered. The Monte Carlo simulation results are shown in Figure 4, where the propagation distance L = 20m and visibility V = 100m.

Figure 2

shows the spatial distribution of the received plane photons. The propagation distance is (a) 10m (b) 50m (c) 70m (d) 100m

Figure 3

shows the spatial distribution of the received photons. The visibility is (a) 10m (b) 30m (c) 70m (d) 100m

Figure 4

shows the spatial distribution of the received photons. The laser’s initial RMS radius is (a) 10m (b) 30m (c) 70m (d) 100m

It can be seen from Figure 2 that with the increase of the propagation distance, the spatial distribution of photons at the receiving plane has not significantly deviated from the Gaussian distribution, but the number of photons reaching the receiving plane has decreased significantly. This is because the farther the propagation distance, the greater the probability of photons being scattered, resulting in a reduction in the number of photons received. It can be seen from Figure 3 that the number of photons at the receiving plane decreases first and then increases with the increase of visibility, but the spatial distribution of photons has not deviated significantly from the Gaussian distribution. This is because the density of droplet particles decreases as the visibility increases, the probability of photons being scattered decreases, and the number of photons reaching the receiving plane increases. It can be seen from Figure 4 that the larger the initial RMS radius of the laser, the more dispersed the spatial distribution of photons, but it has not deviated significantly from the Gaussian distribution. Figures 2 and 4 are consistent with the conclusions of reference ^[4]. When the visibility is less than 20m in Figure 3, the law is slightly different from reference ^[4], showing the randomness of photon propagation in dense fog.

To further analyze the quantitative relationship between the spatial distribution of photons and various parameters, Figure 5 shows the correspondence between the number of photons reaching the receiving plane and the propagation distance, visibility, and initial RMS radius. As shown in Figure 5, as the propagation distance increases, the number of photons arriving at the receiving plane decreases approximately linearly; as the visibility increases, the number of photons reaching the receiving plane decreases slightly at first, and then increases rapidly; As the root radius increases, the number of photons reaching the receiving plane increases slowly, but the change is not obvious and is roughly linear.

Figure 5

shows the relationship between the number of photons arriving at the receiving plane and the three parameters (a) propagation distance (b) visibility (c) initial RMS radius

Figure 6 shows the normalized photon time distribution received at different distances. It can be seen from Figure 6 that for a certain propagation distance, the number of normalized photons received at the receiving plane increases rapidly with time and then decreases slowly; while with the increase of the propagation distance, the peak time of receiving photons at the receiving plane gradually increases.

Figure 6

shows the time distribution of photons received at the receiving plane at different distances

In order to further understand the relationship between the photon peak time at the receiving plane and the propagation distance, we extracted the relevant data, and the results are shown in Figure 7. Figure 7 shows the relationship between the peak time of the photon received by the receiving plane and the propagation distance. It can be seen that this relationship basically shows a linear change rule.

Figure 7

shows the relationship between the peak time of the photon received by the receiving plane and the propagation distance.

In addition, the parameter visibility V = 1000m, the propagation distance L = 1000m, and the attenuation coefficient μ_t1.064 = 0.002 are substituted into the atmospheric transmittance formula T = exp(–μ · L) to calculate the atmospheric transmittance of 0.1353.

Take the urban surface albedo 0.15 ^[16], substitute MODTRAN online calculation results to get T = 0.1253, the relevant conclusions of the two are basically the same.

4.

Conclusion

The numerical simulation by Monte Carlo method shows that as the propagation distance increasing, the number of photons reaching the receiving plane decreases approximately linearly, but the spatial distribution does not deviate significantly from the Gaussian distribution. It begins to decrease slowly at first and then increases, but the spatial distribution has not significantly deviated from the Gaussian distribution; with the increase of the initial RMS radius, the number of photons reaching the receiving plane does not change initially and then increases slowly. In addition, for a certain propagation distance, the number of normalized photons received at the receiving plane increases rapidly with time and then decreases slowly; while with the increasement of the propagation distance, the peak time of receiving photons at the receiving plane gradually increases.