# Modeling of Scattering Cross Section for Mineral Aerosol with a Gaussian Beam

Authors: Wenbin Zheng; Hong Tang
Journal: Journal of Nanotechnology
 (2018)

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## Abstract

Based on the generalized Lorenz Mie theory (GLMT), the scattering cross section of mineral aerosol within the Gaussian beam is investigated, and an appropriate modeling of the scattering cross sections for the real mineral aerosols including the feldspar, quartz, and red clay is proposed. In this modeling, the spheroid shape is applied to represent the real nonspherical mineral aerosol, and these nonspherical particles are randomly distributed within the Gaussian beam region. Meanwhile, the Monte Carlo statistical estimate method is used to determine the distributed positions of these random nonspherical particles. Moreover, a method for the nonspherical particles is proposed to represent the scattering cross section of the real mineral aerosols. In addition, theT matrix method is also used to calculate the scattering cross sections of the spheroid particles in order to compare the scattering properties between the plane wave and the Gaussian wave. Simulation results indicate that fairly reasonable results of the scattering cross sections for the mineral aerosols can be obtained with this proposed method, and it can provide a reliable and efficient approach to reproduce the scattering cross sections of the real randomly distributed mineral aerosols illuminated by the Gaussian beam.


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## Body

## 1. Introduction

The problem of light scattering by particles has been an important topic of research interest in the wide areas of applications [1–5]. Since the light scattering has been established, some researches have studied the electromagnetic light scattering of particles for the plane wave case [1]. And some common theories and methods have been utilized to analyze this problem. When the plane light wave is incident into the particle, the classical Mie theory, the discrete dipole approximation (DDA), the T matrix method, and the finite difference time domain (FDTD) method can be used for sphere and nonsphere particles [6–9]. Nevertheless, none of those methods can be applied to analyze and calculate the scattering of particles for the nonplane waves such as the Gaussian beam incidence or a top-hat beam.In recent years, with the development of laser technique and the expansion of its application areas, the laser has been used for the measuring of particle sizing and other particle prosperities. It is well known that the generalized Lorenz Mie theory (GLMT) proposed by Gouesbet is a generalization of the Lorenz Mie theory for an arbitrary incident-shaped beam such as the Gaussian beam (laser in the fundamental mode TEM00) and the light sheet [10–12].In this paper, the scattering cross sections for the practical nonspherical mineral aerosols are investigated, and modeling of the scattering cross sections for the feldspar, quartz, and red clay is conducted. Actually, the nonspherical calculations and measurements show significant differences from the sphere particles [13–16]. Here, we choose the spheroid to represent the nonspherical mineral aerosol in order to study the scattering cross sections of nonspherical particles within the Gaussian beam incidence.


## 2. Theoretical Method and Calculations

In the GLMT framework, the incident Gaussian beam field including the electric and the magnetic fields can be described by the Bromwich Scalar Potentials (BSP) in the spherical coordinate system (r, θ, φ) [10–12]. The field components are then found to be(1)Eri=∂2UTM∂r2+k2UTM,Eθi=1r∂2UTM∂r∂θ−iωμrsinθ∂UTE∂φ,Eφi=1rsinθ∂2UTM∂r∂φ+iωμr∂UTE∂θ,Hri=∂2UTE∂r2+k2UTE,Hθi=iωεrsinθ∂UTM∂φ+1r∂2UTE∂r∂θ,Hφi=1rsinθ∂2UTE∂r∂φ−iωεr∂UTM∂θ,where UTM and UTE are the transverse magnetic (TM) and transverse electric (TE) BSP, respectively; i=−1; Ei and Hi are called electric field and magnetic introduction field, respectively; r,θ,φ is the spherical coordinate system; k is the wave number; ω is the angular frequency of the electromagnetic wave; and μ and ε are the permeability and the permittivity of the medium, respectively.The scattering cross section and extinction cross section of particle are evaluated by(2)Csca=λ2π∑n=1∞∑m=−nn2n+1nn+1n+m!n−m!an2gn,TMm2+bn2gn,TEm2,Cext=∫0π∫02π12ReEφiHθs∗+EφsHθi∗−EθiHφs∗−EθsHφi∗r2sinθdθdφ,where gn,TMm and gn,TEm are the generalized functions of GLMT, an and bn are the scattering coefficient of Mie theory, and E and H are the magnetic and electric energy.According to the GLMT, the particle is randomly located in the Gaussian beam, and the scattering properties are also determined by the location information in the beam. In Figure1, the beam propagates alone the z axis from negative z to positive z, and the electric field component is essentially vibrating in the x axis. The coordinate origin o is the beam waist center, and its waist radius is w0. The unit of the incident wavelength λ and the particle radius r, as well as the beam waist radius w0, is μm [17].Figure 1
Geometry of coordinate of the incident Gaussian beam.For a measured particle system, the particles can be distributed anywhere within the Gaussian beam [18]. Here, we define the particles in the semicircular region, that is, 0≤z≤3πw02/λ,x2+y2≤2w02.Table1 shows the scattering cross section of the sphere particle with different location in the Gaussian beam. The particle radius r is 0.1, the incident wavelength λ is 0.5, the waist radius w0 is 1, and the relative complex reflective index of particle m is 1.33. The six positions in the Cartesian coordinate are (0, 0, 0), (0, 1, 0), (0, 2, 10.88), (0, 0, 10.88), (0.1, 0.1, 0.1), and (0.1, 0.1, 1). It is very obvious that the scattering cross section of sphere particle is also different when the position of particle is different. In Table 2, the scattering cross section and absorbing cross section of the sphere particle are calculated in the Gaussian beam. The relative complex reflective index of particle m is 1.33 + 0.1i, the particle radius r is 0.1, the incident wavelength λ is 0.5, and the waist radius w0 is 1. We can see that the absorbing cross section of sphere particle is still different with the different position in the Gaussian beam.Table 1
Scattering cross section of sphere particle with the different position (r = 0.1, λ = 0.5, m = 1.33).
Position
(0, 0, 0)
(0, 1, 0)
(0, 2, 10.88)
(0, 0, 10.88)
(0.1, 0.1, 0.1)
(0.1, 0.1, 1)
C
sca
0.601405E−14
0.828364E−15
0.206664E−15
0.153682E−14
0.578179E−14
0.564966E−14Table 2
Scattering and absorbing cross section of sphere particle with the different location. (r = 0.1, λ = 0.5, m = 1.33 + 0.1i).
Position
(0, 0, 0)
(0, 1, 0)
(0, 2, 10.88)
(0, 0, 10.88)
(0.1, 0.1, 0.1)
(0.1, 0.1, 1)
C
sca
0.579785E−14
0.798824E−15
0.199261E−15
0.148166E−14
0.557396E−14
0.544659E−14
C
abs
0.105321E−13
0.147600E−14
0.365064E−15
0.270385E−14
0.101278E−13
0.989770E−14
(r = 0.1, λ = 0.5, m = 1.33 + 0.1i).In order to investigate the general average scattering cross sections of nonspherical mineral particles, three aerosols, that is, the feldspar, quartz, and red clay, are studied, and the spheroid model is used to represent the real nonspherical mineral particle. Since the scattering prosperities of particles within the Gaussian beam are related with the position information, the Monte Carlo statistical estimate method is used to determine the distributed positions of these random nonspherical particles [19, 20]. After making the average of the position information, we can obtain the general location for the random particles, and then the average scattering cross section of particles is also calculated with the GLMT framework.Figure2 gives the scattering cross sections of spheroid quartz particles with different sizes. For a spheroid, there are two parameters representing its shape, that is, the aspect ratio and the radius of rotation axis. Here, we use the equal-surface area sphere to represent the spheroid particle, and the equal-surface diameter can be calculated by the spheroid particle. In Figure 2, w0 is equal to 1 and 10, respectively, and the T matrix method is also used to obtain the scattering cross sections of spheroid particles for the plane wave as a comparison [21]. The relative complex reflective index of quartz particle m is 1.54, and the incident wavelength λ is 0.5. We can see that serious differences are occurred between the Gaussian beam and the plane wave, and the differences are decreasing with larger w0. That is because the Gaussian beam gradually becomes the plane wave when w0 is infinite. Figure 3 describes the scattering cross sections of spheroid quartz particles with aspect ratio = 1/3 and w0 = 1. The parameter in the horizontal axis is the diameter of rotation axis, and the T matrix method is also used to obtain the scattering cross sections of spheroid particles for the plane wave as a comparison.Figure 2
Scattering cross sections of quartz particles (λ = 0.5, m = 1.54).Figure 3
Scattering cross sections of quartz particles (aspect ratio = 1/3,λ = 0.5, m = 1.54).Figure4 gives the scattering and absorbing cross sections of feldspar particles. Figure 5 is the scattering and absorbing cross sections of feldspar particles with aspect ratio = 2. For the feldspar particles, the imaginary part of the complex reflective index is not zero, and then the absorbing cross sections of feldspar particles can be calculated. With the increasing equal-surface diameter or the diameter of rotation axis, the differences between the Gaussian beam and plane wave are enlarged.Figure 4
Scattering and absorbing cross sections of feldspar particles (λ = 0.5, m = 1.5 + 0.001i).
(a)
(b)Figure 5
Scattering and absorbing cross sections of feldspar particles (aspect ratio = 2,λ = 0.5, m = 1.5 + 0.001i).
(a)
(b)Figure6 gives the scattering and absorbing cross sections of red clay particles. Figure 7 is the scattering and absorbing cross sections of red clay particles with aspect ratio = 1/2. For the red clay particles, the real part of the complex reflective index is larger than that of the feldspar particles and the quartz particles. The differences are relatively small between the Gaussian beam and plane wave with w0 = 10, compared with the differences of scattering and absorbing cross sections of the feldspar particles and the quartz particles.Figure 6
Scattering and absorbing cross sections of red clay particles (λ = 0.5, m = 1.7 + 0.001i).
(a)
(b)Figure 7
Scattering and absorbing cross sections of red clay particles (aspect ratio = 1/2,λ = 0.5, m = 1.7 + 0.001i).
(a)
(b)Figures8 and 9 show the scattering and absorbing cross sections of spheroid particles with 1.5 + 0.01i. In Figure 9, the incident wavelength is 1.2. According to this simulation, there are still differences between the Gaussian beam and plane wave, and the differences are smaller with larger incident wavelength.Figure 8
Scattering and absorbing cross sections of spheroid particles (λ = 0.5, m = 1.5 + 0.01i).
(a)
(b)Figure 9
Scattering and absorbing cross sections of spheroid particles (λ = 1.2, m = 1.5 + 0.01i).
(a)
(b)


## 3. Conclusions

In this paper, the scattering cross sections of nonspherical mineral particles are investigated with in the Gaussian beam based on the GLMT. In the framework of GLMT, the general location information is statistic by the Monte Carlo statistical estimate method, and the scattering cross sections of spheroid particles including the feldspar, quartz, and red clay are calculated. Actually, the spheroid shape can represent the nonspherical feldspar, quartz, and red clay particles with good accuracy. In order to research the scattering of spheroid particle with more efficiency, a sphere of the same surface area as the spheroid is used to calculate the scattering cross sections of spheroid particles within the Gaussian beam, and then the scattering cross sections of spheroid particles are also calculated within the plane wave incidence. The results show that the scattering cross sections and absorbing cross sections of spheroid are different from the plane wave, and the differences are more obvious with the increasing diameters. Meanwhile, the incident wavelength and the complex reflective index of mineral particles also have effects on the scattering cross sections of nonspherical mineral particles.



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*Source: 6513634-2018-03-28.xml*