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

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, the T 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.


Introduction
e problem of light scattering by particles has been an important topic of research interest in the wide areas of applications [1][2][3][4][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][7][8][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][11][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][14][15][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.

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][11][12].
e field components are then found to be where U TM and U TE are the transverse magnetic (TM) and transverse electric (TE) BSP, respectively; i −1 √ ; E i and H i are called electric eld and magnetic introduction eld, 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.e scattering cross section and extinction cross section of particle are evaluated by where g m n,TM and g m n,TE are the generalized functions of GLMT, a n and b n are the scattering coe cient 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 Figure 1, the beam propagates alone the z axis from negative z to positive z, and the electric eld component is essentially vibrating in the x axis.e coordinate origin o is the beam waist center, and its waist radius is w 0 .e unit of the incident wavelength λ and the particle radius r, as well as the beam waist radius w 0 , is μm [17].
For a measured particle system, the particles can be distributed anywhere within the Gaussian beam [18].Here, we de ne the particles in the semicircular region, that is, 0 1 shows the scattering cross section of the sphere particle with di erent location in the Gaussian beam.e particle radius r is 0.1, the incident wavelength λ is 0.5, the waist radius w 0 is 1, and the relative complex re ective index of particle m is 1.33.e 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 di erent when the position of particle is di erent.In Table 2, the scattering cross section and absorbing cross section of the sphere particle are calculated in the Gaussian beam.e relative complex re ective 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 w 0 is 1.We can see that the absorbing cross section of sphere particle is still di erent with the di erent position in the Gaussian beam.
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  2 Journal of Nanotechnology 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.
Figure 2 gives the scattering cross sections of spheroid quartz particles with di erent 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 equalsurface diameter can be calculated by the spheroid particle.In Figure 2, w 0 is equal to 1 and 10, respectively, and the Tmatrix method is also used to obtain the scattering cross sections of spheroid particles for the plane wave as a comparison [21].
e relative complex re ective index of quartz particle m is  1.54, and the incident wavelength λ is 0.5.We can see that serious di erences are occurred between the Gaussian beam and the plane wave, and the di erences are decreasing with larger w 0 .at is because the Gaussian beam gradually becomes the plane wave when w 0 is in nite.Figure 3 describes the scattering cross sections of spheroid quartz particles with aspect ratio 1/3 and w 0 1. e 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 4 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 re ective 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 di erences between the Gaussian beam and plane wave are enlarged.
Figure 6 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 re ective index is larger than that of the feldspar particles and the quartz particles.e di erences are relatively small between the Gaussian beam and plane wave with w 0 10, compared with the di erences of scattering and absorbing cross sections of the feldspar particles and the quartz particles.Figures 8 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 di erences between the Gaussian beam and plane wave, and the di erences are smaller with larger incident wavelength.

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 e ciency, 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.e results show that the scattering cross sections and absorbing cross sections of spheroid are different from the plane wave, and the di erences are more obvious with the increasing diameters.Meanwhile, the incident wavelength and the complex re ective index of mineral particles also have e ects on the scattering cross sections of nonspherical mineral particles.

Figure 1 :
Figure 1: Geometry of coordinate of the incident Gaussian beam.