## 1. Introduction

The investigation of light propagation and scattering by various complex particles is of great importance in a wide range of scientific fields, and it has lots of practical applications, such as detection of atmospheric pollution, optical diagnostics for aerosols, remote sensing of disasters [1–3]. Over the past few decades, some theories and numerical methods have been developed to study the light wave propagation and scattering through various particles. For the particles with special shape, such as spheres, spheroids, and cylinders, the generalized Lorenz-Mie theory (GLMT) [4–15] can obtain an analytic solution in terms of a limited linear system of equations using the method of separation of variables to solve the Helmholtz equation in the corresponding coordinate system. For the complex particles of arbitrary shapes and structure, some numerical methods, such as the discrete dipole approximation (DDA), the method of moments (MOM), the finite element method (FEM), and the finite-difference time-domain (FDTD), have been utilized. For random media composed of many discrete particles, the T-matrix method, the sparse-matrix canonical-grid (SMCG) method, and the characteristic basis function method (CBFM) can be applied to obtain simulation results.

This chapter discusses the light propagation and scattering through particles. Without loss of generality, the incident light is assumed to be Gaussian laser beam, which can be reduced to conventional plane wave. The detailed description of the solution methods, numerical results, and potential application of the light scattering by systems of particles is introduced.

## 2. Light scattering by regular particles

### 2.1. Light scattering by a homogeneous sphere

The geometry of light scattering of a Gaussian beam by a homogeneous sphere is illustrated in **Figure 1**. As shown in **Figure 1**, two Cartesian coordinates

Within the framework of the GLMT, the electromagnetic field components of the illuminating beam are described by partial wave expansions over a set of basic functions, e.g., vector spherical wave functions in spherical coordinates, vector spheroidal wave functions in spheroidal coordinates, and vector cylindrical wave functions in cylindrical coordinates. The expansion coefficients or sub-coefficients are named as beam-shape coefficients (BSCs) are denoted as *X* is TE, transverse electric, or TM, transverse magnetic, with

where the superscript “

The

(5) |

the index corresponds to the spherical Bessel functions of the first, second, third, or fourth kind (

The electric component of the internal field and the scattered field can be expanded in terms of vector spherical wave functions in the particle coordinate system

To solve the scattering problem, light scattering, the scattering coefficients

where

where

**Figure 2** presents the normalized DSCS for the scattering of a Gaussian beam by a homogeneous spherical dielectric particle. The radius of the spherical particle is

### 2.2. Light scattering by a spheroidal particle

Light scattering by a spheroid has been of great interest to many researchers in the past several decades since it provides an appropriate model in many practical situations. For example, during the atomization processes, the shape of fuel droplets departs from sphere to spheroid when it impinges on the wall and breaks. Due to the inertial force, the raindrop also departs from the spherical particle to the near-spheroidal one. A rigorous solution to the scattering problem concerning a homogeneous spheroid illuminated by a plane wave was first derived by Asano and Yamamoto [16]. It was extended later to the cases of shaped beam illumination [17], a layered spheroid [18], and a spheroid with an embedded source [19]. Nevertheless, only parallel incident of the shaped beam, including on-axis and off-axis Gaussian beam scattered by a spheroid was studied, that is to say, the propagation direction of the incident beam is assumed to be parallel to the symmetry axis of the spheroid. An extension of shaped beam scattering with arbitrary incidence was developed within the framework of GLMT by Han et al. [20–22] and Xu et al. [23, 24].

To deal with the shaped beam scattering of a spheroidal particle within the framework of GLMT, the incident Gaussian beam is required to be expanded in terms of the vector spheroidal wave functions in spheroidal coordinates, which can be achieved using the relationship between the vector spherical wave functions and the spherical wave functions. The geometry of shaped beam scattering by a prolate spheroid is illustrated in **Figure 3**. According to the expansion of shaped beam in unrotated spherical coordinates in Eq. (1), we can rewrite it as

(13) |

where we have

(14) |

where

Considering the vector spheroidal wave functions in the spheroidal coordinates, whose explicit expressions are the same as the ones used in Refs. [14, 25], the relationship between the vector spherical wave functions and vector spheroidal wave functions is given as:

From Eq. (15), we can obtain the expansion of Gaussian beam in spheroidal coordinates. Accordingly, the electric component of the internal field and the scattered field can be expanded in terms of vector spheroidal wave functions. The unknown scattered coefficients can be determined by applying the boundary conditions of continuity of the tangential electromagnetic fields over the surface of the particle. Thus, the solution of scattering for Gaussian beam by a spheroidal particle can be obtained.

For the purpose of demonstration, **Figure 4** shows angular distributions of the DSCS for a spheroid with a semimajor axis and a semiminor axis being

### 2.3. Light scattering by a circular cylindrical particle

The geometry of shaped beam scattering by a circular cylinder is illustrated in **Figure 5**. Similarly to a spheroidal particle, due to the lack of spherical symmetry, the arbitrary orientation is also compulsory in the case of GLMTs for cylinders. The expansion of the case of an arbitrary-shaped beam propagating in an arbitrary direction, based on which an approach to expand the shaped beam in terms of cylindrical vector wave functions natural to an infinite cylinder of arbitrary orientation is given below.

The vector cylindrical wave functions in the cylindrical coordinates

(16) |

where

The relationship between the vector spherical wave functions and the vector cylindrical wave functions is defined as

(17) |

where

Based on Eqs. (17) and (1), we can obtain the expansion of the incident shaped beam in terms of the cylindrical vector wave functions in cylindrical coordinates as

where

Accordingly, the electric component of the internal field and the scattered field can be expanded in terms of cylindrical vector wave functions. The unknown scattered coefficients can be determined by applying the boundary conditions of continuity of the tangential electromagnetic fields over the surface of the particle. Thus, the solution of scattering for Gaussian beam by a cylindrical particle can be obtained.

## 3. Light scattering by complex particles of arbitrary shapes and structure

### 3.1. Surface integral equation method

Many particles encountered in nature or produced in industrial processes, such as raindrops, ice crystals, biological cells, dust grains, daily cosmetics, and aerosols in the atmosphere, not only have irregular shapes but also have complex structures. The study of light scattering by these complex particles is essential in a wide range of scientific fields with many practical applications, including optical manipulation, particle detection and discrimination, design of new optics devices, etc. Here, we introduce the surface integral equation method (SIEM) [26–28] to simulate the light scattering by arbitrarily shaped particles with multiple internal dielectric inclusions of arbitrary shape, which can be reduced to the case of arbitrarily shaped homogeneous dielectric particles. For SIEM, the incident Gaussian beam can be described using the method of combing Davis-Barton fifth-order approximation [29] in combination with rotation Euler angles [30].

Now, let us consider the problem of Gaussian beam scattering by an arbitrarily shaped particle with multiple dielectric inclusions of arbitrary shape. As illustrated in **Figure 6**, let

where the integral operators

in which the subscript “0” represents the medium in which the scattered fields are computed and the superscript “

Also based on the surface equivalence principle, the fields

where

By enforcing the continuity of the tangential electromagnetic fields across each surface, the following integral equations may be established

(31) |

(32) |

where the subscripts “

(33) |

The resultant matrix Eq. (33) can also be solved iteratively by employing the multilevel fast multipole algorithm (MLFMA). Once obtained the unknown equivalent electromagnetic currents, the far-zone scattered fields and DSCS can be calculated.

### 3.2. Numerical results

First, we consider the reduced case of arbitrarily shaped homogeneous dielectric particles. To illustrate the validity of the proposed method, the scattering of a focused Gaussian beam by a homogeneous spherical dielectric particle is considered. The radius of the spherical particle is **Figure 7** shows the computed DSCS as a function of the scattering angle in the E-plane. For comparison, the results obtained using the GLMT are given in the same figure. Excellent agreements are observed between them.

To illustrate the validity of the proposed method for composite particles with inclusions, we consider the scattering a Gaussian beam by a spheroidal particle with a spherical inclusion at the center, as shown in **Figure 8**. The semimajor axis and the semiminor axis of the host spheroid are **Figure 9**. For comparison, the result obtained using the analytical theory GLMT is given in the same figure. Excellent agreements are observed between them.

Finally, the scattering of an obliquely incident Gaussian beam by a cubic particle containing 27 randomly distributed spherical inclusions is considered to illustrate the capabilities of the proposed method. The center of the host cube is located at the origin of the particle system and the side length of the cube is **Figure 10**. The host cube is characterized by refractive index **Figure 11** presents the simulated DSCSs as a function of the scattering angle in both the E-plane and the H-plane.

## 4. Light scattering by random discrete particles

Due to the wide range of possible applications in academic research and industry, the problem of light scattering by random media composed of many discrete particles is a subject of broad interest. Over the past few decades, some theories and numerical methods have been developed to study the light scattering by random discrete particles [32–48]. In this section, we introduce a hybrid finite element-boundary integral-characteristic basis function method (FE-BI-CBFM) to simulate the light scattering by random discrete particles [49]. In this hybrid technique, the finite element method (FEM) is used to obtain the solution of the vector wave equation inside each particle and the boundary integral equation (BIE) is applied on the surfaces of all the particles as a global boundary condition. To reduce computational burdens, the characteristic basis function method (CBFM) is introduced to solve the resultant FE-BI matrix equation. The incident light is assumed to be Gaussian laser beam.

### 4.1. FE-BI-CBFM for random discrete particles

Now, let us consider the scattering of an arbitrarily incident focused Gaussian beam by multiple discrete particles with a random distribution, as depicted in **Figure 12**. For simplicity, the background region, which is considered to be free space, is denoted as

where

where

Since Eq. (35) is independent of the excitation, we can remove the interior unknowns to derive a matrix equation that only includes the unknowns on

where

For the convenience of description, we write the relation between

where

It is worth to notice that the calculations of Eqs. (37) and (39) in each particle are independent and can be completely parallelized. Furthermore, since the particles are uniform, the coefficient matrices are the same for each particle. This implies that only one particle needs to be dealt with to obtain all the matrices

To formulate the field in region

Enforcing boundary condition on

and a magnetic field integral equation (MFIE)

where the subscript “

where the subscript

(46) |

The expressions of the elements for matrices

(47) |

The above equation can be written in a more compact form as

where

### 4.2. Numerical results and discussion

In what follows, some numerical results are presented. First, we consider the scattering of Gaussian beam by 125 randomly distributed conducting spherical particles with a radius of **Figure 12**. The particle positions are generated randomly in a cubic box with which the fractional volume is **Figure 13** as a function of the scattering angle in the E-plane. As can be seen from the figure, the DSCS for Gaussian beams is smaller than that for a plane wave. In addition, for a Gaussian beam incidence with a relatively large waist radius of

We then consider the multiple scattering of an obliquely incident Gaussian beam by 512 randomly distributed inhomogeneous spherical particles with which the fractional volume is **Figure 14** presents the simulated DSCSs as a function of the scattering angle.

Finally, we use the present numerical method to simulate the multiple scattering of Gaussian beam by 1000 randomly distributed homogeneous dielectric spherical particles with which the fractional volume is **Figure 15**. Furthermore, the DSCS of the ensembles of randomly distributed particles for the independent scattering is also calculated. Specifically, an individual particle is assumed to scatter light without interactions with other particle in the ensemble. The computed DSCS for the independent scattering is displayed in **Figures 16** and **17**. Comparisons between independent scattering and the multiple scattering are made. The results show that the interactions of the particles lead to a reduction in the scattering intensities, which are identical to the general idea of scattering theory.