Invariant Imbedding T-matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles

By Bingqiang Sun, Lei Bi, Ping Yang and 2 more

Invariant Imbedding T-matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles propels atmospheric research forward as a resource and a tool for understanding the T-Matrix method in relation to light scattering. The text explores concepts ranging from electromagnetic waves and scattering dyads to the fundamentals of the T-Matrix method. Providing recently developed material, this text is sufficient to aid the light scattering science community with current and leading information.

Enriched with detailed research from top field experts, Invariant Imbedding T-matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles offers a meaningful and essential presentation of methods and applications, with a focus on the light scattering of small and intermediate particles that supports and builds upon the latest studies. Thus, it is a valuable resource for atmospheric researchers and other earth and environmental scientists to expand their knowledge and understanding of available tools.

• Systematically introduces innovative methods with powerful numerical capabilities
• Thoroughly presents the rudimentary principles of light scattering and the T-matrix method
• Offers a condensed and well-ordered arrangement of text, figures and formulas that are serviceable for both students and researchers

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 18, 2019
• ISBN: 9780128180914

Bingqiang Sun

Bingqiang Sun is an Assistant Research Scientist in the Department of Atmospheric Sciences at Texas A&M University, College Station, TX, USA. He has published 10 journal papers. He received a B.S. degree from Shandong Normal University, Shandong, China, in 2007, M.S. degree from Peking University, Beijing, China, in 2010, and Ph.D. degree from the Department of Physics & Astronomy at Texas A&M University in 2014. He received the Yuxiang Young Scholar Award by Chinese-American Oceanic and Atmospheric Association (COAA) in 2017.

Book preview

Invariant Imbedding T-matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles

First Edition

Bingqiang Sun

Assistant Research Scientist, Department of Atmospheric Sciences, Texas A&M University, College Station, Texas, United States

Lei Bi

Assistant Research Scientist, Department of Atmospheric Sciences, Texas A&M University, College Station, Texas, United States

Ping Yang

Professor, Department of Atmospheric Sciences Texas A&M University College Station, Texas, United States

Michael Kahnert

Adjunct Professor, Department of Space, Earth and Environment Chalmers University of Technology, Gothenburg, Sweden

Research Department, Swedish Meteorological and Hydrological Institute, Norrköping, Sweden

George Kattawar

Professor Emeritus, Department of Physics and Astronomy & Institute for Quantum Science & Engineering, Texas A&M University, College Station, Texas, United States

Table of Contents

Cover image

Title page

Copyright

Preface

1: Introduction

1.1 Particle shape and inhomogeneity

1.2 Size parameter

1.3 Random orientations

1.4 Invariant-imbedding principle

2: Fundamentals

2.1 Electromagnetic waves

2.2 Basic physical quantities for quantifying light scattering

2.3 A brief summary of rigorous methods for light scattering by nonspherical and inhomogeneous particles

3: T-matrix concept

3.1 A brief history

3.2 Expansion of the incident and scattered fields

3.3 A common approach to compute the T-matrix

3.4 Unitarity, symmetries, and random orientations

4: Invariant-imbedding T-matrix method

4.1 Electromagnetic volume integral equation

4.2 Concept of the invariant-imbedding T-matrix method

4.3 Application of the IITM to arbitrary particle morphologies

5: Application examples of optical properties of small-to-moderate size particles

5.1 Spherical and multilayered spherical particles

5.2 Axially symmetric particles: Spheroids and cylinders

5.3 Finite-fold rotationally symmetric particles—Hexagonal prisms

5.4 Asymmetric particles: Aggregates and hexahedra

5.5 Inhomogeneous particles

References

Index

Copyright

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Preface

The theories and numerical simulations of electromagnetic wave scattering by nonspherical and inhomogeneous particles have found diverse applications in atmospheric sciences, astronomy, engineering, chemistry, and biophysics. The subject of electromagnetic wave scattering has evolved from early studies of simple nonspherical geometries such as an infinite circular cylinder or a spheroid to arbitrarily shaped nonspherical and inhomogeneous particles. However, it is well known that obtaining the analytical solution of Maxwell's equations for an arbitrary particle is unlikely possible, although the mathematical formulation for light scattering by a homogeneous sphere was clearly and well established with the development of the Lorenz-Mie theory. From an application perspective, two associated aspects are critical to the computational capabilities of an electromagnetic-scattering solver: the stability of the algorithm and an advanced numerical implementation. The intent of this book is to present a new powerful computational tool, namely, the invariant imbedding T-matrix method (IITM), for computing the scattering and absorption properties of arbitrarily shaped nonspherical and inhomogeneous particles.

The motivation of writing this book is twofold. First, the T-matrix method may be the most accurate and efficient method for solving the scattering of an electromagnetic wave by a nonspherical particle. Second, for about three decades, the T-matrix method was considered to be practically applicable to axially symmetric and homogeneous particles, while significant efforts were devoted to applying the T-matrix method to nonsymmetric and inhomogeneous particles. At present, an accurate and versatile T-matrix implementation based on the extended boundary condition method (EBCM) (Waterman, 1971) for generally nonspherical and inhomogeneous particles is not available, particularly in the case of large size parameters. Our research efforts to apply the invariant imbedding T-matrix method started in 2013, although the invariant imbedding principle was introduced to obtain the T-matrix recurrence relation by Johnson (1988) in the framework of an electromagnetic volume integral equation. By combining several advances of the T-matrix method and solving relevant instability issues, the power of this technique allows for computing the optical properties of randomly oriented nonspherical and inhomogeneous particles with sizes much larger than the incident wavelength. This book is expected to be useful for active researchers and graduate students who have interests in light scattering and its applications in many disciplines.

The invariant imbedding technique was developed to handle the diffuse reflection of scattering and absorbing semiinfinite media by Ambarzumian (1943). Chandrasekhar (1960) generalized the technique to handle both the semi-infinite and finite medium cases and thus coined the name principles of invariance in Chapter IV of his classic book Radiative Transfer. Bellman and Wing (1975) then used invariant imbedding to coin the name of the technique and applied it to neutron transport theory. In particular, Bellman and Wing (1975) systematically described the invariant imbedding technique in their book An Introduction to Invariant Imbedding. In addition to radiative transfer, the invariant imbedding technique was first applied to electromagnetic wave scattering by Johnson (1988). With an advanced numerical implementation (Bi and Yang, 2014), the IITM was shown to be a powerful tool with far-reaching capabilities in numerical simulations of electromagnetic scattering by arbitrarily shaped particles and recently attracted a lot of attention in the electromagnetic scattering and atmospheric radiation communities. To better share this technique with these communities, we organize the principles and present applications of the IITM in this book.

Basically, the IITM is one of the most efficient methods to compute the T-matrix of a dielectric particle. Consequently, all properties with respect to the T-matrix, such as its symmetry and the analytical scattering phase matrix based on the T-matrix, are available to the IITM. Thus, the T-matrix method is a semianalytical method to solve light scattering by a scattering particle. This book gives a systematic introduction of the IITM in five chapters: terminology highlights, light scattering representation, the T-matrix concepts, the IITM algorithm, and IITM applications.

To better demonstrate the potential advantages of the IITM in later chapters, Chapter 1 highlights four topics: particle shape and inhomogeneity, size parameter, random orientation, and the invariant imbedding principle. Chapter 2 gives a systematic introduction to the representation of an electromagnetic wave and quantifying electromagnetic scattering by a dielectric particle. Maxwell's equations and their boundary conditions are reviewed in Section 2.1.1. Sections 2.1.2 and 2.1.3 present the energy density and the Poynting vector of a propagating electromagnetic wave. The polarization representation of an electromagnetic wave is given in Section 2.1.4. Section 2.2 systematically describes polarization effects caused by a dielectric particle. The topics include the amplitude scattering matrix and the scattering phase matrix and their symmetry properties and the extinction and scattering cross sections associated with single scattering and bulk scattering properties. A brief review of rigorous scattering computational methods is given in Section 2.3.

Chapter 3 gives a full introduction to the T-matrix method. A brief history and development of the T-matrix method is given in Section 3.1. Sections 3.2.1 and 3.2.2 describe how, in the T-matrix method, the electromagnetic field can be expanded by using vector spherical wave functions. The corresponding definition of the T-matrix is shown in Section 3.2.3. Section 3.2.4 implements rotations and translations of the T-matrix in terms of the vector spherical wave functions. The EBCM is briefly reviewed in Section 3.3. The symmetric relations associated with reciprocity and morphology are given in Sections 3.4.1 and 3.4.2, respectively. Light scattering by a dielectric particle in completely and partially random orientations is derived in detail in Sections 3.4.3–3.4.5.

Chapter 4 presents a comprehensive development of the IITM. The IITM is a volume integral method, while the EBCM is a surface integral method. Consequently, the volume integral equation is introduced in Section 4.1.1, and Section 4.1.2 proves the equivalence between the volume and surface integral equations. The dyadic Green's function expansion using the vector spherical wave functions is derived in Section 4.1.3. The expansion using the vector spherical wave functions is rearranged for convenience, and further derivation is in Section 4.1.4. The differential and difference forms of the T-matrix using the invariant imbedding technique are derived in Sections 4.2.1–4.2.3. In these sections, the equivalence between the two forms is proved by allowing the radial difference to approach zero. The verification of the differential form in a spherical situation is given in Section 4.2.4. Section 4.3 discusses issues related to the IITM, such as numerical stability and memory requirements; how to determine the starting point, the processing points, and the ending point; and factors affecting the truncation and imbedding steps.

Chapter 5 describes applications of the IITM to different morphologies. Section 5.1 discusses the effect of different quadrature rules and step sizes on spherical particles. Spheroids and cylinders are used as examples of axially symmetric particles to show the accuracy and the efficiency of the IITM in Section 5.2. Section 5.3 shows IITM simulations of finite-fold rotationally symmetric particles by focusing on hexagonal ice crystals. Section 5.4 summarizes IITM applications to asymmetric particles by focusing on aggregates and irregular hexahedra. Similarly, Section 5.5 focuses on inhomogeneous particles such as nested hexahedra.

In summary, not only are the chapters arranged in progressive order, but also each chapter is relatively independent. Chapter 2 gives the basic knowledge of light scattering. Chapter 3 is a thorough introduction to the T-matrix method. The systematic description of the IITM is in Chapter 4. Application examples of the IITM are presented in Chapter 5. In organizing the theoretical formulation of light scattering process in Chapter 2 and the T-matrix concept in Chapter 3, we have referred to several classical light scattering books, such as Light Scattering by Small Particles by van de Hulst (1957); Absorption and Scattering of Light by Small Particles by Bohren and Huffman (1983); Scattering of Electromagnetic Waves: Theories and Applications by Tsang et al. (2000); Scattering, Absorption, and Emission of Light by Small Particles by Mishchenko et al. (2002); and Electromagnetic Scattering by Particles and Particle Groups by Mishchenko (2014), and acknowledge these books here together.

We are very grateful to several individuals whose help directly impacted the final production of this book: Dr. Steven Schroeder carefully edited the manuscript and offered a number of insightful suggestions to improve the book; Dr. Jiachen Ding made significant contributions to Chapter 5; Adam Bell proofread the manuscript; and Ms. Devlin Person, an editorial project manager at Elsevier, patiently worked with the authors to ensure that the book manuscript was delivered within the originally proposed time frame.

Last but not least, we would like to take this opportunity to thank our families for supporting our effort in writing this book, which required a significant amount of quality time after our normal work hours and during weekends.

References☆

Ambarzumian V.A. Diffuse reflection of light by a foggy medium. In: Dokl. Akad. Nauk SSSR. 257–261. 1943;vol. 38 (8).

Bellman R., Wing G.M. An Introduction to Invariant Imbedding. New York: Wiley; 1975.

Bi L., Yang P. Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method. J. Quant. Spectrosc. Radiat. Transf. 2014;138:17–35.

Bohren C.F., Huffman D.R. Absorption and Scattering of Light by Small Particles. New York: John Wiley & Sons; 1983.

Chandrasekhar S. Radiative Transfer. New York: Dover Publications; 1960.

Johnson B.R. Invariant imbedding T matrix approach to electromagnetic scattering. Appl. Opt. 1988;27:4861–4873.

Mishchenko M.I. Electromagnetic Scattering by Particles and Particle Groups. Cambridge: Cambridge University Press; 2014.

Mishchenko M.I., Travis L.D., Lacis A.A. Scattering, Absorption, and Emission of Light by Small Particles. Cambridge: Cambridge University Press; 2002.

Tsang L., Kong J.A., Ding K.-H. Scattering of Electromagnetic Waves: Theories and Applications. New York: John Wiley & Sons; 2000.

van de Hulst H.C. Light Scattering by Small Particles. New York: John Wiley & Sons; 1957.

Waterman P.C. Symmetry, unitarity, and geometry in electromagnetic scattering. Phys. Rev. D. 1971;3:825–839.

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☆ To view the full reference list for the book, click here

1

Introduction

Keywords

Invariant imbedding T-matrix; Electromagnetic wave; Refractive index; Particle shape; Inhomogeneity; Random orientation

Since van de Hulst published his classical book entitled Light Scattering by Small Particles in 1957, a number of monographs have been published that summarize theoretical and computational developments in light scattering research from their own unique perspectives (e.g., Absorption and Scattering of Light by Small Particles by Bohren and Huffman, 1983; Scattering of Electromagnetic Waves: Theories and Applications by Tsang et al., 2000; Scattering, Absorption, and Emission of Light by Small Particles by Mishchenko et al., 2002; Light Scattering by Systems of Particles by Doicu et al., 2006; and Electromagnetic Scattering by Particles and Particle Groups: An Introduction by Mishchenko, 2014). Such tremendous efforts were motivated by the urgent need to apply the theory of light scattering to multiple scientific disciplines, including particle characterization, biomedical sciences, atmospheric remote sensing, the atmospheric radiant energy budget in climate science, ocean optics, astronomy, and optical engineering. However, the computational problems of the scattering and absorption of electromagnetic waves by nonspherical and inhomogeneous particles have not yet been satisfactorily solved when the particle size becomes large relative to the wavelength of incident radiation and when the particle is randomly oriented. Due to the complexity involved in mathematical physics, van de Hulst (1957) summarized slow progress of research efforts in developing the Lorenz-Mie theory by stating that it is a long way from the formula containing the solution to reliable numbers and curves. Similarly, significant efforts in the light scattering research community have been continually devoted to expanding computational capabilities that solve Maxwell's equations for the solution of light scattering by nonspherical and inhomogeneous particles. As a complementary contribution to previous accomplishments, this book is a treatise on this subject, in which we document recent advances achieved by exploring the concepts and application of the invariant-imbedding T-matrix (IITM) method (Johnson, 1988; Bi et al., 2013; Bi and Yang, 2014; Doicu and Wriedt, 2018).

The IITM is a rigorous computational technique, which has been demonstrated to be most suitable for computing the optical properties of randomly oriented, arbitrarily shaped, and inhomogeneous particles. Although our work is primarily motivated by applications of light scattering research to remote sensing of cirrus clouds and atmospheric aerosols, the concept and developed technique are essentially applicable to electromagnetic wave problems in other research areas including biomedical optics and marine sciences. The main purpose of this book is to provide a self-consistent summary of the IITM and to highlight some canonic IITM simulations. Similar to the subject limitation in most previous research, this book restricts its focus to dielectric particles in a nonabsorbing medium under the illumination of a polarized plane wave. The solution of light scattering by dielectric particles is determined by the value(s) of refractive index related to permittivity and its spatial distribution characterized by shape and the size parameter, which is defined in Section 1.2.

To obtain an overall understanding of the features of the IITM, in this chapter, we briefly recapture the basic definitions related to the modeling capabilities of computational techniques.

1.1 Particle shape and inhomogeneity

In the literature, nonspherical and inhomogeneous are frequently used to describe the particle shape and refractive index distribution involved in the light scattering computation. In the context of Maxwell's equations, the shape of the particle is essentially the spatial distribution of refractive indices. Many techniques developed for analyzing light scattering problems have varying ranges of applicability to particles with differing shapes and inhomogeneity. In general, the complexity of analytical treatment involved in the computation is inversely proportional to the suitable range of applicability to particle shapes and inhomogeneity. For example, the method of separation of variables is only applicable to spheres, spheroids, and infinite cylinders with smooth regular cross-sectional shapes. Classical numerical methods, such as the finite-difference time-domain (FDTD) (Yee, 1966) and the discrete-dipole approximation (DDA) (Purcell and Pennypacker, 1973) methods, are sufficiently flexible to handle a wide range of particle shapes and inhomogeneity. Semianalytical methods, such as the extended boundary condition method (EBCM) (Waterman, 1965, 1971; Mishchenko et al. 2002) and the superposition method (Mackowski and Mishchenko, 1996, 2011; Mackowski, 2014), have a domain of applicability between the separation of variables and purely numerical methods.

The key feature of the IITM is that it treats an arbitrary homogeneous nonspherical particle as an inhomogeneous spherical particle, namely, a spherical volume enclosing the nonspherical particle embedded in the surrounding medium such as air. The refractive index of the nonspherical particle is different from that of the remaining portion in the sphere. Inhomogeneous is a more generalized term than overall shape to describe the spatial distribution pattern of refractive indices. Therefore, the light scattering problem of an arbitrary nonspherical particle is a special scenario of the light scattering of an arbitrarily inhomogeneous sphere. Understanding the scattering of light by a particular nonspherical particle, such as an inhomogeneous sphere, might not lead to new insights into general computational techniques. However, it will be shown that this feature makes the IITM a universal tool for computing light scattering by arbitrarily shaped nonspherical particles. Such universality in handling particle geometry is similar to that of the DDA and FDTD methods, but specific advantages of IITM are further summarized in the sections later.

1.2 Size parameter

The size parameter (kR; k = 2π/λ) of a particle is defined as 2π times the ratio of characteristic particle size (R) to the wavelength (λ) of the incident radiation in the medium. In particular, here, we understand R as the minimum radius of the circumscribed sphere of a nonspherical particle. On the one hand, this parameter is closely related to the scattering mechanism; on the other hand, the modeling capabilities of exact computational techniques highly depend on this parameter. First, as the size parameter approaches practical infinity (the size of the particle is much larger than the incident wavelength), scattering characteristics associated with geometric optics become more evident, as can be predicted from a ray tracing process, although the validity of geometric optics is vague under most scenarios. Second, if we expand the incident plane wave in terms of vector wave harmonics, the infinite series must be truncated to include a sufficient number of terms to guarantee the convergence of the plane waves in the spatial region within the radius R. This number is proportional to kR + (kR)¹/³, which can be understood from the localization principle of geometric waves (van de Hulst, 1957). From the computational perspective, the size parameter also determines the number of volume or surface elements needed to discretize the particle volume or surface. The number of unknowns has an impact on the computational demand on computer memory and central processing unit (CPU) time and also affects the numerical stability of the algorithm. For methods based on the expansion of the incident and scattered waves, the size parameter determines the truncation number.

Based on current modeling capabilities, the solution of light scattering by an arbitrarily shaped nonspherical particle for small size parameters (e.g., <~20) can be easily obtained by using several different computational methods. Due to the increasing power of supercomputers, it is now feasible to obtain reliable solutions for irregular particles with sizes much larger than the incident wavelength. For example, by using the Amsterdam DDA (ADDA) software package, a solution for a sphere with size