Table Of ContentInvariant Imbedding T-matrix
Method for Light Scattering by
Nonspherical and Inhomogeneous
Particles
Invariant Imbedding T-matrix
Method for Light Scattering
by Nonspherical and
Inhomogeneous Particles
Bingqiang Sun
AssistantResearchScientist,DepartmentofAtmosphericSciences,
Texas A&M University, College Station, Texas, United States
Lei Bi
AssistantResearchScientist,DepartmentofAtmosphericSciences,
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, Norrko¨ping, Sweden
George Kattawar
Professor Emeritus, Department of Physics and Astronomy &
Institute for Quantum Science & Engineering, Texas A&M
University, College Station, Texas, United States
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Preface
The theories and numerical simulations of electromagnetic wave scattering by
nonsphericalandinhomogeneousparticleshavefounddiverseapplicationsinatmo-
sphericsciences,astronomy,engineering,chemistry,andbiophysics.Thesubjectof
electromagnetic wave scattering has evolved from early studies of simple non-
spherical geometries such asan infinite circular cylinderora spheroid toarbitrarily
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:
thestabilityofthealgorithmandanadvancednumericalimplementation.Theintent
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 ofarbitrarily shaped nonspherical andinhomogeneous particles.
Themotivationofwritingthisbookistwofold.First,theT-matrixmethodmaybe
themostaccurateandefficientmethodforsolvingthescatteringofanelectromagnetic
wavebyanonsphericalparticle.Second,foraboutthreedecades,theT-matrixmethod
was considered to be practically applicable to axially symmetric and homogeneous
particles, while significant efforts were devoted to applying the T-matrix method
tononsymmetricandinhomogeneousparticles.Atpresent,anaccurateandversatile
T-matriximplementationbasedontheextendedboundaryconditionmethod(EBCM)
(Waterman, 1971) for generally nonspherical and inhomogeneous particles is not
available, particularly in the case of large size parameters. Our research efforts to
applytheinvariantimbeddingT-matrixmethodstartedin2013,althoughtheinvariant
imbedding principle was introduced to obtain the T-matrix recurrence relation by
Johnson (1988) in the framework of an electromagnetic volume integral equation.
BycombiningseveraladvancesoftheT-matrixmethodandsolvingrelevantinstabil-
ityissues,thepowerofthistechniqueallowsforcomputingtheopticalpropertiesof
randomlyorientednonsphericalandinhomogeneousparticleswithsizesmuchlarger
thantheincidentwavelength.Thisbookisexpectedtobeusefulforactiveresearchers
and graduate students who have interests in light scattering and its applications in
manydisciplines.
Theinvariantimbeddingtechniquewasdevelopedtohandlethediffusereflection
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
viii Preface
used“invariantimbedding”tocointhenameofthetechniqueandappliedittoneutron
transporttheory.Inparticular,BellmanandWing(1975)systematicallydescribedthe
invariantimbeddingtechniqueintheirbookAnIntroductiontoInvariantImbedding.
Inadditiontoradiativetransfer,theinvariantimbeddingtechniquewasfirstappliedto
electromagnetic wave scattering by Johnson (1988). With an advanced numerical
implementation(BiandYang,2014),theIITMwasshowntobeapowerfultoolwith
far-reaching capabilities in numerical simulations of electromagnetic scattering by
arbitrarilyshapedparticlesandrecentlyattractedalotofattentionintheelectromag-
neticscatteringandatmosphericradiationcommunities.Tobettersharethistechnique
with these communities, we organize the principles and present applications of the
IITM inthis book.
Basically,theIITMisoneofthemostefficientmethodstocomputetheT-matrixof
adielectricparticle.Consequently,allpropertieswithrespecttotheT-matrix,suchas
its symmetry and the analytical scattering phase matrix based on the T-matrix, are
availabletotheIITM.Thus,theT-matrixmethodisasemianalyticalmethodtosolve
lightscatteringbyascatteringparticle.Thisbookgivesasystematicintroductionof
theIITMinfivechapters:terminologyhighlights,lightscatteringrepresentation,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,
randomorientation,andtheinvariantimbeddingprinciple.Chapter2givesasystem-
atic 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
dielectricparticle.Thetopicsincludetheamplitudescatteringmatrixandthescatter-
ingphasematrixandtheirsymmetrypropertiesandtheextinctionandscatteringcross
sections associated with single scattering and bulk scattering properties. A brief
review ofrigorous scattering computational methods is given inSection 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
3.2.2describehow,intheT-matrixmethod,theelectromagneticfieldcanbeexpanded
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 transla-
tions of the T-matrix in terms of the vector spherical wave functions. The EBCM
isbrieflyreviewedinSection3.3.Thesymmetricrelationsassociatedwithreciprocity
andmorphologyaregiveninSections3.4.1and3.4.2,respectively.Lightscatteringby
adielectricparticleincompletelyandpartiallyrandomorientationsisderivedindetail
in Sections 3.4.3–3.4.5.
Chapter 4 presents a comprehensive development of the IITM. The IITM is a
volumeintegralmethod,whiletheEBCMisasurfaceintegralmethod.Consequently,
thevolumeintegralequationisintroducedinSection4.1.1,andSection4.1.2proves
the equivalence between the volume and surface integral equations. The dyadic
Preface ix
Green’s function expansion using the vector spherical wave functions is derived in
Section4.1.3.Theexpansionusingthevectorsphericalwavefunctionsisrearranged
forconvenience,andfurtherderivationisinSection4.1.4.Thedifferentialanddiffer-
ence 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
differentialforminasphericalsituationisgiveninSection4.2.4.Section4.3discusses
issuesrelatedtotheIITM,suchasnumericalstabilityandmemoryrequirements;how
todeterminethestartingpoint,theprocessingpoints,andtheendingpoint;andfactors
affectingthe truncationand imbedding steps.
Chapter5describesapplicationsoftheIITMtodifferentmorphologies.Section5.1
discussestheeffectofdifferentquadraturerulesandstepsizesonsphericalparticles.
Spheroidsandcylindersareusedasexamplesofaxiallysymmetricparticlestoshow
theaccuracy andtheefficiencyoftheIITMinSection5.2.Section5.3showsIITM
simulationsoffinite-foldrotationallysymmetricparticlesbyfocusingonhexagonal
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 particlessuch asnestedhexahedra.
Insummary,notonlyarethechaptersarrangedinprogressiveorder,butalsoeach
chapterisrelativelyindependent.Chapter2givesthebasicknowledgeoflightscat-
tering.Chapter3isathoroughintroductiontotheT-matrixmethod.Thesystematic
descriptionoftheIITMisinChapter4.ApplicationexamplesoftheIITMarepres-
entedinChapter5.Inorganizingthetheoreticalformulationoflightscatteringprocess
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 Applica-
tionsbyTsangetal.(2000);Scattering,Absorption,andEmissionofLightbySmall
Particles by Mishchenko et al. (2002); and Electromagnetic Scattering by Particles
and Particle Groups by Mishchenko (2014), and acknowledge these books here
together.
Weareverygratefultoseveralindividualswhosehelpdirectlyimpactedthefinal
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;
andMs.DevlinPerson,aneditorialprojectmanageratElsevier,patientlyworkedwith
the authors to ensure that the book manuscript was delivered within the originally
proposed time frame.
Lastbutnotleast,wewouldliketotakethisopportunitytothankourfamiliesfor
supporting our effort in writing this book, which required a significant amount of
qualitytime after ournormal work hours and during weekends.
1
Introduction
Since van de Hulst published his classical book entitled Light Scattering by Small
Particlesin1957,anumberofmonographshavebeenpublishedthatsummarizethe-
oreticalandcomputationaldevelopmentsinlightscatteringresearchfromtheirown
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
bySmallParticlesbyMishchenkoetal.,2002;LightScatteringbySystemsofParti-
clesbyDoicuetal.,2006;andElectromagneticScatteringbyParticlesandParticle
Groups:AnIntroductionbyMishchenko,2014).Suchtremendouseffortsweremoti-
vatedbytheurgentneedtoapplythetheoryoflightscatteringtomultiplescientific
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
ofthescatteringandabsorptionofelectromagneticwavesbynonsphericalandinho-
mogeneous particles have not yet been satisfactorily solved when the particle size
becomeslargerelativetothewavelengthofincidentradiationandwhentheparticle
israndomlyoriented.Duetothecomplexityinvolvedinmathematicalphysics,vande
Hulst(1957)summarizedslowprogressofresearcheffortsindevelopingtheLorenz-
Mietheorybystatingthat“itisalongwayfromtheformulacontainingthesolutionto
reliable numbers and curves.” Similarly, significant efforts in the light scattering
research community have been continually devoted to expanding computational
capabilitiesthatsolveMaxwell’sequationsforthesolutionoflightscatteringbynon-
sphericalandinhomogeneousparticles.Asacomplementarycontributiontoprevious
accomplishments,thisbookisatreatiseonthissubject,inwhichwedocumentrecent
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).
TheIITMisarigorouscomputationaltechnique,whichhasbeendemonstratedto
bemostsuitableforcomputingtheopticalpropertiesofrandomlyoriented,arbitrarily
shaped,andinhomogeneous particles.Althoughourworkis primarilymotivated by
applicationsoflightscatteringresearchtoremotesensingofcirruscloudsandatmo-
spheric 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
thesubjectlimitationinmostpreviousresearch,thisbookrestrictsitsfocustodielec-
tric 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
InvariantImbeddingT-matrixMethodforLightScatteringbyNonsphericalandInhomogeneousParticles
https://doi.org/10.1016/B978-0-12-818090-7.00001-2
©2020ElsevierInc.Allrightsreserved.
2 InvariantImbeddingT-matrixMethod
value(s)ofrefractiveindexrelatedtopermittivityanditsspatialdistributioncharac-
terizedbyshape andthe sizeparameter, which is defined in Section 1.2.
ToobtainanoverallunderstandingofthefeaturesoftheIITM,inthischapter,we
brieflyrecapturethebasicdefinitionsrelatedtothemodelingcapabilitiesofcompu-
tational techniques.
1.1 Particle shape and inhomogeneity
Intheliterature,“nonspherical”and“inhomogeneous”arefrequentlyusedtodescribe
the particle shape and refractive index distribution involved in the light scattering
computation.InthecontextofMaxwell’sequations,theshapeoftheparticleisessen-
tiallythespatialdistributionofrefractiveindices.Manytechniquesdevelopedforana-
lyzinglightscatteringproblemshavevaryingrangesofapplicabilitytoparticleswith
differing shapes and inhomogeneity. In general, the complexity of analytical treat-
ment involved in the computation is inversely proportional to the suitable range of
applicabilitytoparticleshapesandinhomogeneity.Forexample,themethodofsep-
arationofvariablesisonlyapplicabletospheres,spheroids,andinfinitecylinderswith
smooth regular cross-sectional shapes. Classical numerical methods, such as the
finite-difference time-domain (FDTD) (Yee, 1966) and the discrete-dipole approxi-
mation (DDA) (Purcell and Pennypacker, 1973) methods, are sufficiently flexible
tohandleawiderangeofparticleshapesandinhomogeneity.Semianalyticalmethods,
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),haveadomainofapplicabilitybetween
the separation ofvariablesand purely numerical methods.
ThekeyfeatureoftheIITMisthatittreatsanarbitraryhomogeneousnonspherical
particleasaninhomogeneoussphericalparticle,namely,asphericalvolumeenclosing
thenonsphericalparticleembeddedinthesurroundingmediumsuchasair.Therefrac-
tiveindexofthenonsphericalparticleisdifferentfromthatoftheremainingportionin
the sphere. “Inhomogeneous” is a more generalized term than overall shape to
describethespatialdistributionpatternofrefractiveindices.Therefore,thelightscat-
tering problemofanarbitrarynonsphericalparticleisaspecial scenario ofthelight
scattering of an arbitrarily inhomogeneous sphere. Understanding the scattering of
lightby a particular nonspherical particle, such as an inhomogeneous sphere, might
not lead to new insights into general computational techniques. However, it will be
shownthatthisfeaturemakestheIITMauniversaltoolforcomputinglightscattering
by arbitrarily shaped nonspherical particles. Such universality in handling particle
geometryissimilartothatoftheDDAandFDTDmethods,butspecificadvantages
of IITM are further summarized inthe sections later.
1.2 Size parameter
Thesizeparameter(kR;k¼2π/λ)ofaparticleisdefinedas2π timestheratioofchar-
acteristicparticlesize(R)tothewavelength(λ)oftheincidentradiationinthemedium.
Introduction 3
Inparticular,here,weunderstandRastheminimumradiusofthecircumscribedsphere
ofanonsphericalparticle.Ontheonehand,thisparameteriscloselyrelatedtothescat-
teringmechanism;ontheotherhand,themodelingcapabilitiesofexactcomputational
techniques highly depend on this parameter. First, as the size parameter approaches
practicalinfinity(thesizeoftheparticleismuchlargerthantheincidentwavelength),
scatteringcharacteristicsassociatedwithgeometricopticsbecomemoreevident,ascan
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
ofvectorwaveharmonics,theinfiniteseriesmustbetruncatedtoincludeasufficient
numberoftermstoguaranteetheconvergenceoftheplanewavesinthespatialregion
withintheradiusR.ThisnumberisproportionaltokR+(kR)1/3,whichcanbeunderstood
fromthelocalizationprincipleofgeometricwaves(vandeHulst,1957).Fromthecom-
putationalperspective,thesizeparameteralsodeterminesthenumberofvolumeorsur-
face elements needed to discretize the particle volume or surface. The number of
unknownshasanimpactonthecomputationaldemandoncomputermemoryandcen-
tralprocessingunit(CPU)timeandalsoaffectsthenumericalstabilityofthealgorithm.
Formethodsbasedontheexpansionoftheincidentandscatteredwaves,thesizeparam-
eterdeterminesthetruncationnumber.
Basedoncurrentmodelingcapabilities,thesolutionoflightscatteringbyanarbi-
trarilyshapednonsphericalparticleforsmallsizeparameters(e.g.,<(cid:2)20)canbeeas-
ilyobtainedbyusingseveraldifferentcomputationalmethods.Duetotheincreasing
powerofsupercomputers,itisnowfeasibletoobtainreliablesolutionsforirregular
particleswithsizesmuchlargerthantheincidentwavelength.Forexample,byusing
the Amsterdam DDA (ADDA) software package, a solution for a sphere with size
parameter 320 (roughly 100-wavelength diameter) and refractive index 1.05 was
reported (Yurkin and Hoekstra, 2011). As another example, the light scattering by
anasymmetricChebyshevparticlehasbeensolvedusingamultilevelfastmultipole
algorithmwithasizeparameterupto600(ErgulandGurel,2014).Thepseudospectral
time-domain (Liu, 1997; Liu et al., 2012) and discontinuous Galerkin time-domain
methodshavebeensuccessfullyappliedtononsphericalparticleswithsizeparameters
upto200(Cockburnetal.,2000;Panettaetal.,2013).Atpresent,becauseoftremen-
dous computational resources required (number of CPUs, total amount of memory,
andcomputationaltime),thesemethodsareappliedtolargesizeparametersforcase
studies rather than extensive simulations in practical applications. In contrast to the
previous methods, the maximum size parameter for which the invariant-imbedding
methodisnumericallyfeasibledependsonthesymmetryoftheparticle.Forparticles
withoutanysymmetry,theapplicablemaximumsizeparameterissmaller.However,
themainadvantageoftheIITMmethodisthatitcanmoreefficientlyhandleparticle’s
randomorientations than the other numerical methods introduced earlier.
1.3 Random orientations
Theconceptofrandomorientationshasbeenwidelyusedinlightscatteringandradi-
ative transfer calculations to describe a hypothetical but quite realistic ensemble of
