Toroidal Multipole Moments in Classical Electrodynamics: An Analysis of their Emergence and Physical Significance

91 Pages·2016·0.939 MB·English

by  Stefan Nanz (auth.)

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Stefan Nanz
Toroidal Multipole
Moments in Classical
Electrodynamics
An Analysis of their Emergence
and Physical Signifi cance
Stefan Nanz
Karlsruhe, Germany
BestMasters
ISBN 978-3-658-12548-6 ISBN 978-3-658-12549-3 (eBook)
DOI 10.1007/978-3-658-12549-3
Library of Congress Control Number: 2015960957
Springer Spektrum
© Springer Fachmedien Wiesbaden 2016
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Preface
IfIhaveseenfurtheritisbystandingontheshouldersofgiants.
Isaac Newton
Thisbookisbasedonmymasterthesis,onwhichIworkedfromJanuary2014untilFebruary
2015 at the Institute for Theoretical Solid State Physics at Karlsruhe Institute of Technology
(KIT).Thetopicregardingthetoroidalmultipolemomentswasoriginallysuggestedbymysu-
pervisor, ProfessorCarstenRockstuhl, whohadjuststartedtoestablishanewworkinggroup
atKITattheendof2013. Pointingoutsomeinconsistenciesinvariousderivationsanddescrip-
tionsoftheelectromagneticmultipoleexpansion,hemotivatedmetodigintothedepthofthe
theoryofelectrodynamicsandtofindoutwhat’sreallygoingoninthemultipoleexpansion.
SoonafterIstartedworking, Ihadthefortunethattheworkinggroupwasjoinedbyanana-
lyticallywell-experiencedpostdoctoralresearcher,Dr. IvanFernandez-Corbaton,whotookthe
roleasmyadvisorandhelpedmetotacklethesometimesdrytheoreticalanalysis. Inthecourse
of time, it became apparent to us that the derivations which argue in favor of the physical
significanceoftoroidalmomentsaresufferingfromsomeassumptions,whichturnouttobenot
justifiedingeneral. Condensingtheseinsightsthenleadtoamasterthesiswhichwasapproved
by the supervisors, Prof. Carsten Rockstuhl and Prof. Martin Wegener, and later on also by
SpringerSpektrum,resultingintheirdecisiontopublishitasabook.
Atthispoint,Iwanttotakethechancetoexpressmycordialthankstoseveralpeoplewithout
them my master thesis and therefore this book would not have been possible. First of all, I
wouldliketothankProf.Dr.CarstenRockstuhlforgivingmethepossibilitytoworkonsuchan
interestingandrewardingtopicandforbeingagreatmentorregardingallaspectsofresearch. I
wouldalsoliketothankDr.IvanFernandez-Corbaton,whodidagreatjobasadvisorwithhis
analyticalskillsandextensiveknowledgeofliterature.
Furthermore,Iwouldliketothank: Dr.ChristophMenzelforfruitfuldiscussionsandproofread-
ingthethesis;IrisSchwenkforprovidingmeanicetemplateforthelayoutofthethesis;Dr.An-
dreasPoenickeforhissupportregardingcomputerproblems;ClemensBaretzkyforcreatingthe
images; my half-brother Philipp for proofreading the thesis; and my roommates Dr. Giuseppe
Toscano and Alexander Kwiatkowski for sharing the office with me and helping me with all
kindsofproblems. IamgratefultoSpringerSpektrumforpublishingmymasterthesis,andI
want to thank my contact persons there, Nicole Schweitzer and Marta Schmidt. Last but not
leastIwouldliketothankmyparentsfortheirsupportduringmyeducation.
Heidelberg, Stefan Nanz
November2015
Contents
List of Figures IX
List of Symbols XI
1 Introduction and Overview 1
2 Why another Multipole Family? 5
2.1 ABriefHistoryofToroidalMoments . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 CharacterizationofGeneralChargeandCurrentDistributions . . . . . . . . . . 7
2.3 NecessityforThreeMultipoleFamilies . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 DistinctionbetweenToroidalMomentandAnapole . . . . . . . . . . . . . . . . . 11
3 Basic Equations and Notations 13
3.1 Maxwell’sEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 WaveEquationandHelmholtzEquation . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Multipole Expansion of the Potentials 21
4.1 MultipoleExpansioninCartesianCoordinates . . . . . . . . . . . . . . . . . . . 21
4.2 CartesianTracelessExpansionincludingToroidalMoments . . . . . . . . . . . . 35
4.3 MultipoleExpansioninSphericalCoordinates. . . . . . . . . . . . . . . . . . . . 41
5 Direct Multipole Expansion of Electromagnetic Fields 51
5.1 DecompositionoftheFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 ExpansionCoefficientsforgivenElectricField. . . . . . . . . . . . . . . . . . . . 54
5.3 ExpansionCoefficientsforgivenSources . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 PhysicalRelevanceoftheLongitudinalPart . . . . . . . . . . . . . . . . . . . . . 61
6 Connection and Comparison of the Different Approaches 63
6.1 ConversionofSphericalintoCartesianMomentsandViceVersa . . . . . . . . . 63
6.2 PropertiesoftheFieldsincludingToroidalMoments . . . . . . . . . . . . . . . . 66
7 Summary and Outlook 69
Bibliography 73
Detailed Calculations 7(cid:26)
Tabulated Functions 8(cid:22)
List of Figures
2.1 Illustrations of the electric, magnetic, polar toroidal and axial toroidal dipole
moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Poloidal–toroidaldecompositionofavectorfield . . . . . . . . . . . . . . . . . . 18
4.1 Two-dimensionalschemeoftheelectromagneticmultipoleexpansion . . . . . . . 22
4.2 Decompositionofthecurrentdensityintothreeorthonormalfunctions . . . . . . 47
List of Symbols
Thislistincludesthemostimportantsymbolswhichareatleastusedonceinthethesis.
• ∇(cid:2) nablaoperator • Pˆ(n) Cartesian electric multipole mo-
mentofordern
• ΔLaplaceoperator
• Mˆ(n)Cartesianmagneticmultipolemo-
• P parityoperator
mentofordern
• T timeinversionoperator • Tˆ(n) Cartesian toroidal multipole mo-
• Langularmomentumoperator mentofordern
• Ddetracingoperator • p(cid:2)electricdipolemoment
• Λdetracingfunctional • m(cid:2) magneticdipolemoment
• Trtraceofatensor • (cid:2)ttoroidaldipolemoment
• (cid:2)kwavevector • Qˆ(e) Cartesian traceless electric
quadrupolemoment
• x˙ timederivativeofx
• a∗ complexconjugationofa • Qˆ(m) Cartesian traceless symmetric
magneticquadrupolemoment
• (cid:2)rpositionvectorofevaluationpoint • Qˆ(t)Cartesiantoroidalquadrupolemo-
• (cid:2)r(cid:3) positionvectorofthesource ment
• (cid:2)ei unitvectorindirectioni • Qlm spherical electric multipole mo-
ment
• μ0 vacuumpermeability
• ε0 vacuumpermittivity • Mmelmntsphericalmagneticmultipolemo-
• cspeedoflightinvacuum • Tlm spherical toroidal multipole mo-
• ωangularfrequency ment
• A(cid:2) vectorpotential • alm spherical electric parity multipole
moment
• ϕelectricscalarpotential
• (cid:2)j electriccurrentdensity • blmsphericalmagneticparitymultipole
moment
• ρelectricchargedensity • jn sphericalBesselfunction
• B(cid:2),H(cid:2) magneticfield • h(n1) spherical Hankel function of first
• E(cid:2) electricfield kind
1 Introduction and Overview
Theanalysisofelectromagneticradiationisanimportantresourcetoinvestigatetheproperties
ofmaterials. Often,unknownmaterialsareirradiatedbyasuitablelightsource,e.g.alaser,and
the scattered electric and magnetic fields are used to obtain information about the materials.
This requires to link the properties of the incident to the scattered radiation. This asks to
solveMaxwell’sequationswithspatiallydistributedmaterialswhosepropertiesareintroduced
onphenomenologicalgrounds. Aprototypicalexampleforsuchapproachisellipsometry. When
designinge.g.antennas,theoppositeisofrelevance: Anincidentandscatteredelectromagnetic
fieldisgivenandthestructure,whichproducesthisfield,shallbeconstructed.
Thelatterproblemalsoappliesinthecontextofmetamaterials. Metamaterialsaremadefrom
small scattering objects. If their optical response is dominated not just by an electric dipole
moment but also by higher order electromagnetic multipole moments, material properties not
availableinnaturecanbereached. Thisispossible,sinceonphenomenologicalgroundsnatural
materials at optical frequencies usually are considered as an ensemble of polarizable entities
withascatteringresponsethatcorrespondstoanelectricdipole. Thisonlyallowstoobservea
dispersionintheopticalpermittivity
Now, by composing metamaterials from strongly scattering unit-cells, also called meta-atoms
[1],othermaterialpropertiesareallowedtobedispersive. Thekeyistomakemeta-atomssuf-
ficientlysmallandtoarrangethemsufficientlydenselyinspace,suchthatlightwillexperience
ahomogeneousmediumwithpropertiesderivedfromthescatteringresponseoftheindividual
meta-atom. Forexample,iftheirscatteringresponseisdominatedbyamagneticdipolemoment,
adispersivepermeabilitycanbeobserved. Iftheirscatteringresponseisdominatedbyanelec-
tromagneticcoupling,astrongopticalactivitycanbeobserved. Studyingthelightpropagation
inmetamaterialsandthescatteringresponsefromindividualmeta-atomsisamajorchallenge
forcontemporarytheoreticaloptics.
From the many interesting properties meta-atoms may exhibit, we discuss in this thesis the
toroidalmultipolemoments. Toroidalmomentsareathirdmultipolefamilybesidestheelectric
and magnetic moments. They have several properties which attract the interest of research.
It has been shown [2] that current distributions, which cause toroidal moments, violate New-
ton’s third law. With toroidal moments it is possible to generate non-zero vector potentials
without electromagnetic fields [3], hereby realizing non-radiating charge-current distributions
[4,5]. Also,anegativeindexofrefractioncanbecausedbytoroidalmoments[6]. Thoughthe
experimental possibility to realize materials with such properties is quite new, there has been
a theoretical interest for such characteristics for long time. Although being considered in the
contextofelectrodynamicsusuallyasexotic,thepropertiesoftoroidalmomentscameagainin
S. Nanz, Toroidal Multipole Moments in Classical Electrodynamics, BestMasters,
DOI 10.1007/978-3-658-12549-3_1, © Springer Fachmedien Wiesbaden 2016