Toroidal Multipole Moments in Classical Electrodynamics: An Analysis of their Emergence and Physical Significance PDF
Preview Toroidal Multipole Moments in Classical Electrodynamics: An Analysis of their Emergence and Physical Significance
BestMasters Springer awards “BestMasters” to the best master’s theses which have been com- pleted at renowned universities in Germany, Austria, and Switzerland. Th e studies received highest marks and were recommended for publication by su- pervisors. Th ey address current issues from various fi elds of research in natural sciences, psychology, technology, and economics. Th e series addresses practitioners as well as scientists and, in particular, off ers guid- ance for early stage researchers. 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 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci(cid:191) cally the rights of translation, reprinting, reuse of illus- trations, recitation, broadcasting, reproduction on micro(cid:191) lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci(cid:191) c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Spektrum is a brand of Springer Fachmedien Wiesbaden Springer Fachmedien Wiesbaden is part of Springer Science+Business Media (www.springer.com) 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
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