Applied Mathematical Sciences Gang Bao Peijun Li Maxwell’s Equations in Periodic Structures Applied Mathematical Sciences Volume 208 SeriesEditors AnthonyBloch,DepartmentofMathematics,UniversityofMichigan,AnnArbor, MI,USA C.L.Epstein,DepartmentofMathematics,UniversityofPennsylvania, Philadelphia,PA,USA AlainGoriely,DepartmentofMathematics,UniversityofOxford,Oxford,UK LeslieGreengard,NewYorkUniversity,NewYork,NY,USA AdvisoryEditors J.Bell,CenterforComputationalSciencesandEngineering,LawrenceBerkeley NationalLaboratory,Berkeley,CA,USA P.Constantin,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ, USA R.Durrett,DepartmentofMathematics,DukeUniversity,Durham,CA,USA R.Kohn,CourantInstituteofMathematicalSciences,NewYorkUniversity, NewYork,NY,USA R.Pego,DepartmentofMathematicalSciences,CarnegieMellonUniversity, Pittsburgh,PA,USA L.Ryzhik,DepartmentofMathematics,StanfordUniversity,Stanford,CA,USA A.Singer,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA A.Stevens,DepartmentofAppliedMathematics,UniversityofMünster,Münster, Germany S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI, USA FoundingEditors F.John,NewYorkUniversity,NewYork,NY,USA J.P.LaSalle,BrownUniversity,Providence,RI,USA L.Sirovich,BrownUniversity,Providence,RI,USA Themathematizationofallsciences,thefadingoftraditionalscientificboundaries, theimpactofcomputertechnology,thegrowingimportanceofcomputermodeling and the necessity of scientific planning all create the need both in education and researchforbooksthatareintroductorytoandabreastofthesedevelopments.The purposeofthisseriesistoprovidesuchbooks,suitablefortheuserofmathematics, themathematicianinterestedinapplications,andthestudentscientist.Inparticular, this series will provide an outlet for topics of immediate interest because of the noveltyofitstreatmentofanapplicationorofmathematicsbeingappliedorlying close to applications. These books should be accessible to readers versed in mathematics or science and engineering, and will feature a lively tutorial style, a focus on topics of current interest, and present clear exposition of broad appeal. AcomplimenttotheAppliedMathematicalSciencesseriesistheTextsinApplied Mathematicsseries,whichpublishestextbookssuitableforadvancedundergraduate andbeginninggraduatecourses. Moreinformationaboutthisseriesathttp://www.springer.com/series/34 · Gang Bao Peijun Li Maxwell’s Equations in Periodic Structures GangBao PeijunLi SchoolofMathematicalSciences DepartmentofMathematics ZhejiangUniversity PurdueUniversity Hangzhou,Zhejiang,China WestLafayette,IN,USA ISSN0066-5452 ISSN2196-968X (electronic) AppliedMathematicalSciences ISBN978-981-16-0060-9 ISBN978-981-16-0061-6 (eBook) https://doi.org/10.1007/978-981-16-0061-6 JointlypublishedwithSciencePress TheprinteditionisnotforsaleinChinaMainland.CustomersfromChinaMainlandpleaseorderthe printbookfrom:SciencePress,Beijing. ©SciencePress2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublishers,whetherthewholeorpartofthe materialisconcerned,specificallytherightsofreprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageand retrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown orhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface This book addresses significant recent developments in mathematical analysis and computational methods for solving Maxwell’s equations in periodic structures. The model problems arise especially in the mathematical modeling of diffractive optics.Particularemphasisisplacedontheformulationofthemathematicalmodel, well-posedness, and regularity analysis of the solutions of Maxwell’s equations in complexmediaincludinglinearandnonlinearmedia,thedesignandanalysisofnew computational approaches, and inverse and optimal design problems in diffractive optics. Diffractive optics is a fundamental and vigorously growing technology which continuestobeasourceofnovelopticaldevices.Significantrecenttechnologydevel- opmentsofhigh-precisionmicromachiningtechniqueshavepermittedthecreation ofdiffractiongratings(periodicstructures)andotherdiffractivestructureswithtiny features.Currentandpotentialapplicationareasincludecorrectivelenses,microsen- sors, optical storage systems, optical computing and communication components, and integrated opto-electronic semiconductor devices. Because of the small struc- tural features, light propagation in micro-optical structures is generally governed by diffraction. In order to accurately predict the energy distributions of an inci- dent field in a given structure, the numerical solution of full Maxwell’s equations is required. Computational models also allow the exciting possibility of obtaining completely new structures through the solution of optimal design problems. A remarkable application of nonlinear optics is to generate powerful coherent radi- ation at a frequency that is twice that of available lasers, which is called second harmonicgeneration.Nonlinearopticsalsohasapplicationsinlasertechnology,spec- troscopy,opticalswitching,parametricamplifiersandoscillators,opticalcomputing, andcommunications. The fundamental importance of Maxwell’s equations is clear. These equations providesolidfoundationformodelingelectromagneticwavepropagationinoptical andvariousothermedia.Asopticalscienceadvancesrapidly,thereisanincreasing demand for modeling of the relevant physical phenomena. Consequently, various forms of Maxwell’s equations must be studied. While some of the broad subject matter,e.g.,standardlinearMaxwell’sequations,isclassical,thetopicsinthisbook v vi Preface are new and represent the latest developments in their respective fields. Each of themodelproblemsgrowsfromnewtechnologicaldevelopments.Forexample,in diffractiveopticsthefocusisonmicro-opticswherestructuresofscalesarecompa- rable to the wavelength of the visible light. Because of the tiny structural scales, wavepropagationcannolongerbepredictedaccuratelybytheclassicalgeometrical opticsapproximation.Instead,onemustsolvetheMaxwellequationsrigorously. InChap.1,thegeneralelectromagnetictheoryisintroduced.Maxwell’sequations alongwithcommonlyusedjumpandboundaryconditionsaregiven.Thegoverning equations for two fundamental polarizations, Transverse Electric (TE) and Trans- verseMagnetic(TM)polarizations,arediscussedforthethree-dimensionalMaxwell equations. Chapter 2 is devoted to the basic diffraction grating theory. The mathematical modelsarederivedforbothperiodicstructures(one-dimensionalgratings)andbiperi- odic structures (two-dimensional gratings). Grating formulas and conservation of energy are shown for perfectly conducting and dielectric gratings in lossless and lossymedia,respectively. Chapter 3 concerns the variational formulations for one- and two-dimensional gratings. The Transparent Boundary Conditions (TBC) are introduced to reduce equivalently the grating problems from open domains into bounded domains. The well-posedness is examined for the associated boundary value problems of the HelmholtzandMaxwellequations. Chapter4dealswiththeadaptivefiniteelementmethodsforsolvingtheboundary valueproblemsintroducedinChap.3.Webeginwithabasicfiniteelementanalysis for TE and TM polarizations, and then describe two different methods to truncate the unbounded physical domains: the Perfectly Matched Layer (PML) techniques andtheDirichlet-to-Neumann (DtN)operator techniques. Convergence analysis is carriedoutforbothoftheadaptivefiniteelementPMLandDtNmethods. Chapter5addressestheinversediffractiongratingproblems.Abriefoverviewis alsogivenfornumericalmethods.Someuniquenesstheoremsarepresentedforone- and two-dimensional lossy and lossless gratings. Local stability results are given fortheHelmholtzandMaxwellequations.Akeystepofshowinglocalstabilityis to investigate the domain derivatives. As a representative example, a continuation methodispresentedtoillustratetheoptimization-basediterativeschemesforsolving theinverseproblem. Chapter6discussesnear-fieldimagingproblemsindiffractiveoptics.Aparticular emphasisisonthesuper-resolvedcapabilityofnear-fieldimaging.Aframeworkis presented to reconstruct the grating surfaces with super-resolution by using either near-fieldorfar-fielddata. Chapter 7 introduces some related and important topics in diffractive optics. Topicsincludethemethodofboundaryintegralequations,thetime-domainproblems in periodic structures, nonlinear optics modeling and analysis, and optimal design problems. Finally,inAppendix,wecollectwithoutproofssomecommonlyusedidentities and basic concepts in functional analysis, which include vector spaces, Sobolev spaces, linear operators, variational formulations, and finite element methods for Preface vii variationalproblems.Thesepreliminariesaregiveninordertomakethebookself- contained. Thisbookoffersresearchersandespeciallyadvancedundergraduatestudentsand graduatestudentsanopportunitytogetabroadexposuretobasicphysicsandmath- ematicaltheoryofMaxwell’sequationsaswellasimportantproblemsindiffractive optics. It is intended to review recent developments in many important areas of mathematicalmodeling,analysis,andcomputationofMaxwell’sequationsinperi- odicstructures.Itisalsointendedtoprovidebeginnerswithintroductorymaterialand moreexperiencedresearcherswithup-to-datereferencesinmathematicalmodeling ofMaxwell’sequationsandapplicationstodiffractiveoptics. The most distinctive feature of the book is that it reflects the interdisciplinary characterofdiffractiveoptics.Eachofthemodelequationsisderivedfromaphysical model with important applications. This book grows out of a desire to foster the communicationbetweenthemathematicsandengineeringcommunitiesonmodeling problemsinoptics.Intheareascoveredinthisbook,modelingandsimulationhave become an important part of the engineering process. We believe that the applied mathematicscommunityhasopportunitiestocontributesignificantlytotheanalysis of these models, as well as the design and analysis of simulation techniques and automateddesigntools.Ontheotherhand,astheappliedmathematicscommunity hasmaderapiddevelopmentinaddressingchallengingproblemsofopticalscience during the last several decades, this book is also intended to provide researchers inappropriateengineeringdisciplineswithrecentmathematicaladvancesintheory, analysis,andcomputationaltechniquesforsolvingMaxwell’sequationsinperiodic structures. ThebookgrewoutoflecturenotesoftheauthorsfortopiccoursesatMichigan State University, Purdue University, and Zhejiang University, as well as special coursesatseveralotherinstitutionsoveraperiodofmorethan15years.Itwouldnot havebeenpossiblewithoutthecollaborationsandtheconversationswithanumber ofoutstandingcolleagues.Wehavenotonlybenefitedfromgeneroussharingoftheir ideas,insights,andenthusiasmbutalsofromtheirfriendship,support,andencour- agement.WefeelspeciallyindebtedtoHabibAmmari,ZhimingChen,AllenCox, DavidDobson,AvnerFriedman,YixianGao,JunLai,MingLi,JunshanLin,Shuai Lu,Jean-ClaudeNédélec,JianliangQian,FaouziTriki,HaijunWu,XiangXu,Hai Zhang,andWeiyingZheng.WewouldalsoliketothankXueJiang,YuliangWang, and Xiaokai Yuan for helping us with some of the numerical experiments. We are verygratefulforthesuggestionsandlistsofmistakesfromearlierdraftsofthisbook whichweresenttousbymanycolleagues,friends,andstudents. Hangzhou,China GangBao WestLafayette,Indiana PeijunLi March2020 Contents 1 Maxwell’sEquations ........................................... 1 1.1 ElectromagneticWaves ...................................... 1 1.2 JumpandBoundaryConditions .............................. 6 1.3 TwoFundamentalPolarizations ............................... 9 References ..................................................... 12 2 DiffractionGratingTheory ..................................... 13 2.1 PerfectlyConductingGratings ................................ 14 2.2 DielectricGratings ......................................... 22 2.3 BiperiodicGratings ......................................... 32 2.3.1 PerfectElectricConductors ............................ 33 2.3.2 DielectricMedia ..................................... 38 References ..................................................... 42 3 VariationalFormulations ....................................... 45 3.1 TheDirichletProblem ...................................... 46 3.2 TheTransmissionProblem ................................... 53 3.3 BiperiodicStructures ........................................ 59 3.3.1 FunctionSpaces ..................................... 60 3.3.2 TheTransparentBoundaryCondition ................... 68 3.3.3 TheVariationalProblem .............................. 76 References ..................................................... 84 4 FiniteElementMethods ......................................... 87 4.1 TheFiniteElementMethod .................................. 89 4.1.1 FiniteElementAnalysisforTEPolarization ............. 90 4.1.2 FiniteElementAnalysisforTMPolarization ............. 94 4.2 AdaptiveFiniteElementPMLMethod ......................... 98 4.2.1 ThePMLFormulation ................................ 99 4.2.2 TransparentBoundaryConditionforthePMLProblem .... 102 4.2.3 ErrorEstimateofthePMLSolution .................... 105 4.2.4 TheDiscreteProblem ................................. 108 ix x Contents 4.2.5 ErrorRepresentationFormula .......................... 109 4.2.6 APosterioriErrorAnalysis ............................ 111 4.2.7 NumericalResults ................................... 114 4.3 AdaptiveFiniteElementDtNMethod ......................... 118 4.3.1 TheDiscreteProblem ................................. 120 4.3.2 APosterioriErrorAnalysis ............................ 122 4.3.3 TMPolarization ..................................... 125 4.3.4 NumericalResults ................................... 126 4.4 Adaptive Finite Element PML Method for Biperiodic Structures ................................................. 130 4.4.1 ThePMLFormulation ................................ 132 4.4.2 TransparentBoundaryConditionforthePMLProblem .... 135 4.4.3 ConvergenceofthePMLSolution ...................... 140 4.4.4 TheDiscreteProblem ................................. 145 4.4.5 APosterioriErrorAnalysis ............................ 148 4.4.6 NumericalResults ................................... 153 References ..................................................... 158 5 InverseDiffractionGrating ..................................... 163 5.1 UniquenessTheorems ....................................... 164 5.1.1 TheHelmholtzEquation .............................. 165 5.1.2 Maxwell’sEquations ................................. 170 5.2 LocalStability ............................................. 175 5.2.1 TheHelmholtzEquation .............................. 176 5.2.2 Maxwell’sEquations ................................. 182 5.3 NumericalMethods ......................................... 193 References ..................................................... 200 6 Near-FieldImaging ............................................. 205 6.1 Near-FieldData ............................................ 208 6.1.1 TheVariationalProblem .............................. 210 6.1.2 AnAnalyticSolution ................................. 215 6.1.3 ConvergenceofthePowerSeries ....................... 219 6.1.4 TheReconstructionFormula ........................... 224 6.1.5 ErrorEstimates ...................................... 228 6.1.6 NumericalResults ................................... 232 6.2 Far-FieldData ............................................. 233 6.2.1 TheReducedProblem ................................ 236 6.2.2 TransformedFieldExpansion .......................... 238 6.2.3 TheReconstructionFormula ........................... 242 6.2.4 ANonlinearCorrectionScheme ........................ 243 6.2.5 NumericalResults ................................... 244 6.3 Maxwell’sEquations ........................................ 245 6.3.1 TheReducedModelProblem .......................... 248 6.3.2 TransformedFieldExpansion .......................... 249 6.3.3 TheZerothOrderTerm ............................... 253