Table Of ContentApplied Mathematical Sciences
Gang Bao
Peijun Li
Maxwell’s
Equations
in Periodic
Structures
Applied Mathematical Sciences
Volume 208
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·
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
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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
