Periodic Structures: Mode-Matching Approach and Applications in Electromagnetic Engineering

305 Pages·2012·11.846 MB·English

by  Ruey?Bing (Raybeam) Hwang(auth.)

PERIODIC STRUCTURES
PERIODIC STRUCTURES
MODE-MATCHING APPROACH AND
APPLICATIONS IN
ELECTROMAGNETIC ENGINEERING
Ruey-Bing(Raybeam)Hwang
DepartmentofElectricalEngineering
NationalChiao-TungUniversity
Hsinchu,Taiwan
Thiseditionfirstpublished2013
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LibraryofCongressCataloging-in-PublicationData
Hwang,Ruey-Bing.
Periodicstructures:mode-matchingapproachandapplicationsin
electromagneticengineering/Ruey-Bing(Raybeam)Hwang.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-18803-3(hardback)
1. Electricfilters. 2. Optoelectronicdevices. 3. Waveguides.
4. Antennas(Electronics) 5. Photoniccrystals. 6. Crystallattices–Electricproperties.
7. Electromagneticwaves. I. Title.
TK7872.F5H932013
621.3–dc23
2012027877
AcataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN:9781118188033
Typesetin11/13ptTimesbyThomsonDigital,Noida,India
Contents
Preface ix
1 Introduction 1
1.1 HistoricalPerspectiveontheResearchinPeriodicStructures 1
1.2 From1DPeriodicStratifiedMediumto3DPhotonicCrystals:
AnOverviewofthisBook 3
1.2.1 Chapter2:WavePropagationinMultipleDielectricLayers 3
1.2.2 Chapter3:One-DimensionalPeriodicMedium 4
1.2.3 Chapter4:Two-andThree-DimensionalPeriodicStructures 6
1.2.4 Chapter5:IntroducingDefectsintoPeriodicStructures 9
1.2.5 Chapter6:PeriodicImpedanceSurface 11
1.2.6 Chapter7:ExoticDielectricsMadeofPeriodicStructures 13
References 14
FurtherReadings 15
2 WavePropagationinMultipleDielectricLayers 17
2.1 Plane-WaveSolutionsinaUniformDielectricMedium 17
2.2 Transmission-LineNetworkRepresentationofaDielectric
LayerofFiniteThickness 21
2.2.1 WavePropagatinginRegularandExoticMediums 25
2.3 ScatteringCharacteristicsofPlaneWavebyMultiple
DielectricLayers 28
2.3.1 Recursive-ImpedanceMethod 30
2.3.2 Transfer-MatrixMethod 32
2.3.3 Scattering-MatrixMethod 37
2.4 TransverseResonanceTechniqueforDeterminingtheGuiding
CharacteristicsofWavesinMultipleDielectricLayers 45
2.4.1 TransverseResonanceTechnique 45
2.4.2 WillSurfaceWavesbeSupportedinaSingleInterface
Environment? 47
vi Contents
2.4.3 SingleDielectricLayerBackedwithaPECorPMC 49
2.4.4 ModeDispersionRelationofaClosedStructureConsisting
ofDielectricLayers 53
Appendix:DyadicDefinitionandProperties 61
References 62
FurtherReading 63
3 One-DimensionalPeriodicMedium 65
3.1 Bloch–FloquetTheorem 65
3.2 Eigenwaveina1DHolographicGrating 66
3.2.1 TwoSpace-HarmonicApproximation 68
3.2.2 SingleInterfacebetweenaSemi-infiniteUniformanda
1DPeriodicMedium 76
3.3 Eigenwavein1DDielectricGratings:ModalTransmission-Line
Approach 81
3.3.1 In-PlaneIncidence:k = 0 88
y
3.3.2 Out-of-PlaneIncidence:k =/ 0 89
y
3.3.3 EigenwaveinaTwo-TonePeriodicMedium 94
3.3.4 Sturm–LiouvilleDifferentialEquationwithPeriodic
BoundaryCondition 96
3.4 Eigenwaveina1DMetallicPeriodicMedium 98
3.4.1 GeneralizedScatteringMatrixattheInterfacebetweena
1DMetallicPeriodicMediumandUniformMedium 99
3.5 Hybrid-ModeAnalysisofa1DDielectricGrating:
Fourier-ModalApproach 102
3.6 Input–OutputRelationofa1DPeriodicMediumof
FiniteThickness 108
3.7 ScatteringCharacteristicsofaGratingConsistingofMultiple
1DPeriodicLayers 111
3.7.1 Building-BlockApproach 111
3.7.2 ScatteringAnalysisof1DDiffractionGratings 112
3.8 GuidingCharacteristicsofWaveguidesConsistingofMultiple
1DPeriodicLayers 119
3.8.1 TransverseResonanceTechnique 119
3.8.2 DispersionRelationofa1DGratingWaveguide 119
References 129
FurtherReadings 130
4 Two-andThree-DimensionalPeriodicStructures 131
4.1 ModalTransmission-LineApproachfora2DPeriodicMetallic
Medium:In-PlanePropagation 131
Contents vii
4.1.1 GeneralizedScatteringMatrixattheInterfacebetweena
1DPeriodicMetallicMediumandUniformMedium 133
4.1.2 PeriodicBoundaryConditionontheUnitCellalong
they-axis 137
4.1.3 ASimpleGraphicalMethod 138
4.1.4 PhaseRelation:TheRelationshipamongk ,k ,andk 138
x y o
4.1.5 DispersionRelation:TheRelationshipbetweenk andk
o x
(ork ) 143
y
4.1.6 BrillouinZoneandBandStructure 146
4.2 ModalTransmissionLineApproachfora2DPeriodicDielectric
Medium:In-PlanePropagation 152
4.2.1 Input–OutputRelationattheInterface:Generalized
ScatteringMatrixRepresentation 156
4.2.2 BrillouinDiagramandPhaseRelation 158
4.3 DoubleFourier-ModalApproachfora2DDielectricPeriodic
Structure:Out-of-PlanePropagation 166
4.3.1 ScatteringAnalysisofa2DGrating:Out-of-Plane
Propagation 171
4.4 Three-DimensionalPeriodicStructures 172
4.4.1 ScatteringAnalysisofa3DPeriodicStructure 174
4.4.2 EigenwaveAnalysisofa3DPeriodicMedium 180
Appendix:Closed-FormSolutionofε andμ 189
pq,mn pq,mn
References 190
5 IntroducingDefectsintoPeriodicStructures 191
5.1 AParallel-PlaneWaveguidehavingaPairof1DSemi-Infinite
PeriodicStructuresasitsSideWalls 191
5.1.1 BlochImpedance 192
5.1.2 SurfaceStatesSupportedattheInterfaceofa
Semi-Infinite1DPeriodicStructure 193
5.1.3 ASemi-Infinite1DPeriodicStructureConsistingof
SymmetricDielectricWaveguides 200
5.2 DispersionRelationofaParallel-PlaneWaveguidewith
Semi-Infinite1DPeriodicStructuresasWaveguideSideWalls 203
5.2.1 NumericalExample 204
5.3 AParallel-PlaneWaveguidewith2DDielectricPeriodic
StructuresasitsSideWalls 208
5.3.1 MethodofMathematicalAnalysis 211
5.3.2 DispersionRelationofaChannelwithaPairof2D
PeriodicStructuresasitsWaveguideSideWalls 214
viii Contents
5.4 ScatteringCharacteristicsofaPeriodicStructurewithDefects 223
5.4.1 Fabry–PerotEtalon 229
5.4.2 TheCorrelationbetweentheScatteringandGuiding
Characteristics 231
5.5 AParallel-PlaneWaveguidewith2DMetallicPeriodicStructures
asitsSideWalls 236
5.6 OtherApplicationsinMicrowaveEngineering 240
References 243
6 PeriodicImpedanceSurface 245
6.1 ScatteringCharacteristicsofPlaneWavebya1DPeriodic
StructureConsistingofaCavitiesArray 246
6.1.1 AnAMCSurfaceMadeofCorrugatedMetalSurface
withQuarter-WavelengthDepth 256
6.2 PeriodicImpedanceSurfaceApproach(PISA) 264
6.3 ScatteringofPlaneWaveby1DPeriodicImpedanceSurface:
Non-PrincipalPlanePropagation 268
6.3.1 GuidingCharacteristicsofWavesSupportedbya1D
PeriodicImpedanceSurface 277
6.4 ScatteringofPlaneWavebyaDyadic2DPeriodic
ImpedanceSurface 277
References 280
7 ExoticDielectricsMadeofPeriodicStructures 283
7.1 SyntheticDielectricsUsinga2DDielectricColumnsArray 283
7.1.1 DescriptionoftheExample 284
7.1.2 Phase-RelationDiagramofaUniformDielectricMedium 285
7.2 RefractiveIndexofa2DPeriodicMedium 287
7.2.1 Conclusion 291
7.3 AnArtificialDielectricMadeof1DPeriodicDielectricLayers 292
7.3.1 EffectiveRefractiveIndexofthe1DDielectric
PeriodicMedium 293
7.3.2 EffectiveWave-Impedanceofthe1DDielectric
PeriodicMedium 293
7.4 Conclusion 295
References 295
Index 297
Preface
My objective in writing this book is twofold. The first objective is to build up a
firm and rigorous mathematical framework, namely a mode-matching approach and
transmission-line network representation, for analyzing the typical problem of wave
processes involved in periodic structures ranging from the one-dimensional to the
three-dimensional. The second objective is to allow the reader to understand that
mostoftheinterestingphenomenaoccurringincontemporaryperiodicstructurescan
beclarifiedusingexistingclassicalelectromagnetism,suchascoupled-modetheory,
phase-matching condition, and so on. I believe that some people will question the
mode-matching method in handling periodic structures in regard to two disadvan-
tages:theslowconvergencerateformetallicstructuresandtheinfinitestructureunder
consideration.Iconfessthathowtosolvetheelectromagneticfieldsinafiniteperiodic
structureisnotmymajorconcerninthisbook;afterall,therearemanywell-developed
commercialsoftwarepackagesavailablenowadaysbasedonseveralnumericalmeth-
ods (e.g., the finite-difference time (frequency) domain, the finite-element method,
theintegralequationwithmomentmethod,thefinite-integrationmethod)fordealing
withreal-worldelectromagneticproblemsinthemicrowaveandopticalcommunities.
Astotheconvergenceproblem,themodaltransmission-lineapproachtobeelucidated
inthisbookcantacklethetask.Themode-matchingmethodhasitsownadvantages
in facilitating the understanding of electromagnetic fields using the concept of the
modalsolution;forexample,theeigenwavesolutioninaperiodicmediumcanreveal
informationconcerningthemodephase-anddispersion-relation;themodedispersion
relation in a gratings-assisted waveguide can be directly determined by solving the
generalizedeigenvalueequationratherthanbyextractingfromelectromagneticfields.
Some of the subject matter in this book has been presented for several years as
a one-semester course in the Graduate School of National Chiao-Tung University,
Hsinchu, Taiwan. The prerequisites for the course are a knowledge of linear algebra
and electromagnetic theory. I have not attempted the task of referring to all relevant
publications. The lists of books and journal and conference articles in the reference
sectionsattheendsofeachchapterarerepresentative,butarebynomeansexhaustive.
IwouldliketothankProfessorSong-TsuenPeng(ProfessorEmeritusatNational
Chiao-Tung University) who guided me to this exciting field during my PhD study.
x Preface
IappreciateProf.Fung-YuelChangforpolishingmyEnglishwritingandencouraging
me to pursue independent research when he was a visiting Professor in National
Chiao-TungUniversity.
I wish to acknowledge with thanks the consistent encouragement and support
I received from Professor Jan-Dong Tseng (National Chin-Yi University of
Technology).
I would like to express my gratitude to Professor Chang-Yu Wu, Professor
Ching-WenHsueatNationalTaiwanUniversityofScienceandTechnology,andPro-
fessorRajMittraatPennsylvaniaStateUniversityfortheirconstructivesuggestions.
Thefollowingwarrantparticularmention:Dr.Cheng-ChiHsiao,ProfessorsT.Tamir
and K. M. Leung at Polytechnic University, Brooklyn, New York, my colleagues
at National Chiao-Tung University, Professor Sin-Horng Chen (Dean), Professor
Li-ChunWang(Chairman),ProfessorJen-TsaiKuo,ProfessorLin-KunWu,Professor
Kuan-Kin Chan, Professor Yu-De Lin, Professor Shyh-Jong Chung, Professor Yi-
Chiu, Professor Malcolm Ng, and Professor Edward-Yi Chang (Dean of Office of
ResearchandDevelopment,NCTU),ProfessorChing-ChengTienatChungHuaUni-
versity,ProfessorChien-JenWangatNationalUniversityofTainan,ProfessorChih-
Wen Kuo and Professor Ken-Huang Lin at National Sun Yat-Sen University, and
ProfessorKitazawaToshihideatRitsumeikanUniversity,ProfessorsMing-ShingLin
and Chung-I G. Hsu at National Yunlin University of Science and Technology, Pro-
fessor Shyue-Win Wei at National Chi-Nan University, Professor Dau-Chyrh Chang
atOrientalInstituteofTechnology.
I would convey special acknowledgement to James Murphy (Publisher at John
Wiley&Sons)forhisfullandconsistentsupport.
Last but not the least, a special word of thanks is also due my wife, Belinda, who
took care of everything, including me, during the writing and preparation of this
book.
Ruey-Bing(Raybeam)Hwang
Hsinchu,Taiwan
March2012
1
Introduction
1.1 Historical Perspective on the Research in Periodic Structures
The class of periodic structures has been a subject of continuing interest in the lit-
erature. The main effort in the past has been on the scattering and guiding of waves
by one-dimensional (1D) periodic structures. In particular, the microwave field has
employed periodic structures in many different applications, of which a few exam-
plesarelinearaccelerators,slow-wavestructuresinmicrowavetubes,filters,artificial
dielectrics, slot arrays, phase-array antennas, frequency-selective structures, leaky-
wave antennas, and so on. On the other hand, the 1D periodic structure also has its
own applications in optical engineering; for example, in dielectric gratings used in
integrated-opticsapplications(Tamir1975,1979)(diffractiongratingsforbeamsplit-
ting,gratingcouplers,andleaky-wavestructures).
In recent years, considerable attention has been focused on the numerical and ex-
perimental studies of wave phenomena associated with two-dimensional (2D) and
three-dimensional (3D) structures, and many applications have been demonstrated.
Most of the potential applications were found and developed in the optics commu-
nity, such as the photonic crystal. A photonic crystal contains dielectric or metallic
inclusions periodically arranged in a 2D or 3D lattice pattern; these mimic a natural
crystalwithasmallandbasicbuildingblockofatomsormoleculesrepeatedinspace.
Theperiodicnatureofthedielectricfunctionresultsinthesimultaneousreflectionof
wavesfromeachperiod,producingastop-bandwherethewavepropagationisforbid-
den.Suchbehaviorisanalogouswiththeelectronicbandgapinelectronicmaterials,
whichiscausedbyintroducingagapintotheenergybandstructureofacrystalsothat
electronsareforbiddentopropagatewithacertainenergyinacertaindirection(Kittle
1986).Asemiconductoristhebestrepresentativehavingacompletebandgapbetween
PeriodicStructures:Mode-MatchingApproachandApplicationsinElectromagneticEngineering,FirstEdition.
Ruey-Bing(Raybeam)Hwang.
©2013JohnWiley&SonsSingaporePte.Ltd.Published2013byJohnWiley&SonsSingaporePte.Ltd.