Adjoint Topology Optimization Theory for Nano-Optics

168 Pages·2022·9.528 MB·English

by  Yongbo Deng

Yongbo Deng
Adjoint Topology
Optimization
Theory
for Nano-Optics
Adjoint Topology Optimization Theory
for Nano-Optics
Yongbo Deng
Adjoint Topology
Optimization Theory
for Nano-Optics
YongboDeng
StateKeyLaboratoryofAppliedOptics
(SKLAO)
ChangchunInstituteofOptics,Fine
MechanicsandPhysics(CIOMP),Chinese
AcademyofSciences
Changchun,China
ISBN978-981-16-7968-1 ISBN978-981-16-7969-8 (eBook)
https://doi.org/10.1007/978-981-16-7969-8
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Preface
Topologyoptimizationcandeterminethetopology,shape,andsizeofthestructures,
simultaneously.Ithasbeenregardedtobeoneofthemostpowerfulapproachesfor
theinversedesignofstructuresinseveralscientificareas,includingelasticity,fluid
dynamics, thermodynamics, wave optics or electromagnetics, etc. In wave optics,
oneofthehotresearchareasisthenano-optics/photonics.Nano-opticshasprovided
new freedom for the manipulation and control of optical wave propagation. The
two most typical materials for the nano-optical devices are dielectrics and noble
metal. Those two material types correspond to the effects of directional scattering
andsurfaceplasmonics,respectively.Severaladvancednano-fabricationprocesses,
includingelectronic-beamlithographyandtwo-photonlithography,havebeendevel-
oped for the manufacturing of nanostructures. Those processes bring new design
freedoms and fabrication constraints for the design of nano-structures. Because
of the specialities, the usually used design approaches encounter the problem on
synthetically considering the typical physical effects of nano-optics and manufac-
turabilitytoachievereasonableopticalperformance.Therefore,theauthorandhis
colleagueswereinspiredtoutilizetopologyoptimizationmethodtoimplementthe
inversedesignfornano-opticsbasedonthedesiredfunctionalperformance.Thereis
stillaplentyofroomintheresearchoftopologyoptimizationfornano-optics.The
authorhopesthatthisbookcaninspiremoreresearchersintherelevantareas.
Thisbookisacompletelyrevised,updated,andexpandedversionoftheresearches
oftheauthorandhiscolleaguesontopologyoptimizationmethodfornano-optics.
The author, was the Ph.D. student of Profs. Yihui Wu and Zhenyu Liu during
the period from 2007 to 2012. Before the author’s Ph.D. defense, he was super-
vised by Profs. Yihui Wu and Zhenyu Liu, to pursue the topology optimization
method for flow problems. During this period, Profs. Yihui Wu, Zhenyu Liu, and
the author together with their related colleagues achieved the topology optimiza-
tion for unsteady flows and flows with body forces. Simultaneously, they have
appliedthedevelopedtopologyoptimizationtoimplementtheinversedesignofthe
typicalmicrofluidiccomponents,includingthepassivemicromixers,no-moving-part
microvalve,andvalvelessmicropump.AfterhisPh.D.study,theauthorstartedhis
v
vi Preface
professional position in the State Key Laboratory of Applied Optics (SKLAO) of
CIOMP,fromJulyof2012.
Sequentially, the author began his attempts on topology optimization of direc-
tionalopticalcloaks.In2014,hesolvedthematerialinterpolationproblemforthe
topologyoptimizationofmetalnanostructuresforsurfaceplasmonics,togetherwith
hiscolleaguesincludingProf.ZhenyuLiu,Dr.ChaoSong,andProfs.PengHaoand
Yihui Wu of CIOMP, Prof. Yongmin Liu of Northeastern University in USA and
Prof.JanG.KorvinkofKarlsruheInstituteofTechnologyinGermany.Basedonthe
solution of the problem on material interpolation, he developed the topology opti-
mizationbasedcomputationaldesignmethodforsurfaceplasmonpolaritonsin2014,
togetherwithhiscolleaguesincludingProf.ZhenyuLiu,Dr.ChaoSong,andProfs.
JunfengWu,YongshunLiuandYihuiWuofCIOMP.Asatypicalapplicationofthe
developedtopologyoptimizationmethod,periodicnanostructureswereinvestigated
for extraordinary optical transmission, and the results were published in 2021 by
theauthortogetherwithMr.YuLinandMr.YemingHanofCIOMP,andDr.Chao
SongofIowaStateUniversityinUSA,basedontheinvitationofProf.YongminLiu
ofNortheasternUniversityinUSA.Duringtheauthor’sworkingintheDepartment
ofMicrosystemsEngineering(IMTEK)ofFreiburgUniversityinGermanyduring
the period from 2014 to 2015, he together with Prof. Jan G. Korvink, moved to
KarlsruheInstituteofTechnologyin2015,begantheresearchontopologyoptimiza-
tionforthree-dimensionalelectromagneticwaves,andtheresultswerepublishedin
2016; during this period, he derived his first Guest Professor Fellowship of Karl-
sruheInstituteofTechnology.In2017,theauthortogetherwithProf.JanG.Korvink
began to develop the self-consistent adjoint analysis for topology optimization of
waveoptics,andtheresultswerepublishedin2018;duringthisperiod,hederived
hissecondGuestProfessorFellowshipofKarlsruheInstituteofTechnology.
In2018,theauthorderivedhisHumboldtResearchFellowshipforExperienced
Researchers, and began his research in Prof. Jan G. Korvink’s Group in the Insti-
tuteofMicrostructureTechnology,KarlsruheInstituteofTechnology,untiltheend
of February of 2020. He was then blocked by COVID-19 in Germany before his
returning to China until May of 2020. In 2020, the topology optimization-based
inversedesignofplasmonicnanodimerwithmaximumnear-filedenhancementwas
published, based on his collaberation with Profs. Yiqin Chen, Yueqiang Hu, and
Huigao Duan of Hunan University, and Prof. Dangyuan Lei of City University
of Hong Kong. In the same year, the author together with Profs. Zhenyu Liu and
Jan G. Korvink published their topology optimization work implemented on two-
dimensionalmanifolds.HederivedhisfullprofessorpositioninSeptemberofthis
year.
Heisgratefultoallhiscolleaguesprovidingvaluablediscussionandsuggestions
on the related topics, and the reviewers’ kind attention and valuable suggestions
derived during the peer review of the related publications. The researches in this
bookweresupportedbytheInnovationGrantofChangchunInstituteofOptics,Fine
Mechanics and Physics (CIOMP), the Humboldt Research Fellowship for Expe-
rienced Researchers (Humboldt-ID: 1197305), the Fund of State Key Laboratory
of Applied Optics (SKLAO), the National Natural Science Foundation of China
Preface vii
(Nos.51875545,51405465,61727813,51275504,51205381),theYouthInnovation
PromotionAssociationoftheChineseAcademyofSciences(No.2018253),Guest
Professor Fellowships of the Karlsruhe Institute of Technology (KIT) by the DFG
Metacoilproject(KO1883/20-1),theNationalHighTechnologyProgramofChina
(No. 2015AA042604), Science and Technology Development Plan of Changchun
CityandJiLinProvince(Nos.15SS12and20140519007JH).
Changchun,China YongboDeng
October2021
Contents
1 Introduction ................................................... 1
1.1 Self-consistency of Adjoint Analysis for Topology
OptimizationinFrequencyDomain ........................... 2
1.2 Dielectric Material Based Topology Optimization
forNano-Optics ............................................ 4
1.3 Metal Material-Based Topology Optimization
forNano-Optics ............................................ 5
1.4 Extension of Design Space: Topology Optimization
onTwo-DimensionalManifoldsforWaveOptics ................ 6
1.5 Conventions ............................................... 7
References ..................................................... 7
2 Self-consistentAdjointAnalysis ................................. 15
2.1 TopologyOptimizationProblems ............................. 15
2.2 SplitofWaveEquations ..................................... 21
2.3 AdjointAnalysis ........................................... 23
2.4 NumericalImplementation ................................... 25
2.5 NumericalExamples ........................................ 27
2.5.1 OpticalCloak ....................................... 27
2.5.2 Nanostructures for Localized Surface Plasmon
Resonances ......................................... 33
2.6 Summary .................................................. 38
2.7 Appendix ................................................. 41
2.7.1 AdjointAnalysisofTopologyOptimizationProblem
forTwo-DimensionalOpticalWaves .................... 41
2.7.2 AdjointAnalysisofTopologyOptimizationProblem
forThree-DimensionalOpticalWaves ................... 44
References ..................................................... 46
3 Dielectric Material-Based Topology Optimization
forNano-Optics ................................................ 49
3.1 Methodology .............................................. 50
ix
x Contents
3.1.1 MagneticFieldFormulation ........................... 50
3.1.2 AdjointAnalysisforMagneticField-BasedTopology
Optimization ........................................ 52
3.1.3 ElectricFieldFormulation ............................. 53
3.1.4 AdjointAnalysisforElectricField-BasedTopology
Optimization ........................................ 55
3.1.5 Numericalimplementation ............................ 56
3.2 Applications ............................................... 59
3.2.1 CloakforPerfectConductor ........................... 59
3.2.2 DielectricResonator .................................. 62
3.2.3 BeamSplitter ........................................ 64
3.2.4 CloakforDielectricResonator ......................... 67
3.2.5 MetalenswithOpticalVortices ......................... 76
3.3 Summary .................................................. 91
References ..................................................... 92
4 MetalMaterial-BasedTopologyOptimizationforNano-Optics ..... 95
4.1 Methodology .............................................. 95
4.1.1 TopologyOptimizationProblem ....................... 96
4.1.2 Adjointanalysis ..................................... 98
4.1.3 Numericalimplementation ............................ 100
4.2 Applications ............................................... 101
4.2.1 Nanostructures for Localized Surface Plasmonic
Resonances ......................................... 101
4.2.2 NanoslitsforExtraordinaryOpticalTransmission ......... 106
4.2.3 Nanoantennas for Coupling Free Space
andMetal-Insulator-MetalWaveguide ................... 117
4.2.4 CloakforSurfacePlasmonicPolaritons ................. 133
4.3 Summary .................................................. 136
4.4 Appendix ................................................. 137
References ..................................................... 140
5 Extension of Design Space: Topology Optimization
onTwo-DimensionalManifoldsforWaveOptics ................... 145
5.1 Methodology .............................................. 145
5.1.1 Two-DimensionalManifolds ........................... 145
5.1.2 PDEsforPhysicalFields .............................. 146
5.1.3 Regularization ....................................... 148
5.1.4 TopologyOptimizationProblem ....................... 148
5.1.5 AdjointAnalysis ..................................... 149
5.1.6 NumericalImplementation ............................ 151
5.2 Applications ............................................... 152
5.3 Summary .................................................. 158
References ..................................................... 162
Chapter 1
Introduction
Currenthottopicsinopticsresearchcoveropticalcloaking[6, 61, 88, 108],surface
plasmonicpolaritons[30, 75, 79, 129],metamaterials[50, 96, 121, 131],metasur-
faces[55,126],andbandgapstructureorphotoniccrystals[54].Inthoseresearches,
thecontrolofopticalwavesisrealizedbythestructureswithcomplexspacialcon-
figurationsusingpre-selectedmaterials.Theincidentwaveshavecomplicatedpolar-
izations.Mostofthosesituationscannotbereducedintotwodimensional,exceptfor
aminorityofcasesinvolvinglinearpolarizedwaves.Forthiscomplexity,intuition-
basedopticalstructuredesignhasitslimitation.Toovercomethislimitation,several
testsofinversedesignhavebeenimplementedbasedonstructuraloptimizationmeth-
ods[7, 8, 10, 26, 28, 29, 35–38, 48, 74, 77, 80, 86, 87, 89, 134, 137],wherethe
mostemployedoneisthetopologyoptimizationmethod.Topologyoptimizationis
currentlyregardedtobeoneofthemostrobustmethodologyfortheinversedetermi-
nationofmaterialdistributioninstructuresthatmeetgivenstructuralperformance
criteria.
Topologyoptimizationisarobustmethodusedtodeterminethestructuralcon-
figuration, which corresponds to the material distribution in a structure [20]. It is
currentlyregardedtobethemostrobustmethodologyfortheinversedetermination
ofmaterialdistributioninthestructuresthatmeetgivenstructuralperformancecri-
teria.Incontrasttodesigningdevicesbytuningahandfulofstructuralparametersin
sizeandshapeoptimization,topologyoptimizationutilizesthefull-parameterspace
todesignastructurebasedontheuser-desiredperformance,anditismoreflexible
androbust,becauseofthelowdependenceontheinitialguessoftheoptimization
procedure.
Material distribution method for modern topology optimization was pioneered
by Bendsøe and Kikuchi for elasticity [18], and this method has been extended to
a variety of areas, e.g., acoustics, electromagnetics, fluid dynamics and thermody-
namics [2, 7, 8, 10, 19, 21, 25, 28, 29, 36, 38, 43, 45, 48, 57, 74, 85, 87, 89,
91, 92, 104, 132, 134, 136, 137],althoughoptimizationofstructuraltopologywas
investigatedasearlyas1904fortrussesbyMichell[68].Severalothermethodsalso
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Y.Deng,AdjointTopologyOptimizationTheoryforNano-Optics,
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