Electromagnetic bound states in the radiation continuum and second harmonic generation in double arrays of periodic diel...

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Title:
Electromagnetic bound states in the radiation continuum and second harmonic generation in double arrays of periodic dielectric structures
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1 online resource (124 p.)
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english
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Ndangali,Remy Friends
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Shabanov, Sergei
Committee Members:
Gopalakrishnan, Jay
Pilyugin, Sergei
Tanner, David B
Klauder, John R
Hebard, Arthur F

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Subjects / Keywords:
amplification -- array -- bound -- continuum -- control -- coupled -- cylinders -- data -- dielectric -- double -- effects -- electromagnetic -- field -- generation -- harmonic -- nanophotonic -- near -- nonlinear -- optical -- periodic -- radiation -- resonance -- scattering -- second -- siegert -- state -- subwavelength -- vanishing -- width
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Mathematics thesis, Ph.D.
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Electronic Thesis or Dissertation

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Abstract:
Electromagnetic bound states in the radiation continuum are studied for periodic double arrays of subwavelength dielectric cylinders. These states are similar to the bound states in the radiation continuum in quantum mechanics discovered by von Neumann and Wigner. For the system studied, these states are shown to exist at specific distances between the arrays in the spectral region where one or more diffraction channels are open. The near field and scattering resonances of the structure are investigated when the distance between the arrays varies in a neighborhood of its critical values at which the bound states are formed. In particular, it is shown that the near field in the scattering process becomes significantly amplified in specific regions of the array as the distance approaches its critical values. This amplification is explained through the Siegert state formalism, which is extended from quantum mechanics to Maxwell's theory of electromagnetism. The said amplification is also used to control the second harmonic generation in a periodic double array of subwavelength dielectric cylinders with a second order nonlinear susceptibility. The conversion efficiency of the incident fundamental flux into the second harmonic flux is shown to be as high as 40% at a distance between the arrays as low as half of the incident radiation wavelength.
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In the series University of Florida Digital Collections.
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by Remy Friends Ndangali.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Shabanov, Sergei.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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ELECTROMAGNETICBOUNDSTATESINTHERADIATIONCONTINUUMANDSECONDHARMONICGENERATIONINDOUBLEARRAYSOFPERIODICDIELECTRICSTRUCTURESByREMYFRIENDSNDANGALIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011RemyFriendsNdangali 2

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Myfamily'sjourneyisamiracleweowetothekindnessofamultitudeoffriends,andthecharityofperfectstrangers.Toallofyouwhoshowedusthewayinourdaysofdespair,Idedicatethiswork. 3

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ACKNOWLEDGMENTS IwouldliketothankmydoctoraladvisorDr.SergeiShabanovforawonderfuladventureoverthecourseofmyPh.Dprogram.Thanksforthemanylessonsinmathematics,physics,andlifeingeneral.Thanksfortheencouragementsinthemomentsofhardships,andthecelebrationsonachievingmilestones.Iwouldalsoliketothankmyotherdoctoralcommitteemembers,namely,Dr.Klauder,Dr.Pilyugin,Dr.Gopalakrishnan,Dr.Tanner,andDr.Hebard,fortheirinterest,questionsandsuggestedimprovementstothiswork.IwouldliketothankespeciallyDr.Klauderforhislessonsonmathematicalmethodsoftheoreticalphysics.IwouldalsoliketothankalltheotherprofessorswhoseclassesItookduringthecourseofthesixyearsIspentinthePh.DprogramattheUniversityofFlorida.SpecialmentionshouldbemadeofDr.BlockandDr.BrooksforteachingmeAnalysis,andDr.TurullforteachingmeAbstractAlgebra.Iamalsoverygratefulfortheadvice,andteachingofDr.Robinsonthroughouttheyears.Lastbutnotleast,IwouldliketothankthestaffattheDepartmentofMathematicsforallthehelpandadvicethroughouttheyears.IthankespeciallyGretchen,Sandy,andMargaretforkeepingmeincheck,andmakingsureIneveroverlookadeadline,orarequirement.IthankalsoConnieandMarie,fortheiravailabilitytoassistatalltimes. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Motivation .................................... 10 1.2HistoricalContext:BoundStatesandSiegertStatesintheTheoryofQuantumMechanics .............................. 12 1.3TheExtensiontoElectromagnetismanditsChallenges .......... 17 1.4MainResults .................................. 19 2ELECTROMAGNETICSIEGERTSTATESINPERIODICSTRUCTURES ... 22 2.1ElectromagneticSiegertStates ........................ 22 2.2RegularPerturbationTheory ......................... 30 2.2.1PerturbationofBoundStatesintheRadiationContinuum ..... 31 2.2.2NearFieldAmplicationMechanism ................. 35 2.3DecayofElectromagneticSiegertStates .................. 40 2.4ProofoftheRegularPerturbationTheorem ................. 44 3ELECTROMAGNETICBOUNDSTATESINTHERADIATIONCONTINUUMFORPERIODICDOUBLEARRAYSOFSUBWAVELENGTHDIELECTRICCYLINDERS ..................................... 50 3.1ScatteringTheoryandClassicationoftheFields .............. 50 3.2BoundStates .................................. 55 3.2.1BoundStatesBelowtheRadiationContinuum ............ 60 3.2.2BoundStatesintheRadiationContinuumI:OneOpenDiffractionChannel ................................. 62 3.2.3Application:ZeroWidthResonancesandNearFieldAmplication 67 3.3BoundStatesintheRadiationContinuumN,N2 ............. 72 3.3.1BoundStatesintheRadiationContinuumII:TwoOpenDiffractionChannels ................................ 76 4ARESONANTGENERATIONOFSECONDHARMONICSINDOUBLEARRAYSOFSUBWAVELENGTHDIELECTRICCYLINDERS ............... 86 4.1TheScatteringTheory ............................. 86 4.2SubwavelengthCylindersApproximation ................... 93 5

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4.3AmplitudesoftheFundamentalandSecondHarmonics .......... 94 4.4FluxAnalysis:TheConversionEfciency .................. 99 APPENDIX ACOMPLEMENTSI .................................. 106 A.1TheLippmann-SchwingerIntegralEquation ................. 106 A.2SolutionoftheLippmann-SchwingerIntegralEquationintheZeroRadiusApproximation ................................. 107 A.3ComplementsonBoundStatesintheContinuumsIandII ......... 111 A.4Approximations ................................. 113 BCOMPLEMENTSII .................................. 116 B.1Estimationofand .............................. 116 B.2ComplementsontheFluxAnalysis:FluxConservation ........... 117 B.3ComplementsontheAmplitudeE1 ...................... 120 REFERENCES ....................................... 122 BIOGRAPHICALSKETCH ................................ 124 6

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LISTOFTABLES Table page 3-1Existenceofsolutionstosystems( 3 ) ...................... 81 7

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LISTOFFIGURES Figure page 2-1ExamplesofstructuresconsideredandtheCkx-plane. .............. 24 2-2The-planefortheintegralinEq.( 2 ) ...................... 44 3-1Periodicdoublearraysofdielectriccylinders,andthecorrespondingspectrumrepresentation. .................................... 51 3-2Valuesoftheparametersa,kx,R,and"csusceptibletoallowtheformationofaboundstate. .................................... 57 3-3Plotsoftheamplitudesofboundstatesintheradiationcontinnum ....... 68 3-4Thespecularcoefcientandtheelectriceldonthecylindersnearaboundstateinthecontinuum ................................ 72 4-1Thescatteringprocessfornormalincidentradiation,andthecorrespondingboundstatesintheradiationcontinuum ...................... 87 4-2Theconversionefciencyanditsregionofvalidity ................ 103 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyELECTROMAGNETICBOUNDSTATESINTHERADIATIONCONTINUUMANDSECONDHARMONICGENERATIONINDOUBLEARRAYSOFPERIODICDIELECTRICSTRUCTURESByRemyFriendsNdangaliAugust2011Chair:SergeiV.ShabanovMajor:Mathematics Electromagneticboundstatesintheradiationcontinuumarestudiedforperiodicdoublearraysofsubwavelengthdielectriccylinders.ThesestatesaresimilartotheboundstatesintheradiationcontinuuminquantummechanicsdiscoveredbyvonNeumannandWigner.Forthesystemstudied,thesestatesareshowntoexistatspecicdistancesbetweenthearraysinthespectralregionwhereoneormorediffractionchannelsareopen.Theneareldandscatteringresonancesofthestructureareinvestigatedwhenthedistancebetweenthearraysvariesinaneighborhoodofitscriticalvaluesatwhichtheboundstatesareformed.Inparticular,itisshownthattheneareldinthescatteringprocessbecomessignicantlyampliedinspecicregionsofthearrayasthedistanceapproachesitscriticalvalues.ThisamplicationisexplainedthroughtheSiegertstateformalism,whichisextendedfromquantummechanicstoMaxwell'stheoryofelectromagnetism.Thesaidamplicationisalsousedtocontrolthesecondharmonicgenerationinaperiodicdoublearrayofsubwavelengthdielectriccylinderswithasecondordernonlinearsusceptibility.Theconversionefciencyoftheincidentfundamentaluxintothesecondharmonicuxisshowntobeashighas40%atadistancebetweenthearraysaslowashalfoftheincidentradiationwavelength. 9

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CHAPTER1INTRODUCTION 1.1Motivation ThisworkisdevotedtothestudyofaspecialclassofsolutionstoMaxwell'sequationsthatdescribetheresonantscatteringoflightonperiodicallystructuredmaterialsandpossibleapplicationstoopticalnonlineareffectsinsuchstructures. Intheasymptoticregionawayfromthescatteringstructure,asolutionofthewaveequationcanalwaysberepresentedasasuperpositionofplanewavescharacterizedbyaspecicsetof(spectral)parameters,namely,thewavenumbers.Spectralparametersdescribingtheasymptotic(orscattered)wavesformacontinuoussetcalledtheradiationcontinuum.Solutionsintheradiationcontinuumarenotlocalizedinspace,i.e.,theyarenotsquareintegrable. Forsomestructures,Maxwell'sequationsalsoadmitlocalizedsolutionswhicharesquareintegrable(e.g.,astandingwaveinametalcavityorpropagatingwavesindefectsofaphotoniccrystal).However,thespectralparametersofalocalizedsolutionarenotusuallyintheradiationcontinuum(thisisthecasefortheaforementionedwaveguidingmodesinmetalcavitiesordefectsinphotoniccrystals). ThereisasimilarclassicationofsolutionsoftheSchrodingerequationthatdescribequantumsystems.Localized(squareintegrable)solutionsoftheSchrodingerequationarecalledboundstates.Consequently,ifspectralparametersofaboundstatelieintheradiationcontinuum,thenitiscalledaboundstateintheradiationcontinuum.Suchboundstateswererstpredictedtoexistin1929byvonNeumannandWigner[ 1 2 ],andtheyareknowntoberatherrareinquantumsystems. ThepresentstudyisdevotedtoboundstatesintheradiationcontinuuminMaxwell'stheoryofelectromagnetismandtheirapplications.Itwillbeshownthat,incontrasttoquantummechanics,thereisasimpleprinciplefordesigningscatteringstructuresthatsupportelectromagneticboundstatesintheradiationcontinuum.The 10

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structuresthathaveboundstatesintheradiationcontinuumexhibitunusualscatteringpropertieswhichcanbeusedinpracticalapplicationstonanophotonics.Oneofthemostimportantapplicationsisthedevelopmentofanovelmechanismtoenhanceandcontrolopticallynonlineareffectsinnanophotonicdevices,theproblemthatshouldbesolvedinonewayoranotherinordertoachieveanultimategoalofnanophotonics:allopticaldataprocessing. Tobespecic,inthesocalledTransverseMagneticPolarization(TMpolarization),Maxwell'sequationsreducetothescalarpartialdifferentialequation, 1 c2@2tD=E(1) wherecisthespeedoflight,andDisaeldthatdependsanalyticallyontheeldE,i.e., D="E+2E2+3E3+...(1) Here,"iscalledthedielectricconstant,whilethen's,n=1,2,...arehigherordernonlineardielectricsusceptibilities.Inthevacuum,E=D.TheregionsofspaceinwhichE6=Drepresentthescatteringstructure.Inthespatiallyasymptoticregion,E=DandsolutionstoEq.( 1 )arelinearcombinationofplanewavescharacterizedbytheirwavenumbersthatformtheradiationcontinuum.Inpractice,thenonlinearsusceptibilitiesareverysmallincomparisontothedielectricconstant",andassuch,theyareusuallyneglected,leadingtothewellknownlinearwaveequation, c2@2tE=E(1) AnelectromagneticboundstateisasquareintegrablesolutiontoEq.( 1 ).Consequently,ifthewavenumberofaboundstateliesintheradiationcontinuumofthescatteringsystem,thesaidstateisanelectromagneticboundstateintheradiationcontinuum.Clearly,suchstatesexistonlyforaspecialchoiceofthefunction". 11

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Itappearsthatifthescatteringsystemadmitsboundstatesintheradiationcontinuum,thenwhenilluminatedbyanincidentradiation,theeldEmaybesignicantlyampliedinsomesubsetsofsupportofthefunction")]TJ /F3 11.955 Tf 12.08 0 Td[(1ascomparedtotheamplitudeoftheincidentradiation.AccordingtoEq.( 1 ),nonlinearsusceptibilitiescannolongerbeneglected,whichleadstoanenhancementofopticallynonlineareffectsinthescatteringstructure.AcompletetheoryoftheseeffectsrequiresolvingafullnonlinearscatteringproblemforEq.( 1 ).Itwillbeshownthatwhenaboundstateintheradiationcontinuumispresentinthesystem,conventionalmethodsofsolvingthisnonlinearscatteringproblemfailbecauseofthenonanalyticityofthesolutioninn.Oneofthegoalsofthepresentstudyisthedevelopmentofarigorousmathematicalformalismtocircumventthisdifcultyandapplythisformalismtosomepracticalproblemssuchashigherharmonicsgeneration. However,beforewemayproceedtogiveanoverviewoftheproposednovelties,itisworthwhiletogiveasimpleexamplefromquantummechanicsthatclariesthesignicanceofboundstatesintheradiationcontinuumaswellastostressthekeymathematicaldifferencesintheverydenitionofquantumandelectromagneticboundstatesintheradiationcontinuum.Thelatterdifferencescompriseamainmathematicalproblemthathasbeensolvedinthepresentstudytoextendtheconceptofboundstatesintheradiationcontinuumtoelectromagnetism. 1.2HistoricalContext:BoundStatesandSiegertStatesintheTheoryofQuantumMechanics Usingsuitablereducedunits,considerthesingleparticleSchrodingerequation, i@ @t(r,t)=bH(r,t)(1) withhamiltonian,bH=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 2+V 12

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ThepotentialVisarealvaluedfunctionthatwewillassumetoberadiallysymmetricintheinnitethreedimensionalspace,i.e.,V(r)=V(r).ItiscustomarytoseekthesolutiontotheSchrodingerequationasasuperpositionofharmonicallytimedependentstates(r,t)=E(r)e)]TJ /F8 7.97 Tf 6.58 0 Td[(iEt,whereEistheenergyoftheparticle.TheamplitudesEaretheneigenstatesofthehamiltonianbH.Specically, bHE(r)=EE(r)(1) WhenthepotentialVisidenticallyzero,i.e.,forafreeparticle,theeigenstateEtakestheform[ 2 3 ],E,f(r)=sin(kr) kr,k=p 2E whereitisunderstoodthatifE<0,thesquarerootwithpositiveimaginarypartistaken.ThechoiceoftheparticularsolutionE,fismadetoensurethattheeigenstateEremainsnitethroughoutspace,andthereforeatthespatialorigininparticular.Therequirementforthefreestatetoremainboundedatthespatialinnity(r!1)impliesthatk>0.ThestatesE,fthusdenedarethefreestatesoftheparticle. IfVisnotidenticallyzero,butV(r)=0forallr>r0forsomer0,thenthequantumscatteringtheoryrequiresthatforr>r0,eacheigenstateofthehamiltonianbHbeasuperpositionofafreestate,andanoutgoingsphericalwaver)]TJ /F6 7.97 Tf 6.59 0 Td[(1eikrthatoccursasaresultofthescatteringofthefreestatebythepotentialV[ 2 3 ].Thatis, E(r)=Isin(kr) kr+Seikr r,k=p 2E,r>r0(1) TheamplitudeSisthencalledthescatteredamplitudeofthestateE,whereasIdesignatestheamplitudeofthefreestate. ForpositiveenergiesE,thestateEremainsboundedthroughoutspace.Itremainsoscillatoryinthespatialinnity(r!1),andtherefore,suchastateisnotsquareintegrableon[0,1).Itiscalledascatteringstate. 13

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Fornegativeenergies,thenumberk=p 2Eispureimaginarywithpositiveimaginarypart,andthereforetheeigenstateEisboundedinspaceifandonlyifI=0.Itfollowsinparticularthatifsuchastateexists,itissquareintegrableon[0,1)sinceitdecaysexponentiallyinthespatialinnity.Suchastateiscalledaboundstate. ThusthespectrumoftheenergiesEofthehamiltonianbHiscomposedofthescatteringstateswhichexistforE>0,andtheboundstatesthatexistforE<0.AnimportantremarkisthatthescatteringstatesdoexistforallE>0.Inparticular,theirspectrumisthecontinuum(0,1),whichinthejargonofquantummechanicsisreferredtoastheradiationcontinuum.Ontheotherhand,thespectrumoftheboundstatesisnecessarilyadiscretesubsetoftheinterval(,0).ThisisbecausetheproblemofndingtheboundstatesreducestoaneigenvalueproblemforthehamiltonianbHontheHilbertspaceL2([0,1)).Asthesaidhamiltonianisself-adjoint,thespectrumoftheboundstatesmustnecessarilybediscrete.Sincefortheboundstatesjustdescribed,theenergyEisnegative,thesestatesarereferredtoasboundstatesbelowtheradiationcontinuum. TheremarkableobservationmadebyvonNeumannandWigneristhatiftheconditiononthepotentialVtovanishforallr>r0isrelaxed,andreplacedbythemoregeneralconditionV(r)!0,asr!1,thensomeboundstatesmaybefoundintheradiationcontinuumofthescatteringstates.Thisisapeculiar,andratherrarefeatureinphysicalsystems.Infact,intheirpaper[ 1 ],vonNeumannandWignerusedaconstructiveapproachtoderiveapotentialVprovidingaboundstateintheradiationcontinuum,andthesoobtainedexamplewasratherarticialandofnophysicalsignicance.Thisledtoboundstatesintheradiationcontinuumbeingconsideredasmathematicalcuriositiesuntilaphysicalexamplewasfoundinatomicphysicsmuchlater[ 4 ].Forthecuriousreader,theactualpotentialproducedbyvonNeumannand 14

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Wigneris1,V(r)=)]TJ /F3 11.955 Tf 34.74 8.09 Td[(64k2A2sin4(kr) [A2+(2kr)]TJ /F3 11.955 Tf 11.95 0 Td[(sin(2kr))2]2+48k2sin4(kr))]TJ /F3 11.955 Tf 11.95 0 Td[(8k2(2kr)]TJ /F3 11.955 Tf 11.96 0 Td[(sin(2kr))sin(2kr) A2+(2kr)]TJ /F3 11.955 Tf 11.96 0 Td[(sin(2kr))2,k>0 ForAanonzeroarbitraryconstant,thispotentialproducesaboundstateofenergyE=1 2k2>0,andthereforethisstateisindeedaboundstateintheradiationcontinuum.Itsexplicitexpressionis[ 2 ],(r)=sin(kr) kr(A2+(2kr)]TJ /F3 11.955 Tf 11.96 0 Td[(sin(2kr))2) Thesignicanceofboundstatesintheradiationcontinuumisunderstoodthroughthetheoryofscatteringresonances.Resonancesarepeaks(maxima)ofthesocalledscatteringcrosssection,,whichissimplytheprobabilityuxperunitsolidangle,perunitincidentux.It'sexpressionforaneigenstateEis,(E)=4S I2 PeaksinthescatteringcrosssectionareexplainedthroughtheBreit-Wignertheory[ 5 ]bythepresenceofcomplexpolesinthescatteringcrosssectionwhenitisextended,asafunctionoftheenergy,tothecomplexplane.Eachsuchpoleiscalledaresonancepole.Itissimple,anditliesinthelowerhalfofthecomplexplane,i.e.,apoleisoftheformEn)]TJ /F5 11.955 Tf 11.96 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n,)]TJ /F8 7.97 Tf 14.25 -1.8 Td[(n>0. Inthevicinityofaresonancepole,thescatteringcrosssectionhasaLorentzianproleofresonancewidth)]TJ /F1 11.955 Tf 10.1 0 Td[(describedbythewellknownBreit-Wignerformula:(E)/)]TJ /F6 7.97 Tf 6.78 4.34 Td[(2n (E)-222(En)2+)]TJ /F6 7.97 Tf 18.73 3.45 Td[(2n 1Theoriginalpaper[ 1 ]byvonNeumannandWignercontainedaminorerror.Theformulaproducedhereisthecorrectedformulaasprovidedin[ 2 ]. 15

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In1939,A.J.F.Siegert[ 3 ]showedthatingeneral,polesofthescatteringcrosssectioncorrespondtoeigenfunctionsofthehamiltonianbHwithspeciccomplexboundaryconditions.InthecaseofapotentialVthatvanishesforallr>r0asdescribedhigher,theSiegertstatesaredenedaseigenfunctionsEofthehamiltonianbHsatisfyingtheboundarycondition, @ @r(rE))]TJ /F5 11.955 Tf 11.95 0 Td[(ikrEr=r0=0,k=p 2E(1a) ThisconditionamountstorequiringthattheincidentamplitudeIinEq.( 1 )bezero.Inparticular,forthisspeciccase,boundstatesbelowtheradiationcontinuumareSiegertstateswithrealnegativeeigenvaluesEn. Forageneralsphericalpotentialvanishingatthespatialinnity,theboundaryconditionfordeningtheSiegertstatesbecomes, @ @r(rE))]TJ /F5 11.955 Tf 11.96 0 Td[(ikrEr=r0,r0!1=0,(1b) Justasinthepreviouscase,aSiegertstatencorrespondstoaresonancepoleEn)]TJ /F5 11.955 Tf 12.09 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n.Whensuchapoleisreal,i.e.,)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n=0,thentheSiegertstatebecomesaboundstatebeloworintheradiationcontinuumdependingonwhetherEn<0orEn>0. ThephysicalsignicanceofSiegertstateswith)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n>0canbeunderstoodthroughtheinitialvalueproblemforthetimedependentSchrodingerequation( 1 )inwhichawavepacketi,initiallypositionedintheasymptoticregionr>r0(i.e.,attheinitialtimet=0thesupportofiliesintheasymptoticregion),propagatesintothescatteringregionr
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Iftheincidentwavepacketpassesthroughthescatteringregionfasterthanthedecaytimen=1=)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(nofaSiegertstate,thentheoutgoingwavecanbeobserved,anditresemblestoastationarystatewiththeenergyEn.Themorenarrowthescatteringresonanceis,thelongerlivesthecorrespondingSiegertstate.So,Siegertstateswith)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n1maybeinterpretedasquasi-stationarystatesofthesystemthatcanbeexcitedbyanincidentwaveandlivelongafterthescatteringprocessisover. 1.3TheExtensiontoElectromagnetismanditsChallenges OneofthegoalsofthisworkistoestablishinMaxwell'stheoryofelectromagnetismapicturesimilartothatofquantummechanicsinthespeciccaseofTMpolarizationforperiodicdielectricstructures. Amongotherdifferencesbetweenthetwotheories,thereisanimportantdistinctionthatarisesonthephysicalsignicancethatshouldbeattributedtotheSiegertstates.WhereaslonglivedquantummechanicalSiegertstatescorrespondtothetrappingofaparticleinaniteregionofspaceforaverylongtime,longlivedelectromagneticSiegertstateswillaccountforanaccumulationofelectromagneticenergyinniteregionsofspaceforaverylongtime.Thesubsequentenergybuildupenhancestheintensityoftheelectriceldinthesaidregions,thustheenhancementofthenonlineareffectsdescribedbyEq.( 1 ). AmajorhurdleinthedenitionofelectromagneticSiegertstatesisintheanalysisoftheeigenvalueproblemthatleadstothedenitionofSiegertstatesinTMpolarization.Followingthetheoryofquantummechanics,ifasolutionE(r,t)tothelinearwaveequation( 1 )isassumedtohaveaharmonictimedependence,i.e.,E(r,t)=E!(r)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!t,thenoneisledtotheequation, bH(k2)[E!]=k2E!,k2=!2 c2(1) wheretheoperatorbH(k2)=)]TJ /F3 11.955 Tf 9.3 0 Td[(+k2(")]TJ /F3 11.955 Tf 10.34 0 Td[(1)istheanalogofthehamiltonian.Notethatthespectralparameter=k2inthisinstancedoesnotcorrespondtotheelectromagnetic 17

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energy,asopposedtothequantummechanicalcasewherethespectralparameteristheenergyoftheparticle.Thespectralparameter=k2issimplythesquareofthemagnitudeofawavevectorkinthedirectionofpropagationoftheelectromagneticenergy. AnimportantobservationisthattheoperatorbH(k2)dependsonthespectralparameter=k2,sothattheproblemofEq.( 1 )isageneralizedeigenvalueproblem,asopposedtoastandardeigenvalueproblemsuchastheoneinEq.( 1 ).Ifthescatteringstructureisperiodicalongaparticulardirectione(e.g.,aperiodicallyperforatedlm),thedependenceonthespectralparameter=k2inthegeneralizedeigenvalueproblem( 1 )isfurthercomplicatedbytheboundaryconditionsimposedbythescatteringofanelectromagneticwaveonaperiodicstructure,i.e.,thedielectricfunction"isperiodic.TheperiodicityimposesontheamplitudeE!aconditionknownasBloch'speriodicitycondition.ItstatesthatifthepotentialV=k2(")]TJ /F3 11.955 Tf 12.48 0 Td[(1)hasperiode,then,E!(r+e)=eikeE!(r) forallrandforsomerealparameterke.TheincorporationofBloch'speriodicityconditionrequiresthatthegeneralizedeigenvalueproblem( 1 )fork2bestudiedintheform,bH0(k2,ke)[E!]=E! wherebH0(k2,ke)isacompactintegraloperatorwithakernelthatisperiodicalongthee-axis.Whileadetaileddiscussionofthisoperatoratthispointisinappropriateandunnecessary,itisworthmentioningthatitishighlynonlinearink2.Inparticular,approachessimilartothoseinEqs.( 1 )cannotbeusedtodeneelectromagneticSiegertstates.Instead,theproblemissolvedinChapter2usingthecomplexanalysisofcompactoperatorsonHilbertspaces. ApeculiaritythatisobservedoncetheSiegertstatesformalismhasbeenextendedtoelectromagnetismisthelackofanalyticityofsolutionstothenonlinearwaveequation 18

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inthevicinityofspectral,physicalandgeometricalparametersthatallowfortheformationofboundstatesintheradiationcontinuum.Tobeprecise,considerthenonlinearwaveequationobtainedbykeepingthersttwotermsinEq.( 1 ).Itis, 1 c2@2t)]TJ /F4 11.955 Tf 5.48 -9.68 Td[("E+2E2=E(1) Asalreadynoted,thesecondorderdielectricsusceptibility2isonlysupportedonthescatterers,meaningthatitiszerointhevacuum,andithasacertainvaluec,assumedconstant,onthescatterers. Sincecisverysmall,thetextbookapproachtosolvingEq.( 1 )istolookforapowerseriessolutionincas,E=E1+cE2+2cE3+... Itturnsoutthatforaperiodicsystemadmittingtheexistenceofboundstatesintheradiationcontinuumsuchaseriesdiverges,indicatingalackofanalyticityofthesolutionEincinthevicinityofaboundstateintheradiationcontinuum.Thislackofanalyticityisactuallymoregeneral.Thesolutionwilllackanalyticityinotherparameterssuchasthewavenumber,andthedistanceseparatingtwoarraysofperiodicstructuresforinstance.AnovelfullynonlinearperturbativeapproachthatreliesonthetheoryofSiegertstatesisdevelopedtohandlethiskindofsingularity.Itisthroughthistheorythatthecontrolofopticalnonlineareffectssuchassecondharmonicgenerationisachieved. 1.4MainResults TheformalismofSiegertstateshasbeenextendedtoperiodicelectromagneticscatteringstructures.Resultsanalogoustothoseofquantummechanicalresonantscatteringtheoryhavebeenestablishedforelectromagneticresonantscatteringtheory.Inparticular,theRegularPerturbationTheoremforelectromagneticSiegertstateshasbeenformulatedandproved.ThisworkappearsinChapter2ofthisDissertation. 19

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BymeansoftheRegularPerturbationTheoremithasbeenprovedthatifphysicalpropertiesofascatteringperiodicstructuredependonaphysicalparameter(e.g.adistancebetweentwoparallelperiodicarrays)sothatthereisaboundstateintheradiationcontinuumataparticularvalueofthisparameter,thentheamplitudeoftheeldcanlocallybeampliedasmuchasdesiredincomparisontotheamplitudeoftheincidentradiationbyadjustingthevalueofthephysicalparameter.Thissuggestsanoveluniversalprincipletoenhanceandcontrolopticallynonlineareffectsinnanophotonicperiodicstructures.ThedevelopedmethodsofelectromagneticSiegertstatesprovideanovelrigorousmathematicalformalismfortheoreticalstudiesofsucheffects.ThisresultappearsinSection 2.2.2 ofthisDissertation. Forperiodicdoublearraysofsubwavelengthdielectriccylinders,ithasbeenshownthatelectromagneticboundstatesintheradiationcontinuumexistinthespectralregionwheremorethanonediffractionchannelsareopen.AnexplicitformofthesesolutionsofMaxwell'sequationshasbeenfound.Qualitatively,thesolutionsdescribewaveguidingmodes,eitherstandingorpropagatingalongthestructure,whicharelocalizedinavicinityofthestructureandwhosewavelengthissmallerthanthestructureperiod(themorediffractionchannelsareopenthesmallerthewavelength).Withincreasingthenumberofopendiffractionchannels,wavenumbersofthesemodesbecomemoresparse.Thekeydifferencewithcommonlyknownwaveguidingmodesthatoccurinmetalcavitiesandphotoniccrystalsisthatthewavenumbersofthestudiedmodeslieintherangeofthoseofradiation(orscattered)modes.TheseresultsarepresentedinChapter3ofthisDissertation. Anovelscatteringtheoryformalismhasbeendevelopedtostudyopticallynonlineareffectsduetothepresenceofboundstatesintheradiationcontinuumwhichresolvestheaforementionedproblemofnon-analyticityofsolutionsofthenonlinearelectromagneticscatteringproblem.ThistheoryisexposedinSection 4.1 ofthepresentDissertation. 20

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Theconceptofenhancingopticallynonlineareffectsinperiodicstructuressupportingboundstatesinradiationcontinuumhasbeenappliedtostudyaresonantgenerationofthesecondharmonicsbyaperiodicdoublearrayofparalleldielectriccylinderswithanonlineardielectricsusceptibility.Ithasbeenshownthatsuchasystemiscapableofaconversionrateofthefundamentalharmonicintothesecondoneashighas44%,i.e.,comparabletoratesachievedbyconventionaldevices.Butastrikingcontrasttoconventionaldevices,suchas,e.g.,opticallynonlinearcrystals,isthat:rst,theconversiondoesnotrequireanyphasematching;second,themaximalconversionratedependsweaklyonthenonlinearsusceptibility(i.e.,thesame44%canbeobtainedforawiderangeofvaluesofthenonlinearsusceptibility);and,third,thedevicewidthatwhichsuchahighconversionrateisachievedcanbeassmallasahalfoftheincidentradiationwavelength.Forexample,inatypicalnonlinearcrystal,asimilarconversionraterequiresacrystalwidthof1-4cm.Foraninfraredlasergeneratingradiationwiththewavelength900nm,thedistancebetweenthetwoarraysofcylindersatwhichtheconversionrateof44%canbeachievedisabout450nm=4.510)]TJ /F6 7.97 Tf 6.59 0 Td[(5cm,i.e.,thesystemwidthcanbereducedbyafewthousandtimes.Thesepropertiesallowforanovelprincipleofminiaturizationofnonlinearphotonicdevicesforfrequencyconversion.AUSpatentapplicationispendingforthisresult(ApplicationSerialNo.:61/488,971).ThedetailsofthisresultaretheobjectofChapter4ofthisDissertation. 21

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CHAPTER2ELECTROMAGNETICSIEGERTSTATESINPERIODICSTRUCTURES Inthischapter,thetheoryofSiegertstatesisextendedtoMaxwell'stheoryofelectromagnetisminTMpolarization.Inquantummechanics,theSiegertstatesareoutgoingsolutionstotheSchrodingerequation,andtheycorrespondtothepositionsofscatteringresonances.Inthetheoryofelectromagnetismforperiodicdielectricstructures,itisfoundthatSiegertstatesaregeneralizedeigenfunctionstoageneralizedeigenvalueprobleminwhichtheanalogoftheHamiltonianofthesystemdependsontheeigenvaluesoftheSiegertstates. 2.1ElectromagneticSiegertStates Considerascatteringproblemforanelectromagneticplanewaveimpingingastructuremadeofanon-dispersivedielectric.Thestructureisdescribedbyadielectricfunction"(r)thatdenesthevalueofthedielectricconstantateverypointroccupiedbythestructureanditequalsthedielectricconstantofthesurroundingmediumotherwise.Withoutlossofgenerality,thesurroundingmediumisassumedtobethevacuum,i.e.,"=1.Thematerialissaidtobenon-dispersiveifitsdielectricconstantdoesnotdependonthefrequencyoftheincidentwave.Thestructureisassumedtohaveatranslationalsymmetryalongaparticulardirection.Inthiscasethedielectricfunctionisindependentofoneofthespatialcoordinates,say,theycoordinate.Iftheincidentwaveispolarizedalongthey)]TJ /F1 11.955 Tf 9.3 0 Td[(axis(theelectriceldisparalleltothey)]TJ /F1 11.955 Tf 9.3 0 Td[(axis),thenthescatteringproblemcanbeformulatedasthescalarMaxwell'sequation: c2@2tE=E,=@2x+@2z(2) wherecisthespeedoflightinthesurroundingmedium(thevacuum),andEistheelectriceld.Foraplanarstructure,thefunction"differsfrom1onlywithinastrip(x,z)2(,1)()]TJ /F5 11.955 Tf 9.3 0 Td[(a,a).Aplanarstructureisperiodicif"(x+Dg,z)="(x,z)whereDgistheperiod.Inwhatfollows,theunitsoflengtharechosensothatDg=1.Itis 22

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furtherassumedthatthestructure'sdielectricconstantexceedsthatofthesurroundingmedium,i.e.,"(r)1forallr,and"isbounded.TwoexamplesofsuchstructuresareshowninFig. 2-1 (Panels(a)and(b))depictingperiodicarraysofinnitedielectriccylinders,allparalleltothey)]TJ /F1 11.955 Tf 9.3 0 Td[(axis.Thefunction"(r)ispiecewisecontinuous.Forgeneralperiodicstructures,"isinvariantundertheactionofasuitableafneCoxetergroupinthexy)]TJ /F1 11.955 Tf 9.3 0 Td[(plane.Inthiscase,vectorMaxwell'sequationsshouldbeusedasthepolarizationoftheincidentwaveisnotpreservedinthescatteringprocess.Thisgeneralproblemwillnotbestudiedhere. Iftheincidentwavehasaxedfrequency,thentheeldEhasaharmonictimedependenceE(r,t)=E!(r)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!tinEq.( 2 )wheretheamplitudeE!satisestheequation: E!(r)+k2"(r)E!(r)=0,k2=!2 c2(2) Letthex,y,andzaxesbeorientedbytheunitvectorse1,e2,ande3,respectively.Inthescatteringtheory,E!issoughtasasuperpositionofanincidentwave,andthecorrespondingscatteredwaveEs!: E!(r)=eikr+Es!(r),k=kxe1+kze3(2) wherekisthewavevectoroftheincidentwave.TheunitsoftheeldEarechosensothattheamplitudeoftheincidentwaveis1.Thescatteredwaveobeystheoutgoingwaveboundaryconditionsatthespatialinnity.TheperiodicityofthescatteringstructurerequiresthattheamplitudeE!satisesBloch'speriodicitycondition, E!(x+1,z)=eikxE!(x,z)(2) Underthespeciedboundaryconditions,theamplitudeE!satisestheLippmann-Scwingerintegralequation: E!(r)=bH[E!](r)+eikr(2) 23

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wherebHistheintegraloperatordenedby,bH[E](r)=k2 4ZR2("(r0))]TJ /F3 11.955 Tf 11.96 0 Td[(1)Gk(rjr0)E(r0)dr0 ThekernelGk(rjr0)istheGreen'sfunctionfortheHelmholtzequation,(+k2)Gk(rjr0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(4(r)]TJ /F12 11.955 Tf 12.22 0 Td[(r0),withtheoutgoingboundarycondition.ItisgivenbyGk(rjr0)=iH0(kjr)]TJ /F12 11.955 Tf -449.5 -23.91 Td[(r0j)whereH0istheHankelfunctionoftherstkindoforder0.ForaproofoftheLippmann-SchwingerintegralequationrefertoSection A.1 Figure2-1. Panel(a):Anexampleofscatteringstructuresconsideredinthiswork.Innitelylongparallelcylindersareplacedparalleltoay-axisperiodicallyalonganx-axisinthevacuum"0=1.Thecylindersarecharacterizedbyadielectricconstant"c>1.TherectangleDinEq.( 2 )isenclosedbythedashedline,andthez-axis.Panel(b):TwoperiodicarrayssuchastheoneonPanel(a)areplacedparalleltoeachotheratadistance2hbetweentheaxesofanytwoopposingcylinders.Panel(c):TheCkxplane.Thecutsrunverticallyfromthediffractionthresholds(indicatedbyemptycirclesontherealaxis)intothelowerhalfofthecomplexplane.ThediffractionthresholdsdividetherealaxisintoacountablesetofintervalsdenotedIl,l0.TheintervalI0liesbelowtheradiationcontinuum.Therestoftheintervalspartitiontheradiationcontinuum. TheexistenceofsolutionstotheLippmann-SchwingerintegralequationthatalsosatisfyBloch'sconditionisestablishedbyextendingtheoperatorvaluedfunctionk27!bH(k2)toasuitablycutcomplexplane,andthereafter,applyingtheFredholmtheoryof 24

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compactoperators.Tothisend,letthex)]TJ /F1 11.955 Tf 9.3 0 Td[(componentkxoftheincidentwavevectorkbexed.Themagnitudek=jkjvariesonlyinitsz)]TJ /F1 11.955 Tf 9.3 0 Td[(component,kz=p k2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x.NowletS"bethesupportofthefunction(x,z)!("(x,z))]TJ /F3 11.955 Tf 12.14 0 Td[(1)inthestripS=[0,1](,1)ofthex,z-plane,andletDbearectangleinScontainingS",i.e., S"D=[0,1][z)]TJ /F3 11.955 Tf 7.09 1.79 Td[(,z+](2) ThenforanamplitudeE!satisfyingBloch'scondition, bH[E!](r)=k2 4Xm2ZeimkxZD("(r0))]TJ /F3 11.955 Tf 11.95 0 Td[(1)Gk(rjr0)]TJ /F5 11.955 Tf 11.95 0 Td[(me1)E!(r0)dr0(2) whereZdenotesthesetofallintegers.ThisequationdenesbHasanoperatoronL2(D),theHilbertspaceofsquareintegrablefunctionsonDwithrespecttotheLebesguemeasure.NotethatbHdoesnotdependontherectangleDwhichtosomeextentisarbitrary.Indeed,theintegralsofEq.( 2 )extendonlyoverthesupportS"ofthefunction(x,z)7!("(x,z))]TJ /F3 11.955 Tf 11.97 0 Td[(1)inthestripS=[0,1](,1)ofthex,z-plane.TherectangleDisonlyintroducedtoobtainaconnectedandcompactregionofintegration,whichisconvenientforthesubsequentanalysis.Ingeneral,thesupportofthefunction(x,z)7!("(x,z))]TJ /F3 11.955 Tf 10.25 0 Td[(1)isnotnecessarilyconnectedas,forexample,inthecaseofmultiplescatterersinthestripSdepictedinFig. 2-1 (b).ThesolutiontotheLippmann-Schwingerintegralequation( 2 )willbesoughtintheHilbertspaceL2(D).Suchasolutionthenextendsnaturallytothefullx,z)]TJ /F1 11.955 Tf 9.3 0 Td[(planebyBloch'speriodicitycondition( 2 ),andtheLippmann-Schwingerintegralequation. ItisnothardtoseethatthesummationandintegrationcanbeinterchangedinEq.( 2 )eventhoughtheunderlyingseriesisonlyconditionallyconvergent.ThePoissonsummationformulaisthenappliedtoyieldthefollowingformfortheintegraloperatorbH, bH[E](r)=ZD("(r0))]TJ /F3 11.955 Tf 11.95 0 Td[(1)H(k2;r)]TJ /F12 11.955 Tf 11.96 0 Td[(r0)E(r0)dr0,E2L2(D)(2a) 25

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wherethefunctionHisgivenby, H(;x,z)=i 2Xm2Zei(xkx,m+jzjp )]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,m) p )]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,m,kx,m=kx+2m,6=k2x,m,m2Z(2b) Thesquarerootsaredenedbychoosingthebranchcutofthelogarithmalongthenegativeimaginaryaxis,i.e.,ifwisanonzerocomplexnumber,then logw=lnjwj+iargw,)]TJ /F4 11.955 Tf 10.5 8.09 Td[( 20forallbutanitenumberofdiffractionthresholdsk2x,msuchthat=k2>k2x,m.ItfollowsthatintheseriesofEq.( 2b )allbutanitenumberoftermsdecayexponentiallyasjzj!1.HencethesaidseriesconvergesuniformlyonD.TheuniformconvergenceonDstillholdswhentherangeofthevariable=k2isextendedtothecutcomplexplaneCkxobtainedbyexcludingalltheverticalhalflinesrunningfromthediffractionthresholdsintothelowerhalfofthecomplexplane,i.e.,Ckx=C)]TJ /F13 11.955 Tf 14.64 11.36 Td[([m2Zfk2x,m)]TJ /F5 11.955 Tf 11.96 0 Td[(is,s0g Thisisbecauseinthiscasetoo,Imp )]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,m>0exceptforanitenumberofterms.Itfollowsthat7!H(;r)extendstoananalyticfunctiononCkx.Figure 2-1 (c)showsasketchofthecutplaneCkx. Sinceforall2Ckx,thekernel(r,r0)7!("(r0))]TJ /F3 11.955 Tf 10.69 0 Td[(1)H(;r)]TJ /F12 11.955 Tf 10.69 0 Td[(r0)issquareintegrableonDD,theintegraloperatorbH()isHilbert-SchmidtinLfL2(D)g,thespaceofboundedlinearoperatorsonL2(D).Inparticular,bH()iscompactforeach2Ckx.Moreover,theoperatorvaluedfunctiondenedonCkxby7!bH()isanalytic.BytheanalyticFredholmtheorem[ 7 ],itthenfollowsthatiftheinverseoperator[1)]TJ /F13 11.955 Tf 13.42 3.15 Td[(bH()])]TJ /F6 7.97 Tf 6.59 0 Td[(1existsatsomepoint2Ckx,itmustbemeromorphicthroughoutCkx.Thatthesaidinverse 26

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existsatsomepointisobvious.Indeed,as!0inCkx,thenormoftheoperatorbH(),i.e.,jjbH()jj=jj("())]TJ /F3 11.955 Tf 12.82 0 Td[(1)H(;)jjL2(DD),alsoconvergestozero.Inparticularfornear0,jjbH()jj<1andtherefore[1)]TJ /F13 11.955 Tf 13.35 3.15 Td[(bH()])]TJ /F6 7.97 Tf 6.58 0 Td[(1existsasaNeumannseries.Thus,thegeneralizedresolvent7![1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH()])]TJ /F6 7.97 Tf 6.59 0 Td[(1ismeromorphicinCkx. Thepolesfngnof[1)]TJ /F13 11.955 Tf 13.69 3.16 Td[(bH()])]TJ /F6 7.97 Tf 6.58 0 Td[(1areisolatedandformadiscreteset.ThesameanalyticFredholmtheoremguaranteesthatateachofthepolesn,thereexistsanonzerosolutiontothegeneralizedeigenvalueproblembH(n)[En]=En ThegeneralizedeigenfunctionsEnwillbereferredtoastheSiegertstates. ItwillbeassumedthroughouttherestoftheworkthattheSiegertstatesarenondegenerate,i.e.,thepolesnareallsimple.Theresultstobederivedintherestoftheworkcouldeasilybegeneralizedtothecasewhenthepolesarenotsimple.Yet,toourknowledge,therehasbeennoreportofresonancepolesofmultiplicitygreaterthanoneineithertheliteratureofelectromagnetismorthatofquantummechanics.However,wedonothavearigorousproofthathighermultiplicitypolescannotoccurinthestudiedscatteringproblem.ForthestructuresdepictedinFig. 2-1 ,perturbationtheory(whentheradiusofcylindersismuchlessthanthestructuresperiod)showsthatallthepolesaresimple(Chapter3). TheassumednondegeneracyoftheSiegertstatesfEngnimpliesthattheresiduesfbHngnof[1)]TJ /F13 11.955 Tf 12.87 3.15 Td[(bH()])]TJ /F6 7.97 Tf 6.58 0 Td[(1atthepolesfngnarerankoneoperatorsonL2(D).Inotherwords,ifhf,gi=RD f(r0)g(r0)dr0istheinnerproductontheHilbertspaceL2(D),then 8n,9'n2L2(D):8 2L2(D),bHn[ ]=h'n, iEn(2) SincetheincidentwaveEi(k2;r)=eikrisanalyticin=k2onCkx,itfollowsthattheamplitudeE!=[1)]TJ /F13 11.955 Tf 13.17 3.16 Td[(bH()])]TJ /F6 7.97 Tf 6.59 0 Td[(1[Ei(;)]extendstoameromorphicfunctionofthevariable 27

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=k2onCkx.Itspartialfractionexpansionreads E!(r)=eikr+Xnan k2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2n+i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nEn(r)+Ea(k2;r),an=h'n,Ei(k2;)ik2=k2n)]TJ /F8 7.97 Tf 6.59 0 Td[(i)]TJ /F9 5.978 Tf 4.82 -.99 Td[(n(2) whereeachpolenhasbeendecomposedinitsrealandimaginarypartsasn=k2n)]TJ /F5 11.955 Tf 12.54 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nowingtothefactthattheimaginarypartsofthepolesarenonpositiveaswillbeshownshortly.TheremainderEaisanalyticink2onthecutplaneCkx.Itdescribesthesocalledbackgroundorpotentialscattering,whiletheSiegertstatesaccountfortheresonantscattering. SincetheSiegertstatesaresolutionstothehomogeneouslinearequationbH()[E]=E,theyshouldbenormalizedinsomewayinorderforthefunctions'ninEq.( 2 )andthecoefcientsaninEq.( 2 )tobeuniquelydened.AnaturalnormalizationconditionwillbeintroducedinSection 2.2.1 forSiegertstatesthatariseasperturbationsofboundstatesintheradiationcontinuum. Thatallthepolesofthegeneralizedresolvent7![1)]TJ /F13 11.955 Tf 13.35 3.16 Td[(bH()])]TJ /F6 7.97 Tf 6.59 0 Td[(1havenonpositiveimaginaryparts,i.e.,)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n0,8n,followsfromaseriesofobservations.First,notethataSiegertstateEncorrespondingtothepolen=k2n)]TJ /F5 11.955 Tf 12.33 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(nsatisesEq.( 2 )fork2=n,andtherefore EnEn)]TJ /F5 11.955 Tf 11.96 0 Td[(En En)]TJ /F3 11.955 Tf 11.96 0 Td[(2i)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n"jEnj2=0 ByGreen'stheorem,itfollowsthat, 2i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nZD0jEn(r0)j2"(r0)dr0=I@D0( En(r0)rEn(r0))]TJ /F5 11.955 Tf 11.95 0 Td[(En(r0)r En(r0))dn0(2) whereD0=[0,1][z1,z2]isanyrectanglecontainingtherectangleD,andn0istheoutwardnormal.ByBloch'speriodicitycondition,theintegralsonthelinesegments(x,z)2f0,1g[z1,z2]of@D0arecanceledoutsothattheintegralontherightsideistobecarriedoutonlyoverthelinesegments(x,z)2[0,1]fz1,z2gof@D0.TheresultcanbeexpressedintermsofthescatteringamplitudesSmassociatedwiththeasymptotic 28

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behavioroftheSiegertstateEn: En(r)=8>>>><>>>>:Xm2ZS)]TJ /F8 7.97 Tf -1.07 -7.9 Td[(mei(xkx,m)]TJ /F8 7.97 Tf 6.58 0 Td[(zp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,m),zz+(2a) Equations( 2a )followimmediatelyfromequations( 2 ).Inparticular,theamplitudesSmaregivenbytheformulas, Sm=in 2p n)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x,mZD("(r0))]TJ /F3 11.955 Tf 11.95 0 Td[(1)En(r0)ei()]TJ /F8 7.97 Tf 6.58 0 Td[(x0kx,mz0p n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,m)dr0,r0=x0e1+z0e3(2b) EvaluatingtherighthandsideofEq.( 2 )yields, )]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n=Pm2ZRep n)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,mjS+mj2e)]TJ /F6 7.97 Tf 6.59 0 Td[(2z2Imnp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,mo+jS)]TJ /F8 7.97 Tf -1.07 -7.3 Td[(mj2e2z1Imnp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,mo RD0jEn(r0)j2"(r0)dr0(2) TheseriesinEq.( 2 )isthensplitintothesumsoverthetwocomplementaryindexsets, Iop(n)=fm2ZjRefngk2x,mg,Icl(n)=fm2ZjRefng0,8m2Icl(n)(2b) Inparticular,intheseriesofEq.( 2 ),thecontributionsfromtermswhoseindiceslieinIcl(n)decayexponentiallyasz2!+1andz1!.Inthesaidlimits,theaforementionedseriesisthereforereducedtoanitesumoverthesetIop(n):)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n=limz1!z2!+1Pm2Iop(n)Rep n)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,mjS+mj2e)]TJ /F6 7.97 Tf 6.58 0 Td[(2z2Imnp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,mo+jS)]TJ /F8 7.97 Tf -1.07 -7.29 Td[(mj2e2z1Imnp n)]TJ /F8 7.97 Tf 6.58 0 Td[(k2x,mo RD0jEn(r0)j2"(r0)dr0 29

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ObservingthatRep n)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,m0forallm2Iop(n),andsincethedielectricfunction"ispositive,itfollowsthat)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n0,andthereforetheimaginarypartofthepolenisindeednecessarilynegativeorzero,i.e.,allthepolesnareinthelowerhalfofthecutcomplexplaneCkx. 2.2RegularPerturbationTheory Supposethatthedielectricfunction"dependsonarealparameterhwhichiscalledacouplingparameter.Thesimplestexampleofsuchacouplingisgivenbytwoperiodicarraysofdielectricscatterersthatareparallelandseparatedbythedistance2h(Fig. 2-1 (b)).Thearraysareembeddedinamediumofdielectricsusceptibility1.Eacharrayischaracterizedbyadielectricfunction"i(x,z),i=1,2suchthat"i1onthescatterers,and"iisofperiod1inthex-direction.Theresultingdielectricfunctiondescribingthescatteringoflightonthestructureisthen, "(h;x,z)=1+("1(x,z)]TJ /F5 11.955 Tf 11.95 0 Td[(h))]TJ /F3 11.955 Tf 11.96 0 Td[(1)+("2(x,z+h))]TJ /F3 11.955 Tf 11.96 0 Td[(1)(2) Inthemostgeneralcase,thedependenceofthedielectricfunction"onthecouplingparameterhimpliesthatthepolesofthegeneralizedresolvent7![1)]TJ /F13 11.955 Tf -422.66 -20.75 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.58 0 Td[(1alsodependonh.Ifthecouplingissufcientlysmoothasindicatedinthetheoremstatedbelow,thenthepolesofthegeneralizedresolventaswellasthecorrespondingSiegertstatesdependcontinuouslyonh. Beforestatingthetheorem,aclaricationmustbemadeontheboundaryoftherectangleDinEq.( 2 ).ThisrectanglewaschosentocontainthesupportS"ofthefunction(x,z)7!("(x,z))]TJ /F3 11.955 Tf 12.87 0 Td[(1)inthestripS=[0,1](,1)ofthex,z-plane.Inthecurrentsituation,thedielectricfunctiondependsonthecouplingparameterh,andthereforethesupportS"(h)inquestioncouldchangewithhresultinginadifferentchoiceforthesetD.Thisisthecase,forinstance,intheexampleofthetwoparallelarraysseparatedbythedistance2h.IfthetwoscatterersinthestripSaretakenfurtherapart,therectangleDisstretchedfurtheraccordingly.So,inwhatfollowsitwillalways 30

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beassumedthathvariesinanopenpossiblyniteintervalJ0suchthatSh2J0S"(h)isbounded,andtherectangleDwillbechosentocontainthelatterset.TheTheoremontheregularperturbationofelectromagneticSiegertstatesisformulatedasfollows: Regularperturbationtheorem. Supposethatthemap(h,)7!bH(h,)fromJ0CkxtoLfL2(D)giscontinuouslyFrechetdifferentiable.Further,supposethatforsomehn2J0,thegeneralizedresolvent7![1)]TJ /F13 11.955 Tf 13.06 3.15 Td[(bH(hn,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1hasasimplepolen2Ckx,andletEnbethecorrespondingSiegertstate.ThenthereexistsanopenintervalJJ0containinghn,andauniquecontinuouslydifferentiablefunctionh7!n(h),h2J,suchthatn(hn)=n,andforallh2J,n(h)isasimplepoleofthegeneralizedresolvent![1)]TJ /F13 11.955 Tf 13.07 3.16 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1.IfEn(h)istheSiegertstatecorrespondingtothepolen(h),thenthefunctionh7!En(h)fromJtoL2(D)iscontinuouslydifferentiable. ThistheoremisanextensionoftheKato-Rellichtheorem[ 5 ]tothepresentcase.Duetoitslengthandcomplexity,theproofislefttotheAppendix.Itshouldbenoted,however,thatifthefunctions(x,z)7!"i(x,z),i=1,2inEq.( 2 )arepiecewisedifferentiable,then(h,)7!bH(h,)isFrechetdifferentiablesothatthetheoremdoesindeedholdfortwoparallelarraysseparatedbythedistanceh. 2.2.1PerturbationofBoundStatesintheRadiationContinuum Thediffractionthresholdsk2x,monthereallinedependonthex)]TJ /F1 11.955 Tf 9.3 0 Td[(componentkxoftheincidentwavevector.Theycanbeorderedindependentlyoftheparameterkxbydeningasequencem(kx)=(2mj[kx]j)2,m2f0,1,2,...g,where[kx]istheargumentofeikxintheinterval()]TJ /F4 11.955 Tf 9.3 0 Td[(,].Thesequencefm(kx)g1m=0coincideswiththesequenceofdiffractionthresholdsf(kx+2m)2gm2Z,andforallkx,0(kx))]TJ /F6 7.97 Tf 6.59 0 Td[(1(kx)1(kx))]TJ /F6 7.97 Tf 6.59 0 Td[(2(kx)2(kx))]TJ /F6 7.97 Tf 6.59 0 Td[(3(kx)... Thethreshold0(kx)iscalledtheradiationcontinuumthreshold,andtheintervalI0=(,0(kx))issaidtoliebelowtheradiationcontinuum.Thisisbecausewheneverk2<0(kx),thenthescatteredamplitudeEs!inEq.( 2 )necessarilydecays 31

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exponentiallyinthespatialinnity,jzj!1,ascanbeinferredfromEqs.( 2 ),( 2 ),and( 2 ).Hence,noelectromagneticuxiscarriedtothespatialinnityinthisspectralrange.Incontrast,ifk2>0(kx),theamplitudeEs!oscillatesandrepresentsoutgoing(scattered)radiationthatcarriesanelectromagneticuxtothespatialinnity.Thespectralrangeabove0(kx)onareallineisthereforereferredtoastheradiationcontin-uum.Itisthedisjointunionoftheintervals,I1=(0(kx),)]TJ /F6 7.97 Tf 6.58 0 Td[(1(kx)),I2=()]TJ /F6 7.97 Tf 6.59 0 Td[(1(kx),1(kx)),I3=(1(kx),)]TJ /F6 7.97 Tf 6.59 0 Td[(2(kx)),... Notethatwhen[kx]is0or,someoftheintervalsIlareempty,owingtothefactthatsomeofthediffractionthresholdsfuse. ReturningtoSiegertstates,recallthatthesestatesaregeneralizedeigenfunctionstothegeneralizedeigenvalueproblembH(n)[En]=EnonL2(D)withcomplexeigenvaluesn.Thesestatesarenaturallyextendedtothewholex,z)]TJ /F1 11.955 Tf 9.3 0 Td[(planebymeansofEqs.( 2 ).Ingeneral,ifthegeneralizedeigenvaluen=k2n)]TJ /F5 11.955 Tf 11.97 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nliesbeloworintheintervalI0intheCkx-plane,thenthecorrespondingSiegertstateissquareintegrableonthestripS=[0,1](,1)ofthex,z-plane.Thisisbecauseforsuchstates,thesetIop(n)ofEq.( 2a )isempty,andthereforebyEqs.( 2a )and( 2b ),thesestatesdecayexponentiallyintheasymptoticregionjzj!1.Inparticular,theSiegertstatesforwhichthepolebisrealandlessthanthecontinuumthreshold0(kx)aretheboundstatesbelowtheradiationcontinuumofthesystem. Onthecontrary,theSiegertstatesEnwhosecorrespondinggeneralizedeigenvaluesn=k2n)]TJ /F5 11.955 Tf 12.38 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(nliebelowanintervalIl,l1oftheradiationcontinuum,i.e.,k2n2Iland)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n>0,arenotnecessarilysquareintegrableonthestripS.Thisisbecauseforthesestates,thesetIop(n)isnotempty,andconsequently,thetermsoftheseriesinEq.( 2a )indexedbythissetareunbounded.However,whenthepolenisinIl,l1,i.e.,)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n=0,itwillbeshownshortlythattheresultingSiegertstateissquareintegrable 32

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onthestripSandtherefore,suchaSiegertstateisaboundstateintheradiationcontinuum.Therestofthissubsectionisdevotedtotheproofofthisassertion. Toproceed,supposethatforsomevaluehbofthecouplingconstanth,thereexistsaSiegertstateEbwhosecorrespondinggeneralizedeigenvaluebisrealandliesinanintervalIl,l1,oftheradiationcontinuum.BytheRegularPerturbationTheoremthereexistcontinuouslydifferentiablefunctionsh7!n(h)andh7!En(h)onanintervalJcontaininghbsuchthatn(hb)=b,andEn(hb)=Eb.Furthermore,En(h)istheSiegertstatecorrespondingtothepolen(h)ofthegeneralizedresolvent7![1)]TJ /F13 11.955 Tf 13.36 3.15 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.58 0 Td[(1forallh2J.Putn(h)=k2n(h))]TJ /F5 11.955 Tf 12.54 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n(h)asbefore.Then)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(hb)=0.Withoutlossofgenerality,theintervalJisassumedtobesufcientlysmallsothatforallh2Jnfhbg,)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n(h)6=0.Equation( 2 )fortheSiegertstateEn(h)isthenrewrittenasZD0jEn(r0)j2"(h;r0)dr0=1 )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nXm2ZRenq n)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x,mo jS+mj2e)]TJ /F6 7.97 Tf 6.58 0 Td[(2z2Imnp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,mo+jS)]TJ /F8 7.97 Tf -1.07 -7.89 Td[(mj2e2z1Imnp n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2x,mo! whereitisunderstoodthattheSiegertstateEn,thepolen,anditsimaginarypart)]TJ /F5 11.955 Tf 9.3 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n,aswellastheamplitudesSmareallfunctionsofh.Notethatthevaluesofz1andz2areassumedtobeindependentofthecouplingparameterh.ThisisbecausehvariesinasmallintervalJ,andthereforethevaluesofzinEq.( 2 ),whichnowdependonharebounded.Theconditionz1z)]TJ /F3 11.955 Tf 7.09 1.8 Td[((h)
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m(h)=k2n(h))]TJ /F5 11.955 Tf 11.96 0 Td[(k2x,m,m2Z.Thenq n(h))]TJ /F5 11.955 Tf 11.96 0 Td[(k2x,m=8>>>><>>>>:q p m(h)2+)]TJ /F9 5.978 Tf 11.41 -1 Td[(n(h)2+m(h) 2)]TJ /F8 7.97 Tf 59.67 5.48 Td[(i)]TJ /F9 5.978 Tf 4.82 -1 Td[(n(h) r 2p m(h)2+)]TJ /F9 5.978 Tf 11.41 -.99 Td[(n(h)2+m(h),ifm2Iop(n(h)))]TJ /F6 7.97 Tf 58.35 5.48 Td[()]TJ /F9 5.978 Tf 4.82 -1 Td[(n(h) r 2p m(h)2+)]TJ /F9 5.978 Tf 11.41 -.99 Td[(n(h)2)]TJ /F17 7.97 Tf 6.59 0 Td[(m(h)+iq p m(h)2+)]TJ /F9 5.978 Tf 11.41 -.99 Td[(n(h)2)]TJ /F17 7.97 Tf 6.59 0 Td[(m(h) 2,ifm2Icl(n(h)) whereIopandIclaretheindexsetsofEq.( 2a ).Inparticular,ifm2Iop(n(h)),thenm(h)0,whereasm(h)<0wheneverm2Icl(n(h)).Ash!hb,)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(h)!0,and, ZD0jEb(r0)j2"(hb;r0)dr0=limh!hb241 )]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n(h)Xm2Iop(b)p m(h))]TJ /F2 11.955 Tf 5.47 -9.68 Td[(jS+m(h)j2+jS)]TJ /F8 7.97 Tf -1.07 -7.89 Td[(mj235)]TJ /F13 11.955 Tf 20.13 11.36 Td[(Xm2Icl(b)1 2p )]TJ /F4 11.955 Tf 9.3 0 Td[(m(hb) jS+m(hb)j2e)]TJ /F6 7.97 Tf 6.59 0 Td[(2z2p )]TJ /F17 7.97 Tf 6.59 0 Td[(m(hb)+jS)]TJ /F8 7.97 Tf -1.07 -7.89 Td[(m(hb)j2e2z1p )]TJ /F17 7.97 Tf 6.59 0 Td[(m(hb)!(2) Inparticular,foreachm2Iop(b),thelimitslimh!hbjSm(h)j2 )]TJ /F9 5.978 Tf 4.83 -.99 Td[(n(h)mustexistandmustbenite.Bycontinuityofthefunctionsh7!Sm(h)onJ,thereexistcomplexnumbersSm,bsuchthat limh!hbSm(h) p )]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n(h)=Sm,b,8m2Iop(b)(2) Asz1!andz2!1,thesecondseriesinEq.( 2 )convergestozero.Thereforethefunctionr!Eb(r)p "(hb;r)issquareintegrableonthestripS=[0,1](,1)ofthex,z)]TJ /F1 11.955 Tf 9.3 0 Td[(plane,and ZSjEb(r0)j2"(hb;r0)dr0=Xm2Iop(b)p m(hb))]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jS+m,bj2+jS)]TJ /F8 7.97 Tf -1.07 -8.28 Td[(m,bj2(2a) Since,"(hb,)1,itfollowsthatEb2L2(S)asclaimed.Thus,ifEbisaboundstateintheradiationcontinuum,itcanalwaysbenormalizedbythecondition ZSjEb(r0)j2"(hb;r0)dr0=1(2b) 34

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ThisresultjustiesthetermboundstateintroducedbyanalogywithquantummechanicswhereboundstatesrepresentsquareintegrableeigenfunctionsoftheHamiltonianoperator,i.e.,anelectromagneticboundstateisalocalizedsolutionofMaxwell'sequations. Finally,bycontinuityEn(h)!Ebash!hb,andthereforethenormalizationcondition( 2b )determinesuniquelytheSiegertstatesEn(h)alongthecurveh7!n(h),h2J. 2.2.2NearFieldAmplicationMechanism ResonantphenomenainthescatteredelectromagneticuxcanbedescribedbytheformalismofSiegertstatesasgiveninEq.( 2 ).However,acompletedescriptionrequirescalculatingtheresiduesaninEq.( 2 ).HeretheseresiduesarecalculatedforSiegertstatesthatariseasperturbationsofboundstatesintheradiationcontinuuminthesenseofSection 2.2.1 ,i.e.,forthestatesEn(h)wherejh)]TJ /F5 11.955 Tf 12.11 0 Td[(hbj=hb1.Inparticular,itisshownthatifthescatteringstructurehasboundstatesintheradiationcontinuum,thenthereexistregions(thesocalledhotspots)inwhichtheeldamplitudecanbeampliedasmuchasdesiredbytakingthevalueofthecouplingparameterhcloseenoughtohb.TheunboundedlocalgrowthoftheeldamplitudeisessentiallyduetothelinearityofMaxwell'sequations.Itdisappearswhenanon-lineardielectricsusceptibility,requiredforlargeeldamplitudes,isincludedintoMaxwell'sequations(Chapter4).Nevertheless,suchalocaleldamplicationenhancesquitesubstantiallyopticalnonlineareffectsinthescatteringstructureasshowninChapter4.Thefollowinganalysisalsosuggeststhattheconcepttoenhanceopticallynon-lineareffectsbyusingboundstatesintheradiationcontinuumisuniversalbecausetheexistenceofhotspotsisprovedtobeacharacteristicfeatureofsuchscatteringstructures. Whenestimatingthecoefcientsan,itisconvenientrsttogiveanotherequationforthemthatisalternativetothatofEq.( 2 ).Next,thisequationwillbeanalyzedinthevicinityofaboundstateintheradiationcontinuum.Inparticular,ifaboundstateinthe 35

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radiationcontinuumEbexistsatthecriticalvaluehbofthecouplingparameter,andJistheintervalproducedbytheRegularPerturbationTheorem,whileEn(h;r)istheSiegertstatethatarisesasacontinuousperturbationoftheboundstateEbforh2J,thenitwillbeprovedthatan(h)p )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n(h)ash!hb.Recallthatn(h)=k2n(h))]TJ /F5 11.955 Tf 12.6 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(h)isthegeneralizedeigenvalueatwhichtheSiegertstateEn(h)exists. Toproceed,puteE!=)]TJ /F17 7.97 Tf 6.59 0 Td[(n anE!whereE!istheamplitudeinEq.( 2 ).Thenbythedecomposition( 2 ),eE!!Enas!n.Fromthesystem,8>><>>:eE!+"(h)eE!=0En+"(h)En=0 itisderivedbyGreen'stheoremthat,Z@D0 EnreE!)]TJ /F13 11.955 Tf 13.08 3.15 Td[(eE!r Endn0+()]TJ ET q .478 w 269.47 -275.95 m 279.65 -275.95 l S Q BT /F4 11.955 Tf 269.47 -285.92 Td[(n)ZD0eE! En"(h)dr0=0 whereD0isthesamerectangleasinEq.( 2 ).Then,bysplittingtheamplitudeeE!intermsoftheincidentandscatteredwaves,eE!(r)=)]TJ /F17 7.97 Tf 6.58 0 Td[(n aneikr+eEs!(r),oneinfersthatan(h)=)]TJ /F3 11.955 Tf 10.5 8.84 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(n)R@D0(ik En)-222(r En)eikr0dn+()]TJ /F4 11.955 Tf 11.95 0 Td[(n)RD0 Eneikr0"(h)dr0 R@D0 EnreEs!)]TJ /F13 11.955 Tf 13.08 3.16 Td[(eEs!r Endn+()]TJ /F4 11.955 Tf 11.95 0 Td[(n)RD0eEs! En"(h)dr0 whereitisunderstoodthatEnandnarefunctionsofh.Sinceanisindependentof,thedesiredexpressionforanisobtainedbytakingthelimit!n: an(h)=)]TJ /F13 11.955 Tf 77.99 18.48 Td[(R@D0)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(ikn En)-221(r Eneiknr0dn R@D0 Enr@eEs!)]TJ /F4 11.955 Tf 11.95 0 Td[(@eEs!r Endn=n+RD0jEnj2"(h)dr0,kn=k=n(2) wherefortherstterminthedenominator,l'Hopital'srulehasbeenapplied.ThisformulawouldnotgenerallybeusefulasthetermeEs!involvesthecoefcientsanimplicitly.However,iftheSiegertstateEn(h)istakennearaboundstateintheradiationcontinuum,asassumedhere,therst-orderperturbativeexpressionofanonlydependsontheSiegertstateEn(h). 36

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SuchaperturbativeexpressionisobtainedbyanalyzingeachoftheintegralsinEq.( 2 )separately.First,itisobservedthatash!hb,thenEn(h)!Eb.Bylettingz1!,andz2!1,itfollowsfromthenormalizationofEq.( 2b )thatthesecondintegralinthedenominatorofEq.( 2 )canbemadearbitrarilycloseto1providedhissufcientlyclosetohb.Hence,iftherstintegralofthesaiddenominatorcanbeshowntoconvergetozerointhelimitsconsidered,thiswouldimplythatintheleadingorderofperturbationtheory,thedenominatorofEq.( 2 )isapproximatedby1.Thattherstintegralofthesaiddenominatordoesindeedconvergetozeroash!hb,z1!,andz2!1canbeestablishedthroughalengthy,butstraightforwardcalculationwhosedetailsareomitted.Itcanbecarriedoutalongthefollowinglines.Asnotedearlier,Bloch'sconditionimpliesthattherstintegralinthedenominatorofEq.( 2 )istobecarriedoutonthesegments(x,z)2[0,1]fz1,z2goftheboundaryofD0.Now,ash!hb,theSiegertstateEn(h)becomesaboundstateEb,andtherefore,itdecaysexponentiallyinthespatialinnityjzj!1.Similarly,thesameexponentialdecaycanbeestablishedforthetermsinvolvingthederivativesofeEs!inthelimith!hb.Itthenfollowsthatinthelimitsh!hb,z1!andz2!1,theintegralinquestionvanishes.ThusthedenominatorofEq.( 2 )remainsindeedcloseto1,providedhissufcientlyclosetohb. Ontheotherhand,thenumeratortoEq.( 2 )canbeevaluatedintermsoftheamplitudesSminEqs.( 2 )toyieldZD0)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(ikn En)-222(r Eneiknr0dn=p )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(h)An(h,z1,z2) whereifkn=kxe1+p n)]TJ /F5 11.955 Tf 11.96 0 Td[(k2xe3,thenAn(h,z1,z2)=2iRenp n(h))]TJ /F5 11.955 Tf 11.95 0 Td[(k2xo S+0(h) p )]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n(h)e)]TJ /F6 7.97 Tf 6.58 0 Td[(2z2Imnp n(h))]TJ /F8 7.97 Tf 6.59 0 Td[(k2xo+2Imnp n(h))]TJ /F5 11.955 Tf 11.95 0 Td[(k2xo S)]TJ /F6 7.97 Tf -1.07 -7.97 Td[(0(h) p )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(h)e2iz1Renp n(h))]TJ /F8 7.97 Tf 6.59 0 Td[(k2xo 37

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Thecasekn=kxe1)]TJ /F13 11.955 Tf 12.63 10.1 Td[(p n)]TJ /F5 11.955 Tf 11.96 0 Td[(k2xe3issimilar.Ash!hb,thenn(h)!b,andsinceb>k2x,itfollowsthatImnp n(h))]TJ /F5 11.955 Tf 11.95 0 Td[(k2xo!0.Hence,inthesaidlimit,An(h,z1,z2)!2i S+0,bp b)]TJ /F5 11.955 Tf 11.95 0 Td[(k2xforS+0,bgiveninEq.( 2 ).Thus,ingeneral,ancanbewrittenas an(h)=ean(h)p )]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n(h)(2) whereean(h)isboundedandhasthepropertyean(h)!2i S+0,bp b)]TJ /F5 11.955 Tf 11.95 0 Td[(k2xash!hb. ThemostimportantconsequenceofthestructureofthecoefcientsannearaboundstateintheradiationcontinuumisalocalamplicationoftheamplitudeE!ascomparedtotheamplitudeoftheincidentradiation.Indeed,ifthewavenumberkoftheincidentradiationandthecouplingparameterharetunedtosatisfytheconditionk=kn(h),8h2J,thentheamplitude( 2 )becomes: E!(r)=iean(h) p )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n(h)En(r)+eEa(h;r)(2) whereh7!jjeEa(h,)jjL2(D)isboundedonJ.Itfollowsthat,jjE!jjL2(D)jean(h)j p )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(h)jjEnjjL2(D))-222(jjeEajjL2(D) Ash!hb,thenEn!EbinL2(D),andthereforejjEnjjL2(D)!jjEbjjL2(D)6=0.HenceiftheconstantS+0,binEq.( 2 )isnonzero,then,limh!hbjjE!jjL2(D)=1 andthiscanonlyhappeniftheamplitudeE!divergesinsomeregionsoftherectangleD.Eventhoughthefactthatean(h)isalwaysnonzerointhelimith!hb(i.e.,S+0,b6=0)couldnotbeveriedforallkindsofcouplings,therearemanysystemsinwhichitholdstrue.Forexample,thisisthecaseofthenormalincidence(kx=0)forasymmetricdoublearraydepictedinFig. 2-1 (b)whenthedielectricfunctions"1and"2inEq.( 2 )areidentical,andsymmetricwithrespecttothereection(x,z)7!(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z).Inthisparticularcase,theparityoperator,bS[E](x,z)=E(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)onL2(D),andthe 38

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Lippmann-SchwingerintegraloperatorbHcommutesothatSiegertstatesarealwayssymmetricorskewsymmetricwithrespecttothereection(x,z)7!(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)inthex,z-plane.Inparticular,theamplitudesSmgiveninEqs.( 2 )aresuchthatS+m=S)]TJ /F8 7.97 Tf -1.06 -7.3 Td[(mdependingonwhethertheSiegertstatetheycorrespondtoissymmetricorskewsymmetric.ItfollowsthatiftheboundstateEbhappenstolieintheintervalI1oftheCkx-plane(caseofoneopendiffractionchannel),thenEqs.( 2 )arereducedtojS+0,bj2=jS)]TJ /F6 7.97 Tf -1.06 -8.28 Td[(0,bj2=1 2p b)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x sothatthecoefcientinquestionisindeednonzeroandtheamplitudeE!isampliedinthevicinityoftheboundstateintheradiationcontinuumEb.NotethattheamplicationoftheamplitudeE!wasalsoobservedperturbativelyinthecaseofamoregeneralincidenceangle(i.e.,kx6=0)whentheboundstateEbliesintheintervalsI1andI2oftheradiationcontinuum(Chapter3). Theaforementionedamplicationcanonlyhappeninaregionnearorwithinthescatteringstructure.ThiscanbeunderstoodbyanalyzingthersttermofEq.( 2 )fromwhichtheamplicationshouldresult.Outsidethescatteringregion,theSiegertstateisexpressedintermsofitsscatteredamplitudesbyEq.( 2a ).Asbefore,theseriesinvolvedinthisexpressionssplitovertheindexsetsIop(b)andIcl(b)(forhnearhb,Iop(n(h))=Iop(b)andIcl(n(h))=Icl(b)).Ononehand,thetermsoftheserieswhoseindiceslieinIcl(b)decayexponentiallyinthespatialinnitysothatnoamplicationoftheeldcanbeobtainedfromthem.Ontheotherhand,thetermswhoseindiceslieinIop(b),disappearneartheboundstateintheradiationcontinuumasthesetermsareproportionaltop )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n(h)asindicatedbyEq.( 2 ).Fromthephysicalpointofview,thenotedlocaleldamplicationresultsfromaconstructiveinterferenceofscatteredeldsfromeachscatterer(anelementarycell)oftheperiodicstructure.Itisimportanttonotethattheamplicationmagnitudecanberegulatedbyvaryingthecouplingparameterh,whichprovidesanaturalphysicalmechanismtocontroloptical 39

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nonlinearphenomenaifthestructurehasanonlineardielectricsusceptibility.Thismechanismhasbeenusedtodemonstratethatadoubleperiodicarrayofdielectriccylinders(depictedinFig. 2-1 (b))withanon-zerosecond-ordernonlinearsusceptibilitycanconvertasmuchas44%oftheincidentuxintothesecondharmonicswhenthedistancebetweenthearraysisproperlyadjusted.Thesmallestdistanceatwhichsuchahighconversionratecanbeachievedisaboutahalfofthewavelengthoftheincidentradiation(Chapter4). 2.3DecayofElectromagneticSiegertStates ASiegertstatecanbeexcitedbyanincidentradiation,e.g.,byawavepacketwhosespectraldistributionpeaksaroundadesiredwavenumberkckn.Whenpassingthroughthestructure,someofthewavepacketenergyistrappedbythestructureandremainsinthereforalongperiodoftime,decayingslowlybyemittingamonochromaticradiation.Thisisestablishedinafashionsimilartothatofthestudyofthedecayofunstablestatesinquantummechanics[ 6 ].Toillustratetheprinciple,onlythecaseofnormalincidenceisconsideredhere(i.e.,kx=0).Otherincidenceanglescanbetreatedinasimilarmanner. Theexactstatementwhichwillbeprovedisasfollows.SupposethataSiegertstateEnexistsatthepolen=k2n)]TJ /F5 11.955 Tf 12.44 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(nsuchthatkn>0and)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n>0(i.e.,theSiegertstateisnotaboundstateintheradiationcontinuum).Thenthisstatewilldecaywithtimebyemittingamonochromaticradiationatawavenumberekn,andwithalifetimen: ekn=s k2n+p k4n+)]TJ /F6 7.97 Tf 18.73 3.45 Td[(2n 2)375()222()375(!)]TJ /F9 5.978 Tf 4.82 -1 Td[(n1kn,n=2ekn c)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n)374()223()374(!)]TJ /F9 5.978 Tf 4.82 -1 Td[(n12kn c)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n(2) Asastartingpoint,consideranincidentwavepacket,Ei(r,t)=Z10A(k)cos(kr)]TJ /F4 11.955 Tf 11.95 0 Td[(!t)dk,!=ck 40

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whereA(k)isthedistributionofwavenumbersinthewavepacket.ThenthesolutionEofthewaveequation( 2 )readsE(r,t)=ReZ10A(k)E!(r)e)]TJ /F8 7.97 Tf 6.58 0 Td[(i!tdk whereE!istheamplitudeofEq.( 2 ).FromthemeromorphicexpansionofE!inEq.( 2 )itthenfollowsthat,E(r,t)=Re(Ei(r,t)+XnEn(r)n(t)+Z10A(k)Ea(k2,r)e)]TJ /F8 7.97 Tf 6.58 0 Td[(i!tdk) wherethetimedependencen(t)oftheSiegertstateEnis, n(t)=eanp )]TJ /F8 7.97 Tf 6.78 -1.8 Td[(nZ10A(k) k2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2n+i)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(ne)]TJ /F8 7.97 Tf 6.59 0 Td[(icktdk(2) foreandenedinEq.( 2 ).Ast!1,thenn(t)!0asrequiredbytheRiemann-Lebesguelemma.Foratypicalphysicalwavepacket,thefunctionA(k)decaysfastask!1andisalsoanalyticinthecomplexk)]TJ /F1 11.955 Tf 9.3 0 Td[(plane.Thesepropertiesallowsforevaluatingtheintegral( 2 )bythestandardmeansofthecomplexanalysis.Itisalsoworthnotingthatboundstatesintheradiationcontinuumcannotbeexcitedbyanincidentradiationbecause)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n=0. Toavoidexcessivetechnicalitiesofthegeneralcase,aspecicformofthedistributionA(k)ischosentoillustratetheprocedure,whichissufcienttoestablishthemainpropertiesofthedecayofSiegertstates.ThesimplestformthatisalsomostcommonlyusedinphysicsisaGaussianwavepacketcenteredaroundawavenumberkc:A(k)=e)]TJ /F16 5.978 Tf 7.78 3.86 Td[((k)]TJ /F9 5.978 Tf 5.75 0 Td[(kc)2 22 p 2 Inthelimit!0,A(k)!(k)]TJ /F5 11.955 Tf 12.47 0 Td[(kc),andthemonochromaticcaseisrecovered.TheanalysiswillbecarriedoutforaSiegertstateEncorrespondingtoapolen=k2n)]TJ /F5 11.955 Tf 12.3 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(nsuchthatk2n>0and)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n>0.Siegertstateswithk2n<0arediscussedattheendofthissection. 41

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Theintegrandin( 2 )ishighlyoscillatoryforlarget,thecontourofintegrationmustbedeformedtoacurvealongwhichthephaseisconstanttoobtainafastconvergingintegral,andtherebytodeterminetheleadingterm.Tothisend,rst,thechangeofvariable=k)]TJ /F8 7.97 Tf 6.59 0 Td[(kc+ict2 p 2ismadetoobtainthefollowingmoreamenableformforn: n(t)=eanp )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(ne)]TJ /F8 7.97 Tf 6.59 0 Td[(ictkc)]TJ /F16 5.978 Tf 7.78 3.26 Td[(1 2c2t22 22p ZC0e)]TJ /F17 7.97 Tf 6.59 0 Td[(2 ()]TJ /F4 11.955 Tf 11.96 0 Td[(+)()]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 7.09 1.79 Td[()d,=)]TJ /F5 11.955 Tf 9.3 0 Td[(kc+ict2p n p 2(2) wherethecontourofintegrationC0isahorizontalrayoutgoingfromthepoint0=)]TJ /F8 7.97 Tf 6.58 0 Td[(kc+ict2 p 2towardtheinnityinthe-planeasshowninFig. 2-2 .Thepositionofthepolesindicatedonthesamegurefollowsfromthefactthatfork2n>0,thenRefp ng>0whileImfp ng<0. Put(u,v)=(Re,Im).Sincethefunction7!e)]TJ /F17 7.97 Tf 6.58 0 Td[(2decaysexponentiallyintheregionRef2g>0,thecontourofintegrationinEq.( 2 )canbedeformedtoacontourthatconsistsoftheconstantphasecurveC1=f(u,v)juv=)]TJ /F8 7.97 Tf 10.49 4.71 Td[(kcct 2,uRe0gextendingfrom0toandanotherconstantphasecontourC2:Imfg=0fromto1.Figure 2-2 showsthemodiedcontour.ByCauchy'stheorem,nbecomesthesumofthreeterms:n(t)=ean(An(t)+Bn(t)+Cn(t)) whereAnistheresidualcontributionatthepole+,whileBnandCnarethecontributionsofthelineintegralsalongthecontoursC1andC2,respectively.ItisprovedshortlythatitistheresidualtermAnthataccountsforanexponentialtimedecayoftheSiegertstateEn.Foralongperiodoftime,thistermdominatesinthedecayradiationofaSiegertstate,anditisonlyafterithasdecayedconsiderablythatthetermBnbecomesdominant.ThetermCnremainssmallincomparisontoAnandBn. Tobegin,theresidualtermAnisevaluatedtoyieldtheformula,An(t)=ip )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n p nA(p n)e)]TJ /F8 7.97 Tf 6.59 0 Td[(ictp n 42

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Inparticular,thewavenumberekn,andthelifetimenoftheSiegertstategiveninEq.( 2 )followfromtheformulaforp nwhichis,p k2n)]TJ /F5 11.955 Tf 11.96 0 Td[(i)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(n=s k2n+p k4n+)]TJ /F6 7.97 Tf 18.73 3.45 Td[(2n 2)]TJ /F5 11.955 Tf 11.95 0 Td[(i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n r 2k2n+p k4n+)]TJ /F6 7.97 Tf 18.73 3.45 Td[(2n Notethatfor)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(n1(anarrowresonance),theamplitudeoftheSiegertstateisproportionaltoA(p n)A(kn).Hence,theSiegertstatecorrespondingtothescatteringresonanceatk=knisexcitediftheGaussianwavepackethasasufcientamplitudeatk=kn,or,ideally,iscenteredatkntoachievethemaximaleffect. ThetermsBnandCnalsodecaynecessarilyintime.ForCn,theintegralalongC2willnecessarilyremainboundedintimesothatthemultiplicativefactorinEq.( 2 )isdominant,i.e.,Cn(t)=O(e)]TJ /F16 5.978 Tf 7.79 3.26 Td[(1 2c2t22) ThesamerelationholdsforBn,providedthat ctkc 2(2) Whenthisconditionisviolated,thepoint0movesintotheregionRef2g<0,andtheexponentiale)]TJ /F17 7.97 Tf 6.59 0 Td[(2startstogrowlargeonapartofthecontourC1.Nonetheless,thetermBn(t)stilldecaystozerointime.Thisisbecause,asnotedearlier,theRiemann-Lebesguelemmaensuresthedecayofn(t)intime,andthetermsAnandCnhavebeenprovedtoconvergetozero.FortheactualdecayrateofBn,itcanbeinferredfromLaplace'smethodfortheasymptoticexpansionofintegralsthatBn=O(1=t)forsomepositivenumber,whichisofnoimportanceherehowever.Indeed,itisobviousthatastincreases,thetermBnshouldbelargerthantheresidueat+,sincethearcofthecurveC1intheregionRef2g<0growslonger,andthereforetheexponentialdecayoftheSiegertstatecannolongerbeobservedabovethebackgrounddescribedbyBn.Thecondition( 2 )showsthatthesmallerthewidthofthewavepacketis,thelonger 43

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theexponentialdecaycanbeobserved.Forinstance,onecanensuretheobservationofthedecayradiationofthestateEnbyrequiringthatthehalf-lifetimeoccursbeforethedecayamplitudebecomessmallerthanthatofthebackground,i.e.,cnkc 2()ln2q k2n+p k4n+)]TJ /F6 7.97 Tf 18.73 3.45 Td[(2n kc)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(n 2 Inotherwords,thewidthofthepacketmustbecomparabletop )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(n,ifnotsmaller. Figure2-2. The-planefortheintegralinEq.( 2 ).TheregionRef2g0isshadowed.Theexponentiale)]TJ /F17 7.97 Tf 6.59 0 Td[(2isboundedinit.Thesketchisrealizedunderthecondition( 2 ),whenthepoint0isstillintheshadowedregion.Whentheconditionisviolated,partofthecurveC1liesintheregionRef2g<0. ForSiegertstateswithk2n<0,thepole+isnolongerenclosedbythecontoursC0,C1,andC2iftissufcientlylarge.ThepolemovestotheleftofthecontourC1andtheexponentialtimedecayisabsent,owingtothefactthattheincidentwavepacketdoesnotcontainanyradiationatwavenumbersclosetoknfortheSiegertstatetobeexcited. 2.4ProofoftheRegularPerturbationTheorem Recallthat8(h,)2J0Ckx,bH(h,)2LfL2(D)g,andthatthemapJ0Ckx!LfL2(D)g(h,)7!bH(h,) 44

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iscontinuouslyFrechetdifferentiable.For(h,)2J0Ckx,letR(bH(h,))=[)]TJ /F13 11.955 Tf -427.88 -20.75 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.58 0 Td[(1,2C,betheresolventofbH(h,)whenitexists,andlet(bH(h,))=f2CjR(bH(h,))existsgbetheresolventsetofbH(h,).TheproofoftheRegularPerturbationTheoremisaidedwiththefollowinglemma: Lemma1. LetS=f(h,,)2J0CkxCj2(bH(h,))g.ThesetSisopeninJ0CkxCandnonempty. Proof. ThatSisnonemptyfollowsfromtheresultsofSection 2.1 .Indeed,ifisnotapoleof7![1)]TJ /F13 11.955 Tf 13.23 3.16 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1,then(h,,1)2S.ToshowthatSisopen,let(h0,0,0)2S.Thenonehastoprovethattherealwaysexistsaneighborhoodof(h0,0,0)inJ0CkxCthatliesinS. Since(bH(h0,0))isopeninC,thereexists12(bH(h0,0)),16=0sothat[1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(h0,0)])]TJ /F6 7.97 Tf 6.58 0 Td[(1exists.Letthenbethefunction,:J0Ckx!LfL2(D)g(h,)7!1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(h,) Themapiscontinuous,and(h0,0)isinvertible.AsthesetofallinvertibleoperatorsinLfL2(D)gisopen,thereexistsanopenneighborhoodUJ0Ckxof(h0,0)suchthat8(h,)2U,(h,)isinvertible. Nowbytherstresolventformula,R0(bH(h0,0)))]TJ /F5 11.955 Tf 11.95 0 Td[(R1(bH(h0,0))=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)R0(bH(h0,0))R1(bH(h0,0)) sothat,R0(bH(h0,0))1)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[(0)R1(bH(h0,0))=R1(bH(h0,0)) Itfollowsthat1)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(0)R1(bH(h0,0))isinvertible.Nowletbethefunction,:UC!LfL2(D)g((h,),)7!1)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()R1(bH(h,)) 45

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Theniscontinuous.Since(h0,0,0)isinvertible,andagain,thesetofallinvertibleoperatorsinLfL2(D)gisopen,itfollowsthatthereexistsanopensetWUCsuchthatforall(h,,)2W,theoperator(h,,)isinvertible. Thus,for(h,,)2W,both1)]TJ /F13 11.955 Tf 12.92 3.16 Td[(bH(h,)and1)]TJ /F3 11.955 Tf 12.01 0 Td[((1)]TJ /F4 11.955 Tf 12 0 Td[()R1(bH(h,))areinvertible.Theircompositionisthereforeinvertible,andthiscompositionis,(1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(h,))(1)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()R1(bH(h,)))=)]TJ /F13 11.955 Tf 12.86 3.16 Td[(bH(h,) ItfollowsthatWS.AsWisaneighborhoodof(h0,0,0)inJ0CkxC,thesetSisopen. TheproofoftheRegularPerturbationTheoremisasfollows. Proof. Letnbeasimplepoleof7![1)]TJ /F13 11.955 Tf 10.63 3.15 Td[(bH(hn,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1forsomexedhn2J0,andletEnbethecorrespondingSiegertstate.Then1isaneigenvalueofbH(hn,n)correspondingtotheeigenfunctionEnintheusualsense.Thetheoremamountstoprovingtheexistenceofacurveh7!(h)inJ0CkxalongwhichthefamilyofoperatorsbH(h,(h))stillhas1asaneigenvalue.ThecorrespondingeigenfunctionswillthenbetheSiegertstatesofthesaidoperatorsalongthecurveinquestion. Theproofstartsbyprovidingageneralformulafortheeigenvalue0(h,)ofbH(h,)whichistheperturbedvalueoftheeigenvalue1whenthepoint(hn,n)isdisplacedto(h,)intheJ0Ckxspace.ThenbytheImplicitFunctionTheorem,acurveh7!(h)isfoundalongwhichtheeigenvalue(h,)remains1,i.e.,(h,(h))=1. Asastartingpoint,notethatasbH(hn,n)isacompactoperatorontheHilbertspaceL2(D),theRiesz-Schaudertheorem[ 7 ]impliesthat1isnecessarilyanisolatedeigenvalue.Therefore,thereexists>0suchthat1istheonlyeigenvalueofbH(hn,n)inthediskf2Cjj)]TJ /F3 11.955 Tf 12.03 0 Td[(1jg.ItfollowsthatV=f(hn,n,)j2C,j)]TJ /F3 11.955 Tf 12.02 0 Td[(1j=gSgwhereSisthesetofthepreviouslemma.SinceSisopen,andViscompact,thereexistsanopenWinJ0CkxCsuchthatVWS.Thereforethereexists 46

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aconnectedneighborhoodUof(hn,n)inJ0Ckxsuchthat2(bH(h,))forall(h,)2U,andforall2Cwithj)]TJ /F3 11.955 Tf 11.95 0 Td[(1j=. Put bP(h,)=1 2iIj)]TJ /F6 7.97 Tf 6.59 0 Td[(1j=[)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(h,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1d,(h,)2U(2) ThenbP(h,)2LfL2(D)g,andbyTheoremXII.6of[ 5 ],bP(h,)isaprojection.Asthemap(h,)7!bP(h,)ofUtoLfL2(D)giscontinuous,itfollowsthatthedimensionoftherangeofbP(h,)isconstantthroughoutU. Sincetheeigenvalue1ofbH(hn,n)isnondegenerate,itfollowsthattherangeofbP(hn,n)hasdimension1,andthereforethedimensionoftherangeofbP(h,))is1throughoutU.ByTheoremXII.6of[ 5 ],itfollowsthatforall(h,)2U,thereexistsauniqueeigenvalue0(h,)ofbH(h,)inf2Cjj)]TJ /F3 11.955 Tf 11.97 0 Td[(1jg,andbP(h,)istheprojectiononthecorrespondingeigenspace.Inparticular,bP(h,)[En]isintheeigenspaceof0(h,),andtherefore,bH(h,)hbP(h,)[En]i=0(h,)bP(h,)[En],(h,)2U Hence,0(h,)=hEn,bH(h,)hbP(h,)[En]ii hEn,bP(h,)[En]i,(h,)2U whereonceagain,h,iistheinnerproductonL2(D).Thisistheformulafortheperturbedeigenvalueannouncedinthepreambleofthisproof. Now,observethatsince(h,)7!bH(h,)iscontinuouslyFrechetdifferentiableinU,sois(h,)7!bP(h,).Therefore,0iscontinuouslyFrechetdifferentiableinU.Now,0(hn,n)=1,andasitwillbeshownshortly,@0(hn,n)6=0.ItfollowsbytheImplicitFunctionTheoremforBanachspacesthatthereexistsanopenintervalJJ0containinghn,andauniquecontinuouslydifferentiablefunctionh7!n(h),h2J,suchthatn(hn)=nand0(h,n(h))=1forallh2J.ThustheproofoftheRegularperturbationtheoremiscomplete,providedtheproperty@0(hn,n)6=0isestablished. 47

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ThelatterisachievedbystudyingtheanalyticityoftheresolventR(bH(hn,))=[)]TJ /F13 11.955 Tf -436.24 -20.75 Td[(bH(hn,)])]TJ /F6 7.97 Tf 6.58 0 Td[(1inthevariablesandseparatelywhenh=hnisxed. Let2Cand2Ckxsuchthatj)]TJ /F3 11.955 Tf 12.56 0 Td[(1j=,(hn,)2U,and6=n.ThentheresolventR(bH(hn,))=[)]TJ /F13 11.955 Tf 12.5 3.16 Td[(bH(hn,)])]TJ /F6 7.97 Tf 6.59 0 Td[(1existsbydenitionoftheneighborhoodU.Also,theresolventR1(bH(hn,))=[1)]TJ /F13 11.955 Tf 13.07 3.16 Td[(bH(hn,)])]TJ /F6 7.97 Tf 6.58 0 Td[(1existsandismeromorphicin.Bytherstresolventformula, R(bH(hn,)))]TJ /F5 11.955 Tf 11.95 0 Td[(R1(bH(hn,n))=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()R(bH(hn,))R1(bH(hn,))(2) SubstitutingEq.( 2 )intoEq.( 2 ),itfollowsthat, bP(hn,)=1 2iIj)]TJ /F6 7.97 Tf 6.58 0 Td[(1j=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()R(bH(hn,))R1(bH(hn,))d(2) Now,themeromorphicexpansionof7!R(bH(hn,))at0(hn,)is, R(bH(hn,))=1 )]TJ /F4 11.955 Tf 11.96 0 Td[(0(hn,)bP(hn,)+eR(hn,)(2) where7!eR(hn,)isanalyticinthediskf2C:j)]TJ /F3 11.955 Tf 12.45 0 Td[(1jg.ThesubstitutionofEq.( 2 )intoEq.( 2 )yields bP(hn,)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(0(hn,))bP(hn,)R1(bH(hn,))(2) Now,atthepolen,thegeneralizedresolvent7!R1(hn,)hasthemeromorphicexpansion: R1(hn,)=1 )]TJ /F4 11.955 Tf 11.95 0 Td[(nbHn+eR(hn,)(2) wherebHnistheresidueof7!R1(hn,)asgiveninEq.( 2 ),and7!eR(hn,)isanalyticinthevicinityofn.SubstitutingEq.( 2 )intoEq.( 2 )andtakingthelimitas!nyieldstheequation, bP(hn,n)=)]TJ /F4 11.955 Tf 9.3 0 Td[(@0(hn,n)bP(hn,n)bHn(2) 48

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Next,theactionofboththesidesofthisoperatorequalityisevaluatedonthestateEn.SincebP(hn,n)projectsonEn,andinlightofEq.( 2 ),itfollowsthat,En=h'n,Eni@0(hn,n)En AstheSiegertstateEnisnotidenticallyzero,itfollowsthat@0(hn,n)6=0,andtheproofofthetheoremiscomplete. AnalnoteworthyremarkisthattheprojectionbP(hn,n)andtheresiduebHnareproportional,whichisestablishedbyapplyingboththesidesofEq.( 2 )toanarbitrarystateF2L2(D):bP(hn,n)[F]=)]TJ /F4 11.955 Tf 9.3 0 Td[(@0(hn,n)bHn[F] 49

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CHAPTER3ELECTROMAGNETICBOUNDSTATESINTHERADIATIONCONTINUUMFORPERIODICDOUBLEARRAYSOFSUBWAVELENGTHDIELECTRICCYLINDERS Inthischapter,boundstatesintheradiationcontinuumarestudiedanalyticallyforasystemoftwoarraysofparalleldielectriccylinders.Theapproachisbasedontheresonantscatteringtheory[ 5 10 ]wheretheboundstatesareidentiedasresonanceswiththevanishingwidth(thedistancebetweenthearraysisaphysicalparameterwhichregulatesthecouplingoftheresonances).Thestudyiscarriedoutforthewholeradiationspectrumrange.Inadditiontotheboundstatesinthezero-orderdiffraction(whichwereknowntoexistfromtheearlynumericalstudies),boundsstatesarefoundinthespectralrangewheretwoormorediffractionchannelsareopen.AnalyticsolutionsofMaxwell'sequationsforalltheboundstatesaregiveninthelimitwhenthecylinderradiusismuchsmallerthantheperiodofthestructure.Thesystemisshowntohaveboundstatesbelowtheradiationcontinuumwhoseexplicitformisalsofound. 3.1ScatteringTheoryandClassicationoftheFields ThesystemconsideredissketchedinFig. 3-1 (a).Itconsistsofaninnitedoublearrayofparallel,periodicallypositioned,dielectriccylinderssuspendedinthevacuum[ 11 ].Thecylindersareassumedtobenon-dispersivewithadielectricconstant"c>1.Thecoordinatesystemissetsothatthecylindersareparalleltothey)]TJ /F1 11.955 Tf 9.29 0 Td[(axis,thestructureisperiodicalongthex)]TJ /F1 11.955 Tf 9.3 0 Td[(axis,andthez)]TJ /F1 11.955 Tf 9.3 0 Td[(axisisnormaltothestructure.Theunitoflengthistakentobethearrayperiodandthearrayshavearelativemismatcha2[0,1 2]alongthex-axis. Thestructureisilluminatedbyalinearlypolarizedmonochromaticbeamwiththeelectriceldparalleltothecylinders(TMpolarization).Inthegivensettings,Maxwell'sequationsarereducedtothescalarwaveequationforasinglecomponentoftheelectriceld,denotedE,inthex,z)]TJ /F1 11.955 Tf 9.3 0 Td[(plane: c2@2tE)-222(4E=0,(3) 50

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Figure3-1. Panel(a):Doublearrayofdielectriccylinders.Theunitoflengthisthearrayperiod.Theaxisofeachcylinderisparalleltothey-axisandisatadistancehfromthex-axis.Panel(b):TheenergyspectrumforaSchrodingerequationwithradiallysymmetricpotentialconsistsofadiscretespectrumofboundstateswithnegativeenergiesandtheradiationcontinuum.Thelattermaycontainadditionalboundstates;thesearetheboundstatesintheradiationcontinuum.Panels(c)and(d):HarmonicallytimedependentsolutionstoEq.( 3 )arelabeledbypointsofthespectralcylinderasexplainedintext.ThediffractionthresholdsEnareringsthatpartitionthecylinderintosectionscorrespondingtoaxednumberofopenchannels.Inparticular,E0isthethresholdfortheradiationcontinuumbelowwhichtherecanbenoscatteringstates. wherethedielectricfunction"ispiecewiseconstant;itisequalto"c>1onthescatterersandto1otherwise.ForaharmonictimedependentelectriceldE(r,t)=E!(r)e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!t,theamplitudeE!(r)satisestheequation [+k2(1)]TJ /F4 11.955 Tf 11.95 0 Td[(")]E!=k2E!,(3) wherek2=!2=c2isthespectralparameter.ThisequationissimilartotheSchrodingerequationwithanattractive(negative)potential1)]TJ /F4 11.955 Tf 13.03 0 Td[(".Ifkisthewavevectoroftheincidentradiation,thentheperiodicityofthestructurerequiresthatsolutionstoEq.( 3 )satisfyBloch'stheorem, E!(r+e1)=eikxE!(r),(3) 51

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wherek=kxe1+kze3andfe1,e2,e3garetheunitvectorsalongthecoordinateaxes.Consequently,thewavevectorsofthescatteredradiationcanonlyhavethefollowingform km=(kx+2m)e1p k2)]TJ /F3 11.955 Tf 11.96 0 Td[((kx+2m)2e3,m=...)]TJ /F3 11.955 Tf 11.95 0 Td[(2,)]TJ /F3 11.955 Tf 9.3 0 Td[(1,0,1,2...(3) withtheconventionp k2)]TJ /F3 11.955 Tf 11.96 0 Td[((kx+2m)2=ip (kx+2m)2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2ifk2<(kx+2m)2.Thevaluesofmforwhichk2(kx+2m)2labelopendiffractionchannels,allothervaluesofmcorrespondtoclosedchannels.If,forinstance,theincidentradiationpropagatesinthepositivez)]TJ /F1 11.955 Tf 9.3 0 Td[(directionasshowninFig. 3-1 (a),thenthetransmittedeldreads E!(r)=eikr+XmTmeik+mr,z>h+R,(3a) whereTmaretheamplitudesofthetransmittedmodesandthersttermdescribestheincidentradiationwhoseamplitudeissettoone.Similarly,intheregiontotheleftofthestructure,theelectriceldisgivenby E!(r)=eikr+XmRmeik)]TJ /F9 5.978 Tf 0 -5.09 Td[(mr,z<)]TJ /F5 11.955 Tf 9.3 0 Td[(h)]TJ /F5 11.955 Tf 11.95 0 Td[(R,(3b) whereRmaretheamplitudesofthereectedmodes.Thechoicebetweenk+mandk)]TJ /F8 7.97 Tf 0 -7.29 Td[(misdictatedbytheoutgoingwaveboundaryconditionatthespatialinnityjzj!1.Inparticular,theeldcorrespondingtoclosedchannelsinEqs.( 3 )decaysexponentiallyintheasymptoticregionjzj!1.Sotheclosedchannelsdonotcontributetotheenergyuxcarriedbythescatteredwave.Thescattereduxiscarriedonlybytheradiationinopendiffractionchannels.Theconditionk2m=(kx+2m)2denesthethresholdforopeningthemthdiffractionchannel.ThecoefcientsRmandTmforopenchannelsarethereectionandtransmissioncoefcients,respectively. 52

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ThetransmissionandreectionamplitudesmaybeinferredfromthesolutionE!oftheLippmann-Schwingerintegralequation[ 10 ] E!(r)=eikr+k2 4Z("(r0))]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!(r0)G(rjr0)dr0,(3) inwhichG(rjr0)=iH0(kjr)]TJ /F12 11.955 Tf 11.96 0 Td[(r0j)isthe2-dimensionalfree-spaceGreen'sfunctionfortheHelmholtzoperator4+k2withoutgoingboundaryconditions,andH0istheHankelfunctionoftherstkindoforder0. Observethat,unlessthereisnoincidentradiation(thetermeikrisomitted),thesolutionstoEq.( 3 )cannotbesquareintegrablealongthez-axis.Bytheanalogywithquantumscatteringtheoryforradiallysymmetricpotentials,suchsolutionsareintheradiationcontinuumoftheenergyspectrum.Inquantumtheory,solutionstotheSchrodingerequationareeigenvectorsoftheenergyoperatorwhosespectrumEformasubsetintherealline.Theupperpart(positive)ofthespectrumiscontinuousandcorrespondstoscatteredstatesthatcancarrytheprobabilityuxtothespatialinnity.Belowthecontinuouspartofthespectrum,i.e.,E<0,theenergyspectrumisdiscrete(Fig. 3-1 (b)).Itcorrespondstoboundstates.BoundstateshaveaniteL2norm(theydecayfastenoughatthespatialinnity).ItwasrstprovedbyVonNeumannandWignerin1929thatunderspecialcircumstances,theremightexistboundstatesintheradiationcontinuum.Acounterintuitivephysicalpeculiarityofsuchstatesisthataboundstateisastandingwaveinapotentialwell(anattractivepotential),whiletheconventionalquantummechanicalwisdomwouldsuggestthatforapotentialboundedaboveastandingwavewiththeenergyintheradiationcontinuumshouldtunnelthroughthepotentialbarriertothespatialinnityand,hence,cannotbestable.Nevertheless,suchstatesdoexistandthetheoryoftheirformationisnowwelldeveloped[ 6 8 12 ].Thegoalhereistoestablishasimilarpictureforelectromagneticexcitationsintheperiodicdoublearrayofdielectriccylinders,and,specically,tondtheconditionsonthephysicalparametersofthissystemunderwhichtheboundstatesintheradiation 53

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continuumexist,theireigen-frequencies,andtheanalyticformofthecorrespondingelectromagneticelds.Theveryexistenceofboundstatesforthissystemwasrstdemonstratedbynumericalsimulations[ 11 ].Hereacompleteanalyticstudyofthesystemisgiven. Anyboundstateisasolutionofthegeneralizedeigenvalueproblem( 3 )(noincidentradiationterm)and,hence,isfullycharacterizedbythespectralparameterE=k2>0andtheBlochphasefactoreikxbecauseoftheboundarycondition( 3 ).Thepair(E,eikx)isviewedasapointonthe(half)cylinderR+S1whereE2R+andeikx2S1.Itwillbecalledaspectralcylinder(orspectralspace)ofaperiodicgrating.Incontrasttoquantummechanicalboundstatesinsphericallysymmetricsystems,theBlochboundaryconditionrequiresamoreadequateclassicationofboundstateshere.Inordertoidentifyboundstatesintheradiationcontinuum,onehasrsttodeterminetheregionofthespectralcylinderoccupiedbytheradiationstates.Intheasymptoticregionjzj!1,aharmonictimedependentsolutionE=E!e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!ttoEq.( 3 )ischaracterizedbythepair(E,kx)whereE=k2=!2=c2.GivenEandkx,theeldoutsidethescatteringregioniscompletelydeterminedbyitsbehaviorinthediffractionchannelsasspeciedinEqs.( 3 ).Consequently,iftworadiationmodeshavethesameE,whiletheirparameterskxdifferbya2-multiple,say,by2m0,thentheyhaveexactlythesameopendiffractionchannelsbecauseintheclassicationintroducedinEq.( 3 )thedifferenceofchannelswouldmerelymeantherelabelingm!m)]TJ /F5 11.955 Tf 12.62 0 Td[(m0(orthesamechangeofthesummationindexinEq.( 3 )).Thereforetheradiationmodescorrespondtopoints(E,eikx)onthespectralcylinderforwhichoneormorediffractionchannelsareopen. Thespectralcylindercanbepartitionedintosectionsassociatedwithaxednumberofopendiffractionchannels.Thediffractionthresholdsappearascurvesseparatingtheseportionsofthecylinder.Indeed,let[kx]designatetheargumentofeikxin()]TJ /F4 11.955 Tf 9.3 0 Td[(,],i.e.,[kx]=kxmod2.Then,uptotheaforementionedreordering,the 54

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diffractionthresholdsonthespectralcylinderareexactly En(kx)=(2nj[kx]j)2,n=0,1,2,3...,(3a) andtheyappearintheorder, E0E)]TJ /F6 7.97 Tf 6.59 0 Td[(1E1E)]TJ /F6 7.97 Tf 6.59 0 Td[(2E2E)]TJ /F6 7.97 Tf 6.59 0 Td[(3E3...(3b)Ifkxisidentiedwiththeangularvariablespanningthecompactieddirectionofthespectralcylinder,thenthediffractionthresholdsarecurvesinaneverrisingorderonthespectralcylinderwithnodesontheaxes[kx]=0and[kx]=.ThecurveE=E0(kx)isthethresholdbelowwhichnoradiationmodesexist,andtherefore,theradiationcontinuumliesimmediatelyabovethiscurve.ThiscontinuumissplitintodistinctregionscorrespondingtoaxednumberofopenchannelsbyconsecutivethresholdsasindicatedbyEq.( 3b ).TheseregionswillbelabeledasradiationcontinuumI,radiationcontinuumII,radiationcontinuumIII,etc.,wheretheRomannumeralindicatesthenumberofopenchannelsineachregion(Fig. 3-1 (b),(c),and(d)). AllthesolutionstoEq.( 3 )belowthethresholdE0,ifany,mustbeboundstates,i.e.,theydecayexponentiallyinalldiffractionchannelsand,hence,haveaniteL2norminthespaceS1Rspannedby(x,z)(xiscompactiedintoacircleS1becauseoftheboundarycondition( 3 )).Incontrast,radiationmodesbehaveasharmonicfunctionsintheasymptoticregionjzj!1and,hence,donothaveaniteL2norm.Sotheproblemistond,ifany,squareintegrablesolutionsonS1RabovethecurveE=E0whicharethesought-forboundstatesintheradiationcontinuum. 3.2BoundStates Asdenedabove,boundstatesaresquareintegrablesolutionsofthehomogeneousLippmann-Schwingerintegralequation E!(r)=k2 4Z("(r0))]TJ /F3 11.955 Tf 11.96 0 Td[(1)E!(r0)G(rjr0)dr0,k2>0,(3) 55

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whichsatisfyBloch'sboundarycondition( 3 ).ThesquareintegrabilityheremeansaniteL2norminS1R(withthex-directioncompactiedintoacircle).Asalsonotedabove,Eq.( 3 )isunderstoodinthedistributionalsense(Appendix A.1 ).ThisisageneralizedeigenvalueproblembecausetheGreen'sfunctionG(rjr0)alsodependsonthespectralparameterk2.Inthisproblem,theBlochparameterkxmayberestrictedtotheinterval[)]TJ /F4 11.955 Tf 9.3 0 Td[(,].Itmustbestressedthatthisrestrictionisonlypermittedbytheabsenceofanincidentwavewhichotherwisedeterminesthephasefactorin( 3 ). Thetaskistodeterminethevaluesofa,kx,handkthatallowfortheexistenceofnontrivialsolutionsE!toEq.( 3 )forxedradiusRandxeddielectricconstant"cinthelimitofthincylinders,i.e.,kR1.Inwhatfollows,theexistenceofaboundstatealwaysmeanstheexistenceofawavenumberkatwhichtheboundstateE!occurs.ThemainresultsestablishedinthepresentstudymaybesplitintothefollowingcaseswhicharealsosummarizedinFig. 3-2 (a): Belowtheradiationcontinuum:Boundstatesexistforallkx,aandforalldistancesbetweenthearrays. ContinuumI(oneopendiffractionchannel):Boundstatesonlyexistif eitherkx=0anda2[0,1 2]isarbitrary ora2f0,1 2gandkx2()]TJ /F4 11.955 Tf 9.3 0 Td[(,)isarbitrary Undertheseconditions,foreachpair(a,kx)thereisadiscretesetofdistancesbetweenthearraysatwhichboundstatesexist. ContinuumII(twoopendiffractionchannels):Boundstatesonlyexistfor a=0ora=1 2andforacertaindensesetofvaluesofkx.Foreachallowablepair(a,kx),thereisexactlyoneortwodistancesbetweenthearraysatwhichtheboundstatesexist. ContinuumN,N3(threeormoreopendiffractionchannels):BoundstatesexistonlyforspecicvaluesoftheradiusRandthedielectricconstant"c. Inalltheabovecases,boundstatesoccurintwotypes,symmetricandskew-symmetricrelativetothetransformationz!)]TJ /F5 11.955 Tf 24.77 0 Td[(zandx!a+x,whentheshiftparametertakesthe 56

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Figure3-2. Panel(a):Valuesoftheparametersaandkxforwhichboundstatesexist.Thebottomlevelrepresentsthemodesbelowtheradiationcontinuum,thesecondlevelrepresentsthemodesintheradiationcontinuumI,andthethirdlevelrepresentsthemodesintheradiationcontinuumII.Panels(b)and(c):Theshadedareasrepresentstheregionofvalidityofinequalities( 3 )and( 3 ),respectively. boundaryvalues,a=0ora=1 2.However,thisclassicationcannotbeestablishedfordoublearrayswithintermediatevaluesoftheshiftparameters,a2(0,1 2). Theaboveclassicationoftheboundstatesinthecasesa=0anda=1 2isrelevantwhenanalyzingthe(discrete)valuesofthedistance2hbetweenthetwoarraysatwhichtheboundstatesoccur.Whenthedoublearrayissymmetric(a=0),thenthesevaluesalwaysexistinthewholerangeofh2(R,1).Onthecontrary,skew-symmetricboundstateswillonlyoccurforthosevaluesofhthatexceedacertainminimalthreshold,andthisthresholdincreasesasR2("c)]TJ /F3 11.955 Tf 11.81 0 Td[(1)!0.Inotherwords,whenthescatteringstructurebecomesmoretransparent,thetwoarrayshavetobetakenfurtherapartinorderforskew-symmetricboundstatestoform.Asimilarphenomenonisobservedfortheskew-symmetricarray(a=1 2).Inthiscase,however,theskew-symmetricmodesbehaveasthesymmetriconesinthepreviouscaseandviceversa,i.e.,theminimalthresholddistanceexistsforthesymmetricmodesandincreasesasthesystembecomesmoretransparent(R2("c)]TJ /F3 11.955 Tf 11.98 0 Td[(1)!0),whiletheskew-symmetricmodesoccuratadiscretesetofvaluesofhinthewholerangeh2(R,1). Theexistenceoftheaboveclassicationcanbeestablishedbystudyingthesymmetryofthefunction"(x,z).ConsidertheoperatorPadenedbyPaE!(x,z)=e)]TJ /F8 7.97 Tf 6.58 0 Td[(iakxE!(x+a,)]TJ /F5 11.955 Tf 9.3 0 Td[(z).Fora=0ora=1 2,theoperatorPacommuteswiththeoperatorH 57

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ofEq.( 3 )andhaseigenvalues1sinceitisaprojection,i.e.,P2a=1.Byvirtueofthecommutativity,eachboundstateE!isaneigenfunctionofPaand,hence,E!(x+a,)]TJ /F5 11.955 Tf 9.29 0 Td[(z)=eiakxE!(x,z). SymmetricstatesmaythenbedenedasthoseforwhichE!(x+a,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)=eiakxE!(x,z)whiletheskew-symmetricstatessatisfyE!(x+a,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)=)]TJ /F5 11.955 Tf 9.3 0 Td[(eiakxE!(x,z).Fora=0thissimplymeansthattherststatesareeveninzwhilethesecondareodd. ThisdescriptioncannotbegeneralizedtoanarbitraryshiftasincePanolongercommuteswithHifa2(0,1 2).Instead,itfollowsfromthesymmetryof"(x,z)thattheoperatorQaE!(x,z)=E!(a)]TJ /F5 11.955 Tf 12.52 0 Td[(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)commuteswithHforallaandalsoQ2a=1.However,theoperatorQaisnotsymmetricinthesubspaceoffunctionsinL2(S1R)satisfyingcondition( 3 )unlesskx=0orkx=.Indeedifitwere,thenanyboundstateE!wouldhavebeenaneigenfunctionofQa,and,therefore,E!(a)]TJ /F5 11.955 Tf 12.93 0 Td[(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z)=E!(x,z).Byreplacingxbyx+1andapplyingcondition( 3 )onegetseikxE!(x,z)=e)]TJ /F8 7.97 Tf 6.58 0 Td[(ikxE!(x,z)sothatkx=0orkx=. Asnotedabove,theboundstatesarefoundbyperturbationtheoryinthelimitofthincylinders.ThetechnicaldetailsaregiveninAppendix A.2 whereitisshownthatnon-trivialsolutionstoEq.( 3 )canonlyexistiftheeldsE!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)andE!(ae1+he3)onthetwocylinderspositionedat(0,)]TJ /F5 11.955 Tf 9.29 0 Td[(h)and(a,h)respectivelydonotsimultaneouslyvanishandsatisfythehomogeneoussystemofequations 8>><>>:0E!(ae1+he3)++E!()]TJ /F5 11.955 Tf 9.29 0 Td[(he3)=0)]TJ /F5 11.955 Tf 7.08 -4.34 Td[(E!(ae1+he3)+0E!()]TJ /F5 11.955 Tf 9.29 0 Td[(he3)=0(3) inwhichthecoefcients0andaregivenbythefollowingexpressions0(k,kx)=1Xm=1 kz,m)]TJ /F3 11.955 Tf 42.26 8.09 Td[(1 2i(jmj+1)+i 21 0(k)+2ln(2R)(a,h,k,kx)=1Xm=ei(a(kx+2m)+2hkz,m) kz,m (3) 58

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withkz,m=p k2)]TJ /F3 11.955 Tf 11.96 0 Td[((kx+2m)2beingthez-componentofthewavevectorinthemthdiffractionchanneland 0(k)=kR 22("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(3) Inparticular0(k)>0as"c>1.Itshouldalsobekeptinmindthroughouttheworkthat0(k)1inthelimitconsideredkR1.IntermsoftheeldsonthecylindersinEqs.( 3 ),theelectriceldstrengtheverywhereoffthescatterersisthengivenby, E!(r)=2i0(k) E!(ae1+he3)Xmei((x)]TJ /F8 7.97 Tf 6.58 0 Td[(a)(kx+2m)+jz)]TJ /F8 7.97 Tf 6.59 0 Td[(hjkz,m) kz,m+E!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)Xmei(x(kx+2m)+jz+hjkz,m) kz,m!(3)ThesystemofEqs.( 3 )admitsnon-trivialsolutionsifandonlyifitsdeterminant (a,h,k,kx)=20)]TJ /F3 11.955 Tf 11.96 0 Td[(+)]TJ /F1 11.955 Tf 135.61 -4.94 Td[((3) vanishesatsomepoint(a,h,k,kx)inthespaceofsystemparameters.Thisistheconditionforboundstatestoexist,nomatteriftheyarebeloworintheradiationcontinuum. Inthefollowingtwosubsections,rootsof(a,h,k,kx)inkforxeda,h,andkxareanalyzedtondboundstatesbelowtheradiationcontinuumaswellasboundstatesinthecontinuumI.Theanalysisofthehighercontinuainthespectrum,whilebeingsimilartothecaseofthecontinuumI,istechnicallymoreinvolved.Toavoidexcessivetechnicalitiesbeforethediscussionofapplicationsofresonanceswiththevanishingwidthtoaneareldamplication,theanalysisofhighercontinuaispostponedtoSection 3.3 .Ineachstudy,thespectralparameterkrangesoveranopenintervalinwhichthefunctions0andareanalyticinkanddivergeattheendpoints.Also,notethat,since(a,h,k,kx)iseveninkx,therangeofthisparametercanbereducedfrom[)]TJ /F4 11.955 Tf 9.3 0 Td[(,]to[0,].ThroughouttherestofthissectionaswellasinSection 3.3 ,the 59

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symbols,c,andsdenote,respectively,thefollowingfunctions:c(a,h,k,kx)=Xmcle)]TJ /F6 7.97 Tf 6.58 0 Td[(2hqz,m qz,mcos(2am),s(a,h,k,kx)=Xmcle)]TJ /F6 7.97 Tf 6.59 0 Td[(2hqz,m qz,msin(2am)(k,kx)=Im(0(k,kx)) (3) wherethesuperscriptclinmclmeansthatthesumsaretakenoverallm'sthatcorrespondtocloseddiffractionchannels,andqz,mistheimaginarypartofkz,mwhenthemthchannelisclosed,i.e.,qz,m=p (kx+2m)2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2ifk2<(kx+2m)2.Recallthatkz,m=p k2)]TJ /F3 11.955 Tf 11.96 0 Td[((kx+2m)2.Byconstruction,thefunctions( 3 )arealwaysreal-valued. 3.2.1BoundStatesBelowtheRadiationContinuum BoundstatesbelowtheradiationcontinuumarenontrivialsolutionstothehomogeneousLippmann-Schwingerintegralequationwhenalldiffractionchannelsareclosed,kx2(0,]and0
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atkxof)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(.Theselimitsare,limk!0+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[((k)=1,limk!kx)]TJ /F3 11.955 Tf 8.25 5.81 Td[()]TJ /F3 11.955 Tf 7.09 -4.94 Td[((k)=)]TJ /F3 11.955 Tf 9.3 0 Td[(4hcos2(a))]TJ /F3 11.955 Tf 23.11 8.09 Td[(sin2(a) p ()]TJ /F5 11.955 Tf 11.96 0 Td[(kx)+1 20(kx)+O(1).Inparticularforxedkx2(0,]anda6=1 2,hcanbechosensufcientlylargeforthelastlimittobenegativeandthereforebytheIntermediateValueTheoremawavenumberk)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(suchthat)]TJ /F3 11.955 Tf 7.08 -4.34 Td[((k)]TJ /F3 11.955 Tf 7.09 -4.34 Td[()=0existsontheinterval(0,kx).Conversely,ifhisxed,then limkx!)]TJ /F3 11.955 Tf 14.55 5.81 Td[(limk!kx)]TJ /F3 11.955 Tf 8.25 5.81 Td[()]TJ /F3 11.955 Tf 10.41 -4.94 Td[(=8>><>>:ifa6=0)]TJ /F3 11.955 Tf 9.29 0 Td[(4h+1 20()+O(1)ifa=0(3) Thusif0
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3.2.2BoundStatesintheRadiationContinuumI:OneOpenDiffractionChannel Whenonlythe0th-orderdiffractionchannelisopen,kx2[0,)andkx>><>>>:sin2(2hkz) k2z+2s=+)]TJ /F3 11.955 Tf -125.6 -39.01 Td[(+sin2(hkz)+)]TJ /F3 11.955 Tf 9.08 -4.34 Td[(cos2(hkz)=0(3) Therstoftheseequationsimpliesthat+)]TJ /F2 11.955 Tf 11.64 -4.34 Td[(0.Ifthisinequalityweretobestrict,thenthesecondequationwouldnothaveheld,and,therefore,+)]TJ /F3 11.955 Tf 10.53 -4.34 Td[(=0.Thus,therstequationimpliesthats=0andsin(2hkz)=0.Inturn,thelatterequationimpliesthateithercos(hkz)=0orsin(hkz)=0,and,therefore,thesystemofEqs.( 3 )splitsintotwosystems,namely, 8>>>>>><>>>>>>:s(a,h,k,kx)=0cos(hkz)=0+(a,h,k,kx)=08>>>>>><>>>>>>:s(a,h,k,kx)=0sin(hkz)=0)]TJ /F3 11.955 Tf 7.08 -4.33 Td[((a,h,k,kx)=0(3) Tosolvetherstequationofeachsystem,theseriesforsisrewrittenas, s=)]TJ /F14 7.97 Tf 16.08 14.94 Td[(1Xm=1cmsin(2am),cm=e)]TJ /F6 7.97 Tf 6.58 0 Td[(2hqz,)]TJ /F9 5.978 Tf 5.76 0 Td[(m qz,)]TJ /F8 7.97 Tf 6.58 0 Td[(m)]TJ /F5 11.955 Tf 13.15 8.08 Td[(e)]TJ /F6 7.97 Tf 6.58 0 Td[(2hqz,m qz,m(3) Recallthatqz,m=p (kx+2m)2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2.Inparticular,ifkx=0thencm=0,8m=1,2,3...andtheequations=0holdstrivially.Similarly,ifa=0ora=1 2,thentheequation 62

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holdstriviallyassin(2am)=0,8m.Itturnsoutthatthesearetheonlypossiblerootsofsifh>ln2 40.055.Thisconclusionstemsfromthefollowingfactorizationofs, s(a,h,k,kx)=)]TJ /F3 11.955 Tf 11.29 0 Td[(sin(2a)1Xm=1" cm)]TJ /F3 11.955 Tf 11.96 0 Td[(2cm+1+21Xn=1(cm+2n)]TJ /F5 11.955 Tf 11.96 0 Td[(cm+2n+1)!sin2(am) sin2(a)#(3) togetherwiththefactsthatforkx6=0, cm>0and1supm=1cm+1 cm=e)]TJ /F6 7.97 Tf 6.59 0 Td[(4h(3) ThesestatementsareestablishedinAppendix A.3 .Whenh>ln2 4,thencm+1<1 2cm,8mandthereforeeachofthesummandsintheseriesofEq.( 3 )isnonnegative.Sincethersttermofthesaidseriesdoesnotvanish,itfollowsthattheseriesdoesnotvanish.Consequently,ifkx6=0thens=0ifandonlyifsin(2a)=0,i.e.,a=0ora=1 2. Itisremarkablethatundertherestrictionh>ln2 4nosolutiontoEqs.( 3 )islost.Indeed,sincesin(2hkz)=0ataboundstate,itfollowsthath=n 2kz,kz=p k2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x forsomepositiveintegern.Intherangesconsidered,kz<2andthereforeh>n 41 4>ln2 4.Thusanecessaryconditionfortheexistenceofboundstatesisthateitherkx=0whilea2[0,1 2]isarbitraryoraiseither0or1 2whilekx2[0,)isarbitrary.Insetnotation, (a,kx)2L=[0,1 2]f0g[f0,1 2g[0,)(3) ThesetLisrepresentedbythesecondlevelofFig. 3-2 (a). 63

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LetusturntosolvingthelasttwoequationsineachofthesystemsinEqs.( 3 ).Forthispurpose,thefunctionnisdenedforeachpositiveintegernby, n(k,kx,a)=8>>><>>>:+(a,n 2kz,k,kx)ifnisodd)]TJ /F3 11.955 Tf 7.09 -4.93 Td[((a,n 2kz,k,kx)ifniseven=1 2 1 0(k)+1+2ln(2R)!+Xm6=0 1 2(jmj+1))]TJ /F3 11.955 Tf 13.15 8.08 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2am)e)]TJ /F8 7.97 Tf 6.59 0 Td[(nqz,mk)]TJ /F16 5.978 Tf 5.75 0 Td[(1z qz,m!(3) wherek2(kx,2)]TJ /F5 11.955 Tf 12.29 0 Td[(kx).ThenthesystemsofEqs.( 3 )splitintothecountablesetofsystems 8>><>>:h=n 2p k2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2xn(k,kx,a)=0n=1,2,3...(3) wherethesystemscorrespondingtooddnarisefromtherstofsystems( 3 )andthosecorrespondingtoevennresultfromthesecondsystem.Ineachofthesystems( 3 ),thesecondequationdeterminesthewavenumbersatwhichboundstatesoccur.Inturn,bytherstequation,thesewavenumbersdeterminethedistanceshthatallowfortheboundstatestoexist.Inthenextparagraphitisshownthat,forxed(a,kx)2L,eachsystemcanadmitatmostonesolution.Hencethesetofdistanceshallowingtheexistenceofboundstatesisdiscrete. Let(a,kx)2Lbexed.ItisshowninAppendix A.3 thatthefunctionk7!nismonotonedecreasingonitsdomain(kx,2)]TJ /F5 11.955 Tf 11.21 0 Td[(kx)andthereforeadmitsatmostonerootinthesaiddomain.Moreover,therootonlyexistsifthelimitsofk7!natkxand2)]TJ /F5 11.955 Tf 12.21 0 Td[(kxareofoppositesign.Specically,thelimitatkxmustbepositivewhilethelimitat2)]TJ /F5 11.955 Tf 12.01 0 Td[(kx 64

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mustbenegative.Therstlimitis,limk!k+xn=8>>>>>>>><>>>>>>>>:1ifkx=01 2 1 0(kx)+1+2ln(2R)!+Xm6=01 2(jmj+1))]TJ /F3 11.955 Tf 56.04 8.09 Td[(1 p 42m2+4mkxifkx6=0 Therequirementthatthislimitbepositivewhenkx6=0putsarestrictiononthevaluesofRand"cthatallowfortheexistenceofboundstates.However,thisisatoocomplicatedconditiontoanalyze.Aweaker,buteasiertoanalyze,conditionisobtainedbyrstrewritingthepositivityconditionas, 2 R2("c)]TJ /F3 11.955 Tf 11.95 0 Td[(1)k2x>1Xm=11 p 42m2)]TJ /F3 11.955 Tf 11.95 0 Td[(4mkx+1 p 42m2+4mkx)]TJ /F3 11.955 Tf 18.53 8.08 Td[(1 m+1 1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(ln(2R)(3) Therearrangementoftheseriesismadetoensurethatallthesummandsintheseriesarenonnegative;theyvanishatkx=0.Also,sincethecylindersarethin,itmaybeassumedthatR

0.ThereforeallthesummandsinEq.( 3 )arenonnegativeandhencethelefthandsidemustbelargerthaneachofthesummandsontherightindividually.ToestimatethethresholdvalueofRp "c)]TJ /F3 11.955 Tf 11.96 0 Td[(1belowwhichboundstatesmayexist,therstterm(m=1)isretainedinthesum( 3 ).Thistermisthenwrittenintheformk2x()]TJ /F5 11.955 Tf 12.68 0 Td[(kx))]TJ /F6 7.97 Tf 6.59 0 Td[(1=2g(kx)toisolateitsbranchpointatkx=anditsrootatkx=0whichisofmultiplicity2.Theminimizationofg(kx)on[0,]producesanestimate: Rp "c)]TJ /F3 11.955 Tf 11.95 0 Td[(1
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toobserveInparticularwhenkxiscloseto,thequantityRp "c)]TJ /F3 11.955 Tf 11.96 0 Td[(1mustbesmallenoughinorderforboundstatestoexistatall.Figure 3-2 (b)showstheregionsinwhichEq.( 3 )isvalid. Asspeciedabove,forboundstatestoexistitisrequiredthatthelimitofnat2)]TJ /F5 11.955 Tf 11.96 0 Td[(kxbenegative.Thislimitis,limk!(2)]TJ /F8 7.97 Tf 6.59 0 Td[(kx))]TJ /F3 11.955 Tf 8.25 5.81 Td[(n=8>>><>>>:ifnisoddanda6=1 2ornisevenanda6=0)]TJ /F5 11.955 Tf 34.45 8.09 Td[(n p ()]TJ /F5 11.955 Tf 11.95 0 Td[(kx)+1 20(2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)+O(1)otherwise Inparticular,thelimitisnegativeexceptpossiblywhennisoddanda=1 2ornisevenanda=0.Eveninthelattercaseshowever,thenegativityconditionmaybeensuredbytakingkxsufcientlyclosetosothatk)]TJ /F6 7.97 Tf 6.58 0 Td[(1zislargeorbychoosingnsufcientlylarge.Thus,iftheparametersRand"cofthescatteringcylindersverifycondition( 3 ),boundstatesinthecontinuumIdoexist.Tobeprecise,givenapositiveintegern;then 8(a,kx)2(0,1 2)f0g[f0g[0,)thereexistsaboundstateatthewavenumberk2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1(a,kx)2(kx,2)]TJ /F5 11.955 Tf 12.94 0 Td[(kx)andatthedistance2h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1(a,kx)=(2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1) p k22n)]TJ /F16 5.978 Tf 5.76 0 Td[(1)]TJ /F8 7.97 Tf 6.59 0 Td[(k2xbetweenthetwoarraysofcylinders.Whena=1 2,thenthewavenumberk2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1existsforsufcientlylargenorforkxsufcientlycloseto. 8(a,kx)2(0,1 2)f0g[f1 2g(0,)thereexistsaboundstateatthewavenumberk2n(a,kx)2(kx,2)]TJ /F5 11.955 Tf 12.45 0 Td[(kx)andatthedistance2h2n(a,kx)=2n p k22n)]TJ /F8 7.97 Tf 6.59 0 Td[(k2xbetweenthetwoarraysofcylinders.Whena=0,thenthewavenumberk2nexistsforsufcientlylargenorforkxsufcientlycloseto. Inthelimitofthethincylindersconsidered,approximatevaluescanbeinferredforthewavenumberskn,n=1,2,3..,byonlykeepingtheleadingtermsintheequationsn(k)=0.ThisisdetailedinAppendix A.4 .Forinstanceifnisoddanda=0ornisevenanda=1 2,thenthewavenumberkn(kx)andthedistancehn(kx)areapproximated 66

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intheleadingorderof0by, kn(kx)2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(8220(2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx) 2)]TJ /F5 11.955 Tf 11.95 0 Td[(kxhn(kx)n 4p ()]TJ /F5 11.955 Tf 11.96 0 Td[(kx)1+220(2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx) )]TJ /F5 11.955 Tf 11.96 0 Td[(kx(3)Finally,fromEqs.( 3 )and( 3 )theexplicitformtheelectriceldfortheboundstatesfEng1n=1isobtainedatthewavenumbersfkng1n=1andthedistancesf2hng1n=1betweenthearraysofcylinders.TheeigenfunctionofboundstatesinthecontinuumcanonlybedetermineduptoamultiplicativeconstantwhichischosentobethevalueoftheelectriceldEn()]TJ /F5 11.955 Tf 9.3 0 Td[(hne3)onthecylinderat(0,)]TJ /F5 11.955 Tf 9.29 0 Td[(hn).Intermsofthisvalue,theelectriceldonthecylinderat(a,hn)isthen,En(ae1+hne3)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+1eiakxEn()]TJ /F5 11.955 Tf 9.3 0 Td[(hne3) andeverywhereoffthescatteringcylindersitis: En(r)=2i0(kn)En()]TJ /F5 11.955 Tf 9.29 0 Td[(hne3)Xmeix(kx+2m) knz,meijz+hnjknz,m+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+1ei(jz)]TJ /F8 7.97 Tf 6.58 0 Td[(hnjknz,m)]TJ /F6 7.97 Tf 6.59 0 Td[(2am)(3) whereknz,m=p k2n)]TJ /F3 11.955 Tf 11.96 0 Td[((kx+2m)2.Itcanbeveriedeasilythatoutsidethescatteringregion,i.e.,jzj>hn,thereisnocontributiontotheeldEnfromthe0)]TJ /F1 11.955 Tf 9.29 0 Td[(orderdiffractionchannel.Withthischannelbeingtheonlyopenchannel,itfollowsthatEndecaysexponentiallyintheasymptoticregionjzj!1andthereforeitissquareintegrableonS1Rasrequired.Figure 3-3 showsexamplesofplotsoftheabsolutevaluesoftheeldsEn. 3.2.3Application:ZeroWidthResonancesandNearFieldAmplication ThestudyofSection 3.2 showsthat,giventwoparallelarraysofsubwavelengthdielectriccylindersofradiusRanddielectricconstant"c,therearepoints(a,kx)forwhichaboundstateexistsifthedistancebetweenthearraysattainsaspecicvalue2hb;thisvaluealsodeterminesthewavenumberkboftheboundstate.Throughoutthefollowingdiscussion,thepair(a,kx)isxed. 67

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Figure3-3. ThemodesEnasdenedinEq.( 3 )fora=0,R=0.1,"c=1.5,andkx=0.ThepanelsshowEnasafunctionofx(verticalaxis)andz(horizontalaxis).ThecolorshowstheabsolutevalueofEnasindicatedontheleftinset.Thepositionsofcylindersareindicatedbyasolidblackcurve.Thevaluesofhnareshownbeloweachpanel.Topleft:E1,thesymmetricmodeforthesmallesth=h1.Topright:E2,theskew-symmetricmodeath=h2.Bottomleft:E3,thesecondsymmetricmodeath=h3.Bottomright:E4,thesecondskew-symmetricmodeath=h4. Considerthescatteringproblemforthedoublearraywhenh=hb.Iftheincidentwavehasthewavenumberkb,thenthesolutiontoEq.( 3 )isnotuniqueasanysolutionofthehomogeneousequationcanalwaysbeadded.Thelatteraretheboundstates.Ofcourse,thisambiguityisrelatedtothefactthattheincidentradiationcannotexcitetheboundstate(whichisawave-guidingmodepropagatingalongthearray),and,hence,anadditionalconditionmustbeimposedthattheboundstateisnotpresentinthesystemattheverybeginningifonewishestohaveauniquesolution.Ontheotherhand,thepresenceofaboundstatehasnoeffectwhatsoeveronthescatteringamplitudesastheyaredenedinthefareldzone(z!)towhichtheboundstategivesnocontributionanyway.Thisdegeneracydisappearsassoonas(h,k)6=(hb,kb).Inthelattercase,thesolutionE!toMaxwell'sequationsisuniquelydeterminedbyEq.( 3 ).Suchirregularitysuggeststhat,asafunctionofhandk,theeldE!isnotanalyticinthevicinityofthepoints(hb,kb).Thisisindeedthecase.Itisshowninwhatfollowsshortlythatthevaluesofthereecteduxaswellasthoseoftheeldsinsidethecylindersnearthepoints(hb,kb)dependonthepathalongwhichthesepointsareapproachedintheh,k)]TJ /F1 11.955 Tf 9.3 0 Td[(plane.Fromamathematicalpointview,thislackofanalyticityisexplainedbythe 68

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presenceofsimplepolesatthewavenumberskbintheeldE!whenconsideredasafunctionofkforxedh=hb.Theobjectivehereistoexploittheexistenceofthesepolestoshowthattheevanescenteldinthescatteringproblemisampliedascomparedtotheamplitudeoftheincidenteldwhen(h,k)iscloseto(hb,kb)insomeregionsofthearray,inparticular,onthecylinders.Theeffectcanthereforebeusedtoamplifyopticalnon-lineareffectsinthestructureinacontrollableway.Thisisillustratedwithanexampleofoneopenchannelforanarraywithouttheshift,i.e.,a=0. Supposethataplanewaveofwavenumberk2(kx,2)]TJ /F5 11.955 Tf 12.57 0 Td[(kx)andunitamplitudeimpingesthedoublearray.Inthiscase,thespecularreectioncoefcientwhichistheratioofthereecteduxtotheincidentuxatthespatialinnityis,R=jR0j2 whereR0isthereectioncoefcientoftheonlyopendiffractionchannel,namely,the0orderchannelasgiveninEq.( 3b ).If(h,k)isnotoneofthepoints(hb,kb),thenthereectioncoefcientR0andtheeldsinsidethecylindersare, R0=)]TJ /F3 11.955 Tf 37.05 8.09 Td[(cos2(hkz) cos2(hkz)+1 2ikz++sin2(hkz) sin2(hkz)+1 2ikz)]TJ ET BT /F1 11.955 Tf 426.8 -382.53 Td[((3a)E!(he3)=ikz 20(k) cos(hkz) cos2(hkz)+1 2ikz+isin(hkz) sin2(hkz)+1 2ikz)]TJ /F13 11.955 Tf 8.28 26.05 Td[(! (3b) forthefunctionsofEq.( 3 )(Appendix A.2 ).Thedenominatorsinbothexpressionsarefactorsofthedeterminant(0,h,kx,k)inEq.( 3 )(Herea=0).Inparticular,thepoints(hb,kb)arerootsofthedenominatorstothespecularcoefcientandtheeldsinsidethecylinders. Fortheillustrationpurpose,thebehaviorofthespecularcoefcientandtheeldsinsidethecylindersarestudiedas(h,k)approachesacriticalpoint(hb,kb)=(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)forsomepositiveintegernintheh,k)]TJ /F1 11.955 Tf 9.29 0 Td[(plane.AsdescribedinSubsection 3.2.2 ; 69

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atthepoint(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)thefollowingsystemholds,8>><>>:cos(hkz)=0+(h,k)=0 Intherest,thecurvescos(hkz)=0and+(h,k)=0willbedenotedbyCcandC+respectivelyandtheirintersectionpoints,i.e.,thepoints(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1),willbedenotedbyP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Therstobservationisthatas(h,k)approachesP2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,thencos(hkz)!0and+(h,k)!0independentlyasthecurvesCcandC+intersectatanonzeroangleatP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thismaybeestablishedthroughthelinearizationsofthefunctions(h,k)7!cos(hkz)and(h,k)7!kz+(h,k)at(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1).Ifh=h)]TJ /F5 11.955 Tf 12.03 0 Td[(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1andk=k)]TJ /F5 11.955 Tf 12.03 0 Td[(k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,theninthevicinityofP2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,cos(hkz)(h,k)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)nh2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 kz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1k+kz,2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1h1 2kz+(h,k)(h,k)=1 2kz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1@k+(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)k+@h+(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)h wherekz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1=p k22n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(k2x.Thefunctionsandarethenlinearlyindependentif,h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 kz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1@h+(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F5 11.955 Tf 11.95 0 Td[(kz,2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1@k+(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)6=0 Thatthisconditionindeedholdscanbeprovedbyexaminingthefunction2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1(k)=+(2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1) 2kz,kintroducedinEq.( 3 ).InAppendix A.3 itisshownthat@k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1(k)<0forallk2(kx,2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx).Consequently,@k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1(k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1 k2z,2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1@h+(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+@k+(h2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)<0 Thisestablishesthelinearindependenceofand.Thus,as(h,k)!(0,0),thereshouldbe!0and!0independently.InthevicinityofthecriticalpointP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1the 70

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principalpartsofR0andE!(he3)arethen, R0(h,k)1 1+ikz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 7.09 -3.45 Td[((h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+2(h,k) 2(h,k)+i(h,k)(3a) E!(he3)ikz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 20(k2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)i()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+1 1+ikz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 7.09 -3.46 Td[((h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+(h,k) 2(h,k)+i(h,k)(3b) Therstsummandsineachoftheseequationsareconstantandobeytheestimate,1 1+ikz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 7.09 -3.45 Td[((h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1,k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0(k2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1) kz,2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 ThesecondsummandsinEqs.( 3 )accountforthelackofanalyticityofthespecularcoefcientRandtheeldsE!(he3)inthevicinityofthecriticalpointP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Inparticular,since0alongthetangentlinetoC+atP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,itfollowsthatalongthistangentlineandhencealongthecurveC+,E!(he3)()]TJ /F3 11.955 Tf 9.29 0 Td[(1)ni 20(k22n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)1)]TJ /F8 7.97 Tf 13.15 5.72 Td[(h2n)]TJ /F16 5.978 Tf 5.75 0 Td[(1k2n)]TJ /F16 5.978 Tf 5.76 0 Td[(1 k2z,2n)]TJ /F16 5.978 Tf 5.75 0 Td[(1@h+ @k)]TJ /F13 11.955 Tf 7.45 15.12 Td[(h,h=h)]TJ /F5 11.955 Tf 11.96 0 Td[(h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1!0 ThustheelectriceldinsidethecylindersdivergesatthepointsP2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1. Forthespecularcoefcient,thesignicanceofthepointsP2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1isthattheyarepositionsofresonanceswiththevanishingwidth,which,inturn,demonstratesthattheboundstateintheradiationcontinuumareinterpretedasresonanceswiththevanishingwidthintheformalscatteringtheory.Indeed,ifh6=h2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1isxedandkr(h)isawavenumbersuchthat+(h,kr(h))=0,thenkrisaresonantwavenumberforthespecularcoefcientR.Forknearkr,theBreit-WignertheoryassertsthatR0willhavetheform,R0i)]TJ ET q .478 w 219.42 -546.66 m 276.37 -546.66 l S Q BT /F5 11.955 Tf 219.42 -557.85 Td[(k)]TJ /F5 11.955 Tf 11.96 0 Td[(kr+i)]TJ ET BT /F1 11.955 Tf 0 -580.04 Td[(where2)]TJ /F1 11.955 Tf 16.37 0 Td[(istheresonancewidthoftheLorentzianproleofR=jR0j2.Thehalf-width)]TJ /F1 11.955 Tf -452.61 -23.9 Td[(maybefoundbyexpandingthefunctionk7!+(h,k)inaTaylorseriesattheresonant 71

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wavenumberkr,itis,)-277(=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(2cos2(hkz) kz@k+k=kr sothatatthepointsP2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1thiswidthvanishes.Figure 3-4 showsplotsofthespecularcoefcientsandtheelectriceldalongthecurveC+. Figure3-4. Thespecularcoefcientandtheelectriceldonthecylindersnearaboundstateinthecontinuumfora=0,R=0.1,and"c=1.5.Panel(a):Showsthespecularreectioncoefcientasafunctionofhandthewavenumberk.Itisplottedforkx= 5nearthethresholdk)]TJ /F6 7.97 Tf 6.59 0 Td[(1=9 5.Thespecularcoefcientisveryclosetoitsmaximumalongthecurves(h,k)=0thatdeterminetheresonancepositions.Thelower(upper)brightregionroughlycorrespondstothecurve+(h,k)=0()]TJ /F3 11.955 Tf 7.08 -4.34 Td[((h,k)=0).Theboundstatescorrespondtothepointsseparatingtwoconsecutivebrightstrips.Attheboundstatesthespecularcoefcientlacksanalyticityanditsvaluedependsstronglyonhowtheboundstateisapproachedinthe(h,k)-plane.Panel(b):ThespecularcoefcientR(k(h),h)(dashedbluecurve)andtheabsolutevalueoftheelectriceldE!(he3)(k(h),h)(solidredcurve)onthecylinderat(x,z)=(0,h)ofthedoublearraywherek=k(h)isimplicitlydenedby+(h,k)=0.Alongthiscurve,theelectricelddivergesneartheboundstates.Panel(c):SamelegendasforPanel(b)inthecaseoftwoopenchannelsandkx=.Inthiscasetoo,theeldsonthecylindersdivergeinthevicinityoftheboundstates. 3.3BoundStatesintheRadiationContinuumN,N2 Whenmorethanonediffractionchannelareopen,theboundstatesmaystillbeshowntoexist.Howevertheybecomerarerasthenumberofdiffractionchannels 72

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increases.Thephysicalreasonforthatissimple.Assuggestedintheintroduction,aboundstateisformedduetoadestructiveinterferenceofthedecayradiationoftwoquasi-stationaryelectromagneticmodeslocalizedinthevicinityofeacharray.Ifmorethanonedecaychannelareopenforthesemodes,thenthedestructiveinterferencemustoccurinallthedecaychannelsinorderforaboundstatetoform,whichputsmorerestrictionsonthesystemparameters.Indeed,consider,forinstance,possiblechoicesoftheparametersaandkxthatallowfortheexistenceofboundstates.Itwasshownthatboundstatesdoexistbelowtheradiationcontinuumforallpairs(a,kx),i.e.,norestrictionsatall.Whenonediffractionchannelisopen,thenboundstatescanformwhenthepairs(a,kx)lieonthesetLdenedinEq.( 3 ).Whentwodiffractionchannelsareopen,thentheboundstatesareshowntoonlyoccuriftheshiftais0or1 2andforaspecicdensesetofvaluesofkx.AsonegoesontohigherlevelsofthespectralcylinderinFig. 3-1 (d),thevaluesofkxatwhichboundstatesmayexistbecomemoresparseandaredeterminedbysolutionsofasystemofdiophantineequations.TheconditionsunderwhichboundstatesexistareformulatedrstforthestudiedsysteminthecontinuumN2. Theaboveassertionsfollowfromtheobservationsthat,ifthediffractionchannels0,)]TJ /F3 11.955 Tf 9.3 0 Td[(1,1,...areopen,thenataboundstate,theparameterskx,k,h,a,R,and"csatisfytherelations, 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:(a,h,kx,k)=0sin(2hkz)=0sin(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=0sin(2hkz,1)=0...(3) where(a,h,kx,k)isthedeterminantofEq.( 3 ).TheadditionalequationsarearesultofthesquareintegrabilityoftheboundstatesonS1R.Indeed,ifasolutionE!to 73

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Eq.( 3 )istobesquareintegrableonS1Rthen,thefunction:z7!Z10jE!(x,z)j2dx isintegrableinzoverR.Now,(z)!4220(k)8>>>><>>>>:Xmop1 k2z,meihkz,mE!()]TJ /F5 11.955 Tf 9.29 0 Td[(he3)+e)]TJ /F8 7.97 Tf 6.58 0 Td[(i(a(kx+2m)+hkz,m)E!(ae1+he3)2,z!1Xmop1 k2z,me)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkz,mE!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)+e)]TJ /F8 7.97 Tf 6.59 0 Td[(i(a(kx+2m))]TJ /F8 7.97 Tf 6.59 0 Td[(hkz,m)E!(ae1+he3)2,z! wherethesuperscriptopinmopindicatesthatthesummationsaretobecarriedoverallm'sthatcorrespondtoopendiffractionchannels.Inparticular,forsquareintegrabilitytohold,eachsummandintheequationsabovemustbezero.Itfollowsthatforeachopenchannelm0thefollowingsystemholds, 8>><>>:eihkz,m0E!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)+e)]TJ /F8 7.97 Tf 6.59 0 Td[(i(a(kx+2m0)+hkz,m0)E!(ae1+he3)=0e)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkz,m0E!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)+e)]TJ /F8 7.97 Tf 6.58 0 Td[(i(a(kx+2m0))]TJ /F8 7.97 Tf 6.58 0 Td[(hkz,m0)E!(ae1+he3)=0(3) Fornontrivialsolutions,thedeterminantoftheabovesystemmustbezero.Thus,sin(2hkz,m0)=0foreachopenchannelm0.Moreover,byconsideringtheratiooftheeldE!(ae1+he3)totheeldE!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)foralltheopenchannelsitisdeducedfromsystem( 3 )that, cos(2hkz)=cos(2hkz,)]TJ /F6 7.97 Tf 6.58 0 Td[(1)e)]TJ /F6 7.97 Tf 6.59 0 Td[(2ia=cos(2hkz,1)e2ia=...(3) Inparticular,sinceforeachopenchannelm0wehavesin(2hkz,m0)=0,thencos(2hkz,m0)=1.Thuse2ia=1,hencea=0ora=1 2. Thersttwoequationsofsystem( 3 )determinekandhasfunctionsofkxwhilethelastequationsdeterminethevaluesofkx.Thusaspointedoutabove,thevaluesofkxatwhichboundstatesmayexistbecomemoresparseasthenumberofopendiffractionchannelsincreases. 74

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Inthecaseoftwoopendiffractionchannels,itisremarkablethatthesevaluesofkxaredensein[0,].AproofofthisstatementisgiveninSection 3.3.1 .Itisshowntherethatthevaluesofkxallowingfortheexistenceofboundstatesoccurinadoublesequencekn,lxwheren,larepositiveintegers.IntheleadingorderofR2("c)]TJ /F3 11.955 Tf 12.63 0 Td[(1),theelementsofthesubsequencek2n+1,lxareshowntohavetheform: k2n+1,lx 2r2)]TJ /F3 11.955 Tf 11.96 0 Td[(1+5(r2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(4r2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)4 4(2r2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)5R4("c)]TJ /F3 11.955 Tf 11.95 0 Td[(1)2,r=l 2n+1(3) Theelementsofthesubsequencek2n,lxarehardertoderiveduetoamoreintricatedependenceonthesystemparameters.Forthesubsequencek2n+1,lx,thecorrespondingwavenumbersanddistancesbetweenthearraysatwhichtheboundstatesoccurareprovedtobeobtainedbysubstitutingk2n+1,lxintothefollowingexpressions: k2n+1(kx)2+kx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(8220(2+kx) 2+kx,h2n+1(kx)(2n+1) 2p 2kx1+20(2+kx) kx(3) AsinthecaseofboundstatesinthecontinuumI,boundstatesinthecontinuumIIareshowntoonlyoccurunderthefollowingrestrictionontheradiusanddielectricconstantofthescatteringcylinders: Rp "c)]TJ /F3 11.955 Tf 11.95 0 Td[(1
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and, kx=n21)]TJ /F5 11.955 Tf 11.95 0 Td[(n22 2n20)]TJ /F5 11.955 Tf 11.96 0 Td[(n21)]TJ /F5 11.955 Tf 11.95 0 Td[(n22,h=1 4p 2q 2n20)]TJ /F5 11.955 Tf 11.96 0 Td[(n21)]TJ /F5 11.955 Tf 11.95 0 Td[(n22,k=p (n21+n22)]TJ /F3 11.955 Tf 11.96 0 Td[(4n20)2+4n21n22 2n20)]TJ /F5 11.955 Tf 11.96 0 Td[(n21)]TJ /F5 11.955 Tf 11.96 0 Td[(n22(3) Whenonlythreechannelsareopen,theintegersn0,n1andn2areonlyrequiredtobeintheordern0>n1n2.Whenfourchannelsareopen,theadditionalequationsin(2hkz,)]TJ /F6 7.97 Tf 6.58 0 Td[(2)=0insystem( 3 )requiresthattheaforementionedintegerssatisfythesystem,8>><>>:3n21+n22=3n20+n23n0n1>n2n3 Asmorediffractionchannelsbecomeavailable,therearemoreandmoreconstraintsontheintegersni,i=0,1,2...Providedintegerssatisfyingthoseconstraintscanbefound,boundstateswillthenbeformedfordoublearraysforwhichtheradiusRandthedielectricconstant"csatisfy(a,h,kx,k)=0withh,kandkxgivenbyEqs.( 3 )anda2f0,1 2g.FromEqs.( 3 )and( 3 ),itfollowsthatRand"cmustbeoncurves,2 k2R2("c)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+ln(2R)=C(n0,n1,...) wherekisgiveninEq.( 3 )andCissomeconstantthatdependsontheintegersni,i=0,1,2,... 3.3.1BoundStatesintheRadiationContinuumII:TwoOpenDiffractionChan-nels Supposethatboththe0thand)]TJ /F3 11.955 Tf 9.3 0 Td[(1stdiffractionchannelsareopen,i.e.,kx2(0,]and2)]TJ /F5 11.955 Tf 12.24 0 Td[(kx
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equations: 8>>>>>>>>><>>>>>>>>>:2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(cos(2a)cos(2hkz)cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)) kzkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1=+)]TJ /F3 11.955 Tf -245.41 -36.74 Td[(+ 1)]TJ /F3 11.955 Tf 11.95 0 Td[(cos(2hkz) kz+1)]TJ /F3 11.955 Tf 11.96 0 Td[(cos(2a)cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1) kz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1!+)]TJ /F13 11.955 Tf 7.09 15.51 Td[( 1+cos(2hkz) kz+1+cos(2a)cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1) kz,)]TJ /F6 7.97 Tf 6.58 0 Td[(1!=0(3) where=cforthefunctionsandcofEqs.( 3 ).Also,recallthataisnecessarily0or1 2asderivedfromEqs.( 3 ). Therstequationofthesystemimpliesthat+)]TJ /F2 11.955 Tf 10.78 -4.34 Td[(0.Ifthisinequalityweretobestrict,thesecondequationwouldnothaveheld,and,therefore,+)]TJ /F3 11.955 Tf 10.96 -4.34 Td[(=0.Thuseither+=0or)]TJ /F3 11.955 Tf 11.55 -4.34 Td[(=0.Notethatthefunctions+and)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(cannotvanishsimultaneously.Thiscanbeveriedbyobservingthat,+2+)]TJ /F6 7.97 Tf 7.08 1.66 Td[(2=2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(2+2c andtherefore,ifthefunctions+and)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(weretovanishsimultaneously;itwouldfollowthatc=0.But,c=8>>>>>>>><>>>>>>>>:Xm6=0,)]TJ /F6 7.97 Tf 6.59 0 Td[(1e)]TJ /F6 7.97 Tf 6.58 0 Td[(2hqz,m qz,m>0ifa=0Xm6=0,)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F3 11.955 Tf 9.3 0 Td[(1)me)]TJ /F6 7.97 Tf 6.59 0 Td[(2hqz,m qz,m<0ifa=1 2andkx6=0ifa=1 2andkx= Thus(a,kx)=(1 2,).Butthenkz=kz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1andtherstequationinsystem( 3 )reads,1+cos2(2hkz)=0 Thisisimpossibleandthereforeataboundstate+and)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(donotvanishsimultaneously.Inparticular,therearenoboundstatescorrespondingtothepair(a,kx)=(1 2,).ThisisthereasonthispointwasremovedfromthethirdlevelofFig. 3-2 (a). 77

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Fortheremainderofthediscussionitisassumedthat(a,kx)6=(1 2,).System( 3 )thensplitsintotwosystems,namely, 8>>>>>><>>>>>>:cos(2hkz)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1cos(2a)cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1+=08>>>>>><>>>>>>:cos(2hkz)=1cos(2a)cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=1)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(=0(3) Thustoeachvalueofacorrespondsapairofsystemswhosesolutions,ifany,giverisetoboundstatesinthecontinuum.Fora=0,thesesystemsare, (A):8>>>>>><>>>>>>:cos(2hkz)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1+=0(B):8>>>>>><>>>>>>:cos(2hkz)=1cos(2hkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=1)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(=0(3a) whilefora=1 2theyare, (C):8>>>>>><>>>>>>:cos(2hkz)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1cos(2hkz,)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=1+=0(D):8>>>>>><>>>>>>:cos(2hkz)=1cos(2hkz,)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(=0(3b) Theexistenceofsolutionstothelasttwoequationsineachsystemisprovedrst.Thentherstequationofeachsystemisaddedtoshowtheexistenceofboundstates. Foreachpositiveintegern,thefunctionnisdenedby, n(k,kx,a)=8>>><>>>:+(a,n 2kz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k,kx)ifnisoddanda=0ornisevenanda=1 2)]TJ /F3 11.955 Tf 7.08 -4.94 Td[((a,n 2kz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1,k,kx)ifnisevenanda=0ornisoddanda=1 2(3a) 78

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wherek2(2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx,2+kx).Theexplicitexpressionofnis, n(k,kx,a)=1 2 1 0(k)+3 2+2ln(2R)!+Xm6=0,)]TJ /F6 7.97 Tf 6.59 0 Td[(1 1 2(jmj+1))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+2a(m+1)e)]TJ /F8 7.97 Tf 6.59 0 Td[(nqz,mk)]TJ /F16 5.978 Tf 5.75 0 Td[(1z,)]TJ /F16 5.978 Tf 5.75 0 Td[(1 qz,m!(3b) Thenthesystemsformedbythelasttwoequationsofeachofsystems( 3 )splitintothecountablesetofsystems 8>><>>:h=n 2p k2)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2n(k,kx,a)=0n=1,2,3...(3) ItisshowninAppendix A.3 thatforeachpositiveintegern,thefunctionk7!nismonotonedecreasingonitsdomain(2)]TJ /F5 11.955 Tf 12.4 0 Td[(kx,2+kx).Itfollowsthatifsystem( 3 )hasasolution,thenthissolutionisunique.Moreover,suchasolutionwillonlyexistifandonlyifthelimitofk7!nat2)]TJ /F5 11.955 Tf 12.56 0 Td[(kxispositivewhilethelimitofnat2+kxisnegative.Therstlimitis, limk!(2)]TJ /F8 7.97 Tf 6.59 0 Td[(kx)+n=1 2 1 0(2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)+3 2+2ln(2R)!+Xm6=0,)]TJ /F6 7.97 Tf 6.59 0 Td[(1 1 2(jmj+1))]TJ /F3 11.955 Tf 81.92 8.09 Td[(1 p (2m+kx)2)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2!(3) AsinthecaseofboundstatesinthecontinuumI,therequirementforthislimittobepositiveputsarestrictiononthevaluesofRand"cthatallowfortheexistenceofboundstatesinthecontinuumII.AneasilyanalyzableconditionontheseparametersisobtainedbyfollowingthesameprocedureasinSection 3.2.2 .First,thepositivity 79

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conditionisrewrittenas, 1Xm=1 1 p (2m+kx)2)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2+1 p (2(m+1))]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2)]TJ /F3 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2)]TJ /F3 11.955 Tf 44.81 8.09 Td[(1 p m(m+1)!+1 s)]TJ /F3 11.955 Tf 13.15 8.09 Td[(3 4)]TJ /F3 11.955 Tf 11.96 0 Td[(ln(2R)<2 R2("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2(3) wheres=1Xm=1 1 p m(m+1))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 21 m+1+1 m+2!0.691 TherearrangementoftheseriesismadetoensurethatallthesummandsintheseriesofEq.( 3 )arenonnegative;theyvanishatkx=.Alsosincethecylindersarethin,itmaybeassumedthatR<1 2es)]TJ /F16 5.978 Tf 7.78 3.25 Td[(3 40.150sothats)]TJ /F6 7.97 Tf 13.89 4.71 Td[(3 4)]TJ /F3 11.955 Tf 12.7 0 Td[(ln(2R)>0.ThusallsummandsinEq.( 3 )arenonnegativeandhencethelefthandsidemustbelargerthaneachindividualsummandontherighthandside.ToestimatethethresholdvalueofRp "c)]TJ /F3 11.955 Tf 11.96 0 Td[(1belowwhichboundstatesmayexist,therstterm(m=1)isretainedinthesum( 3 ).Thistermisthenwrittenintheform()]TJ /F5 11.955 Tf 12.29 0 Td[(kx)p kxg(kx)toisolateitsbranchpointatkx=0anditssimplerootatkx=.Theminimizationofg(kx)on[0,]producestheestimate( 3 )whereCisexactly,C=25 4)]TJ /F16 5.978 Tf 7.78 3.26 Td[(3 4min0t1(2)]TJ /F5 11.955 Tf 11.95 0 Td[(t)2 p 3)]TJ /F5 11.955 Tf 11.95 0 Td[(tp 3)]TJ /F5 11.955 Tf 11.96 0 Td[(t 1+p t)]TJ 39.02 17.75 Td[(p t p 2+p 3)]TJ /F5 11.955 Tf 11.95 0 Td[(t)]TJ /F16 5.978 Tf 7.79 3.26 Td[(1 22.016 Asmentionedabove,inadditiontotherequirementthatthelimitat2)]TJ /F5 11.955 Tf 12.98 0 Td[(kxofk7!nbepositive,onemustalsorequirethatthelimitat2+kxbenegativeinorderforsystem( 3 )tohaveasolution.Thelatterlimitis,limk!(2+kx))]TJ /F3 11.955 Tf 8.24 5.81 Td[(n=8>>>>>>>><>>>>>>>>:ifnisodd)]TJ /F5 11.955 Tf 20.86 8.08 Td[(n p 2kx+O(1)ifnisevenanda=0)]TJ /F3 11.955 Tf 41.22 8.09 Td[(1 p 3()]TJ /F5 11.955 Tf 11.96 0 Td[(kx)+O(1)ifnisevenanda=1 2 80

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Inparticular,thelimitisnegativeifnisodd.Ifnisevenanda=0,thenegativityconditionmaybeensuredbychoosingkxsufcientlyclosetozeroorbychoosingnsufcientlylarge.Ifnisevenanda=1 2,thenkxmustbeveryclosetoforthelimittobenegative.ForparametersRand"csatisfyingcondition( 3 ),theconditionsofexistenceofthesolutions)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(kn(kx),hn(kx)tosystem( 3 )aresummarizedinTable 3-1 .Intheleadingorderof0(2+kx),approximatevaluesofk2n+1andh2n+1aregivenbyEq.( 3 )(Appendix A.4 ). Theapproximatevaluesforthewavenumbersk2n(kx)aremoredifculttondduetothedependenceoftheirexistenceonthephysicalparametersofthesystem. Table3-1. Existenceofsolutionstosystems( 3 ).Inparticular,systems(A)and(D)in( 3 )alwayshavesolutionswhereas(B)and(C)mightnot. a=0a=1 2 nodd(kn,hn)exists8kx2(0,](kn,hn)exists8kx2(0,)neven(kn,hn)existsfornlargeorkxsmall(kn,hn)existsonlyforkxverycloseto Sincetheconditionsofexistenceofsolutionstosystem( 3 )arenowestablished,theexistenceofsolutionstosystems( 3 )canbeinvestigated.Sofaronlythelasttwoequationsineachofthelattersystemshavebeenusedtodeterminethevalueskn(kx)andhn(kx)foreachkx2(0,]thataresusceptibletopermittheexistenceofboundstates.Itfollowsthattherstequationsineachofsystems( 3 )determinethevaluesofkxthatallowfortheexistenceofboundstates.Inthecomingparagraphsthesetofthesevaluesisshowntobediscreteanddensein[0,]. Letnbeapositiveintegerforwhichkn(kx)existsforallkx2(0,](kx2(0,)ifa=1 2).Considerthefunction'ndenedby,'n(kx)=2hn(kx)p k2n(kx))]TJ /F5 11.955 Tf 11.96 0 Td[(k2x=ns k2n(kx))]TJ /F5 11.955 Tf 11.95 0 Td[(k2x k2n(kx))]TJ /F3 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2,kx2(0,) wherethevaluekx=ispurposelyleftoutandwillbediscussedlater. Collectively,therstequationsofsystems( 3 )arecos('n(kx))=1andtherefore,theyhavesolutionsiftherangeof'ncanbeshowntocontainevenandodd 81

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integermultiplesof.Thatthisisindeedthecasefollowsfromthecontinuityof'nanditslimitsat0and.Theseare,limkx!0+'n(kx)=1andlimkx!)]TJ /F4 11.955 Tf 8.25 5.81 Td[('n(kx)=n sothattherangeof'ncontainstheinterval(n,1).Inparticular,foreachpositiveintegerl>n,thereexistsapointkn,lx2(0,)suchthat'n(kn,lx)=landthereforecos)]TJ /F4 11.955 Tf 5.48 -9.68 Td[('n(kn,lx)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)l.ThisestablishestheexistenceofboundstatesinthecontinuumII.Asclaimed,theyexistforadiscretesetofkxvaluesin(0,),namely,thepointskn,lx.Toeachofthesepointscorrespondsaspecicwavenumberkn,l=kn(kn,lx)andaspecicdistancehn,l=hn(kn,lx)atwhichaboundstateinthecontinuumexists.Notethatthepointskn,lxdependonR,"candtheshiftaasisillustratedforinstanceforthepointsk2n+1,lxinEq.( 3 )(Appendix A.4 ).Notealsothatbydenition,thevaluesk2n+1,2l+1xaresolutionstosystem(A)ofEqs.( 3 )whilethevaluesk2n+1,2lxaresolutionstosystem(B)ofthesamesetofsystemsandthatthetwosetsofpointsdonotoverlap. Asfarasthecasekx=isconcerned,itwasestablishedinthebeginningofthissectionthattherecanbenoboundstatesifa=1 2.However,theydoexistifa=0.Thisisbecausekz=kz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1atkx=sothattherstequationsinsystems( 3a )aresuperuousandhencetheexistenceofsolutionstosystems( 3 )alonesufcestoguaranteetheexistenceofboundstatesinthecontinuumII(Notethatsystems( 3b )becomeinconsistentasexpected).Thusifa=0,thelistofpointsfkn,lx,l>ngistobecompletedbyaddingtoitthepointkx=.Keepingwiththenotation,thispointiskn,nxforeachnsince'ncanbeextendedtobydening'n()=nsothatcos('n())=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n.Thusthepointkx=isofinnitemultiplicityinthelistfkn,lx,lngandhenceisassociatedwithaninnitesetofboundstates.Indeedforeachpositiveintegern,sufcientlylargeifitiseven,thereexistsaboundstateatthewavenumberkn,n=kn()andatthedistance2hn,n=2hn()betweenthetwoarraysofcylinders.All 82

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theotherpointskn,lxchangewiththephysicalparametersofthecylindersandtheindicesn,landthereforenoneofthemiscertaintoberepeated. Itisremarkablethatthesetofpointskn,lxisdensein[0,]asissuggestedbyEq.( 3 ).Todemonstratethisfact,consideranarbitraryintervalI=(,)(0,).Itwillbeshownshortlythat, limn!1'2n+1())]TJ /F4 11.955 Tf 11.96 0 Td[('2n+1()=1(3) Therefore,howeversmalltheintervalImaybe,theinterval'2n+1(I)containspositiveintegermultiplesofforsufcientlylargen.Iflissuchamultiple,thenk2n+1,lx2I.Thusthepointsfk2n+1,lxgaredensein(0,]. Beforeestablishingthelimit( 3 ),recallthatkn=kn(kx)designatesthesolutiontotheequationn(kn(kx),kx,a)=0.Thersttaskistoshowthatthesequencefk2n+1(kx)g1n=1convergesforeachxedkx.Tothisend,let1bethefunctiondenedby, 1(k,kx)=1 20(k)+Xm6=0,)]TJ /F6 7.97 Tf 6.59 0 Td[(11 2(jmj+1))]TJ /F3 11.955 Tf 20.39 8.09 Td[(1 qz,m+1 3 4+ln(2R)(3) fork2(2)]TJ /F5 11.955 Tf 12.13 0 Td[(kx,2+kx).Forxed(a,kx),thefunctionk7!1isthepointwiselimitofthesequenceoffunctionsk7!n;itisalsocontinuousandmonotonedecreasingonitsdomain(2)]TJ /F5 11.955 Tf 12.12 0 Td[(kx,2+kx)asisshowninAppendix A.3 .Ask!(2+kx))]TJ /F1 11.955 Tf 7.08 -4.33 Td[(,1!whilethelimitof1at(2)]TJ /F5 11.955 Tf 12.34 0 Td[(kx)+isexactlylimit( 3 ).Inparticular,thelatterlimitispositiveasrequiredbycondition( 3 ).Itfollowsthatforeachkx2(0,],thereexistsauniquepointk1(kx)2(2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx,2+kx)suchthat1(k1(kx),kx)=0. 83

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Nowforxedkx, 1(k2n+1)=1(k2n+1))]TJ /F3 11.955 Tf 11.96 0 Td[(2n+1(k2n+1)=8>>>>><>>>>>:1Xm=1 e)]TJ /F6 7.97 Tf 6.59 0 Td[((2n+1)qz,mk)]TJ /F16 5.978 Tf 5.76 0 Td[(1z,)]TJ /F16 5.978 Tf 5.76 0 Td[(1 qz,m+e)]TJ /F6 7.97 Tf 6.59 0 Td[((2n+1)qz,)]TJ /F9 5.978 Tf 5.76 0 Td[(m)]TJ /F16 5.978 Tf 5.76 0 Td[(1k)]TJ /F16 5.978 Tf 5.76 0 Td[(1z,)]TJ /F16 5.978 Tf 5.76 0 Td[(1 qz,)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1!ifa=01Xm=1 e)]TJ /F6 7.97 Tf 6.59 0 Td[((2n+1)qz,mk)]TJ /F16 5.978 Tf 5.76 0 Td[(1z,)]TJ /F16 5.978 Tf 5.76 0 Td[(1 qz,m)]TJ /F5 11.955 Tf 13.15 8.09 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[((2n+1)qz,)]TJ /F9 5.978 Tf 5.76 0 Td[(m)]TJ /F16 5.978 Tf 5.76 0 Td[(1k)]TJ /F16 5.978 Tf 5.76 0 Td[(1z,)]TJ /F16 5.978 Tf 5.76 0 Td[(1 qz,)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1!()]TJ /F3 11.955 Tf 9.3 0 Td[(1)m+1ifa=1 2(3) whereitisunderstoodthatinthetwoseriesontheright,thetermsqz,mandkz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1aretobeevaluatedatk=k2n+1.Thesumoftherstseriesisobviouslypositive.Thesumofthesecondseriesisalsononnegativeasitisthesumofanalternatingserieswhosetermsdecreaseinabsolutevalue.Thus1(k2n+1)0. Ask7!1isadecreasingfunctionitfollowsthat, k2n+1(kx)k1(kx)<2+kx,8n=1,2,3...and8kx2(0,](3) Inparticular,qz,1=p (2+kx)2)]TJ /F5 11.955 Tf 11.96 0 Td[(k22n+1p (2+kx)2)]TJ /F5 11.955 Tf 11.95 0 Td[(k21>0andthereforeqz,1doesnotconvergeto0asn!1.Hence,1(k2n+1)=O e)]TJ /F6 7.97 Tf 6.59 0 Td[((2n+1)qz,1k)]TJ /F16 5.978 Tf 5.75 0 Td[(1z,)]TJ /F16 5.978 Tf 5.75 0 Td[(1 qz,1!)381()222()381(!n!10=1(k1) Sincethefunctionk7!1iscontinuousandbijectiveforeachxedkx2(0,],itfollowsthatk2n+1(kx)!k1(kx)asn!1.Notethatwithoutcondition( 3 ),qz,1couldconvergeto0causingthesequencef1(k2n+1)g1n=1todivergeasindicatedbyEqs.( 3 ).Thisisthereasonthesubsequencefk2ng1n=1hadtobeexcluded.Forthissubsequence,itcanbeshownthroughananalysissimilartotheabovethatk2n(kx)k1(kx),8n=1,2,3...andthereforemoreworkwouldbeneededtoshowthatnosubsequenceofthesequencefk2ng1n=1convergesto2+kx. 84

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Fromtheconvergenceofthesequencefk2n+1(kx)g1n=1itisdeducedthat, limn!11 (2n+1)'2n+1())]TJ /F4 11.955 Tf 9.3 0 Td[('2n+1()='1())]TJ /F4 11.955 Tf 9.3 0 Td[('1(),'1(kx)=s k21(kx))]TJ /F5 11.955 Tf 11.96 0 Td[(k2x k21(kx))]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2(3) InAppendix A.3 thefunction'1isshowntobestrictlydecreasingsothat'1())]TJ /F4 11.955 Tf -428.2 -23.91 Td[('1()>0.Thisestablisheslimit( 3 )andtherebythedensityofthepointsfk2n+1,lxgintheinterval(0,].Moreover,ifjAn(,)jisthecardinalityofthesetAn(,)=fl:kn,lx2(,)gthen,jA2n+1(,)j=(2n+1)'1())]TJ /F4 11.955 Tf 11.96 0 Td[('1()+o(n))347()222()222()223()222()222()348(!R2("c)]TJ /F6 7.97 Tf 6.59 0 Td[(1)!0(2n+1) p 2r 1+ )]TJ /F13 11.955 Tf 11.95 16.34 Td[(r 1+ +o(n) ThelastlimitfollowsfromthefactthatasR2("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)!0thenk1(kx)!2+kx. Lastly,fromEqs.( 3 )and( 3 ),theanalyticexpressionsoftheboundstatesEn,lcanbeobtainedforpositiveintegersnandlsuchthatlnifa=0andl>nifa=1 2.Asmentionedinpriorsections,eachofthesestatescanonlybedetermineduptoamultiplicativeconstantwhichischosentobethevalueEn,l()]TJ /F5 11.955 Tf 9.3 0 Td[(hn,le3)oftheelectriceldonthecylinderat(0,)]TJ /F5 11.955 Tf 9.3 0 Td[(hn,l).Intermsofthisvalue,theelectriceldonthecylinderat(a,hn,l)isthen,En,l(ae1+hn,le3)=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+2a+1eiakn,lxEn,l()]TJ /F5 11.955 Tf 9.29 0 Td[(hn,le3) andeverywhereoffthescatteringcylindersitis, En,l(r)=2i0(kn,l)En,l()]TJ /F5 11.955 Tf 9.3 0 Td[(hn,le3)Xmeix(kn,lx+2m) kn,lz,meijz+hn,ljkn,lz,m+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)n+2a(m+1)+1eijz)]TJ /F8 7.97 Tf 6.58 0 Td[(hn,ljkn,lz,m(3) wherekn,lz,m=q k2n,l)]TJ /F3 11.955 Tf 11.96 0 Td[((kn,lx+2m)2. 85

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CHAPTER4ARESONANTGENERATIONOFSECONDHARMONICSINDOUBLEARRAYSOFSUBWAVELENGTHDIELECTRICCYLINDERS Inthischapter,anonlinearelectromagneticscatteringproblemisstudiedfortwoparallelperiodicarraysofdielectriccylinderswithasecondordernonlinearsusceptibility.Forawiderangeofvaluesofthenonlinearsusceptibility,theconversionrateoftheincidentfundamentalharmonicintothesecondoneisshowntobeashighas40%atthedistancebetweenthearraysaslowasahalfoftheincidentradiationwavelength.Intheframeworkofresonantscatteringtheory,theeffectisattributedtotheexistenceofelectromagneticboundstatesintheradiationcontinuum. 4.1TheScatteringTheory ThesystemconsideredissketchedinFig. 4-1 (a).Itconsistsofaninnitedoublearrayofparallel,periodicallypositionedcylinders.Thecylindersaremadeofanonlineardielectricmaterialwithalineardielectricconstant"c>1,andasecondordersusceptibilityc1.Thecoordinatesystemissetsothatthecylindersareparalleltothey-axis,thestructureisperiodicalongthex-axis,andthez-axisisnormaltothestructure.Theunitoflengthistakentobethearrayperiod,andthedistancebetweenthetwoarraysrelativetotheperiodis2h.Thethreecoordinateaxesx,y,andz,areorientedbytheunitvectorsfe1,e2,e3g.Inthecasewhenthestructureisilluminatedbyaplanewavewithelectriceldparalleltothecylinders(TMpolarization),Maxwell'sequationsarereducedtothescalarwaveequation, 1 c2@2t"E+ 4E2=E(4) wherethedielectricfunction"hasaconstantvalue"c>1onthedielectriccylinders,andequals1elsewhere.Similarly,thesecondordersusceptibilitytakesaconstantvaluec1onthecylindersand0elsewhere.Duetothetranslationinvarianceofthescatteringstructureinthey)]TJ /F1 11.955 Tf 9.3 0 Td[(directionandtheTMpolarization,asolutiontoEq.( 4 )isafunctionofxandzalone. 86

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Figure4-1. Panel(a):Doublearrayofdielectriccylinders.Theunitoflengthisthearrayperiod.Theaxisofeachcylinderisparalleltothey-axis,andisatadistancehfromthex-axis.Panel(b):Thescatteringprocessforthenormalincidentradiation(kx=0).Thescatteredfundamentalharmonicissymbolizedbyasingleheadedarrowwhilethe(generated)secondharmonicradiationissymbolizedbyadoubleheadedarrow.Theincidentradiationwavelengthissuchthatonlyonediffractionchannelisopenforthefundamentalharmonicwhilethreediffractionchannelsareopenforthesecondharmonic.TheuxmeasuredthroughthefacesL1=2:x=1=2cancelsoutduetotheBlochperiodicityconditionasexplainedinAppendix B.2 .Panel(c):Thesolidanddashedcurvesshowtheposition(frequency!r=ckr)ofscatteringresonancesasfunctionsofthedistancebetweenthearrays,k=kr(h).Thedotsonthecurvesindicatepositionsofboundstatesintheradiationcontinuum(i.e.,thevaluesofhatwhicharesonanceturnsintoaboundstate).Thesolidcurveconnectsboundstatessymmetricrelativetothereectionz!)]TJ /F5 11.955 Tf 24.58 0 Td[(z.Thedashedlineconnectstheskewsymmetricboundstates.ThecurvesarerealizedforR=0.08,"c=2,andkx=0(normalincidence). ItiscustomarytoassumethatthesolutionEisanalyticincsothathigherharmoniceffectsmaybestudiedthroughapowerseriesexpansion, E=2ReE1e)]TJ /F8 7.97 Tf 6.59 0 Td[(i!t+cE2e)]TJ /F6 7.97 Tf 6.59 0 Td[(2i!t+2c)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(E3,1e)]TJ /F8 7.97 Tf 6.58 0 Td[(i!t+E3,3e)]TJ /F6 7.97 Tf 6.59 0 Td[(3i!t+...(4) whereE1istheamplitudeofthefundamentalharmonicsinthezeroorderofc,E2istheamplitudeofthesecondharmonicsintherstorderofc,andsoon.SuchaseriesisobtainedbyperturbationtheoryincwhichentailsrstsolvingEq.( 4 )inthelinearcase(c=0)toproduceasolutionEL.ThegeneralsolutionEtothenonlinearwaveequationisthensoughtintheformE=EL+ENL,whereENListhecorrectiondueto 87

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nonlineareffects.IfbGistheGreen'sfunctionoftheoperator" c2@2t)]TJ /F3 11.955 Tf 12.32 0 Td[(withappropriate(scattering)boundaryconditions,then,ENL=)]TJ /F4 11.955 Tf 16.71 8.09 Td[(c 4c2bG@2t(EL+ENL)2 wherethesecondordersusceptibilityhasbeenwrittenintheform=ctoisolatetheperturbationparameterc.Thefunctionissimplytheindicatorfunctionoftheregionoccupiedbytheinnitedoublearray,i.e.,itsvalueis1onanyofthecylinders,and0elsewhere.ApowerseriesexpansionforthecorrectionENLand,hence,forthefullsolutionEisthenobtainedbythemethodofsuccessiveapproximations. Accordingtoscatteringtheory[ 6 10 ],thekerneloftheGreen'sfunctionbGwillbemeromorphicink2=!2 c2.Ifthekernelhasarealpolek2=k2b>0,itisnotsummable,andthereforethesuccessiveapproximationsproduceadivergingseries,thusindicatinganon-analyticbehaviorofthesolutioninc.Asisclariedshortly,realpolescorrespondtoboundstatesintheradiationcontinuumsothatthegoalistoinvestigatethenonlinearwaveequation( 4 )inasmallneighborhoodofarealpoleoftheGreen'sfunctionbG,whichamountstondinganon-analyticalsolutioninc.Thisisacrucialdifferencebetweenconventionaltreatmentsofopticallynonlineareffectsandthepresentstudyfromboththephysicalandmathematicalpointsofview.Theconclusionssummarizedintheintroductionstemdirectlyfromthenon-analyticityofthesolutionofthenon-linearscatteringprobleminaspectralregionthatcontainsboundstatesintheradiationcontinuum. Ingeneral,theproblemisposedasascatteringproblem.Anincidentradiation,Ein(r,t)=2cos(kr)]TJ /F4 11.955 Tf 11.96 0 Td[(!t),k=kxe1+kze3,ck=!, isscatteredbythedoublearrayofdielectriccylinders.ThegeneralsolutiontoEq.( 4 )shouldthenbeoftheform, E(r,t)=1Xl=El(r)e)]TJ /F8 7.97 Tf 6.58 0 Td[(il!t(4) 88

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whereE00,andforalll,E)]TJ /F8 7.97 Tf 6.59 0 Td[(listhecomplexconjugateofEl(asEisreal).ThereforeitissufcienttodetermineonlyEl,l1.TheamplitudesElsatisfytheBloch'speriodicityconditionderivedbyrequiringthatthefullsolutionEtothewaveequationsatisesthesameperiodicityconditionastheincidentwaveEin,namely,Ein(r+e1,t)=Einr,t)]TJ /F5 11.955 Tf 13.15 8.09 Td[(kx Itthenfollowsthat, El(r+e1)=eilkxEl(r)(4) ThisistheBloch'sconditionfortheamplitudeEl.ByEq.( 4 ),theamplitudesofdifferentharmonicssatisfytheequations,El+l2k2"El=)]TJ /F4 11.955 Tf 9.29 0 Td[(l2k2(")]TJ /F3 11.955 Tf 11.96 0 Td[(1)XpEpEl)]TJ /F8 7.97 Tf 6.59 0 Td[(p,=c 4("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1) Foreaseofnotation,theparameterisoftenusedinlieuofc.Wheneverthisisthecase,itshouldbekeptinmindthatisproportionaltoc,andisthereforeverysmall. Thescatteringtheoryrequiresthatforl6=1,thepartialwavesEle)]TJ /F8 7.97 Tf 6.59 0 Td[(il!tbeoutgoinginthespatialinnity(jzj!1).ThefundamentalwavesE1ei!tareasuperpositionofanincidentplanewaveei(kr)]TJ /F17 7.97 Tf 6.58 0 Td[(!t)andascatteredwavewhichisoutgoingatthespatialinnity.Inall,theaboveboundaryconditionsleadtoasystemofLippmann-SchwingerintegralequationsfortheamplitudesEl: 8><>:E1=bH(k2)[E1+PpEpE1)]TJ /F8 7.97 Tf 6.59 0 Td[(p]+eikrEl=bH((lk)2)[El+PpEpEl)]TJ /F8 7.97 Tf 6.58 0 Td[(p],l2(4) andE)]TJ /F8 7.97 Tf 6.58 0 Td[(l= Elforl)]TJ /F3 11.955 Tf 21.92 0 Td[(1,wherebH(q2)istheintegraloperatordenedbytherelation bH(q2)[ ](r)=q2 4Z("(r0))]TJ /F3 11.955 Tf 11.95 0 Td[(1)Gq(rjr0) (r0)dr0(4) inwhichGq(rjr0)istheGreen'sfunctionofthePoissonoperator,(q2+)Gq(rjr0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(4(r)]TJ /F12 11.955 Tf 12.3 0 Td[(r0),withtheoutgoingwaveboundaryconditions.Fortwospatialdimensions, 89

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asinthecaseconsideredherer=(x,z)andr0=(x0,z0),theGreen'sfunctionisknown[ 13 ]tobeGq(rjr0)=iH0(qjr)]TJ /F12 11.955 Tf 11.97 0 Td[(r0j)whereH0isthezeroorderHankelfunctionoftherstkind. When=0,theamplitudesofallhigherharmonics(l2)vanish.ThereforeitisnaturaltoassumethatjE1jjE2jjE3jforasmall.Notethatthisdoesnotgenerallyimplythatthesolution,asafunctionof,isanalyticat=0.Underthisassumption,thesolutiontothesystem( 4 )canbeapproximatedbykeepingonlytheleadingtermsineachoftheseriesinvolved.Inparticular,therstequationin( 4 )isreducedto E1eikr+bH(k2)[E1]+2bH(k2) E1E2(4) whilethesecondequationbecomes E2bH((2k)2)[E2]+bH((2k)2)E21(4) Itthenfollowsthatarstorderapproximationtothesolutionofthenonlinearwaveequation( 4 )maybefoundbysolvingthesystemformedbytheequations( 4 )and( 4 ).Tofacilitatethesubsequentanalysis,thesystemisrewrittenas 8>><>>:[1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(k2)][E1]=eikr+2bH(k2) E1E2[1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH((2k)2)][E2]=bH((2k)2)E21(4) ThesolutionoftherstofEqs.( 4 )involvesinvertingtheoperator1)]TJ /F13 11.955 Tf 13.21 3.16 Td[(bH(k2),andthereforenecessitatesastudyofthepolesoftheresolvent[1)]TJ /F13 11.955 Tf 12.37 3.15 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.58 0 Td[(1.Suchpolesaregeneralizedeigenvaluestothegeneralizedeigenvalueproblem, bH(k2)[E]=E(4) forxedkx.ThecorrespondingeigenfunctionsE=EsarereferredtoasSiegertstates.IncontrasttoSiegertstatesinquantumscatteringtheory[ 6 ],electromagneticSiegertstatessatisfythegeneralizedeigenvalueproblem( 4 )inwhichtheoperator 90

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isanonlinearfunctionofthespectralparameterk2.TheirpropertieswerestudiedinChapter2.Ingeneral,forxedh,apoletotheresolvent[1)]TJ /F13 11.955 Tf 12.98 3.15 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.58 0 Td[(1willhavetheformk2r(h))]TJ /F5 11.955 Tf 12.35 0 Td[(i\(h).Scatteredmodessatisfytheconditionk>kx(theradiationcontinuum).Hence,ifkr(h)>kx,then,accordingtoscatteringtheory,suchapoleisaresonancepole.Inthecaseofthelinearwaveequation,i.e.,c==0,thescattereduxpeaksatk=kr(h)indicatingtheresonanceposition,whereastheimaginarypartofthepoledenesthecorrespondingresonancewidth(oraspectralwidthofthescattereduxpeak;asmall)]TJ /F1 11.955 Tf 10.1 0 Td[(correspondstoanarrowpeak).Itappearsthatthereisadiscretesetofvaluesofthedistanceh=hbatwhichthewidthofsomeofresonancesvanishes,\(hb)=0,thatis,theresolvent[1)]TJ /F13 11.955 Tf 12.41 3.16 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.59 0 Td[(1hasarealpoleatkb=kr(hb)>kx(Chapter3).Thecorrespondingeigenfunctions(Siegertstates)satisfyingEq.( 4 )areknownasboundstatesintheradiationcontinuum.Itisthepresenceofsuchstatesthatmakesthescatteringproblemdenedbythesystem( 4 )impossibletoanalyzeforrealknearkr(hb)byexpandingthesolutionintoapowerseriesinc.Itshouldbenotedthatthereexistsanotherclassofboundstatesinthissystemforwhichk2
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nonlinearintegralequation, E1=1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(k2))]TJ /F6 7.97 Tf 6.59 0 Td[(1eikr+22bH(k2) E11)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH((2k)2))]TJ /F6 7.97 Tf 6.58 0 Td[(1hbH((2k)2)E21i(4) where,inaccordwiththenotationintroducedin( 4 ),thefunctiononwhichanoperatoractsisplacedinthesquarebracketsfollowingtheoperator.Theoperator2(1)]TJ /F13 11.955 Tf 10.92 3.16 Td[(bH(k2)))]TJ /F6 7.97 Tf 6.59 0 Td[(1thatdeterminesthenonlinearpartofEq.( 4 )cannotbeviewedassmallwhenk2!k2bnomatterhowsmall22cis,whichprecludestheuseofapowerseriesrepresentationofthesolutionin2.Thebehaviorofthesolutionoftheintegralequation( 4 )willbeanalyzedfork2neark2binthelimitofsubwavelengthdielectriccylinders.Inthislimit,theactionoftheoperatorbHwillbeprovedtobedeterminedbytheactionofa22matrixsothatEq.( 4 )becomesasystemoftwoquadraticequations.Theanalysisofthissystemwillbeconsiderablysimpliedbyestablishingrstaparticularpropertyoftheeldratiodenedas (x,z)=E1(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z) E1(x,+z)(4) Thesaidpropertyisaresultofthefactthatboundstates,andmoregenerallySiegertstates,haveaspecicparityrelativetothereection(x,z)!(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z),i.e.,theyareeitheroddorevenfunctionsofz.ThisisanimmediateconsequenceofthecommutativityoftheoperatorbHandtheparityoperatorbPdenedbybP[E](x,z)=E(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z).Now,nearapolek2r(h))]TJ /F5 11.955 Tf 12.17 0 Td[(i\(h),itfollowsfromthemeromorphicexpansionof[1)]TJ /F13 11.955 Tf 12.87 3.16 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.59 0 Td[(1that, E1=iC(h) k2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2r(h)+i\(h)Es+O(1)(4) whereC(h)issomeconstantdependingonh,andEsisanappropriatelynormalizedSiegertstate(RefertoChapter2).ConsiderthecurveofresonancesC:k=kr(h)inthe(h,k)-plane.AlongC, E1(x,z)=C(h) \(h)Es(x,z)+O(1)(4) 92

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Now,ash!hb,thewidth\(h)goesto0,andtheSiegertstateEsbecomesaboundstateEbintheradiationcontinuum.ItthenfollowsfromEq.( 4 )thatifC(h)doesnotgotozerofasterthan\(h)ash!hb,i.e.,thepolestillgivestheleadingcontributiontoE1inthislimit,then(x,z)!Eb(x,)]TJ /F5 11.955 Tf 9.3 0 Td[(z) Eb(x,+z)=1 dependingonwhethertheboundstateEbisevenoroddinz.Inthecaseofthelinearwaveequation(=0),theconstantC(h)isshowntobeproportionaltop \(h).Therefore,forasmall,theassumptionthatC(h)doesnotgotozerofasterthan\(h)ash!hbisjustied. ThefollowingapproachisadoptedtosolveEq.( 4 )nearaboundstate.First,thecurveofresonancesCinthe(h,k)-planeisfound.Next,theequationissolvedintermsoftheratiowiththepair(k,h)beingonthecurveC.TheprincipalpartoftheamplitudeE1relativetoh=h)]TJ /F5 11.955 Tf 12.55 0 Td[(hbisthenevaluatednearacriticalpoint(hb,kb)onCbytakingtoitslimitvalue.Thisapproachrevealsanon-analyticdependenceoftheamplitudeE1onthesmallparametershandcofthesystem,whichiscrucialforthesubsequentanalysisofthesecondharmonicgeneration. 4.2SubwavelengthCylindersApproximation FollowingtheworkofChapter3,theactionoftheintegraloperatorbH(q2)inEq.( 4 )isapproximatedinthelimitofsubwavelengthdielectriccylinders.Theapproximationisdenedbyasmallparameter 0(q)=(qR)2 4("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)1(4) whichisthescatteringphaseofaplanewavewiththewavenumberqonasinglecylinderofradiusR.ForsufcientlysmallR,thisapproximationisjustied.InthisapproximationtheintegralkernelofbH(q2)isdenedby( 4 )andhassupportontheregionoccupiedbycylinders.Thecondition( 4 )impliesthatthewavelengthismuchlargerthantheradiusR,andthereforeeldvariationswithineachcylindermaybe 93

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neglected,sothat (x,z) (n,h)where(n,h)arethepositionsoftheaxesofthecylinders(nisaninteger).TheintegrationinbH(q2)[ ]yieldsthenaninnitesumoverpositionsofthecylinders.ByBloch'scondition, (n,h)=einqx (0,h),sothatthefunctionbH(q2)[ ](x,z)isfullydeterminedbythetwovalues (0,h).Inparticular, bH(q2)[ ](0,h) (0,h)+ (0,h)(4) wherethecoefcientsandareshowntobe (q,qx)=2i0(q) 1Xm=1 qz,m)]TJ /F3 11.955 Tf 42.26 8.09 Td[(1 2i(jmj+1)+i ln(2R)!(q,qx,h)=2i0(q)1Xm=e2ihqz,m qz,m whereqz,m=p q2)]TJ /F3 11.955 Tf 11.96 0 Td[((qx+2m)2withtheconventionthatifq2<(qx+2m)2,thenqz,m=ip (qx+2m)2)]TJ /F5 11.955 Tf 11.95 0 Td[(q2.Toobtaintheenergyuxscatteredbythestructure,theactionoftheoperatorbH(q2)on mustbedeterminedintheasymptoticregionjzj!1.Itisfoundthatforjzj>h+R, bH(q2)[ ](x,z)2i0(q)1Xm=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[( (0,h)eijz)]TJ /F8 7.97 Tf 6.59 0 Td[(hjqz,m+ (0,)]TJ /F5 11.955 Tf 9.3 0 Td[(h)eijz+hjqz,meix(qx+2m) qz,m(4) 4.3AmplitudesoftheFundamentalandSecondHarmonics NowthattheactionoftheoperatorbH(q2)hasbeenestablishedin( 4 )and( 4 ),theamplitudesE1andE2ofthefundamentalandsecondharmonicscanbedeterminedbysolvingthesystem( 4 ).Asnotedearlier,thiswillbedonealongacurveCintheh,k-planedenedbyk=kr(h)wherekr(h)istherealpartofapoleof[1)]TJ /F13 11.955 Tf 12.89 3.15 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.59 0 Td[(1,orequivalently,whentheincidentradiationhastheresonantwavenumberk=kr(h).Thiscurveisobtainedbystudyingthesingularitiesink2oftheoperator1)]TJ /F13 11.955 Tf 13.57 3.16 Td[(bH(k2).Withthispurpose,Eq.( 4 )thatdenesSiegertstates,iswritteninthesubwavelengthapproximationaccordingtotherule( 4 ),i.e.,theactionofbH(k2)is 94

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takenatthepoints(0,h).Thisproducesthesystem, [1)]TJ /F24 11.955 Tf 11.95 0 Td[(H]0B@Eb+Eb)]TJ /F13 11.955 Tf 7.09 43.16 Td[(1CA=0B@001CA,H=0B@1CA(4) whereEb=Eb(0,h)andthefunctions=(k,kx)and=(k,kx)havebeendenedintheprevioussection.Inparticular,boundstatesoccuratthepoints(hb,kb)atwhichthedeterminantdet(1)]TJ /F24 11.955 Tf 11.95 0 Td[(H)vanishes,det0B@1)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1CA=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[()(1)]TJ /F4 11.955 Tf 11.95 0 Td[(+)=0 ItfollowsfromEq.( 4 )thattheboundstatesforwhich1)]TJ /F4 11.955 Tf 12.54 0 Td[()]TJ /F4 11.955 Tf 12.53 0 Td[(=0areeveninzbecauseEb+=Eb)]TJ /F1 11.955 Tf 10.41 1.79 Td[(inthiscase.Similarly,theboundstatesforwhich1)]TJ /F4 11.955 Tf 12.54 0 Td[(+=0areoddinz.Moregenerally,thepolesoftheresolvent[1)]TJ /F13 11.955 Tf 13.91 3.16 Td[(bH(k2)])]TJ /F6 7.97 Tf 6.59 0 Td[(1arecomplexzerosofdet(1)]TJ /F24 11.955 Tf 12.41 0 Td[(H).Theyarefoundbytheconventionalscatteringtheoryformalism.Specically,theresonantwavenumbersk2=k2r(h)areobtainedbysolvingtheequationRef1)]TJ /F4 11.955 Tf 11.94 0 Td[(g=0forthespectralparameterk2.Accordingtotheconventionadoptedintherepresentation( 4 ),thecorrespondingresonancewidthsaredenedby\(h)=)]TJ /F1 11.955 Tf 19.16 8.08 Td[(Imf1)]TJ /F4 11.955 Tf 11.95 0 Td[(g @k2Ref1)]TJ /F4 11.955 Tf 11.96 0 Td[(gk2=k2r(h) where@k2denotesthederivativewithrespecttok2.ThisdenitionofthewidthcorrespondstothelinearizationofRef1)]TJ /F4 11.955 Tf 12.79 0 Td[(gneark2=k2r(h)asafunctionofk2inthepolefactor[1)]TJ /F4 11.955 Tf 12.43 0 Td[(])]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thecurvesofresonancesk=kr(h)>kxcomeinpairs.Thereisacurveconnectingthesymmetricboundstatesintheh,k-plane,andanothercurvethatconnectstheoddones. Inwhatfollows,onlythecurveconnectingsymmetricboundstateswillbeconsidered.Theothercurvecanbetreatedsimilarly.Panel(c)ofFig. 4-1 showsthattherstsymmetricboundstateoccurswhenthedistance2hisabouthalfthearrayperiod,whiletheskew-symmetricboundstatesemergeonlyatlargerdistances. 95

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ThisfeatureisexplainedindetailinChapter3.So,thesolutionobtainedneartherstsymmetricboundstatecorrespondstothesmallestpossibletransversedimensionofthesystem(roughlyahalfofthewavelengthoftheincidentradiation)atwhichasignicantenhancementofthesecondharmonicgenerationcanbeobserved.Thus,fromnowonthecurveofresonancesCreferstothecurveintheh,k-planedenedbytheequationRef1)]TJ /F4 11.955 Tf 12.01 0 Td[()]TJ /F4 11.955 Tf 12.01 0 Td[(g=0.Tosimplifythetechnicalities,itwillbefurtherassumedthatonlyonediffractionchannelisopenforthefundamentalharmonics,i.e.,kx
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Tothisend,letE1=E1(0,h)bethevaluesoftheeldE1ontheaxesofthecylindersat(0,h),andE2=E2(0,h)bethevaluesoftheeldE2onthesamecylinders.Inthesubwavelengthapproximation,thesevaluesfullydeterminethescatteredeldasisshownlaterand,hence,havetobefoundrst.Applyingtherule( 4 )toevaluatetheactionoftheoperatorbH(q2)inthesystem( 4 ),therstequationofthelatterbecomes, [1)]TJ /F24 11.955 Tf 11.95 0 Td[(H]0B@E1+E1)]TJ /F13 11.955 Tf 7.08 43.16 Td[(1CA=2H0B@ E1+E2+ E1)]TJ /F5 11.955 Tf 7.09 1.8 Td[(E2)]TJ /F13 11.955 Tf 7.09 43.16 Td[(1CA+0B@e+ihkze)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkz1CA(4a) Similarly,thesecondofEqs.( 4 )yields, [1)]TJ /F24 11.955 Tf 11.96 0 Td[(H2]0B@E2+E2)]TJ /F13 11.955 Tf 7.08 43.15 Td[(1CA=H20B@E21+E21)]TJ /F13 11.955 Tf 7.08 44.31 Td[(1CA,H2=H(2k,2kx)(4b) Asstatedabove,theresolvent[1)]TJ /F13 11.955 Tf 13.52 3.15 Td[(bH((2k)2)])]TJ /F6 7.97 Tf 6.59 0 Td[(1isregularinaneighborhoodofkbsothatEq.( 4b )canbesolvedforE2,whichdenesthelatterasfunctionsofE1.ThesubstitutionofthissolutionintoEq.( 4a )givesasystemoftwononlinearequationsfortheeldsE1.AddingtheseequationsandreplacingtheeldE1)]TJ /F1 11.955 Tf 10.41 1.8 Td[(byitsexpressionE1)]TJ /F3 11.955 Tf 12.68 1.79 Td[(=(0,h)E1+intermsoftheeldratioofEq.( 4 )yieldsthefollowingimplicitrelationbetweentheeldE1+anditsamplitudejE1+j: E1+=)]TJ /F4 11.955 Tf 43.84 8.09 Td[(' 2 jE1+j2+'2(4) where'=cos(hkz)and=c 4("c)]TJ /F6 7.97 Tf 6.58 0 Td[(1)aresmalland,intermsoftheeldratio(0,h),thevaluesofandread, =i20(k) kz(1+),1 =1+i40(k) kz'2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(a+b2+ )]TJ /F5 11.955 Tf 5.48 -9.69 Td[(b+a2(4) 97

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withaandbbeingdenedbytherelation, [1)]TJ /F24 11.955 Tf 11.95 0 Td[(H2])]TJ /F6 7.97 Tf 6.59 0 Td[(1H2=0B@abba1CA(4) Inparticular,andarecontinuousfunctionsofand'.InAppendix B.1 itisshownthatifbandbaretherespectivelimitsofandash!hbalongthecurveofresonancesC,thentheselimitsarenonzero.ItfollowsthenthatEq.( 4 )forE1+issingularinbothand'whentheseparametersaresmall,i.e.,inthelimit(,')!(0,0).Furthermore,thereisnowaytosolvethesaidequationperturbativelyineitheroftheparameters.Afullnon-perturbativesolutioncanbeobtainedusingCardano'smethodforsolvingcubicpolynomials.Indeed,bytakingthemodulussquaredofbothsidesoftheequation,itisfoundthat, X3+2'2 2RefgX2+'4 4jj2X)]TJ /F4 11.955 Tf 13.15 8.09 Td[('2 4jj2=0,X=jE1+j2(4) ThesolutiontothiscubicequationisobtainedinAppendix B.3 .ItisprovedtherethatEq.( 4 )admitsauniquerealsolutionsothatthereisnoambiguityonthechoiceofE1+.Inthevicinityofapoint(hb,kb)alongtheresonancecurveC,theeldE1+isfoundtobehaveas, jE1+j=jhj1=3 c(h,c)(4) Recallthath=h)]TJ /F5 11.955 Tf 12.34 0 Td[(hb.Anexplicitformofthefunction(h,c)isgiveninAppendix B.3 (Eq.( B )).Itinvolvescombinationsofthesquareandcuberootsoffunctionsinhandcandhasthepropertythat(h,c)!0as(h,c)!(0,0)(inthesenseofthetwo-dimensionallimit).Inthelimith!hb,theeldratioapproaches1forasymmetricboundstateasarguedearlier.ThereforeitfollowsfromEqs.( 4b )thatE2E21+becausethematrix( 4 )existsath=hb.Sincec,relation( 4 )leadstotheconclusionthatE2 E1+=O(jhj1=3) 98

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Thus,theapproximationjE1jjE2jusedtotruncatethesystem( 4 )remainsvalidforhclosetothecriticalvaluehbdespitethenon-analyticityoftheamplitudesat(h,c)=(0,0). 4.4FluxAnalysis:TheConversionEfciency Forthenonlinearsystemconsidered,eventhoughPoynting'stheoremtakesaslightlydifferentformascomparedtolinearMaxwell'sequations,theuxconservationforthetimeaveragedPoyntingvectorholds.Thescatteredenergyuxcarriedacrossaclosedsurfacebyeachofthedifferentharmonicsaddsuptotheincidentuxacrossthatsurface.TheuxconservationtheoremisstatedinAppendix B.2 .Consideraclosedsurfacethatconsistsoffourfaces,L=f(x,z)j)]TJ /F6 7.97 Tf 21.03 4.71 Td[(1 2x1 2,z!gandL1=2=f(x,z)jx=1=2gasdepictedinFig. 4-1 (b).AsarguedinAppendix B.2 ,thescattereduxofeachlth-harmonicsacrosstheunionofthefacesL1=2vanishesbecauseoftheBlochcondition(andsodoestheincidentuxforanykx).ThereforeonlytheuxconservationacrosstheunionofthefacesLhastobeanalyzed.Ifldesignatestheratioofthescattereduxcarriedbythelth-harmonicsacrossthefacesLtotheincidentuxacrossthesamefaces,thenPl1l=1.Thus,forl2,ldenestheconversionratiooffundamentalharmonicsintothelth-harmonics. Intheperturbationtheoryusedhere,onlytheratios1and2maybeevaluated.Bylaboriouscalculationsitcanbeshownthat1+21asonewouldexpect(Appendix B.2 ).Hence,theefciencyofconvertingthefundamentalharmonicintothesecondharmonicissimplydeterminedbythemaximumvalueof2asafunctionoftheparameterhatagivenvalueofthenonlinearsusceptibilityc. Theratio2isdenedintermsofthescatteringamplitudesofthesecondharmonic,i.e.,bytheamplitudeofE2intheasymptoticregionjzj!1: E2(r)!8>>>><>>>>:Xmop,shRshmeirk)]TJ /F9 5.978 Tf 0 -6.42 Td[(m,sh,z!Xmop,shTshmeirk+m,sh,z!+1(4) 99

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wherekm,sh=(2kx+2m)e1kshz,me3isthewavevectorofthesecondharmonicinthemthopendiffractionchannel.Recallthatthemthchannelisopenprovided(2k)2>(2kx+2m)2andinthiscasekshz,m=p (2k)2)]TJ /F3 11.955 Tf 11.95 0 Td[((2kx+2m)2,whileifthechannelisclosed,thenkshz,m=ip (2kx+2m)2)]TJ /F3 11.955 Tf 11.95 0 Td[((2k)2.Intheasymptoticregionjzj!1,theeldinclosedchannelsdecaysexponentiallyand,hence,theenergyuxcanonlybecarriedinopenchannelstothespatialinnity.ThesummationinEqs.( 4 )istakenonlyoverthosevaluesofmforwhichthecorrespondingdiffractionchannelisopenforthesecondharmonic,whichisindicatedbythesuperscriptop,shinthesummationindexmop,sh.Notethatthereismorethanoneopendiffractionchannelforthesecondharmoniceventhoughonlyonediffractionchannelisopenforthefundamentalone.Forinstance,ifthex)]TJ /F1 11.955 Tf 9.3 0 Td[(componentofthewavevectork,i.e.,kx,islessthan 2,thereare3opendiffractionchannelsforthesecondharmonic,thechannelsm=0,m=)]TJ /F3 11.955 Tf 9.3 0 Td[(1,andm=1.Thesethreedirectionsofthewavevectorofthesecondharmonicpropagatingineachoftheasymptoticregionsz!aredepictedinFig. 4-1 (b)bydouble-arrowrays.Thus,intermsofthescatteringamplitudesintroducedinEqs.( 4 ),theratioofthesecondharmonicuxacrossLtotheincidentuxis2=1 2kzXmop,shkshz,m)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jRshmj2+jTshmj2 ThescatteringamplitudesRshmandTshmareinferredfromEq.( 4 )inwhichtherule( 4 )isappliedtocalculatetheactionoftheoperatorbH((2k)2)inthefar-eldregionsjzj!1:8>>><>>>:Rshm=2i0(2k) kshz,mh(E2++E21+)eihkshz,m+(E2)]TJ /F3 11.955 Tf 9.74 1.8 Td[(+E21)]TJ /F3 11.955 Tf 7.08 2.96 Td[()e)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkshz,miTshm=2i0(2k) kshz,mh(E2++E21+)e)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkshz,m+(E2)]TJ /F3 11.955 Tf 9.74 1.79 Td[(+E21)]TJ /F3 11.955 Tf 7.08 2.96 Td[()eihkshz,mi Since!1ash!hbalongC,theprincipalpartof2inavicinityofaboundstatealongthecurveCisobtainedbysettingE1)]TJ /F3 11.955 Tf 11.06 1.8 Td[(=E1+inEq.( 4b ),solvingthelatterfor 100

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E2,andsubstitutingthesolutionintotheexpressionfor2.Theresultreads 2=Cb2jE1+j4(4) whereCbisaconstantobtainedbytakingallnonsingularfactorsintheexpressionof2totheirlimitash!hb,whichgivesCb="(160(k))2 kzj1+a+bj2Xmop,shcos2(hkshz,m) kshz,m#(h,k)=(hb,kb) foraandbdenedinEq.( 4 ).UsingtheidentityjE1+j4=jE1+j2jE1+j2,andsubstitutingEq.( 4 )intooneofthefactorsjE1+j2,theconversionratio2isexpressedasafunctionofasinglerealvariable, 2(u)=C0bu ju+bbj2,u=jE1+j '2(4) whereC0b=Cbjbj2isaconstant,andandinEq.( 4 )havebeentakenattheirlimitsash!hbtoobtaintheprincipalpartof2.Thefunctionu7!2(u)on[0,1)isfoundtoattainitsabsolutemaximumatu=jbbj.Thisconditiondeterminesthedistance2hbetweenthearraysatwhichtheconversionrateismaximalforgivenparametersR,"candcofthesystem.Indeed,sinceujE1+j2shouldalsosatisfythecubicequation( 4 ),thesubstitutionofu=jbbjintothelatteryieldsthecondition 2 '4=2jbj2(jbbj+Refbbg)(4a) Inparticular,intheleadingorderin0(k),theoptimaldistance2hbetweenthetwoarraysisgivenbytheformula, (h)]TJ /F5 11.955 Tf 11.95 0 Td[(hb)4=2c 85kz,b(kbR)6("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)5(4b) whereaspreviously,kz,b=p k2b)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x. Themaximumvalue2,maxoftheconversionratio2isthesought-forconversionefciency.Aninterestingfeaturetonoteisthat2,max=2(jbbj)isindependent 101

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ofthenonlinearsusceptibility2becausetheconstantsC0b,b,andbarefullydeterminedbythepositionoftheboundstate(kb,hb).Inotherwords,ifthedistancebetweenthearraysischosentosatisfythecondition( 4a ),theconversionefciency2,maxisthesameforawiderangeofvaluesofthenonlinearsusceptibility2.Thisconclusionfollowsfromtwoassumptionsmadeintheanalysis.First,thesubwavelengthapproximationshouldbevalidforboththefundamentalandsecondharmonics,i.e.,theradiusofcylindersshouldbesmallenough.Second,thevaluesofh)]TJ /F5 11.955 Tf 12.32 0 Td[(hband(orc)mustbesuchthattheanalysisoftheexistenceanduniquenessofjE1+jgiveninAppendix B.3 holds,thatis,Eq.( 4 )shouldhaveauniquerealsolutionunderthecondition( 4a ).ThegeometricalandphysicalparametersofthestudiedsystemcanalwaysbechosentojustifythesetwoassumptionsasillustratedinFig. 4-1 Panels(a)and(b)ofFig. 4-2 showtheconversionefciency2,maxfortherstthreesymmetricboundstatesn=1,2,3,as,respectively,afunctionofthecylinderradiusRwhenkx=0andofkxwhenR=0.15.Forallcurvespresentedinthepanels,"c=1.5.Thevaluesof2,maxareevaluatednumericallybyEq.( 4 )whereu=jbbj.ThesolidpartsofthecurvesinPanel(a)correspondtothescatteringphase0(k)<0.25withk=kb.Notethatthewavelengthatwhichthesecondharmonicgenerationismostefcientistheresonantwavelengthdenedbyk=kr(h)wherehsatisesthecondition( 4a ).Forasmallc,thescatteringphaseattheresonantwavelengthcanwellbeapproximatedas0(k)0(kb).Thecondition0(kb)<0.25ensuresthatthescatteringphaseforthesecondharmonicsatisestheinequality0(2kb)=40(kb)<1otherwisethevalidityofthesubwavelengthapproximationcannotbejustied.ThedashedpartsofthecurvesinPanels(a)ofFig. 4-2 correspondtotheregionwhere0(kb)>0.25.Panels(a)and(b)ofthegureshowthattheconversionefciencycanbeashighas40%forawiderangeoftheincidentanglesandvaluesofthecylinderradius.Suchaconversionefciencyiscomparablewiththatachievedinopticallynonlinearcrystalsatatypicalbeampropagationlength(activelength)ofafewcentimeters,whereas 102

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Figure4-2. Panel(a):TheconversionefciencyisplottedagainstthecylinderradiusRforthecriticalpoints(hb(n),kb(n)),n=1,2,3,and"c=1.5andkx=0(thenormalincidence).Thedashedpartsofthecurvesindicatetheregionswhere0(k)>0.25and,hence,0(2k)>1,i.e.,thesubwavelengthapproximationbecomesinapplicableforthesecondharmonic.Panel(b):Theconversionefciencyisplottedagainstkxforthecriticalpoints(hb(n),kb(n)),n=1,2,3.ThecurvesarerealizedforR=0.15and"c=1.5.Panel(c):Theregionofvalidityofthedevelopedtheoryfortherstboundstatehb(1)0.259.Theshadowedpartofthe(,h))]TJ /F1 11.955 Tf 9.3 0 Td[(planeisdenedbythecondition++)]TJ /F4 11.955 Tf 10.4 1.79 Td[(>0underwhich,accordingtoEq.( B ),theamplitudejE1+jexistsanduniqueasexplainedinAppendix B.3 .TheplotisrealizedforR=0.1,"c=2andkx=0.Theparabola-likecurveisanactualboundaryoftheshadowedregion;thetophorizontallinerepresentsnorestriction.Sothereisawiderangeofthephysicalandgeometricalparameterswithintheshadowedregionwhichsatisfy( 4a ).Theregionsofvalidityforotherboundstateslookssimilar. herethetransversedimension2hofthesystemstudiedherecanbeaslowasahalfofthewavelength,i.e.,foraninfraredincidentradiation,2hisaboutafewhundrednanometers.Indeed,asonecanseeinFig. 4-2 (c),therstboundstateoccursathb(1)0.259andkb2whichcorrespondstothewavelengthb=2=kb1. Thestatedconversionefciencycanbefairlywellestimatedintheleadingorderof0(k): 2,max=2(jbbj)80(k)Xmop,shcos2(hkshz,m) kshz,m(h,k)=(hb,kb)(4) 103

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Supposethatonlyonediffractionchannelisopenfortheincidentradiation.Thenm=0,1inEq.( 4 )(threeopenchannelsforthesecondharmonics).Let02,maxdenotethetermm=0,i.e.02,maxisthesecondharmonicuxinwhichthecontributionofthechannelswithm=1isomitted.Inparticular,02,max<2,max.OneinfersfromEq.( 4 )that,02,max40(kb) kz,bcos2(2hbkz,b) Asthepair(hb,kb)atwhichasymmetricboundstateisformedsatisestheequationcos(hbkz,b)=0,itfollowsthatcos(2hbkz,b)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1.Hence,02,maxk2b kz,bR2("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1) Thewavenumberskbatwhichtheboundstatesoccurliejustbelowthediffractionthreshold2)]TJ /F5 11.955 Tf 12.05 0 Td[(kx,i.e.,kb/2)]TJ /F5 11.955 Tf 12.05 0 Td[(kx(Chapter3).Sothatinthecaseofnormalincidence(kx=0),theaboveestimatebecomes,02,max2(R2)("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1) with0(2)1.If,forinstance,R=0.15and"c=2,then,02,max44% for0(2)0.22. Itisnoteworthytoemphasizethefollowingfeaturesoftheproposedmechanismtogeneratehigherharmonicsthataretobecontrastedwiththeconventionalmethods.First,thenecessitytofulllthephasematchingcondition,muchneededinopticallynonlinearcrystals,hasbeeneliminated.Thereasonisthatthesecondharmonicisgeneratedinregions(cylinders)ofdimensionsmuchsmallerthanthewavelengthand,hence,thephasemismatchinpropagationoftherstandsecondharmonicsduetothedifferenceinthecorrespondingrefractionindicesdoesnotevenoccur.Second,theprocessdoesnotrequireanyfocusingoftheincidentbeam.Instead,ifthegeometrical 104

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parametersofthesystemaresettomaximizetheconversionrate,thefocusingoccurswithinthestructureautomatically,andthemaximalconversionrateofabout40%isachieved,eventhoughcEi1fortheincidentradiation.Third,themaximumvalueoftheconversionratedependsweaklyonthenonlinearsusceptibility.Thisprovidesapossibilityforthefrequencyconversioninlowerpowerlightbeams.Fourth,anactivelengthatwhichtheconversionrateismaximaliscloseto2hbwhosesmallestvalueforthesystemstudiedisroughlyahalfofthewavelengthoftheincidentlight,whileaneffectiveconversioninaslabofanopticalnonlinearmaterialrequiresalengthvaryingbetweenafewmillimeterstoafewcentimeters.Thismeansthattheconversioncaneffectivelybedoneatnanoscalesforavisiblelight.Theveryexistenceofalltheaforementionedfeaturesisessentiallyattributedtotheexistenceofresonanceswiththevanishingwidthorboundstatesintheradiationcontinuumforthescatteringsystemstudied,andwouldnotbepossibleotherwise. 105

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APPENDIXACOMPLEMENTSI A.1TheLippmann-SchwingerIntegralEquation Inthissectionoftheappendix,itisprovedthatthesolutiontotheLippmann-SchwingerintegralequationinEq.( 2 )solvesEq.( 2 ).Itshouldbestressedthatthisequationistobeunderstoodinthedistributionalsense.Thisisbecausethepotentialunderconsiderationisneithercompactlysupportednordoesitvanishatinnitysothattheusualmethodsthatestablishthisequationcannotbeused[ 10 ].Thus,theproofwillbedonebyconsideringthefunctionsinvolvedasdistributionsactingonsmoothfunctionsofcompactsupport. Todoso,onlylocallyintegrablesolutionstoEq.( 2 )aresought.Inthissetting,allthefunctionsinvolvedinEq.( 2 ),namely;eikr,E!,"E!,(")]TJ /F3 11.955 Tf 12.06 0 Td[(1)E!,Gkand((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,arelocallyintegrable. Letthen'beasmoothfunctionofcompactsupportandE!bealocallyintegrablesolutiontoEq.( 2 ).Then hE!+k2"E!,'i=k2h"E!,'i)]TJ /F5 11.955 Tf 19.26 0 Td[(k2hEi,'i+k2 4h((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,'i(A) whereEi(r)=eikr.Sincenoneofthedistributions(")]TJ /F3 11.955 Tf 13.05 0 Td[(1)E!andGiscompactlysupported,theconvolutionusedhereistobeunderstoodinthesenseoftheusualconvolutionoffunctions.Therefore,thelastintegralofEq.( A )maybeinterpretedas h((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,'i=Z("(r0))]TJ /F3 11.955 Tf 11.96 0 Td[(1)E!(r0)hGr0,'idr0(A) whereGr0(r)=Gk(rjr0)andsatisesthedistributionalHelmholtzequation Gr0+k2Gr0=)]TJ /F3 11.955 Tf 9.3 0 Td[(4r0(A) ThereforehGr0,'i=hGr0,'i=)]TJ /F5 11.955 Tf 9.3 0 Td[(k2hGr0,'i)]TJ /F3 11.955 Tf 19.26 0 Td[(4hr0,'i 106

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Itfollowsthat h((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,'i=)]TJ /F5 11.955 Tf 9.3 0 Td[(k2h((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,'i)]TJ /F3 11.955 Tf 19.26 0 Td[(4h(")]TJ /F3 11.955 Tf 11.96 0 Td[(1)E!,'i(A) ThushE!+k2"E!,'i=0foralltestfunctions'. NotethattheonlydifferencebetweentheproofthatEq.( 2 )solvesEq.( 2 )inthecaseofthenitearrayandthatofaninnitearrayisthewayEq.( A )isderivedfromEq.( A ).Indeed,ifthearrayofcylindersisnitethentheconvolutionofEq.( A )isinthedistributionalsenseas(")]TJ /F3 11.955 Tf 12.62 0 Td[(1)E!wouldbeacompactlysupporteddistribution.ThereforeonecouldestablishEq.( A )immediatelyfromEq.( A )andtheidentityofdistributionalconvolution h((")]TJ /F3 11.955 Tf 11.96 0 Td[(1)E!)Gk,'i=h((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)Gk,'i(A) Thisidentityisnotimmediateinthecaseoftheinnitearraybecause,asmentionedabove,noneoftheconvolutedfunctionshascompactsupport.Infact,thefunction((")]TJ /F3 11.955 Tf 11.95 0 Td[(1)E!)GkisaconditionallyconvergentseriessothatintegrationbypartscannotbeappliedtoestablishEq.( A ). A.2SolutionoftheLippmann-SchwingerIntegralEquationintheZeroRadiusApproximation HereanapproximatesolutiontotheLippmann-Schwingerintegralequationisestablishedinthesmallradiusapproximation.Intheusualtheoryofscatteringfromsmallparticles[ 15 18 ],theproblemissolvedwithhighaccuracybyassumingthatfarfromthescatteringregionthesolutionisalinearsuperpositionofthewavesscatteredbyeachindividualparticle.Duetothecylindricalgeometryofthedielectricscatterers,itturnsoutthatsimilarapproximationsmaybemadetosolvethescatteringproblemonthedoublearraybutwithsolutionswhicharevalideverywhereoffthescatterersevenintheregionbetweenthetwogratingstructures.Thevalidityofsolutioninthisextended 107

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regioncanbeusedtondtheso-calledhotspots(Fig. 3-3 )wherethemagnitudeoftheelectromagneticeldspeaks. ThestructureconsideredisshowninFig. 3-1 (a).ThecylindersinthestructurearelabeledasCm,nwhereniseither1or)]TJ /F3 11.955 Tf 9.3 0 Td[(1dependingonwhetherthecylinderisontherightorleftarray.Theintegermreferstothex-coordinateofthecylinder'saxis.Inparticular,fortherightarraycylinders,theaxesarepositionedatrm,1=(m+a)e1+he3andthoseoftheleftarrayareatpositionsrm,)]TJ /F6 7.97 Tf 6.58 0 Td[(1=me1)]TJ /F5 11.955 Tf 11.96 0 Td[(he3. TheLippmann-Schwingerintegralequationmaythenbewrittenasasumoverallcylindersas E!(r)=eikr+ik2("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1) 4Xm,nZCm,nE!(r0)H0(kjr)]TJ /F12 11.955 Tf 11.95 0 Td[(r0j)dr0(A) Farfromthescatterers,eachoftheintegralsiswellapproximatedthroughthemeanvaluetheorembyZCm,nE!(r0)G(rjr0)dr0i2R2eimkxH0(kjr)]TJ /F12 11.955 Tf 11.96 0 Td[(rm,nj)E!(r0,n) sothatthefareldmaybeexpressedintermsoftheeldsonthecylindersC0,1as E!(r)=eikr+i0(k)Xn=1E!(r0,n)1Xm=eimkxH0(kjr)]TJ /F12 11.955 Tf 11.95 0 Td[(rm,nj)for0(k)=1 4k2R2("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)(A) Theclaimisthatthisapproximationremainsvalidinthenearregiontoo.ThismaybeestablishedbymeansoftheBesselfunctionexpansionsoftheeldinsidethecylindersandtheHankelfunctionH0.Tothisend,letr0beapositionvectoronthecylinderCm,n,thenr0=rm,n+uwithu=jujR.Ifrisapositionvectoroffthescatterers,thenjr)]TJ /F12 11.955 Tf 11.96 0 Td[(rm,nj>RandthereforeH0(kjr)]TJ /F12 11.955 Tf 11.95 0 Td[(r0j)=1X=ei(m,n)]TJ /F17 7.97 Tf 6.58 0 Td[()J(ku)H(kjr)]TJ /F12 11.955 Tf 11.95 0 Td[(rm,nj) 108

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whereandm,naretheanglesbetweenthex-axisandthevectorsuandr)]TJ /F12 11.955 Tf 12.98 0 Td[(rm,nrespectively.Ontheotherhand,theeldinsidethecylinderCm,nisgivenby E!(r0)=eimkx1X=,neiJ(k0u)(A) withk0=nckfortheindexofrefractionncofthecylindersandthecoefcients,ngivenby,n=1 2J(k0R)Z20e)]TJ /F8 7.97 Tf 6.59 0 Td[(iE!(Rer+r0,n)d Hereer=e1cos+e3sinistheusualradialvectorofpolarcoordinates.Inparticular,,nisatmostoftheorderof(kR)jjinthelimitofthincylinders. ItfollowsthatZCm,nE!(r0)G(rjr0)dr0=22ieimkx1X=,neim,nH(kjr)]TJ /F12 11.955 Tf 11.96 0 Td[(rm,nj)ZR0J(k0u)J(ku)uduAstheintegralintheaboveseriesisoftheorderof(kR)2jj+2,itisthenjustiedtoapproximatetheseriesbyits0thsummandinthelimitkR1,sothat ZCm,nE!(r0)G(rjr0)dr0i2R2eimkx0,nH0(kjr)]TJ /F12 11.955 Tf 11.96 0 Td[(rm,nj)(A)Thevalueof0,nmaythenberecoveredbysettingu=0inEq.( A ).ItisE!(r0,n).ThisestablishesEq.( A )everywhereoffthescatterers.Inparticular,theeldsaredeterminedbytheknowledgeoftheirvaluesE!(r0,1)onthecylindersC0,1alone. TodeterminethevaluesE!(r0,1),letnbeeither1or)]TJ /F3 11.955 Tf 9.3 0 Td[(1and,r0,nbesubstitutedforrinEq.( A ).Thelatterequationbecomes, E!(r0,n)=eikr0,n+ik2("c)]TJ /F3 11.955 Tf 11.95 0 Td[(1) 4ZC0,nE!(r0)H0(kjr0,n)]TJ /F12 11.955 Tf 11.95 0 Td[(r0j)dr0+Xm6=0ZCm,nE!(r0)H0(kjr0,n)]TJ /F12 11.955 Tf 11.96 0 Td[(r0j)dr0+XmZCm,)]TJ /F9 5.978 Tf 5.75 0 Td[(nE!(r0)H0(kjr0,n)]TJ /F12 11.955 Tf 11.95 0 Td[(r0j)dr0(A) 109

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wheretheintegraloverC0,nhasbeenisolatedduetothesingularityofitsintegrandatr0=r0,n.Toapproximatethisparticularintegral,Eq.( A )isusedtoobtainintheleadingorderofkR;ZC0,nE!(r0)H0(kjr0,n)]TJ /F12 11.955 Tf 11.95 0 Td[(r0j)dr0=2E!(r0,n)ZR0J0(k0u)H0(ku)uduR2E!(r0,n)1+2i +lnkR 2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2 whereistheEulerconstant.AlltheotherintegralsinEq.( A )obeytheestimate( A ).Bytakingnsuccessivelyequalto1thento)]TJ /F3 11.955 Tf 9.3 0 Td[(1,thefollowingsystemisobtained: 8>>><>>>:0E!(ae1+he3)++E!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)=i 20(k)ei(akx+hkz))]TJ /F5 11.955 Tf 7.08 -4.33 Td[(E!(ae1+he3)+0E!()]TJ /F5 11.955 Tf 9.3 0 Td[(he3)=i 20(k)e)]TJ /F8 7.97 Tf 6.58 0 Td[(ihkz(A) Thefunctions0,+and)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(are0(k,kx)=i 20(k)+1 2 Xm6=0eimkxH0(kjmj)+1+2i +lnkR 2)]TJ /F3 11.955 Tf 13.16 8.08 Td[(1 2!(a,h,k,kx)=1 2XmeimkxH0(kj(ma)e1+he3j) ThevariantsofthesefunctionsinEqs.( 3 )areobtainedthroughtheformulas, 1 21Xm=eimkxH0(kjr)]TJ /F5 11.955 Tf 11.95 0 Td[(me1j)=1Xm=ei(x(kx+2m)+jzjkz,m) kz,m,r6=0(Aa) 1 2Xm6=0eimkxH0(kjmj)=1Xm=1 kz,m)]TJ /F3 11.955 Tf 40.32 8.09 Td[(1 2(jmj+1))]TJ /F3 11.955 Tf 11.4 8.09 Td[(1 2)]TJ /F5 11.955 Tf 12.99 8.09 Td[(i +lnk 4)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2(Ab) Relation( Aa )canbeprovedbysubstitutingtheplanewaverepresentationoftheHankelfunctionH0(kr)=i 2ZZei(xKx+zKz) k2)]TJ /F5 11.955 Tf 11.95 0 Td[(K2x)]TJ /F5 11.955 Tf 11.96 0 Td[(K2z+idKzdKx,!0+ intotheleftsideof( Aa )andcarryingouttheintegrationwithrespecttoKzfollowedbyanapplicationofthePoissonsummationformulatoevaluatetheintegralwithrespect 110

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toKx.Relation( Ab )istheobtainedfrom( Aa )inthelimitr!0,whereforthetermm=0intheleftsideofEq.( Aa ),theasymptoticexpansionoftheHankelfunctionforasmallargumenthastobeused. Whenthedeterminantofthesystem( A )isnonzero,Eq.( 3 )hasauniquesolution.Otherwise,thehomogeneousLippmann-Schwingerequationadmitsnonzerosolutions:theboundstates.Thesestatescanthenbearbitrarilysuperposedtoobtainthegeneralsolution.ThevariousexpressionsfortheseboundstatesinSections 3.2 and 3.3.1 areobtainedbyapplyingformula( Aa )toEq.( A )intheabsenceoftheincidentwave,i.e.,byomittingthetermeikr. A.3ComplementsonBoundStatesintheContinuumsIandII ThissectionoftheAppendixgivessomeofthetechnicaldetailsomittedinsections 3.2.2 and 3.3.1 .First,Eq.( 3 )isestablishedforthesequencefcmg1m=1denedinEq.( 3 ).Inthecaseunderconsideration,thesequencefqz,mgisintheorder,qz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1qz,10,m=1,2,3....TocompletetheproofofEq.( 3 ),itsufcestoshowthat, cm+1 cme)]TJ /F6 7.97 Tf 6.58 0 Td[(4handlimm!1cm+1 cm=e)]TJ /F6 7.97 Tf 6.59 0 Td[(4h(A) Toestablishtherstofconditions( A ),theratioofcm+1tocmisrewrittenas, cm+1 cm=)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1 )]TJ /F8 7.97 Tf 6.59 0 Td[(m1)]TJ /F17 7.97 Tf 16.43 5.42 Td[(m+1 )]TJ /F9 5.978 Tf 5.75 0 Td[(m)]TJ /F16 5.978 Tf 5.75 0 Td[(1 1)]TJ /F17 7.97 Tf 16.03 4.7 Td[(m )]TJ /F9 5.978 Tf 5.75 0 Td[(m,m=e)]TJ /F6 7.97 Tf 6.59 0 Td[(2hqz,m qz,m(A) 111

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Next,thefollowingchainofconclusionsholds:8>><>>:qz,m+qz,)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1qz,)]TJ /F8 7.97 Tf 6.58 0 Td[(m+qz,m+1qz,mqz,)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1qz,)]TJ /F8 7.97 Tf 6.58 0 Td[(mqz,m+1))]TJ /F8 7.97 Tf 6.59 0 Td[(mm+1 )]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F6 7.97 Tf 6.58 0 Td[(1m1)1)]TJ /F17 7.97 Tf 16.43 5.43 Td[(m+1 )]TJ /F9 5.978 Tf 5.75 0 Td[(m)]TJ /F16 5.978 Tf 5.76 0 Td[(1 1)]TJ /F17 7.97 Tf 16.03 4.71 Td[(m )]TJ /F9 5.978 Tf 5.75 0 Td[(m1 Therefore,cm+1 cm)]TJ /F8 7.97 Tf 6.58 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1 )]TJ /F8 7.97 Tf 6.58 0 Td[(me2h(qz,)]TJ /F9 5.978 Tf 5.76 0 Td[(m)]TJ /F8 7.97 Tf 6.59 0 Td[(qz,)]TJ /F9 5.978 Tf 5.76 0 Td[(m)]TJ /F16 5.978 Tf 5.75 0 Td[(1) Therstofconditions( A )thenfollowsasqz,)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F5 11.955 Tf 12.11 0 Td[(qz,)]TJ /F8 7.97 Tf 6.58 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 22.31 0 Td[(2.Thelimitin( A )followsfrom( A )andthelimits,limm!1)]TJ /F8 7.97 Tf 6.58 0 Td[(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1 )]TJ /F8 7.97 Tf 6.58 0 Td[(m=e)]TJ /F6 7.97 Tf 6.59 0 Td[(4hlimm!1m )]TJ /F8 7.97 Tf 6.59 0 Td[(m=e)]TJ /F6 7.97 Tf 6.59 0 Td[(4hkx6=1Second,theformula( 3 )isproved.ThisisdonebyarepetitiveapplicationofAbel'spartialsummationformula.Letunbedenedforeachn=0,1,2,...by,un=1Xm=1cm+nsin(2am) Theobjectiveistoshowthatanotherexpressionof)]TJ /F5 11.955 Tf 9.29 0 Td[(u0is( 3 ).ByAbel'spartialsummationformula,un=1Xm=1(cm+n)]TJ /F5 11.955 Tf 11.95 0 Td[(cm+n+1)sin(am)sin(a(m+1)) sin(a)=cot(a)1Xm=1(cm+n)]TJ /F5 11.955 Tf 11.95 0 Td[(cm+n+1)sin2(am)+1 21Xm=1(cm+n)]TJ /F5 11.955 Tf 11.96 0 Td[(cm+n+1)sin(2am)=cot(a)1Xm=1(cm+n)]TJ /F5 11.955 Tf 11.95 0 Td[(cm+n+1)sin2(am)+1 2un)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2un+1 Thus,u0=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)N+1uN+1+2cot(a)1Xm=1 cm+2NXn=1()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncm+n+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)N+1cm+N+1!sin2(am) ForallN=1,2,3,...Byusingtherstofconditions( A )itisstraightforwardthatuN+1!0asN!1,andEq.( 3 )follows. 112

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Third,thefunctionsk7!n(k,kx,a)denedinEq.( 3 )areprovedtobemonotonicallydecreasing,i.e.,@kn<0.Thisderivativereads@n @k=)]TJ /F3 11.955 Tf 27.72 8.09 Td[(1 k0(k))]TJ /F13 11.955 Tf 12.35 11.36 Td[(Xm6=0k q3z,m1)]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2am)e)]TJ /F8 7.97 Tf 6.59 0 Td[(nqz,mk)]TJ /F16 5.978 Tf 5.75 0 Td[(1z1+nqz,m kz+q3z,m k3z Now,ift>0andnisapositiveinteger;thene)]TJ /F8 7.97 Tf 6.59 0 Td[(t)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1+t)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1+(n))]TJ /F6 7.97 Tf 6.58 0 Td[(2t21.Settingt=nqz,mk)]TJ /F6 7.97 Tf 6.58 0 Td[(1zshowsthatallsummandsarepositiveandhence@kn<0,8n=1,2,3,....Thefunctionsk7!n(k,kx)andk7!1(k,kx)denedinEqs.( 3 )and( 3 )areshowntobedecreasinginasimilarfashion. Lastly,thefunctionkx7!'1denedinEq.( 3 )isshowntobemonotonicallydecreasingon(0,).Toestablishthisfact,notethatsince@k1<0,theimplicitfunctiontheoremimpliesthatthefunctionkx7!k1(kx)iscontinuouslydifferentiableandk01(kx)=)]TJ /F4 11.955 Tf 9.3 0 Td[(@kx1(@k1))]TJ /F6 7.97 Tf 6.59 0 Td[(1.Now,@1 @kx=1Xm=1 2m+kx p (2m+kx)2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2)]TJ /F3 11.955 Tf 39.69 8.09 Td[(2m+2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx p (2m+2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2!>0 Hencek01(kx)>0.Bylogarithmicdifferentiation,itfollowsthat,'01(kx) '1(kx)=k1k01)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(k2x)]TJ /F3 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(k21)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2)]TJ /F3 11.955 Tf 11.96 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(k21)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x (k21)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x)(k21)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2)<0 since0
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Sincethedielectriccylindersformingthedoublearrayareassumedtobethinincomparisontothewavelength,i.e.,kR1,thequantity(20(k)))]TJ /F6 7.97 Tf 6.59 0 Td[(1intheexpressionofnislarge.Consequently,thewavenumberknsuchthatn(kn,kx,a)=0mustbeclosetothediffractionthreshold2)]TJ /F5 11.955 Tf 12.66 0 Td[(kxsothatthetermq)]TJ /F6 7.97 Tf 6.58 0 Td[(1z,)]TJ /F6 7.97 Tf 6.58 0 Td[(1innislargeenoughtocompensateforthemagnitudeof(20(k)))]TJ /F6 7.97 Tf 6.58 0 Td[(1.Thusarstapproximationforthewavenumberknmaybefoundbysolvingtheequation,1 20(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a)e)]TJ /F8 7.97 Tf 6.59 0 Td[(nqz,)]TJ /F16 5.978 Tf 5.75 0 Td[(1k)]TJ /F16 5.978 Tf 5.75 0 Td[(1z qz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1=0 whichisobtainedbykeepingonlytheleadingtermsintheexpressionofnnearkn.When()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a)=1,thentheequationbecomes, 1 20(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(e)]TJ /F8 7.97 Tf 6.59 0 Td[(nqz,)]TJ /F16 5.978 Tf 5.75 0 Td[(1k)]TJ /F16 5.978 Tf 5.75 0 Td[(1z qz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1=0(A) Inparticularifnisnotlarge,thisequationhasnorootssinceasqz,)]TJ /F6 7.97 Tf 6.58 0 Td[(1becomessmaller,thenthesecondsummandgetsclosertonk)]TJ /F6 7.97 Tf 6.59 0 Td[(1zandhenceismuchsmallerthantherstsummand.Thiswastobeexpectedsinceinthecase()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a)=1,itwasalreadyestablishedthatboundstatesexistonlyforsufcientlylargen.Also,theinitialintegernatwhichthewavenumberknexistsfor()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a)=1growsaskR!0.ThismakesitimpossibletoprovideagoodapproximationfortheexponentialterminEq.( A )thatwouldallowaperturbativesolutionofthesaidequation.Thiscomplexitydisappearswhen()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a)6=1.Inthiscaseandfornnottoolarge,theapproximatione)]TJ /F8 7.97 Tf 6.58 0 Td[(nqz,)]TJ /F16 5.978 Tf 5.76 0 Td[(1k)]TJ /F16 5.978 Tf 5.75 0 Td[(1z1isvalidandEq.( A )becomes,1 20(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a) qz,)]TJ /F6 7.97 Tf 6.59 0 Td[(1=0 Thus, knp (2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)2)]TJ /F3 11.955 Tf 11.95 0 Td[(42(1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)ncos(2a))220(2)]TJ /F5 11.955 Tf 11.96 0 Td[(kx)(A) 114

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TherstofEqs.( 3 )followsbykeepingthersttwotermsinaseriesexpansionoftheright-handsideofEq.( A )inpowersof0(2)]TJ /F5 11.955 Tf 12.02 0 Td[(kx).ThedistancehninEqs.( 3 )canthenbederivedasindicatedbysystem( 3 ). Eqs.( 3 )and( 3 )areobtainedthroughsimilartreatmentsofthefunctionsdenedinEqs.( 3 )and( 3 )respectively.Inparticular,fortheboundstatesbelowthecontinuum,itappearspossibletogiveonlytheapproximatevalueofthewavenumberk+whilethewavenumberk)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(eludestheperturbationmethodduetoacomplicateddependenceofitsexistenceconditiononthesizeofthecylinders.Similarly,inthecaseoftwoopenchannels,itturnsouttobeonlypossibletosolveforthewavenumbersk2n+1whoseexistenceisnotsubjecttochangesincylindersizes.ExpressionsanalogoustoEq.( A )forthewavenumbersk+andk2n+1are, k+p k2x)]TJ /F3 11.955 Tf 11.96 0 Td[(16220(kx)(Aa) k2n+1p (2+kx)2)]TJ /F3 11.955 Tf 11.95 0 Td[(16220(2+kx)(Ab)ToestablishEq.( 3 ),werecallthatthepointk2n+1,lxissolutiontotheequation,s k22n+1)]TJ /F5 11.955 Tf 11.96 0 Td[(k2x k22n+1)]TJ /F3 11.955 Tf 11.95 0 Td[((2)]TJ /F5 11.955 Tf 11.95 0 Td[(kx)2=l 2n+1 Hence,4kx(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2r2)+42=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(r2)k,r=l 2n+1 wherek22n+1=(2+kx)2)]TJ /F3 11.955 Tf 11.96 0 Td[(kforkgivenbyEq.( Ab ).Thus,4kx(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2r2)+42=2u(1)]TJ /F5 11.955 Tf 11.96 0 Td[(r2)(2+kx)4,u=R4("c)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2 Onecanthenlookforaseriessolutionkx=a0+a1u+a2u2+...ThisleadstoEq.( 3 ). 115

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APPENDIXBCOMPLEMENTSII B.1Estimationofand HerethelimitvaluesbandbofthefunctionsanddenedinEq.( 4 )areestimatedash!hbalongtheresonancecurveC.Forbthisisimmediate.Indeed,intheaforementionedlimit,theeldratio!1forasymmetricboundstateand,therefore,b=i40(k) kzk=kb Forb,theestimatefollowsfromthatwavenumberskbatwhichboundstatesexistareclosetothediffractionthreshold2)]TJ /F5 11.955 Tf 12.43 0 Td[(kxwhenonlyonediffractionchannelisopenforthefundamentalharmonic,i.e.,kx
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Theseexpressionsarethenusedtoestimate)]TJ /F6 7.97 Tf 6.59 0 Td[(1b=2(a+b).Intherstorderof0(kb)oneinfersthat b)]TJ /F3 11.955 Tf 23.12 8.09 Td[(1 4)]TJ /F3 11.955 Tf 11.96 0 Td[(2i0(k)Xmop,shcos2(hkshz,m) kshz,m(h,k)=(hb,kb)(B) B.2ComplementsontheFluxAnalysis:FluxConservation Forthenonlinearwaveequation( 4 ),thePoyntingTheorembecomes, 1 8d dtZV"E2+B2+ 3E3dr=)]TJ /F13 11.955 Tf 11.29 16.27 Td[(Z@VSdn(B) whereVisaclosedregion,and@Visitsboundary.ThevectorS=EBisthePoyntingvector(forsimplicity,itisassumedthat@Vliesinthevacuumsothat="=1,and=0inasmallneighborhoodof@V).Inthecaseofamonochromaticincidentwave,theuxmeasuredisthetime-averageofSoveratimeintervalT!1.ByaveragingEq.( B ),itthenfollowsthat,Z@VhSidn=0,hSi=1 TZT0S(t)dt,T!1 Thisistheuxconservation.IntermsofthedifferentharmonicsofEq.( 4 ),thetimeaveragedPoyntingvectorbecomes,hSi=)]TJ /F5 11.955 Tf 15.24 8.08 Td[(c2 2!Im 1Xl=1ElrE)]TJ /F8 7.97 Tf 6.58 0 Td[(l l! OfinterestistheuxofthePoyntingvectoracrosstherectangledepictedinFig. 4-1 (b).ByBloch'scondition( 4 ),thecontributionstotheuxfromthefacesL1=2:x=1 2canceloutsothattheuxmeasuredisthroughtheverticalfacesL=f(x,z)j)]TJ /F6 7.97 Tf 18.91 4.71 Td[(1 2x1 2,z!g.NotethatthevanishingoftheuxacrossthefacesL1=2isaconsequenceofthefactthattheincidentwaveisuniformlyextendedoverthewholex)]TJ /F1 11.955 Tf 9.29 0 Td[(axis.Forexample,considerthenormalincidence(kx=0)withonediffractionchannelopenfortheincidentradiation.ThenthePoyntingvectorofthereectedandtransmittedfundamentalharmonicisnormaltothestructureand,hence,carriesnouxacross 117

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L1=2.Thesecondharmonic(l=2)hasthreeforwardandthreebackwardscatteringchannelsopen,m=0,1,relativetothez)]TJ /F1 11.955 Tf 9.29 0 Td[(axis.Thewavewithm=0propagatesinthedirectionnormaltothestructureanddoesnotcontributetotheuxacrossL1=2.Sincetheincidentwavehasaninnitefrontalongthex)]TJ /F1 11.955 Tf 9.29 0 Td[(axis,sodothescatteredwaveswithm=1.Thewaveswithm=1andm=)]TJ /F3 11.955 Tf 9.3 0 Td[(1carryoppositeuxesacrosseachofthefacesL1=2asthecorrespondingwavevectorshavethesamez)]TJ /F1 11.955 Tf 9.3 0 Td[(componentsandoppositex)]TJ /F1 11.955 Tf 9.3 0 Td[(componentsand,hence,thetotaluxvanishes.Foranitewavefront(butmuchlargerthanthestructureperiod),thesecondharmonicwouldcarrytheenergyuxinallthedirectionsparalleltothecorrespondingwavevectorsineachopendiffractionchannel. IflisasdenedinSection 4.4 ,thentheuxconservationimpliesthatP1l=1l=1.Therefore,intheperturbationtheoryused,i.e.,whenthesystem( 4 )istruncatedtoEqs.( 4 ),theinequality1+21mustbeveriedtojustifythevalidityofthetheory. Theconversionratio2isgiveninSection 4.4 .Ifonlyonediffractionchannelisopenforthefundamentalharmonic,thentheratio1ofthescatteredandincidentuxesofthefundamentalharmonicreads,1=j1+T0j2+jR0j2 whereT0andR0arethetransmissionandreectioncoefcientswhichareobtainedfromthefar-eldamplitudeofE1as,E1!8>><>>:eirk+R0eirk)]TJ /F3 11.955 Tf 6.76 -7.15 Td[(,z!(1+T0)eirk,z!+1 118

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wherek=kxe1+kze3istheincidentwavevectorandk)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(=kxe1)]TJ /F5 11.955 Tf 11.01 0 Td[(kze3isthewavevectorofthereectedfundamentalharmonic.ItthenfollowsfromEqs.( 4 )and( 4 )that8>><>>:R0=i20(k) kzh(E1++2E2+ E1+)eihkz+(E1)]TJ /F3 11.955 Tf 9.74 1.79 Td[(+2E2)]TJ ET q .478 w 344.36 -61.41 m 353.25 -61.41 l S Q BT /F5 11.955 Tf 344.36 -71.39 Td[(E1)]TJ /F3 11.955 Tf 7.08 1.79 Td[()e)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkziT0=i20(k) kzh(E1++2E2+ E1+)e)]TJ /F8 7.97 Tf 6.59 0 Td[(ihkz+(E1)]TJ /F3 11.955 Tf 9.74 1.79 Td[(+2E2)]TJ ET q .478 w 351.59 -91.49 m 360.48 -91.49 l S Q BT /F5 11.955 Tf 351.59 -101.47 Td[(E1)]TJ /F3 11.955 Tf 7.09 1.79 Td[()eihkzi Inthevicinityofacriticalpoint(hb,kb),thecoefcientsR0andT0obeytheestimate,R0T0i40(kb) kz,b'E1+1+2jE1+j2 b Aftersomealgebraicmanipulations,itisfoundthat,1+2=1+80(kb) kz,b2jE1+j4Ab+40(kb) kz,b'2Re1 b+2jE1+j2 jbj2 whereAbistheconstantdenedas,Ab=2Scja+b+1j2)]TJ /F1 11.955 Tf 11.96 0 Td[(Im1 (h,k)=(hb,kb) andSc=Imf2+2gisintroducedinEqs.( B ).Expressingaandbdenedby( 4 )viathecoefcients2and2ofthesymmetricmatrixH2,onealsoobtainsSc=Ima+b 1+a+b Sinceatthepoint(hb,kb)thevalueofisb=(2(a+b)))]TJ /F6 7.97 Tf 6.58 0 Td[(1j(h,k)=(hb,kb),itfollowsthat,Ab=2Ima+b 1+a+bja+b+1j2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Imfa+bg(h,k)=(hb,kb) Forgeneralcomplexnumbersaandb,theexpressioninsquarebracketsisalwayszero.ThereforeAb=0,and, 1+2=1+240(kb) kz,b'jE1+j22Re1 b+2jE1+j2 jbj2(B) 119

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ByEq.( B ),Ref)]TJ /F6 7.97 Tf 6.59 0 Td[(1bg)]TJ /F3 11.955 Tf 35.18 0 Td[(4.InAppendix B.3 itisprovedthat2jE1+j2=O('2=3).Consequently,nearthecriticalpoint(hb,kb),therighthandsummandinEq.( B )isnegativesothat1+21asrequired. B.3ComplementsontheAmplitudeE1 TheamplitudeoftheeldE1+isarootofthecubicpolynomialinEq.( 4 )whichcanbesolvedbyCardano'smethod.PutY=X+2 3)]TJ /F17 7.97 Tf 6.68 -4.42 Td[(' 2Refg.ForthenewvariableY,Eq.( 4 )assumesthestandardform, Y3+pY+q=0(B) where,p='4 34)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jj2)]TJ /F3 11.955 Tf 11.96 0 Td[(2Ref()2g,q=2'6 276Refg)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(4Ref()2g)]TJ /F3 11.955 Tf 20.59 0 Td[(5jj2)]TJ /F4 11.955 Tf 13.15 8.09 Td[('2 4jj4 AstheamplitudeE1isuniquelydenedbythesystem( 4 ),itisthereforeexpectedthatthecubicinEq.( B )shouldhaveauniquerealsolutioninorderforthetheorytobeconsistent.ThelatterholdsifandonlyifthediscriminantD3=4 27p3+q2 isnonnegative.ToprovethatD30,noterstthatjj2)]TJ /F3 11.955 Tf 12.17 0 Td[(2Ref22g>0inthevicinityofacriticalpoint(hb,kb).ThisfollowsfromtheestimatesestablishedinAppendix B.1 .Indeed,intherstorderof0(kb), jbbj2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Ref(bb)2g32 k2z20(k)k=kb>0(B) Next,considerthecomplexnumber=4 272Refg2Ref()2g)]TJ /F3 11.955 Tf 21.78 8.09 Td[(5 2jj2+i(jj2)]TJ /F3 11.955 Tf 11.95 0 Td[(2Ref()2g)3 2 120

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Thepositivitycondition( B )ensuresthatthecoefcientofthecomplexnumberiintheexpressionofisindeedreal.Aftersomealgebraicmanipulations,itcanbeshownthat,D3='4 12jj22)]TJ /F4 11.955 Tf 11.95 0 Td[('42 ThusD30asrequired.TheonlyrealsolutionYtoEq.( B )isthen,Y=3s )]TJ /F5 11.955 Tf 9.3 0 Td[(q+p D3 2+3s )]TJ /F5 11.955 Tf 9.29 0 Td[(q)]TJ 11.96 9.29 Td[(p D3 2 Itthenfollowsthat, jE1+j=j'j1 3 p ++)]TJ /F3 11.955 Tf 7.09 1.8 Td[(,=3s 1 22jj2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2'4Refgjj22)]TJ /F4 11.955 Tf 11.96 0 Td[('4)]TJ /F4 11.955 Tf 13.15 8.09 Td[('4 3 3Refg(B) provided++)]TJ /F2 11.955 Tf 13.78 1.79 Td[(0.Thelatterconditionimposesalimitonthevalidityoftheperturbationtheorydevelopedinthepresentstudy,i.e.,thereductionofthesystem( 4 )to( 4 )isjustiedif++)]TJ /F2 11.955 Tf 11.78 1.8 Td[(0.Thisistobeexpectedbecauseofthelackofanalyticityincofthesolutiontothenonlinearwaveequation( 4 )thatcanonlyoccuratthecriticalpoints(hb,kb)atwhichboundstatesintheradiationcontinuumexist.Asonegetsawayfromthesecriticalpointsinthe(h,k)-plane,thesolutiontothenonlinearwaveequationbecomesanalyticinc,meaningthatallthetermsthatwereneglectedinndingtheprincipalpartsoftheamplitudesmustnowalsobetakenintoaccounttondasolutionbettingtheseriesofEq.( 4 ).TheshadowedregiondepictedinFig. 4-2 (b)showstheregionofthe(h,k)-planeinwhichthecondition++)]TJ /F2 11.955 Tf 12.1 1.79 Td[(0holdsfortherstsymmetricbound.Thepresentedanalysisoftheefciencyofthesecondharmonicgenerationisvalidforanychoiceofthegeometricalparameters,h=h)]TJ /F5 11.955 Tf 12.53 0 Td[(hbandR,andthephysicalparameters,"candc>0,whichsatisfytheconditions( 4a )and++)]TJ /F2 11.955 Tf 10.41 1.79 Td[(0. 121

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BIOGRAPHICALSKETCH RemyFriendsNdangaliwasbornin1980inRwanda.Aftertherwandancivilwar,hesuccessivelytookrefugeintheDemocraticRepublicofCongo,Kenya,andnallySenegal.ItisinthelattercountrythathegraduatedHighSchoolatCoursSecondaireSacreCoeurin1999.HethenstudiedMathematicalSciencesatCheikhAntaDiopUniversityinSenegal,andgraduatedwithanAEAin2004.IntheFallof2005,RemyenrolledintheGraduateSchooloftheUniversityofFlorida,andheearnedhisdoctorateinmathematicsin2011undertheguidanceofDr.SergeiShabanov. 124