ADIRICHLET-TO-NEUMANNAPPROACHFORTHEEXACTCOMPUTATION OFGUIDEDMODESINPHOTONICCRYSTALWAVEGUIDES SONIAFLISS† Abstract.Thisworksdealswithonedimensionalinfiniteperturbation-namelylinedefects-inperiodicmedia. Inoptics,suchdefectsarecreatedtoconstructan(open)waveguidethatconcentrateslight. Theexistenceandthe computationoftheeigenmodesisacrucialissue. Thisisrelatedtoaself-adjointeigenvalueproblemassociated toaPDEinanunboundeddomain(inthedirectionsorthogonaltothelinedefect),whichmakesboththeanalysis andthecomputationsmorecomplex. UsingaDirichlet-to-Neumann(DtN)approach,weshowthatthisproblemis 2 equivalenttoonesetonasmallneighborhoodofthedefect. Oncontrarytoexistingmethods,thisoneisexactbut 1 thereisapricetobepaid: thereductionoftheproblemleadstoanonlineareigenvalueproblemofafixedpoint 0 nature. 2 b Keywords.Periodicmedia,linedefect,guidedwaves,spectralanalysis,Dirichlet-to-Neumannoperator e F AMSsubjectclassifications.78A10,78A48,47A70,35P99 2 2 1. Introduction. Periodic media play a major role in applications, in particular in op- ticsformicroandnano-technology[20,21,27,38]. Fromthepointofviewofapplications, ] oneofthemaininterestingfeaturesisthatitcanexistintervalsoffrequenciesforwhichthe P propagativewavescannotexistinthemedia. Thisphenomenonisduetothefactthatawave S onthemediaismultiplyscatteredbytheperiodicstructure,whichcanlead,dependingonthe . h characteristicsofthemediaandthefrequency,topossiblydestructiveinterferences. Itseems t a then that an appropriate choice of the structure and the dielectric materials of the photonic m crystalcancreateparticularbandgapandthen, fromapracticalpointofview, banishsome [ monochromatic electromagnetic waves. Thus, the periodic media could be used to several potentialapplicationssuchasintherealizationoffilters,antennasandmoregenerally,com- 1 ponentsusedintelecommunications. v 8 2 Mathematically, this property is linked to the gap structure of the spectrum of the under- 9 lyingdifferentialoperatorappearinginthemodel. Foracomplete, mathematicallyoriented 4 presentation,wereferthereaderto[27,28].Evenifthenecessaryconditionsfortheexistence . 2 ofbandgapsarenotknown-exceptfortheonedimensionalcasein[4]whereauthorsstate 0 that the absence of gaps implies that the coefficients of the media are constant-, sufficient 2 conditions exist. Figotin and Kuchment have given examples of high contrast medium for 1 : whichatleastonebandgapexistsandcanbecharacterized[13,14]. Usingasymptoticargu- v ments,Nazarovandco-workershaveestablishedthatasmallperturbationofanhomogeneous i X waveguidecanopenagapinthecontinuousspectrumoftheoperator[31,7].Moreover,other r bandgapstructureshavebeencharacterizedthroughnumericalapproachesin[8]. a Besides, Sommerfeld and Bethe in 1933 made the conjecture that the number of gaps has tobefinitein2dand3D.ThisconjecturehasbeenprovenbySkriganovforSchro¨dingerop- erator in 2D [40, 36] and in 3D [41, 42] - the last article goes further: if the contrast is too small, gapscannotexistandonsuitableassumptionsonthelattice, theconjectureholdsfor higherdimensions. Parnovski[32]hasgeneralizedthislastresultprovidedthatthepotential issufficientlysmoothandin[33]theconjecturewasproveninfullgenerality. Anotherproof †LaboratoirePOEMS,UMR7231CNRS/ENSTA/INRIA, ENSTAParisTech, 32,boulevardVictor,75739ParisCedex15,France [email protected] 1 2 S.Fliss can be found in the book of Yu Karpeshina [24]. For Maxwell equations, it is much more complicated. Nevertheless, thereisnowevidencethatgapsexists. Experimentallythiswas firstobservedbyYablonovitchetal[47]. Therearealsomanynumericalresultsfordifferent structures (see [43, 44] for references). Recently, Vorobets in [46] has proven the conjec- turefor2Dperiodicmaxwelloperator(whichcorrespondstoa2Dperiodicphotoniccrystal whichishomogeneousinthethirddirection)withseparabledielectricfunction. Moreover,if thedielectricfunctioniscloseenoughtoaconstant,thereisnogapsatall. Thesemediacanpresentperturbationsordefectswhichareintroducedinthemediatochange their properties. Thus, in optic, in order to produce lasers, fibers or waveguides in general, it is necessary to have authorized frequencies inside the forbidden intervals of frequencies. This property can be obtained, for example, introducing localized defect or line defect and corresponds,fromamathematicalpointofview,toisolatedeigenvaluesoffinitemultiplicity insidethegapsoftheunderlyingdifferentialoperatorappearinginthemodel. Figotin and Klein have proven rigorously that introducing a defect in a periodic structure, namelyaperturbationofcompactsupport, cancreatedefectmodes, whichareeigenvectors associated to eigenvalues inside a gap [10, 11, 12]. More precisely, these defect modes are stationarywavesexponentiallydecreasingfarfromtheperturbation,thentheyseemsto”live” aroundthedefectwhichexplainstheirname.Usingasymptoticarguments,Nazarovhasstud- iedin[30]sufficientconditionsforexistenceofeigenvaluesbelowtheessentialspectrumfor elasticwaveguide. Hereagain,itseemsdifficulttoobtainnecessaryconditionsfortheexis- tenceofdefectmodes. There are very few works in the same spirit in the mathematical literature for the case of linedefect,thatmeansaonedimensionalperturbationoftheperiodicmedium.Afirstnatural question concerns the spectrum of the perturbed operator compared to the spectrum of the perfectlyperiodicoperator. Moreprecisely,ifagapisinthespectrumoftheperfectlyperi- odicoperator,canapartofthespectrumoftheperturbedoneariseinthisgap? Moreover,ifa partofaspectrumarises,isitofabsolutelycontinuousorsingulartype? Doesitcorresponds toguidedmodes,meaningthatthewavesareconfinedtotheguide(evanescentintheperiodic media)andtheyarepropagatingalongthelinedefect? KuchmentetOng[29]haveanswered partiallytothesequestionsforthecaseofhomogeneouslinedefect.Indeed,foranygap(a,b) oftheperfectlyperiodicoperator,theyfoundasufficientconditionona, b,thecharacteristics of the line defect (its size and its coefficients) such that spectrum of the perturbed operator arises in the gap. Moreover, they have shown that the associated eigenvector is confined to the guide. However, they have not concluded on the existence or not of bound states - this question is linked to the nature of the spectrum. Vu Hoang and Maria Radosz in [19] have shownforwaveguidein2Dperiodicstructurestheabsenceofboundstates.Itseemsthenthat ifthespectrumarisesinthegap,itcorrespondstoguidedmodes. Letusremarkthatforthe particularcaseofalineperiodicperturbationinahomogeneousmedia,Bonnet-BenDhiaand Starling[3]havegivennecessaryandsufficientconditionsfortheexistenceofguidedmodes. Finally,fewworks([1]andthepresentone)characterizedprecisely(respectivelyusingGreen’s functionorDtNoperator)theguidedwavesiftheyexist,forgenerallinedefect. Thesepar- ticularmodescanhelpforthedeterminationofthespectrumoftheperturbedoperator. The advantageisthatthestudycanbereducedtoaband(boundedinthedirectionoftheperiod- icityandinfiniteintheotherdirection). Waveguidesinperiodicmedia 3 From a numerical point of view, there exist only few methods. The most known is the Su- percellmethod. Itconsistsinmakingcomputationsinaboundeddomainoflargesizewith periodicboundaryconditions,theresultingsolutionconvergingtothetruesolutionwhenthe sizetendstoinfinity. Theconvergenceforthecomputationofdefectmodeshasbeenshown for 2D problems and compact perturbations in [45] and generalized to 3D problems and to exponentially decreasing perturbations in [6]. In this case, as the localized modes are ex- ponentiallydecreasing, thisconvergenceisexponentiallyfastwithrespecttothesizeofthe truncated domain. In practice, this approach replaces the eigenvalue problem set in an un- bounded domain to an approximated one set in a bounded domain. See [39] for numerical results. Themaindrawbackofthisstrategyreliesontheincreaseofthecomputationalcost, especially when a mode is not well confined. We can mention also the fictitious source su- perpositionmethod[5]andthereflectivescatteringmatrixmethod. ByadaptingtoeigenvalueproblemstheconstructionofDirichlet-to-Neumannoperatorsorig- inallydevelopedforscatteringproblems[22,16,15],wewanttoofferarigorouslyjustified alternativetoexistingmethods. Comparedtothesupercellmethod, theDtNmethodallows ustoreducethenumericalcomputationtoasmallneighborhoodofthedefectindependently fromtheconfinementofthecomputedguidedmodes. Moreover,asthemethodisexact,we improvetheaccuracyfornonwell-confinedguidedmodes. Obviously,thereisapricetobe paid: thereductionoftheproblemleadstoanonlineareigenvalueproblem,ofafixedpoint nature. However, this difficulty has been already overcome for homogeneous open waveg- uidesforwhichtheDtNapproachiswellknown[23,34,35]. 2. Model problem. In order to describe the medium of study, let us first consider a two-dimensional periodic medium - the photonic crystal - characterized by a coefficient ρ p (typicallythesquareoftherefractionindexofthemedium)whichis • aL∞function: ∃ρ ,ρ , 0<ρ ≤ρ (x,y)≤ρ ; − + − p + • periodicinthetwodirections ∃L ,L >0, ∀n,m∈Z, ∀(x,y)∈R2 ρ (x+nL ,y+mL )=ρ (x,y) x y p x y p wheretheperiodsL andL arenotnecessarilyequal.Weintroduceaonedimensionalinfi- x y niteperturbation-calledthelinedefect-inthey−directionin Ω =]−a,a[×R 0 andcharacterizedbyacoefficientρ whichis 0 • aL∞functionsatisfying0<ρ ≤ρ ≤ρ ; − 0 + • periodicinthey−direction ∃L0 >0, ∀m∈Z, ∀(x,y)∈Ω0 ρ (x,y+mL0)=ρ (x,y) y 0 y 0 whereL andL0 arecommensurate. Withoutlackofgenerality,wesupposeL = y y y L0. y Thepropagationmediumisthencharacterizedbythefunctionρdefinedby ρ(x,y)= ρ (x,y) in R2 ⊂Ω p 0 (2.1) ρ (x,y) in Ω . 0 0 REMARK2.1(Someextensions). 1. WecouldconsideramoregeneralmediumofpropagationthanΩ = R2. Itcouldconsist forexampleofR2 minusaperiodicsetofholes. Theonlyassumptionisthattheperiodicity 4 S.Fliss Ω Ω Ω − 0 + FIG.2.1. Domainofpropagation: typicallyρ = 1inthewhiteregion,ρ = 2inthedarkgreyregionsand ρ=3inthelightgreyregions. propertyofΩhastothebethesameasρ. 2. Evenifthespectralpropertiesaredifferent,themethodcanapplytocoefficientdefinedby ρ(x,y)= ρ−(x,y) in ]−∞,−a[×R p ρ (x,y) in Ω 0 0 ρ+(x,y) in ]a,+∞[×R. p whereρ− andρ+ havethesamepropertiesthanρ withthesameperiodinthey−direction p p p (L )andnotnecessarilythesameinthex-direction(L−andL+). y x x REMARK 2.2 (Someopenquestions). Thecasewherethelinedefecthasnotthesame periodicitypropertiesthanthephotoniccrystalandthecasewherethelinedefectisnotin- troducedalongoneofthedirectionofperiodicityofthephotoniccrystalareoutofthescope ofthepresentpaper. Thepropagationmodelisasimple2Dspace-(x = (x,y))timeharmonicscalarwaveequa- tion (corresponding for example to transverse electric (TE) mode electromagnetic wave in twodimensions) −(cid:52)w−ρω2w =0, inΩ, (P) whereω ∈R+isthefrequency.Intherestofthearticle,ω2willbeconsideredasthespectral parameter. REMARK 2.3. Theresultsdevelopedinthisarticlecanbeeasilyextendedtomoregen- eralellipticoperatoru(cid:55)→∇·(µ∇u)whereµislineperturbationofaperiodicfunction. Using the Floquet-Bloch theory, we could show that the spectrum of the operator −(cid:52)/ρ couldbededucedfromthestudyoftheguidedmodes. Aguidedmodeofthisproblemisby definitionasolutionw (cid:54)=0to(P)whichcanbewrittenintheform w(x,y)=v(x,y)eıβy (2.2) Waveguidesinperiodicmedia 5 where,infullgenerality • β -calledthequasiperiod-isin]−π/L ,π/L [; y y • visperiodicinthey−directionwithperiodL and y v(cid:12)(cid:12) ∈H1(B) whereB =R×]− Ly,Ly[ B 2 2 WewillidentifyinthefollowinganyperiodicfunctioninΩtoitsrestrictiontotheperiodB. REMARK 2.4. Infullgenerality,thecoefficientβ isinRbutitisenoughtoconsiderit onlyin]−π/L ,π/L [. Indeed,itiseasytoseethatifthereexistsamodeforβ ∈R, y y w(x,y,t)=v(x,y)eıβy, withvL -periodicandω ∈R+, y itcorrespondstoamodeforβ±π/L oftheform y w(x,y)=v˜(x,y)eı(β±π/Ly)y withv˜(x,y)=v(x,y)e∓ıπ/Lyy whichisL -periodic. y Replacingtheexpression(2.2)in(P)weseeeasilythatfindingtheguidedmodescorresponds Γ− Γ+ Σ a a B− B0 B+ Σe FIG.2.2.ThebandB-oneperiodofthedomaininthey-direction. tofindingcouples(ω2,β)suchthatthereexistsv ∈H1(B),v (cid:54)=0solutionof 1 2ıβ β2 − (cid:52)v− ∇v+ v =ω2v, inB ρ ρ ρ (2.3) v(cid:12)(cid:12)Σ =v(cid:12)(cid:12)Σ(cid:101), ∂yv(cid:12)(cid:12)Σ =∂yv(cid:12)(cid:12)(cid:12)Σ(cid:101) where(seefigure2.2)Σ=R×(cid:8)Ly(cid:9)andΣ(cid:101) =R×(cid:8)− Ly(cid:9). 2 2 Wecanrewritethepreviousproblemusingu(x,y)=v(x,y)eıβyandthenfindingtheguided modes is equivalent to finding couples (ω2,β) such that there exists u ∈ H1(B), u (cid:54)= 0 solutionof 1 − (cid:52)u=ω2u, inB ρ (2.4) u| =eıβu| , ∂ u| =eıβ∂ u| Σ Σ(cid:101) y Σ y Σ(cid:101) Tocharacterizetheguidedmodes,thereexisttwodifferentformulations:theω−formulation inwhichω isfixedandβ islookedforandtheβ−formulationinwhichβ isfixedandω is lookedfor. ThefirstoneleadstoaquadraticeigenvalueprobleminthebandB (itisobvious with the formulation (2.3) of the problem) and the second one to an eigenvalue problem in thebandB. The classical approach for solving this problem set in an unbounded domain is the Super- cellMethodwhichisbasedontheexponentiallydecreasingpropertyofthemodeinthex− 6 S.Fliss direction(seeTheorem3.5whereweremindthisresult)andconsistsintruncatingtheband B farenough. One(artificial)parameterofthemethodisthenthepositionofthetruncature position. The main advantage is that this approach leads to an eigenvalue problem (for the β-formulation)oraquadraticone(fortheω−formulation)setonaboundeddomain(thetrun- cated one). The main drawback of this strategy relies on the increase of the computational costwhenamodeisnotwellconfined. HereweproposeanovelmethodbasedonaDtNapproach,whichisoriginallydevelopedfor scattering problems. This method offers a rigorously justified alternative to existing meth- ods.Thisapproachallowsustoreducethenumericalcomputationtoasmallneighborhoodof thedefectindependentlyfromtheconfinementofthecomputedguidedmode. Themainad- vantageisthatthemethodisexactsoweimprovetheaccuracyfornonwell-confinedguided modesandcoulddeducethebehaviorofthedispersivecurves-β (cid:55)→ ω(β)where(β,ω(β)) issolutionof(2.3)-neartheedgesofthegaps. Themaindrawbackisthatthereductionof theproblemleadstoanonlineareigenvalueproblem,ofafixedpointnature. Tosimplifythe presentation,wechooseheretheω−formulationbutthemethodextendstotheotherformu- lation-whichcouldbemoreadaptedfordispersivemediaforexample. ρ=ρ(ω)). The article is organized as follows. We will remind in Section 3 the spectral properties of the locally perturbed periodic operator involved in the study. Section 4 deals with the DtN approachandpresentthenonlineareigenvalueproblemwhichhastobesolvedtocompute theguidedmodes.Itisthemostimportantandoriginalpartofthearticle.Section5isdevoted tonumericalresultsandSection6togivesomeconclusionsandperspectives. 3. Spectraltheoryresultsfortheproblem. Wefocusontheguidedmodesdefinedin (2.4)andthenreducetheproblemtothefollowingone: π π Foranyβ ∈]− , [, L L y y findω2 ∈R+, s.t.∃u∈H1(B), u(cid:54)=0, A(β)u=ω2u (E ) β where 1 A(β)=− (cid:52) ρ D(A(β))=(cid:8)u∈H1((cid:52),B), u| =eıβu| and ∂ u| =eıβ∂ u| (cid:9). Σ Σ(cid:101) y Σ y Σ(cid:101) withH1((cid:52),B)={u∈H1(B), ∆u∈L2(B)}. The problem is then reduced to the determination of eigenvalues of the locally perturbed periodicoperatorA(β),foranyβ ∈]−π/L ,π/L [. y y Weintroduceforthefollowingtheoperatorwithperfectlyperiodiccoefficients 1 A (β)=− (cid:52), D(A (β))=D(A(β)) (3.1) p ρ p p Using[26,10]andtheWeyl’stheoremwedescribeinthefollowingpropositiontheessential spectrumoftheoperatorA(β). Waveguidesinperiodicmedia 7 PROPOSITION 3.1 (Essential spectrum of A(β)). The operator A(β) is selfadjoint in H =L2(B,ρdxdy),positiveanditsessentialspectrum,denotedσ (β),satisfies ess (cid:91) σ (β)=σ(A (β))=R\ ]a (β),b (β)[ ess p n n n∈ 1,N(β) (cid:74) (cid:75) where A (β) is the operator with periodic coefficient defined by (3.1), and N(β) (0 ≤ p N(β)≤+∞)isthenumberofgaps,0≤a (β)<b (β). n n Proof. TheresolvantoperatorofA(β)isacompactperturbationoftheoneofA (β),so p theessentialspectrumofA(β)coincidewiththeessentialspectrumofA (β) p σ ((β))=σ (A (β)) (3.2) ess ess p Moreover, one of the main result of the Floquet-Bloch theory is (see [26] for more details) thatthespectrumofA (β),σ(A (β))isreducedtoitsessentialspectrum,σ (A (β))and p p ess p isgivenby (cid:91) σ(A (β))=σ (A (β))= σ(A (β,k)) (3.3) p ess p p k∈]−π/Lx,π/Lx] where 1 A (β,k)=− (cid:52), p ρ p  u| =eıkLxu| ,   ∂ ux=|Lx/2 =eıkLx∂x=u−L|x/2  D(A (β,k))= u∈H1((cid:52),C), x x=Lx/2 x x=−Lx/2 . p u| =eıβLyu| ,  ∂ uy=|Ly/2 =eıβLy∂y=u−L|y/2  y y=Ly/2 y y=−Ly/2 withC =]−L /2,L /2[×]−L /2,L /2[andH1((cid:52),C)={u∈H1(C), ∆u∈L2(C}. x x y y Moreover, for any k in ]−π/L ,π/L ], A (β,k) is a self-adjoint positive operator with x x p compactresolventsoitsspectrumispurelydiscrete 0<ω (β,k)≤ω (β,k)≤...≤ω (β,k)<... with lim ω (β,k)=+∞ 1 2 n n n→+∞ andwecanfind(e (β,k))acorrespondingfamilyofeigenvectorswhichisanhilbertianba- n sisofL2(C). Usingthemin-maxprincipletoA˜ (β,k),wecouldshowthatthesocalleddispersivecurve p (β,k)(cid:55)→ω (β,k)iscontinuous. (3.4) n Finally,wededucefrom(3.2),(3.3)and(3.4)that (cid:91) σ (β)=σ(A (β))= ω ([−π,π[,β) ess p n n∈N REMARK3.2. InSection4.4,wewillgiveanothercharacterizationoftheessentialspec- trumwithaby-productofthemethod. 8 S.Fliss AsdescribedintheIntroduction,onlysufficientconditionsofexistenceareknownandfinite- nessofthenumberofgapsisconjecturedinthegeneralcaseandprovenforparticularcases. Letussupposeinthefollowingthatatleastonegapexists(N(β) ≤ 1). Usingthetheoryof selfadjointoperators([37]),wededucethat PROPOSITION 3.3 (DiscretespectrumofA(β)). ThespectrumofA(β)insidethegaps consistsonlyofisolatedeigenvaluesoffinitemultiplicity, whichcanaccumulateonlyatthe edgesofthegap. One should ask if there is a way to ensure the rise ot at least one eigenvalue in gaps of A(β). FigotinandKleinhavegivenin[10]sufficientconditiononthedefect(supposedho- mogeneous) and the gaps to introduce eigenvalues in any gaps of A(β). Using asymptotic arguments, Nazarov and co-workers [30] have studied sufficient conditions for existence of eigenvalueforelasticwaveguides. Weareinterestednow,incharacterizingandthencomputingtheeigenvalues(λ (β)) , m m 0≤λ (β)≤λ (β)≤...≤λ (β), 0≤M(β)≤+∞ 1 2 M(β) iftheyexist(wesupposethenM(β)≥1),whichareinthegapsoftheessentialspectrum(see [2]-opticalwaveguides-and[29]-photoniccrystalwaveguides-forexistenceofeigenvalues insidegaps): PROPOSITION 3.4 (Propertiesofeachλm(β)). Thedispersioncurvesβ (cid:55)→ λm(β)are 2π/L -periodic,evenandcontinuous. y Proof. • β (cid:55)→λ (β)is2π/L -periodic: bydefinition m y A(β+2π/L )=A(β) and D(A(β+2π/L ))=D(A(β)). y y • β (cid:55)→λ (β)isevenbecause m A(β)=A(−β) and D(A(β))=D(A(−β)). andA(β)isaself-adjointpositiveoperator. • Thecontinuitywithrespecttoβ oftheeigenvaluessortedinascendedorderisdue totheanalyticityoftheoperatorA(β)withrespecttoβ [25]. Wededuceinparticularthatitissufficienttostudythedispersivecurvesforβ ∈[0,π/L ]. y We study now the properties of the eigenvectors associated to the eigenvalues. By defini- tion,theyareinL2(B)butwecanbemoreprecise: theydecayexponentiallyfastfarfrom thedefect,witharatedependingonthedistancefromtheeigenvaluetotheedgesofthegap. Itisaresultbasedon[9]andshownin[10]. THEOREM 3.5 (Exponentialdecayoftheeigenvectors). Letβ ∈]−π/Ly,π/Ly]. For anyeigenvalueλ (β)ofA(β),theassociatedeigenvectorsϕ (β;·)satisfies n n ∀n∈ 1,M(β) , ∃C ,C >0, ∃a>0,∀(x,y)∈B, |x|<a, 1 2 (cid:74) (cid:75) C |ϕ (β;x,y)|≤ 1 e(−C2dist(λn(β),σess(A(β)))|x|) (3.5) n dist(λ (β),σ (A(β))) n ess Waveguidesinperiodicmedia 9 This propriety is exactly the one which encourages to use the Supercell method. Indeed, this method consists in approaching the eigenvalues and the associated eigenvectors by the ones of a truncated problem in the x−direction with periodic conditions. If the eigenvec- torsareexponentiallydecreasinginthex−direction,theywillsatisfyalmostperiodiccondi- tionsfarenoughfromtheperturbationandthenbealmost-eigenvectorsforthealmost-same eigenvalues of an operator defined from the truncated domain. The main advantage of this approach it that it leads to eigenvalue problem for a β-formulation and a quadratic one for a ω−formulation, both of them set on a bounded domain. However, we could note several drawbacks. 1. TheessentialspectrumofA(β)hastobecomputedinitially. 2. Forafixedβ,itseemsimportanttohaveanestimationofthedistancebetweenthe notyetcomputedeigenvalueandtheessentialspectrumofA(β)tochoosearelevant truncateddomain. 3. The size of the truncated domain depends on β and on the distance between the eigenvalueandtheessentialspectrum. Iftheeigenvalueapproachesmoreandmore the essential spectrum of A(β) when β varies, the corresponding eigenvector be- comes less and less confined (less and less exponentially decreasing) and then the truncated domain has to be bigger and bigger. This can increase dramatically the computationalcost. 4. Because,thismethodisbasedontheexponentialdecayoftheeigenvectors,itcannot describe the behavior of the dispersive curves (the eigenvalues as functions of β, β (cid:55)→λ (β))neartheedgesoftheessentialspectrum. n We propose now a novel method based on a DtN approach which overcome these disad- vantages because it is an exact method (in the sense which is precised in the next section). However,itsmaindrawbackisthatitleadstoanonlineareigenvalueproblemofafixedpoint nature. 4. Thenonlineareigenvalueproblem. 4.1. The DtN approach. Let β ∈]−π/L ,π/L ] and let us suppose from this point y y thatα2 ∈/ σ (β). Wewantheretowriteanequivalentproblemtotheproblem(E )which ess β issetonaboundeddomain. Forthesequel,itisessentialtointroducefunctionalspacesappearingnaturallyinthestudy. Let us remind that B± = B ∩Ω±, Σ± = Σ∩Ω±, Σ(cid:101)± = Σ(cid:101) ∩Ω± and Γ± = {±a}×]− a L /2,L /2[(seefigure2.2). y y Westartfromsmoothquasi-periodicfunctionsinB±: C∞(B±)=(cid:110)u=u˜(cid:12)(cid:12) , u˜∈C∞(Ω±), u˜(x,y+L)=eıβLy u˜(x,y)(cid:111). β B± LetH1(B±)betheclosureofC∞(B±)inH1(B±). β β H1(B±)=(cid:110)u∈H1(B±), u(cid:12)(cid:12) =eıβLy u(cid:12)(cid:12) (cid:111). β Σ± Σ(cid:101)± where in the last equality we have identified the spaces H1/2(Σ±) and H1/2(Σ(cid:101)±). As H1(B±) is a closed subspace of H1(B±), we equip it with the norm of H1(B±). Let β H1((cid:52),B±)betheclosureofH1((cid:52),B±)={u∈H1(B±), ∆u∈L2(B±)}. β (cid:110) ∂u(cid:12) ∂u(cid:12) (cid:111) H1((cid:52),B±)= u∈H1((cid:52),B±)∩H1(B±), (cid:12) =eıβLy (cid:12) . β β ∂y(cid:12)Σ± ∂y(cid:12)Σ(cid:101)± 10 S.Fliss whereinthelastequalitywehaveidentifiedthespacesH1/2(Σ±)(cid:48)andH1/2(Σ(cid:101)±)(cid:48). 00 00 ThespaceH1/2(Γ±)isdefinedby β a (cid:16) (cid:17) H1/2(Γ±)=γ± H1(B±) β a 0 β whereγ± ∈L(H1(B±),H1/2(Γ±))isthetracemaponΓ±:∀u∈H1(B±), γ±u=u| . 0 a a 0 Γ±a H1/2(Σ ) is then a dense subspace of H1/2(Σ ) and the injection from H1/2(Γ±) onto β 0 0 β a H1/2(Γ±)iscontinuous. a WedefineH−1/2(Γ±)asthedualofH1/2(Γ±). β a β a Finally,thetraceapplicationγ± ∈L(H1((cid:52),B±),H1/2 (Γ±)(cid:48))definedby: 1 (a,a) a ∂u(cid:12) ∀u∈H1((cid:52),B±), γ±u= (cid:12) 1 ∂x(cid:12)Γ±a isacontinuousapplicationfromH1((cid:52),B±)ontoH−1/2(Γ±)andwecanshowthat β β a (cid:16) (cid:17) H−1/2(Γ±)=γ H1((cid:52),B±) . β a 1 β Let us now introduce the two half-band problems: for any β and α and for any given ϕ in H1/2(Γ±) β a −(cid:52)u±−ρ α2u± =0 inB± Findu± ∈ H1((cid:52),B±), p (P±) β u| =ϕ. Γ±a THEOREM4.1(Well-posednessoftheproblem(P±)). Ifα2 ∈/ σ (β),theproblem(P±)iswell-posedinH1exceptforacountablesetoffrequen- ess β cieswhichdependsonβ. If the periodicity cell is symmetric with respect to the axis x = 0 and if α2 ∈/ σ (β), ess theproblem(P±)isalwayswell-posedinH1. β Proof. Itisenoughtoshowtheresultfor(P+). Weintroducethefollowingoperators AD,+(β)=−ρ1p(cid:52), D(AD,+(β))=(cid:110)u+ ∈Hβ1(B+), u+(cid:12)(cid:12)Γ+a =0(cid:111) AD,−(β)=−ρ1p(cid:52), D(AD,−(β))=(cid:110)u− ∈Hβ1(Ω\B+), u−(cid:12)(cid:12)Γ+a =0(cid:111) AD(β)=−ρ1p(cid:52), D(AD(β))=(cid:110)u∈Hβ1(Ω), u(cid:12)(cid:12)Γ+a =0(cid:111) It is easy to see that the spectrum of A is the union of the spectrums of A and A . D D,+ D,− Moreover the resolvant of A is a compact perturbation of the resolvant of A so A and D D A has the same essential spectrum. These two properties implies that A has its essen- D,+ tial spectrum included in the spectrum of A but this operator may have a countable set of eigenvalues. In conclusion, α2 ∈/ σ (β), the problem (P+) is well posed if α2 is not an ess eigenvalueofA . D,+