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Bunc Compressor for the TESLA Linear Collider W

Dec king G Hostaetter T Lim erg DESY Notkestr ae 85 22603 Hambur g Germany Septem er 2000 Abstract TESLA200040 2000 discuss dieren bunc compression systems for the TESLA collider The est alternativ is wiggler yp compressor where lis

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Bunc Compressor for the TESLA Linear Collider W






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BunchCompressorfortheTESLALinearColliderW.Decking,G.Hostaetter,T.LimbergDESY,Notkestrae85,22603Hamburg,GermanySeptember2000AbstractTESLA-2000-40(2000)WediscussdierentbunchcompressionsystemsfortheTESLAcollider.Thebestalternativeisawigglertypecompressor,wherewelisttheimportantparameters.Systemswhichallowamanipulationofhigherordereectshavebeenanalyzedindetailandtheirlimitationsarederived.1IntroductionInlinearcolliders,thebeamshavetobereducedinemittanceindampingrings.ThelowerlimitonDRbunchlengthbecauseofwake-eldinducedinstabilitiesorpowerconsiderationsfortheRFsystemisafewmillimeters,toolongforoptimaloperationofthecollider.Abunchcompressionsystemhastoshortenthebunchbeforeinjectionintothemainlinac.BunchcompressionforrelativisticparticlescanonlybeachievedbyinducingacorrelationbetweenlongitudinalpositionandenergyosetwithanRFsystemandmakinguseofthepathlengthdierencesinafollowingdispersivebeamlinesection(e.g.,amagnetchicane)tobringheadandtailofthebunchclosertogether.Toovercometheinitialenergyspreadi,theRFinducedenergycorrelationmustincreasetheenergyspreadaftercompressionto:f=izizf(1)withzithebunchlengthoutofthedampingringandzfthenalbunchlength.Fromtheinducedenergyspreadthenfollowsthenecessary'longitudinaldispersion'R56=z=:R56needed=q2zi2zff:(2)ThebasicparametersfortheTESLAbunchcompressoraregivenintable1.Therequiredbigcompressionratio(zi=zf18)andthenon-negligibleincominguncorrelatedmomentumspreadnecessitatealargemomentumspreadofnearly3%1 Table1:TargetparametersfortheTESLAbunchcompressorhorizontalinputemittance"x8106mverticalinputemittance"y0:02106minputbunchlengthzi6103mDRejectionmomentumspreadi0:13%DRejectionenergyE05GeVnalbunchlengthzf0:3103mnalmomentumspreadf2:7102mR56ofcompressorchicane0:23mTotalRFvoltage@zerocrossingneeded725MVRMSinthedispersivesection.Higherordertermslikethesecondordermomentumcompactionhavetobetakenintoaccount.Chapter2describesapossiblecancelationofhigherordermomentumcompactionandnonlinearitiesintheRFwaveform.ThebeamlinesectionsupplyingthenecessaryR56mustnotincreasetransversebeamemittanceduetoincoherentorcoherentsynchrotronradiation.Bothsetcon-straintsontheusablestrengthofbendingmagnets.Soevenifthecompressionisdoneinonestage,thetotallengthofthecompressorisoftheorderofahundredmeters.Asimplesolutionisawigglerchicaneaspresentedinchapter3.Itsmaindisad-vantageisthatitssecondorderlongitudinaldispersionR566curvesthelongitudinalphasespacesomuchthattherequiredbunchlengthcannotbeachievedifnotcom-pensatedforbytheupstreamRFsystem.Thisnecessarycompensationschemecausesadecelerationofabout0.4GeV.Inchapter4wepresenteortstoavoidthisdecelerationbyusingacompressortypewheretheR566canbeadjusted.Intheso-calledFODOtypecompressor,thedispersionisshapedwithquadrupolemagnetsandtheR566canbeadjustedbysextupolemagnets.Uptonowwecouldnotndasextupoleschemewithatolerabletransverseemittancegrowth.Attachment1givesbasicideasandconceptstondopticalschemeswheretheR566iszeroandtheopticsdistortionbythenecessarysextupolemagnetsdocancelandwhyitishardtondthem.SincethepreservationofemittanceissuchacentralquestionfortheTESLAcolliderthesolutionofchoiceatthismomentisthesimplewigglerchicane.2ASecondOrderCompensationSchemeInthecaseofnon-zerosecondorderlongitudinaldispersionR566theresultinglongi-tudinalphasespacedistortioncanbecanceledbyproperchoiceoftheR56andtheacceleratingvoltagephaseandamplitude.ConsideraparticlepassingthroughanRF-systemwithphaseandacceleratingvoltageV.Thenalrelativeenergydeviationis2 in2ndorder:f=Ai+Bzi+Cz2i=EiEfi2VsinEfzi22Vcos2Efz2i(3)withEitheinitialenergy,Efthenalenergy,itheinitialrelativeenergydeviation,andzitheinitiallongitudinalparticleposition.Thenallongitudinalcoordinatezfoftheparticleafterpassagethroughadownstreamdispersivebeamlinesectionisin2ndorder:zf=zi+R56f+R5662f:(4)Combiningthesetwoequationsyields:zf=AR56i+(1+BR56)zi+(R56C+R566B2)z2i+A2R5662i+2ABR566izi+2BCR566z3i+2ACR566iz2i+C2R566z4i(5)Figure1showsthenecessaryacceleratingphase,gradient,andR56tocancelthetermsinzto2ndorderforvariousratiosr=R566=R56.Thenecessaryacceleratingvoltageissmallestforr=0.ForpositivertheRF-phasehastobetunedtoacceleratethebunch,whilefornegativeratiosrthebunchisdecelerated.Forlargesecondorderdispersionsotherhigherordertermscannotbeneglected.Forratiosrbeloworabove1:5thesecondordereectswilldistortthenallongitudinalphasespaceandleadtoalargernalbunchlength.3WigglercompressorThewigglercompressorconsistsofbendingmagnetchicanes(wiggler)embeddedinaFODOstructure.Noadditionalopticalelementsareincludedbetweenthebendingmagnetsofeachwigglersection.Inthiscasethedispersioniszerouptoanyorderattheendofeachwigglersection.Thedrawbackisthattheratiorcannotbeinuenced.Foranywigglerorchicanebasedbunchcompressorris1:5.Thesecondordereectsofthisnon-zeroR566canbecompensatedwiththebunchcompressorRFtunedtoadeceleratingphaseasdescribedabove.Therequiredacceleratingvoltagebeforethewiggleris890MVataphase=113deg.Thenalenergyis4:6GeV.Thetotallengthofthebunchcompressor(includingsomematchingand4acceleratingmoduleswithanaveragegradientof25MV=m)amountsthento165m.Theopticsofthewigglercompressorandthegeometricallayoutisshowningure2.Themaximumdeviationfromthemiddleaxisis0:3m,whichshouldteasilyinthemainlinactunnel.Trackingofparticleswhicharerandomlydistributedinthesix-dimensionalphasespacewithamaximumamplitudeof3usingthecodeMAD[5]showsnosignicantemittancegrowth.Figure3showsthelongitudinalphasespace,whilegures4and5showthehorizontalrespectivelyverticalphasespacebeforeandafterthebunchcompressor.3 -50510050100150f [deg]ratio R566/R56-5051000.10.20.30.4R56ratio R566/R56-5051012345sdf [%]ratio R566/R56-50510-2-1012ratio R566/R56V [GV];DE [GeV]V [GV]DE [GeV]Figure1:Acceleratingphase,R56,nalenergyspread,andeectiveacceleratingvoltage(indescendingorder)forabunchlengthof300mandvariousratiosr.4 Table2:ParametersoftheWigglerBunchCompressorR560:215mf2:8%VRF890VRF113degtotallength86:4m"x;sync:rad:2:2108mbendangle3:23deg,6:46degbendeld0:44Tnumberofbends12,6quadrupolelength0:2mquadrupolegradient6:5T=mnumberofquadrupoles7ThenalcheckforthewigglerperformanceisacalculationofemittancegrowthduetoCoherentSynchrotronRadiationeects.ThecodeTraFiC4modelstheincomingbunchasalineofGaussian3-dsub-buncheswhicharetrackedthroughthedispersivebeamlinesection,generatingtheelectromagneticelds.Theactionoftheeldonthegeneratingsub-bunchesleadstocorrelatedosetsoftheircenters.Inaddition,anen-sembleofafewhundredtestparticleswhicharelongitudinallypositionedclosetothebunch-centeraretrackedthroughtheeldstoprobeforuncorrelatedtransverseemit-tancegrowth.Frombothensembles,theprojectedemittanceiscalculated(seegure6).Thecorrelatedemittanceoscillatesbecauselineardispersiveosetsaresubtractedbutnotquadraticones.Atthecompressorexit,thecorrelatedemittance(normal-ized)isabout2:5107m,theuncorrelatedemittanceispreservedandtheprojectedemittancegrowsbylessthan5%.Theparametersforthewigglertypecompressorarepresentedinthetable2.4FODOcompressorMostecientuseoftheRFsystemrequiresacompressorwithapositiver.AsanexampleservesasocalledFODOcompressor.ItconsistsofaFODOchannelwithbendingmagnetsformingachicane.Theopticsandthelayoutforthisdesignisshowningure7.Notethatthatthecompactdesign(40mlength)requiresatransversedeviationofabout2.5matthemidpoint;morereasonablevaluesoflessthanameterwouldresultinoveralllengthcomparabletothewigglerchicane.TheFODOcompressorhasapositiveratior10.ThisallowsinprincipletooperatethebunchcompressorRFwithanacceleratingphase.Unfortunately,accord-ingtochapter2higherordertermsdistortthe2ndordercompensationifrassumesvaluesabove3=2.However,sextupolescanbeemployedtotuner.Atavalueof3=2,compressioncouldbedonewithanacceleratingvoltageof900MVandanenergygain5 0.020.40.60.80.100.s (m)dE/p0c=0.Table name = TWISSHP/UX version 8.23/002/11/:0 10.28.4710.15.20.25.30.35.40.45.50.b(m)-0.4-0.3-0.2-0.10.00.10.20.30.4Dx(m)bxbyDx-101020406080100x [m]Figure2:Opticalfunctionsofthewigglerbunchcompressor.beforeaftersDz [mm]sDE/Eemittance0.0059980.000302880.00131920.0284867.908e-068.6336e-060.9% of particles outside 3 sfresulting core emittance:7.9495e060.020.0100.010.020.150.10.0500.05Dz [m]DE/E00.020.0100.010.02050100150200250300#Dz [m]01002003000.150.10.0500.05DE/E0#Figure3:Longitudinalphasespacebefore(circles)andafter(crosses)thebunchcom-pressor.6 beforeaftersx [mm]spx [mrad]emittance0.000115330.000105068.6167e068.6295e069.9329e109.9457e100.8% of particles outside 3 sfresulting core emittance:9.5829e10420246x 10442024x 105x [mm]px [mrad]505x 104050100150200250300350#x [mm]010020030042024x 105px [mrad]#Figure4:Horizontalphasespacebefore(circles)andafter(crosses)thebunchcom-pressor.7 beforeaftersy [mm]spy [mrad]emittance6.4903e067.0556e061.4844e071.3049e079.6274e139.6438e130.7% of particles outside 3 sfresulting core emittance:9.3773e1342024x 105505x 107y [mm]py [mrad]42024x 105050100150200250300350#y [mm]0100200300505x 107py [mrad]#Figure5:Verticalphasespacebefore(circles)andafter(crosses)thebunchcompressor.02e-064e-066e-068e-061e-05020406080100Emittances [m-rd] Distance [m] 02e-064e-066e-068e-061e-05020406080100 Distance (m)Bunch Length [mm]02e-064e-066e-068e-061e-05020406080100 Distance (m)EnergySpread []Figure6:TraFiC4-calculationofemittancegrowthduetoCoherentSynchrotronRa-diationeects.Drawnline:sliceemittance;dottedline:projectedemittance.Thebendingmagnetsareindicatedbythe'x'dots.8 0.07.014.021.028.035.0s (m)dE/p0c=0.Table name = TWISSHP/UX version 8.23/002/11/:0 13.32.570.01.2.3.4.5.6.b(m)-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0Dx(m)bxbyDx-1012305101520253035x [m]Figure7:LayoutoftheFODObunchcompressor.of400MeV,whichcorrespondstotwoRFmodules.Inthiscasethenonlineartermswhichareintroducedbythesextupolesdistortthetrajectoriesofo-energyparticlesandcauseemittancegrowth.Toavoidthisproblemanachromatinveryhighorderforp=p0=10%isneeded.5CorrectionofR566Incaseofthedescribedwigglercompressor,R566=R561:5isnegativeandthereforethebeamwouldhavetobedeceleratebyabout400MeVinthecompressor.IntheFODOcompressorthisratioispositivebutverylargeandthereforenonlinearparticlemotionleadstoablowupofthelongitudinalemittance.WethereforetriedtominimizeR566fortherequiredjR56j0:24.Forthisoptimizationsomebasicformulaeforwillbederived.Thereferenceparticletravelsonthedesigncurveandhasthecoordinates(x;y)=0.InordertondR56andR566,oneneedstoknowthetrajectory(x();y())ofaparticlethatstartsonthedesigncurvewitharelativemomentumdeviationfromthedesignmomentump0.Sincethepathlengthofatrajectory~r(l)isgivenbyjd~rj=q(1+x=0)2+x02+y02dl;(6)thedierenceintraveledpathwithrespecttothereferenceparticleatpositionLalong9 02e-064e-066e-068e-061e-0505101520253035Emittances [m-rd] Distance [m] 02e-064e-066e-068e-061e-0505101520253035 Distance (m)Bunch Length [mm]02e-064e-066e-068e-061e-0505101520253035 Distance (m)EnergySpread []Figure8:TraFiC4-calculationofemittancegrowthduetoCoherentSynchrotronRa-diationeects.Drawn:sliceemittance;dottedline:projectedemittance.Thebendingmagnetsareindicatedbythe'x'dots.theacceleratoris=ZL0f1q[1+0x()]2+x0()2+y0()2gdl;(7)where0=1=0isthecurvatureofthedesigncurveandtheprimedenotesd=dl.Forapowerexpansionx()=x(l)+2x(l)+:::,andassumingaatbeamlinesothaty()=0,thecomponentsR56andR566ofabunchcompressoraregivenby=R56+2R566+:::=ZL00xdl2ZL0[0x+12x02+12(0x)2]dl:(8)Itisthereforeimportanttoconsiderwhatgivesrisetothetermx.ToexpresstheLorentzforceequationintermsofderivativeswithrespecttothedesignpathlengthlratherthant(indicatedbyadot),oneusesacoordinatesystemwhichisco{movingwiththereferencecurve~R(l)andcontainsthetangentialvector~t=@l~R.Hereaatreferencecurveisassumedwithacurvature0whichispiecewise0orconstant.Thisleadstotherepresentationofaparticle'scoordinatesas~r=x~ex+y~ey+~R;~r0=x0~ex+y0~ey+(1+0x)~t;(9)~r00=[x00(1+0x)0]~ex+y00~ey+20x0~t;(10)where@l~t=0~exand@l~ex=0~tisused.Fromm~r=q_~r~Boneobtainstheequationofmotionbyusingv=j_~rjandddl=j~r0jvddt;~r0=j~r0jv_~r;(11)10 ~r00=(ddlj~r0j)_~rv+(j~r0jv)2~r=(ddlj~r0j)~r0j~r0j+(j~r0jv)2qm_~r~B(12)=(~r00~r0j~r0j)~r0j~r0j+j~r0jqp~r0~B(13)(14)Withh=1+0xandj~r0j2=h2+x02+y02,thisleadstotheequation1j~r0j20h2+y02x0y0x0hx0y0h2+x02y0hx0hy0hx02+y021~r00=j~r0jqp0y0BlhByhBxx0Blx0Byy0Bx1(15)Theequationofthethirdrowisobsolete,sinceitcanbeproducedfromthersttworows.TakingonlythersttworowsandassumingthelongitudinaleldBltovanish,leadstox000hy00!=1h2h2+x02x0y0x0y0h2+y02!fj~r0jhqpáByBx!+h20x0j~r0j2x0y0!g(16)Inordertohavethedesigncurveasareferencecurve,(x;x0)=0and(y;y0)=0mustleadtox00=0.Thisleadstotherelation0=qpB0,whereB0istheverticaleldonthereferencecurve.Thisnallyleadstothetwodimensionalequationofmotionx00y00!=j~r0jhqph2+x02x0y0x0y0h2+y02!áByBx!+20x0hx0y0!+h÷00!:(17)Inthecomplexnotationw=x+iyandB=Bx+iBythisleadstothesimpliedequationw00=pw0w0+h2hqp[ih2B+w0=(w0B)]+20(w0)hw0+0h:(18)Themagneticeldofnon-skewdipoles,quadrupoles,andsextupolesisgivenbyBxBy!=p0q[00!+kqyx!+ksxy12(x2y2)!];Bl=0:(19)Forthecomputationofthesecondorderdispersionxtheycoordinateissetto0andoneobtainsx00=ph2+x023hp0p(0+kqx+ks12x2)+20x02h+h0:(20)Tondtheequationsoflinearmotion,onelinearizeswithrespecttoxandx0andobtainsx(1)00+(kq+2)x(1)=0;(21)11 withtwoindependentsolutions.Inordertoeasilyspecifyatrajectorybyitsinitialconditionsxi=x(0)andx0=x0(0),onetypicallyusesthesocalledcos{likerayxcandthesin{likerayxswithxc(0)=1;x0(0)=0;xs(0)=0;x0s(0)=1;x(1)=xixc(l)+x0ixs(l):(22)Thesolutionofthenonlinearequationofmotioncanbefoundbyvariationofconstants.Whenthenonlinearequationofmotionhastheformx(n)00+(kq+2)x(n)=f(n);(23)witharighthandsidewhichdependsonsmallquantitiesinordern,thenvariationsofconstantsoftheexpressionx(n)=A(l)xc+B(l)xsleadstoxcxsx0x0s!A0B0!=0f(n)!:(24)Thematrixhasaconstantdeterminantsinced=dl(xcx0sxsx0c)=xcx00sxsx00c=0anditisgivenbyxcx0sxsx0c=1.ThereforetheinverseoftheabovematrixissimplyobtainedbyreorderingthematrixelementsandoneobtainsA0B0!=f(n)áxsxc!;x(n)=xsZl0f(n)xcdlxcZl0f(n)xsdl:(25)Tondthedispersionxbyinsertingx=x+:::,oneusesp=p0(1+)andlinearizesinxand,x00+(kq+2)x=0:(26)Thereforethedispersionxisgivenbythewellknownformulax=xsZl00xcdlxcZl00xsdl:(27)Insertingx=x+2x+:::andtakingintoaccountallsecondordersin,oneobtainsx00+(kq+2)x=f(2);(28)f(2)=0[112x020x(20x)]+kqx(120x)ks12x2:(29)Thesecondorderdispersionisthengivenbyx=xsZf(2)xcdlxcZf(2)xsdl:(30)Forseparatedfunctionmagnets,wherethedesigncurvehasnocurvatureinquadrupoles,thetermkq0vanishes.Forthebunchcompressorswhichareconsideredhere,thedipolelengtharearoundld=1mandtheirbendingangleisbetween1and5,therefore02[0:017;0:087]1m.Theslopeofxaftertherstdipoleisx0ld0and12 theopticwillnotallowthisslopetobecomemuchlarger,sothat12x021canbeneglectedwithanerroroflessthan0:4%.Ifweconsiderupto10mspacebetweenthedipoles,thenxwillbebelow1mand0x2canbeneglectedwithanaccuracyofbetterthan5%.Ifwealsoneglect20x,whichisanapproximationwithanerroroflessthan20%,oneobtainsf(2)=0+kqxks12x2:(31)Infreespaceandindipoles,thesecondorderdispersionthereforefollowsx.InQuadrupoles,however,xobtainstheextrakickkqx.Whenxx,thesecondorderdispersionisfocusedtwotimesstrongerthanx.Thischangeofthefocusingstrengthforthesecondorderdispersionwithrespecttoxmadeithardtondabunchcompressorwhichhasnosecondorderdispersionatapositionwherealsotherstorderdispersionvanishes.Equation(8)canbeapproximateduptoanerrorofpossibly10%byR566=ZL00xdl:(32)InthefollowingitwillbedemonstratedhowsystemscanbeconstructedforwhichtheratioR566=R56iszeroorslightlypositive.Thelatterwouldbeevenbetter,becauseitwouldleadtoanaccelerationofthebeam.6FODOCompressorwithMatchedDispersionWhenaFODOcellhasasinglepassdispersion~x=(x;x0)T,thetransportmatrixMforhorizontalphasespacevectors~xleadstoM~xi+~x=~xf:(33)Sincetherstorderperiodicdispersion~satisesM~+~x=~,itcanbecomputedby~=(1M)1~x:(34)Once~isknown,thetermkqcanbecomputed,whichtheninturnleadstothesec-ondordersinglepassdispersion~xandnallytothesecondorderperiodicdispersion~.ForsimplethinlensFODOcellswithfocalstrengthkforthetwofamiliesofquadrupoles,withbendanglesb,andwithalengthwhichleadstoaphaseadvancexands=sin(x2),onecancomputetherstandsecondorderperiodicdispersionatthefocusingandthedefocusingquadrupoleas~max=bk(1s+12)0!;~min=bk(1s12)0!;~max=~min=bk1s0!:(35)13 WithR56=RL00xdlandwiththeapproximationR566=RL00dlthisleadstothetimeofighttermsR56=b24ks[7+cos(x)];R566=b24ks[9cos(x)]:(36)Theratior=R566=R561leadustotrytobaseabunchcompressoronaFODOlatticewithperiodicdispersion.ComingfromalinearacceleratorintothebunchcompressorFODOcells,thedispersionwillnotbetheperiodicdispersionbutaftertherstFODOcellitwillbethesinglepassdispersion~x1.Afterncells,thedispersionwouldbe~x=Mn1~x1+Mn2~x1+:::+~x1=(1Mn)(1M)1~x1=(1Mn)~:(37)Thecorrectnessofthesecondequivalenceischeckedbyapplying(1M)fromtherighthandside.WheneverMn=1,whichcorrespondstoabetatronphaseadvanceof,~xis2~.WhenrectangularbendsareusedinaFODO,thenthetransportmatrixdoesnotdependonthebendingangle.Thedispersion~x1inequation(27)ishoweverlinearlyrelatedtothebendingangle.IfthereforenFODOcellsarebuildwithhalfthebendingangles,then~xafterthesecellswillbe(1Mn)(1M)1~x112andforaphaseadvanceofthisisequivalenttotheperiodicdispersion~.Thisconstitutesashortproveofthewellknownhalfstrengthdispersionmatchingdescribedin[7,8].FODOcellswithreversedbenddirectionhaveastheirperiodicdispersion.IftheFODOcellsandthecellswithreducedbendingstrengthshouldbendinoppositedirection,thenthelatterwouldneedtohaveadispersionof.Butthiscannotbeachievedforanynumbernofcells.Inpassingwenotethatitwasalsotriedtousethemissingmagnetschemeforcreatingtheperiodicdispersion,butthiswaslessadvantageousforthesecondorderdispersion.ForthisschemenFODOcellsarefollowedbymFODOcellswithoutbendingmagnets.Thetotaldispersionafterthisschemewillbe~x=Mm(1Mn)(1M)1~x1:(38)Thisistheperiodicdispersion~whenever1Mn=Mm.WhenthetransportmatrixisexpressedinthenormalformspaceoftheFODO,itdescribessimplyarotationbythephaseadvancexandleadsto1cos(nx)sin(nx)sin(nx)1cos(nx)!=cos(mx)sin(mx)sin(mx)cos(mx)!:(39)Theodiagonalconditionimplieseithermx=nx+mod2,whichisincom-patiblewiththediagonalequations,ormx=nxmod2,whichtogetherwiththeequationsonthediagonalleadstomx==3mod2.Thisschemecouldbeusedtomatchtothenegativedispersion,since(1Mn)canbeMm,whichisthecasewhen1cos(nx)sin(nx)sin(nx)1cos(nx)!=ácos(mx)sin(mx)sin(mx)cos(mx)!:(40)14 Thisrequireseithermx=nxmod2ormx=nx+mod2.Therstequationisincompatiblewiththediagonalequations.Thesecondleadstonx==3mod2.Thisoptionhoweverturnedouttobenotuseful,sinceitrequiresatleast4FODOcellsforthedispersionmatch.TheFODOcompressorwhichperformedbestusesthehalfbendmatchingschemeoftwoFODOswithx=90.Theopticsisshowningure9.TherstorderdispersionxcxsxxFigure9:Top:Thecos{like(xc)andthesin{like(xs)ray.Bottom:therstandsecondorderdispersionintheFODOcompressorwithmatcheddispersion.isperiodicin4ofthe12FODOcells.Atafocusingquadrupolewherethesecondorderdispersionhasamaximum,aweaksextupolewasinsertedtomakex0=0andamirrorsymmetricsystemwasusedtobringthesecondorderdispersionbacktozero.Inordertomakethetotalbeamdirectionparallel,thesystemof6FODOcellswasrepeatedinamirrorsymmetricway.ThisFODObunchcompressorhasaratioR566=R56=2:4whichavoidsthelongi-tudinalemittanceincreaseobservedwiththepreviouslymentionedFODOcompressor.Additionallythebeamisacceleratedbyabout500MeV.Theresultinglongitudinalphasespacefor12ellipsesaroundemittancesbetween0and3afterthebunchcom-pressorareshowningure11.Theslightdeformationathighamplitudeswouldbetolerable.However,itturnsoutthat,evenafter~xiseliminated,thehigherorderdispersionsleadtoanon{negligibleincreaseoftheemittance.Thehigherorderdispersionsuptoorder15werecomputedwithCOSYINFINITY[9]andaregivenbellow.Itis15 Figure10:Twelveellipsesaroundlongitudinalemittancesofupto3afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.Herethe7thorderdispersionwascorrected,butthelongitudinalphasespaceisnotinuencedsignicantlybythehigherorderdispersioncorrection.normalizedtoa3energydeviationof10%with3==0:1,x()=mm=1933+4:243+16538:9631:273+2:7831:193+0:1103+0:091130:06123+0:021330:005143+0:0004153:(41)Aftermultipolesareinsertedinsidethe3rdand9thquadrupoletocorrectthedispersionuptoorder7,therst15ordersofthedispersionaregivenbyx()=mm=7:683+10933:01035:1113+0:9123+0:81330:11430:1153:(42)Theremaininghorizontalemittanceblowupafterthecorrectionofthe7thorderdis-persionisshowningure11(left).Theellipsearoundthe1transverseemittancehasbeentransportedthroughtheillustratedFODObunchcompressorforparticleswithupto2energydeviation.Thisgureshowsthatothernonlinearaberrationsarenotcriticalandthatthecorrectionofthe7thorderdispersionissucient.Correctingonlyuptoorder6washowevernotsucient.Ingure11(right)energydeviationsofupto2havebeenassumedforparticlesonthe1ellipse.Thisshowsthatthehigherorder16 Figure11:Left:Theellipsearoundthehorizontal1emittanceforenergydeviationsofupto2afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2emittanceforenergydeviationsofupto1afterbeingtrackedthroughthisFODOcompressor.dispersionprohibitstheuseofaFODObunchcompressorforthelargeenergyspreadofupto10%at3oftheenergydistribution.Forsmallerenergyspreads,FODOcompressorsofthistypemightbefeasible.7360CompressorEveryquadrupoleatadispersivesectioncontributestothehigherorderdispersionduetothekickxkqp0pinequation(20).Thehigherorderdispersionsmightthereforebereducedbyreducingthehorizontalphaseadvancefrom3360intheFODOcompressortoasmallervalue.WhiletheratiobetweenR566andR56issmallandpositiveinthediscussedFODOcompressorevenwithouttheaidofsextupoles,nosuchsolutionwasfoundforadevicewithsignicantlysmallerphaseadvance.However,sextupolescanbeusedtomanipulatetheintegralR566RL00xdl.Thishastobedonewhileguaranteeing~x=0.Inaddition,suchsextupolesintroducehigherorderaberrations,whichshouldbekeptsmall.Tomeettheserequirements,asymmetricarrangementwassoughtwherex=0andx0=0areguaranteedbytwomirrorsymmetricsectionsinthedevice.Additionallythegeometricaberrationsintroducedbythesextupolescanbecanceledbyaphaseadvanceof180betweensextupoles[10,11].Asshowningure12thetworequirements:(a)ofhavingtwosymmetryplanesand(b)ofhavingeachsextupolecompensatedbyasecondequivalentsextupolewhichis180apartinbetatronphase,canonlybesatisedwhenthetotalbetatronphaseadvanceisatleast360.Asuitableopticwithoutsextupolesisshowningure13.Attherstsymmetryplanethedispersioniszero,butitsslopex0isnotzero.Ananti-symmetricarrangementofthedipolemagnetsleadsto~x=0atthecentralsymmetryplane.Furthermorexsandxchavesymmetrypropertieswithrespecttotherstandtothecentralsymmetryplane.Theproductkqxandthecurvature0areantisymmetric17 180180ABFigure12:Thesymmetryplan(greencentralverticalline)requiresthateveryinsertedsextupole(redboxes)hastobeaccompaniedbyasecondsextupole.Theconditionofhavingsextupolsinpairswithaphaseof180inbetweenleadstotwomoresextupoles.Therequirementofasecondsymmetryplane(tworedverticallines)leadstoA=Bandthereforetoatotalphaseadvanceof360.withrespecttotherstplane,andthereforethesecondorderdispersionx=xsZL0xc(0+kqx)dlxcZL0xs(0+kqx)dl=xsZL0xc(0+kqx)dl:(43)Sincexs=0inthecentralplane,alsox=0.Theslopex0isnotzero,sincex0s=1inthecentralplane.Anantisymmetricarrangementofthebendingangleswithrespecttothecentralplanetheneliminates~xinthenalplane.Inordernottodestroythisproperty,sextupoles,ifapplied,needtohavethesamesymmetryasthedipoleelds.Thefollowinglistillustratesthesymmetryproperties:Symmetryat1.planecentralplane(a)0--(b)kq++(c)ks--(d)xc-+(e)xs+-(f)x--(g)xnone-ThisarrangementhasaratioR566=R56=2:8andsextupoleshavetobeusedtobringthelargeexcursionsofxtotheothersideofthedesigncurve.Thesextupolekicks18 xcxsxxFigure13:Top:Thecos{like(xc)andthesin{like(xs)ray.Bottom:therstandsecondorderdispersioninthe360compressorwithmatcheddispersion.12ksx2areeectivewhenthedispersionislarge,andthereforetherstsextupolewasplacedafterthethirdquadrupole.Theantisymmetricrequirementsleadtothreemoresextupoles,onewithreversedsignbeforethefthquadrupoleandtwomoresextupoles,eachshiftedby180withrespecttothersttwosextupoles.Therequirementofcancelinggeometricsecondorderaberrationsbythis180phaseadvanceisonlypossiblewitheithertwosymmetricorwithtwoantisymmetricplanes.Withmixedsymmetries,thesextupoleswouldalsobeapartbuttheywouldneedtohaveoppositesignswhichwouldnotleadtocancelationbuttoabuildupofaberrations.Foraberrationsweusethenotationxf=xixc+x0xs+x+x2ixxx+xix0ixxx0+x0i2xx0x0+:::(44)x0f=xix0c+x0ix0s+x0+x2ix0xx+xix0ix0xx0+x0i2x0x0x0+:::(45)(46)Thefollowinglistshowsthatnearlyallsecondorderaberrationsduetothesextupolesarecompensatedbythissetup,19 AberrationIntegralform0duetosymmetryinabovetablexxx12RL0ksxsx2dl0dueto(c1),(d1),and(e1)xxx0RL0ksx2xcdl0dueto(c2),(d2),and(e2)xx0x012RL0ksx3dl0dueto(c1)and(e1)x012RL0ksx3dl0dueto(c2)and(d2)x0xx0RL0ksxsx2cdl0dueto(c1),(d1),and(e1)x0x0x012RL0ksx2xcdl0dueto(c2),(d2),and(e2)x12RL0ksxsx2dl0dueto(c1),(e1),and(f1)x012RL0ksxcx2dl0dueto(c2),(d2),and(f2)xxRL0ksxsxcxdl0dueto(c1/2),(d1/2),(e1/2),and(f1/2)x00RL0ksxsxcxdl0dueto(c1/2),(d1/2),(e1/2),and(f1/2)Theonlysecondorderaberrationswhicharenotcanceledarexx0=ZL0ksx2xdl;x0x=ZL0ksx2cxdl:(47)Theseaberrationscannotbecompensatedbyanantisymmetry,sincex2andx2caresymmetricwithrespecttoanysymmetryplane,andalsoksxissymmetric.Buttheseaberrationswouldbeavoidedifthesextupoleswerechoseninasymmetricratherthaninanantisymmetricway.Thefollowingtwocasescanbedistinguished:(a)ksischosensymmetricatbothsymmetryplanes.Allsecondordergeomet-ricaberrationswouldstillcancelduetothephaseadvanceof180betweenequalsextupoles.Alsothesecondorderdispersionwouldstillvanishattheendsincetheintegralsintheabovetablewouldstillcancelduetotheantisymmetrypropertiesofxsandxc.However,whileksxisantisymmetricandtheintegralsinequation(47)bothvanish,theaberrationsxx0andx0xwouldnotcancel.ButtheterminaldisadvantageofthisstrategyisthefactthatR566cannotbealteredinthisway.Infact,R566ofanarrangementwithsymmetryplaneand~x=0atitsendisnotinuencedbyanarrangementofsextupolesforwhichxksisantisymmetricandforwhichthefundamentalraysxsandxcaresymmetricorantisymmetric.Wedenotethesymmetricraybyx+andtheantisymmetriconebyx.Thecontributionofthesextupolestoxisthengivenby(x+Zl0xksx2dlxZl0x+ksx2dl)=x+S(l)+x(K+A(l))orxS(l)+x+(K+A(l));(48)Therstpossibilityfollowsforasymmetricarrangementsofksandthesecondforanantisymmetricarrangement.HereS(l)issomesymmetricandA(l)someantisymmetricfunctionoflwithrespecttothesymmetryplane;Kisaconstant.Thisfollowsfromthefactthattheintegraloveranantisymmetricfunctionisasymmetricfunctionandtheintegraloverasymmetricfunctionisaconstantplusanantisymmetricfunction.ThecontributionofthesextupolestoR566isgivenbyRks566=ZL0(0xS(l)+0x(K+A(l))dl;(49)20 wheretheuppersigncorrespondstosymmetricsexutpolearrangements.Since~x=0attheendofthedevice,thecurvature0inthedipoleshasthesamesymmetryasx.Sinceksxisantisymmetric,0xS(l)isantisymmetricsincefortheuppersignksissymmetricand0antisymmetricandforthelowersignviceversa.Twopartsoftheintegralthereforehaveanantisymmetricintegrandandvanish,leavingRks566=KZL00xdl:(50)Accordingtoequation(27)isproportionaltoeitherxorx0afterthebunchcompressor,whicharebothzero.Thisprovesthatinthegiven360compressorasymmetricarrangementofsextupoles,whilenotcratingtheaberrationxx0andx0cannotbeusedtoinuenceR566.(b)Whenthesextupolesaresymmetricwithrespecttoonlyoneofthesymmetryplanes,butantisymmetricwithrespecttotheother,thenallfourchromaticaberrationsxx0,x0,xx,andx0x0wouldnotbecreatedbythesextupoles.Butsomegeometricsecondorderaberrationswouldbecreated.However,inthegivenarrangementthismightnotbeverysignicantsinceallproblemsarerelatedtotheverylargeenergyspreadofthebeam.ButagainR566couldnotbemanipulatedandeitherxorx0wouldbecomenon{zeroduetothesextupoles.TheonlypossibilitytoadjustR566isthereforetheappliedantisymmetricarrange-mentsofsextupoleswithrespecttobothsymmetryplanes.Thecriticalaberrationsofsucha360compressorarethereforetheenergydependentfocusingerrorsandthehigherorderdispersions.The15thorderpowerexpansionofthedispersionrelativetoanenergyspread=0:13witha3valueof10%,whichcorrespondsroughlytothedesiredvalueofthebunchcompressor,attheendofthesystemisx()=mm=14033+5043+4653+7163+9773+8383+1593+17103321130:7123+6:21331:2143+1:4153:(51)Inthis360bunchcompressorevencorrecting~xuptoorder7isnotsucient.Thedispersioncoecientsuptoorder15arethengivenbyx()=mm=519583+80109396571031856113+1206123+836213316401143696652153:(52)Figure14(left)showsthehorizontal1ellipseforparticleswithupto2energyspreadafterpassingthebunchcompressor.Obviouslythedispersionisstillfartoolarge.Figure14(right)showsthehorizontal2ellipseforparticleswithupto1energydeviation.Thedispersioneecthasdisappearedcompletely,duetooitshighorderdependenceon,buttheenergydependentopticsbecomesdominant.Figure15showsthelongitudinalphasespaceofupto3afterhavingbeentrackedthroughthe360bunchcompressor.SinceR566hasbeenmatchedto0,thelongitudinal21 Figure14:Left:Theellipsearoundthehorizontal1emittanceforenergydeviationsofupto2afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2emittanceforenergydeviationsofupto1afterbeingtrackedthroughthisFODOcompressor.Theellipsecorrespondstoonenergyparticles(=0).Figure15:Twelveellipsesaroundlongitudinalemittancesofupto3afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.dynamicsissatisfactory.ThelongitudinalphasespacemotionissucientlylinearduetoR566=0.Butalsoherehigherordereectscanbeobserved.Fortheheredescribedapplicationwitha3energyspreadof10%thisbunch22 compressorisnotsuitable.Forapplicationswithlessenergyspreaditmightbefeasiblehowever.8RequiredAccuracyforaWigglerCompressorExceptforthewigglercompressordescribedinthispaper,allotherarrangementswhichhavebeenanalyzedhavelargehigherorderdispersioncoecients.Thiscouldindicatethatthesymmetricmagneticeldsofthewigglercompressor,whichleadstoacancelationofallhigherorderterms,areverysensitivetomisalignments.Thecancelationwouldbemostlyviolatediftherewasanedgefocusingatoneoftheparallelfacedmagnets.Thepowerexpansionofthedispersionaftersuchasystemwhichhasapolefaceangleof0:5mradattheexitofthesecondmagnetisgivenbyx()=mm=0:02830:0087230:002233+0:0018430:0006553+0:00017630:00003773:(53)Theedgesoftheparallelfacedmagnetsofthewigglerbunchcompressorthereforehastobeparalleluptolessthanhalfamrad.Thisaccuracyof0.1mmovera50cmmagnetpolefaceshouldbeachievable.ReferencesR.Brinkmann,G.Materlik,J.Robach,A.Wagner(editors),ConceptualDesignofa500GeVe+e-LinearColliderwithIntegratedX-rayLaserFacility,ReportDESY{97{048andReprotECFA{97{182(1997)[2]R.Brinkmann,BasicAssumptionsfortheTESLATDR,DESYinternalnote(2000)[3]P.Emma,BunchCompressorBeamlinesfortheTESLAandS-BandLinearCol-liders,ReportTESLA{95{17(1995)[4]P.Emma,BunchCompressorOptionsfortheNewTESLAParameters,ReportTESLA{98{31(1998)[5]H.Grote,F.Iselin,TheMADProgram,ReportCERN/SL/90{13(AP)(1996)[6]M.Dohlus,A.Kabel,T.LimbergNIMA445,pp.338-342(2000)[7]F.Willeke,VerboteneQWertebeiPETRA,ReportDESYPET{81/28(1981)[8]K.Steen,PeriodicDispersionSuppressors,ReportDESYHERA{81/19(1981)[9]M.Berz,K.Makino,K.Shamseddine,G.H.Hostaetter,andW.Wan,COSYINFINITYanditsapplicationsinnonlineardynamics,inComputationalDieren-tiation,Techniques,Applications,andTools,pp.363{367,SIAM(1996)23 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