Bunc Compressor for the TESLA Linear Collider W - PDF document

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BunchCompressorfortheTESLALinearColliderW.Decking,G.Ho®staetter,T.LimbergDESY,Notkestra¼e85,22603Hamburg,GermanySeptember2000AbstractTESLA-2000-40(2000)Wediscussdi®erentbunchcompressionsystemsfortheTESLAcollider.Thebestalternativeisawigglertypecompressor,wherewelisttheimportantparameters.Systemswhichallowamanipulationofhigherordere®ectshavebeenanalyzedindetailandtheirlimitationsarederived.1IntroductionInlinearcolliders,thebeamshavetobereducedinemittanceindampingrings.ThelowerlimitonDRbunchlengthbecauseofwake-¯eldinducedinstabilitiesorpowerconsiderationsfortheRFsystemisafewmillimeters,toolongforoptimaloperationofthecollider.Abunchcompressionsystemhastoshortenthebunchbeforeinjectionintothemainlinac.Bunchcompressionforrelativisticparticlescanonlybeachievedbyinducingacorrelationbetweenlongitudinalpositionandenergyo®setwithanRFsystemandmakinguseofthepathlengthdi®erencesinafollowingdispersivebeamlinesection(e.g.,amagnetchicane)tobringheadandtailofthebunchclosertogether.Toovercometheinitialenergyspread¾±i,theRFinducedenergycorrelationmustincreasetheenergyspreadaftercompressionto:¾±f=¾±i¢¾zi¾zf(1)with¾zithebunchlengthoutofthedampingringand¾zfthe¯nalbunchlength.Fromtheinducedenergyspreadthenfollowsthenecessary'longitudinaldispersion'R56=¢z=±:R56needed=q¾2zi¡¾2zf¾±f:(2)ThebasicparametersfortheTESLAbunchcompressoraregivenintable1.Therequiredbigcompressionratio(¾zi=¾zf¼18)andthenon-negligibleincominguncorrelatedmomentumspreadnecessitatealargemomentumspreadofnearly3%1 Table1:TargetparametersfortheTESLAbunchcompressorhorizontalinputemittance"x8£10¡6mverticalinputemittance"y0:02£10¡6minputbunchlength¾zi6£10¡3mDRejectionmomentumspread¾±i0:13%DRejectionenergyE05GeV¯nalbunchlength¾zf0:3£10¡3m¯nalmomentumspread¾±f2:7£10¡2mR56ofcompressorchicane0:23mTotalRFvoltage@zerocrossingneeded725MVRMSinthedispersivesection.Higherordertermslikethesecondordermomentumcompactionhavetobetakenintoaccount.Chapter2describesapossiblecancelationofhigherordermomentumcompactionandnonlinearitiesintheRFwaveform.ThebeamlinesectionsupplyingthenecessaryR56mustnotincreasetransversebeamemittanceduetoincoherentorcoherentsynchrotronradiation.Bothsetcon-straintsontheusablestrengthofbendingmagnets.Soevenifthecompressionisdoneinonestage,thetotallengthofthecompressorisoftheorderofahundredmeters.Asimplesolutionisawigglerchicaneaspresentedinchapter3.Itsmaindisad-vantageisthatitssecondorderlongitudinaldispersionR566curvesthelongitudinalphasespacesomuchthattherequiredbunchlengthcannotbeachievedifnotcom-pensatedforbytheupstreamRFsystem.Thisnecessarycompensationschemecausesadecelerationofabout0.4GeV.Inchapter4wepresente®ortstoavoidthisdecelerationbyusingacompressortypewheretheR566canbeadjusted.Intheso-calledFODOtypecompressor,thedispersionisshapedwithquadrupolemagnetsandtheR566canbeadjustedbysextupolemagnets.Uptonowwecouldnot¯ndasextupoleschemewithatolerabletransverseemittancegrowth.Attachment1givesbasicideasandconceptsto¯ndopticalschemeswheretheR566iszeroandtheopticsdistortionbythenecessarysextupolemagnetsdocancelandwhyitishardto¯ndthem.SincethepreservationofemittanceissuchacentralquestionfortheTESLAcolliderthesolutionofchoiceatthismomentisthesimplewigglerchicane.2ASecondOrderCompensationSchemeInthecaseofnon-zerosecondorderlongitudinaldispersionR566theresultinglongi-tudinalphasespacedistortioncanbecanceledbyproperchoiceoftheR56andtheacceleratingvoltagephaseandamplitude.ConsideraparticlepassingthroughanRF-systemwithphaseÁandacceleratingvoltageV.The¯nalrelativeenergydeviationis2 in2ndorder:±f=A±i+Bzi+Cz2i=EiEf±i¡2¼VsinÁ¸Efzi¡2¼2VcosÁ¸2Efz2i(3)withEitheinitialenergy,Efthe¯nalenergy,±itheinitialrelativeenergydeviation,andzitheinitiallongitudinalparticleposition.The¯nallongitudinalcoordinatezfoftheparticleafterpassagethroughadownstreamdispersivebeamlinesectionisin2ndorder:zf=zi+R56±f+R566±2f:(4)Combiningthesetwoequationsyields:zf=AR56±i+(1+BR56)zi+(R56C+R566B2)z2i+A2R566±2i+2ABR566±izi+2BCR566z3i+2ACR566±iz2i+C2R566z4i(5)Figure1showsthenecessaryacceleratingphase,gradient,andR56tocancelthetermsinzto2ndorderforvariousratiosr=R566=R56.Thenecessaryacceleratingvoltageissmallestforr=0.ForpositivertheRF-phasehastobetunedtoacceleratethebunch,whilefornegativeratiosrthebunchisdecelerated.Forlargesecondorderdispersionsotherhigherordertermscannotbeneglected.Forratiosrbeloworabove§1:5thesecondordere®ectswilldistortthe¯nallongitudinalphasespaceandleadtoalarger¯nalbunchlength.3WigglercompressorThewigglercompressorconsistsofbendingmagnetchicanes(wiggler)embeddedinaFODOstructure.Noadditionalopticalelementsareincludedbetweenthebendingmagnetsofeachwigglersection.Inthiscasethedispersioniszerouptoanyorderattheendofeachwigglersection.Thedrawbackisthattheratiorcannotbein°uenced.Foranywigglerorchicanebasedbunchcompressorris¼¡1:5.Thesecondordere®ectsofthisnon-zeroR566canbecompensatedwiththebunchcompressorRFtunedtoadeceleratingphaseasdescribedabove.Therequiredacceleratingvoltagebeforethewiggleris890MVataphaseÁ=113deg.The¯nalenergyis4:6GeV.Thetotallengthofthebunchcompressor(includingsomematchingand4acceleratingmoduleswithanaveragegradientof25MV=m)amountsthento¼165m.Theopticsofthewigglercompressorandthegeometricallayoutisshownin¯gure2.Themaximumdeviationfromthemiddleaxisis0:3m,whichshould¯teasilyinthemainlinactunnel.Trackingofparticleswhicharerandomlydistributedinthesix-dimensionalphasespacewithamaximumamplitudeof3¾usingthecodeMAD[5]showsnosigni¯cantemittancegrowth.Figure3showsthelongitudinalphasespace,while¯gures4and5showthehorizontalrespectivelyverticalphasespacebeforeandafterthebunchcompressor.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,ande®ectiveacceleratingvoltage(indescendingorder)forabunchlengthof300¹mandvariousratiosr.4 Table2:ParametersoftheWigglerBunchCompressorR560:215m¾±f2:8%VRF890VÁRF113degtotallength86:4m¢"x;sync:rad:2:2£10¡8mbendangle3:23deg,6:46degbend¯eld0:44Tnumberofbends12,6quadrupolelength0:2mquadrupolegradient6:5T=mnumberofquadrupoles7The¯nalcheckforthewigglerperformanceisacalculationofemittancegrowthduetoCoherentSynchrotronRadiatione®ects.ThecodeTraFiC4modelstheincomingbunchasalineofGaussian3-dsub-buncheswhicharetrackedthroughthedispersivebeamlinesection,generatingtheelectromagnetic¯elds.Theactionofthe¯eldonthegeneratingsub-bunchesleadstocorrelatedo®setsoftheircenters.Inaddition,anen-sembleofafewhundredtestparticleswhicharelongitudinallypositionedclosetothebunch-centeraretrackedthroughthe¯eldstoprobeforuncorrelatedtransverseemit-tancegrowth.Frombothensembles,theprojectedemittanceiscalculated(see¯gure6).Thecorrelatedemittanceoscillatesbecauselineardispersiveo®setsaresubtractedbutnotquadraticones.Atthecompressorexit,thecorrelatedemittance(normal-ized)isabout2:5¢10¡7m,theuncorrelatedemittanceispreservedandtheprojectedemittancegrowsbylessthan5%.Theparametersforthewigglertypecompressorarepresentedinthetable2.4FODOcompressorMoste±cientuseoftheRFsystemrequiresacompressorwithapositiver.AsanexampleservesasocalledFODOcompressor.ItconsistsofaFODOchannelwithbendingmagnetsformingachicane.Theopticsandthelayoutforthisdesignisshownin¯gure7.Notethatthatthecompactdesign(40mlength)requiresatransversedeviationofabout2.5matthemidpoint;morereasonablevaluesoflessthanameterwouldresultinoveralllengthcomparabletothewigglerchicane.TheFODOcompressorhasapositiveratior¼10.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-diatione®ects.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.Toavoidthisproblemanachromatinveryhighorderfor¢p=p0=§10%isneeded.5CorrectionofR566Incaseofthedescribedwigglercompressor,R566=R56¼¡1:5isnegativeandthereforethebeamwouldhavetobedeceleratebyabout400MeVinthecompressor.IntheFODOcompressorthisratioispositivebutverylargeandthereforenonlinearparticlemotionleadstoablowupofthelongitudinalemittance.WethereforetriedtominimizeR566fortherequiredjR56j¼0:24.Forthisoptimizationsomebasicformulaeforwillbederived.Thereferenceparticletravelsonthedesigncurveandhasthecoordinates(x;y)=0.Inorderto¯ndR56andR566,oneneedstoknowthetrajectory(x(±);y(±))ofaparticlethatstartsonthedesigncurvewitharelativemomentumdeviation±fromthedesignmomentump0.Sincethepathlengthofatrajectory~r(l)isgivenbyjd~rj=q(1+x=½0)2+x02+y02dl;(6)thedi®erenceintraveledpathwithrespecttothereferenceparticleatpositionLalong9 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-diatione®ects.Drawn:sliceemittance;dottedline:projectedemittance.Thebendingmagnetsareindicatedbythe'x'dots.theacceleratoris¿=ZL0f1¡q[1+·0x(±)]2+x0(±)2+y0(±)2gdl;(7)where·0=1=½0isthecurvatureofthedesigncurveandtheprimedenotesd=dl.Forapowerexpansionx(±)=±x±(l)+±2x±±(l)+:::,andassuminga°atbeamlinesothaty(±)=0,thecomponentsR56andR566ofabunchcompressoraregivenby¿=±R56+±2R566+:::=¡±ZL0·0x±dl¡±2ZL0[·0x±±+12x02+12(·0x±)2]dl:(8)Itisthereforeimportanttoconsiderwhatgivesrisetothetermx±±.ToexpresstheLorentzforceequationintermsofderivativeswithrespecttothedesignpathlengthlratherthant(indicatedbyadot),oneusesacoordinatesystemwhichisco{movingwiththereferencecurve~R(l)andcontainsthetangentialvector~t=@l~R.Herea°atreferencecurveisassumedwithacurvature·0whichispiecewise0orconstant.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+2·0x0~t;(10)where@l~t=¡·0~exand@l~ex=·0~tisused.Fromm°Ä~r=q_~r£~Boneobtainstheequationofmotionbyusingv=j_~rjandddl=j~r0jvddt;~r0=j~r0jv_~r;(11)10 ~r00=(ddlj~r0j)_~rv+(j~r0jv)2Ä~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+y02¡x0y0¡x0h¡x0y0h2+x02¡y0h¡x0h¡y0hx02+y021~r00=j~r0jqp0y0Bl¡hByhBx¡x0Blx0By¡y0Bx1(15)Theequationofthethirdrowisobsolete,sinceitcanbeproducedfromthe¯rsttworows.Takingonlythe¯rsttworowsandassumingthelongitudinal¯eldBltovanish,leadstoÃx00¡·0hy00!=1h2Ãh2+x02x0y0x0y0h2+y02!fj~r0jhqpáByBx!+h2·0x0j~r0j2Ãx0y0!g(16)Inordertohavethedesigncurveasareferencecurve,(x;x0)=0and(y;y0)=0mustleadtox00=0.Thisleadstotherelation·0=qpB0,whereB0isthevertical¯eldonthereferencecurve.This¯nallyleadstothetwodimensionalequationofmotionÃx00y00!=j~r0jhqpÃh2+x02x0y0x0y0h2+y02!áByBx!+2·0x0hÃx0y0!+h÷00!:(17)Inthecomplexnotationw=x+iyandB=Bx+iBythisleadstothesimpli¯edequationw00=pw0¹w0+h2hqp[ih2B+w0=(w0¹B)]+2·0(w0)hw0+·0h:(18)Themagnetic¯eldofnon-skewdipoles,quadrupoles,andsextupolesisgivenbyÃBxBy!=p0q[·0Ã0!+kqÃyx!+ksÃxy12(x2¡y2)!];Bl=0:(19)Forthecomputationofthesecondorderdispersionx±±theycoordinateissetto0andoneobtainsx00=¡ph2+x023hp0p(·0+kqx+ks12x2)+2·0x02h+h·0:(20)To¯ndtheequationsoflinearmotion,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)xsleadstoÃxcxsx0x0s!ÃA0B0!=Ã0f(n)!:(24)Thematrixhasaconstantdeterminantsinced=dl(xcx0s¡xsx0c)=xcx00s¡xsx00c=0anditisgivenbyxcx0s¡xsx0c=1.ThereforetheinverseoftheabovematrixissimplyobtainedbyreorderingthematrixelementsandoneobtainsÃA0B0!=f(n)áxsxc!;x(n)=xsZl0f(n)xcdl¡xcZl0f(n)xsdl:(25)To¯ndthedispersionx±byinsertingx=±x±+:::,oneusesp=p0(1+±)andlinearizesinxand±,x±00+(kq+·2)x±=·0:(26)Thereforethedispersionx±isgivenbythewellknownformulax±=xsZl0·0xcdl¡xcZl0·0xsdl:(27)Insertingx=±x±+±2x±±+:::andtakingintoaccountallsecondordersin±,oneobtainsx±±00+(kq+·2)x±±=f(2);(28)f(2)=¡·0[1¡12x02¡·0x±(2¡·0x±)]+kqx±(1¡2·0x±)¡ks12x2:(29)Thesecondorderdispersionisthengivenbyx±±=xsZf(2)xcdl¡xcZf(2)xsdl:(30)Forseparatedfunctionmagnets,wherethedesigncurvehasnocurvatureinquadrupoles,thetermkq·0vanishes.Forthebunchcompressorswhichareconsideredhere,thedipolelengtharearoundld=1mandtheirbendingangleisbetween1±and5±,therefore·02[0:017;0:087]1m.Theslopeofx±afterthe¯rstdipoleisx0¼ld·0and12 theopticwillnotallowthisslopetobecomemuchlarger,sothat12x02¿1canbeneglectedwithanerroroflessthan0:4%.Ifweconsiderupto10mspacebetweenthedipoles,thenx±willbebelow1mand·0x±2canbeneglectedwithanaccuracyofbetterthan5%.Ifwealsoneglect2·0x±,whichisanapproximationwithanerroroflessthan20%,oneobtainsf(2)=¡·0+kqx±¡ks12x2:(31)Infreespaceandindipoles,thesecondorderdispersionthereforefollows¡x±.InQuadrupoles,however,x±±obtainstheextrakickkqx±.Whenx±±¼¡x±,thesecondorderdispersionisfocusedtwotimesstrongerthanx±.Thischangeofthefocusingstrengthforthesecondorderdispersionwithrespecttox±madeithardto¯ndabunchcompressorwhichhasnosecondorderdispersionatapositionwherealsothe¯rstorderdispersionvanishes.Equation(8)canbeapproximateduptoanerrorofpossibly10%byR566=¡ZL0·0x±±dl:(32)InthefollowingitwillbedemonstratedhowsystemscanbeconstructedforwhichtheratioR566=R56iszeroorslightlypositive.Thelatterwouldbeevenbetter,becauseitwouldleadtoanaccelerationofthebeam.6FODOCompressorwithMatchedDispersionWhenaFODOcellhasasinglepassdispersion~x±=(x±;x0)T,thetransportmatrixMforhorizontalphasespacevectors~xleadstoM~xi+±~x±=~xf:(33)Sincethe¯rstorderperiodicdispersion~´±satis¯esM~´±+~x±=~´±,itcanbecomputedby~´±=(1¡M)¡1~x±:(34)Once~´±isknown,thetermkq´±canbecomputed,whichtheninturnleadstothesec-ondordersinglepassdispersion~x±±and¯nallytothesecondorderperiodicdispersion~´±±.ForsimplethinlensFODOcellswithfocalstrength§kforthetwofamiliesofquadrupoles,withbendanglesb,andwithalengthwhichleadstoaphaseadvanceÁxands=sin(Áx2),onecancomputethe¯rstandsecondorderperiodicdispersionatthefocusingandthedefocusingquadrupoleas~´max±=Ãbk(1s+12)0!;~´min±=Ãbk(1s¡12)0!;~´max±±=~´min±±=Ãbk1s0!:(35)13 WithR56=¡RL0·0x±dlandwiththeapproximationR566=¡RL0·0´±±dlthisleadstothetimeof°ighttermsR56=¡b24ks[7+cos(Áx)];R566=¡b24ks[9¡cos(Áx)]:(36)Theratior=R566=R56¼1leadustotrytobaseabunchcompressoronaFODOlatticewithperiodicdispersion.ComingfromalinearacceleratorintothebunchcompressorFODOcells,thedispersionwillnotbetheperiodicdispersionbutafterthe¯rstFODOcellitwillbethesinglepassdispersion~x1.Afterncells,thedispersionwouldbe~x±=Mn¡1~x1+Mn¡2~x1+:::+~x1±=(1¡Mn)(1¡M)¡1~x1=(1¡Mn)~´±:(37)Thecorrectnessofthesecondequivalenceischeckedbyapplying(1¡M)fromtherighthandside.WheneverMn=¡1,whichcorrespondstoabetatronphaseadvanceof¼,~x±is2~´±.WhenrectangularbendsareusedinaFODO,thenthetransportmatrixdoesnotdependonthebendingangle.Thedispersion~x1inequation(27)ishoweverlinearlyrelatedtothebendingangle.IfthereforenFODOcellsarebuildwithhalfthebendingangles,then~x±afterthesecellswillbe(1¡Mn)(1¡M)¡1~x112andforaphaseadvanceof¼thisisequivalenttotheperiodicdispersion~´±.Thisconstitutesashortproveofthewellknownhalfstrengthdispersionmatchingdescribedin[7,8].FODOcellswithreversedbenddirectionhave¡´±astheirperiodicdispersion.IftheFODOcellsandthecellswithreducedbendingstrengthshouldbendinoppositedirection,thenthelatterwouldneedtohaveadispersionof¡´±.Butthiscannotbeachievedforanynumbernofcells.Inpassingwenotethatitwasalsotriedtousethemissingmagnetschemeforcreatingtheperiodicdispersion,butthiswaslessadvantageousforthesecondorderdispersion.ForthisschemenFODOcellsarefollowedbymFODOcellswithoutbendingmagnets.Thetotaldispersionafterthisschemewillbe~x±=Mm(1¡Mn)(1¡M)¡1~x1:(38)Thisistheperiodicdispersion~´±whenever1¡Mn=M¡m.WhenthetransportmatrixisexpressedinthenormalformspaceoftheFODO,itdescribessimplyarotationbythephaseadvanceÁxandleadstoÃ1¡cos(nÁx)¡sin(nÁx)sin(nÁx)1¡cos(nÁx)!=Ãcos(mÁx)¡sin(mÁx)sin(mÁx)cos(mÁx)!:(39)Theo®diagonalconditionimplieseithermÁx=¡nÁx+¼mod2¼,whichisincom-patiblewiththediagonalequations,ormÁx=nÁxmod2¼,whichtogetherwiththeequationsonthediagonalleadstomÁx=§¼=3mod2¼.Thisschemecouldbeusedtomatchtothenegativedispersion,since(1¡Mn)canbe¡M¡m,whichisthecasewhenÃ1¡cos(nÁx)¡sin(nÁx)sin(nÁx)1¡cos(nÁx)!=ácos(mÁx)sin(mÁx)¡sin(mÁx)¡cos(mÁx)!:(40)14 ThisrequireseithermÁx=¡nÁxmod2¼ormÁx=nÁx+¼mod2¼.The¯rstequationisincompatiblewiththediagonalequations.ThesecondleadstonÁx=§¼=3mod2¼.Thisoptionhoweverturnedouttobenotuseful,sinceitrequiresatleast4FODOcellsforthedispersionmatch.TheFODOcompressorwhichperformedbestusesthehalfbendmatchingschemeoftwoFODOswithÁx=90±.Theopticsisshownin¯gure9.The¯rstorderdispersionxcxsx±x±±Figure9:Top:Thecos{like(xc)andthesin{like(xs)ray.Bottom:the¯rstandsecondorderdispersionintheFODOcompressorwithmatcheddispersion.isperiodicin4ofthe12FODOcells.Atafocusingquadrupolewherethesecondorderdispersionhasamaximum,aweaksextupolewasinsertedtomakex0±=0andamirrorsymmetricsystemwasusedtobringthesecondorderdispersionbacktozero.Inordertomakethetotalbeamdirectionparallel,thesystemof6FODOcellswasrepeatedinamirrorsymmetricway.ThisFODObunchcompressorhasaratioR566=R56=2:4whichavoidsthelongi-tudinalemittanceincreaseobservedwiththepreviouslymentionedFODOcompressor.Additionallythebeamisacceleratedbyabout500MeV.Theresultinglongitudinalphasespacefor12ellipsesaroundemittancesbetween0¾and3¾afterthebunchcom-pressorareshownin¯gure11.Theslightdeformationathighamplitudeswouldbetolerable.However,itturnsoutthat,evenafter~x±±iseliminated,thehigherorderdispersionsleadtoanon{negligibleincreaseoftheemittance.Thehigherorderdispersionsuptoorder15werecomputedwithCOSYINFINITY[9]andaregivenbellow.Itis15 Figure10:Twelveellipsesaroundlongitudinalemittancesofupto3¾afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.Herethe7thorderdispersionwascorrected,butthelongitudinalphasespaceisnotin°uencedsigni¯cantlybythehigherorderdispersioncorrection.normalizedtoa3¾energydeviationof10%with±3¾=±=0:1,x(±)=mm=¡19±33¾+4:2±43¾+16±53¾¡8:9±63¾¡1:2±73¾+2:7±83¾¡1:1±93¾+0:1±103¾+0:09±113¾¡0:06±123¾+0:02±133¾¡0:005±143¾+0:0004±153¾:(41)Aftermultipolesareinsertedinsidethe3rdand9thquadrupoletocorrectthedispersionuptoorder7,the¯rst15ordersofthedispersionaregivenbyx(±)=mm=7:6±83¾+10±93¾¡3:0±103¾¡5:1±113¾+0:9±123¾+0:8±133¾¡0:1±143¾¡0:1±153¾:(42)Theremaininghorizontalemittanceblowupafterthecorrectionofthe7thorderdis-persionisshownin¯gure11(left).Theellipsearoundthe1¾transverseemittancehasbeentransportedthroughtheillustratedFODObunchcompressorforparticleswithupto2¾energydeviation.This¯gureshowsthatothernonlinearaberrationsarenotcriticalandthatthecorrectionofthe7thorderdispersionissu±cient.Correctingonlyuptoorder6washowevernotsu±cient.In¯gure11(right)energydeviationsofupto2¾havebeenassumedforparticlesonthe1¾ellipse.Thisshowsthatthehigherorder16 Figure11:Left:Theellipsearoundthehorizontal1¾emittanceforenergydeviationsofupto2¾afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2¾emittanceforenergydeviationsofupto1¾afterbeingtrackedthroughthisFODOcompressor.dispersionprohibitstheuseofaFODObunchcompressorforthelargeenergyspreadofupto10%at3¾oftheenergydistribution.Forsmallerenergyspreads,FODOcompressorsofthistypemightbefeasible.7360±CompressorEveryquadrupoleatadispersivesectioncontributestothehigherorderdispersionduetothekickxkqp0pinequation(20).Thehigherorderdispersionsmightthereforebereducedbyreducingthehorizontalphaseadvancefrom3¢360±intheFODOcompressortoasmallervalue.WhiletheratiobetweenR566andR56issmallandpositiveinthediscussedFODOcompressorevenwithouttheaidofsextupoles,nosuchsolutionwasfoundforadevicewithsigni¯cantlysmallerphaseadvance.However,sextupolescanbeusedtomanipulatetheintegralR566¼¡RL0·0x±±dl.Thishastobedonewhileguaranteeing~x±±=0.Inaddition,suchsextupolesintroducehigherorderaberrations,whichshouldbekeptsmall.Tomeettheserequirements,asymmetricarrangementwassoughtwherex±±=0andx0±=0areguaranteedbytwomirrorsymmetricsectionsinthedevice.Additionallythegeometricaberrationsintroducedbythesextupolescanbecanceledbyaphaseadvanceof180±betweensextupoles[10,11].Asshownin¯gure12thetworequirements:(a)ofhavingtwosymmetryplanesand(b)ofhavingeachsextupolecompensatedbyasecondequivalentsextupolewhichis180±apartinbetatronphase,canonlybesatis¯edwhenthetotalbetatronphaseadvanceisatleast360±.Asuitableopticwithoutsextupolesisshownin¯gure13.Atthe¯rstsymmetryplanethedispersioniszero,butitsslopex0isnotzero.Ananti-symmetricarrangementofthedipolemagnetsleadsto~x±=0atthecentralsymmetryplane.Furthermorexsandxchavesymmetrypropertieswithrespecttothe¯rstandtothecentralsymmetryplane.Theproductkqx±andthecurvature·0areantisymmetric17 180±180±ABFigure12:Thesymmetryplan(greencentralverticalline)requiresthateveryinsertedsextupole(redboxes)hastobeaccompaniedbyasecondsextupole.Theconditionofhavingsextupolsinpairswithaphaseof180±inbetweenleadstotwomoresextupoles.Therequirementofasecondsymmetryplane(tworedverticallines)leadstoA=Bandthereforetoatotalphaseadvanceof360±.withrespecttothe¯rstplane,andthereforethesecondorderdispersionx±±=xsZL0xc(¡·0+kqx±)dl¡xcZL0xs(¡·0+kqx±)dl=xsZL0xc(¡·0+kqx±)dl:(43)Sincexs=0inthecentralplane,alsox±±=0.Theslopex0±isnotzero,sincex0s=¡1inthecentralplane.Anantisymmetricarrangementofthebendingangleswithrespecttothecentralplanetheneliminates~x±±inthe¯nalplane.Inordernottodestroythisproperty,sextupoles,ifapplied,needtohavethesamesymmetryasthedipole¯elds.Thefollowinglistillustratesthesymmetryproperties:Symmetryat1.planecentralplane(a)·0--(b)kq++(c)ks--(d)xc-+(e)xs+-(f)x±--(g)x±±none-ThisarrangementhasaratioR566=R56=¡2:8andsextupoleshavetobeusedtobringthelargeexcursionsofx±±totheothersideofthedesigncurve.Thesextupolekicks18 xcxsx±x±±Figure13:Top:Thecos{like(xc)andthesin{like(xs)ray.Bottom:the¯rstandsecondorderdispersioninthe360±compressorwithmatcheddispersion.12ksx2aree®ectivewhenthedispersionislarge,andthereforethe¯rstsextupolewasplacedafterthethirdquadrupole.Theantisymmetricrequirementsleadtothreemoresextupoles,onewithreversedsignbeforethe¯fthquadrupoleandtwomoresextupoles,eachshiftedby180±withrespecttothe¯rsttwosextupoles.Therequirementofcancelinggeometricsecondorderaberrationsbythis180±phaseadvanceisonlypossiblewitheithertwosymmetricorwithtwoantisymmetricplanes.Withmixedsymmetries,thesextupoleswouldalsobe¼apartbuttheywouldneedtohaveoppositesignswhichwouldnotleadtocancelationbuttoabuildupofaberrations.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)x0¡12RL0ksx3dl0dueto(c2)and(d2)x0xx0¡RL0ksxsx2cdl0dueto(c1),(d1),and(e1)x0x0x0¡12RL0ksx2xcdl0dueto(c2),(d2),and(e2)x±±12RL0ksxsx2dl0dueto(c1),(e1),and(f1)x0±±¡12RL0ksxcx2dl0dueto(c2),(d2),and(f2)xx±RL0ksxsxcx±dl0dueto(c1/2),(d1/2),(e1/2),and(f1/2)x00±¡RL0ksxsxcx±dl0dueto(c1/2),(d1/2),(e1/2),and(f1/2)Theonlysecondorderaberrationswhicharenotcanceledarexx0±=ZL0ksx2x±dl;x0x±=¡ZL0ksx2cx±dl:(47)Theseaberrationscannotbecompensatedbyanantisymmetry,sincex2andx2caresymmetricwithrespecttoanysymmetryplane,andalsoksx±issymmetric.Buttheseaberrationswouldbeavoidedifthesextupoleswerechoseninasymmetricratherthaninanantisymmetricway.Thefollowingtwocasescanbedistinguished:(a)ksischosensymmetricatbothsymmetryplanes.Allsecondordergeomet-ricaberrationswouldstillcancelduetothephaseadvanceof180±betweenequalsextupoles.Alsothesecondorderdispersionwouldstillvanishattheendsincetheintegralsintheabovetablewouldstillcancelduetotheantisymmetrypropertiesofxsandxc.However,whileksx±isantisymmetricandtheintegralsinequation(47)bothvanish,theaberrationsxx0±andx0x±wouldnotcancel.ButtheterminaldisadvantageofthisstrategyisthefactthatR566cannotbealteredinthisway.Infact,R566ofanarrangementwithsymmetryplaneand~x±=0atitsendisnotin°uencedbyanarrangementofsextupolesforwhichx±ksisantisymmetricandforwhichthefundamentalraysxsandxcaresymmetricorantisymmetric.Wedenotethesymmetricraybyx+andtheantisymmetriconebyx¡.Thecontributionofthesextupolestox±±isthengivenby§(x+Zl0x¡ksx2dl¡x¡Zl0x+ksx2±dl)=x+S(l)+x¡(K+A(l))orx¡S(l)+x+(K+A(l));(48)The¯rstpossibilityfollowsforasymmetricarrangementsofksandthesecondforanantisymmetricarrangement.HereS(l)issomesymmetricandA(l)someantisymmetricfunctionoflwithrespecttothesymmetryplane;Kisaconstant.Thisfollowsfromthefactthattheintegraloveranantisymmetricfunctionisasymmetricfunctionandtheintegraloverasymmetricfunctionisaconstantplusanantisymmetricfunction.ThecontributionofthesextupolestoR566isgivenby§Rks566=¡ZL0(·0x§S(l)+·0x¨(K+A(l))dl;(49)20 wheretheuppersigncorrespondstosymmetricsexutpolearrangements.Since~x±=0attheendofthedevice,thecurvature·0inthedipoleshasthesamesymmetryasx±.Sinceksx±isantisymmetric,·0x§S(l)isantisymmetricsincefortheuppersignksissymmetricand·0antisymmetricandforthelowersignviceversa.Twopartsoftheintegralthereforehaveanantisymmetricintegrandandvanish,leavingRks566=¡KZL0·0x¨dl:(50)Accordingtoequation(27)isproportionaltoeitherx±orx0afterthebunchcompressor,whicharebothzero.Thisprovesthatinthegiven360±compressorasymmetricarrangementofsextupoles,whilenotcratingtheaberrationxx0±andx0cannotbeusedtoin°uenceR566.(b)Whenthesextupolesaresymmetricwithrespecttoonlyoneofthesymmetryplanes,butantisymmetricwithrespecttotheother,thenallfourchromaticaberrationsxx0±,x0,xx±,andx0x0±wouldnotbecreatedbythesextupoles.Butsomegeometricsecondorderaberrationswouldbecreated.However,inthegivenarrangementthismightnotbeverysigni¯cantsinceallproblemsarerelatedtotheverylargeenergyspreadofthebeam.ButagainR566couldnotbemanipulatedandeitherx±±orx0±wouldbecomenon{zeroduetothesextupoles.TheonlypossibilitytoadjustR566isthereforetheappliedantisymmetricarrange-mentsofsextupoleswithrespecttobothsymmetryplanes.Thecriticalaberrationsofsucha360±compressorarethereforetheenergydependentfocusingerrorsandthehigherorderdispersions.The15thorderpowerexpansionofthedispersionrelativetoanenergyspread±=0:1¢±3¾witha3¾valueof10%,whichcorrespondsroughlytothedesiredvalueofthebunchcompressor,attheendofthesystemisx(±)=mm=¡140±33¾+50±43¾+46±53¾+71±63¾+97±73¾+83±83¾+15±93¾+17±103¾¡32±113¾¡0:7±123¾+6:2±133¾¡1:2±143¾+1:4±153¾:(51)Inthis360±bunchcompressorevencorrecting~x±±uptoorder7isnotsu±cient.Thedispersioncoe±cientsuptoorder15arethengivenbyx(±)=mm=5195±83¾+8010±93¾¡9657±103¾¡1856±113¾+1206±123¾+8362±133¾¡16401±143¾¡696652±153¾:(52)Figure14(left)showsthehorizontal1¾ellipseforparticleswithupto2¾energyspreadafterpassingthebunchcompressor.Obviouslythedispersionisstillfartoolarge.Figure14(right)showsthehorizontal2¾ellipseforparticleswithupto1¾energydeviation.Thedispersione®ecthasdisappearedcompletely,duetooitshighorderdependenceon±,buttheenergydependentopticsbecomesdominant.Figure15showsthelongitudinalphasespaceofupto3¾afterhavingbeentrackedthroughthe360±bunchcompressor.SinceR566hasbeenmatchedto0,thelongitudinal21 Figure14:Left:Theellipsearoundthehorizontal1¾emittanceforenergydeviationsofupto2¾afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2¾emittanceforenergydeviationsofupto1¾afterbeingtrackedthroughthisFODOcompressor.Theellipsecorrespondstoonenergyparticles(±=0).Figure15:Twelveellipsesaroundlongitudinalemittancesofupto3¾afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.dynamicsissatisfactory.Thelongitudinalphasespacemotionissu±cientlylinearduetoR566=0.Butalsoherehigherordere®ectscanbeobserved.Fortheheredescribedapplicationwitha3¾energyspreadof10%thisbunch22 compressorisnotsuitable.Forapplicationswithlessenergyspreaditmightbefeasiblehowever.8RequiredAccuracyforaWigglerCompressorExceptforthewigglercompressordescribedinthispaper,allotherarrangementswhichhavebeenanalyzedhavelargehigherorderdispersioncoe±cients.Thiscouldindicatethatthesymmetricmagnetic¯eldsofthewigglercompressor,whichleadstoacancelationofallhigherorderterms,areverysensitivetomisalignments.Thecancelationwouldbemostlyviolatediftherewasanedgefocusingatoneoftheparallelfacedmagnets.Thepowerexpansionofthedispersionaftersuchasystemwhichhasapolefaceangleof0:5mradattheexitofthesecondmagnetisgivenbyx(±)=mm=0:028±3¾¡0:0087±23¾¡0:0022±33¾+0:0018±43¾¡0:00065±53¾+0:00017±63¾¡0:000037±73¾:(53)Theedgesoftheparallelfacedmagnetsofthewigglerbunchcompressorthereforehastobeparalleluptolessthanhalfamrad.Thisaccuracyof0.1mmovera50cmmagnetpolefaceshouldbeachievable.ReferencesR.Brinkmann,G.Materlik,J.Ro¼bach,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.Ste®en,PeriodicDispersionSuppressors,ReportDESYHERA{81/19(1981)[9]M.Berz,K.Makino,K.Shamseddine,G.H.Ho®staetter,andW.Wan,COSYINFINITYanditsapplicationsinnonlineardynamics,inComputationalDi®eren-tiation,Techniques,Applications,andTools,pp.363{367,SIAM(1996)23 [10]K.L.Brown,ASecond{OrderMagneticOpticalAchromat,ReportSLAC{PUB{2257(February1979)[11]K.L.BrownandR.V.Servranckx,First{andSecond{OrderChargedParticleOptics,ReportSLAC{PUB{3381(1984)24