Bunc Compressor for the TESLA Linear Collider W - Pdf

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BunchCompressorfortheTESLALinearColliderW.Decking,G.Ho�staetter,T.LimbergDESY,Notkestra�e85,22603Hamburg,GermanySeptember2000AbstractTESLA-2000-40(2000)Wediscussdi�erentbunchcompressionsystemsfortheTESLAcollider.Thebestalternativeisawigglertypecompressor,wherewelisttheimportantparameters.Systemswhichallowamanipulationofhigherordere�ectshavebeenanalyzedindetailandtheirlimitationsarederived.1IntroductionInlinearcolliders,thebeamshavetobereducedinemittanceindampingrings.ThelowerlimitonDRbunchlengthbecauseofwake-�eldinducedinstabilitiesorpowerconsiderationsfortheRFsystemisafewmillimeters,toolongforoptimaloperationofthecollider.Abunchcompressionsystemhastoshortenthebunchbeforeinjectionintothemainlinac.Bunchcompressionforrelativisticparticlescanonlybeachievedbyinducingacorrelationbetweenlongitudinalpositionandenergyo�setwithanRFsystemandmakinguseofthepathlengthdi�erencesinafollowingdispersivebeamlinesection(e.g.,amagnetchicane)tobringheadandtailofthebunchclosertogether.Toovercometheinitialenergyspread��i,theRFinducedenergycorrelationmustincreasetheenergyspreadaftercompressionto:��f=��i��zi�zf(1)with�zithebunchlengthoutofthedampingringand�zfthe�nalbunchlength.Fromtheinducedenergyspreadthenfollowsthenecessary'longitudinaldispersion'R56=�z=�:R56needed=q�2zi��2zf��f:(2)ThebasicparametersfortheTESLAbunchcompressoraregivenintable1.Therequiredbigcompressionratio(�zi=�zf�18)andthenon-negligibleincominguncorrelatedmomentumspreadnecessitatealargemomentumspreadofnearly3%1 Table1:TargetparametersfortheTESLAbunchcompressorhorizontalinputemittance"x8�10�6mverticalinputemittance"y0:02�10�6minputbunchlength�zi6�10�3mDRejectionmomentumspread��i0:13%DRejectionenergyE05GeV�nalbunchlength�zf0:3�10�3m�nalmomentumspread��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-systemwithphase�andacceleratingvoltageV.The�nalrelativeenergydeviationis2 in2ndorder:�f=A�i+Bzi+Cz2i=EiEf�i�2�Vsin��Efzi�2�2Vcos��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.Foranywigglerorchicanebasedbunchcompressorris��1:5.Thesecondordere�ectsofthisnon-zeroR566canbecompensatedwiththebunchcompressorRFtunedtoadeceleratingphaseasdescribedabove.Therequiredacceleratingvoltagebeforethewiggleris890MVataphase�=113deg.The�nalenergyis4:6GeV.Thetotallengthofthebunchcompressor(includingsomematchingand4acceleratingmoduleswithanaveragegradientof25MV=m)amountsthento�165m.Theopticsofthewigglercompressorandthegeometricallayoutisshownin�gure2.Themaximumdeviationfromthemiddleaxisis0:3m,whichshould�teasilyinthemainlinactunnel.Trackingofparticleswhicharerandomlydistributedinthesix-dimensionalphasespacewithamaximumamplitudeof3�usingthecodeMAD[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)forabunchlengthof300�mandvariousratiosr.4 Table2:ParametersoftheWigglerBunchCompressorR560:215m��f2:8%VRF890V�RF113degtotallength86:4m�"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.TheFODOcompressorhasapositiveratior�10.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=R56��1:5isnegativeandthereforethebeamwouldhavetobedeceleratebyabout400MeVinthecompressor.IntheFODOcompressorthisratioispositivebutverylargeandthereforenonlinearparticlemotionleadstoablowupofthelongitudinalemittance.WethereforetriedtominimizeR566fortherequiredjR56j�0:24.Forthisoptimizationsomebasicformulaeforwillbederived.Thereferenceparticletravelsonthedesigncurveandhasthecoordinates(x;y)=0.Inorderto�ndR56andR566,oneneedstoknowthetrajectory(x(�);y(�))ofaparticlethatstartsonthedesigncurvewitharelativemomentumdeviation�fromthedesignmomentump0.Sincethepathlengthofatrajectory~r(l)isgivenbyjd~rj=q(1+x=�0)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=�0isthecurvatureofthedesigncurveandtheprimedenotesd=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��~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+y02�x0y0�x0h�x0y0h2+x02�y0h�x0h�y0hx02+y021~r00=j~r0jqp0y0Bl�hByhBx�x0Blx0By�y0Bx1(15)Theequationofthethirdrowisobsolete,sinceitcanbeproducedfromthe�rsttworows.Takingonlythe�rsttworowsandassumingthelongitudinal�eldBltovanish,leadsto�x00��0hy00!=1h2�h2+x02x0y0x0y0h2+y02!fj~r0jhqpáByBx!+h2�0x0j~r0j2�x0y0!g(16)Inordertohavethedesigncurveasareferencecurve,(x;x0)=0and(y;y0)=0mustleadtox00=0.Thisleadstotherelation�0=qpB0,whereB0isthevertical�eldonthereferencecurve.This�nallyleadstothetwodimensionalequationofmotion�x00y00!=j~r0jhqp�h2+x02x0y0x0y0h2+y02!áByBx!+2�0x0h�x0y0!+h÷00!:(17)Inthecomplexnotationw=x+iyandB=Bx+iBythisleadstothesimpli�edequationw00=pw0�w0+h2hqp[ih2B+w0=(w0�B)]+2�0(w0)hw0+�0h:(18)Themagnetic�eldofnon-skewdipoles,quadrupoles,andsextupolesisgivenby�BxBy!=p0q[�0�0!+kq�yx!+ks�xy12(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)xsleadsto�xcxsx0x0s!�A0B0!=�0f(n)!:(24)Thematrixhasaconstantdeterminantsinced=dl(xcx0s�xsx0c)=xcx00s�xsx00c=0anditisgivenbyxcx0s�xsx0c=1.Thereforetheinverseoftheabovematrixissimplyobtainedbyreorderingthematrixelementsandoneobtains�A0B0!=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�rstdipoleisx0�ld�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����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,andwithalengthwhichleadstoaphaseadvance�xands=sin(�x2),onecancomputethe�rstandsecondorderperiodicdispersionatthefocusingandthedefocusingquadrupoleas~�max�=�bk(1s+12)0!;~�min�=�bk(1s�12)0!;~�max��=~�min��=�bk1s0!:(35)13 WithR56=�RL0�0x�dlandwiththeapproximationR566=�RL0�0���dlthisleadstothetimeof�ighttermsR56=�b24ks[7+cos(�x)];R566=�b24ks[9�cos(�x)]:(36)Theratior=R566=R56�1leadustotrytobaseabunchcompressoronaFODOlatticewithperiodicdispersion.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,whichcorrespondstoabetatronphaseadvanceof�,~x�is2~��.WhenrectangularbendsareusedinaFODO,thenthetransportmatrixdoesnotdependonthebendingangle.Thedispersion~x1inequation(27)ishoweverlinearlyrelatedtothebendingangle.IfthereforenFODOcellsarebuildwithhalfthebendingangles,then~x�afterthesecellswillbe(1�Mn)(1�M)�1~x112andforaphaseadvanceof�thisisequivalenttotheperiodicdispersion~��.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,itdescribessimplyarotationbythephaseadvance�xandleadsto�1�cos(n�x)�sin(n�x)sin(n�x)1�cos(n�x)!=�cos(m�x)�sin(m�x)sin(m�x)cos(m�x)!:(39)Theo�diagonalconditionimplieseitherm�x=�n�x+�mod2�,whichisincom-patiblewiththediagonalequations,orm�x=n�xmod2�,whichtogetherwiththeequationsonthediagonalleadstom�x=��=3mod2�.Thisschemecouldbeusedtomatchtothenegativedispersion,since(1�Mn)canbe�M�m,whichisthecasewhen�1�cos(n�x)�sin(n�x)sin(n�x)1�cos(n�x)!=ácos(m�x)sin(m�x)�sin(m�x)�cos(m�x)!:(40)14 Thisrequireseitherm�x=�n�xmod2�orm�x=n�x+�mod2�.The�rstequationisincompatiblewiththediagonalequations.Thesecondleadston�x=��=3mod2�.Thisoptionhoweverturnedouttobenotuseful,sinceitrequiresatleast4FODOcellsforthedispersionmatch.TheFODOcompressorwhichperformedbestusesthehalfbendmatchingschemeoftwoFODOswith�x=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.Theresultinglongitudinalphasespacefor12ellipsesaroundemittancesbetween0�and3�afterthebunchcom-pressorareshownin�gure11.Theslightdeformationathighamplitudeswouldbetolerable.However,itturnsoutthat,evenafter~x��iseliminated,thehigherorderdispersionsleadtoanon{negligibleincreaseoftheemittance.Thehigherorderdispersionsuptoorder15werecomputedwithCOSYINFINITY[9]andaregivenbellow.Itis15 Figure10:Twelveellipsesaroundlongitudinalemittancesofupto3�afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.Herethe7thorderdispersionwascorrected,butthelongitudinalphasespaceisnotin�uencedsigni�cantlybythehigherorderdispersioncorrection.normalizedtoa3�energydeviationof10%with�3�=�=0:1,x(�)=mm=�19�33�+4:2�43�+16�53��8:9�63��1:2�73�+2:7�83��1:1�93�+0:1�103�+0:09�113��0:06�123�+0:02�133��0:005�143�+0:0004�153�:(41)Aftermultipolesareinsertedinsidethe3rdand9thquadrupoletocorrectthedispersionuptoorder7,the�rst15ordersofthedispersionaregivenbyx(�)=mm=7:6�83�+10�93��3:0�103��5:1�113�+0:9�123�+0:8�133��0:1�143��0:1�153�:(42)Theremaininghorizontalemittanceblowupafterthecorrectionofthe7thorderdis-persionisshownin�gure11(left).Theellipsearoundthe1�transverseemittancehasbeentransportedthroughtheillustratedFODObunchcompressorforparticleswithupto2�energydeviation.This�gureshowsthatothernonlinearaberrationsarenotcriticalandthatthecorrectionofthe7thorderdispersionissu�cient.Correctingonlyuptoorder6washowevernotsu�cient.In�gure11(right)energydeviationsofupto2�havebeenassumedforparticlesonthe1�ellipse.Thisshowsthatthehigherorder16 Figure11:Left:Theellipsearoundthehorizontal1�emittanceforenergydeviationsofupto2�afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2�emittanceforenergydeviationsofupto1�afterbeingtrackedthroughthisFODOcompressor.dispersionprohibitstheuseofaFODObunchcompressorforthelargeenergyspreadofupto10%at3�oftheenergydistribution.Forsmallerenergyspreads,FODOcompressorsofthistypemightbefeasible.7360�CompressorEveryquadrupoleatadispersivesectioncontributestothehigherorderdispersionduetothekickxkqp0pinequation(20).Thehigherorderdispersionsmightthereforebereducedbyreducingthehorizontalphaseadvancefrom3�360�intheFODOcompressortoasmallervalue.WhiletheratiobetweenR566andR56issmallandpositiveinthediscussedFODOcompressorevenwithouttheaidofsextupoles,nosuchsolutionwasfoundforadevicewithsigni�cantlysmallerphaseadvance.However,sextupolescanbeusedtomanipulatetheintegralR566��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,thesextupoleswouldalsobe�apartbuttheywouldneedtohaveoppositesignswhichwouldnotleadtocancelationbuttoabuildupofaberrations.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��3�witha3�valueof10%,whichcorrespondsroughlytothedesiredvalueofthebunchcompressor,attheendofthesystemisx(�)=mm=�140�33�+50�43�+46�53�+71�63�+97�73�+83�83�+15�93�+17�103��32�113��0:7�123�+6:2�133��1:2�143�+1:4�153�:(51)Inthis360�bunchcompressorevencorrecting~x��uptoorder7isnotsu�cient.Thedispersioncoe�cientsuptoorder15arethengivenbyx(�)=mm=5195�83�+8010�93��9657�103��1856�113�+1206�123�+8362�133��16401�143��696652�153�:(52)Figure14(left)showsthehorizontal1�ellipseforparticleswithupto2�energyspreadafterpassingthebunchcompressor.Obviouslythedispersionisstillfartoolarge.Figure14(right)showsthehorizontal2�ellipseforparticleswithupto1�energydeviation.Thedispersione�ecthasdisappearedcompletely,duetooitshighorderdependenceon�,buttheenergydependentopticsbecomesdominant.Figure15showsthelongitudinalphasespaceofupto3�afterhavingbeentrackedthroughthe360�bunchcompressor.SinceR566hasbeenmatchedto0,thelongitudinal21 Figure14:Left:Theellipsearoundthehorizontal1�emittanceforenergydeviationsofupto2�afterbeingtrackedthroughtheFODOcompressorafterthe7thorderdispersioniscorrected.Right:Theellipsearoundthehorizontal2�emittanceforenergydeviationsofupto1�afterbeingtrackedthroughthisFODOcompressor.Theellipsecorrespondstoonenergyparticles(�=0).Figure15:Twelveellipsesaroundlongitudinalemittancesofupto3�afterbeingtrackedthroughtheFODObunchcompressorwithmatcheddispersion.dynamicsissatisfactory.Thelongitudinalphasespacemotionissu�cientlylinearduetoR566=0.Butalsoherehigherordere�ectscanbeobserved.Fortheheredescribedapplicationwitha3�energyspreadof10%thisbunch22 compressorisnotsuitable.Forapplicationswithlessenergyspreaditmightbefeasiblehowever.8RequiredAccuracyforaWigglerCompressorExceptforthewigglercompressordescribedinthispaper,allotherarrangementswhichhavebeenanalyzedhavelargehigherorderdispersioncoe�cients.Thiscouldindicatethatthesymmetricmagnetic�eldsofthewigglercompressor,whichleadstoacancelationofallhigherorderterms,areverysensitivetomisalignments.Thecancelationwouldbemostlyviolatediftherewasanedgefocusingatoneoftheparallelfacedmagnets.Thepowerexpansionofthedispersionaftersuchasystemwhichhasapolefaceangleof0:5mradattheexitofthesecondmagnetisgivenbyx(�)=mm=0:028�3��0:0087�23��0:0022�33�+0:0018�43��0:00065�53�+0:00017�63��0:000037�73�:(53)Theedgesoftheparallelfacedmagnetsofthewigglerbunchcompressorthereforehastobeparalleluptolessthanhalfamrad.Thisaccuracyof0.1mmovera50cmmagnetpolefaceshouldbeachievable.ReferencesR.Brinkmann,G.Materlik,J.Ro�bach,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