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 ID: 8253 Download Pdf
KILC12 / . Daegu. Jonathan . Bagger. Chair, ILCSC. Johns . Hopkins University. 4/23/12. LHC: 125 . GeV. ?. Since Granada, evidence is accumulating that there is a 125 . GeV. Higgs-like object being discovered at the LHC.
David . Neuffer. November 2013. 2. Outline. Front End. “. Muon. Collider” versions. 325 MHz. Add Chicane/Absorber. rematch. . Fermilab. PIP X 4 . Linac. , Accumulator, Compressor run at 15 Hz – 4 bunches .
DATASHEET Tesla TESLA C1060 COMPUTING PROCESSORS ARE THE CORE OF THE TESLA PERSONAL SUPERCOMPUTER To learn more about NVIDIA Tesla Personal Supercomputer, please visit www.nvidia.com/personal_supercom
LHC, MAP, and LEMC. David . Neuffer. July 2018. 2. Outline. Next Heavy Lepton Collider . up to ~14 . TeV. in LHC tunnel. Needs muon source. PS or new MW proton . Linac. /storage ring. cooling …. or LEMC??.
Automobile Industry Analysis. Mic . Dabulskis. & Skye Galley. Quick Facts. Forecasting. Financial Ratios. 2014. Conclusion. Founded in 2003. Tesla Roadster 2008. Model S 2012. Updated Model S in 2014.
Danielle . Darisse. Michael Easton. Nicholas Lawrence. Julian Saldana. Amed. . Tovi. Tesla commercial. Mission/Vision. To produce a fully electric car, rather than a hybrid.. To create a better looking tomorrow by producing efficient, zero emission electric vehicles without too many compromises. .
Substantial activities related to accelerator work and now also detectors. Based on CERN-KEK agreement and CLIC – ILC common working groups . Plans at CERN the common years . Accelerator collaborations.
America. (cooperation with France. ). Presentation at IRFU Linear Collider Days prepared by: . Marc Ross . (. SLAC. ) . November 29, 2013. 1. 2013 IRFU Linear Collider Days. TDP Goals:. R & D to enable Project Proposal and updated Value estimate – with Cost Containment.
Telsa’s. Life. Nikola . Telsa. was born in Serbia in 1856. In 1875 he attended university in Austria however he dropped out after three years. Telsa. worked in a telegraph company and as an electrical engineer before emigrating to the united states in 1884 to work for Thomas Edison.
The car of tomorrow . By: Rachel Politi & Linda Yang. The Idea. There are dozens of eco-friendly cars and dozens of luxury vehicles in the market today, however, Tesla has is the first to combine the two. Tesla utilizes both the luxury market and green market in the car market, by providing the world’s first luxury electric car. It’s eco-friendly capacities and sharp aesthetics attract many sectors of the car-buying market. Tesla also provides the electric powertrain equipment, such as the Tesla Supercharger Station and their backup batteries. .
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
Download Pdf - The PPT/PDF document "Bunc Compressor for the TESLA Linear Col..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
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:7.9495e 06 0.02 0.0100.010.02 0.15 0.1 0.0500.05Dz [m]DE/E0 0.02 0.0100.010.02050100150200250300#Dz [m]0100200300 0.15 0.1 0.0500.05DE/E0#Figure3:Longitudinalphasespacebefore(circles)andafter(crosses)thebunchcom-pressor.6 beforeaftersx [mm]spx [mrad]emittance0.000115330.000105068.6167e 068.6295e 069.9329e 109.9457e 100.8% of particles outside 3 sfresulting core emittance:9.5829e 10 4 20246x 10 4 4 2024x 10 5x [mm]px [mrad] 505x 10 4050100150200250300350#x [mm]0100200300 4 2024x 10 5px [mrad]#Figure4:Horizontalphasespacebefore(circles)andafter(crosses)thebunchcom-pressor.7 beforeaftersy [mm]spy [mrad]emittance6.4903e 067.0556e 061.4844e 071.3049e 079.6274e 139.6438e 130.7% of particles outside 3 sfresulting core emittance:9.3773e 13 4 2024x 10 5 505x 10 7y [mm]py [mrad] 4 2024x 10 5050100150200250300350#y [mm]0100200300 505x 10 7py [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 [10]K.L.Brown,ASecond{OrderMagneticOpticalAchromat,ReportSLAC{PUB{2257(February1979)[11]K.L.BrownandR.V.Servranckx,First{andSecond{OrderChargedParticleOptics,ReportSLAC{PUB{3381(1984)24
© 2021 docslides.com Inc.
All rights reserved.