High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
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<strong>High</strong><strong>Brightness</strong><strong>Electron</strong><strong>Beam</strong><strong>Diagnostics</strong><strong>and</strong><strong>their</strong> SuperconductingEnergy-RecoveringFree-<strong>Electron</strong> Applicationsto<strong>Beam</strong>Dynamicsina<br />
12000,JeersonAve.,NewportNewsVA23606,USA ThomasJeersonNationalAcceleratorFacility, AcceleratorDivision,Free-<strong>Electron</strong>LaserTeam Ph.Piot, Laser<br />
UniversiteJoseph-Fourier-Grenoble-I,Sciencesetgeographie DomaineUniversitairedeSaintMartind'Here 38000,Grenoble,France February8,2000<br />
<strong>and</strong>
REPORTJLAB-TN-00-0699<br />
ThisWorkhasbeenpreparedundertheUS-DOEcontractnumberDE-AC05-84ER40150,theOce ofNavalResearch,theCommonwealthofVirginia<strong>and</strong>theLaserProcessingConsortium.<br />
correspondingelectronicaddress:philippe.piot@desy.de
eratedupto48MeVpriortothelasingsystemconsistinginaplanarwigglerwhosespontaneousingacceleratorhasbeenbuilt<strong>and</strong>commissionedatThomasJeersonNationalAcceleratorFacility. Thesystemisprincipallycomposedofa10MeVphotoinjectorcapableofdeliveringahighcharged (60pC)shortbunch(1ps)electronbeamwhichisinjectedinasuperconductinglinac<strong>and</strong>accel- Abstract<br />
radiationisampliedwitharesonantcavity. Thepresentreportdetailsthediagnosticsthathavebeendeveloped<strong>and</strong>implementedintheIRFEL driver-acceleratorforcharacterizingbothtransverse<strong>and</strong>longitudinalphasespace.Wealsoreport Ahighpowerinfraredfree-electronlaser(IRFEL)facilitydrivenbyarecirculatingsuperconduct-<br />
ontheuseofthedevelopedinstrument<strong>and</strong>relatedtechniquestostudy<strong>and</strong>trytounderst<strong>and</strong>some <strong>Beam</strong>Dynamicsproblemsinthedriver-accelerator.Whenpossible,wehavetriedtobenchmark laser,Bunchlengthcharacterization,Coherentradiation,Opticaltransitionradiation. measurementswithnumericalsimulations. Keywords:<strong>Electron</strong>beamdynamics,Phasespace,Emittance,<strong>High</strong>-brightnessbeam,Free-electron<br />
temederecuperationd'energieaeteconstruitetrecemmentmisenrouteaThomasJeerson Resume Unlaseraelectronslibresinfrarouge(IRFEL)utilisantunaccelerateursupraconducteuravecsys- |||||{<br />
NationalAcceleratorFacility.Lesystemesecomposeprincipalementd'uninjecteur,dontlasource<br />
lesapplicationsdecesdiagnosticsaquelquesproblemesdedynamiquedefaisceau.Qu<strong>and</strong>celafut phasetransversauxetlongitudinaldufaisceaud'electrondel'accelerateur.Nousdecrivonsaussi pouvantatteindre48MeV.Lesystemedeproductiondelumieresecomposed'unonduleurplan dontl'emissionspontaneeestamplieegr^aceaunecaviteoptiqueresonnante. Lepresentrapportdecritlesdiagnosticsquiontetedeveloppes<strong>and</strong>ecaracteriserlesespacesde tronssontensuiteinjectesdansunaccelerateurlineaireouilssontacceleresjusqu'auneenergie d'electronsestbaseesurl'eetphotoelectrique.Cettesourcepeutproduiredespaquetsd'electron<br />
possiblenousavonstentedecomparerlesresultatsdenosmesuresavecdessimulationsnumeriques. fortementcharges(60pC)ultra-court(1ps),ayantuneenergiedel'ordrede10MeV.Ceselec-<br />
Mots-Clefs:Dynamiquedefaisceaud'electrons,Espacedephase,Emittance,Faisceaud'electrons afortebrillance,Laseraelectronslibres,Mesuredelalongueurdepaquetsd'electrons,Rayonnementcoherent,Rayonnementdetransitionoptique. |||||{
Acknowledgments<br />
metoworkononeofhisfavoritetopics:thecharacterizationofultra-shortbeam,healsogave suchastransition<strong>and</strong>diractionradiationfortheCEBAFaccelerator.GeoreyKratconded manyadvicesin<strong>Beam</strong>Dynamics.CourtBohnoeredmeapositionintheFELdepartmentto experimentallycharacterizebunchselfinteractionduetocoherentsynchrotronradiation. Ithasrequiredmanypeopletoachievetheworkthatconstitutespartofthisthesis.FirstIwishto thanksthethreepeoplethathavechronologicallydirectedmyworkatJeersonLab:Jean-Claude Denard,GeoreyKrat,<strong>and</strong>Courtl<strong>and</strong>tBohn.Jean-ClaudeDenardoeredmetocomeatCE-<br />
InFrancethisworkwasdirectedbyJean-MarieLoiseaux,Ithankhimforlettingmetotalfreedom BAFasaphDstudent<strong>and</strong>withhimIworkedonthedevelopmentofnon-interceptivediagnostics<br />
acceptedtheroleofrefereeforthepresentreport,<strong>their</strong>carefulreading,suggestions<strong>and</strong>comments havemadethismanuscriptclearer.ManythanksalsotoFern<strong>and</strong>Merchezwhochairedtheexaminationcommittee,butalsoforhisinterest<strong>and</strong>encouragementforthepresentwork. NextIwishtothanksthecolleaguesfromJeersonLabwhospenttimetoeducatemeinvarious inmyresearchwork<strong>and</strong>Iamgratefulforhistotalcondence<strong>and</strong>advises.Iwouldliketoexpress<br />
areaof<strong>Beam</strong>Physics:DavidDouglas,DaveKehne,PeterKloeppel,RobbertLegg,LiaMerminga, mygratitudetoPascalElleaumefromESRF<strong>and</strong>LouisRinolfromCERN,bothofthemhave<br />
ByungYunnhaveprovidedgeneralAcceleratorPhysicshelp<strong>and</strong>insights.RuiLihassharedher mademanycontribution,heespeciallycodedthedatareductionalgorithmforthemultislitsothat wecouldhaveresultson-line.UweHappek<strong>and</strong>MichaelJamesbuilttheinterferometersthatare workonCSR<strong>and</strong>Ihavefreelyborrowsomepassageofherpublicationsinthepresentmanuscript.<br />
dataacquisition<strong>and</strong>diagnosticscontrolsystem.SueWitherspoonhelpsettingupthemodelserver GeorgeNeil<strong>and</strong>MichelleShinnhavetaughtmethefewthingsIknowaboutFree-electronlasers.<br />
<strong>and</strong>xedtheproblemsIencouteredveryeciently. versionoftheparmelacode.AliciaHoer<strong>and</strong>AlbertGrippomademanycontributionstothe usedforbunchlengthmeasurement.ThankstoHongxiuLiuforlettingmeusehiscustomized SteveBensonhelpedmeunderst<strong>and</strong>ingtheimpactofelectronbeamparametersontheFELphoton<br />
IwouldliketoexpressmygratitudetoKevinJordan<strong>and</strong>histeam(RichHill<strong>and</strong>RichEvans), beam.ThankstoJinhuSongwhoinitiatedmetotheworldofEPICS<strong>and</strong>MEDMbutwhoalso<br />
KarelCapek,<strong>and</strong>ErichFeldlfor<strong>their</strong>excellenttechnicalsupport,withoutthemmostofthework presentedinthismanuscriptcouldnothavehappened.IoughtalottoTimothySiggins,\The Creator"(ofelectrons):despitesmany\event"weencounteredwiththeelectronsource,he<strong>and</strong> JoeGubeliwereabletoprovide"good-enough-photocathode"torunhighenoughchargeelectron beam;withoutelectronsthismanuscriptwon'texist! MyspecialthanksalsogotoDikOepts,visitorfromFOMinstituutvoorPlasmafysicaRijnhuizen formanyhelpfulcomments<strong>and</strong>suggestionshemadeontheo-lineanalysiscodeofthepolarizing ging<strong>and</strong>troubleshootingdiagnostics(<strong>and</strong>seatingmanyshiftswithme!).interferometer,<strong>and</strong>PeterMichelfromtheForschungszentrumRossendorfe.V.forhishelpdebug
IthanktheCEBAFmachineoperators,withwhomIsharedendlessnightsinthemachinecontrol center"theyhavetaughtmehowtobea\smoothoperator".<br />
Ialsothankthem! MicheleBlancfromISNtookcareofalltheadministrativepaperworkatUniversiteJosephFourier; herkindhelpisgreatlyappreciated.IalsowouldliketothankKateRoss,fromCERN,forprovidingenglishsyntax<strong>and</strong>grammarcorrections. ThisworkwassponsoredbyUStaxpayersviatheDepartmentofEnergygrantnumberDE-AC05- 84ER40150,theOceofNavalResearch,theCommonwealthofVirginia<strong>and</strong>theLaserProcessing Consortium.<br />
Lastly,IwouldliketothankmyfellowgraduatestudentswithwhomIspentmanyenjoyabletimes intheUnitedStates.IapologizeforthosethatIhavenotexplicitlymentionedhere<strong>and</strong>ofcourse
Contents<br />
2<strong>Electron</strong>Radiation<strong>and</strong>Free-<strong>Electron</strong>Lasers 1Introduction 2.2SingleParticle<strong>and</strong>Multi-particleEmission.......................4 2.1Introduction.........................................4 41<br />
2.3TransitionRadiation....................................6 2.4SynchrotronRadiation...................................10 2.5RudimentsonFEL-oscillatorTheory...........................12<br />
2.6CharacteristicsoftheIRFELdriver-accelerator.....................17 2.5.2AmplicationoftheSpontaneousUndulatorRadiation.............15 2.5.3FELGain......................................16 2.5.1UndulatorRadiation................................12<br />
3TheFELdriveraccelerator:LatticeStudy 2.7TheJeersonLabIRproject...............................19<br />
3.3MeasurementoftheTransverseResponse........................25 3.2ABriefOverviewoftheFELOpticalLattice......................22 3.1Introduction.........................................22 22<br />
3.3.2ExperimentalMethod...............................27 3.3.1TheoreticalBackground..............................26 v
3.4MeasurementoftheLongitudinalResponse.......................32 3.3.5SummaryoftheTransverseResponseMeasurements..............32 3.3.4ResultsonDispersionMeasurement.......................30 3.3.3ResultsonTransverseResponse..........................28<br />
3.4.4Simulationofhinjoutitransfermap.......................38 3.4.3ExperimentalMethod...............................37 3.4.2TheoreticalBackground..............................35 3.4.1Motivation.....................................32<br />
3.4.8ConcludingRemarksontheLongitudinalResponseMeasurement.......44 3.4.7Measurementofhinjoutitransfermap.....................43 3.4.6Simulationofhinjoutitransfermap.......................42 3.4.5Measurementofhinjoutitransfermap.....................39<br />
4TransversePhaseSpaceCharacterization 4.1Introduction.........................................49 4.1.1<strong>Beam</strong>,HamiltonianDynamics<strong>and</strong>Liouville'sTheorem.............49 4.1.2PhaseSpace<strong>and</strong>Emittance............................51 49<br />
4.2Measurementof<strong>Beam</strong>ProleUsingTransitionRadiation...............52<br />
4.3ThePossibleUseofCarbonasTRradiator.......................61 4.2.1ThelimitationofTransitionRadiationMonitor.................53<br />
4.3.1Anon-interceptiveTRbeamprolemonitor...................62 4.2.2ThermalStudies..................................54 4.2.3StudyofMultipleScatteringinAluminumfoil.................56<br />
4.4MeasurementofEmittanceinthe38+MeVRegion...................69 4.4.1GeneralConsiderations..............................69<br />
4.3.2ProleMonitorCongurationintheFEL....................66
4.4.4SimulationofEmittanceMeasurementintheIRFEL..............73 4.4.2Thequadrupolescanmethod...........................71 4.4.5EectofspuriousDispersiononEmittanceMeasurement...........74 4.4.3Themulti-monitormethod............................72<br />
4.5MeasurementofEmittanceintheInjectionTransferLine...............77 4.5.2MechanicalConsiderations............................83 4.5.1Designoftheslitsassembly............................81 4.4.6ExperimentalMethod...............................75<br />
4.5.3EmittanceCalculation&Trace-SpaceReconstruction.............84<br />
5LongitudinalPhaseSpaceCharacterization 4.6Summary..........................................90 4.5.4ErrorAnalysis...................................87 4.5.5FirstExperimentintheInjectorTestSt<strong>and</strong>...................88<br />
5.3TheoryofBunchLengthMeasurementusingFrequencyDomain...........95 5.2TheLongitudinalPhaseSpaceManipulationintheIRFEL..............93 5.1Introduction.........................................93 93<br />
5.4ObservationofCoherentTransitionRadiation......................104 5.5TheMichelsonPolarizingInterferometer.........................106 5.3.2RetrievaloftheBunchDistributionbyHilbertTransformingtheBFF....102 5.3.1TheuseoftheBFFtocompute<strong>and</strong>monitorthebunchlength........100<br />
5.5.2TheoryofOperation................................108 5.5.3RelatinganInterferogramMeasurementtoaBunchLengthMeasurement..112 5.5.1Overviewoftheexperimentalsetup.......................106<br />
5.5.5ExperimentalResultsUsinganAutocorrelationTechnique...........114<br />
5.5.4Extractingthebunchformfactor.........................113
5.6ZerophasingTechniqueforBunchLengthMeasurement................117<br />
5.7IntrinsicEnergySpreadMeasurement..........................121 5.6.3ExperimentalResults...............................120 5.6.2NumericalSimulationoftheMethod.......................120 5.6.1BasisoftheMethod................................117<br />
6<strong>Beam</strong>DynamicsStudies 5.8EstimateofLongitudinalEmittanceintheUndulatorVicinity.............122 5.7.1Method.......................................121<br />
6.1StudyofthePhotoinjector.................................137 5.8.1Conclusion.....................................124 6.1.1Introduction....................................138 6.1.2The350keVregion................................138 137<br />
6.3<strong>Beam</strong>ParametersMeasurementPriorto\Firstlasing".................153 6.2BunchCompressionStudiesintheLinac.........................151 6.1.4The10MeVRegion................................144 6.1.3Thehighgradientstructure............................140<br />
6.4StudyofPotentialEmittanceGrowth..........................154 6.4.1Chromaticity....................................155<br />
6.5PreliminaryExperimentalResultsonEmittance<strong>and</strong>EnergySpreadMeasurements.161 6.4.4BunchSelfInteractionviaCoherentSynchrotronRadiation..........159 6.4.2RF-eects......................................156<br />
6.5.1EmittanceMeasurement..............................162 6.4.3EnergySpreadinducedinaDispersiveregion..................156<br />
6.5.3Conclusion.....................................164<br />
6.5.2EnergySpreadMeasurement...........................163
B<strong>Beam</strong>Dynamics:Notes&Tools AAbbreviations 7Conclusion 172 174 165<br />
B.1Linear<strong>and</strong>SecondOrderTransport:Convention....................174 B.2Anoteonspacecharge...................................175 B.3TheSimulationTools...................................176 B.1.1TransferMatrix...................................174<br />
B.3.1DIMAD.......................................176 B.1.2<strong>Beam</strong>Matrix....................................174<br />
CDispersionRelationsforBunched<strong>Beam</strong>Distributions B.3.3PARMELA.....................................177 C.0.4Introduction....................................180 B.3.2TLIE........................................176<br />
C.0.5Background.....................................180 C.0.6TheDispersionRelationsfor^S(!)........................181 180<br />
DTheRadio-FrequencyControlSystem C.0.7TheDispersionRelationsforlog[^S(!)].....................181 185
ListofTables<br />
3.2Comparisonofcoecientsobtainedfromthenon-lineartofthemeasured<strong>and</strong> 2.1ParametersofthechosenwigglerfortheIR-DemoFEL.................17<br />
4.1PhysicalpropertiesoftheconsideredmaterialforOTRscreens.............56 3.1Twissparametersdownstreamthecryomoduleexpectedfromsimulationswiththe parmela-simulatedphase-phasetransfermap......................42 codeparmela........................................24<br />
4.3Comparisonoftheprolemeasurementswiththewirescanner<strong>and</strong>OTR-monitor..65 4.4Simulationoftheemittancemeasurementusingthequadrupolescanmethodprior 4.2SurveyofmaterialscommerciallyavailableformonitoringintensebeamswithTR<br />
ofthequadrupolebeingusedduringthemeasurement..................74 totherstrecirculationarc.Theparameterspresentedareallattheentranceface radiators;weexcludedthematerialswithlowthermalconductivity<strong>and</strong>meannumber ofcollisiongreaterthan30in<strong>their</strong>smallestthickness..................61<br />
4.7Typicalsystematicerroronemittance<strong>and</strong>Twiss-parametersforthenominalemit- 4.6Typicalerrorinpercentonthecomputedemittancefordierentsetofparameters 4.5Simulationofemittancemeasurementusingthemulti-monitormethodintheundutancevalue<strong>and</strong>twoextremecases.............................87<br />
decompressorchicane....................................74 (d,w)<strong>and</strong>forvariousemittance..............................83 latorregion.Theparameterspresentedareallattheexitfaceoflastdipoleofthe<br />
5.1Relationshipsbetween\equivalent","RMS"<strong>and</strong>"FWHM"lengthsforagaussian 4.8Comparisonofthermstransverseemittancemeasurementperformedwiththemul- <strong>and</strong>squarelongitudinalbunchdensity...........................112 tislits<strong>and</strong>theone-slit<strong>and</strong>one-harptechniques......................89 x
5.3rmsbeamhorizontalsizesimulatedwiththeparmelaparticlepushingcodeonthe 6.1Nominalinjectorsettingsbeforetherstseriesmeasurement..............150 5.2Measuredbunchlength<strong>and</strong>transversebeamdimensionforthecasesreportedin energyrecoverytransferlineprolemeasurementstation................120 Fig.5.16...........................................116<br />
6.3Comparisonoftheachieved,required<strong>and</strong>simulatedbeamparameters(thebeam 6.2Nominalinjectorsettingsbeforethesecondseriesofmeasurement...........151 parametersarespeciedforachargeperbunchof60pC)................155
ListofFigures 1.1ComparisonoftheexpectedpoweroftheIRcwfree-electronlaserofJeersonLab 2.1Geometryoftheproblem.Inthecaseofsynchrotronemission(A),theoptical withcommonhighaveragepowersource.........................2<br />
2.2Denitionoftheanglesusedinequations(2.9)<strong>and</strong>(2.10)................8 2.3DistributionofforwardTRradiationfordierentvalueof(mentionedclosetothe pulsereferencecoordinatearetheoneoftheelectronbunchattheretardedtime. Forbackwardtransitionradiation(B),thereferencecoordinatesarethespecular reectionoftheelectronbunchcoordinateasitstrokethealuminumradiator.....5 appropriatecurve)asanelectronpassesfromtheinterfacevacuum-aluminum(A).<br />
2.4Polarplotofthenormalizedradiationpatternforanaluminumfoilwithanelectron ComparisonoftherenormalizedTRforwardangulardistributionemittedbyanelec-<br />
undernormalincidence(i.e. crystaldirection.The5.7valueisthesmallestpermittivity.Privatecommunication fromGoodfellowInc.,London,U.K.)...........................9 tronpassingthroughavacuum-aluminum(solidline)<strong>and</strong>vacuum-carbon(dashed moreexactlygraphitehastwodierentelectricpermittivityforitstwodierent line)interface(B).Forcarbonthepermittivityisassumedtobe5.7.(Carbonor<br />
2.5Fractionofthetotaltransitionpoweremittedintothehemispherethatisconcen- 1=(B)............................................11 (2.9)<strong>and</strong>(2.10)wererenormalizedto<strong>their</strong>maximumvalue...............10 tratedwithinaconeofsemi-angle(A)<strong>and</strong>containswithinaconeofsemi-angle 45degincidence(B)(i.e. Lorentzfactorwaschosentobe=10forclarityofthegure,<strong>and</strong>theequations =45deginEqns.(2.9))<strong>and</strong>(2.10).Fortheseplotsthe =0deginEqns.(2.9)<strong>and</strong>(2.10))(A)<strong>and</strong>witha<br />
2.6Angular<strong>and</strong>frequencydistributionofsynchrotronradiationforthe(A)<strong>and</strong>the 2.7PlotoftheUniversalfunctionS(!=!c).Thefrequencydistributionofthetotal 2.8FEL-oscillatorprinciple(CourtesyJ.Martz,JeersonLab)...............13 synchrotronradiationisproportionaltotheUniversalfunction.............12 (B)polarization......................................11<br />
xii
2.10Anexampleofvariationofthegainversusthebunchlength(A)<strong>and</strong>thetransverse 2.11Anactualtopviewofthe\asbuilt"IRFELdriveraccelerator.............19 2.9Normalizedpoweroftheon-axisundulatorradiationforthetwodierentvalueof emittance(bothx<strong>and</strong>yplane)(B)(theseplotswerecomputedusingtheverysimple 1Dmodelexposedintheprevioussection)........................18 KconsideredfortheIRFEL................................15<br />
3.1Dispersedoverviewofthemainringofthedriveracceleratorcorrespondingtog2.12Simpliedschematicoftheelectronsource:A527nmlaserbeamthatcanbemodulatedbytwoelectro-opticscell(EO1<strong>and</strong>EO2)<strong>and</strong>attenuatedbyarotationalpolarizerilluminatestheGaAsphotocathode.Ejectedphoto-electronsareaccelerated 3.2HorizontalDispersion(topgraph)<strong>and</strong>transversebetatronfunctions(bottom ure2.11.Thepathoftheelectronbeamisindicatedwitharrows............23 throughtotheacceleratingvoltageofnominally350kVbetweenthephotocathode<br />
graph)forthenominalsettingsofthemagneticoptics.................25 <strong>and</strong>theanode........................................21<br />
3.4Exampleofcalibrationofacorrector.Theslopeofthelinearinterpolationis 3.3schematiccutofabeampositionmonitor(BPM)....................27 3.5Betatronphaseadvancebetweeneachcorrectorusedtoperturbtheorbitalongthe 0:0183mm=(G:cm)whichcorrespondstoanangulardeectionof6:54rad=(G:cm).29<br />
3.7Comparisonbetweentheexperimentaldataaftercorrection<strong>and</strong>thesimulatedlattice 3.6Comparisonbetweenthemeasured<strong>and</strong>simulatedlatticeresponseforthesixcorrec- latticeinthehorizontal(leftplot)<strong>and</strong>vertical(rightplot)plane(100meters correspondsapproximatelytotheendofthebacklegbeamline)............30<br />
3.8Comparisonbetweenanenergychangeinducedbeamdisplacementalongthelattice responseforthesixcorrectors(100meterscorrespondsapproximatelytotheendof thebacklegbeamline)....................................32 torsusedduringthedierenceorbitmeasurement(100meterscorrespondsapprox<br />
<strong>and</strong>theresponsetoanangularperturbation.......................33 imatelytotheendofthebacklegbeamline).......................31<br />
3.9<strong>Beam</strong>displacementinthebacklegtransportforarelativevariationof1%ofthe magneticeldoftherstrecirculationarc........................34
3.11Momentumcompaction<strong>and</strong>nonlinearmomentumcompactionforonearcversusthe 3.10Energycompressionscheme:Therstrow(fromlefttoright)presentsthelongitu- Theresultforthethreecasesafterdecelerationareshowninthethirdrow......35 dinalphasespaceatthelinacexit,afterthecompressionchicane,<strong>and</strong>justafterthe wigglerinteractionhastakenplace;thesecondrowshowlongitudinalphasespaceat theentranceofthelinacjustpriortodecelerationforthreedierentchoiceofR56 secondfamilyoftrimquadrupolesexcitationstrength(kq).nonlinearmomentum <strong>and</strong>T556(for(A)-0.2<strong>and</strong>0.m,for(B)0.2<strong>and</strong>0m<strong>and</strong>for(C)0.2<strong>and</strong>3.0m).<br />
3.12LocationofthepickupcavitiesalongthetransportlineintheFELdriveraccelerator.38 3.13BlockdiagramofthecompressioneciencyR55<strong>and</strong>momentumcompactionR56 compactionversusthesecondfamilysextupolesstrengthks(forthiscalculation,the<br />
3.14phase-phasebeamtransferfunctionbetweenthephotocathodesurface<strong>and</strong>thethree trimquadrupolesareunexcited)..............................36 measurement.........................................39<br />
3.16Demonstrationoflongitudinaldierence-orbit:phase-phasetransfermapmeasured 3.15Comparisonofthephase-phasebeamtransferfunctionbetweenthephotocathode (A)<strong>and</strong>simulated(B)forthreedierentsettingsofthetrimquadrupole.Plot(C) row)withtheonesimulatedusingparmela(toprow)................41 surface<strong>and</strong>thethreedierentpickupcavities(pickup#2,#3,<strong>and</strong>#4)(bottom dierentpickupcavities:pickup#2,#3,<strong>and</strong>#4.....................40<br />
3.17Phase-phasetransferfunctionsfordierentsettingsoftheopticalcavitylength.In respectivelytheexperiment<strong>and</strong>thesimulation......................43 (A)theopticalcavityiscompletelydetunedsothatthelaserdoesnotoperate.In <strong>and</strong>(D)correspondtodierenceofthemeasuredmappresentedinthetoprawfor<br />
3.18EvolutionoftheR56alongthebeamtransportintherecirculationtransport,from thelinacexittoarc3Fexit.MeasuredR56arealsopresentedaslledsquares....47 thephase-phasetransfermapismeasuredfordierentdetuningoftheopticalcavity case(B)<strong>and</strong>(E)......................................46 cavityistunedsothatthelaseroperateatthelimitofitsturno.In(C)<strong>and</strong>(D), (B)thecavityispreciselytunedtomaximizetheFELoutputpower.In(E)the<br />
3.19Comparisonofthelineartermextractedbyttingthemeasuredenergy-phasetrans-<br />
4.1Schematicsofprincipleforthedierenttypeofbeamproledensitymeasurement 3.20EectofthesextupolesintheArc3Fontheenergy-phasecorrelation.(Simulations <strong>and</strong>experimentaldataareosetforclarity)........................48 fermapwiththeexpectedmomentumcompactionR56fromthedimadcode,for<br />
devices............................................54<br />
dierenttrimquadrupolessettings.............................47
4.3Maximumaveragecurrentthatcanwithst<strong>and</strong>aTRradiatorasafunctionofthe 4.2MethodologytocomputetemperatureriseinacylindricallysymmetricTRradiator.55 4.4SteadystatetemperatureversusaveragebeamcurrentforthreedierentTRradiator equivalentbeamradius.ThreetypesofTRradiatorhavebeenconsidered:Aluminum,Gold<strong>and</strong>Carbon.Thethreeradiatorare0:8mthick.............574.6Angularscatteringdistributionexperimentallymeasuredforthreedierentthick- 4.5Anexampleoftheeectofa0:8mthickaluminumfoilonthebeamprole.The nessesofAluminumfoil...................................60 beamismeasuredusingawirescannerlocateddownstreamthefoil..........59 thickness<strong>and</strong>abeamequivalentradiusof2mm.....................58<br />
4.9OverviewofthecarbonfoilbasedOTRexperiment(CourtesyfromS.Spata)(A) 4.7ComparativeresultsofexperimentwithKeil'ssemi-empiricaltheory<strong>and</strong>geant<br />
4.10ComparisonoftheMissingmassspectraobtainedusingoneoftheexperimentalhall 4.8geantcomputationforthincarbonfoils.........................63 <strong>and</strong>atypicalbeamdensitymeasuredwithsuchdevice(B)...............64 computationforthinaluminumfoils............................62<br />
spectrometerwith(A)<strong>and</strong>without(B)thebeambeinginterceptedbythecarbon<br />
4.13Comparisonofbeamspotsizevariationversusquadrupolestrengthfortwodierent 4.11St<strong>and</strong>ardcongurationoftheOTR-basedprolemonitorintheFEL-driveraccelerator.66 4.12Rawdatabeamprole(topgraph)<strong>and</strong>beamproleafterprocessing(background foil(CourtesyofP.Gueye,HamptonUniversity,VAUSA)...............65<br />
4.14Relativeerroroncomputedemittanceforthetwocasespresentedingure4.13 versustherelativeerroronbeamsizemeasurement...................76 <strong>and</strong>6m............................................75 settingoftheupstreamopticstoachievetwodierentminimumbetatronvalue,3 subtracted,ghostpulsecontributionremoved,...).....................67<br />
4.16Relativeemittanceerrorversusdimensionlessspuriousdispersioncontributionto 4.15Monte-Carlosimulationof200emittancemeasurements.Theplots(fromtop)are theun-normalizedrmsemittance,the-function,theparameter...........77 beamsize..........................................78
4.18Overviewofthephasespacesamplingtechnique.AnincomingAmultislitmask 4.17AnexampleoftransverseemittancemeasurementinthehighenergyregionoftheIRinterceptstheincomingspace-charge-dominatedbeam.Thebeamletsissuedfrom<br />
numberarethebeamparametersdeducedfromthet.Thechargeperbunchwas Thedashedlinesareobtainedwiththeleastsquarettechnique.Thereported approximatelysetto40pC.................................79 (top)<strong>and</strong>vertical(bottom)rmsbeamsizeversustheexcitationofthequadrupole. FELusingquadrupolescanmethod.Thetwoplotspresentvariationofthehorizontal<br />
4.20Comparisonoftheexpectedphase-space,generatedviaparticleretracing(eachgrey 4.19Simulatedmulti-beamletspatternontheopticaltransitionradiationradiator....84 dotsrepresentamacroparticle),withtheretrievedphase-space(representedwith backlineiso-contours)usingthesimulatedbeampatternontheopticaltransition radiationpresentedgure4.19..............................85 theslitsareemittance-dominated.............................80<br />
4.21Fractionofincidentelectronthatscattersontheslitsedgeversusthebeamincident<br />
4.23Anexampleofreconstructedphasespaceiso-contourdensity..............90 4.22Anexampleof2Dbeamdistributionontheanalyzerscreendownstreamthemultislit mask.Theprojectionontothex-axisisalsodisplayed..................89 anglewithrespecttothenormalaxisofthemultislitmask.Thedepthoftheslits approximatelytheinteractionof10%oftheincidentelectronwiththematerial....86 is5mm.Amisalignmentofthemaskof1:2mradcomparetothebeamaxisyields<br />
4.24Emittance<strong>and</strong>betatronfunctionversusthesolenoidexcitation.............91 5.1SequencesofparmelarunsdemonstratingthebunchingprocessintheIRFEL. 4.25Emittanceasafunctionofthechargeperbunchfortwodierentmacropulsewidth (10sec<strong>and</strong>50sec)....................................92<br />
5.2AngularBFFforthreedierentvalueoftheRMSbeamdivergence..........97 (H),thearc#1(I).Notethatelectronswithpositivebelongtothebunchtail SRF-linac(F),thebunchcompressorchicane(G),thebunchdecompressorchicane whiletheonewithnegativeareinthebunchhead..................95 theSRF-cavity#1(C),theSRF-cavity#2(D),theachromaticchicane(E),the Thelongitudinalphasespaceisplottedattheexitofthegun(A),thebuncher(B),<br />
5.3Bunchformfactorcomputedfora300m(RMS)square(dashline)<strong>and</strong>gaussian 5.4Monte-Carlosimulatedbunchformfactor(right)with106macroparticleforthree (solidline)bunches.....................................98 typesofbunchlongitudinaldistribution(left)......................99
5.6EectofdierentkeyelementsinthebunchingprocessoftheelectronsontheBFF. 5.5Eectoftransversebeamsizeonthe3D-BFF.Forthreedierenttypeofbunch TheBFFcorrespondingtothenominalsettingsfortheRFelements(solidline)is width.............................................100 (ellipsoidal,pancake<strong>and</strong>linechargebunch),theBFF(A),theCTR(B)<strong>and</strong>CSR assuminganangularacceptanceof0:3rad<strong>and</strong>aregivenfora20%frequencyb<strong>and</strong>- (C)powerspectrumarenumericallycomputed.Thepowerspectrumarecomputed<br />
5.7SimpliedschematicsofaGolaycell(A)<strong>and</strong>theassociatedsignalacquisitionelec- 5.8Scalingofcoherenttransitionradiationpowerversuschargeperbunch.Thecharge tronics(B)..........................................104comparedwiththecaseswherethebuncher(dottedline),therstcavityinthein- perbunchischangedbyvaryingtheintensityofthephotocathodedrivelaser.The jector(greyline)<strong>and</strong>thephotocathodedrivelaser(dottedline)areoperated+3deg<br />
circlesaretheexperimentaldatapoint<strong>and</strong>thedashlineistheresultofaquadratic o<strong>their</strong>nominalsettings..................................102<br />
5.10CTRsignalversusbeamequivalenttransversespotsizepxy.Theerrorbars 5.9CTRsignalversustheSRFlinacoverallphase.Asthelinacphaseisvariedthe bunchlengthattheundulatorvicinityischanged....................106 interpolationoftheexperimentalpoint..........................105<br />
5.11Overviewoftheopticstoguidethecoherenttransitionradiationemittedfroman 5.12SimpliedschematicsoftheMichelsonpolarizinginterferometer............109 5.13LimitationofsomeopticalcomponentsintheMichelsonpolarizinginterferometer. correspondtothevarianceofveconsecutivemeasurements..............107 aluminumfoiltotheentranceoftheinterferometer...................108<br />
5.14Completeinterferogramstakenfewminutesapart:In(A),theghost-pulsewasnot totallysuppressedwhilein(B)itwas.(C)givesthedierencebetweenthetwo theGolaycellentrancewindow...............................114 previousplots(A)-(B)..................................125 componentperpendiculartothewires,<strong>and</strong>T[E]gcisthetransmissioncoecientof R[Ek]wgisthereectioncoecientofthewiregridfortheelectriceldcomponent paralleltothewires,R[E?]wgisthereectioncoecientofthewiregridforthe<br />
5.15FinescanofthecentralpartoftheinterferogrampresentedinFig.5.14(A).The FWHMof110m(B).Thebaselineis2.7V.......................126<br />
aninterferogramofthesumofasquare<strong>and</strong>gaussi<strong>and</strong>istributionbothhavinga fromagaussi<strong>and</strong>istribution(solidline)<strong>and</strong>asquaredistribution(dashline)bothof thesedistributionhavethesameFWHM(A).Theinterferogramiscomparedwith experimentalinterferogram(circle)iscomparedwithaninterferogramgenerated
5.16InterferogrammeasuredfordierenttransversebeamsizeontheTRradiator.The 5.17Symmetrizationoftheautocorrelationbydierentmethods(A)(seetextfordetail) 5.18Energyspectrumobtainedbyconsideringdierentnumbersofdatapointsfrom leftplotsarethemeasuredinterferogramswhereastherightplotsshowthebeam transverse(horizontal<strong>and</strong>vertical)distributions.....................127<br />
5.19EnergyspectrumobtainedbyFouriertransforminganautocorrelationwith256data alongwiththecorrespondingpowerspectrum(B)....................128 theautocorrelation.Thenumbersareallpowersoftwo,asrequiredbytheFFT algorithmweused......................................128<br />
5.21ExperimentalsetuptomeasurebunchlengthwithzerophasingmethodintheIR-FEL 5.20Recoveredlongitudinalbunchdistributionforthedierentextrapolationofthepower driveraccelerator......................................130 spectrumshowningure5.19...............................129 sidered............................................129points(solidlinewithsquares)<strong>and</strong>thedierentlowfrequencyextrapolationscon- 5.24Longitudinalphasespaceslopeevolutionalongthedriftbetweentheentranceofthe 5.23picturaleectoflongitudinalspacechargeforceonthephasespacedistribution...131 5.22Horizontalbeamenvelopeevolutionfromtheexitofthefourcavitiesinthecry- parmelaisturnedon<strong>and</strong>o...............................130 ferentnumbersofzerophasingcavities.Foreachcase,thespacechargeroutineinomoduleuptothebeamprolestationinthespectrometertransportlinefordif 5.25RMShorizontalbeamsizeattheprolemeasurementstationversusnumberof thespacechargeroutineinparmelaturnedon(uppercurve)<strong>and</strong>o(lowercurve).131 numberofcavitiesusedforthezerophasingisindicatedbeloweachcurve.Theslope isnormalizedtotheinitialenergy.Foreachcasethesimulationisperformedwith zerophasingcavities....................................132 rstzero-phasingcavityuptotheentranceplaneofthespectrometerdipole.The<br />
5.26<strong>Beam</strong>spotonthedispersiveOTRmonitorforthethreephasesofthezero-phazing 5.27Typicalzerophasingbasedmeasurement.The2Dfalsecolorimagerepresentsthe whereallthe\zerophasing"cavitiesareo(topline),arephased-90degw.r.t. themaximumaccelerationphase(middleline)<strong>and</strong>arephased+90degw.r.t.the measurement.........................................133 beamspotmeasuredonthedispersiveviewerinthespectrometerlinewhereasthe maximumaccelerationphase(bottomline)........................134<br />
rightplotistheprojectionontothehorizontalaxis.Measurementforthecase
5.28Comparisonfor=0:2%(topleft)<strong>and</strong>for=2%(topright)ofthebeamrmssize<br />
5.29Schematicsoftheavailablediagnosticsintheundulatorregionformeasuringthe dashedlineontheseplotsisthebeamsizecontributionduetodispersiononly. variationinthecenterofthebunchcompressorversustheexcitationofanupstream quadrupoleusingdimadsimulation(solidlines),usingEqn.(5.57)(crosses).The<br />
5.30ComparisonofthelongitudinalphasespaceattheCTRfoil(greydots)<strong>and</strong>the right)............................................135 Thebottomplotspresenttheratioofthebeamsizecontributionduetodispersion<br />
decompressormid-point(blackdots).Theenergydistributionareexactlythesame longitudinalemittance....................................135 onlywiththetotalbeamsizefor=0:2%(bottomleft)<strong>and</strong>=2%(bottom<br />
5.31rmsenergyspreadinducedbywakeeldsgeneratedasa60pCbunchofelectron 6.1SimpliedschematicoftheIRFELphotoinjector(seetextforexplanation)......138 travelsthroughthevacuumchambertransitionattheundulatorlocation.......136 whilethebunchlengthattheCTRfoilissmaller(dashedlines)thanatthechicane mid-point(solidlines)....................................136<br />
6.3\spacechargeoveremittanceratioforthetransverse(Rr)<strong>and</strong>longitudinal(Rz) 6.2RMStransverse(x<strong>and</strong>y)<strong>and</strong>longitudinal(z)beamsizesalongtheinjector. Thebottomschematicslocatestheopticalelementsalongthebeamline(thegun acceleratingvoltageis350keV)..............................139<br />
6.5Focallengthoftherst(cavity#4)<strong>and</strong>second(cavity#3)cryounitcavitiesversus 6.4Reducedenergygain,,alongtherst(cavity#4)<strong>and</strong>second(cavity#3)cryounit cavity.............................................142 direction...........................................141<br />
6.7Comparisonofthetransversebeamdensitymeasuredattheexitoftheaccelerating 6.6SteeringeectduetoFP<strong>and</strong>HOMcouplerinthecryounitversus,thephase structure(A)withparticlepushingnumericalsimulation(B)..............146 dierencew.r.t.themaximumenergyphase(so-called\crestphase").........145 ,thephasedierencew.r.t.themaximumenergyphase(so-called\crestphase")144<br />
6.8Measured(squares)<strong>and</strong>predictedtransverseemittances(solid<strong>and</strong>dashlines)atthe 6.9<strong>Beam</strong>densitymeasuredonthehighdispersionOTRmonitorforninedierentbunch cryounitexitversustheemittancesolenoidmagneticeld................147 gradientsettings(theimages(A)to(I)correspondstothepointspresenteding.6.10 startingfromthelowgradientvalues)..........................148
6.10Comparisonofthermstransversehorizontalbeamsizemeasuredinthehighdisper- 6.11\R55"transfermapgeneratedwithparmelafor(1)dierentcavitymodel(3Dmaa 6.12\R55"transfermapfordierentexperimentaloperatingpointsofthebunchergrasionOTRmonitorwiththeoneexpectedfromparmela................149 numericalsimulations.(Firstseriesofmeasurement)..................151 dient.Themeasurementsarealsocomparedwiththe\ideal"injectordevisedfrom thecasewherethepickupcavityislocateddownstreamthecryounitafteradriftin adispersion-freeregion...................................150 <strong>and</strong>2Dsupersh)atthepickupcavitylocatedintheinjectionchicane<strong>and</strong>(2)for<br />
6.13ComparisonofthemeasuredR55patternaftertheFELwasoptimizedwiththe 6.14rmsbunchlengthevolutionalongtheIRFELfromthephotocathodeuptotheexit 6.15Variationofthebunchlengthversusthelinacacceleratingphaseperparmelasim- \ideal"R55patterngeneratedwithparmela(usingthesecondsetup).........152 ofthesecondchicane....................................153<br />
6.16Phasespacedistortionduetochromaticaberrationatthedecompressorchicaneexit (topplots)<strong>and</strong>thearc1exit(bottomplots).......................156 coveredfromtheCTRautocorrelationusingthetechniquementionedinChapter5]. Comparisonbetweentheexperimental<strong>and</strong>simulatedbunchlongitudinaldistribution (C).TotalCTRpowersignalmeasuredduringplot(B)experiment(D).......154 ulation(A)<strong>and</strong>measured(B)["measured"meansthebunchdistributionwasre<br />
6.17Emittancegrowthduetochromaticaberrationversusthemomentumspread<strong>and</strong> 6.19SchematicsofCSRselfinteractionofabunch.......................160 6.18Evolutionofbeamparameters(bunchlengthz,rmsenergyspreadE,transverse emittances~"x;y,<strong>and</strong>rmsbeamsizesx;y)versustheoperatingacceleratingphaseof thelinac...........................................158 energyosetofthebeam..................................157<br />
6.20AnalyticalcomputationforCSR-inducedenergylossalongagaussianbunch<strong>and</strong> 6.21TransverseHorizontalemittance<strong>and</strong>totalpowerCTRsignalmeasuredasafunction 6.22Energydistributionmeasuredalongthebeamline,atthechicanemidpoint(A)<strong>and</strong> chicane(macroparticlewith>0areinthebunchtail)................161 predictionusingasimplenumericalmodelinamodiedversionofthescparmela codefor200m(A)<strong>and</strong>100m(B).Thesystemconsideredisasimpleachromatic ofthelinacgangphase...................................162 (B)<strong>and</strong>entranceofthearcs(C)<strong>and</strong>(D)[thehorizontalaxisoftheseplotrepresent therelativeenergyspread(nounits)].Thebottomplotpresentsthermsrelative energyspreadcomputedfromthedistributions......................163
C.1Exampleofphaseretrievalforabi-gaussian-likebunchdistribution.Threetypeof B.1Parmelasimpliedalgorithm................................179<br />
D.1OverviewoftheRF-controlsystemfortheIRFEL....................185<br />
<strong>their</strong>Fouriertransform(dashlinesonplot(b),(d),<strong>and</strong>(f))<strong>and</strong>therecoveredphase usingthedispersionrelation(crosses)onthesameformerplots.............184 bi-modaldistributions(a),(c)<strong>and</strong>(e)arepresentedalongwiththeexactphaseof
Chapter1<br />
UVdomain<strong>and</strong>areplannedtogenerateXrays.Theyhavefoundmanyapplicationsranging positron)accelerators.Suchlightsourceshaveproventobecapableofgeneratingphotonindeep Introduction<br />
fromfundamentalsciences(biology,crystallography,etc...)toindustrialapplications(e.g.nanoelectronics). Thegenericcongurationconsistsof,e.g.,anelectronbeamacceleratorthatgenerates<strong>and</strong>prepares (i.e.accelerates,bunches<strong>and</strong>transverselyshapes)theelectronbeambeforesendingitinaperiodic magneticeldcreatedbyanundulatormagnetwhichcausestheelectronstooscillatetransver- Inrecentyears,therehasbeenagrowinginterestincoherentlightsourcesdrivenbyelectron(or<br />
sally.Asanelectronbunchoscillates,itcreatesa(spontaneous)synchrotronradiationpulsethat lightsources:thestoragering<strong>and</strong>thefree-electronlaser(FEL).Intheformercase,theparticle mirrorsitscharacteristics(i.ebunchlength,shape,..).Twoschemesaregenerallyusedforsuch beamisstoredinaring<strong>and</strong>periodicallygoesthroughanundulatormagnet,whileinthelatter casethebeamisgeneratedbyalinac<strong>and</strong>passesthoughtheundulatoronce.Thefree-electron producehighbrightnessphotonbeamcomparedtostoragering.Furthermorebrightnessofstorage ringtendstoworsenatlowenergy,duetoTouschekintra-beamcollision,<strong>and</strong>highenergybecause oftheimportanceofquantumuctuations.Inaresonator-basedFELtheundulatormagnetis laserhasgeneratedmuchmoreinterestedintherecentyearsbecauseof<strong>their</strong>unequaledabilityto<br />
oftheincoherentsynchrotronemission.ThislattertypeofFELisespeciallysuitedtogenerate neousradiationcreatedviasynchrotronemission.InaSASE(selfampliedspontaneousemission) FEL,theundulatorismadelongenoughsothatlasingarise\naturally"fromselfamplication ultra-shortwavelength(e.g.X-rays)coherentlightsinceforsuchwavelengthsamirrormightnot quitesimilartoconventionallaser,the\free"electronsactsasmediumthatampliesasponta- coherentlysuperimposedtotheformerphotonpulse.InsuchanoscillatorFELthemechanismis insertedbetweentwomirrorswhichconstitutesaresonantopticalcavitythatrecirculatesthepho-<br />
beavailabletherebypreventingtheresonatorconguration.AmainfeatureoftheFEL-basedlight tonpulsecoincidentlywiththenextincomingelectronbunch.Itideallygeneratesaphotonpulse<br />
electronbeam. Generallythedriveracceleratorconsistsinroomtemperaturecavitiesthatdonotallowsimultane- sourceis<strong>their</strong>abilitytoprovidelaserlightoveracontinuoustunablerangeofwavelength,thatcan besubstantial,byvaryingthemagnetostaticeldoftheundulatorortheenergyoftheincoming 1
walllossesviaJouleseecttherebyallowingtheoperatingofthecavitiesathighcontinuouswave (cw)gradient.Acomparisonoftheaveragepowerthatcanbeproducedbysuchsuperconduct- consideredthroughoutthisthesis,istousesuperconductingacceleratingcavitieswhichoerlow ouslyhighacceleratinggradient<strong>and</strong>highelectronbunchrepetitionrate.Hencethecommonlight sourcecanproducehighpeakpowerbecauseofthehighchargethatcanbestoredinabunch,but cannoteasilyproducehighaveragepowerlightneededbycertainapplicationsuchaspowerbeaming,micro-machining,etc...Analternativescheme,thathasbeenusedinthedriver-acceleratoringlinearaccelerators(e.g.theIRFELfromJeersonLab),withconventionalhighaveragepower cavitydirectlysupplementstheinputpowerprovidedbytheklystrontoacceleratehigheraverage alightsourceforindustrialapplication,iscosteciency.Thiscostismainlyimpactedbytheinput laser,<strong>and</strong>recentlyweachieved1.7kWexperimentally. Anotherconcernthathasarisen,especiallyinourprojectwherethemainmotivationistodevelop powerdem<strong>and</strong>toacceleratethebeam.Thisdem<strong>and</strong>wasreducedintheJLABIRFELbyrecir- sourcegenerallyused(e.g.excimer<strong>and</strong>carbonlasers)isdepictedinFigure1.1.Amaximumoutput<br />
currentelectronbeamforagivenbeamenergytherebyreducingthedem<strong>and</strong>oninputpowerfrom poweroftheorderof2kWcanbeexpectedfromtheJeersonLabsuperconductingfree-electron<br />
klystron. culating<strong>and</strong>deceleratingthebeamusingthesamelinac.Thedeceleration-inducedvoltageinthe<br />
Figure1.1:ComparisonoftheexpectedpoweroftheIRcwfree-electronlaserofJeersonLabwith commonhighaveragepowersource. ThesuperconductingFELthatservedasexperimentalplatformforthisthesisisdedicatedtostudy thepossibleapplicationofhigh-powercwFEL's.ItisaninfrareddemoFELcapableofprovidingcontinuousphotonbeamsintheinfraredspectrum(3m-6m)withhigh-averagepowerof thedriver-accelerator. Thedriver-acceleratorneedstoprovideahigh-brightness,ultra-shortbunch,lowtransverse-emittance, electronbeam.Becauseofthehigh-chargeconcentratedinthebunchofelectrons,eectsuchas<br />
technologiesrequiredforhigh-powerfree-electronlasersespeciallythebeamdynamicsaspectsin approximately1kW.ThoughthisFELisanuser-orientedfacility,itisalsodevotedtostudythe<br />
Laser Average Power (kW)<br />
2<br />
1<br />
0<br />
Excimers<br />
Alex<strong>and</strong>rite<br />
Nd:YAG<br />
IRFEL<br />
HF<br />
10 kW<br />
100 1000<br />
Wavelength (nm)<br />
10000<br />
CO 2
space-chargeinthelowenergyregime,<strong>and</strong>wakeeld(forshortbunch)canleadtobeaminstabil-<br />
density,thetransverseemittance<strong>and</strong>thebeamenergyspread.Thepresentreportdealswithseveral phase-spaceyieldinganemittancegrowthinthebendingplane. Thebeamparametersweneedtomeasureacuratelyare:thebunchlength,thebeamtransverse CSRleadstoanincreaseinenergyspreadwhichinturncoupleviadispersiontothetransverse ity.Anotherpotentialproblemthatcanarisewithsuchbeamparametersistheselfinteractionof<br />
aspectsconcerningthedevelopmentofthe<strong>Beam</strong>Instrumentationrequiredtoproperlycharacterize abunchviacoherentsynchrotronradiation(CSR)emittedasthebunchpassesthroughdipoles.<br />
opedtocharacterizebeaminbothemittance<strong>and</strong>space-charge-dominatedregime.TheChapter theelectronbeam<strong>and</strong>performsome<strong>Beam</strong>PhysicsstudiesintheJeersonLabIRFEL.InChapter<br />
willpresentsomebeamdynamicsstudiesoftheinjector,<strong>and</strong>intherecirculatorwithanattempt electronbeamparameters.ChapterThreepresentstheFEL-driveracceleratoralongwithsomeopticallatticecharacterizationthatwerecrucialforunderst<strong>and</strong>ing<strong>and</strong>settinguptheenergyrecoveryscheme.InChapterFour,wedescribethetransversephasespaceinstrumentationwehavedevel- Fivepresentsourworkforcharacterizingultra-short(sub-picosecond)bunch.InChapterSixwe Twowewillreviewradiationsemittedbyelectrons<strong>and</strong>explaintheprincipleofFELoscillatorby usingasanexampletheIRFEL<strong>and</strong>trytounderst<strong>and</strong>whatarethespecicationontheIRFEL tomeasuretheemittancegrowthintherecirculationarcoftheIRFEL.Wewillthenconcludein aChapterSeven.
<strong>Electron</strong>Radiation<strong>and</strong>Free-<strong>Electron</strong> Chapter2 Lasers<br />
canbeusedtogenerateintenselightpulseoverlargedomainofwavelength,theycanalsobeused toinfercertaincharacteristicsoftheelectronbeamthatproducedthem.InthepresentChapter <strong>and</strong>diractionradiationshavebeenwidelystudiedinliterature.Thoughthesetypesofradiation changeintheelectronenvironment.Tonamefewofthem,synchrotron,transition,Smith-Purcel, 2.1Introduction<br />
wewillrecallfewpropertiesofthetwotypesofradiationthatwillbeconsideredinthisreport: Therearemanyprocessesamongwhichelectronscanemitradiation.Mostofthemaredueto<br />
2.2SingleParticle<strong>and</strong>Multi-particleEmission synchrotron<strong>and</strong>transitionradiation.Wewillthendiscussundulatorradiation<strong>and</strong>itsamplication infree-electronlaserssuchastheoneusedasexperimentalplatforminthereport.Inalastsection wewillpresenttherequiredelectronbeamparametertodrivethedesiredFEL. Inthissectionwederivetheexpressionforthetotalradiationemittedbyanensembleofparticle <strong>and</strong>introducethebunchformfactor. Radiationemittedbyelectronsdependsontheelectron'sdensitydistribution.Forpurecontinuous beam(DC),noradiationistheoreticallyemitted(theeldFouriertransformRE(t)exp(i!t)dt thatinducestemporaluctuationdependenceontheelectronmotion.Inhighenergyparticle bunches<strong>and</strong>thereforeelectromagneticwavescanberadiated. Whenanelectromagneticeldisradiatedbyacollectionofelectron,thetotalelddetectedbyan observerlocatedinP(seegure2.1)isthesuperpositionoftheeldatthispointgeneratedby iszero).Indeed,experimentallythereisanincoherentradiationresultingfromSchottkynoise accelerator,accelerationisprovidedbyradio-frequencywave:thebeammustconsistofaseriesof 4
!Vjitsthevelocity,<strong>and</strong>!Xjisthepositionvectorthatlocatesthej-thelectronwithrespecttothe bunchcenter. eachelectron[2,3]1:!ET(P)=X wherekjVj<strong>and</strong>Xjarerespectivelythewavevectoroftheelectriceldemittedbythej-electron, Underthefar-eldapproximation,wecan,withoutsignicantlychangingtheresults,replacethe j=1:::N!kj^(!kj^!Vj) j!kj^(!kj^!Vj)jj!E1e(k)jei!kj!Xj, (2.1)<br />
observationP.Introducingthenormalizedvelocity,j,theformerequationtakestheform: <strong>and</strong>bnistheunitvectorpointingfromthecenterofthechargedistributiontowardthepointof Figure2.1:Geometryoftheproblem.Inthecaseofsynchrotronemission(A),theopticalpulse referencecoordinatearetheoneoftheelectronbunchattheretardedtime.Forbackwardtransition radiation(B),thereferencecoordinatesarethespecularreectionoftheelectronbunchcoordinate asitstrokethealuminumradiator. wavevectorkjby2bn==!=cwhere(resp.!)isthewavelength(resp.frequency)ofobservation ThetotaldensitypowerradiatedatthelocationPisthenETET;itwrites: d2P(!) d!d=E1e(!)E1e(!)8
thenthecoherentpowersimplyrewritesas:<br />
wherewehaveusedthepropertiesoftheDirac-functions.Let'sconsiderthespatial<strong>and</strong>angular distributionsrespectivelyS(!X)=(1=N)Pj=1:::N(XXj)<strong>and</strong>A(!)=(1=N)Pj=1:::N(j). d2P(!) d!d="d2P(!) k=1:::N;k
equation boundarybetweenvacuum<strong>and</strong>amediumofrelativedielectricpermittivity=absolute=0(where 0isthevacuumelectricpermittivity),theproblemconsistsofsolvingthescalar<strong>and</strong>vectorpotential<br />
atthemediainterface:thefollowingcomponentsoftheelectromagneticeldmusthavecontinuity: Ek,B?,Hk,<strong>and</strong>D?(\k"<strong>and</strong>\?"correspondstothecomponentsparallel<strong>and</strong>perpendicularto theinterfacesurface).Moreovertheelectriceldsolutionofthehomogeneousequation(i.e.the electromagneticeldsolutionofEqn.(2.8)mustbematchedwiththeproperboundarycondition inthetwomediai.e.vacuum(byletting=1)<strong>and</strong>inthemediawithpermittivity.Thehomogeneoussolutionofthelatterequationgivestheradiationeld(photon;!Aphoton).Theobtained r2"!A#1c2@t"!A#=10e"(!r;t) !(!r;t)# (2.8)<br />
typesofradiationarefound:aforwardradiationwhichisemittedinthedirectioncenteredaround thedirectionofmotionoftheelectron,<strong>and</strong>abackwardradiationemittedaroundthespecularaxis ofreectionoftheinterface.Themostgeneralexpressionforthetransitionradiationemittedin is[7]: radiationpotentials)mustsatisfyr:!Eradiation=0everywhere.Whensolvingthisproblemtwo anangleofincidence thebackwarddirectionbyanelectronmovingfromvacuumtoamediumofpermittivitywith d!d= d2Wk j12z+zqsin2()2zxcos(x)sin2()xzcos(x)qsin2() 43[(12xcos2(x))22xcos2()]2sin() (denedintheplanexz)withrespecttotheinterfacenormaldirection Z0e22zcos2()j1j2<br />
d2W? d!d= 43[(1xcos(x))22zcos2()]sin2() 1+zqsin2()xcos(x)cos()+qsin2() e22x4zcos2(y)cos2()j1j2 1 (2.9) j2<br />
Z0=120isthevacuumfreespaceimpedance,<strong>and</strong>thedierentanglesarepresenteding- thedirectionofobservation<strong>and</strong>thexoryaxis.Theseanglesaredenedbycos(x)=sin()cos() Aprioritransitionradiationspectrumhasnodirectdependenceonthefrequency!ofobservation; ure2.2.Thedependenceon referencedw.r.t.thezaxis. <strong>and</strong>cos(y)=sin()sin(),istheazimuthalangleinthex-yplane<strong>and</strong> j1zqsin2()xcos(x)qsin2()+cos()j2(2.10)<br />
inrealitythisdependenceiscomingfromtheelectricpermittivity=(!). isinx=sin( )<strong>and</strong>z=cos( )<strong>and</strong>x;yaretheanglesbetween<br />
spectralenergydistributionemittedinthebackwarddirectionviatransitionradiationreducesto: Undernormalincidence,i.e. =0(x=y=0),onlythe"k"componentremains,<strong>and</strong>the istheincidenceangle<br />
d!d=e22sin2()cos2() d2W2c(12cos2())2j<br />
1+qsin2()cos()+qsin2()j2(2.11)<br />
(1)12+qsin2()
y<br />
electron<br />
ψ<br />
O<br />
z<br />
densitymeasurementwhereweusedcarbontoproducetransitionradiation.<br />
θ<br />
Finally,anotherimportantcaseistheoneofaperfectconductor(i.e.(!)!1,8!).Forsuch<br />
φ<br />
classofmaterial,<strong>and</strong>undernormalincidence,thelatterEqn.(2.11)reducestothewellknown ThiscaseisofimportanceforourdiscussionontheelaborationofanoninterceptiveTR-based Figure2.2:Denitionoftheanglesusedinequations(2.9)<strong>and</strong>(2.10).<br />
x<br />
relation3:<br />
Observation<br />
d!d=Z0e22sin2() d2W<br />
Point<br />
referencedwithrespecttothespecularaxis. wheretheelectronincomesontheinterfacewitha45degincidence;theangle,inthiscase,being TheEqn.(2.12),inthelimitofanultra-relativisticelectron(i.e.!1)isalsovalidinthecase intercepttheelectronbeamwithverythinfoil.Inourcase,thefoilismadeofaluminumorcar- Thecongurationgenerallyusedtogeneratetransitionradiationinaparticleacceleratoristo 44c(12cos2())2(1;1) !Z0e2 43(2+2)2 2 (2.12)<br />
atsmalleranglesinceitis1=()2.Intheextremecasewhere=p2themaximumoccursat angleof90degw.r.t.thespecularaxis.Ingure2.3(B),wecomparetherenormalized(compared toitsmaximumvalue)TRangulardistributionemittedbyintheforwarddirectionbyanelectron bon.Thistypeofcongurationallowstogeneratebothbackward(atthevacuum-to-aluminum<br />
normallyincidentonacarbon<strong>and</strong>aluminumfoil.Typicalradiationpatternarepresentedinthe gulardistributionofbackwardTR,forthecaseofanaluminuminterface,generatedbyanelectron undernormalincidenceispresentedingure2.3(A)forthreedierentvaluesoftheLorenzfactor interface)<strong>and</strong>forward(atthealuminum-to-vacuuminterface)transitionradiation.Atypicalan-<br />
antennadiagramingure2.4forthecaseofnormal<strong>and</strong>45degincidenceoftheelectronbeamon thefoil.Inthecaseofnormalincidence,thepatternissymmetricwithrespecttotheelectronaxis. .Astheelectronenergyincrease,themaximumoftheangulardistributiongetlarger<strong>and</strong>occur<br />
chargeusuallyusetorendereasierthetreatmentofboundaryvaluesproblem.Inthepresentcase,theproblemofan electronmovingtowardaninniteperfectlyconductingplanecanbereducetoanelectron<strong>and</strong>itselectromagnetic imagetravelingtowardeachother.Thepassagefromtheelectronintotheperfectconductoristhenequivalenttothe collisionoftheelectronwithitsimage,formalismtotreatsuch\collapsingdipole"isreadilyavailable(seereference[8] Chap.(15)).<br />
3Infactthisrelationcanbederiveddirectly,withoutsolvingthewaveequation,byusingthemethodofimage
10<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5<br />
−4<br />
10 −3<br />
10 −2<br />
10 −1<br />
10 0<br />
(A) (B)<br />
Aluminum<br />
100<br />
therenormalizedTRforwardangulardistributionemittedbyanelectronpassingthroughavacuumaluminum(solidline)<strong>and</strong>vacuum-carbon(dashedline)interface(B).Forcarbonthepermittivity appropriatecurve)asanelectronpassesfromtheinterfacevacuum-aluminum(A).Comparisonof<br />
10<br />
twodierentcrystaldirection.The5.7valueisthesmallestpermittivity.Privatecommunication isassumedtobe5.7.(Carbonormoreexactlygraphitehastwodierentelectricpermittivityforits fromGoodfellowInc.,London,U.K.). Figure2.3:DistributionofforwardTRradiationfordierentvalueof(mentionedclosetothe<br />
2<br />
Carbon<br />
Howeverinthecaseofnon-normalincidencethereisadis-symmetryinthelobesamplitude.This inthecaseof45degincidence,forultra-relativisticelectrons.Let'sstudyhowtheradiation,in<br />
θ (rad)<br />
termofenergy,isdistributedarounditsmaximum.Forsuchapurposeweneedtoevaluatethe dis-symmetrytendstobereducedastheelectronenergyisincreased,<strong>and</strong>becomesinsignicant,<br />
θ (rad)<br />
θ (rad)<br />
integrals:dW<br />
aboveangularintegralto==2: Thereforethetotalenergyradiatedinthehemisphereisobtainedsettingtheupperlimitofthe d!=Zdd2W =+(12)argtanh() 2(21)cos()(1+2)argtanh(cos())(2+22+2cos(2)) d!d=Z2 2 0dZ0dd2W 2(2+2+cos(2)) + d!d<br />
dW d!tot=+(1+2)2log1+ 1 (2.13)<br />
Firstlywenotethatforultra-relativisticelectronthetotalenergyemittedinthehemispherehasa logarithmicdependenceontheenergyinlog(42). 1=-coneversustheenergyoftheincidentelectron.Wenotethatforultra-relativisticelectrons, Ingure2.5(B)wepresentthedependenceofthefractionofthetotalenergyencompassedinthe mostoftheenergyislocatedoutsidethis1=-cone.Despitetransitionradiationhasasharp maximumlocatedatthe1=-cone,itspowerisnot,likeforinstanceforsynchrotronradiation,<br />
TR Spectral Angular Intensity (a.u)<br />
TR Spectral Angular Intensity (a.u)<br />
OTR Distribution (a.u.)<br />
2 (2.14)
(A)<br />
120<br />
90<br />
1<br />
60<br />
(B)<br />
120<br />
90<br />
1<br />
60<br />
Figure2.4:Polarplotofthenormalizedradiationpatternforanaluminumfoilwithanelectron<br />
0.8<br />
0.8<br />
Specular Axis<br />
undernormalincidence(i.e.<br />
0.6<br />
0.6<br />
(B)(i.e.<br />
150<br />
30<br />
150<br />
30<br />
0.4<br />
0.4<br />
0.2<br />
0.2<br />
electron<br />
180 0<br />
180 0<br />
weshouldbecarefulwhendetectingtransitionradiationtooptimizetheangularacceptanceofthe =0deginEqns.(2.9)<strong>and</strong>(2.10))(A)<strong>and</strong>witha45degincidence<br />
detectionsystemasafunctionoftheelectronsenergythatproducetheradiation.Forsuchpurpose =45deginEqns.(2.9))<strong>and</strong>(2.10).FortheseplotstheLorentzfactorwaschosento<br />
210<br />
330<br />
210<br />
330<br />
wehaveplottedingure2.5(A)thefractionofthetotalenergyversustheangularacceptancefor be=10forclarityofthegure,<strong>and</strong>theequations(2.9)<strong>and</strong>(2.10)wererenormalizedto<strong>their</strong> maximumvalue. locatedwithinthiscone:mostofthepowerisinfactinthetailofthedistribution.Therefore thedierentelectronenergywewillconsiderinthepresentdissertation.<br />
240<br />
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240<br />
300<br />
electron<br />
270<br />
270<br />
2.4SynchrotronRadiation Theelectromagneticradiationemittedbyachargedparticlewithnon-zeroaccelerationishistoricallytermed\synchrotronradiation"afteritsrstvisualobservationnearlyftyyearsagoina synchrotronaccelerator.Suchradiationistypicallyemittedinpresenceofamagneticdeecting beenknownasalimitationtooperatecircularacceleratorsabovethe100GeVregimetoaccelerate electrons.Nowadaysitisacommonmechanismonwhichaccelerator-basedlightsourceareworking.Also,sinceSRisgenerated\forfree"inaccelerator,examinationoftheSRpropertiesemitted byaelectronbunchcanrevealinformationonthebunchpropertiesaswewillseeinChapter5. withuniformcircularorbit.Synchrotronradiation(SR)<strong>and</strong>theassociatedenergylosshasrst ThepurposeofthissectionistoexposefewbasicpropertiesofSR.Sinceithasbeenwidelytreated inmanytextbook(e.g.see[8]),wewillnotderiveanyofitsproperties<strong>and</strong>onlyreproducethe eldsuchastheonegeneratedbydipolemagnets,becauseofthecentrifugalaccelerationassociated<br />
resultswefeelnecessaryforthepresentdiscussion. Animportantquantityisthespectralangulardistributionofthesynchrotronradiationwhichis givenby(extendedfrom[8]): "d2W d!d#1e=3rcmec2Z02 163c!2 !2c(1+22)2"K2=3()+22 1+22K21=3()# (2.15)<br />
Al foil<br />
Al foil
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-1/2 Figure2.5:Fractionofthetotaltransitionpoweremittedintothehemispherethatisconcentrated<br />
(A)<br />
(B)<br />
withinaconeofsemi-angle(A)<strong>and</strong>containswithinaconeofsemi-angle1=(B). where=1=2!=!c(1+22)3=2,!cisthecriticalfrequencyasdenedinthesynchrotron electriceldcomponentparalleltothedeectingmagneticeld.Asonecannoticefromgure2.6, theelectrontrajectorywhilethesecondterm(/K21=3())correspondstothe-polarizationi.e.the thepowerassociatedwiththe-polarizationoftheelectriceldi.e.thecomponentintheplaneof radiationformalism(!c=2=3c3=),Z0isvacuumfreespaceimpedance,metheelectronmass <strong>and</strong>rc=e2=(40mec2)istheclassicalelectronradius.Therstterminthebracket(/K2=3())is<br />
(A) (B)<br />
<strong>and</strong>thereforewhenmostofthelobecanbecapturedbyadetector,onequantityofinterestisthe<br />
1.5<br />
totalsynchrotronradiationpowerspectrumintegratedoverthewholesolidanglewhichisgiven Figure2.6:Angular<strong>and</strong>frequencydistributionofsynchrotronradiationforthe(A)<strong>and</strong>the<br />
0.2<br />
1<br />
by[10]: Becausewegenerallydealwithrelativisticbeam(1),i.e.whentheradiationisverycollimated (B)polarization. theangulardistributionofthetwopolarizationsverystronglydependonthefrequency.Forinstance theangularopeningofthe-moderadiationgetsnarrowerasthefrequencyisincreased.<br />
0.1<br />
-2<br />
0.5<br />
-2<br />
0<br />
0<br />
-2<br />
0<br />
-2<br />
0<br />
dP<br />
0<br />
0<br />
γθ<br />
γθ<br />
2<br />
d!1e=Ptot !cS(!=!c) (2.16)<br />
2<br />
2<br />
2<br />
Log (ω/ω )<br />
Log (ω/ω )<br />
10 c 10 c<br />
(no unit)<br />
(no unit)
R10S(x)dx=R11S(x)dx=1=2R10S(x)dx. ingure2.7.Itisworthwhiletomentionthatthepointx=1isthemid-totalintegralpoint: 105cE4=(22),wherethenumericalfactoristheS<strong>and</strong>'sdenitionoftheradiationconstantE wherePtotistheinstantaneoustotalSRpoweremitted:inpracticalunits(GeV/s),Ptot=8:8575 <strong>and</strong>aretheelectronenergyinGeV<strong>and</strong>theradiusoftrajectorycurvatureinmeters.S(x) inEqn.(2.16)istheso-calledUniversalfunction,S(x)=9p3 8xR1xK5=3(x)dx,whichisplotted<br />
chrotronradiationisproportionaltotheUniversalfunction. 2.5RudimentsonFEL-oscillatorTheory Figure2.7:PlotoftheUniversalfunctionS(!=!c).Thefrequencydistributionofthetotalsyn-<br />
Despitethefactthepresentreportdoesnotspecicallydealwiththephotonbeamgenerated free,indeeditmeanstheyareunbounded(contrarytoconventionallaser)butthereareconnedin amagnetostaticregionsincethefreeelectronswillnotradiateunlesstheyareexperiencingsome kindofacceleration. Asinaconventionallaser,FELconsistsinthreemainprocesses:(i)aspontaneousemissionis providedbysynchrotronradiationemittedaselectronswiggleinamagnet;(ii)theso-generated bytheIRFEL,webrieyexplainthebasesofFELtheorysincetheywillenablethereaderto all,weshouldnotethatthewordfreeinfree-electronlaserdoesnotmeanthattheelectronsare underst<strong>and</strong>bettertherequirementsonthedriver-acceleratorelectronbeamparameters.Firstof radiationisrecirculatedinaresonator;(iii)<strong>and</strong>isampliedasitcopropagateswiththeelectron<br />
amagnetthatgeneratesaspatiallyperiodicmagnetostaticeld.InthecaseoftheIRFELof InaFEL,thespontaneousemissionisgeneratedastheelectronsareinjectedintoawiggler, beam(stimulatedemission).<br />
JeersonLab,theundulatorisaplanarone:itconsistsintworowsofNupermanentmagnets ofoppositepolaritiesstackedtogetherwithaperiodu;therowareseparatedbyaxgapas<br />
2.5.1UndulatorRadiation<br />
S( ω/ω c )<br />
1<br />
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0.001 0.01 0.1 1 10<br />
ω/ω c
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<strong>Electron</strong><br />
<strong>Beam</strong><br />
Resonant Cavity Length = 8.02 m<br />
Photon <strong>Beam</strong><br />
λ u<br />
Figure2.8:FEL-oscillatorprinciple(CourtesyJ.Martz,JeersonLab).<br />
schematicallydescribedingure2.8.Insuchcongurationthegeneratedmagnetostaticeldis<br />
transversewithrespecttotheelectronvelocity.Aselectronstravelinthewiggler,theyareslightly<br />
deectedalternativelyup<strong>and</strong>down(seeFigure2.8)<strong>and</strong>therebyspontaneouslyemitsynchrotron<br />
radiationthatislinearlypolarized(inthecaseofaplanarwiggler).<br />
InthecaseofIRFEL,theundulatorproducesaweakmagnetostaticeldoftypically0.4Tesla.<br />
Theelectrontrajectorywhenitislocatedwithintheundulatorpolesisdescribedby:<br />
y=acos(2z=u) (2.17)<br />
dy<br />
dz=2a<br />
ucos(2z=u)<br />
Theforceontheelectronatthemaximumcurvaturecorrespondstothepeakvalueofthe<br />
magneticeld!B:=mec=(eB).Itiscommontocharacterizetheundulatormagnetbytheso<br />
calleddeectionparameterKdenedasK=dy=dzjmax=2ua.Togetherwiththerelation<br />
(2=)2a=eB=(mec),Ktakestheform:K=eBu<br />
2mec (2.18)<br />
thisdeectionparameteristhemaximumangularexcursionofthebeaminunitsof1=.Itis<br />
interestingtocomputethemaximumamplitudeinthecaseoftheIRFEL:a=Ku=(2)'60m<br />
whichissmallerthattheelectronbeamsizesatthislocation(x'y'200m).<br />
Thewavelengthoftheradiationemittedbytheundulatorisdeterminedbythetimecontraction<br />
factordt=dt0=1cos,tbeingthetimereferenceinthemovingframewhereast0isthelaboratory<br />
(i.e.undulator)time.Intheelectronrestframe,theelectron\sees"theNuperiodsofthewiggler
adiationpulseoflengthNu0u.centeredonthewavelength0s'0u.Inotherterms,theelectron asanNucounter-propagatingradiationeldwithaLorentz-contractedwavelength0u=uu.Thus itoscillatesNutimesalongaverticallineperpendiculartothewiggleraxis,therebyemittinga actsasarelativisticmirror<strong>and</strong>reecttheincomingradiationviaComptonback-scattering.Infact providedweobservewavelengththatarelargerthantheComptonwavelengthCompton=hc 0sisalsoshiftedbytheComptonwavelength,butthisshiftisnegligibleforrelativisticelectrons<br />
Usingtheaveragez-velocityhcosi=(1K2 whereistheangleofobservationreferencedtotheaxisoftheundulator. agoodassumptioninthecaseofIRFEL.Thereforethefundamentalwavelengthoftheundulator radiationis: 1=u(1hicos) 42+O(K4))(isthetrajectorydeectionangle), (2.19) mc2,<br />
onendsthatthefundamentalwavelengthis:<br />
widthwillhavethelimit=!0SincetheelectrononlymakesNuoscillationsintheundulator thegeneratedradiationcontainsthesamenumberofwavelengths<strong>and</strong>thereforethedurationof wavelengthsaresuppressed.Iftheundulatorwouldhaveainnitenumberofperiod,theline wavelengthrepresentsthewavelengthoftheeldcomponentthatinterfereconstructively.Other Infactalltheharmonicarealsopresenti.e.thewavelengthn=1=nwithn2N.The 1=u 22(1+K2 2+22) (2.20)<br />
thepulseisT=Nu=c.TheFouriertransformofaplanewavetruncatedafterNuoscillations frequencydependence: issinc-function4,hencethefrequencyspectrumofthespontaneousundulatorradiationhasthe<br />
to: on-axis(=0)componentisofinterested,i.e.isamplied,thefundamentalwawelengthreduces Whichmeanstheradiationispeakedatthefrequency!n=2c=n.Thewidthofthespectrum isabout! !=1 Nu.ItisimportanttonotethatinthecaseoftheFEL-oscillator,sinceonlythe d!d/sinc2Nu!!n d2W !n (2.21)<br />
reference[9])<strong>and</strong>areworthmentioninginthepresentdiscussion.Theopticalwavegeneratedfrom anundulatorcanbewellapproximated,ifNuislargeenough,byapureTEwave.Insuchcase, theform[9]: theon-axisradiationonlycontainsoddharmonic.Thepowerspectralangulardistributionisof Finallyweneedtoelaboratethepowerdensityspectrum.Wehavequalitativelyexplainedthe sincdependencebuttherearemanyotherpropertiesthathavebeenderived(seeforinstance 1=u 22(1+K2 2) (2.22)<br />
Thelatterequationisplottedingure2.9forthetwodierentvaluesofKthatareconsidered withKdef 4Thecardinalsinusfunctionisdenedas:sinc(x)=sin(x) =K=p1K2=2. d!d/mK(1K2=2)hJ(m+1)=2(mK2=4)J(m1)=2(mK2=4)i2def d2W x<br />
=Q (2.23)
harmonicwithproperchoiceoftheKvaluewhichcanbesetbychangingtheundulatorgap5. emissionassociatedwiththethirdharmoniccanbealmostaspowerfulastheemissionatrst Suchfeatureisveryinterestingforproducingshorterwavelengthlight.IntheIRFELoperationat thethird(=1=3)<strong>and</strong>fth(=1=5)harmonichasbeenachieved. fortheIRFELoperation(K=1:00<strong>and</strong>optionally1:39).Inthisgureoneseesthatspontaneous<br />
Figure2.9:Normalizedpoweroftheon-axisundulatorradiationforthetwodierentvalueofK consideredfortheIRFEL.<br />
oftheorderofthewavelength.Thespacingbetweenthetwomirrorsischosensothatthepulse OncearadiationpulseatthewavelengthgiveninEqn.(2.22)hasbeenestablishedviatheinteractionofanelectronbunchwiththemagneticeldoftheundulator,itisrecirculatedinaresonator cavitythatconsistsintwosphericaldielectricmirrors.Oneofthemisatotalreectorwhilethe otherisapartiallyreector<strong>and</strong>out-coupleradiationthroughasmallaperturewithadiameter 2.5.2AmplicationoftheSpontaneousUndulatorRadiation<br />
willcopropagatewiththenextincomingelectronbunch<strong>and</strong>thereforethelengthofthecavityis beasub-multipleof8:003minthecaseoftheIRFEL.Forelectronsinjectedattheresonantenergy L=c=(2)(whereisthetemporalspacingbetweentwoconsecutiveelectronbunches)<strong>and</strong>should<br />
phase,eachelectroninthebunchcan(i)giveenergytotheeld<strong>and</strong>decelerate,thatis\stimulated emission";(ii)takeenergyfromtheradiationeld<strong>and</strong>accelerate,thatis,\absorption".Thusifwe considerabunchofelectronwhosecenterenergyisresonantenergy,wecouldeasilyimaginethat ofenergybetweentheelectron<strong>and</strong>theradiation.Hence,dependingonthevalueoftherelative radiationelectriceld(!v:!E)isnon-zero<strong>and</strong>slowlyvaryingsothattherecanbeanetexchange ponentparalleltotheopticalpulseelectroneld,thescalarproductofanelectronvelocity<strong>and</strong>pulsewill,inprinciple,remainconstant.Sincetheelectronbeamvelocityhasanonzerocom- r=12qus(1+K2),therelativephasebetweentheelectrons<strong>and</strong>thecopropagatingradiation<br />
5TheKvaluedependenceonthegapdisoftheformK/exp(d=u)<br />
Intensity (a.u)<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0 2 4 6 8 10<br />
Harmonic Number<br />
K=1.00<br />
K=1.39
halfoftheelectronisdeceleratedwhiletheotherhalfisacceleratedresultinginanullamplication. Thisistrueintherstperiodofthewigglerbutinveryshorttime,themoreenergeticelectrons catchuptotheless-energetic,introducinganenergymodulationwithintheelectronbunchwhich, inturn,leadstoalongitudinaldensitymodulationormicro-bunching:theelectronbeamthat hasaninitialdistributiondependingonthepreviousdynamics,soonconsistsinasub-bunchesof electronsspacedatthespontaneouswavelength.Itturnsoutthatiftheelectronenergyisslightly higherthantheresonantenergyitresultsinanetgaini.e.anamplicationofthelightpulses. Thissimplemodelassumesthatelectronswithinabunchdonotinteracteachwithother,thatis single-particle-dynamicsmodelisvalid.Whensuchamodelisvalid,likeinthecaseoftheIRFEL, theFELissaidtooperateintheComptonregime. 2.5.3FELGain Theamplicationofthespontaneousemissionisquantiedbythegainthatcorrespondstothe ratiooftheenergytransmittedbytheelectronbeamtothecopropagatingelectromagneticwave totheinitialenergyofthecopropagatingelectromagneticwave.Thegain6isdenedas:<br />
beendiscussedbyS.Benson[12].Thegaincanbeparametrizedas: electronbeamparametersontheFEL-gainbyconsideringtheapproximate1Dmodelthathas whereisthereducedenergytransmittedbytheelectronbeamtotheelectriceldoftheoptical mode<strong>and</strong>W0istheenergyoftheopticalmodeconsidered(i.e.theonewhichissupposedtobe amplied).ThederivationofanalyticformulafortheFELgainhasbeenperformedinreference[11] <strong>and</strong>isbeyondthescopeofthisthesis.Howeveritseemsworthwhiletostudytheeectsofthe Gdef =mc2W0 (2.24)<br />
inEqn.(2.23).The'scoecientsinEqn.(2.25)representdegradationfactorsofthegain. oftheincomingelectronbeam:sinceelectronsinabunchdonotallhaveanenergyexactly where<strong>and</strong>Iaretheelectronbeamenergy<strong>and</strong>peakcurrent.Qisafactorthathasbeendened correspondingtotheresonantenergy;thisfactorisdenedas: isthegaindegradationduetoenergyspread<strong>and</strong>isaresultofthenon-mono-energeticcharacter g=0:0004IQNuN2~"f (2.25)<br />
onthecoecient: theharmonicnumber. whereisthereducedrms-energyspreadoftheincomingelectronbunches,<strong>and</strong>h,asbefore,is ~"isthedegradationduetobeamnon-zerotransverseemittance.Thisdegradationalsodepends = 1+4p2hNu=2 1 (2.26)<br />
ofimportancewhenconsideringthestartupofFELinteraction,ourprimaryconcerninthepresentsection.<br />
6Inthisdissertation,gaindesignatestheso-called\smallsignalgain"intheFELliteraturesinceitisthequantity ~"= q1+(42~"N=)2 1 (2.27)
Bu(rms)0.28T ParameterValueUnit Nu u 40<br />
where~"isthebeamtransverserms-emittance<strong>and</strong>Nisthenumberofbetatronoscillationsalong Table2.1:ParametersofthechosenwigglerfortheIR-DemoFEL. K2 Gap 0.5 12mm 2.7cm<br />
thewiggler.fisthegainreductionduetothellingfactorfortheopticalmode;itsimplyresults fromthenonintegraloverlapoftheelectronbunch<strong>and</strong>theradiationpulse<strong>and</strong>isapproximatedby: isthegainreductionduetoslippage:=1 f=1 1+4~"= 1+hNu 3z (2.28)<br />
beamparametersrequiredtodrivetheIRFEL. wherezistherms-longitudinalbunchlength. tobemoreecient,requirehigh-charge,ultrashort-bunchelectronbeam.Wenowpresentthe FromEqn.(2.25)weseethatthegainisproportionaltopeakcurrentwhichinturnisproportional tothechargeperbunch<strong>and</strong>inverselyproportionaltothelongitudinalbunchlength.HenceFELs, (2.29)<br />
2.6CharacteristicsoftheIRFELdriver-accelerator TodiscussthecharacteristicsrequiredfortheelectronbeamgeneratedbytheIR-Demodriver accelerator,welistinTable2.1thespecicationsonthewigglermagnetwhichhavebeenderived fromtherequirementsonphotonbeamparametersdenebytheexperimentalists.IntheJLabFEL IR-demo[15](seegure2.11),thechargeperbunchwasinitiallychosentobe60pC,themaximum valuethatyieldsatolerableemittancegrowthduetospace-charge.Theaveragecurrentshouldbe ashighaspossibletomaximizetheaveragepowerofthelaser.Itisafunctionofthechargeper bunch,<strong>and</strong>thebunchrepetitionthatdependsonthephotocathodedriver-laserwhichinturnmust beasub-harmonicofthesuperconductinglinacoperatingfrequency(1497MHz).Themaximum<br />
arbitrarywavelengthbychoosingtheproperenergy.Experimentally,theoutputwavelengthislimitedtoacertain bunchrepetitionrateis74:85MHz<strong>and</strong>itislimitedbytheelectronsource.Therepetitionrate<br />
astheoneusedtolasewithanoutputwavelengthof6m<br />
rangethatdependsontheFEL-opticalcavitymirror.Forinstancethemirrorusedtolaseat3marenotthesame currentthatcanbereachisapproximately5mA. Sincetheenergydoesnotaectthegain,<strong>and</strong>itsonlyimplicationisonFELwavelength:theIR- oftheelectronbunchwasindeedinitiallysetto37:425MHzwhichmeanthemaximumaverage demoisforeseentoinitiallylaseintheregion3m-6mrange7(laterthisrangewillbeextended 7Theoreticallytheoutputwavelengthonlydependsonenergy.Hencewecould,intheIRFELoperateatany
to100mwavelengths);thisimpliesthemaximumelectronbeamenergyshouldbeintherange<br />
Thefactor=(4)isthetransversephasespaceareaoftheopticalbeamassumingitcanbewell 38-48MeV.<br />
describedbyGaussianoptics. Thetransversenormalizedemittancespecicationaresetbythewavelengthatwhichwewishto<br />
Forthelaserwavelength<strong>and</strong>energyofoperation(i.e.1'3m<strong>and</strong>'77)theEqn.(2.30)yields operatethelaser:theelectronbeamemittanceshouldbelessthantheopticalbeamemittance:<br />
value:itwasassessedfromnumericalsimulationoftheFELgainthattheemittanceshouldbeless anormalizedemittanceforbothtransverseplanethatmustbesmallerthan~"n'19mm-mradto enabletheoperationofthelaseratthefundamentalwavelength.Infactthisvalueisan\edge" ~"n
2.7TheJeersonLabIRproject usedasexperimentalplatformthroughoutthepresentreport. InthebuiltIRFEL(seetopviewingure2.11),theelectronbeamisgeneratedbya350keVphotoemissionelectrongun<strong>and</strong>acceleratedbytwosuperconductingRFCEBAF-typecavities(5-cells ThepurposeoftheconsortiumgatheredaroundJeersonLabistodevelopthetechnologiesneeded<br />
cavityoperatingon-acceleratingmode)mountedasapairintheso-called\quartercryounit" torealizehigh-powerfree-electronlaserinacosteectiveframe.Thelongtermprojectistobuild a20kWinfraredfree-electronlaserAsastartingpointitwasdecidedtobuildtheIRFELthatis<br />
paction<strong>and</strong>pathlengthuptotheentranceofthecryomodulewiththepropertimeofarrivalso\spent"beamisrecirculatedwithaquasi-isochronousrecirculatorwithvariablemomentumcom- inTable2.1)islocatedbetweenthetwoaforementionedchicanes.AftertheFELinteraction,the 48MeV.Thiscryomoduleisfollowedbytwo4-bendschicanesthatbypasstheFELopticalcavities <strong>and</strong>provideadditionallongitudinalphasespacemanipulation.TheIRundulator(seeparameters superconductingRFcavities.Thelinaccancurrentlyacceleratethebeamuptoapproximately themainlinacwhichiscomposedofonecryomodule,containing8superconductingCEBAF-type whichprovidesabeamenergygainofapproximately10MeV.Thebeamistheninjectedinto<br />
thattheelectronbunchesareonthedeceleratingphaseoftheradio-frequencywave.Thesecondary beamistherebydecelerateddownto10MeV<strong>and</strong>separatedfromtheprimarybeambeforebeing dumpedinthe\energyrecoverydump"bythemeanofthe\extractionchicane".Theenergy recoveryschemeallowtherecoveryofalmostalltheenergyprovidedtothebeambythemainlinac duringtheaccelerationphase. TheFELlightisdirectedinanopticalroomwhereitcanbediagnosed<strong>and</strong>senttooneofthe experimenttomeasurethegoldreectivityintheIRregion,theobservationoflaserlighteecton aplasma,<strong>and</strong>somepreliminarytestsinmicro-machining. sixuserlaboratories.Theexperimentsthathavebeenruntodate,includest<strong>and</strong>ardpump-probe<br />
Theelectronsource[14]isaDC-photo-emissiongunpresentedingure2.12.Itconsistsofa GalliumArsenide(GaAs)photocathodeilluminatedbyalasersystemcapableofproviding5W.<br />
Figure2.11:Anactualtopviewofthe\asbuilt"IRFELdriveraccelerator.<br />
~ 5 meters<br />
Recirculation Arc<br />
Straight Ahead Dump<br />
Extraction Chicane<br />
Decompressor<br />
Compressor<br />
Energy Recovery Dump<br />
Return Transport Line<br />
Injector Dump<br />
Recirculation Arc
Reflected Laser <strong>Beam</strong><br />
Anode Plate<br />
Figure2.12:Simpliedschematicoftheelectronsource:A527nmlaserbeamthatcanbemodulatedbytwoelectro-opticscell(EO1<strong>and</strong>EO2)<strong>and</strong>attenuatedbyarotationalpolarizerilluminates Solenoidal Lens<br />
Photocathode<br />
Insulating Ceramics<br />
theGaAsphotocathode.Ejectedphoto-electronsareacceleratedthroughtotheacceleratingvoltage ofnominally350kVbetweenthephotocathode<strong>and</strong>theanode.<br />
Incoming Laser <strong>Beam</strong><br />
Tunnel<br />
Optical Room<br />
Transport System<br />
EO2<br />
Attenuator<br />
EO1<br />
Nd:YLF laser<br />
01<br />
01<br />
01<br />
01<br />
01<br />
01
Study TheFELdriveraccelerator:Lattice Chapter3<br />
experimentalresultsobtainedaswetriedtocharacterizethelattice<strong>and</strong>compareitwithanumerical model. reviewingtheacceleratormagneticopticswiththehelpofanumericalmodel,wepresentfew 3.1Introduction<br />
3.2ABriefOverviewoftheFELOpticalLattice ThepresentChapterdealswiththeopticallatticeoftheIRFELdriver-accelerator.Afterbriey<br />
thepurposeofthepresentdiscussion,thedriver-acceleratorcanbedividedintoveparts: InthissectionwedescribetheFELopticallatticeforthemainacceleratoronly.Ithasbeendesigned byD.Douglas<strong>and</strong>itisalsodescribedinnumerousreference(seeforinstancereference[16]).For 1.A10MeVinjector<strong>and</strong>theinjectiontransferline 2.A48MeVsuperconductingradio-frequency(SRF)linearaccelerator(itisusedbothtoac- 3.Awigglerinsertionregion, 4.Arecirculationring, 5.Areinjectiontransferline. celeratetherstpassbeam<strong>and</strong>todeceleratethesecondpassbeam);thislinacisalsotermed \cryomodule"hereafter,<br />
<strong>and</strong>5inthelistabove. Inthissectionwewillonlyconcentrateonthehighenergylattice(E'48MeV)thatisitems3,4, TherstregionencounteredbythebeamattheexitoftheSRFlinacisa\matching"region.It 22
To Linac entrance<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
Figure3.1:Dispersedoverviewofthemainringofthedriveracceleratorcorrespondingtog-<br />
0000000000000<br />
1111111111111<br />
Quadrupole<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
0000000000000<br />
1111111111111<br />
Sextupole<br />
0000000000000<br />
1111111111111<br />
ure2.11.Thepathoftheelectronbeamisindicatedwitharrows.<br />
0000000000000<br />
1111111111111<br />
Quadrupole<br />
Sextupole<br />
180 deg<br />
dipole<br />
Reinjection Line<br />
180 deg<br />
dipole<br />
Sextupole<br />
Quadrupole<br />
Sextupole<br />
Quadrupole<br />
To Backleg To Arc 2 (5F)<br />
innitelylongwigglerw=w=(2p2K),withwthewigglerperiod<strong>and</strong>Ktheundulator thebetatronfunctionsarematchedtothewiggler\natural"betatronfunctioneigenvalueofan insuresthebeamlatticefunctionsareproperlymatchedtothedesiredvalueattheundulatorcenter.<br />
parameter(seeChapter2).The\upstream"by-passchicaneprovidefurtherlongitudinalphase Thisregionconsistsoftwoquadrupoletelescopesdisposedupstream<strong>and</strong>downstreamanachromatic chicane.Thetelescopesconsisteachofthreequadrupoles.Thesetwotelescopesprovidesixfree whileinsuringthebeamsizecanstillbecontainedwithinthevacuumchamberwithacceptablylow particlelossviascraping.The-functionsaresupposedtobezeroatthewigglercenterwhereas parameters(thestrengthofthequadrupoles)toadjustthefourTwissparameters(x,y,x,y)<br />
Quad Quad Quad Quad Quad Quad Quad Quad Quad Quad Quad Quad Quad<br />
Afterthewiggler,twoadditionaltelescopeseachcomposedofquadrupoletripletsareusedtomatch theopticallatticefunctionstothedesiredvaluesattheinjectionpoint.Longitudinally,further spacerotationbecauseofitsnonisochronicity(momentumcompaction1R56=28cm).<br />
whereitisdenedastheratioofrelativepathlengthchangeforarelativeenergychange=L=L transportformalism.Thisdenitionisdierentfromtheusualdenition(generallyforclosedorbitaccelerator) phasespacerotationisprovidedbythe\downstream"by-passchicanewhichisidenticaltothe upstreamchicane. Therecirculationloopiscomposedoftwoarcslinkedbyastraightlinesection,termed\backleg" 1ThroughoutthisreportthemomentumcompactionisdenedasthetransfermatrixelementR56=z E=EE=Einthe<br />
Sextupole<br />
Quadrupole<br />
Sextupole<br />
Quadrupole<br />
first Telescope<br />
To dump<br />
Fourth Telescope<br />
To Arc 1 (3F)<br />
SRF Linac<br />
<strong>Beam</strong> Direction<br />
Third Telescope<br />
Undulator Magnet<br />
Second Telescope<br />
<strong>Beam</strong> from Injector<br />
Quadrupole<br />
Sextupole<br />
Quadrupole<br />
Sextupole
ParameterValue<br />
x -0.178<br />
x(m) 8.331<br />
y -0.124<br />
y(m) 3.979<br />
Table3.1:Twissparametersdownstreamthecryomoduleexpectedfromsimulationswiththecode<br />
parmela.<br />
thatconsistsofsixperiodofaFODOlattice.<br />
ThearcsarebasedontheMIT-Batesacceleratordesign[17];theyprovideeachatotalbendingangle<br />
of180deg.Theyincludefourwedge-typedipoles,eachbendingthebeambyanangleof'28deg<br />
alternatively,installedinpairsymmetricallyarounda180degdipole.Furthermoreprovidingthe<br />
desiredrotation,thearcsarealsousedtoadjustthetotalbeampathlengthoftherecirculated<br />
beaminasuchwaythattheelectronbuncheshavethepropertimingtobeonthedecelerating<br />
phaseoftheSRFlinac,averyimportantparameterforenergyrecovery.Forsuchapurpose,<br />
thearcisinstrumentedwithapairofhorizontalsteererslocatedupstream<strong>and</strong>downstreamthe<br />
180degdipoletovarythereferenceorbitpathlengthinsidethe180degmagnet.Twofamilies<br />
ofquadrupoles(thetrimquadrupoles)<strong>and</strong>sextupolesareinstalledinthearctoprovideboth<br />
linear<strong>and</strong>quadraticenergydependentpathlengthvariationthatarenecessaryinthe\energy-<br />
compression"schemeneededtoproperlyenergyrecoverthebeam[18]i.e.topreciselyadjustthe<br />
momentumcompactionR562(lineardependenceoflongitudinalpositionwithrelativeenergy)<strong>and</strong><br />
thenonlinearmomentumcompactionT566(quadraticdependence).Whenthequadrupoles<strong>and</strong><br />
sextupolesarenotpowered,thearcisoperatinginanon-isochronousmode(R56=13:124cm).<br />
Howeverundernominaloperation,i.e.whentheFELisoperating<strong>and</strong>thelinacisinenergyrecovery<br />
mode,becauseoftheneedofenergycompression,thesextupoles<strong>and</strong>quadrupolesofonefamily<br />
areexcitedtopropervaluesinordertoprovidetherequiredR56betweenthewigglerexitupto<br />
thelinacentrance.<br />
Thebacklegtransportlineconsistsofthirteenquadrupoles.NominallyitisoperatedasaF0D0<br />
latticewithaphaseadvancepercell=90deg,butwehavedemonstrateditsoperationwith<br />
aphaseadvanceof=60degneededduringemittancemeasurementbasedonmulti-monitor<br />
technique.Itstotaltransfermatrixis-Iforbothtransverseplane:itimagesthelatticeoptical<br />
functionsattherstarcexitintotherstarcentranceaccordinglyto!<strong>and</strong>!.<br />
Thereinjectionlineconsistsofatelescopecomposedoffourquadrupolesthatisusedtoadjustthe<br />
beamlatticefunctiontoachievereasonablebeamenvelopeattheentranceoftheSRFlinac.<br />
AstheelectronbunchesgoforthesecondtimesthroughtheSRFlinac,theyaredecelerated<strong>and</strong><br />
inducevoltageinthecavityviabeamloading.Thisinduced-voltageisatthepropermodetoserve<br />
toacceleratethenextbunchintheaccelerationphasewhichisintheneighboringRFbucket.The<br />
wastedbeam,oncedecelerated(i.e.atapproximately10MeV),bifurcatesintoadump.<br />
Typicallatticefunctions,computedwiththesecondorderopticscodedimad,areshownin<br />
gure3.2.Theinitialconditions(seeTable3.2)thatareused,havebeencomputedusingthe<br />
parmelacodesinceitincludespacechargeeectsinthelowenergyregion<strong>and</strong>alsohaveavery<br />
detailedmodeloftheSRFcavities.Forthegurepresented,theundulatormagnetisinstalled.<br />
2Rs0!s<br />
56=Rs<br />
s0=R16(x)<br />
(x)dx,where(x)isthelocalbendingradius.Sosincethepresentlymentionedquadrupole<br />
arelocatedinadispersiveregion,i.e.R166=0,sothatR56canbevaried.
Horizontal Dispersion (m)<br />
Arc 1 ‘‘back−leg’’<br />
extr. chic.<br />
Arc 2<br />
−1<br />
20<br />
Figure3.2:HorizontalDispersion(topgraph)<strong>and</strong>transversebetatronfunctions(bottomgraph)<br />
βx 15<br />
βy 10<br />
manyfactorsthatcancausetherealmachinetobedierentfromthenumericalmodel.Amongthe forthenominalsettingsofthemagneticoptics.<br />
5<br />
causesofthesediscrepancies,themostusualarealignmenterrors,magnetdisfunctions,etc...Hence itisofprimeimportance,intherstphaseofcommissioningoftheaccelerator,todiagnosethese defects<strong>and</strong>potentiallyxorunderst<strong>and</strong>themtomatchascloseaspossiblethenumericalmodel Thoughthedriver-acceleratordesignisessentiallyspeciedvianumericalsimulations,thereare 3.3MeasurementoftheTransverseResponse<br />
0<br />
0 20 40 60 80 100<br />
tothe\asbuilt"accelerator.Inthissectionwereportonmeasurementofthetransverselattice<br />
Distance From Cryomodule Exit (m)<br />
responseofthedriveracceleratoralongtherecirculator.Inthisreport,wehaveonlyconcentrated ontwotypesofmeasurements:(1)trytoverifythattherstordertransfermatrixusedinthe modelisclosetothemachine,(2)measurepreciselythedispersion(i.e.thetransverseposition dependenceonenergy)functionintheback-legtransferline.<br />
Betatron Functions (m)<br />
2<br />
1<br />
0<br />
Compressor<br />
Undulator<br />
Decompressor<br />
reinj chic.
Thepurposeofmeasuringthetransverseresponseoftheopticallatticeistogetsomeinsightsonthe 3.3.1TheoreticalBackground<br />
validity,accordinglyto: traryposition0,ispropagatedtoanotherposition,undertheassumptionofrstordertransportcitation,opticalelementsmisaligned.Wehaveperformedtwotypesofmeasurement:(1)response ofthelatticeforagivenangularexcitationbymeanofcorrectors<strong>and</strong>(2)energydependenceofthe latticei.e.measurementofthedispersion. Inadispersion-freeregion,theposition<strong>and</strong>divergencecentroidofabeam(x(0);y(0))atanarbirstordertransport<strong>and</strong>potentiallyndoutproblemswiththelatticei.e.magnetscorruptedex- anevent,the\perturbed"beampositiondownstreamnowwrites: bativeangularkickonthebeamtrajectoryatthepositionx(0)i.e.x0(0)!x0(0)+x0.Insuch Theobservablethatcaneasilybemeasuredisabeamcentroidpositionatthedownstreamlocation x(s).Henceatechniquetomeasurethetransverseresponseisdistorttheorbitbyusingapertur- x0(s)!= x(s) R0!s 11R0!s 21R0!s 12 22! x0(0)! x(0) (3.1)<br />
Moreover,foradipolewecanrelatetheangularkicktothebeammomentum:x0=1=(3:3356 ontheangularperturbationat0: Thatisthedisplacementofthebeampositionats,x(s)def xperturb(s)=R0!s x(s)=R0!s 11x(0)+R0!s 12x0 12(x0(0)+x0) =xperturb(s)x(s)isonlydependent (3.3) (3.2)<br />
ofR11requiresmoreelaboration:oneneedstocreateaperturbationthatisexactly90degoutof betatronphasewithrespecttothekickintendedtomeasuretheR12;insuchwayonecancompute p)RBdlwhereRBdlrepresentstheeldintegral(inT.m),<strong>and</strong>pisthebeammomentum(in<br />
towhatpositiondisplacementitcorresponds. ordertransfermatrixmodel,istomeasurethebeampositionalongthebeamlinefordierentperturbations(angularkickvaluesbutalsopositionofthekickermagnetused).Itisworthmentioning thatthistechniqueallowsdirectlytomeasuretheR12transfermatrixelement.Themeasurement GeV). Henceaverysimplewaytocheckifthe\realworld"machineisperformingaspredictedbyrst<br />
wecangetanestimateofthedispersionfunction.<br />
totheorbit.Notethat(s)R0!s asx(s)=x(s)+(s)(0),wherex(s)isapurebetatroninducedposition<strong>and</strong>theproductof Theothertypeofmeasurementistheenergydependenceoftheopticallattice.Thismeasurementis verysimilartothetransfermatrixresponseaforementioned:itisknowthataftermagneticelement systemssuchasdipolestherecanbepositiondependenceonenergy,i.e.thepositionx(s)writes (s),dispersionfunctionats,with(0),energyosetat0,representthedispersivecontribution !+)<strong>and</strong>measuringtheassociatedpositionchangedownstreamthebeamlinex= 16.Thereforebyvaryingtheenergyofthebunchcentroid(i.e.
3.3.2ExperimentalMethod Todetermineexperimentallythelatticeresponseduetoeitherkickexcitation(puretransverse pickupcalledbeampositionmonitors(BPMs)[25].TheseBPMsconsistofanumberofpick-up response)orenergychange(dispersion),weonlyhavetomeasurethebeamposition,theonly observablewecaneasilyaccess.Thebeampositionismeasuredbymeansofelectromagnetic antennaedistributedaroundthediameterofthevacuumchamberthatdetecttheCoulombeld associatedwithanelectronbunchasitpropagatesalongthebeamline.Basedontheasymmetry ofthesignaloneachantenna,thebeamcentroidcoordinatescanbeinferred.TwotypesofBPMs areinstalledintheIRFELaccelerator.Inthemeasurementpresentedhereafter,wehaveonly wakeeld.Thebeamcentroidcoordinateisasimplefunctionoftheelectricpotential(VR,VL,VD, usedtheso-calledstriplineBPMswhosecrosssectionisdepictedingure3.3.ThistypeofBPM consistsoffourstrip-likepickupantennaeorientedat90degfromeachother.Thistypeofgeometry forthepickupantennaehastheadvantagetominimizethebeamqualitydeteriorationbecauseof<br />
VU)inducedbythebeamoneachantenna[19]oftheBPM(R,L,D,Uongure3.3): Figure3.3:schematiccutofabeampositionmonitor(BPM).<br />
Theelectronicsystemusedtoprocessthesignalistheso-calledSwitchedElectrode<strong>Electron</strong>ics system.Andthebeampositionisinferredusingthest<strong>and</strong>ardtechniquefromthissignal.An (SEE)[24].Thesignalofperiphericalelectrodesismultiplexedtothesameprocessingelectronics y/VUVD x/VRVL VU+VD VR+VL<br />
typically30Hztostudypotentialtimedependentbeampositionuctuation. advantageofthiselectronicisthatitcanbeusedtoacquirebeampositionatveryhighrate (3.4)<br />
Thepracticalmethodtomeasuretheresponseofthelatticeisasfollows:(1)Forthenominal condition,measurethebeampositiononallthedesiredBPMsfx0;y0gi=1:::N(thesubscriptiisthe angularkickforR12measurement);(3)measurethenewbeampositionontheNBPMsfx;ygi=1:::N BPMindex);(2)impressthedesireddistortion(i.e.energychangefordispersionmeasurementor <strong>and</strong>computethedisplacement(ordierenceorbit)ofthebeamcentroidalongthebeamlineatthe<br />
L<br />
Vacuum Chamber<br />
U<br />
y<br />
y<br />
O<br />
D<br />
x<br />
<strong>Beam</strong> cross-section<br />
R<br />
Pickup Antenna<br />
x
computedusingthelatticeset-upusedduringthemeasurement;thebeamcentroidinducedbythe locationofeachBPMsfx;ygi=1:::N:<br />
angularperturbationarecomputedusingtheR12transfermatrixelement. Foratransverseresponselatticemeasurement,thetransversematrixelementsarenumerically xy!i=1:::N= xy!i=1:::N x0 y0!i=1:::N (3.5)<br />
ofthetransfermatrixbetweenthedispersiongeneratorexit<strong>and</strong>theconsideredlocation: Fordispersionmeasurement,theenergychangeisimpressedusingthelastpairofcavitiesinthecryomodule.Unfortunatelythismethoddoesnotprovidevaluableinformation(seetheexperimental sectionformoreexplanation)<strong>and</strong>wehadtouseanothertechniquetoperformsuchmeasurement. Intherstplacedispersionalwaysresultsinanon-zerolocalR16transfermatrixelemente.g.due tothepresenceofadipolemagnet.Afterithasbeengenerated,itcanpropagateinregionwithno magneticeld;forinstanceifattheexitofthedispersiongeneratorthedispersion<strong>and</strong>itsderivative are0<strong>and</strong>00,thenatadownstreamlocation,thedispersioncanbecomputedfromtheknowledge<br />
duetoarelativeenergychangeisequivalenttoarelativemagneticeldvariation: dipoleB(B=ecp),wehaveafterdierentiation(p)=p=(B)=B.Henceatransverseoset Becauseinthedispersiongeneratorthebeammomentumpisrelatedtothemagneticeldofthe Notethatinthecasethedispersiongeneratorisachromatic,wehave0=0<strong>and</strong>00=0sothat 0. =R110+R1200 (3.6)<br />
wherethesubscript0indicatetheenergychangedisimpressedbeforethemagneticsystem,<strong>and</strong> thehdBiistherelativemagneticeldvariationofthemagneticsystemdownstreamwhichthe dispersionismeasured. x(s)=(s)dp p0(s)dBB (3.7)<br />
beampositionmonitor.Thequadrupolesinbetweenthesetwoelementswerenotpowered.So Fromtheaforementionedtechniquetoassesswhethertheopticallatticeisperformingaccordingly tothemodel,weneedtohaveanaccurateknowledgeoftheangularexcitationprovidedbyacorrectormagnet.Inordertoestimatesuchangularkick<strong>and</strong>sinceallthecorrectormagnetsareofthe sametype,wehaveuseacorrectorlocatedinthebacklegtransportlinewiththenextupstream 3.3.3ResultsonTransverseResponse<br />
thetransfermatrixbetweentheBPM<strong>and</strong>thecorrectoryieldsanangularkickprovidedbythe thetransfermatrixbetweentheelementistheoneofadriftspaceoflength2.80m.Fordierent correctorexcitation,wemeasuredthebeampositionaspresentedingure3.4.Thebeamposition islinearlydependentonthecorrectorstrengthintherangeinpositiontheBPMwasused[-4mm, +4mm].Alinearinterpolationofthedatapresentedinthegure,alongwiththeknowledgeof correctormagnetofapproximately0:64mrad=(100Gauss:cm)thisvalueisveryclose,within3%,
5<br />
4<br />
3<br />
2<br />
Figure3.4:Exampleofcalibrationofacorrector.Theslopeofthelinearinterpolationis<br />
1<br />
0:0183mm=(G:cm)whichcorrespondstoanangulardeectionof6:54rad=(G:cm)<br />
0<br />
−1<br />
−2<br />
−3<br />
−4<br />
alongthebeamline;typicalvalueusedduringtheacquisitionofdierenceorbitmeasurementare ofthecalculatedvaluededucedfromthecorrectormagneticeldmapmeasurement3whichgivea<br />
−5<br />
excitationofthecorrector)alltheBPMreadbacksalongthebeamlineareacquiredthreetimes BPMvaluealongthebeamline,insuchawaythatthekickprovideasignicantpositionchange approximately100G:cm.ForagivencorrectorchangeBnom+B,(Bnomisthenominalmagnetic Practically,thecorrectorstrengthissetdevisobylookingattheon-linehistogramplotofthe kickof0:65mrad=(100Gauss:cm)<br />
−300 −200 −100 0 100 200 300<br />
Corrector Field Integral (G.cm)<br />
oflinearityofthesystem,theBPMreadbackshouldbetheoppositeoftheonemeasuredforthe latteroperationallowtovalidatethemeasurement,sinceforthelattercorrectorsetting,because (toquantifythebeampositionjitter).ThenthecorrectorissettothevalueBnomB.This<br />
2F04V<strong>and</strong>2F08V).Thecorrectorsarechosensothattherelativebetatronphaseadvancebetween forthehorizontalplane(2F00H,2F04H<strong>and</strong>2F08H)<strong>and</strong>threefortheverticalplane(2F00V, probedierentpartofthelattice.Inourpresentstudy,weusesixdierentcorrectors:three zeroforalltheBPMs. Theuseofonlyonecorrectortostudytheresponseofthelatticeisnotsucientsinceitonly\probe" thelatticeatlocationthathavearelativebetatronphaseadvanceofapproximately90deg4.Hence itispreferabletouseatleasttwocorrectorsseparatedbytheproperphaseadvancesothatthey rstmeasurement.Thereforethecomputationofthesumofthetwomeasurementsshouldgive<br />
istherelativebetatronphaseadvancebetweenthepointss<strong>and</strong>s0.<br />
3G.H.Biallas,privatecommunication,June99 4InthegeneralTwisstransfermatrixformalismonehas:Rs0!s 12=p(s)(s0)sin()where=(s)(s0)<br />
∆ x 2.80 meters downstream (mm)
etatronexcitationaspicturedingure3.5.Anexampleofmeasurementforthesixcorrector eachotherisapproximately60degsothatonecanaccuratelyprobethewholeperiodofthe<br />
0F00H<br />
0F00V<br />
300 0F08H<br />
0F04V<br />
0F04H<br />
0F08V<br />
Figure3.5:Betatronphaseadvancebetweeneachcorrectorusedtoperturbtheorbitalongthe latticeinthehorizontal(leftplot)<strong>and</strong>vertical(rightplot)plane(100meterscorrespondsap-<br />
200<br />
proximatelytotheendofthebacklegbeamline).<br />
100<br />
theoneeectivelyproducedaccordingtothedierenceorbitanalysisisapproximately10%.After aforementionedispresentedingure3.6.Itisexperimentalresponsewiththelattice<strong>and</strong>simulated<br />
0<br />
0 20 40 60 80 100 0 20 40 60 80 100<br />
inclusionofthisdiscrepancyinthemodel,thenewlycomputedpattern(seegure3.7)areingood fortheverticalplane).Inthepresentcase,itwasfound5thatoneofthequadrupoleswasnot varyinthemodeldierentmagneticelements<strong>and</strong>trytominimizea2-typequantitiesdenedas: producingthemagneticgradientthatitwassetto;thediscrepancybetweenthesetgradient<strong>and</strong> response.Althoughthetwopatternsgenerallymatchquitewellonecanseeinthecaseofcorrector 2F04Hthattherearelargediscrepancies.Atechniqueusedtondoutthediscrepanciesisto 2x=PNi=1(xmeasured i xsimu i)2forthehorizontalplane(thesamekindofquantityisdened<br />
Distance from Cryomodule Exit (m)<br />
beusedtodescribeaccuratelythelattice. 3.3.4ResultsonDispersionMeasurement Aswehavealreadymentioned,thedispersionmeasurementtheoreticallyreducestothemeasure- agreementwiththemeasurementindicatingthemodel(secondorderbasedtransfermatrix)can<br />
mentoftheorbittransversedisplacementforagivenenergychange.Theeasywayofvaryingthe energyintheacceleratoristochangetheacceleratinggradientortheinjectionphaseofonecavity. Unfortunately,therearetransverseeldsintheCEBAFcavitythatcansignicantlydeectthe 5D.R.Douglasrstnotedthisfact<br />
Betatron Phase Advance (Deg)
∆ x (mm), Corrector 0F00H<br />
∆ x (mm), Corrector 0F04H<br />
x 10−3<br />
5<br />
0<br />
−5<br />
0<br />
x 10−3<br />
5<br />
20 40 60 80 100<br />
∆ y (mm), Corrector 0F00V<br />
x 10−3<br />
5<br />
0<br />
−5<br />
0<br />
x 10−3<br />
5<br />
20 40 60 80 100<br />
0<br />
0<br />
−5<br />
−5<br />
0 20 40 60 80 100<br />
0 20 40 60 80 100<br />
x 10−3<br />
x 10−3<br />
5 5<br />
problemfortheoperatingenergiesoftheIRFEL:simulationsindicatestheinduceddeectiondueto gradientchangeissignicant<strong>and</strong>canbeoftheorderoffewtensofmrad.Wehaveexperimentally Figure3.6:Comparisonbetweenthemeasured<strong>and</strong>simulatedlatticeresponseforthesixcorrectors<br />
veriedsuchresultinourpreliminarymeasurementofdispersionintheback-legtransferlineby usedduringthedierenceorbitmeasurement(100meterscorrespondsapproximatelytotheendof thebacklegbeamline). beam6.BecausethisRF-induceddeectionisinverselyproportionaltothebeamenergyitcanbea<br />
0<br />
0<br />
<strong>and</strong>compareitwiththecasewhereweoperatethecavityat<strong>their</strong>nominalgradient<strong>and</strong>useda<br />
−5<br />
−5<br />
0 20 40 60 80 100<br />
0 20 40 60 80 100<br />
(whichisinfacttheR12-inducedpattern)isreproduced.Thisresultconrmsoursuspicionthat varyingthecavitygradient.Wepresentingure3.8thebeampositionosetalongthebeamline correctoratthelinacexittosimulatepotentialRF-inducedsteering:thesametypeofpattern<br />
Distance (meters)<br />
Distance (meters)<br />
approximately500100mforamagneticeldvariationof1%insuringthespuriousdispersion therecirculationarc.Wenotethatthemaximumtraversedisplacementofthebeamcentroidis islowerthan5cminabsolutevalue.Thisisanimportantresultforemittancemeasurementaswe shallseeinChapter4. dispersionmeasurementsperformedusingRF-gradientvariationisnotvalid.Nextwepresent,in gure3.9,ameasurementofdispersionobtainedbyvaryingthemagneticeldinallthedipolesof 6AdetailedstudyofthiseectispresentedinChapter6<br />
∆ x (mm), Corrector 0F08H<br />
∆ y (mm), Corrector 0F04V<br />
∆ y (mm), Corrector 0F08V
∆ x (mm), Corrector 0F00H<br />
∆ x (mm), Corrector 0F04H<br />
x 10−3<br />
5<br />
0<br />
−5<br />
0<br />
x 10−3<br />
5<br />
20 40 60 80 100<br />
∆ y (mm), Corrector 0F00V<br />
x 10−3<br />
5<br />
0<br />
−5<br />
0<br />
x 10−3<br />
5<br />
20 40 60 80 100<br />
0<br />
0<br />
Figure3.7:Comparisonbetweentheexperimentaldataaftercorrection<strong>and</strong>thesimulatedlattice responseforthesixcorrectors(100meterscorrespondsapproximatelytotheendofthebackleg beamline).<br />
−5<br />
−5<br />
0 20 40 60 80 100<br />
0 20 40 60 80 100<br />
x 10−3<br />
x 10−3<br />
5 5<br />
3.3.5SummaryoftheTransverseResponseMeasurements Wehavedemonstratedthatthenumericalmodeldescribeswithhighcondencethetransverse<br />
0<br />
0<br />
propertiesoftheas-builtrecirculatingaccelerator.Wehavemeasuredthedispersioninthebackleg transferlinewhenthetrimquadrupoles<strong>and</strong>sextupolesinarc1arenotexcited<strong>and</strong>havediscovered thespuriousdispersionhasanamplitudesmallerthan5cm.<br />
−5<br />
−5<br />
0 20 40 60 80 100<br />
0 20 40 60 80 100<br />
Distance (meters)<br />
Distance (meters)<br />
compressionschemewas.Itenablestoh<strong>and</strong>lethelargemomentumspreadinducedasthelaser 3.4.1Motivation ItwasdemonstratedinReference[18]howimportant,forenergyrecoverypurpose,theenergy 3.4MeasurementoftheLongitudinalResponse<br />
operates.Thebasicsideaistosetthelongitudinallattice,intherecirculation,insuchaway<br />
∆ x (mm), Corrector 0F08H<br />
∆ y (mm), Corrector 0F04V<br />
∆ y (mm), Corrector 0F08V
2<br />
1<br />
Figure3.8:Comparisonbetweenanenergychangeinducedbeamdisplacementalongthelattice<br />
0<br />
thelinac,theenergyosetiforanelectroncanbewrittenasafunctionofitsenergyoset0<strong>and</strong> longitudinalpositionz0atthewigglerexitas: reduced.Undertheassumptionofsingledynamicsrstorderopticsthisissimplybecauseafter <strong>and</strong>theresponsetoanangularperturbation. thatduringthedecelerationphase,forenergyrecoverypurpose,therelativemomentumspreadis<br />
−1<br />
R12 pattern<br />
Cavity Gradient Pattern<br />
−2<br />
properenergyforlasing.Thereforetheonlyparameterthatcanaecttheenergyspreadatthelinac TheR65transfermatrixelementofthelinacisgenerallyxed:sincethelinacissetuptoprovidethe i=Rlinac 65z0+Rlinac 65Rwiggler!linacentrance 56 0 (3.8)<br />
0 20 40 60 80 100<br />
Distance from Cryomodule exit (m)<br />
exitafterenergyrecoveryistheR56oftherecirculationfromthewigglerexittothelinacentrance. Anexampleoftheimportanceinsettingstheenergyrecoverytransportisshowningure3.10 wherewepresentsimulationoftheenergyrecoveryscheme:theparmelacodeisusedasaskeleton<br />
Forsakeofsimplicity<strong>and</strong>inordertoexpeditenumericalsimulations,wehaveturnedospace transportissimplysimulatedbytrackingparticlesusingthelongitudinaltransformation7. slippageduetothenon-relativisticnatureofthebeamaretakenintoaccount);therecirculation tosimulatetheacceleration<strong>and</strong>decelerationofthebeam(i.e.RF-inducedcurvature<strong>and</strong>phase<br />
chargeroutineinparmela.Theproperphasetoachievetheshortestbunchatthewigglerin- cavityviabeamloadingisdirectlyusedtosupplementtheavailablepowerfortheaccelerating sertionisapproximately8degocrest.Thebeamisrecirculated<strong>and</strong>re-injected8+180deg ocrestinthecryomodulesothatitisdecelerated<strong>and</strong>theelectromagneticenergystoredinthe zi=zi+R56i+T5662i+O(3i) i=i (3.9)<br />
linecomponent,norspacechargecollectiveforcearetakenintoaccount<br />
mode.Wepresenttheobtainedresultsin3.10,thelongitudinalphasespaceatthelinacexit(after 7Inthisnumericalanalysis,neitherlongitudinalwakeeldinducedintheacceleratingstructure<strong>and</strong>variousbeam-<br />
∆x (mm)
1<br />
0.5<br />
0<br />
spreadoftheparticlewassetto:i!i+1=2Er<strong>and</strong>(1;1)where\r<strong>and</strong>"r<strong>and</strong>omlygenerate ofthewigglerwasnumericallysimulatedbygeneratinganenergyosetof:5MeV<strong>and</strong>theenergy anumberin[-1,1]interval.Thelongitudinalphasespaceaftertherecirculationfordierentvalues Figure3.9:<strong>Beam</strong>displacementinthebacklegtransportforarelativevariationof1%ofthemagnetic<br />
ofR56<strong>and</strong>T566.Thecorrespondingphasespaceafterthedecelerationinthelinacarepresented acceleration),thedownstream<strong>and</strong>upstreamthewigglerarepresentedintherstrow.Theeect eldoftherstrecirculationarc.<br />
−0.5<br />
energyspreadcanbegreatlyreduced,yieldingabeamthatcanbetransportedthroughthedump transferline. inthebottomrow.Itisseen(bottomright)thatwithproperchoiceofR56<strong>and</strong>T566,theresulting<br />
−1<br />
35 45 55 65 75<br />
Distance From the Cryomodule Exit (m)<br />
sigmamatrixelements).Thismatchingconditionisrequiredtoinsureonecanproducetheshortest possiblebunchlengthatthepointwheretheFEL-interactiontakesplace.Thisminimumbunch length,whenthematchingconditionisveried,isMIN Theoptimumpointforoperationofthelinacintermsofbothphase<strong>and</strong>acceleratingvoltagemust berelatedtothemomentumcompactionofthetransportfromthelinacexittothewigglerinorder toachievetherightlongitudinalphasespaceslopeattheentranceofthelinac.Infactthelinac ofthecompressorchicanebyfulllingtherelation:56=55=1=R56(56<strong>and</strong>55arethebeam voltage<strong>and</strong>phasearedictatedbythefactthatoneneedstomatchthelongitudinalphasetotheR56<br />
quadrupolesaswehavealreadymentioned.Anexampleofvariationofthemomentumcompaction ofonearcwithrespecttothesecondfamilyquadrupolesexcitationisshowningure3.11.Inthe freesystemtoadjusttheR56aretheend-looparcsoftherecirculationtransportbyvaryingthetrim about38MeV<strong>and</strong>isoperatedatapproximately-8dego-crest. emittance~"z<strong>and</strong>thebunchlength0zaretakenatthelinacexit. Fromparmelaoptimization8<strong>and</strong>recentexperimentaloperation,thelinacacceleratingvoltageis Theby-passchicaneshave<strong>their</strong>momentumcompactionxedbydesignto28:8cm<strong>and</strong>theonly z=R56~"z=0zwherethermslongitudinal<br />
gurewealsoplottedthenonlineartermT566forthequadrupolebutalsofordierentsextupoles 8B.C.Yunn,updatedparmelainputlesfortheIRFELdriver-accelerator,privatecommunications<br />
∆x (mm) for |∆E/E|=0.01
Energy Spread (keV)<br />
After Acceleration Before Wiggler After Wiggler<br />
Figure3.10:Energycompressionscheme:Therstrow(fromlefttoright)presentsthelongitudinal<br />
(A) (B) (C) After<br />
Recirculation<br />
phasespaceatthelinacexit,afterthecompressionchicane,<strong>and</strong>justafterthewigglerinteraction<br />
(A)<br />
(B)<br />
(C)<br />
After<br />
Deceleration<br />
compactionasafunctionofthequadrupolestrengthcanbeparameterizedbyaquadraticregression hastakenplace;thesecondrowshowlongitudinalphasespaceattheentranceofthelinacjust<br />
togivethe\h<strong>and</strong>y"formula: thethirdrow. excitation(inthislattercasethetrimquadrupolesareunexcited).Notethatthemomentum priortodecelerationforthreedierentchoiceofR56<strong>and</strong>T556(for(A)-0.2<strong>and</strong>0.m,for(B)0.2 <strong>and</strong>0m<strong>and</strong>for(C)0.2<strong>and</strong>3.0m).Theresultforthethreecasesafterdecelerationareshownin R56(kq)=0:14360:5496kq0:05255k2q (3.10)<br />
Phase (RF-deg)<br />
Thelongitudinallatticecharacterizationisverysimilartothetransverseresponsemeasurement measurement. 3.4.2TheoreticalBackground ExperimentallythisformulacanbeusedtoquicklychecktheR56ofanarc<strong>and</strong>comparewith<br />
condition<strong>and</strong>measuringthecorrespondingresponsedownstreamthesectiononewishestochar- previouslydetailed:themethodagainconsistsofimpressingaknownvariationofthebeaminitial IRFEL:thecompressioneciency(orphase-phasecorrelation)<strong>and</strong>themomentumcompaction(or<br />
acterize.Therearetwotypesoflongitudinalmeasurementthatcanbedoneveryeasilyinthe
2<br />
1<br />
−5 −3 −1 1 3 5<br />
k (m q −2<br />
) or k (m s −3<br />
0<br />
Figure3.11:Momentumcompaction<strong>and</strong>nonlinearmomentumcompactionforonearcversusthe secondfamilyoftrimquadrupolesexcitationstrength(kq).nonlinearmomentumcompactionversus<br />
−1<br />
thesecondfamilysextupolesstrengthks(forthiscalculation,thetrimquadrupolesareunexcited).<br />
M from trim quadrupoles<br />
56<br />
−2<br />
phase-energycorrelation).Inthesemeasurements,thebunchisconsideredasamacroparticle<strong>and</strong> thetime-of-ight(TOF)ofitscentroidismeasuredbetweenthelocationswherethevariationis<br />
T /10 from trim quadrupoles<br />
556<br />
−3<br />
−4<br />
Thephase-phasecorrelation,hinjoutirequiresthemeasurementoftime-of-ight(TOF)ofthe impressed<strong>and</strong>apointwherethetimeofarrivalcanbeestimated. CompressionEciency<br />
)<br />
FELdriver-accelerator,thisquantityisonlymeasuredstartingfromthephotocathodesinceitis bunchcentroidversusavariationofits\injectionphase"inthesectionwewishtostudy.Inthe theonlylocationthe\injectionphase"incanbevariedorthogonallywithrespecttotheinjection withrespecttothemasteroscillator(<strong>and</strong>alltheRF-elementsettingsarekeptconstant).Thephase variationanarbitrarylocationdownstreamthephotocathodeoutasafunctionoftheinjection phaseperturbationinwrites:out=@out energybyvaryingthephaseofthephoto-cathodedrivelaserthatilluminatesthephotocathode<br />
Thereforebyvaryinginaknownfashiontheinjectionphase<strong>and</strong>measuringtheassociatedresponse downstream,weobtainthephase-phaseresponseofthesectioncomprisedbetweentheperturbation <strong>and</strong>themeasurementstation.Fromthisphase,usingthest<strong>and</strong>ardnotationofthetransport<br />
@inindef =hinjoutiin (3.11)<br />
M 56 or T 556 /10 (meters)<br />
T 566 /10 from sextupoles
latticecode9,wehave:<br />
MomentumCompaction thisphase-phasetransfermapyieldsthearbitraryordertransfermatrixelementhinjouti. whererkisk-thco-ordinateofthevectorr=(xin;x0in;yin;y0in;in;in)Therefore,nonlineartof hinjoutiin=(R55+Xk
Experimentallythecalibrationcoecientsarestoredinsoftware,thevoltageVoutisdigitizedby Inthissectionwewillonlyconsidermeasurementperformedbythepickupcavities2,3,<strong>and</strong>4, Inthedriveracceleratorfourcavitieshavebeeninstalled;<strong>their</strong>locationsareshowningure3.12. anADCcard,<strong>and</strong>adataanalysisprogramdirectlyoutputtheTOFinunitofRF-deg(1RF-Deg is556mat1.497GHz).<br />
frequencyof60Hz). amacropulsethatconsistsof4675electronbunches(thecharacteristicsofthemacropulseusedfor themeasurementare:awidthof250sec,amicro-bunchfrequencyof18:7MHz<strong>and</strong>amacropulse themeasurementfromthecavity#1requiringsomedeeperanalysisasweshallseeinchapter6. Itisworthnotingthatcontrarytowhatwehaveimplicitlyassumedinthepreviousdiscussion goodenoughsignalovernoiseinthetimeofightmeasurement,thecavitysignalcorrespondsto themeasurementisnotasinglebunchmeasurement.Becauseoftheneededsignaltoachieve Toexpeditethemeasurements,thequantityvaried(i.e.laserphaseforhinjoutitransfermap<br />
<strong>and</strong>cavitygradientforhinjoutitransfermap),isindeedmodulatedwithafrequencyof60Hz Figure3.12:LocationofthepickupcavitiesalongthetransportlineintheFELdriveraccelerator. (seeagaing.3.13).Themodulationisperformedwithatriangularwaveformgeneratedbya cavitygradient).Thechoiceofatriangularmodulationwasdonetouniformlypopulatethetransfer map.Moreoversinusoidalmodulationisalsoplanned(butwasnotusedduringtheworkreported G.KratfortheCEBAFinjector[22]atJeersonLab. 3.4.4Simulationofhinjoutitransfermap hereafter)inordertousetheTchebytchev-polynomial-basedanalysisthathasbeendevelopedby functiongeneratorusedtovarytheproperquantity(phaseofthephoto-cathodedrivelaseror<br />
Asaforementionedthemeasurementofphase-phasetransfermapprovideimportantinformationon howthebunchingprocessisperforming<strong>and</strong>cangivesomeinsightsonthebunchlength.Because themapismeasuredbetweenthephotocathode<strong>and</strong>thepickupcavities(seeg.3.12for<strong>their</strong>locations),wecannotusest<strong>and</strong>ardsingledynamicsrelativisticcodessuchasTLieordimadbutneed touseparticletrackingcodesuchasparmelawhichincludenonrelativisticeectssuchasphase Weusetheparmelacodetogenerateuniformmacroparticledistributionoveragivenextentin<br />
slippageeectsinacceleratingcavities.Thetechniquewehaveusedtocomparemeasurementwith numericalsimulationisasfollows.<br />
Pickup #2 Pickup #1<br />
Pickup #3 Pickup #4<br />
Gun
R<br />
55<br />
R<br />
56<br />
Drive Laser<br />
Phase<br />
Cavity<br />
Gradient<br />
Function<br />
Generator<br />
1.5 GHz<br />
Voltage Controlled<br />
Phase Shifter<br />
Pickup Cavity<br />
measurement.<br />
X Y<br />
DAC DAC DAC<br />
phaseatthephotocathodesurface.Thecorrespondingphaseofemissioninofthei-thmacropar- Figure3.13:BlockdiagramofthecompressioneciencyR55<strong>and</strong>momentumcompactionR56<br />
VME VME VME<br />
ticleatthephotocathodesurfaceisrecorded<strong>and</strong>themacroparticlespopulatingthisuniformdistributionare\pushed"alongthebeamlines.Duringtrackingthespacechargesubroutineinparmela GP-IB<br />
isturnedo,<strong>and</strong>eachmacroparticleisassimilatedtoabunchcentroidofbunchesemittedatdifferentdrive-laserphase.Wethenrecordthephaseofarrivalioutatthedesiredpickupcavitiesin IOC/ VxWorks<br />
thesimulation.Thecouplefin,ioutgi=1;:::;Ngivesthephase-phasetransfermap<strong>and</strong>canreadily becomparedwiththeexperimentaldata.<br />
Ethernet<br />
Display<br />
EPICS Database<br />
Workstation<br />
bepresentedinChapter6.Thegure3.14presentsameasurementofphase-phasebeamtransfer 3.4.5Measurementofhinjoutitransfermap functionbetweenthedrivelaserphotocathode<strong>and</strong>thethreedierentpickupcavitiesaforemen-<br />
Wewillnottreatmeasurementofphase-phasecorrelationfunctionintheinjectorsincetheywill NominalSet-upMeasurement<br />
Mixer
tioned.FromthetransferfunctioninFig.3.15wec<strong>and</strong>educedbyperformingnon-lineart,an<br />
40<br />
Photcathode Drive Laser Phase (RF−deg)<br />
20<br />
Figure3.14:phase-phasebeamtransferfunctionbetweenthephotocathodesurface<strong>and</strong>thethree<br />
pickup #2<br />
dierentpickupcavities:pickup#2,#3,<strong>and</strong>#4.<br />
0<br />
pickup #3<br />
pickup #4<br />
estimateofthetransfermatrixelementtheresultsofthenon-lineartaregatheredinTable3.4.5. Themeasurementofthelinearpartareinverygoodagreementwiththesimulationexceptfor<br />
−20<br />
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−2 0 2 4<br />
Wehaveexperimentallyinvestigatedtheeectsofthesecondfamilytrimquadrupolesonthe thecavity#4,webelievethediscrepancycomesfromabadcenteringoftheelectronbeamonthe magneticaxisofthetrimquadrupoles<strong>and</strong>sextupolesinthearc2. EectsoftheQuadrupoles<br />
Time of Flight (RF−deg)<br />
to<strong>their</strong>nominalvalues.Thenweiteratethemeasurementforboththetrimquadrupolesturnedo <strong>and</strong>with<strong>their</strong>valueoppositetonominal.Themeasurementsaregathered3.16(A).Sincenotime wasspenttore-measurethetransfermapalongthelinac<strong>and</strong>iteratethesameprocedureasbefore, i.e.setthemodelsuchthatthesametransfermapisachievedatthelinacexit,<strong>and</strong>thenusethe measurementwehavemeasuredthebeamphase-phasecorrelationwiththetrimquadrupolesset fordierentcases.Thestudyhasonlybeencarriedoutusingthepickupcavity#3.Forsuch modeltopredictthechangeatcavities#3<strong>and</strong>#4,thedisagreementisquiteimportant.However, phase-phasetransfermap.Ingure3.16wepresentmeasurementofthephase-phasecorrelation<br />
ifwelookatwhatistherelativechange,i.e.bycalculatingthedierencehoutjiniQuadSettings houtjiniNominalSettingsforboththeexperiment(seeg.3.16(C))<strong>and</strong>thenumericalmodel(see toextractquantitativenumberforR55<strong>and</strong>T555,itshowsthatthismapdoesevolvethesameway<br />
g.3.16(D))theagreementissatisfactory.Thoughinthepresentcasethemethoddoesnotallow
Time of Flight out (RF-deg)<br />
Drive Laser Phase in (RF-deg)<br />
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-5.0 -2.5 0.0 2.5 5.0<br />
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Figure3.15:Comparisonofthephase-phasebeamtransferfunctionbetweenthephotocathode<br />
surface<strong>and</strong>thethreedierentpickupcavities(pickup#2,#3,<strong>and</strong>#4)(bottomrow)withthe<br />
onesimulatedusingparmela(toprow).<br />
bothexperimentally<strong>and</strong>inthemodelwhenwetrytoperformaperturbation-typemeasurement.<br />
Thisalsosuggestthatthesame\dierenceorbit"techniqueweuseforthetransverseplanecould<br />
beperformedinthepresentcase,ifthemotivationwasnottoextractmatrixelements.<br />
EectsoftheFELonthePhase-Phasemap<br />
AssessingtheeectoftheFELonthebeamhasbeenstudiedonlyexperimentally.Wewillattempt<br />
toprovideaqualitativeexplanation.Inthisexperimentwehavesetthelaserphasemodulation<br />
amplitudeto40deg,sinceitcorrespondsthefullbunchlengthatthecathodesurface.Firstlywe<br />
preventedthelasertorunbydetuningtheopticalcavity<strong>and</strong>recordedthephase-phasetransfer<br />
function.Thenwetunedtheopticalcavity,i.e.adjustthelengthtomatchthetimeofight<br />
ofelectronbunchesinsidethecavity,<strong>and</strong>recordedthephase-phasetransfermap.Theother
Pickup Experiment #2 #3 #4 Simulation #2 #3 LinearCoecientQuadraticCoecient<br />
#4 -0.0801 -0.0834 0.1172 0.0911 0.1070 0.0256 0.0016 0.0003 0.0008 0.0006 0.0007<br />
obtainedtransfermapsaregatheredingure3.17.Forthecasewherethelaseristurnedo(see parmela-simulatedphase-phasetransfermap. measurementswereperformedatvariousstagesofthedetuningcurveoftheopticalcavity.The Table3.2:Comparisonofcoecientsobtainedfromthenon-lineartofthemeasured<strong>and</strong> 0.0004<br />
fold-overduetothenon-lineareectintroducedbythelaserinteractionislessimportant.We tothe\nominalphase"havealargertimeofightbecausenowtheyarecontributingtothelaser process<strong>and</strong>thereforearelessenergetic.Atvariousstageofdetuningcurvesthephase-phasemap extractamaximumoutputpower,thefold-overissubstantial,bunchesemittedwithphaseclose havesucceededinoperatingthelaseratthelimitofitsturn-opointbyproperlyadjustingthe toT555contribution.Howeverwhenthelaseristurnedon(seeg.3.17(B)),<strong>and</strong>optimizedto cavity,inthisregion,(seeg.3.17(E))wecannoticethatthephase-phasetransfermaphastwo g.3.17(A)),thetransfermaplooksasusual,mainlylinearwithasmallparabolicbehaviordue<br />
transfermaphasthesamefold-overaswhenthelaserisoptimizedformaximumoutputpower(see contributions:forbunchcenteredaroundthezero-crossingphase(i.e.10
φ OUT (RF−Deg)<br />
5<br />
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Figure3.16:Demonstrationoflongitudinaldierence-orbit:phase-phasetransfermapmeasured<br />
5<br />
5<br />
(A)<strong>and</strong>simulated(B)forthreedierentsettingsofthetrimquadrupole.Plot(C)<strong>and</strong>(D)correspondtodierenceofthemeasuredmappresentedinthetoprawforrespectivelytheexperiment Off−Nominal<br />
<strong>and</strong>thesimulation.<br />
Off−Nominal<br />
0<br />
0 Reversed−Nominal<br />
Reversed−Nominal<br />
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themomentumcompaction,R56.Sincethequadrupoleintroduceapathlengthvariationlinearly 3.4.7Measurementofhinjoutitransfermap EectsoftheTrimQuadrupoles Thetrimquadrupolefamilyhastwoeectsontheenergy-phasecorrelation.Firstlyitmodies<br />
φ (RF−Deg)<br />
φ (RF−Deg)<br />
IN IN<br />
Theresultsarepresentedingure3.18:alineartofthemeasuredtransfermaphasbeenper-<br />
Wehavemeasured,usingthenominalopticallatticesetuptheenergy-phasetransfermapatboth pickupcavitieslocateddownstreamarc1<strong>and</strong>2(pickupcavity#3<strong>and</strong>#4)onthegure3.12. dependentontheenergyoset.Alsoviaitssecondordercoecient,thetrimquadrupolesalso introducedaquadraticallyenergyosetdependentpathlengthvariationwhichresultsinamodicationofthenonlinearmomentumcompactionT566=houtj20i. φ OUT (RF−Deg)
FromthesebothmeasurementitispossibletodeducetheR56oftheby-passchicanes:Usingthe nominalsettingsmeasurementwend:Rchic trimquadrupoleinarc1bothpowered<strong>and</strong>turnedo.Itisseenthelevelofagreementisexcellent. TypicalR56measured<strong>and</strong>expectedforthewholerecirculation,i.e.fromthecryomoduleexitup toitsentranceisapproximately-20cmforthenominalsetupusedatthattime(February1999). themagneticopticcodedimad.Wehaveperformedthemeasurementatbothlocationwiththe formed<strong>and</strong>iscomparedinthisgurewiththeexpectedmomentumcompactioncomputedusing<br />
mentumcompactionfromthelinacexituptothepickupcavity#3.Theresultsarepresentedinexcitationbysystematicallyvaryingthequadrupolesstrength<strong>and</strong>eachtimemeasuringthemo- valuesof28cm.WehaveattemptedtoquantifytheR56dependenceonthetrimquadrupoles thecodeseemstobeaverygoodtooltopredicttheR56evolutionaroundtherecirculation. theverygoodagreement,with2cmbetweenthemeasuredR56forthechicanes<strong>and</strong>itsdesign gure3.19wherewecomparedthemeasurementwithnumericalsimulationusingthedimadcode; 56'29:60cm<strong>and</strong>Rarc 56'23:51cm,againwecannote<br />
Wehavecarriedaqualitativestudyofthesextupoleeectontheenergy-phasetransferfunction. EectsoftheSextupoles Theexperimentconsistedofmeasuringthehinjoutitransferfunctionusingthepickupcavity number3.Duringthemeasurementthetrimquadareun-powered.Ingure3.20,wepresentthe<br />
3.4.8ConcludingRemarksontheLongitudinalResponseMeasurement tointroducedapositivenon-linear(quadratic)curvature.Againonecanusethesameschemeas weusedbefore<strong>and</strong>comparenottheabsolutetransfermap,butrelativetransfermapi.e.compare thealgebraicdierencehinjoutionhinjoutioffforthesimulated<strong>and</strong>measuredset. measuredtransferfunctions.Quantitativelythereissomedisagreementbetweenthesimulated<strong>and</strong> measureddata.Howeveritisseenthatthesextupolehavethesameeect:whenturnontheytend<br />
Inthissectionwehaveshowedthat:<br />
2.phase-phasemapcanbeusedtosetthelatticei.e.tooperateinisochronousmode,e.g.by 1.phase-phaserstordermap<strong>and</strong>nonlinearitiesmeasuredcanberatherwellreproducedwith spacechargeforce. makingsurethemapbefore<strong>and</strong>afterasectionisapproximatelythesame latticecompressionrate,practically,<strong>and</strong>especiallyinthelowenergyregion,wherethebeam isinaspace-charge-dominatedregime,thecompressionisstronglyinuencedbycollective theparmelacode.Alsoitcanbeusedtodeduceboththecompressionratebetweenthepoint ofmeasurement(pickupcavitylocations)<strong>and</strong>thephoto-cathodesurface;ofcoursethisisa<br />
4.energy-phasetransfermapcanalsobeusedtocharacterize,withhighaccuracytheeect 3.energy-phasetransfermap,cangivewithfairlygoodaccuracythemomentumcompactionofa ofthesextupoleonthelongitudinaldynamics,alsointhepresentworkwewerenotable<br />
R56'12cmfromthecryomoduleexittothearc3F.Whichgivetheforthesectionwiggler tolinacentranceR56'+16cmveryclosetothedesiredvalueof20cm. section.Wehavemeasuredthemomentumcompactionoftherecirculationtoapproximately
topreciselyextracttheT566termprobablybecauseofmis-centeringinthearcstransport,<br />
itcouldbeuseforsuchpurposetoeasethepathlengthcorrectionrequiredbyintroducing<br />
linear<strong>and</strong>highorderenergychirp.
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Figure3.17:Phase-phasetransferfunctionsfordierentsettingsoftheopticalcavitylength.In<br />
(A)theopticalcavityiscompletelydetunedsothatthelaserdoesnotoperate.In(B)thecavity<br />
ispreciselytunedtomaximizetheFELoutputpower.In(E)thecavityistunedsothatthelaser<br />
operateatthelimitofitsturno.In(C)<strong>and</strong>(D),thephase-phasetransfermapismeasuredfor<br />
dierentdetuningoftheopticalcavitycase(B)<strong>and</strong>(E).
1<br />
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thelinacexittoarc3Fexit.MeasuredR56arealsopresentedaslledsquares. Figure3.18:EvolutionoftheR56alongthebeamtransportintherecirculationtransport,from<br />
Meas. Arc 1<br />
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Meas. w Arc 1<br />
Both<br />
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Quads Off<br />
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All trim Quad On<br />
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Numerical Simulation<br />
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quadrupolessettings.<br />
fermapwiththeexpectedmomentumcompactionR56fromthedimadcode,fordierenttrimFigure3.19:Comparisonofthelineartermextractedbyttingthemeasuredenergy-phasetrans- −0.45<br />
−0.5<br />
−40 −30 −20 −10 0 10 20 30 40<br />
Trim Quadrupole Gradient Integral (G)<br />
Momentum Compaction (m)<br />
Experiment
10<br />
5<br />
Sext. ON<br />
Figure3.20:EectofthesextupolesintheArc3Fontheenergy-phasecorrelation.(Simulations<br />
Sext. OFF<br />
<strong>and</strong>experimentaldataareosetforclarity).<br />
0<br />
Sext. ON<br />
Sext. OFF<br />
−5<br />
−10<br />
−0.01 −0.005 0 0.005 0.01<br />
δE/E (no unit)<br />
φ OUT (RF−Deg)
Characterization TransversePhaseSpace Chapter4<br />
4.1Introduction ThepresentChapterisintendedtodiscusstheemittancemeasurementthatwehavedeveloped intheIRFEL.Techniquestomeasurebothemittance-dominated1<strong>and</strong>space-charge-dominated2 beamaredescribed.BecausebeamprolemeasurementisanintegralpartofanemittancemeasurementwedescribetheOTR-basedbeamdensitymonitorthathavebeeninstalledintheIRFEL. Beforediscussingingreatdetailthetechniquesweusetomeasurethetransverseemittance<strong>and</strong> phasespaceparameters,<strong>and</strong>becauseofthedierentdenitionsthatvaryfromsourcetosourcein thecontemporarybeamdynamicsliterature,wenditimperativetosettlethedenitionofbeam<br />
Bydenition,abeamisacollectionofparticlesthatarecontainedwithinaniteregionofthephase 4.1.1<strong>Beam</strong>,HamiltonianDynamics<strong>and</strong>Liouville'sTheorem emittancethatwewillusethroughoutthisdissertation.<br />
space.Inthemostgeneralcase,thephasespaceisa6-dimensionalspace[26]<strong>and</strong>theparticles representationconcernsthesimplestcase:inothercases,additionalcoordinatesuchasspin,for (assumedtobepoint-like)arerepresentedby<strong>their</strong>positionvector(x;y;z)<strong>and</strong>kineticmomentum vector(px;py;pz),<strong>and</strong>occupiesasix-dimensionalhyper-volumegenerallyreferredas6.This<br />
repulsion) polarizedbeams,orcharge<strong>and</strong>mass,formultiple-speciesbeams,mightberequired.Thenotion whichthekineticmomentumismuchgreaterthanthemomentuminthetwootherdirections.The choiceof(x;y;z)<strong>and</strong>(px;py;pz)ascoordinateissimplycomingfromtheHamiltoni<strong>and</strong>escription oftheparticlesystemwhichrequirescanonicallyconjugatevariables.Inthesix-dimensionalphase ofabeamalsoentailstheexistenceofaprivilegeddirection,thedirectionofpropagation,along 1emittancedominatedbeammeansthatthebeamevolutionisdrivenbyexternalforces(e.g.externalfocussing) 2emittancedominatedbeammeansthatspace-chargeforcesdominatesthebeamevolution(i.e.Coulombian 49
spaceabeamthatconsistsofNparticles,isbestdescribed,atagiveninstant,intermsofa d6=dxdpxdydpydzdpzinthevicinityofthepoint(x;y;z;px;py;pz)is: densityfunctionn6(x;y;z;px;py;pz).Thenumberofparticlesinanelementofphasespacevolume Thetotalvolumein6occupiedbythebeam,atagiveninstantis: Thisquantity,generallyreferredas6D-hyper-emittance,iswelldenedprovidedthedensityfunctionn6isacompactfunction. d6N=n6(x;y;z;px;py;pz)d6 V6=ZZZZZZd6 (4.1)<br />
Ausefulsimplication,whendescribingabeam,occurswheneachdegreeoffreedomisindependent ofthetwootherdegreesoffreedom.ThenthesinceHamiltonianwriteasthesumofuncoupled (4.2)<br />
factorizesas: summarizedasfollow: projectedphasespace.Themainpropertiesoftheparticletrajectoriesintheseplanescanbe Insuchcase,thebeamdynamicscanbestudiedseparatelyineachofthethreetwo-dimension sub-hamiltoniancorrespondingforeachofthethreedegreeoffreedom,thedensitydistribution<br />
Thetrajectoriesdependontheinitialvaluesofthecoordinate<strong>and</strong>thetime.Animportant consequenceisthattwotrajectorieswithdierentinitialconditioncannotintersect.Also n6(x;y;z;px;py;pz)=n2;x(x;px)n2;y(y;py)n2;z(z;pz) (4.3)<br />
Inlinearphasespacetransformations,ellipsesmaptoellipses,straightlinestostraightlines. mapintoaboundaryatatimet0whichenclosethesamegroupofparticles. Aboundaryinthephasespacethatencloseagivennumberofparticleatagiventimetwill bifurcation. notethatatrajectoryatagiventimecanhaveseveralvaluetherebyyieldingphase-space<br />
ville'stheoremwhichstatesthatthedensityofparticleintheappropriatephasespaceisinvariantalongthetrajectoryofanygivenpoint.Thistheoremcanalsobeexpressedintermsoftheinvarianceofthephasespacehyper-volumeenclosingachosengroupofpointsastheymoveinthephase space.Theensembleofparticlebehavesasanincompressibleuid: Generallythephasespacedensityfunction,n6(x;y;z;px;py;pz)isLiouvilliani.e.itsatisesLiou- Suchgeometrymightbeappropriatetolimitphasespacedensity.<br />
WeshouldinsistthatLiouville'stheoremappliestoconservativeHamiltoniansystemsi.e.systems theoremcannotbeappliedwhen: inwhichtheforcescanbederivedfromapotential.Inthecaseofchargedparticlebeam,Liouville Emissionofelectromagneticradiation(e.g.synchrotron) dV dt=0 (4.4)<br />
Quantumexcitationeectarenonnegligible<br />
Nonnegligibleselfinteraction(e.g.spacechargeforce,coherentsynchrotronradiation,...)
istherenormalizedhorizontalmomentumortheparticledivergence.Thevariablesx<strong>and</strong>x0areno Firstlyitisalwayspreferabletoworkinthetracespacewhichistheplanexx0wherex0=px=pz thereisnocouplingbetweenthesetwosubphase-spaces.Thereforewewillconsiderthehorizontal phasespacexpx,similardiscussionisvalidfortheverticalphasespaceypy. Henceforth,wewillonlyconcentrateonthetransversephase-space,xpx<strong>and</strong>ypy<strong>and</strong>assume 4.1.2PhaseSpace<strong>and</strong>Emittance<br />
parameters,theemittance",thebetatronfunctionT<strong>and</strong>theTfunction;itsequationisgiven morecanonicallyconjugatebutthephasespacepropertiesexposedpreviouslyarestillapplicablein thetracespace.Forsakeofsimplicity,thephasespacedistributionisgenerallyarbitrarilybounded<br />
whereTisdenedasT=1+2T byellipsessincetheyhavethegoodpropertiestomapintoellipsesundercanonicaltransformation. Suchellipseisgenerallyreferredasthephasespaceellipse.Itcanbefullyspeciedwiththree by3: geometricemittance<strong>and</strong>correspondstotheareaoftheellipse: T.Inthispreviousequation,theemittanceisgenerallynamed Tx2+2Txx0+Tx02=" Zellipsedxdx0def =" (4.5)<br />
<strong>and</strong>beingthebeammatrix: ThebilinearformexpressedinEqn.(4.5),canberewritteninamatrixform!x!xTwith!x=(x;x0) def = TT!def TT = 1112 1222! (4.6)<br />
Despitethisdenitionofemittanceistheonegenerallyusedbyexperimentalist,itsuersfrommany problemespeciallyinpresenceofnon-gaussianphasespacedistributionorwhennonlineareects nonlinearprocessesgenerallyyieldnonlineardistortionsofphasespacewhichrenderthegeometric emittanceconceptdiculttoquantifyaphasespacewhichshowsagreatdealofstructure(e.g. arepresentinthetransportchannel(chromaticaberration,wakeeld,spacecharge,...).These (4.7)<br />
order,hx2i,hx02i,hxx0i,momentsofthephasespacedistribution.Thenwec<strong>and</strong>enearootmean Aconvenientwayistostatisticallycharacterizethephasespaceusingtherst,hxi,hx0i<strong>and</strong>second squareemittance[27]as: lamentation).<br />
thermsemittance.Also,mostofthetimeonerathernormalizedtheemittancewithrespecttothe momentum<strong>and</strong>denethenormalizedrmsemittanceas: Itisalsocommontondintheliteraturetheeectiveemittancewhichisthedenedas4times 3InthisSectiontheTwissparametersareindexedwiththesubscriptTtoavoidconfusionwithothervariables.<br />
~"x=hh(xhxi)2ih(x0hx0i)2ih(xhxi)(x0hx0i)i2i1=2 ~"nx=~"x (4.9) (4.8)<br />
Later,whereconfusioncannotoccur,wewillomitthissubscript.
Asforthegeometricemittanceonec<strong>and</strong>eneTwissparametersfromtherst<strong>and</strong>secondorder Thisdenitionofemittanceispracticalsinceitdoesnotvaryiftheforcesarelinear. moments:<br />
forthermsemittance: Introducingthermsbeamsize,x,<strong>and</strong>divergence,0x,itispossibletohaveasimpleexpression 8>:T=h(xhxi)2i T=hxx0ihxihx0i T=1+2T ~"x<br />
~"x=x0xq1r212=x0x 2T ~"x1+2<br />
(4.11) (4.10)<br />
Aswewillsee,bothofthesemeasurementsindeedreducetothemeasurementofabeamtrans- Wherer12isacorrelationcoecientdenedasr12=hxx0i Therefore,measuringatransverseemittance~"xalwaysreducestothemeasurementofbeamdensity alongx-axis(i.e.beamsizemeasurement)<strong>and</strong>alongx0-axis(i.e.beamdivergencemeasurement). versedensityprole.Thereforeweshallrstconcentrateourdiscussiononthislattertypeof measurement.Wewillthendiscusstheemittancecomputation. x0x;itisameasureofthetracespaceslope.<br />
theIRFELdriver-accelerator,thetransversebeamdistributionaremeasuredby: Aswehaveseenintheprevioussection,measurementoftransversetracespacegenerallyrequires measuringthebeamprolesi.e.thetransverseparticledensityalongthehorizontalorverticalaxis. Severaltechniquesarecommonlyusedforsuchapurposedependingonthebeam.Forinstancein 4.2Measurementof<strong>Beam</strong>ProleUsingTransitionRadiation<br />
linearresponse<strong>and</strong>cansaturateresultinginerroneousbeamdensitymeasurements. measurementofhigheraveragebeamcurrentisnotpossible:theceramicdoesnothavea auorescentscreen:aceramicplateisinsertedinthebeampath,<strong>and</strong>thelightemittedviathe oftheinjectortoobservelowcurrentbeam.Theuseofthesetypescreensforquantitative ofthespectrum(seeg.4.1b).Thesetypesofbeamprolemonitorareusableforquantitative measurementonlyforextremelylowaveragebeamcurrentoftypically10nA.Itisonlyused asaqualitativebeamtransversesectionmeasurementinthelowenergy(350keV)region uorescenceeectisobservedwithacamerasincetheuorescenceoccursintheopticalregion<br />
thepositionofthewiregivesthetransversebeamdensityalongthedirectionperpendicular wirescanthebeaminthetransverseplane,thepotentialacrossitsendsisproportionalto awirescanner:thebeamisinterceptedbyathin(20m)movingtungstenwire.Asthe tothewire(seeg.4.1b).Despitethistypeoftechniquecanachievedveryhighresolution, dependingonthediameterofthewire<strong>and</strong>thestepsofthescan,itasfewinconvenient:it thebeamcurrentintercepted.Thereforethemeasurementofthiselectricpotentialversus isaveryslowmeasurement,becauseofitsgenerallylargediameter(morethan20m),the
severalsynchrotronradiationmonitors:asthebeamisbentindipolemagnets,itemits components<strong>and</strong>potentiallydamageselectronicssystem. synchrotronradiation.IntheIRFELthisradiationisemittedintheinfra-redregionofthe wirecanyieldsalargelossofparticlethatcanhitthevacuumchamberorotherbeamline<br />
manytransitionradiationmonitors:thinaluminumfoilareinsertedintothebeampath electromagneticspectrum<strong>and</strong>isimagedonaverysensitiveCCDdetector(seeg.4.1c).This prolemonitorhastheadvantageofbeingnon-invasive(itdoesnotyieldbeamdegradation) <strong>and</strong>itcanbeusedtomeasurebeamdistributionatveryhighcurrent.Howeveritisnotwell<br />
radiationisdetected(seegure4.1d).Thiscongurationrequiresthefoilmaterialtohave <strong>and</strong>transitionradiation(seetheIntroductionchapter)isdetectedwithaCCDdetector. Generallythefoilmakesa45deganglewiththebeamdirection<strong>and</strong>backwardtransition suitedforemittancemeasurement:thebeamproleismeasuredinabendi.e.atadispersive<br />
agoodreectioncoecient.Sinceverythinfoilareavailable,thistypeofdevicescanst<strong>and</strong> location<strong>and</strong>thereforetheemittancecomputation,requiresasomewhattediousanalysissince<br />
highcurrentbeamwithoutyieldingsignicantbeamdegradation. weneedtodeconvolvedthedispersioncontributiontothebeamprole.<br />
Suchstudies,experimentallycarriedintheCEBAFmachineatJeersonLab,helpedwiththechoice ofthetypeofscreen(aluminum).Duringthesestudieswealsodevelopedaquasinon-interceptive screenthatwasusedtomeasurethebeamproleofthehighpowerbeamoftheCEBAFaccelerator; choicewasprincipallydrivenbythereliability,thespeed<strong>and</strong>thelowcostofthistypeofinstrument. BeforetheIRFELwasbuiltwestudymanyaspectofthistypeofapparatus:whataretheaverage beamcurrentlimit,whatarethebeamtransversephasespacedegradationafteranOTRscreen. measurementoftransversedistributionrequiredformeasuringtheemittanceintheIRFEL.This Amongthetechniqueslistedabove,transitionradiationwaschosentoprovidethequantitative<br />
alsowedidnotimplementthistypeofmonitorintheIRFELshorttermplan,itmightbesometime implementedtocontinuouslymonitortheelectronbeamqualitywithoutsignicantimpacton<br />
AswehavementionedinChapter2,whenwediscussedelectromagneticradiationemittedbymoving diagnostics<strong>and</strong>thedevelopmentofthenoninterceptiveprolemonitor. 4.2.1ThelimitationofTransitionRadiationMonitor thebeamitself.Inthefollowingsectionswediscussthelimitationsoftransitionradiation-based<br />
methodallowstheobservationoftheTRproducedasthebeamcrossestheboundaryvacuum/foil (backwardTR)orfoil/vacuum(forwardTR).Asthebeamisinterceptedbythefoilsomeconcerns mightarise: FirstlybecauseofthedE=dxofthematerialthebeamdepositssomeenergyintheTRradiator therebyincreasingitstemperature.Thereforewemuststudythethermaleectofthebeamonthe chargedparticles,thattransitionradiationcanbeobservedwheneverachargedparticleexperience<br />
TRradiator. adiscontinuityintheelectricpropertiesofitsenvironment.Acommonwayofobservingtransition<br />
Secondlywhentheelectronsthatconstitutethebeampassthroughthefoilmaterial,theyundergo radiationistointerceptthebeamtrajectorywithathinmetallicfoil(orTRradiator).Such<br />
resultinanon-interceptivediagnostics.Ontheotherh<strong>and</strong>,thedivergenceinducedviascattering<br />
scatteringonthenucleithatcanpotentiallydegradesthebeamemittance<strong>and</strong>thereforewillnot
(a)<br />
ceramic radiator<br />
beam<br />
x-wire beam transverse<br />
x+y<br />
section<br />
2<br />
y-wire<br />
wire<br />
(b)<br />
fluorescence emission<br />
vacuum chamber (spherical wave in<br />
optical region)<br />
Figure4.1:Schematicsofprincipleforthedierenttypeofbeamproledensitymeasurement<br />
vacuum window<br />
optical system<br />
direction of motion<br />
+ detector<br />
(c)<br />
(d)<br />
Metallic foil<br />
forward transition radiation<br />
synchrotron radiation<br />
beam<br />
mirror optical system<br />
anypieceofhardware(vacuumpipe,electronics,...)thatislocatedinthetunnelenclosure.Inthe devices.<br />
+ detector<br />
Magnetic Induction<br />
followingweconsiderthebehaviorofthreekindsofTRradiator:aluminum,gold<strong>and</strong>carbonfoil. acertainthresholditcantriggertheMachineProtectionSystem(MPS)whichwillturnothe canbesolargethatsomeoftheelectroncanbelost.Ifthepercentageoflostparticlesexceeds machine,alsoeveniftheMPSisnotarmed,losingalargefractionofthebeamisalwaysaconcern. Theprotectionsystemthresholddependsonthemachine,<strong>and</strong>insuresthatonecannotdamage<br />
backward transition radiation<br />
vacuum chamber<br />
vacuum window<br />
optical system<br />
4.2.2ThermalStudies<br />
+ detector<br />
foil.Theenergydepositionismainlyduetoionizationlossesoftherelativisticelectronsminusthe energycarriedonbysecondaryelectrons;itwascomputedusingtheEGS4(<strong>Electron</strong><strong>and</strong>Gamma Shower4)code4distributedbyStanfordLinearAcceleratorCenter.Iftheonlymechanismofheat heattransferequation: Theprimaryconcern,aswepreviouslyemphasized,isduetotheenergythebeamdepositsinthe transferisconduction,thetemperatureofabodyinwhichpowerisdepositedisdescribedbythe whereTisthetemperature,thedensity,cp,thespecicheat,thethermalconductivity,Vthe bodywithhighemissivity,heatevacuationviaradiationisanimportantmechanismthatshouldbe volumeofthebody,<strong>and</strong>thedepositedpower. Thisequationcanbenumericallyintegratedbyseveralnite-elementprogram.Inthecaseof 4PrivateCommunicationfromP.K.Kloeppel,November1995<br />
cp@T @tkappa52T=@P @V (4.12)
elation: willresult.AgivensurfaceofthebodydSwhoselocaltemperatureisTswillradiate<strong>and</strong>losepower atarateproportionaltoT4dS.Moreprecisely,thepowerradiatedisgivenbyStefan-Bolztman incorporatedinthecomputations.Typicallyforagivendepositedpower,atemperaturegradient<br />
atatemperatureassumedtobe300Khenceforth). Tocomputethetemperatureriseduetopowerdepositioninthesteadystatecase,weusethe T0istheambienttemperatureofthecanonicalsystemthebodyislocatedin(inourcasevacuum whereistheStefanconstant(=5:670108Wm2K4),istheemissivityofthebody,<strong>and</strong> followingnumericaliterativemethod.Firstly,let'sassume(<strong>and</strong>thisisindeedthecase)thatthe TRradiatorconsistsinacircularfoilofradiusr<strong>and</strong>thicknesstwhosenormalmakesanangle dPS=2dS(T4ST40) (4.13)<br />
withthebeamaxis.Inasuchcase,thepowerisevacuatedradiallyviatheconductionmechanism. Wec<strong>and</strong>ividethefoilbyaseriesofannuliofouter<strong>and</strong>innerradiiri<strong>and</strong>ri+1(seeg.4.2). Thereforethetemperatureoftheithannulusisrelatedtothetemperatureofthei-1thcrownby:<br />
whereRi=1 <strong>and</strong>titsthickness,<strong>and</strong>Piisthetotalpowercomingfromtheithcrown.Partofthepowerisheat radiated<strong>and</strong>theremainingistransmittedviaconductiontothenextelement.Theradiatedpower Figure4.2:MethodologytocomputetemperatureriseinacylindricallysymmetricTRradiator.<br />
is: 2tlnri+1 riisthethermalresistance(isthethermalconductivityofthefoilmaterial Pi=Pi1+2(T43004)(r2ir2i1) Ti=Ti1+RiPi (4.14)<br />
Tocomputethetemperatureriseatthebeamedgecrown,weintroducedtheparameterPext current)usingtheequations[29]:Tcenter=Tn+1 thatrepresenttheevacuatedpower.Hencevaryingthevalueofthisparameter<strong>and</strong>iteratingthe Eqns.(4.14)<strong>and</strong>(4.15)fromtheedgeofthefoiluptotheedgeofthebeamallowstodetermine thetemperatureatthebeamcenter<strong>and</strong>thedepositedpower(fromwhichwecangetthebeam (4.15)<br />
4tPn (4.16)<br />
beam <strong>and</strong> screen center<br />
rxry<br />
rn-1<br />
beam edge<br />
R n R n-1 R n-2<br />
T n<br />
rn-2<br />
T n-1 T n-2<br />
R 3 R 2 R 1<br />
T 3 T 2<br />
screen edge<br />
T=300K<br />
P ext
<strong>and</strong> validundertheassumptionofauniformlypopulatedbeam.Usingthismethodwehavecomputed Eqn.4.16relatesthetemperatureatthebeamcenter<strong>and</strong>thetemperatureatthebeamedge.Itis Aluminum MaterialMeltingPointThermalConductivitydE=dx (K) 933I=PnEtcos W=m:K 237 eV=m 410 (4.17)<br />
themaximumbeamcurrentdierentfoilscanst<strong>and</strong>versustheequivalentbeamsizedenedas Table4.1:PhysicalpropertiesoftheconsideredmaterialforOTRscreens. Gold Carbon 1337 3700 317 333 6380<br />
Theconsideredmaterialwith<strong>their</strong>propertiesaregatheredinTable4.1.Theresultsarepresented prxrywhererx<strong>and</strong>ryarethefullbeamsizerespectivelyinthehorizontal<strong>and</strong>verticaldirection. 130<br />
typicalbeamsizeexpectedinthefree-electronlaser,aluminumfoilcaneasilywithst<strong>and</strong>average currentupto500Aevenwithabeamof300mradius.Eveninthelow-emittanceCEBAF radiusassumingtheTR-radiatorhasa0:8mthickness5.Forinstance,wecanseethatwiththe acceleratorthemaximumdesignof200Acanbereachedwithoutmeltingthefoilfortypical intheg.4.3whichdepictsthemaximumaveragecurrentthatcanbereachedforagivenbeam envelopeof200m. notst<strong>and</strong>higherbeamaveragecurrentduetoitshigherdE=dxcoecient.Ingure4.4wepresent thesteadystatetemperatureversustheincomingbeamaveragecurrentfordierentaluminumfoil thickness<strong>and</strong>abeamof2mm. observebackwardTR. Finallywenotethatdespiteitshigherthermalmeltingpointcomparedtoaluminum,golddoes onlyareconsidered.Unfortunatelythemaindrawbackofcarbon,aswewillseelater,isitslow coecientofreectionincomparisontoaluminumorgoldwhichdoesnotfacilitateitsuseto Obviously,wecannoticeingure4.3thatcarbonisthebestchoiceasfarasthermalaspects<br />
aluminumfoils.TheexperimentwasperformedintheCEBAFinjectorregion(foradescriptionof theinjectorseeReference[28]).Wecomparedthedatawithasemi-empiricalmodel<strong>and</strong>numerical simulations. WenowturntothestudyofbeamdegradationduetoscatteringintheTRradiator.Inthissection, wepresentanexperimentdevotedtostudyscatteringofa45MeVelectronbeamonverythin 4.2.3StudyofMultipleScatteringinAluminumfoil<br />
degradation)stillhavingagoodsurfacereectioncoecient.Alsothicknessislimitedbytheframeonwhichthefoil ismounted:toothinfoilcouldnotbemountedonourholdingsupportbecausetheywouldanymoreself-support.<br />
5thisthicknessisanoptimumvalue:itcorrespondstothethinnestfoilwecanhave(inordertominimizebeam
Maximum Average <strong>Beam</strong> Current ( A)<br />
10 4<br />
5<br />
2<br />
C foil<br />
Al foil<br />
Au foil<br />
10 -1<br />
2 5 10 0<br />
2 5 10 1<br />
2<br />
Equivalent <strong>Beam</strong> Size (rx ry) 1/2 10<br />
5<br />
2<br />
(mm)<br />
2<br />
10<br />
5<br />
2<br />
3<br />
Figure4.3:Maximumaveragecurrentthatcanwithst<strong>and</strong>aTRradiatorasafunctionofthe equivalentbeamradius.ThreetypesofTRradiatorhavebeenconsidered:Aluminum,Gold<strong>and</strong> Carbon.Thethreeradiatorare0:8mthick. Experiment WehaveperformedanexperimentintheCEBAFinjectorat45MeVtostudyscatteringeect asthebeampassthroughaluminumfoilofdierentthickness.Theexperimentalsetupisas scannerhasthreewires(seeg.4.1b)thatrespectively(fromrighttoleftinthegure)givesthe beamprolealongthehorizontal,vertical<strong>and</strong>45degaxes.Thefactthatthebeamsizeinthe prolemeasurementstation:awire-scanner.Allthequadrupole<strong>and</strong>correctormagnetsbetween thefoils<strong>and</strong>thewirescannerareturnedo.Ingure4.5wecomparetwotypicalwirescanner tracesobtainedwith<strong>and</strong>withouta0:8maluminumfoilinsertedinthebeampath.Thewire follow:dierentaluminumfoilaremountedonasupport,<strong>and</strong>canbeinsertedintothebeampath remotely.Thescatteredbeamthendriftsthroughalengthof7:43muptoatransversebeam horizontaldirectionismuchsmallerthantheverticaldirectiononeissimplyduetotheopticstune upstream.Asonecanexpectthebeamproleislargerasthe0:8mfoilisinserted.Moregenerally itgetswiderasthefoilthicknessincreases.Forquantitativeanalysisofthescattering,weonly theconvolutionofthebeamproleBwiththefoil\scatteringtransferfunction"T(orscattering<br />
considerthetailofthebeamprolelocatedontherightsideofthehorizontalprole(rightpeak) becausetheotherpeaksoverlap<strong>and</strong>wouldyieldatediousdeconvolution.Wealsoassumethatwe havescanned100%ofthebeam.LetS()bethefunctionassociatedtothetail.S()isindeed
1000<br />
900<br />
800<br />
700<br />
Figure4.4:SteadystatetemperatureversusaveragebeamcurrentforthreedierentTRradiator thickness<strong>and</strong>abeamequivalentradiusof2mm.<br />
600<br />
distribution):<br />
500<br />
00000<br />
11111<br />
00000<br />
11111<br />
00000<br />
11111<br />
00000<br />
11111<br />
400<br />
2 m 00000<br />
11111 0.5 micron<br />
00000<br />
11111<br />
00000<br />
11111 2 1micron m 00000<br />
11111<br />
00000<br />
11111<br />
RewritingtheaboveequationintheFourierspaceyieldsthescatteringdistributionfunctionofthe whereistheconvolutionproduct. S()=(TB)() (4.18)<br />
00000<br />
11111 1 0.5 micron m 00000<br />
11111<br />
00000<br />
11111<br />
300<br />
0 100 200 300 400 500 600 700<br />
Average <strong>Beam</strong> Current ( A)<br />
Fromthisgurewecanseethatthelargerthethicknessis,thelargerthermsscatteringangle(i.e. varianceofthescatteringdistribution)is. ComparisonwiththeTheory TheF1istheinverseFouriertransformation.Wehavenumericallyperformedthisdeconvolution, <strong>and</strong>thecalculatedscatteringfunctionsforthreedierentfoilthicknessareshowningure4.6.<br />
Itisusefultocomparethepreviousresultswiththetheory,inordertoseehowaccuratelyweare abletopredicttheeectsofanOTRradiatoronthebeam.Thereareseveraltheoreticalmodel<br />
Temperature ( o K)<br />
Melting point<br />
foil: T=F1SB (4.19)
Wire Scanner Readback (mV)<br />
1200<br />
1000<br />
Foil out<br />
800<br />
600<br />
400<br />
whichdependsonthemeanaverageofcollisionanelectronexperiencesasitpassthroughthe foil:simplescattering(1), beamismeasuredusingawirescannerlocateddownstreamthefoil. describingscatteringinsideverythintargetseachofthemhave<strong>their</strong>owndomainofapplicability Figure4.5:Anexampleoftheeectofa0:8mthickaluminumfoilonthebeamprole.The<br />
0.8 m Foil in<br />
200<br />
0<br />
0 10 20 30 40 50 60<br />
pluralscattering(1 20),<br />
Position (mm)<br />
thatdescribesmultiplescattering.ThedetailedstudyofKeilmodelisoutofthescopeofthis thesis.Inbrief,KeilusedtheFouriertransformofthescatteringdistributionderivedbyMoliere Inthecaseofaluminumfoilwiththicknessthinnerthan5m,themeannumberofcollisionbeing lessthen20weareinthepluralscatteringregime.Thistypeofscatteringiswelldescribedby thesemi-empiricalmodelelaboratedbyKeil[30]whichisanextrapolationoftheMolieremodel multiplescattering(20).<br />
<strong>and</strong>onlyconsideredthetworsttermsofthisseries,avalidapproximationifthemeannumber ofcollisionisbelow20.Hethenempiricallycalculatedthenumericalcoecientoftheseriesusing theexperimentalmeasurementsperformedbyLeisegang[31]ashewasexperimentallystudying scatteringthroughverythingoldfoils.Therelationthatgivestheangularscatteringdistribution
Fraction of <strong>Beam</strong> enclosed within (%)<br />
100<br />
90<br />
80<br />
t=0.25 m<br />
t=0.8 m<br />
t=1.5 m<br />
70<br />
60<br />
Figure4.6:Angularscatteringdistributionexperimentallymeasuredforthreedierentthicknesses<br />
50<br />
ofAluminumfoil.<br />
40<br />
30<br />
20<br />
10<br />
0<br />
numberofcollisionisafunctionoftheatomicweightA,theatomicnumberZ,<strong>and</strong>thethickness foragivenmeannumberofcollisionsasafunctionofthereducedanglesisgivenby: wherethecoecentsc1<strong>and</strong>c2areconstantthatwheredeterminedfromexperiment.Themean F(;s)=eXk=0mk Xl=0CkmCmk l bk1Bl2((c1l+c2k)2+2s)3=2 lc1kc2 (4.20)<br />
0.0 0.0002 0.0004 0.0006 0.0008 0.001<br />
(rad)<br />
ofthefoil: whereisthereducedvelocity. Thereducedangle,s,inthelatterequationisrelatedtotheprojectedangleviatherelation: =8:83103tAZ4=3 2 (4.21)<br />
inMeV.ItisworthwhiletonotethattheworkofKeil(asMoliere)wastoshowthattheangular scatteringdistributioninthepluralscatteringregimedoesnotexactlyfollowagaussi<strong>and</strong>istribution<br />
wherethecriticalangleisdenedas=4:23Z1=3 E,Ebeingtheenergyoftheincidentelectrons<br />
s= (4.22)
Table4.2:SurveyofmaterialscommerciallyavailableformonitoringintensebeamswithTRra- corresponding #ofcollisionperm7.58.78.313.628516290127 material thinnestfoilavailable0.250.250.50.510.50.250.250.1 1.92.24.26.828261522.622 BeCMgAlTiFeCuAgAu<br />
diators;weexcludedthematerialswithlowthermalconductivity<strong>and</strong>meannumberofcollision greaterthan30in<strong>their</strong>smallestthickness. (sincetheaveragenumberofcollisionistoolowtofulllthevalidityofthecentrallimittheorem). NumericalSimulations Thegaussiancharacterofthedistributionespeciallydeterioratesatlargeangle,wherelargetail<br />
Tocompletearestudies,wehaveperformednumericalsimulationusingthemonte-carlocode behaviorcorrespondingtothecasek=0<strong>and</strong>l=0intheEqn.(4.20). tendtodevelop.Italsobreaksdownforsmallanglewherethescatteringdistributionhasadirac-like<br />
experiment. AComparisonbetweenExperiment,Theory<strong>and</strong>Simulation scatterings.Wehaveusedthiscodetosimulateeachofthefoilsusedinthepreviouslydescribed GEANTfromtheCERNLIBwhichisapopularsimulationtoolintheParticlePhysicscommunity.ThiscodehasascatteringroutinethatusestheMolieremodel.Howeveriftheparameters aresothattheMolieremodelisnotapplicable,GEANTwillsimulateaseriesofsingleCoulomb<br />
withasafetymargin,thefractionofthebeamwemayloseasthebeampassthroughathinfoilof material. worstcase.ThereforeitseemswecanusethenumericalsimulationsortheKeilmodeltopredict, theworstcase.Howeverbothofthemareoverestimationsofthemeasurementbyfactorof5inthe 70%,95%ofthebeam.Clearly,theKeilmodel<strong>and</strong>thenumericalsimulationagreewithin50%in Wesummarizetheresultsgivenbytheexperiment,thetheory<strong>and</strong>thenumericalsimulationin gure4.7wherethedierentcurvesshowtheeectofthefoilthicknessonthesemi-anglecontaining<br />
carbon;theyareequivalentbutwepreferthelatterbecauseofthechemicaltoxicityofberyllium. 4.3ThePossibleUseofCarbonasTRradiator WehavesurveyedthecommerciallyavailableverythinmaterialthatmaybeusedasTRradiator. Intable4.3wegatherseveralmaterialthatcouldbeused<strong>and</strong>canselfsupportona10mmdiameter holder,withthecorrespondingmeannumberofcollision.Thebestc<strong>and</strong>idatesareberyllium<strong>and</strong> veryhighmeltingtemperature.UsingtheGEANTcodeweestimatedtheangularscattering<br />
Anotheradvantageofcarbonisitscapabilityofwithst<strong>and</strong>ingveryhighcurrentbecauseofits
5<br />
2<br />
2 3 4 5 6 7 8 9 10 0<br />
10<br />
5<br />
2<br />
Foil Thickness ( m)<br />
-1<br />
10<br />
5<br />
2<br />
0<br />
Keil 70%<br />
Geant 70%<br />
Figure4.7:ComparativeresultsofexperimentwithKeil'ssemi-empiricaltheory<strong>and</strong>geantcom-<br />
Meas 95%<br />
putationforthinaluminumfoils.<br />
Meas 70%<br />
distributionforcarbon.Ingure4.8wepresenttheprojectedscatteringangle(normalizedtothe energy)containing70%,95%<strong>and</strong>99.5%ofthebeamversusthemeannumberofcollisionforin thecaseofrelativisticelectrontraversingverythinfoils.Usingthisplots<strong>and</strong>knowingthemachine dynamicsacceptance,wecancomputethefractionofthebeamwecanloseasthefoilisinserted intothebeampath.<br />
transferlinestooneofthenuclearphysicsexperimentalendstation. beamprole.TheexperimentwasperformedintheCEBAFaccelerator<strong>and</strong>locatedinoneofthe Inanattempttocheckourpreviousestimationswehavedevelopedaprototypeformeasuring 4.3.1Anon-interceptiveTRbeamprolemonitor<br />
Experimentalsetup ThebackwardTRdependsonthereectioncharacteristicsofthesurface.Thereareseveralproblemsthatarisewithverythinfoils:surfaceinhomogeneitymakes<strong>their</strong>coecientofreection nonuniform,<strong>and</strong>itisalsodiculttostretchthemenoughtoobtainaveryatsurface.These<br />
(mrad)<br />
Keil 95%<br />
Geant 95%
Prjected Semi-Angle Energy (mrad MeV)<br />
100<br />
90<br />
80<br />
=2.18 for 0.25 m C-foil<br />
99.5 %<br />
70<br />
60<br />
50<br />
95 %<br />
40<br />
30<br />
70 %<br />
20<br />
abeamnormallyincidentontheradiatorhasashorterpathinthematerialwhichinturnreduces thescatteringangle.Basedontheseconsiderations,webuiltaprototypethatusestheforward problemsdisappearwithforwardTRsinceitisemittedinconecenteredonthebeamaxisregardlessoftheangleorreectionpropertiesoftheTRradiator.Therearetwoadditionaladvantagesto theforwardoverbackwardTRwitha45degfoil:(i)thedepthofeldeectbecomesnegligible,(ii) Figure4.8:geantcomputationforthincarbonfoils.<br />
10<br />
TRemittedfroma0:25mthickcarbonfoilpresentedingure4.9(A).Thefoilismountedona supportwhichisU-shaped<strong>and</strong>openonthesidecrossingthebeampath,sothatitcanbeinserted<br />
0<br />
0 2 4 6 8 10 12 14 16 18 20<br />
withoutobligingthebeamtobeturnedo.AmirrorcollectspartoftheTRradiation.With<br />
Mean Number of Collision<br />
edge4mmclosetothebeamtrajectory.Themirrorsendsthecollectedlighttoachargecoupled amagnicationof1=2whichyieldsapixelsizeofapproximately20mintheobjectplane. device(CDD)viatwoachromaticlenses.ThelensesimagethefoilplaneontotheCCDarraywith asthefoil.Themirroris175mmdownstreamfromthefoil;thisinsertionmechanismbringsits anglesfromthebeamaxis.Wedidthisbylocatingthemirroronthesameinsertionmechanism theTRthatisstronglydirectionalina1=cone,weneedtocollectthelightemittedatsmall<br />
ExperimentalResults WetestedourprototypeatthehighestbeamcurrentdeliverableinCWmodebytheCEBAF accelerator:200Aatanenergyof3:2GeV(i.e.beampowerof640kW).Thecarbonfoilwasnot
Actuator Inserts<br />
Foil <strong>and</strong> Mirror<br />
CID<br />
Camera<br />
10000<br />
9500<br />
Lens<br />
9000<br />
8500<br />
Lens<br />
8000<br />
4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200<br />
x (µm)<br />
e- ~ 1 mm (full width)<br />
atypicalbeamdensitymeasuredwithsuchdevice(B). Figure4.9:OverviewofthecarbonfoilbasedOTRexperiment(CourtesyfromS.Spata)(A)<strong>and</strong> damaged,aspredictedbyourthermalstudies. Ongure4.9(B)weshowtypicalmeasuredbeamdensity.Thebeamsize(deneastherms<br />
Transition<br />
Radiation<br />
rmsbeamsizewiththeoneobtainedusingthewirescannerincloseproximity<strong>and</strong>obtainedthe<br />
Mirror<br />
samebeamwidthwithintheuncertaintytoleranceasshowninTable4.3.1.<br />
Horizontal axis<br />
Carbon Foil<br />
<strong>Beam</strong><br />
TothebestofourknowledgethisisthersttimeTRwasusedtomeasuredbeamsizeofhundredsof compatiblewiththeoneexpectedusingthemagneticopticscodeDIMAD.Wealsocomparethese value),obtainedperforminganonlineartofthetransverseproleswithaGaussi<strong>and</strong>istribution, are255m<strong>and</strong>130mforrespectivelythehorizontal<strong>and</strong>verticaldirections.Thesevaluesare<br />
(A) (B)<br />
FORWARD OTR MONITOR<br />
resolutionthathadbeenclaimedwasthatforagivenreducedenergytheminimumrmsbeamsize thatcouldberesolvedbydetectingTRatthewavelengthisoftheorderoftheproduct=(4). beusetomeasuremicron-sizedbeamprolesforrelativisticbeams.Thehypotheticallimitin ofacommonargumentinthebeaminstrumentationcommunityaccordingtowhichTRcannot micronsforanultrarelativisticbeam('6300).Thismeasurementistheproofofthenonvalidity Inourcasesuchcriterionwouldset,atawavelengthofobservationof500m,thesmallestrms thesmallestbeamsizewemeasured.ResolutionissuesconcerningOTRhavebeendiscussedin beamsizewecouldobservedtoapproximately250mrms,i.e.approximately2timeslargerthan toSR,isnotcollimatedwithina1=-cone:forinstancetheTRemissionassociatedwitha1GeV numerouspaper[33].Inbrief,theaforementionedcriterionconcerningtheminimumrmsbeamsize source<strong>and</strong>itsextentareboundedbytherelation0>=(4).InthecaseofTR,thecommon mistakeistowrite0'1=whichnallyyieldtotherelation0
ofP.Gueye,HamptonUniversity,VAUSA). spectrometerwith(A)<strong>and</strong>without(B)thebeambeinginterceptedbythecarbonfoil(Courtesy Figure4.10:ComparisonoftheMissingmassspectraobtainedusingoneoftheexperimentalhall<br />
Table4.3:Comparisonoftheprolemeasurementswiththewirescanner<strong>and</strong>OTR-monitor. Simulation0:250mm OTR-monitor0:2550:060mm0:1300:060mm wirescanner0:2040:050mm0:0820:050mm horizontaldirectionverticaldirection 0:114mm<br />
(A)<br />
(B)<br />
incomingelectronbeamthathitsthetarget,thispeaksasanonzerowidth.Onestraightforward experimentwiththecarbonfoilistodetermineifthefactofinsertingthecarbonfoilinthe lookatthemissingmassspectrum.Inthismissingmassspectrum,wehaveapeakcenteredon theprotonmass(945MeV=c).Duetotheniteresolutionofthedetector,theemittanceofthe todeterminewhetherthefoilhassignicantimpactonthenuclearphysicsmeasurementwasto Namelywewereobservingthereactione+p!e0+p0intheelasticscattering.Acriterion observetherecoilelectronissuedfromthescatteringoftheelectronbeamonahydrogentarget. accelerator.Theexperimentconsistedofsettingtheangle<strong>and</strong>thedipolesofthespectrometerto<br />
800MeV,theprototypewebuiltconstitutesanoninvasivebeamprolemonitor.<br />
theelasticpeakwassimilarwellwithinexperimentalnoise.Thismeasurementwereperformedat 3GeV.LatterincollaborationwithanotherexperimentalHall,thesimilarexperimentwasiterated atlowerenergy800MeVyieldingasimilarconclusion.Therefore,atleastatenergyhigherthan foil.Figure4.10showsthemissingmassspectrainthetwocases.Inbothcases,thermswidthof datawiththespectrometerwhileinasecondsetdatawereacquiredwithoutinsertingthecarbon measurementswereperformed:InarstoneweinsertedthecarbonTRradiator<strong>and</strong>acquired beampathdownstreamthetargetyieldanenlargementofthepeakwidth.Hencetwosetsof
shortterm,touseasbeamdensitymonitorintheFELdriver,aluminumfoilsinthepopularcon- ExperimentalSetup Althoughthecarbonfoilmonitorisveryusefulasanoninterceptivedevice,wedecided,forthe 4.3.2ProleMonitorCongurationintheFEL<br />
serted/withdrawnfromthebeampathbythemeansofanaircylinderactuator.Inthiscongurationthebackwardtransitionradiationemittedat90degwithrespecttothebeamaxisshinesout ofthevacuumchamberthroughasilicaopticalwindow.Itisthencollectedbyaplanarmirror<strong>and</strong> diameterthatmakesa45deganglew.r.t.thebeamdirection.Thefoilcanberemotelyin- diagnosethebeamqualityinacontinuous<strong>and</strong>nonintrusivefashion. Thesystemweuseconsistsof0:8mthickaluminumfoilsmountedonacircularframesof19mm verythincarbonfoilonlargesupport.AlsointhecaseoftheFEL,thelaseritselfcanserveto gurationaspicturedingure4.11.Thischoiceisinpartduetothedicultytoreliablymount<br />
thehighestpossibleresolutionwithareasonableeldofview.Thecircularframeonwhichthefoil ismounteddeterminestheeldofview:this19mmdiameterframeisusedtoaccuratelycalibrate theimage<strong>and</strong>therebydeterminewhatistheconversionfactorbetweentheCCDarraypixels<strong>and</strong> triggered.ThesystemissettoimagethefoilplaneontheCCDarraydetectorwithmagnication ofapproximately1=3.Thechoiceofthemagnicationratioisdictatedbytheneedofachieving senttoanopticalsystemcomposedofacommerciallyavailabletelephotolens(optimizedtoreduce chromatic<strong>and</strong>sphericalopticalaberration)<strong>and</strong>ahighresolutionCCDcamerawhosevideooutput signalisdigitizedbyaVME-basedDATACUBEimageprocessingboardthatcanbeexternally realdistancesinthefoilplane. Toavoiddamagingthealuminumfoil,<strong>and</strong>sincethebeamdynamicsisonlydominatedbysinperbunch,thebunchrepetitionrateinamacropulse,themacropulsewidth<strong>and</strong>themacropulse<br />
repetitionrate.Sinceonlythechargeperbunchaectsthephasespace<strong>and</strong>consequentlythebeam<br />
Therearefourindependentparametersthatcaninuencethebeamaveragecurrent:thecharge Figure4.11:St<strong>and</strong>ardcongurationoftheOTR-basedprolemonitorintheFEL-driveraccelerator. glebuncheect,i.e.thebeamphasespacedensityonlydependsonthechargeperbunch<strong>and</strong> notonthebunchrepetitionrate,thebeamaveragecurrentisdecreasedtoapproximately0:5A.<br />
Incident <strong>Beam</strong><br />
OTR Radiator<br />
<strong>Beam</strong> Sync<br />
Mirror<br />
Telephoto Lens<br />
Charge Coupled Device Camera<br />
VIDEO IN<br />
EXT TRIGG<br />
To EPICS<br />
control system
densityprole,byactingontheotherthreeparametersitispossibletoreducethebeamcurrent withoutaectingonthephasespacedistribution.Naturallythefactwehavetouseareduced beamcurrentduringthemeasurementofthebeamparameterimpactsontheFELoperation<strong>and</strong> subsequentexperiments.Thereforesuchameasurementisinvasive.AswehaveseeninChapter2,<br />
Intensity (a.u)<br />
2<br />
7<br />
6<br />
5<br />
8<br />
109 4<br />
4<br />
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-6 -3 0 3 6<br />
. .. ..... ...<br />
. ........ .. ..<br />
.<br />
... . .. . ..<br />
. .... ....................................................................................... .. .. .......<br />
.. .... .... ..<br />
.... .. ..<br />
. .<br />
. ..<br />
.. . .. .. . .. ... .. . ... ..<br />
. . ..<br />
. ..<br />
. .. . .<br />
. ..<br />
.. ...<br />
....<br />
.... .<br />
. .<br />
10<br />
5<br />
2<br />
-6 -3 0 3 6<br />
Distance (mm)<br />
12<br />
10<br />
5<br />
22<br />
10<br />
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32<br />
10<br />
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42<br />
Figure4.12:Rawdatabeamprole(topgraph)<strong>and</strong>beamproleafterprocessing(background whichalsosynchronouslygeneratea5Vrectangularpulsewhosewidth<strong>and</strong>repetitionratearethe replicaofthemacropulse.Suchsignalcalledthe\beamsync"isavailablefromthecontrolsystem totriggerdataacquisitionsystems. Inthepresentcase,wetriggertheimageprocessingboardtograbonlythevideoframethatcon- atarepetitionrateupto74:85MHz.Allthebeamstructureisgeneratedbyamasteroscillator subtracted,ghostpulsecontributionremoved,...).<br />
tainsthebeamimage.Forsuchapurposeweusethefollowingsetup:the\beamsync"isdelayed by0:435msbeforebeingsenttothe\externaltrigger"inputoftheimageprocessingboard.The eachmacropulsecontainsbunches(whosechargecanbearbitraryvariedfrom0pCto135pC6)<br />
SomeprimaryoperationsonthisdatastreamcanbedirectlyperformedbytheCPUoftheimage processingboardinrealtime[39].Suchoperationsincludethecalculationof: delayissetsothatthedigitizeristriggeredatthenextincomingpulse. Thedigitizeddatastreamattheimageboardoutputconsistsofa660484arraymatrix,I(x;y). 6Intheresultspresentedinthisreportthemaximumchargeusedisapproximately60pC<br />
Intensity (a.u)
Alongwiththeseimplementedoperations,somesignalprocessingfunctionssuchasltering,back- thebeamcentroidpositiondenedasx0=RxdxI(x;y)<br />
groundsubtraction,etc...,areavailable. thebeamspotthermswidthdenedas=R(xx0)2I(x;y)dx<br />
Duringourpreliminarymeasurementswehavenoticedthatdirectlycomputingthevarianceofthe beamprolecomputedontherawdatamatrixwasyieldingerroneousvaluesforthermsbeam <strong>and</strong>theHoughtransforms(i.e.projection)alonghorizontal<strong>and</strong>verticalaxis(i.e.P(x)= RI(x;y)dy)<br />
photo-electronemittedasthedrivelaserpulseifo.These\ghostbunches"areindeedduetothe inabilitytohaveaperfectextinctionratioof0betweendrivelaserpulsethatservetocreate\real" bunches.Physicallythisisduetotheelectro-opticscellthatareusedtoswitchthelaserpulseon size.Thiswastracedbacktobeduetotheso-called\ghostpulse"eect:alongwiththemain interestforourmeasurement,thereareparasiticbunchescalled\ghostbunches"thatconsistsof <strong>and</strong>oonthephotocathode.Thereforewemodiedouracquisitionalgorithmtotakeintoaccount thiseectbyusingthefollowingstepsduringameasurement: macropulsethatcontainsaseriesofelectronbunches,whosetransversedensityisthequantityof<br />
5.mathematicallyperformtheoperationMbeam=MghostMbeam+ghost 3.increasethemacropulsetotheappropriatewidth 4.acquirethepixelmatrixMbeam+ghost<strong>and</strong>storeitinthecurrentbuer 1.reducemacropulsewidthto100nstoonlydetect\ghost"pulses, 2.acquirethepixelmatrixMghost<strong>and</strong>storeitinamemorybuer,<br />
methodyieldsreasonableresults. Intheabovestep3,weneedtoclarifythemeaningof\appropriate"macropulseduration:it TheresultsofthisoperationisgraphicallyshowninFigure4.12.Wehavealsoveriedthatthis dependsonthebeamsize,<strong>and</strong>itisexperimentallydeterminedbyinsuringtheCCDarrayisstill operatedinitslineardomain.Typicallythepowersurfacedensity,dP=dS,onthepixelarray,scales 6.computeparameterusingthematrixMbeam<br />
withthetransversermsbeamsizeimage,x<strong>and</strong>y,<strong>and</strong>withthepoweroftheemittedradiation, P,accordingtotherelation:<br />
conditionchangeswecanusethescalinglawofthelatterequationtore-adjustthemacropulse widthdynamically.<br />
determinedfortypicalbeamsizeoneachofthebeamprolemeasurementstation.Itisthenautomaticallyrecalledwhenameasurementisperformed.Alsotoaccommodatepotentialoperating Duringourexperiments,foreachmeasurementstation,themacropulsewidthwasexperimentally dP dS/P1xy (4.23)
Resolutionofbeamsizemeasurement Itisveryimportanttohaveapreciseknowledgeofthesystematicerroronabeamsizemeasurement sincewewillhavetoincludetheseerrors<strong>and</strong>propagatethemtondtheerrorbarduetosystematic<br />
function,istoimageviathissystemasharpedge[43].Forsuchapurposeatargetimagethat errorsonthetransverseemittancecomputation. Thereareprincipallytwotypesofeectsthatenterintheresolutionofthistypeofimagingdevices systemweuse:opticalresolution<strong>and</strong>electronicresponse.Theformereectcanbeevaluatedin consistsofansharpedgebetweenanopticallyblack<strong>and</strong>opticallywhiteregionispositionedat waytocharacterizetheresolutionofthewholesystemi.e.includingoptical<strong>and</strong>electronictransfer ourcasesinceoursystemisoptimizedtominimizespherical<strong>and</strong>chromaticaberration,theoptical degradationofresolutionisessentiallyduetothedepthofeldeectthatresultsbecausethe planeweareimaging,thefoil,makesa45deganglewithrespecttotheCCDarray.Thebest TRradiatorlocation.Thederivativeofthecorrespondingdigitizedimagewillprovideinformation ontheimpulseresponseofthesystem<strong>and</strong>itswidthcanbeusedtoquantifytheresolutionof thesystem.Typicalresolutionmeasuredwereatmaximum1:5timesthepixelsizeintheobject plane.Fortypicalmagnicationweuse,thepixelsizeintheobjectplaneisabout40mwhich givesatypicalrmsresolutionof60mwellbelowthetypicalbeamsizemeasured(oftheorderof approximately1mmrms).<br />
ThemethodtomeasureemittanceinthehighenergyregionoftheFEListheusualenvelope 4.4.1GeneralConsiderations 4.4MeasurementofEmittanceinthe38+MeVRegion<br />
betweenthehorizontal<strong>and</strong>verticalplanes,<strong>and</strong>thatthedispersionisnegligibleatthelocationof themeasurement.Ifthelatterassumptionsarefullledthenonecanusetransportformalismto transfermatrixbetweenthemR ttingtechnique.Itassumesthatthebeamcanberstordertransport,thatthereisnocoupling<br />
Exp<strong>and</strong>ingtheabovematrixrelation(recallisthebeammatrix)wecanrelatetheRMSbeam ndtherelationbetweenbeamparametersattwodierentlocationsinthebeamlineknowingthe<br />
systemofNequations(correspondingtoNdierenttransfermatrices)withonly3unknowns.Such sizeatthelocationiwiththeRMSdivergence,beamsize<strong>and</strong>beamcorrelationofthebeamatthe location0.Hencevaryingthetransfermatrixforagivensetofinitialvaluesin0,providesdierent beamsizeatthestationlocationi.Thereforeonecaneasilygeta(generallyoverdetermined) (i)=R(0)RT<br />
systemistraditionallyinvertedbythemeansoftheleastsquaremethod:GiventheNsquared- (4.24)<br />
the2: beamsizemeasurements,oneneedstondthesetofparameter((0) 2=NXi=1h(i)(R211(i)(0) 11+R212(0) (i)2 22+R11R12(0) 11,(0) 12)i212,(0)<br />
22)thatminimizes (4.25)
whereiistheerrorontheithbeamsizemeasurement. Itiswellknown[40]thatthevalueof0thatminimizedthe2satises: 0B@PNi=1R211(i)(i) PNi=12R11(i)R12(i)(i) PNi=1R212(i)(i) 2(i) 1CA=0B@PNi=1R411(i) PNi=12R311(i)R12(i) 2(i)PNi=12R311(i)R12(i) 2(i)PNi=14R211(i)R212(i) 2(i)PNi=1R211(i)R212(i) 2(i)PNi=12R11(i)R312(i) 2(i) matricesequation.Itisworth(forsoftwareimplementation)tonotethatthecurvaturematrixhas The3x3matrixisnamedcurvaturematrix.Thesolutionfor0isobtainedinvertingtheprevious PNi=1R211(i)R212(i) c,b=C0 2(i)PNi=12R11(i)R212(i) 2(i) PNi=1R412(i) 2(i) 2(i)1CA0B@(0) (4.26) 11 12 221CA (0)<br />
thefollowingform: (0)<br />
whichyieldsaverytractableformforitsinverse,theerrormatrix: E=1 jCj0B@4(CED2)2(BECD)4(BDC2) 2(BECD)(AEC2)2(ADBC) 4(BDC2)2(ADBC)4(ACB2)1CA C=0B@A2BC 2B4C2B C2BE1CA (4.28) (4.27)<br />
matrixelementnamely: Theelementsoftheerrormatrixarethevariance<strong>and</strong>covariancenumbersonthecomputed(0) wherejCjdenotesthedeterminantofC: jCj=4(ACEAD2B2E+2BCDC3) 8>:11=pE11 11;12=E12 12=pE22 22=pE33 (4.29)<br />
Twissparametersatthelocation(0).Letbethecomputedparameteri.e.theemittanceorthe Twissparameters.Thentheerroronisgivenby: Usingthest<strong>and</strong>arderrorpropagationtheory[40]onecanestimatetheerrorsonthecomputed 11;22=E13 12;22=E23 (4.30)<br />
Thepartialderivativeintheerrorpropagationformulaearegivenby: ()2= +2@ @11@ @T @112(11)2+@ @12(11;12)2+2@ @ @11=T2,@~" @~" @122(12)2+@ @11@ @12=T,@~" @22(11;22)2+2@ @22=T2, @222(22)2+ @12@ @22(12;22)2(4.32)<br />
(4.31)<br />
@11=TT @T@11=2TT<br />
2~",@T 2~",@T @12=1+2T @12=TT ~",@T ~",@T @22=TT @22=2T 2~". 2~", (4.33)
Finallyafterabitofalgebraonecancomputetheuncertaintiesontheemittance,<strong>and</strong>theT<strong>and</strong> ()2= Tparameters:(~")2= +21221131111 1~"6212211(12)2+1112 12112211+1122 1~"2212(12)2+212 22122(11)2+411 2112212221112 4(11)2+211 4(22)2 4(22)2 (4.34)<br />
()2=1~"6222211(12)2+212211 12112121211+2121122 221222211+212211222312111112123111122 24(11)2+212222 11221221122 4(22)2 2 1112 (4.36) (4.35)<br />
ment,istakentobeequaltothemeasuredresolutionofthebeamprolemeasurementsystemi.e. computedelementsofthe-matrix<strong>and</strong>thenpropagatetheseuncertaintiesontheestimatedvalues fortheemittance<strong>and</strong>Twissparameters.Lastlythevalueof(i),theerroronbeamsizemeasure- Thereforebybuildingthecurvaturematrixwecangetanestimateoftheuncertaintiesonthe<br />
Onewayofvaryingthetransfermatrixbetweenareferencepoint<strong>and</strong>themeasurementistochange thestrengthofaquadrupole<strong>and</strong>observethevariationofbeamsizeonanOTRscreenupstream. 4.4.2Thequadrupolescanmethod (i)=60m:<br />
Althoughthismethodisgenerallyeasytoimplementspecialcaremustbetaken: theminimumbeamsizeshouldbechosentobelargewithrespecttotheresolutionofthe themaximumbeamspotsizeinbothdirectionmustbesmallerthatthedimensionofthe OTRscreen<br />
casewherethequadrupole<strong>and</strong>theprolemonitorareseparatedbyadrift(seereference[34]).In amoregeneralcasewherethetransfermatrixbetweenthequadrupoleexit<strong>and</strong>theprolemonitor Onequestionthatarisesishowtosettheopticsdownstreamthequadrupolethatisbeingvariedto getthewantedbeamsizevariationontheprolemonitor?Suchproblemhasbeenstudiedinthe beamsizemeasurement<strong>and</strong>largeenoughnottoproduceanysaturationontheCCDcamera<br />
isR,wec<strong>and</strong>eriveasimilarcriteriononthelatticefunctionsattheprolemonitorlocation:the thatisusedtomeasurethebeamsize.<br />
by: x;y<strong>and</strong>x;yTwissparametersattheentranceofthequadrupolebeingvariedshouldberelated x=R11 y=R33 R12x R34y<br />
(4.37)
Let'sassumethethinlensapproximationtobevalid7.Thenthetransfermatrixbetweenthe entranceofthequadrupole<strong>and</strong>theprolemonitoris: Thex;yinthepreviousrelationaretheminimum-functiononewishestoachieveattheprole monitorstation.Thechoiceoftheminimum-functionhastwoimplications.<br />
betatronfunctionatthequadrupoleentrance(forinstanceinthehorizontalplane): HencethebetatronfunctionattheOTRlocationcanbeexpressedasafunctionoftheinitial R= x(k)=R212 R21R22! R11R12 f2x;0+R212 1=f1! 10 x;0 (4.38)<br />
wherewehaveusethefactthatx;0=x;0=Lwhenonehastakencareofsettingtheupstream opticstosatisfytherelationderivedpreviouslyinEqn.(4.37). Introducingthefocallength(1=f=k1l)<strong>and</strong>recallingthatR12=x;0=x(k=0)yields: x(k)=x(0)+R412k21l2 x(0) (4.39)<br />
whosederivativewithrespecttothequadrupolestrengthk1is: dx(k) dk=2R212k21l2 x(0) (4.40)<br />
latterequationshowsthatthechoiceofx;0,whichwehavesuggestedearliertobeaslargeas possibletoreducetheerroronthebeamsizemeasurement,directlyaectstheslopeofthebeam sizevariationontheprolemonitor:atoolargex;0willgivea\atlooking"variation.Therefore thereisanoptimumbeamvalueforx;0;thisoptimumshouldbedeterminedviaaniterative withthesamekindofrelationintheverticalplane(replacingxindexbyy<strong>and</strong>R12byR34).The processusingnumericalsimulations. (4.41)<br />
4.4.3Themulti-monitormethod<br />
theTwissparametersatthereferencepointbyEqns.(4.24).Indeed,weneedtomakesurethat Anotherwayofvaryingthetransfermatrixbetweenthereferencepointwhereonewishestocompute<br />
thesetwoequationsarenotredundant,namelythat: ofthismeasurementisthatnoelementhastobevaried.Howevertogetaprecisemeasurementa dedicatedopticallatticesettinggenerallyneedtobeelaborated. Letanalyzequantitativelythemethod.Thebeamsizeonaprolestationk<strong>and</strong>larerelatedto requiresatleastthreemonitorsbutoneshouldusemoreofthemforredundancy.Anadvantage thebeamparameters<strong>and</strong>thebeamprolemeasurementstationistomeasurethebeamproleat dierentpositionalongthebeamlinewhichareseparatedbynon-dispersiveoptics.Thismethod<br />
20)butitprovideseasieranalyticalresults<strong>and</strong>doesnotchangesignicantlythephysicsofthepresentdiscussion. Thetreatmentofthefullproblemincludingthethicklenstransfermatrixisdonevianumericalmodeling.<br />
7Thisisafalsestatementifweconsiderthewholerangeofthemagneticstrengthforthequadrupoles(20
zero): Thesethreedeterminantsyieldthesameequation(assumingtheR11<strong>and</strong>R12tobedierentfrom<br />
Henceinorderthelatterequationtobeveried,onemusttakecaretosettheopticallatticeso advancebetweenk<strong>and</strong>l: whichimplies,usingthegeneralformulationofabeamtransfermatrixintermofbetatronphase R11;kR12;lR12;kR11;i6=0 sin()6=0 (4.44) (4.43)<br />
Anothercarethathastobetakenistomakesurethatattheprolemeasurementstationthebeam isnotatawaist;thiswillenlargetheerrorbarsonthemeasurement(oneshouldmakethebeam aslargeaspossiblecomparedtotheerroronthebeamsizemeasurement). 4.4.4SimulationofEmittanceMeasurementintheIRFEL (withn2N). thatthebetatronphasebetweentheviewersbeingusedinthemeasurementisdierentfromn<br />
Afterthedecompressorchicane,thebeamlineconsistsofaquadrupoletriplet<strong>and</strong>isinstrumented<br />
theOTRlocation(suchoptimizationwillbediscussedinmoredetailinChapter5).Afterhaving makesurewecanhavea\right"beamsizevariationoverthequadrupoleexcitationrange.Typically theprolemonitor.Toperformsuchmeasurementoptimallyweneedtosettheupstreamopticsto weusethemagneticopticscodedimadtottheupstreamquadrupolestosatisfyEqn.(4.37)at scanmethod,weusetheOTRmonitorlocatedinthedumpbeamline3.43mdownstreamtheexitof thelastquadrupoleofthistripletquadrupole.Thereforeinthiscasewehaveinvestigatedwhether thisquadrupolecouldbeusetovarythetransfermatrixwhileobservingbeamsizevariationon withtwoOTRviewers.Forthesimulationoftheemittancemeasurementusingthequadrupole<br />
minimumbetatronvalueattheOTRlocationminimizestheerrorbarsonthededucedemittance (<strong>and</strong>ontheotherdeducedTwissparameters).Itisseenthat6misareasonablenumberforwhich thesystematicerrorsachievedonemittancemeasurementcanbewellwithinthedesired10%level. plottedinFigure4.14.Fromthisgureonecanobviouslynoticethatthechoiceofthelargest sizevariationispresentedingure4.13.Thededuceduncertaintiesontheemittanceforthesetwo dierentvaluesoftheminimumbetatronfunctionversustheerrorsonbeamsizemeasurementis properlytunedtheseupstreamquadrupoles,wehavenumericallystudiedthevariationofbeam<br />
Tofullysimulatethewholemeasurement<strong>and</strong>benchmarkourdataanalysisalgorithm,wepropagate sizefortwodierentminimum-functionsatthelocationoftheOTRviewer.Atypicalbeam<br />
wesimulatethemeasurement:wevarythequadrupolestrength<strong>and</strong>foreachsettingpropagatethe parametersuptothelocationoftheOTRmonitorwherewecompute<strong>and</strong>recordthebeamsize. theentranceofthevaryingquadrupole(i.e.lastquadrupoleofthetripletaforementioned).Then usingthedimadcodetheexpectedparameteratthelinacexit(ascomputedwithparmela)upto Intable4.4wecomparetheresultsobtainedonthecomputedbeamparametersatthequadrupole entrancefacewiththedimadinitialparameters:theresultsareinexcellentagreement.Wealso comparetheerrorbarsobtainedwithourerroranalysiswiththeerrorbarsstatisticallycomputed onasetof200simulationsofthemeasurementinwhichthebeamsizeisr<strong>and</strong>omlygeneratedalong anormaldensitycenteredonthebeamsizecomputedwiththeopticscodewithavarianceequalto thermsresolution(60m).Theconclusionisthattheerrorpropagationagreeswiththevariances
obtainedviathestatisticalanalysis.Theuctuationfrommeasurementtomeasurementusingthe MonteCarlotechniqueispresentedingure4.15. Whentheundulatormagnetisinstalledontothebeamline,thetworsttripletareusetomatch Parameter ~"x(mm-mrad)0.170000:169930:004790:169900:00431 x(m) x DIMADErrorPropagationMonte-CarloSimulation 5.56 1.39 5:560:18 1:390:05 5:560:17<br />
therstrecirculationarc.Theparameterspresentedareallattheentrancefaceofthequadrupole Table4.4:Simulationoftheemittancemeasurementusingthequadrupolescanmethodpriorto ~"y(mm-mrad)0.149600:149640:004490:149690:00441 y(m) y 1.39 5.56 1:390:05 5:560:19 1:390:05 5:560:19 1:390:04<br />
energy<strong>and</strong>thewigglerparameter)inbothplanes.Suchmatchingisrealizedbythemeanofthe centerwithavalueforbetatronfunctionofapproximately0.5meters(dependingonthebeam beingusedduringthemeasurement. thelatticefunctionatthemiddleofthewigglerinsuchawaytoobtainawaistattheundulator tomeasureemittanceinthisregion:unfortunatelywefounddicultinsimulationtoachievelarge betatronfunctionontheviewerusedintheundulatorchamber.Thispointisillustratedinthe theerrorbarsaremuchlargerthanthosegenerallyobtainedwithquadrupolescantechnique. tablebelowwhere,asbefore,wevalidatetheerrorpropagationforthemulti-monitortechnique: twoupstreamquadrupoletripletsthatcanadjustthefourbeamparameters(x,x<strong>and</strong>y,y) whilekeepingthebeamenvelopewithinthemachineaperture.Afterthewigglertwoothertriplets areusetomatchthebeamtotherecirculationtransport.Wehaveimplementedanopticstotry Parameter ~"x(mm-mrad)0.170000:170050:012330:167530:01268 x(m) DIMADErrorPropagationMonte-CarloSimulation<br />
Table4.5:Simulationofemittancemeasurementusingthemulti-monitormethodintheundulator ~"y(mm-mrad)0.149600:146600:011860:147260:01214 y(m) x 5.48 3.09 1.25 5:480:46 3:080:30 1:250:19 5:580:53<br />
region.Theparameterspresentedareallattheexitfaceoflastdipoleofthedecompressorchicane. y -0.13 0:130:13 0:150:14 3:170:31 1:270:20<br />
ofmagneticelements,thedispersionmaynotexactlyvanishafterthearcs.Henceitisvery<br />
Uptonowwehaveassumedthebeamprolemeasurement,forthesubsequentestimationof operateinachromaticmode)intheIRFEL.Insuchasystembecauseofpotentialmisalignment 4.4.5EectofspuriousDispersiononEmittanceMeasurement transverseemittance,isperformedinadispersionfreeregion.Practicallythisassumptionisnot afortioritrue:especiallyafterlargebendingsystemssuchastherecirculationarcs(whensetto
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-15 -10 -5 0 5 10 15<br />
k1MQG2F09 (1/m 2 0.006<br />
quadscan with min=3m -x direction<br />
quadscan with min=3m -y direction<br />
quadscan with min=6m -x direction<br />
0.005<br />
quadscan with min=6m -y direction<br />
0.004<br />
0.003<br />
0.002<br />
intheseregions.Thenpriortomeasuringemittance,thedispersionshouldbemeasured<strong>and</strong> insignicantimpactonthetransversemeasurementperformedwithbeamprolemonitorlocated eventuallycorrectedsothatspuriousdispersioniswithinthetoleratedvalue.Toestablishsuch settingoftheupstreamopticstoachievetwodierentminimumbetatronvalue,3<strong>and</strong>6m. importanttoassesswhatarethetoleranceonthevalueofthis\spurious"dispersiontohavean Figure4.13:Comparisonofbeamspotsizevariationversusquadrupolestrengthfortwodierent<br />
0.001<br />
criterion,wewritethebeamrmsspotsizeasx=p~"(1+2)whereisadimensionless<br />
0.0<br />
spuriousdispersionfordierentvaluesof,<strong>and</strong>thencomputetheemittance.Ingure4.16we beamprolemeasurementstationbyusingamagneticopticscode<strong>and</strong>superimposetheeectsof constant(2(E=E)2)=(~")(Eisthermsenergyspread).Wesimulatethebeamsizeatagiven<br />
)<br />
giveanupperlimitonthespuriousdispersion
20<br />
18<br />
16<br />
.<br />
14<br />
.<br />
12<br />
.<br />
Figure4.14:Relativeerroroncomputedemittanceforthetwocasespresentedingure4.13versus<br />
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10<br />
.<br />
8<br />
.<br />
6<br />
.<br />
. 4 .<br />
quadscan with min=3m -x direction<br />
.<br />
quadscan with min=3m -y direction<br />
theemittancemeasurementbycodingthemeasurementprocedure[35],dataacquisition<strong>and</strong>data therelativeerroronbeamsizemeasurement.<br />
2<br />
quadscan with min=6m -x direction<br />
analysisinaCprogramwithaTcl/Tkuserinterface.Fromthisprogramtheuserdenethe ticetoachievedesiredbetatronfunctionvariationforoptimizingtheerrorbarsonthemulti-monitor measurements<strong>and</strong>sincewedesiredtohavethesamemethodtomeasureemittanceeverywherein theaccelerator,wedecidedtoonlyusethequadrupolescantechnique.Wetotallyautomated betatronfunctionvariation.Becausewendwecouldnotreliablysettheacceleratoropticallat-<br />
quadscan with min=6m -y direction<br />
0<br />
20 40 60 80 100 120 140 160 180 200<br />
programthenautomaticallyscanthequadrupolestrength.Foreachquadrupolesetting,thebeam quadrupole<strong>and</strong>viewershe/hedesirestouseforthemeasurement<strong>and</strong>fewotherparameters.The<br />
x,y ( m)<br />
prolemeasurementstationarecomputed<strong>and</strong>storedinale.Oncetheprogramhascompleted sizeontheOTRprolemonitor,thetransfermatrixbetweenthequadrupoleentrance<strong>and</strong>the thequadrupolescan,itcomputestheemittanceusingthealgorithmdetailedabove.Thisprogram canalsobeusedtopropagatethebeamparametersalongtheaccelerator<strong>and</strong>observethebeam envelope,usefulinformatione.g.toquantifylatticemismatch. Theprogramcanbedividedintothreeparts:(1)auserinterfacefromwhichtheuserentersparameters<strong>and</strong>readresultsofdataanalysis,(2)amachinemodel,Artemis8,thatisautomatically updatedtoreectthecurrentacceleratorsettings(magnetstrength,cavitygradient,...);<strong>and</strong>(3) subsystems(i.e.varyquadrupoles,inserttransitionradiationscreenintothebeampath,...). Atypicalemittancemeasurement,performedinthebacklegtransportlineforachargeperbunch of40pC,usingthequadrupolescanmethodispresentedingure4.17.Itshowsthevariationof acontroltoolboxthatcontainsaseriesofepics-protocolsequencesusedtocontrolthemachine 8theon-lineModelServerArtemiswasimplementedintheFELbySueWitherspoon<strong>and</strong>BruceBowling<br />
(%)<br />
.
0.3<br />
0.25<br />
0.2<br />
0.15<br />
0.1<br />
0.05<br />
Emittance (mm−rad)<br />
0<br />
10<br />
un-normalizedrmsemittance,the-function,theparameter. Figure4.15:Monte-Carlosimulationof200emittancemeasurements.Theplots(fromtop)arethe<br />
5<br />
β (meters)<br />
0<br />
2<br />
1.5<br />
1<br />
0.5<br />
α (no units)<br />
ticledynamics.IntheInjectiontransferline,thebeamenergyisapproximately10MeVinthisTheenvelopettingtechniqueexposedintheprevioussectionreliesonthevalidityofsinglepar- beamspotsizeversusthequadrupoleexcitation. 4.5MeasurementofEmittanceintheInjectionTransferLine<br />
0<br />
0 25 50 75 100 125 150 175 200<br />
regime,thespacechargecollectiveforce(i.e.Coulombianrepulsionbetweenelectroninthebunch)<br />
Measurement Number<br />
envelope.Inanycasetheenvelopettingaredicultforcharacterizing,e.g.measuringemittance, aresignicantthebeamenvelopemustbedescribedwithadierentialequation:theSachererrms envelopeequation[37].Propagationofthebeamthroughamagneticelementthenrequiresthe usingthesingleparticleformalismbasedontransfermatrix.Inthecasewherespacechargeforces integrationofthisequation;generallyspeakingthisintegrationhastobeperformedusingnumericalmethodsbutinsomecaseonecanuseperturbativetheorytondgoodapproximationofthe aresignicantfora60pCchargeperbunch.Therefore,thebeamenvelopecannotbepropagated<br />
trace-spacedensity. samplingtechniques[36].Thislattertypeofmeasurementcanalsobeusedtodirectlymeasurethe ofsuchspace-charge-dominatedbeam.Analternativemethodisbasedonthesocalledphasespace tures4.18.Thebeamletgeneratedbyeachapertureretainsthetransversetemperatureofthe beam.Itisdriftedthroughafreespaceuptoabeamprolemonitor.Thedriftlengthischosenso<br />
Thetechniqueconsistsofinterceptingthespace-charge-dominatedbeambyaseriesofaper
100<br />
10 −3<br />
10 −2<br />
10 −1<br />
10<br />
thatthetransversemomentumimpartsasignicantcontributiontothetransverseprole.Hence Figure4.16:Relativeemittanceerrorversusdimensionlessspuriousdispersioncontributiontobeam size.<br />
1<br />
0<br />
1<br />
toperformveryfastmeasurementwithasimple<strong>and</strong>robustdatareductionalgorithm. thebeaminthedirectionwewishtoperformthemeasurement.Thischoicewasessentiallydone themeasureofthebeamletprolesontheuncorrelatedtransversemomentumspread. Thedierencesamongthevariousapparatusbasedonthisinterceptivetechniqueistheshape downstreamtoanalyzethebeamlets(wire-scanner,uorescentviewer,opticaltransitionradiation screen). Inthepresentcase,theselectingaperturewechoseiscomposedofparallelslits[38]thatsamples oftheselectingaperture(hole,slitsormatrixofaperture),<strong>and</strong>thekindofprolemonitorused<br />
ξ (no unit)<br />
Mathematically,theeectoftheslitscanbeseenasasampling:Ifbeforetheslitsthedensityin<br />
IftheprojectionisobservedafteradriftoflengthL,themulti-beamletproleis: wherenisthenumberofslits. spacedensity,thentheprojectioninthex-x0-planeaftertheslitsis: thespatialplane(x;y)isRR4(x;x0;y;y0)dx0dy0,where4(x;x0;y;y0)isthefourdimensionaltrace i=n Xi=1Zxi+w=2 xiw=2dx2(x;x0)'i=n ()=i=n Xi=1w2(xi;=L) Xi=1w2(xi;x0) (4.45)<br />
thesimplecaseofanormaldistributioninthetrace-space: whereisthehorizontalcoordinateinthebeamletobservationplane.Itisinstructivetoconsider 2(x;x0)=1 p2Nexp"Tx2+2Txx0+Tx02 2~"2 # (4.47)<br />
(4.46)<br />
(ε(ξ)−ε(0))/ε(0) (%)
σ x (mm)<br />
1.5<br />
Figure4.17:Anexampleoftransverseemittancemeasurementinthehighenergyregionofthe<br />
1<br />
<strong>and</strong>vertical(bottom)rmsbeamsizeversustheexcitationofthequadrupole.Thedashedlines<br />
2.5<br />
deducedfromthet.Thechargeperbunchwasapproximatelysetto40pC. areobtainedwiththeleastsquarettechnique.Thereportednumberarethebeamparameters IRFELusingquadrupolescanmethod.Thetwoplotspresentvariationofthehorizontal(top)<br />
2<br />
1.5<br />
ε =8.70 mm−mrad<br />
y<br />
α =1.02 y<br />
1<br />
β =1.44 m<br />
y<br />
0.5<br />
−1000 0 1000 2000 3000<br />
Henceeachbeamletwidthyieldsameasureofthewidthofthetransversedivergenceatthecor- whereNisthenumberofparticlesinthebeam.Forsuchadistribution,theprojectionwrites: ()=Nw 2i=n Xi=1exp2442 2L2~" 2xi+!235exp1~"L22 2 (4.48)<br />
Quadrupole Excitation B’dl (Gauss)<br />
respondingslit(i.e.uncorrelateddivergence),whereasthebeamletscentroidsgiveinformationon theslopeofthetransversephasespace(i.e.thecorrelateddivergencedistribution). Asweunderlinedpreviously,oneadvantageofsuchdeviceistobeabletomeasuretheemittanceof<br />
Iftheincomingbeamonthemultislitmaskisemittance-dominated,SCwillbeinsignicantwith shouldincludetheangularspreadSCinducedbyspace-chargeforce: aspace-charge-dominatedbeam.Infact,intheEqn.(4.46)thereplacementofthedivergencex0by<br />
respecttox0.However,inthecaseofaspace-charge-dominatedincomingbeam,theslitswidth =Lispermittedprovidedthebeamletscanberst-ordertransported.Otherwisethedivergence<br />
shouldbeoptimizesothatthebeamletsbecomeemittance-dominated. AcriteriontodeterminatetheneededslitwidthcanbederivedbyintroducingtheDebyelength D,afundamentalparameterinPlasmaPhysicsthatcanalsobebyappliedto<strong>Beam</strong>Physics:<br />
=Lx0+ZdsdSC ds(s) (4.49)<br />
σ y (mm)<br />
2.5<br />
2<br />
ε x =10.41 mm−mrad<br />
α x =−0.704<br />
β x =1.12 m
multislits mask (copper)<br />
Aluminum Foil<br />
Incoming<br />
Emittance-Dominated<br />
<strong>Beam</strong><br />
<strong>Beam</strong>lets<br />
D<br />
w<br />
Foranelectronbeam,Dwrites tentialintroducedbythistestparticleisscreenedduetoreorganizationoftheneighborcharges. interceptstheincomingspace-charge-dominatedbeam.Thebeamletsissuedfromtheslitsare Figure4.18:Overviewofthephasespacesamplingtechnique.AnincomingAmultislitmask emittance-dominated.<br />
Tele-photo lens<br />
whereistheLorenzfactor,kBtheBoltzmanconstant,Tbthebeamtemperature,measuredinthe qualitatively,thislengthcharacterizetheregioncenteredaroundatestparticleinwhichthepo- D=s0kBTb<br />
5 mm L=620 mm<br />
CCD detector<br />
onthemagnitudeofDversustheinter-distanceparticleswithinthebeamlinter<strong>and</strong>thebeam beamreferenceframe,ntheparticledensity<strong>and</strong>0isthedielectricconstantofvacuum.Depending size,therearetwomainregimes: InthecaseDtheDebyescreeningwillbeineective<strong>and</strong>singleparticledynamicswilldominate<br />
Thereforetransversespace-chargecontributionisinsignicantifD>>eq,eqbeingtheequivalentbeamradius(eq=[xy]1=2).ForthesimplecaseofaK-Vdistributionthetransverse temperatureisgivenby[41]: toitsnearestneighborsthantothecollectiveeldofthebeamdistributionasawhole. forthecharge<strong>and</strong>theself-eldmaybeused;IfD'linterthenaparticleismoresensible ontheDebyelengthcomparedtotheinter-particlesdistance:IfD>linter,smoothfunction<br />
IntroducingtheAlfvencurrentIA=40mc3 Where~"ndenotesthenormalizedemittance. kBTb=8mc2~"2n e,thepeakcurrentIp=Nec z(withzbeingthebunch<br />
e2n (4.50)<br />
2eq (4.51)
length<strong>and</strong>NthenumberofparticleswithinabunchN'n2eqz)yieldsfortheexpressionofthe Henceameasureofthedegreeofself-elddominanceoversingleparticledynamicsisdeducedfrom theratioR=2eq=2D: Debyelength:<br />
R'I 2D=2~"2nIA 2IA()(~"2n) IP 2 (4.52)<br />
CollimatingthebeamwithaslitwillscaleRby,denedastheratiooftheslitrms-width(=p12) totherms-beamsize: whereisthebeamsize(assumedtoberound). ! (4.53)<br />
Therefore,inthecaseofaroundbeam,<strong>and</strong>undertheassumption
TheReductionoftheSpaceChargecondition: ory),theratioR0uaftertheslitswrites: Thenonoverlappingcondition: Thisconditionisdirectlydeducedfromratioofspacechargeterm<strong>and</strong>emittancecontribution intheK-Vequationsexposedpreviouslyinthisnote.Ifuistheconsidereddirection(u=x Theobservedpatterninthescreen,aswealreadyshow,consistsinpeaksassociatedtoeach slits.WemustinsureinchoosingthedriftlengthL<strong>and</strong>thatthepeaksdonotoverlap,i.e.: R0u=Ru 4uL
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-6 -4 -2 0 2 4 6<br />
x (mm)<br />
-6<br />
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0<br />
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6<br />
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Figure4.19:Simulatedmulti-beamletspatternontheopticaltransitionradiationradiator<br />
4.5.3EmittanceCalculation&Trace-SpaceReconstruction<br />
FromtheOTRimageofthemulti-beamletpattern,aprojectionisgenerated.Thisprojection<br />
consists,aswehavepreviouslydiscussed,inasuiteofpeaks.Eachpeakisautomaticallyidentied<br />
usinga\recognitionalgorithm"[39].<br />
Fromtheprojection,thebeamaveragepositionhxBicanbecalculated<strong>and</strong>usedasthereference<br />
forx-axis.Thebeamletsarethenreferencedtoaslit<strong>and</strong>therebytoaposition(withrespecttothe<br />
beamaverageposition)accordinglyto:xi=wihxBi (4.58)<br />
whereiisanindexthatcanbepositiveornegative<strong>and</strong>identifytheslits<strong>and</strong>wis,aspreviously,<br />
theslitswidth.<br />
Measuringtheaveragepositionofeachbeamletalsogiveinformationonthecorrelatedspread<br />
inthedivergencewhichinturngiveinformationonthe-Twissparameter.Formthebeamlet<br />
distributionwi;jwec<strong>and</strong>educedthebeamdivergencedistributionx0jatthespecicpositionxi<br />
usingtherelation: x0j=jxi<br />
Lhx0Bi (4.59)
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1.0<br />
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90%<br />
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contours)usingthesimulatedbeampatternontheopticaltransitionradiationpresentedgure4.19<br />
90%<br />
50%<br />
-0.5<br />
wherehx0Biistheaveragedivergenceofthebeamcomputedfromthebeamlet:x0B=PiPjwi;jx0j Figure4.20:Comparisonoftheexpectedphase-space,generatedviaparticleretracing(eachgrey dotsrepresentamacroparticle),withtheretrievedphase-space(representedwithbacklineiso-<br />
10%<br />
Fromallthepreviouscalculationsitisthenstraightforwardtocomputetheemittance<strong>and</strong>Twiss parameters:<br />
-1.0<br />
-8 -6 -4<br />
hx2i'Pix2iPjwi;j PiPjwi;j.<br />
-2 0 2 4 6 8<br />
x<br />
PiPjwi;j<br />
(mm)<br />
hxx0i'PixiPjx0jwi;j hx02i=PiPjx02 PiPjwi;j jwi;j<br />
thenumberofsampleinpositiondoesnotexceed5<strong>and</strong>thereforesomenedetailonthedistributioncouldbemissed.Indeeditispossibletomovethebeamonthemultislitmaskbymeansoftrace-spacedistributioniso-contourcanbededucedfromthebeamletprolesincethislattercorrespondstosampleofthedistributioninposition(2(xi;x0)).Unfortunatelyundernormaloperation Aswealreadypointedoutitisalsointerestingtohaveaccesstothetrace-spacedistribution.The (4.60)<br />
upstreammagneticsteerers,<strong>and</strong>foreachsettingofthismagnetrecordthebeamlets'projection.In suchacaseitispossibletollthetrace-spacecompletely;ofcoursethisrelyonaperfectstability<br />
x (mrad)
30<br />
25<br />
20<br />
Figure4.21:Fractionofincidentelectronthatscattersontheslitsedgeversusthebeamincident<br />
15<br />
anglewithrespecttothenormalaxisofthemultislitmask.Thedepthoftheslitsis5mm.A of10%oftheincidentelectronwiththematerial.<br />
10<br />
oftheelectronbeam.<br />
5<br />
Theacquireddata,inourcaseaprojectionthatcontainsthebeamletproles,isdigitizedby misalignmentofthemaskof1:2mradcomparetothebeamaxisyieldsapproximatelytheinteraction<br />
0<br />
theframegrabber<strong>and</strong>thentransferredtoanIOConwhichVxWorksroutineshavebeenimple-<br />
0.0 0.5 1.0 1.5 2.0 2.5 3.0<br />
mented[39].Afteridentifyingeachbeamletprole<strong>and</strong>theslititcomesfrom,thecodecomputes<br />
<strong>Beam</strong> Incident Angle (mrad)<br />
theEPICSchannel-accessprotocol.WedevelopedX-windowbasedscreensthatdisplayemittance, Twiss-parameters<strong>and</strong>possiblyphase-spaceisocontours.Theachievedspeedsare,respectively, theemittance<strong>and</strong>Twissparameters.TheresultscanthenbeaccessedfromanyX-stationvia phasespaceparametersinrealtimewhiletuningtheinjector.Storingrawdata<strong>and</strong>projectionsis alsopossibleateachstageoftheprocessformoredetailedo-lineanalysis,e.g.using(time<strong>and</strong> about1<strong>and</strong>2secforupdatingparameters<strong>and</strong>plotrefresh,aspeedthatallowsobservingthe CPUconsuming)powerfulimageprocessingtools.<br />
N Edge/N IN (%)<br />
= Parmela<br />
=0.5 Parmela<br />
=3 Parmela
Table4.7:Typicalsystematicerroronemittance<strong>and</strong>Twiss-parametersforthenominalemittance value<strong>and</strong>twoextremecases. 1.15972.5 0.39569.9 0.204019.9 ~" ~"=~"(%)=(%)=(%) 2.6 9.9 19.9 2.6 10.4 20.9<br />
priorifollowanykindofanalyticalfunction.Forthesereasonsweperformthiserrorpropagation alotofapproximation<strong>and</strong>assumptions,especiallysincethetrace-spacedistributiondoesnota 4.5.4ErrorAnalysis ErrorPropagation Theerrorpropagationisquitetedioustoperformanalyticallysincedirectcalculationsrequire errorontherms-emittanceasafunctionofthesecond-ordermoments: Theerroronthehxx0iisgivenby: numerically.Followingpreviousderivation[45],itisstraightforwardtocomputethesystematic (~")2=1~"20@hxx0i2hxx0i2+ (hxx0i)2=PihPjw2i;jx2i(hx0i)2+Pjw2i;jhx0i2i(x)2i hx02i2 PiPjwi;j 4!2hx0i2 hx2i2 4!2(hxi)21A (4.61)<br />
Similarly,theerroronhx02iwrites:hx02i=PjhPi2wi;jx0ji2 Wheretheuncertaintyontheaveragethedivergenceissimplyhx0i'x0.Theerroronhx2iis: hx2i=Pih2Pjwi;j(x0j)i2 PiPjwi;j (4.63) (4.62)<br />
Wheretheerroronthedivergenceisestimatedtox0=1Lq2+D2(L)2 oftheOTRmonitor('60m),hasbeenaddedinquadrature.Theuncertaintyonthedriftlength tocomputeerrorsondierentsetsofdata.Typicaluncertaintiesassociatedwiththeemittance<strong>and</strong> TwissparametersarepresentedonTable4.7forthenominalexpectedemittance<strong>and</strong>twoextreme L,isapproximately5mm.Allthepreviousformulaehavebeengatheredinaprogramthatallows PiPjwi;j L2where,theresolution (4.64)<br />
chargeeld.thiseectisduetothefactthatwhenanelectronbunchgetveryclosetotheslits<br />
OtherSourceofErrors cases;asexpected,thiserrorincreasesastheemittancevaluedecreases. Asmentionedinreference[44],theslits(directedalongyaxis)willreducethex-transversespace
(sayonebunchlength),thetransverseselfeldisshort-circuited.Thiseectisconsideredtobe Anothersourceoferroristheeectofnon-zerospace-chargeforceinthebeamlets.Sucheecthas insignicantinourexperiment. beenstudiednumericallyforthemaximumchargeperbunch(135pC)<strong>and</strong>wasobservedtoenlarge<br />
Wechosetocommissionthemultislitassemblyintheinjectortestst<strong>and</strong>(ITS)ofJeersonLab thantheresolutionofthemonitor<strong>and</strong>thereforeisneglected. 4.5.5FirstExperimentintheInjectorTestSt<strong>and</strong> thebeamletswidthontheOTR-monitorbyapproximately12m(4-).Thisenlargementisless<br />
photocathodegun,asolenoid,<strong>and</strong>adiagnosticbeamlinethanincorporatesatransverseemittance measurementbasedontheone-slit<strong>and</strong>wire-scannermethod[14].Thegunenergycanbevaryup to500keV<strong>and</strong>themaximumavailablechargecanbesettoapproximately135pC.Sincethemask acceptanceisrangingfrom0:6mm-mradto1:1mm-mrad(unormalizedrmsemittance)wehadto couldusetocomparetheresultsobtainedwiththemultislitmask.Thecongurationconsistsina sincethiso-linefacilitywasinstrumentedwithanotheremittancemeasurementsystemthatwe<br />
wasarbitrarysetto250keV. Preliminarytest<strong>and</strong>crosscheckwiththemonoslitmethod emittance;initiallythechargewasvaryfrom5pCupto10pCtoperformourtest.Thegunenergy lowerthechargeperbunchaccordinglytoparmelanumericalsimulationstoachieveanadequate<br />
generatedemittance-dominatedbeamletisanalyzeddownstreambythemeanofawirescanner themultislits,basedonphasespacesamplingmethod:amovableslitselectsaposition<strong>and</strong>the prolemonitor.Theadvantageofthis\oneslit<strong>and</strong>collector"techniqueisitsabilitytoresolvethe phasespacedistributionforawidedynamicalrangeinemittancebyadjustingtheslitspositions toperformveryaccurateemittancemeasurementforawiderangeofcharge.Thetechniqueis,as steps.Suchsystemhasbeensuccessfullyusedtofullycharacterizetheemittanceofthebeam producedoutofthephotoemissiongun.Unfortunatelythismethodistimeconsuming:thetime Asmentionedabove,theinjectortestst<strong>and</strong>isequippedwithone-slit<strong>and</strong>wire-scannerapparatus<br />
requiredtoperformoneemittancemeasurementistypically45mins<strong>and</strong>thereforerelyonthe assumptionofperfectbeamstabilityoverthistime.Duringourtestswendthebeamnotso could.Unfortunatelybecauseoftechnicalproblemwewereonlyabletoilluminateuptofourslits. stableoverthislargetime. Forarsttest,wesetthechargeto10pC<strong>and</strong>actedonthesolenoidstrength(theonlyparameter onwhichwecanplayon-line)totrytoilluminatewiththeelectronbeamasmanyslitsaswe Atypicalbeamletsproleobtainedperformingourtestsispresentedingure4.22alongwitha agreesatthe15percentlevel.<br />
typicalreconstructionoftrace-spacewhoseiso-contourdensityplotispicturedingure4.23. Thetable4.8presentstheresultsofourcrosscheckbetweenthetwomethods.Bothtechnique
180<br />
8<br />
160<br />
6<br />
Figure4.22:Anexampleof2Dbeamdistributionontheanalyzerscreendownstreamthemultislit<br />
4<br />
140<br />
mask.Theprojectionontothex-axisisalsodisplayed.<br />
2<br />
120<br />
0<br />
100<br />
−2<br />
80<br />
−4<br />
60<br />
−6<br />
40<br />
Table4.8:Comparisonofthermstransverseemittancemeasurementperformedwiththemultislits ~"x(mm-mrad)~"x(mm-mrad) Dierence(%)<br />
−8<br />
<strong>and</strong>theone-slit<strong>and</strong>one-harptechniques. 0.4669 0.5607 0.5594 multislitmethodone-slit-oneharpmethod 0.5071 0.5070 0.4859 811<br />
15<br />
−10<br />
20<br />
−5 −4 −3 −2 −1 0 1 2 3 4 5<br />
x (mm)<br />
MeasurementofEmittanceintheInjectorTestSt<strong>and</strong> Wethenvariedthesolenoidstrengthfrom237:5Gupto307:5Gtoseehowwasevolvingthe around280G(seeFig.4.24)asobservedinnumericalsimulation.The-functionpresentsas emittancevaluefordierentsettingsoftherstsolenoid.Theemittancepresentsagapatvalues<br />
Chapter1).Thenumberofproducedelectronsdependsontheselectedwidthforthemacropulse.<br />
thenumberofphotonwasstillhighenoughtoproducedunwantedelectrons(seeourcommentin theoutputoftheswitchwhenitiscloseoropen)wasnotoptimized<strong>and</strong>thenevenwhenclosed, factthiswasduetoproblemwiththeopticalswitchofthephotocathodelaseryieldingalightleak creatinglowemittance\ghostpulses":extinctionratio(ratiobetweentheintensityofthelightat expectedaminimumcorrespondingtothebeamwaist.Chargewasvariedusingthelaserattenuator <strong>and</strong>thebunchchargewasmeasuredusingabeamdumpedequippedwithaFaradaycup.Asitcan beseeningure4.25below,theemittancewasfoundtobedependentonthemacropulsewidth.In<br />
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200<br />
the38+MeVlinac<strong>and</strong>recirculationregion.Wehaveshownthatundertheexpectedexperimental<br />
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Figure4.24:Emittance<strong>and</strong>betatronfunctionversusthesolenoidexcitation.<br />
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SOLENOID FIELD 275 G<br />
Figure4.25:Emittanceasafunctionofthechargeperbunchfortwodierentmacropulsewidth<br />
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Characterization LongitudinalPhaseSpace Chapter5<br />
5.1Introduction Asmentionedearlier,inChapter2,theFELgainstronglydependsonthebunchlength<strong>and</strong> Thebunchingprocess,alongthebeamtransport,iscontrolledbyseveralelements:awarmbuncher energyspreadachievedinthevicinityoftheundulatormagnet.Henceitisofprimeimportanceto properlyinstrumentthedriver-acceleratorinordertomeasurethesequantitiesatcriticalpointof thebunchingstage.<br />
whichconsistsinestimatingthebunchpropertiesbydetectingcoherentradiationemittedfromthe bunch,<strong>and</strong>time-basedmethods. thereforecanallowtoisolateaproblemormonitordriftsinthesystem.Forsuchapurpose severaldiagnosticshavebeendeveloped.Thesediagnosticsincludefrequencydomainmethods, chicanes.Inordertomakesurethebunchingprocessisperformingadequately,itisworthwhileto havemanylongitudinaldiagnosticsthatcanprovideinformationonhowthebunchingisperforming. Alsotheycanbeusefultoidentifyatwhichstagethebunchdynamicsisnotasexpected<strong>and</strong> cavity,SRF-cavitieslocatedintheinjector<strong>and</strong>inthemainlinac,<strong>and</strong>theinjection<strong>and</strong>by-pass<br />
Alongwithbunchlengthmeasurement,thelongitudinalphasespaceemittancecanbeestimated, 5.2TheLongitudinalPhaseSpaceManipulationintheIRFEL manipulationinthedriveraccelerator. undercertainconditions,providedonecanmeasuretheintrinsicenergyspread.<br />
IntheIRFEL,bunchformationstartsattheelectrons'emissionfromthephotocathodewhichis Bothmeasurementsaredescribedinthischapter,afterdiscussingthelongitudinalphasespace<br />
illuminatedbyadriver-laserwhoseopticalpulseisapproximately47psec(FWHM),asmeasuredby autocorrelationtechnique1.Thereforeatthecathodesurface,theelectronsaregatheredinbunch ofapproximately47psec(FWHM),ifweignorethebunchlengtheningduetotheGaAswafertime 1M.D.Shinn,privatecommunication 93
theendofthemagneticdecompressorisbestdescribedintermsofsequencesofparmelaruns response.Theevolutionofthelongitudinalphasespacefromtheelectronbunchemissionupto showningure5.1.Inthefollowingweonlyconcentrateonthecaseof60pCchargeperbunch withlasingturnedo. 2.Therstelementthatsignicantlyaectsthelongitudinalbunchdistributionisthebuncher 1.Thelengthoftheelectronbunchafteremissionviaphotoelectriceect<strong>and</strong>accelerationto cavity.Thiscavityisoperatedatzero-crossingsothattheaveragearrivaltimeofthebunch coincideswithazeroacceleratingeld.Theelectronsarrivingsooner(i.e.thatbelongtothe bunchhead)aredeceleratedwhereasthelateelectrons(i.e.thatarelocatedinbunchtail)are accelerated(seeFig.5.1(B)).Hencetheprincipaleectofthiscavityistoprovideanenergy rampacrossthebunch.Thisenergymodulationtranslatesasthebunchpropagatesthough 350keVintheDC-gunstructureisapproximately15ps(RMS)(seeFig.5.1(A)).<br />
3.Afterdriftingthroughalongitudinallyfreespace,thebunchenterstherstaccelerating adriftspacetoa\bunching"oftheelectrons:becauseoftheelectrons'averageenergyof ofappropriatelengthwillbunchtheelectrons(mathematicallythistraducestothenonzero approximately350keV,whichmakethemnonrelativistic,<strong>their</strong>propagationinadriftspace valueofthemomentumcompactionofadriftspaceoflengthL:R56=L=2).<br />
4.Approximately7cmaftertheexitofthepreviouscavity,thebunchentersasecondSRF- constantuptothecavityexitwhiletherelativeenergyspreadisgreatlyreduced. ofthecavity(thatactsasacapturesection),thenthebunchlengthisfrozen<strong>and</strong>remains acceleratingelectriceld).Thereisastrongcompressionoccurringinthersttwocells Thecavityisoperatedformaximumenergygain(whichdoesnotmean,becauseofthenonrelativisticnatureoftheelectron,thatthebunchisinjectedinphasewiththemaximum ve-cellCEBAF-typeSRF-cavitywithanominalaverageacceleratinggradientof11MV=m.<br />
themaximumenergygainphase,sothatitprovidesfurtherbunchcompression.Indeedthe choiceofthephaseismadetoimpressthelongitudinalphasespacewiththeproperslope cavitywithanominalaverageacceleratinggradientof9MV=m.Thiscavityisoperatedo neededtomatchtheslopedesiredattheentranceoftheupstreamachromaticchicanefor optimumbunchingthroughthischicane.Atthecavityexit,theparametersare:1.2psfor<br />
6.ThenthebunchisinjectedintheSRFlinac.Thegradientofeachcavity<strong>and</strong>theoverallphase 5.Theelectronsthendriftthroughanachromaticthree-bendchicane.Thislattercanreduce thebunchlengthbymeansofmagneticcompressionthatisbasedonthefactthatpathlength insidebendsisenergydependent. thebunchlength,4%fortherelativeenergyspread<strong>and</strong>approximately10MeVforthebeam<br />
ofthelinacisadjustedtogivepreciselythedesiredenergy(whichwilldeterminetheFEL averageenergy.<br />
7.Thecompressorchicanewillcompressthebunchdownto120m(RMS)toachievethe 8.Afterthewigglerasecondchicanethatactsasadecompressorchicanelengthensthebunch chicane. minimumbunchlengthatthewigglerlocation. wavelength)<strong>and</strong>toadjusttheincomingbunchlength<strong>and</strong>energyspreadinthecompressor length.
concentrateonthebeamparametersintheundulatorvicinity,whichareofimportancetostartup oftheFELprocess<strong>and</strong>quantifyfewofitsproperties. Thebeamdynamicsintherecirculationwillbedescribedlater.Inthepresentchapterweonly 9.Thebeamisthenrecirculated.<br />
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(D) (E) (F)<br />
longitudinalphasespaceisplottedattheexitofthegun(A),thebuncher(B),theSRF-cavity#1 (C),theSRF-cavity#2(D),theachromaticchicane(E),theSRF-linac(F),thebunchcompressor chicane(G),thebunchdecompressorchicane(H),thearc#1(I).Notethatelectronswithpositive Figure5.1:SequencesofparmelarunsdemonstratingthebunchingprocessintheIRFEL.The<br />
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WepresentedinChapter2theformalismassociatedtotheemissionofelectromagneticwaves 5.3TheoryofBunchLengthMeasurementusingFrequencyDomain belongtothebunchtailwhiletheonewithnegativeareinthebunchhead.<br />
(RF-deg.)<br />
possiblefromthelattertoextractinformationonthelongitudinalbunchdistribution.<br />
namedthespatial<strong>and</strong>angularBFF.InthenextsectionwestudybothBFFs<strong>and</strong>showhowitis (d2P=(d!d))bysuchsystemhasacontributionthatisproportionaltoN2,whereNisthe byamulti-particledistribution.Wehaveseenthatthetotalspectralangularpoweremitted calledbunchformfactor(BFF)f(!;bn)thatinturncanbewrittenastheproductoftwofactors numberofelectronsinthemulti-particlesystem.Thiscontributionisalsoproportionaltotheso<br />
E (keV)<br />
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(A) (B) (C)
TheAngularBFF ThesecondterminEqn.(2.5)inChapter2isthedependenceofthecoherentemissionwithrespect totheangularbeamproperties.Itisinterestingtonotethatthistermiswavelengthindependent <strong>and</strong>thereforeisnotgoingtoinuencethespectrumoftheradiation:itactsasamultiplicative is: factorthatcanreducethetotalpoweremittedbyabunchedbeam.ItsexpressionfromEqn.(1.5) Itisrelativelydiculttoevaluatethisfactorforanarbitrarybunchdistribution.Howeverunderthe assumptionoftransversecylindricallysymmetricbunches,<strong>and</strong>introducingtheangles=6(bn;b), =6(bnyz;bz), =6(bn;dbz),<strong>and</strong>=6(bnxy;dbz)itreducesto[2]: ~A()=jZd ~A(bn)def ZdA( =jZA(!) )cossin (!)d!j2 cossincos sin j2 (5.2) (5.1)<br />
whichinturncanbeexpressedasacompleteellipticintegral(extendedfromRef.[2])ifweassume theangulardistributionwritesasaGaussi<strong>and</strong>istribution:A( ~A()=j202Z=(2) 0x[(1x)K(2x1=2 1+x)+(1+x)E(2x1=2 1+x)]exp[2 )=1=p202exp( 202x2]j2dx 2=(202)<br />
divergenceisoftheorderof0'1mrad,wesatisfytherelation0'1=forthenominal radiation,thespectralpowerhasitsmaximumatanglesoftheorderof'1=<strong>and</strong>sincetheRMS angulardistributioningure5.2.Itisnoticedthattypicallythisintegralisunityinthecasewhere thebeamdivergence0ismuchsmallerthantheangleofobservation.Inthecaseoftransition introduced2.ThenumericalintegrationofEqn.(5.3)ispresentedfordierentRMSwidthofthe wherethecompleteellipticintegraloftherstkind,K(u),<strong>and</strong>secondkind,E(u),havebeen (5.3)<br />
energyof38MeV(i.e.'77).Henceforthwewillassume,exceptwhenexplicitlymentioned,that<br />
equationmoreexplicit.Ifbnyzistheprojectionofthebnunityvectorinthe(y;z)plane,let thisfactorisalwaysunityforourtypicalbeamparameters. TheSpatialBFF TheEqn.(2.5)iswritteninavectorform.WewillworkinCartesiancoordinatestomakethis<br />
wherewehaveassumedwecouldfactorthe3D-spatialbeamdensitydistributionSastheproduct ofthe1DprojectionsSx,Sy<strong>and</strong>Sz. bn!X=(xsinsin+ysincos+zcos),<strong>and</strong>thisequationrewrites: =6(bn;bz)<strong>and</strong>=6(bnyz;bx)thentheargumentoftheexponentialfunctioninEqn.(2.5)writes<br />
Inordertousefrequency-domainanalysistodeduceinformationonthebunchlongitudinaldistribution,itisnecessarythatthe!-dependencecomeonlyfromlongitudinalcoordinatez.Fromthe 2Theellipticintegraloftherst<strong>and</strong>secondkindarerespectivelydenedasK(u)=R=2 ~S(!;bn)def =jZSx(x)Sy(y)Sz(z)expi!c(xsinsin+ysincos+zcos) 0[1u2sin2()]1=2d (5.4)<br />
<strong>and</strong>E(u)=R=2 0[1u2sin2()]1=2d
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mainlyduetothelongitudinaldistribution: zeff=zcos;<strong>and</strong>deriveacriteriononthermsbeamsizetoensurethewavelengthdependenceis latterequation,wec<strong>and</strong>enetheeectivecoordinatesxeff=xsinsin,yeff=ysincos<strong>and</strong> Figure5.2:AngularBFFforthreedierentvalueoftheRMSbeamdivergence.<br />
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whichcanbeexpressedintermofRMSquantitieswithoutlossofgenerality: or ztanhx2sin2+y2cos2i1=2 zeffhx2eff+y2effi1=2 (5.6) (5.5)<br />
θ (rad)<br />
Ifthislattercriterionisfullledwecanusethelinechargeapproximation,i.e.,treatabunchasaline witha1Dchargedistribution.Insuchcase,analysisofthecoherentemissionofthebunchreveals informationonthebunchlongitudinaldistribution<strong>and</strong>isnotcontaminatedbythetransverseeect aforementioned,<strong>and</strong>wecanwritetheBFFasitisgenerallywrittenintheliterature: ~S()=jZ+1 ztanh2xsin2+2ycos2i1=2 (5.7)<br />
ThecomputationoftheBFFforanormaldistributionorasquaredistributionissimple;theresults arepresentedinFig.5.3wherewehaveassumedthebunchisacontinuum<strong>and</strong>itsRMSextentis withthebunch.Wehaveintroducedthewavenumber=1==!=(2c)forconvenience. 300m.Forbothtypesofdistribution,theBFFssuddenlytakeoatwavelengthoftheorderof<br />
whereasbeforeSz(z)isthelongitudinalbunchdensityalongthelongitudinalaxiszmovingalong 1Sz(z)exp(2iz)dzj2 (5.8)<br />
Angular Bunch Form Factor (a.u.)
structure<strong>and</strong>length.Alsowecannoticethatthesquaredistribution,<strong>and</strong>generallyalltypeof distributionwithsharpedge,inducesBFFwithlowwavelength(highfrequency)components.It thebunchlength.Hencemeasuringthecoherentradiationpoweratwavelengthcomparabletothe bunchlength,i.e.wherethecoherentenhancementoccurs,canprovideinformationonthebunch<br />
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(solidline)bunches. isinterestingtonumericallycomputetheBFFusingaMonte-Carlosimulationtechnique,fora factorfor106macro-particles.Inthecaseofthebunchchargeweareinterestedini.e.60pC,the Figure5.3:Bunchformfactorcomputedfora300m(RMS)square(dashline)<strong>and</strong>gaussian<br />
distribution: macro-particlerepresents375electrons.Thechoicetosimulateonly106macro-particleinsteadof thewholenumberofelectroni.e.3:75108wasimposedbythedesiretoeconomizeCPUtime<strong>and</strong> expeditesimulations.TheMonte-CarlogenerateddistributioncanbewrittenasaKlimontovich nitenumberofmacro-particlesinthebunch.Ingure5.4wepresentcomputationofbunchform<br />
<strong>and</strong>theassociatedbunchformfactor,underthelinechargeassumption,reducestoasum: S(z)=i=N Xi=1(zzi) Weseeingure5.4thatbecauseofthenitenumberofparticles,thebunchdistribution<strong>and</strong> ~S()=i=N Xi=1jsin2zij2+jcos2zij2 (5.10) (5.9)<br />
theBFFcannotbetreatedascontinuum.Thesefeaturesshouldbekeptinmindevenifinthe followingwewillassumethebunchdistributiontobecontinuum. Ifoneusesst<strong>and</strong>ardbeamparametersexperimentallyachievedintheIRFELaccelerator,i.e.transversebeamsizeofapproximately1mm,minimumbunchlengthof140m,Eqn.(5.7)isnotafortiori P tot/P 1e - (no unit)
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coherentradiationsetupthecollectingopticshasanangularacceptanceof0:3rad.Westudythe synchrotronradiationmostofthepowerresidesinaconethatisoftheorderof1=:IntheFEL natelywearealsohelpedbythedirectionalityoftheradiation:inthecaseofbothtransition<strong>and</strong> typesofbunchlongitudinaldistribution(left). satised:transverseeectcanyieldnon-negligiblecontributiontothebunchformfactor.Fortu- Figure5.4:Monte-Carlosimulatedbunchformfactor(right)with106macroparticleforthree<br />
eectoftransversebeamspotsizenumericallybyperformingtheintegration:<br />
(r=10z),<strong>and</strong>a\pancake"bunch(z=0).<strong>Beam</strong>sizesignicantlyaectstheregionofcoherent resultsofthenumericalintegrationofEqn.(5.11)isdepictedinFig.5.5wherewecomparethe Forsimplicity,let'sassumethatthebunchiscylindrically-symmetrici.e.x=ydef eectofthetransversebeamsizeonthebunchformfactor.ThetotalTR<strong>and</strong>SRpowerspectral densityiscomputedforthreetypicalbunchshape:alinechargebunch(r=0),anellipsoidalbunch P(!)=Z1:1! 0:9!d!Zdd2P1e(!) d!d =r.The (5.11)<br />
enhancementintheBFF:ifoneusethetherebycomputedBFFtoretrievethebunchlength,the<br />
Population<br />
BFF (a.u.)<br />
10 12<br />
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10 10<br />
10 0<br />
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transversebeamsizeeectleadstoanunderestimateofthebunchlengthbyaapproximatelya factorof2.However,theeectofthebeamspotsizeisverysmallontheCSR<strong>and</strong>CTRspectrum. Inthecaseofellipsoidalbunch,i.e.theworstcasethatcanhappenintheIRFEL,theerrorisat the10%level. Hence,wewillassumethemeasurementofCTRorCSRintheIRFELisdirectlyrelatedtothe<br />
BFF (no unit)<br />
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Figure5.5:Eectoftransversebeamsizeonthe3D-BFF.Forthreedierenttypeofbunch(ellipsoidal,pancake<strong>and</strong>linechargebunch),theBFF(A),theCTR(B)<strong>and</strong>CSR(C)powerspectrum (B) (C)<br />
sizeeectorfromtheangularbunchformfactor. 5.3.1TheuseoftheBFFtocompute<strong>and</strong>monitorthebunchlength 0:3rad<strong>and</strong>aregivenfora20%frequencyb<strong>and</strong>width. longitudinalbunchdistribution,withoutany\contamination"comingfromthetransversebeam arenumericallycomputed.Thepowerspectrumarecomputedassuminganangularacceptanceof<br />
Underthe\linecharge"assumptionmentionedearlier,theBFFonlydependsonthebunchlength. Inthissection,wederiveasimplerelation,withoutmakinganyassumptionsonthebunchdensity function,thatallowsonetocomputethermsbunchlengthfromthebunchformfactor.Let'sstart withtheBFFdenition,byintroducingthewavelengthnumber=1=forconvenience,<strong>and</strong>by replacingtheexponentialfunctionintheFouriertransformbyitsTaylorexpansion:<br />
Power W/ A/20%BW<br />
(A)<br />
Power W/ A/20%BW<br />
z=100 m, r=1000 m<br />
(ellipsoidal bunch)<br />
z=0 m, r=1000 m<br />
(pancake bunch)<br />
z=100 m, r=0 m<br />
(line charge bunch)
Theequation5.8yields: f()=jZ+1 11Xn=0(2iz)n n!(z)dzj2=j1Xn=0(2i)n exp()=1Xn=0n n! n!Z+1 1znS(z)dzj2 (5.12)<br />
Deningthen-ordermomentnasn=Z+1 Eqn.(5.13)becomes: 1znS(z)dz (5.14) (5.13)<br />
thepreviousequationreducesto: Incaseweareathighwecanapproximatetheseriesbyitsrstthreetermsonly.Insuchcase, ~Sz()=j1+2i1+(2i)2 ~Sz()=j1Xn=0(2i)n n!nj2 22+O(3)j2 (5.15)<br />
exp4222whoseTaylorexpansionatsmallfrequencyisalsogivenbyEqn.(5.16).Itis veryinformativetodeveloptheBFFtohigherordertoseewhetherwecanextractinformationon ofthegaussi<strong>and</strong>istributionexpz2=(22)case:forsuchadistributiontheformfactorwrites wherewehaveintroducedthevariance2=221(def thebunchformfactorwithaparabolicfunctionathighfrequency.Thisresultisageneralization FromEqn.(5.16)wenotethatitisstraightforwardtoextractthebunchlength,,bytting =14222+O(3) =z). (5.16)<br />
thehighermomentsofthebunchlongitudinaldistribution.Performingsuchderivationyieldsthe generalformoftheBFF[46](with=2): ~Sz()=j1Xn=0(1)n2n = =1Xn=04n 1Xn=0(1)n2n2n (2n)!2n+i1Xn=0(1)n2n+1 (2n)!!2+ 1Xn=0(1)n2n+12n+1 (2n+1)!2n+1j2<br />
+1Xn=04n+2 ((2n)!)22n+21Xn=0Xm
powermeasurement,yieldingthelossofphaseinformation. Wehavealsonoticedfromexperimentusingaparticlepushingcodethatthenestructureofthe BFFisaecteddierentlydependingonhowthebunchingprocessisperformed.Forinstance ingure5.6weplottheBFFcomputedfromnumericalsimulationfordierentsettingsofradiofrequencyelementsthatplayakeyroleinthebunchingprocess.Thoughthebunchlengthdoesnot varysignicantly,wecannoticethateachelementaectstheBFFatdierentwavelength.Such parametricstudiesbysystematicallyvaryingeachRF-elements. featurescanbeexperimentallyusedtodeterminewhichelementisnotoperatedatitsnominaloperationpoint(e.g.becauseofdrift,...).Beforeitsapplicationwewillneedtoperformexperimental 2 5 10 -3<br />
2 5 10 -2<br />
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Wavelength (m)<br />
-5<br />
10 -4<br />
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10 -2<br />
10 -1<br />
10 0<br />
Nominal<br />
Buncher +3 deg<br />
Cavity 1 +3 deg<br />
Laser +3 deg<br />
Figure5.6:EectofdierentkeyelementsinthebunchingprocessoftheelectronsontheBFF.<br />
5.3.2RetrievaloftheBunchDistributionbyHilbertTransformingtheBFF photocathodedrivelaser(dottedline)areoperated+3dego<strong>their</strong>nominalsettings. TheBFFcorrespondingtothenominalsettingsfortheRFelements(solidline)iscomparedwith thecaseswherethebuncher(dottedline),therstcavityintheinjector(greyline)<strong>and</strong>the<br />
tiveindexofamaterialfromfromknowledgeoftherealpart.Thistechniquehasbeenrstapplied tothepresentproblembyLaietal.[48];howeverintheliteraturethereisnoclearderivationthat provestheuseofthedispersionrelationforretrievingthephaseoftheBFFislegitimate.We arecommonlyusedinSolidStatePhysicse.g.forreconstructingtheimaginarypartoftherefrac- possibletogetsomeinsightonthephaseinformationusingtheso-calleddispersionrelations3that Aswementionedearlier,theonlyobservablewecanmeasureisthepowerofthecoherentradiation i.e.thesquareamplitudeoftheelectriceld;allthephaseinformationislost.However,itisstill 3AlsoreferredasKramers-Kronig'srelations<br />
Bunch Form Factor (no units)
presentanoutlineofthisproofbelow,<strong>and</strong>adetailedproofinAppendixC.Fromthedenitionof<br />
where canbewrittenas: thebunchformfactorwecanwrite:~S()=^S()^S() mathematicstextbooks(seeforinstance[50])<strong>and</strong>canbeappliedtoS()tocalculateitsimaginary partknowingitsrealpartbecauseS()isasquareintegrablefunction.Inthepresentcase,the isthewavenumber<strong>and</strong>^S()istheFouriertransformofthebunchlongitudinaldistribution;it ()isthephaseassociatedtotheFouriertransform.Themethodisdiscussedinst<strong>and</strong>ard ^S()=q~S()exp(i ()) (5.19) (5.18)<br />
(=+i0isthecomplexwavenumber): Now,log(^S())isnotsquareintegrable<strong>and</strong>theCauchyintegralonlog(^S())doesnotconverge problemisslightlydierent:weknowthemodulusofS()<strong>and</strong>needtocomputethephase.By takingthelogarithmofthelatterequation,wecomebacktothedeterminationoftheimaginary partofthefunctionlog[S()]fromtheknowledgeofitsrealpartlog[jS()j]: log(^S())=log(q~S())+i Ilog(^S())log(^S()) ()=1=2log(j~S()j)+i jj!1 !Z0log(^S())!1 () (5.20)<br />
dispersionrelationsfor^S(seeAppendixCforadetailedderivation),<strong>and</strong>nallythephaseof^S Let'sintroducethefunction()denedas: ()isnotsingularat=<strong>and</strong>issquareintegrable.Wecanthenderiveasetof\modied" ()def =log[^S()]log[^S()] (5.22) (5.21)<br />
takestheform: Letting0=0<strong>and</strong>usingthefact^S()=^S()wenallynd: Thislatterequationiswidelyknown,intheliterature,<strong>and</strong>issometimesreferredasdispersion wherePdesignatestheCauchyprincipalvaluefortheintegral. ()= (0)1(0)PZ+1 ()= (0)2PZ+1 1log[j^S()j]log[j^S()j] 0log[j^S()j] ()(0)d 22d (5.24) (5.23)<br />
relation.Oncethephase inverseFouriertransform:S(z)=Z1 ()iscomputedwecanrecovertheinitialdistributionbyusingthe<br />
discussionisprovidedinAppendixC).<br />
contributiontothephasethatmustbeconsidered.Inthefollowingwewillnotconsidersuchcases byassumingthest<strong>and</strong>ardbunchdistributionisanalyticintheupperhalf-plane(afullydetailed thecomplexplane.Ifithassingularitiesthen,invirtueoftheresiduetheorem,thereareother TwofactsshouldbeemphasedaboutEqn.(5.24)(1)the bezero,<strong>and</strong>(2)thisequationisapplicableprovidedlog[S()]isanalyticintheupperhalf-partof 0^S()cos(2z (0)termisunknown<strong>and</strong>isassumedto ())d (5.25)
tectorconsistsofagas-lledcellenclosedbytwomembranes.Theincomingradiationisabsorbed IntheearlystageoftheIRFELcommissioning,theexperimentalsetupdescribedingure5.12was usedtoimagetheCTRbeamemissionsourceproducedbytheelectronbeamasitpassesthrough analuminumfoilontoaGolaycelldetector.TheGolaycell(seeFig.5.7(A))isathermaldetector withanearlyuniformenergyresponsefromtheultravioletuptothemicrowaveregion.Thede- 5.4ObservationofCoherentTransitionRadiation<br />
overabroadrangeofwavelengths.Theabsorbedradiationheatsthegaswhichinturnincreases chosensothatthecorrespondingsurfaceimpedanceyieldsthemaximumabsorptionofradiation thepressureinsidethecell<strong>and</strong>distortsthesecond(exible)membrane.Thedistortionissensedby aphotodetectorcellthatdetectsalightbeamreectedfromthemembrane.Thiseectisamplied bytwolargegridsarrangedsothatinitially,i.e.whennoradiationisdetected,thelinesofonegrid areimagedonthespaceofthesecondgridresultinginnolightdetectionbythephotodetector. byathinaluminumlayerdepositedonthe\input"membrane.Thealuminumlmthicknessis Whenthemembraneisdistorted,theimageoftherstgridshifts,allowingsignicantlymorelight toreachthephotodetector.Thisgridsystemisinfactamechanicalmeansofenhancingtheeect<br />
workstationcommunicatingtothecontrolsystem. factor10usingasimpleoperationalamplierlocatedintheacceleratortunnelenclosure,thenthe signalisbroughtintheservicebuildinglocatedupstairsthetunnelwhereitisshapedusinganoise ofsmalldisplacementsofthemembrane.Theoutputanalogsignalfromthephotodetectorinthe samplerateequaltothebeammacropulserepetitionrate.Thedigitalsignalisbroadcastviaa VMEinput/outputcontroller(IOC)totheEthernetnetworksothatitcanbeaccessedfromany GolaycellsetupiselectronicallyprocessedasdescribedinFig.5.7(B):First,itisampliedbya lter.Afterward,itisfedintoaAnalogtoDigitalConverter(ADC)whereitisdigitized,ata Itispossibletoverifythenonlinearityoftheradiationversusthebunchchargeaspicturedin<br />
Figure5.7:SimpliedschematicsofaGolaycell(A)<strong>and</strong>theassociatedsignalacquisitionelectronics(B). pronouncedcomparedtotheonemeasuredintheCEBAFaccelerator[20]forexample.Thisis<br />
gure5.8.ThequadraticdependenceoftheCTRsignalversusthechargeperbunchisnotvery<br />
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01 01 01 01 Amplification System <strong>Electron</strong>ic Filter<br />
(B)
contrastintheCEBAFmachine,varyingaslitapertureopeninginachoppercavityvariesthe chargeperbunch.Forallthepracticalslitsopenings,thebeamisneverspace-chargedominated <strong>and</strong>sotheslitopeningdoesnotsignicantlyaectthebunchingprocess. duetothemethodweusetovarythecharge:weusearotationalpolarizertoattenuateorincrease thedrivelaserpoweronthephotocathodethatalsosignicantlyaectsthebeamdynamicsinthe machine(especiallyinthelowenergyregionwherethebeamdynamicsisstronglydependent,via spacechargeforces,onthecharge).Thereforevaryingthechargealsoaectsthebunchlength, <strong>and</strong>thereforetheCTRsignalsinceitisdependentonboththecharge<strong>and</strong>thebunchlength.By Figure5.9depictsthedependenceoftheCTRsignalversustheoverallphaseofthemainlinac:<br />
4.0<br />
3.5<br />
3.0<br />
2.5<br />
perbunchischangedbyvaryingtheintensityofthephotocathodedrivelaser.Thecirclesare<br />
2.0<br />
1.5<br />
1.0<br />
ergy,is4deg<strong>and</strong>theexpectedphaseformaximumbunching(i.e.minimumbunchlength)is Figure5.8:Scalingofcoherenttransitionradiationpowerversuschargeperbunch.Thecharge theexperimentaldatapoint<strong>and</strong>thedashlineistheresultofaquadraticinterpolationofthe<br />
0.5<br />
approximately6degwhichisingoodagreementwiththephasevalueforwhichthemaximumCTR signalisobservedingure5.9('6:3deg). experimentalpoint. duringthisexperimentthemachinesettingswerekeptconstant<strong>and</strong>onlythelinac\gang"phase4 wasvaried.Themaximumaccelerationphase,determinedexperimentallybymaximizingtheen-<br />
0.0<br />
0 10 20 30 40 50 60 70 80 90 100<br />
AlastexperimentconsistedofvaryingthebeamspotsizeontheTRradiator<strong>and</strong>eachtime<br />
Charge per Bunch (pC)<br />
recordingtheCTRsignal.Thevariationofthissignalversustheequivalentbeamradiuswedene aspxyispresentedingure5.10.Atthispointitisdiculttotellwhetherthedecreaseoftotal otherradio-frequencyelementsintheIRFEL(seeAppendixD)<br />
powerdetectedisduetothespatialortheangularBFF. 4the\gangphase"knoballowstoshiftalltheacceleratingcavitiesinthelinacbythesamephasecomparetothe<br />
CTR Signal (Volts)
3<br />
Crest Phase is −4 deg<br />
2<br />
lengthattheundulatorvicinityischanged. Figure5.9:CTRsignalversustheSRFlinacoverallphase.Asthelinacphaseisvariedthebunch<br />
1<br />
performaninterferometricmeasurement.Inadditiontoprovidingtheenergyspectrumofthe 5.5TheMichelsonPolarizingInterferometer Onewayofaccessingthefrequencyspectrumofanelectronbunchlongitudinaldistributionisto radiationemitted,itcanalsogiveanestimateofthebunchlengthdirectlyfromtheinterferogram.<br />
0<br />
2 4 6 8 10<br />
Suchestimatesmustbetakenwithcareaswewillseeinthefollowing.WeequippedtheIRFEL 5.5.1Overviewoftheexperimentalsetup<br />
SRF−Linac Gang Phase (RF−deg)<br />
acceleratorwithtwo\polarizing"MichelsoninterferometerbuiltbytheDepartmentofPhysics<strong>and</strong> AstronomyoftheUniversityofGeorgia.Thelocationofthedevicesare:<br />
thatisparalleltothispreferreddirection<strong>and</strong>transmitsitsorthogonalcomponent. adichroicpolarizerthathasapreferreddirection.Itreectsthepolarizationoftheincomingeld Theadjective\polarizing"referstothenatureofthebeamsplitterusedintheinterferometer:itis thewigglerinsertionregion,tocheckthebunchlengthisadequatetogettheFELlasing. theinjectorfrontend,toverifythebunchingprocessintheinjectoriscorrect,<br />
Becauseelectronsinanacceleratorarer<strong>and</strong>omlydistributedfrombunchtobunch,autocorrelation ofradiationatwavelengthsmallerthanthebunchlengthwillnotprovideanyinformationonthe thanthebunchlength:thisistheregimeofcoherentemission<strong>and</strong>thisinsuresbunch-to-bunch coherenceoftheradiation. IntheIRFEL,thisinterferometerisusedtomeasuretheautocorrelationfunctionofcoherent<br />
bunchstructure.Hencethewavelengthofobservationmustbechosentobecomparableorlarger<br />
Coherent Transition Radiation Signal (Volt)
1.5<br />
1<br />
correspondtothevarianceofveconsecutivemeasurements. transitionradiation(CTR)pulsesemittedastheelectronbunchespassthrougha0:8mthick, Figure5.10:CTRsignalversusbeamequivalenttransversespotsizepxy.Theerrorbars<br />
0.5<br />
0<br />
0 0.5 1 1.5 2<br />
Thewindowthicknessis4:826mm.Afterthewindow,aplano-convexlenswithafocallength ThebackwardCTRemittedfromthefoildirectlyshinesoutofthevacuumchamberthroughan <strong>and</strong>thereforeissomewhatmorediculttoanalyze. opticalwindowlocatedat90degwithrespecttothebeamtrajectory.Thisopticalwindowismade ofsinglecrystalquartzsothatitcantransmitfarinfraredradiationwithoutsignicantlosses. independentintheregionofinterestwhereastheCSRspectrumdependsonthefrequency(/!3=2) havepreviouslydiscussed,itwaspreferredtoCSRbecausetheTRpowerspectrumisfrequency <strong>and</strong>50:8mmdiameteraluminumfoil.ThoughtheuseofCTRisadestructivemeasurementaswe<br />
Equivalent Transverse <strong>Beam</strong> Size (mm)<br />
intheFIRdomainof125mmisusedtocollimatetheCTRbeamtoparallelrays(thelensis bywaterabsorptioninthemicrowaveregionofthespectrum.Inthepolarimetercomponentsare: twobeamsplitters,twoplanarmirrors,oneo-axisparabolicreector,<strong>and</strong>aGolaycelldetector: <strong>and</strong>theinterferometercanbelledwithnitrogensothatthemeasurementisnotcontaminated approximatelylocated125mmfarfromthepointofemissiononthefoil).Thecollimatedbeamis senttotheMichelsonpolarimeterviaoneplanarmirrorM0(seeFig.5.11).Theopticalbeamline Thebeamsplittersaremadeofparalleltungstenwiresof20mdiameterspacedby50m. convertedtoenergyofthecurrent.Thelatteristhenconvertedtoheat,becauseofthesmall butsignicantelectricalresistanceofthewires.Hencetoobeytheboundaryconditionat spacesbetweenthewires,nocurrentcanowperpendiculartothem.Hencetheelectriceld componentperpendiculartothewiresproducenocurrents<strong>and</strong>losenoenergy,thereforeitis transmitted.<br />
thewires.Sucheldsproduceelectriccurrentsinthewires,<strong>and</strong>theenergyoftheeldsis Beingmetallic,thetungstenwiresprovidehighconductivityforelectriceldsparallelto thewire,theeldparalleltothewiresisreected.However,becauseofthenon-conducting<br />
CTR signal (Volts)
Parabolic<br />
off-axis<br />
reflector<br />
~ 300 mm<br />
Wire Grid Polarizer<br />
Figure5.11:Overviewoftheopticstoguidethecoherenttransitionradiationemittedfroman aluminumfoiltotheentranceoftheinterferometer.<br />
Air Actuator<br />
to insert TR radiator<br />
Plano-convex<br />
into the beam path<br />
Light shield<br />
lens (f<br />
IR<br />
=125 mm)<br />
mirror(M1)ismountedonamicropositionerthatcantranslateby1msteps.Twopicomo-<br />
<strong>Beam</strong><br />
direction<br />
optical port<br />
Aluminum Foil<br />
sitivearea.Itisan-oaxisgold-sputteredreectorwithafocallengthof10cm. Theplanarmirrorsarest<strong>and</strong>ardopticalcircularmirrorsof50:8mmdiameter.Themovable Theparabolicreectorisusedtofocusanincomingcollimatedbeamontothedetectorsentorsarealsomountedonthemirrorgimbalmountsthatcanbeusedremotelytoadjustthe horizontal<strong>and</strong>verticalinclinationofthemirrortomakesureitiscoplanarwiththeimageof thexedmirror(M2)throughthebeamsplitter.<br />
"Switcher" Mirror<br />
~ 600 mm 125 mm<br />
ThepolarizinginterferometerisdepictedinFig5.12:Let'srstanalyzehowapolarizinginterfer- 5.5.2TheoryofOperation ometerworksinthesimplistic(usual)caseofaplaneTEMwave.Afterward,wewillrenethis TheGolaycell(seeFig.5.7)<strong>and</strong>itsacquisitionsystemhavebeendescribedpreviously.<br />
analysisincludingtheeectduetoTRelectriceld. Let!E(t)betheelectriceldincomingintotheinterferometer.Whenthiseldentersthe<br />
~ 400 mm<br />
~ 300 mm<br />
Fixed Mirror<br />
Wire Grid Polarizer<br />
Movable Mirror<br />
Golay Cell Detector
Ε’<br />
2<br />
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w<br />
Ω<br />
Ε Ε’<br />
1 1<br />
v^<br />
Ε’’<br />
1<br />
Parabolic<br />
u^<br />
Movable Mirror M<br />
Reflector<br />
Ε’’<br />
1<br />
2<br />
M<br />
3 EG Golay Cell<br />
Direction of propagation<br />
E0 Polarizer P<br />
1<br />
O<br />
TR from Al foil<br />
M<br />
0<br />
^<br />
polarizercanbewrittenas: vertical,bu<strong>and</strong>bv,coordinatesystem(seegure5.11fordetail),theelectriceldafterthe withrespecttothehorizontalplane.Theeectofthispolarizerisonlytotransmittheeld polarizationwhosedirectionisparalleltothewires.Thereforeinthest<strong>and</strong>ardhorizontal- MichelsoninterferometeritrstencountersapolarizerP1whosewiresareorientedat45deg Figure5.12:SimpliedschematicsoftheMichelsonpolarizinginterferometer.<br />
!E0(t)=bu+bv<br />
Inthevariablelengtharm(1):thereectedeldwrites: verticalpolarizationistransmittedtothexedarm(2),(!E2). polarizationoftheelectriceldgetreectedinthevariablelengtharm(1)(!E1)whereasthe ThenasecondpolarizerP2,locatedatapproximately130mmfromP1,thatplaystherole ofbeamsplitter,interceptstheopticalbeam.Sinceitswiresarehorizontal,thehorizontal p2E0(t) (5.26)<br />
<strong>and</strong>propagatesuptothemobilemirrorM1whereitisreected.Thereectedeld!E01is (thereectionintroducesafactorexp(i): !E01(t)=Rp2E0(t+)bu !E1(t)=Rp2E0(t)bu (5.28) (5.27)<br />
theelectriceldwrites: whereisatimedelayintroducedbythemirror.Byconvention,=0whenthemirrorM1 isatthesamedistancefromthebeamsplitterasM2.Theelectriceld!E01back-propagates tothebeamsplitter<strong>and</strong>isreectedasecondtimeonP2.Finallyattheexitofthearm(1) !E00 1(t)=R2 p2E0(t+)bu (5.29)
Inthexedlengtharm(2):thetransmittedeldwrites: writes: <strong>and</strong>propagatesuptothexedmirrorM2whereitisreected.Theeld!E02afterreection !E02(t)=Tp2E0(t)bv !E2(t)=Tp2E0(t)bv (5.31) (5.30)<br />
totalelectriceldwrites: Afterre-combinationoftheelectriceldissuedfromthetwoarmsoftheinterferometer,the throughP2.Finally,attheexitofthearm(2),theresultingelectriceldis: Theelectriceld!E02back-propagatestothebeamsplitter<strong>and</strong>istransmittedasecondtime<br />
E0(t)=1 p2R2E0(t+)buT2E0(t)bv !E00 2(t)=T2 p2E0(t)bv (5.33) (5.32)<br />
ThiseldisthetotalelectriceldincidentontheP1polarizer.Thispolarizerreectselectriceld withcomponent/(bubv)=p2<strong>and</strong>transmitthecomponents/(bu+bv)=p2.Henceonlythelatter polarizationcomponentistransmittedtotheGolaycellafterreection<strong>and</strong>focusingontheo-axis parabolicmirror;ittakestheform:<br />
IndeedtheGolaycellissensitivetotheaveragepowerhjEg(t)j2itwhichis: Ifweassumethepolarizertobeperfectconductor,thenR2=T2=1=2<strong>and</strong>theelddetected reducedto: EG(t)=R EG(t)=R 2p2R2E0(t+)T2E0(t) 2p2[E0(t+)+E0(t)] (5.35) (5.34)<br />
whichcanbeexpressedintheintegralform: I()/2hjE0(t)j2it[E0(t)E0(t+)+E0(t)E0(t+)] I()/2Z+1 | 1jE0(t)j2dt baseline2
FinallyitisinterestingtonotethatFourier-transformingEqn.(5.37)gives: Inthefollowingwewillreplacethetimedelay,,byanopticalpathdierence(OPD),,(=c).<br />
whichmeanthattheautocorrelationistheFouriertransformoftheenergyspectrumoftheincom- Thereforetheautocorrelationpartcanbewritten(sinceeE(!)=eE(!)): I()/2Z+1 1jfE0(!)j2d!2
Withr1=r+c(==cwhereisthemirrorsrelativeposition),r2=r,Eqn.(5.42)takesthe Itisimportanttonotethatthequantity,intheaboveequation,isoftheorderofthebunch form: image<strong>and</strong>thexedmirror,oftheorderof1m;hencereplacingr+cinEqn.(5.43)withrisa 5104eect.Therefore,withoutintroducingsignicanterrortheintensityonthedetectorcan planewithrespectofthepointofemissionwillresultinavalueofr,thedistancebetweenTR length(i.e.atmaximum500mforthepresentcase).A10%errorinthelocationofthefocal I()/1 r+c(0)()+cr((0)()) (5.43)<br />
abovecanbeusedtoextractinformationfromatransitionradiationinterferogram. 5.5.3RelatinganInterferogramMeasurementtoaBunchLengthMeasurement Bymeasuringthefull-widthhalf-maximum(FWHM)oftheautocorrelation,onecanveryeasily bewrittenasforaTEMwave,thatisI()/((0)());<strong>and</strong>thest<strong>and</strong>ardanalysispresented<br />
getanestimateofthebunchlength.Let'sanalyzehowtheFWHMisrelatedtotheRMSvalues oftwotypicalparticledistributions:anormaldistributionthatischaracterizedbyitsrmsvalue<br />
Table5.1:Relationshipsbetween\equivalent","RMS"<strong>and</strong>"FWHM"lengthsforagaussian<strong>and</strong> relatesFWHM,RMS<strong>and</strong>equivalentlengthforthesedistributions. (variance)z<strong>and</strong>asquaredistributionwhosecharacteristiclengthisitsfullwidthw.Table5.1<br />
squarelongitudinalbunchdensity. DistributionEqu.LengthRMSFWHM Gaussian Square p2z w w=p12 z2zpln(2) w<br />
CaseofaGaussi<strong>and</strong>istribution:<br />
CaseofaSquaredistribution: HencetheRMSvalueoftheconvolutionisp2timestheRMSvalueofthedistribution.Using Table5.1,wededucethat S(z)=1=p22zexp(z2=(2z)).Forsuchdistribution,theautocorrelationis()=1=(2pz)exp(2=(42z)).<br />
Wewritethisdistributionas:S(z)=((1=w)ifw=2
Itsautocorrelationis:S(z)=8>:(1=w2)[+w]ifw
Reflection <strong>and</strong> Transmission Coefficents<br />
10 0<br />
10 0<br />
2 5 10 1<br />
2 5 10 2<br />
2 5 10 3<br />
2 5 10 4<br />
Wavenumber (cm -1 10<br />
)<br />
-2<br />
10<br />
5<br />
2<br />
-1<br />
5 R[E ||] wg<br />
2<br />
R[Ek]wgisthereectioncoecientofthewiregridfortheelectriceldcomponentparalleltothe<br />
T[E] gc<br />
TheenergyspectrumcanbederivedbyFouriertransformingtheautocorrelationdeducedfromthe wires,<strong>and</strong>T[E]gcisthetransmissioncoecientoftheGolaycellentrancewindow. wires,R[E?]wgisthereectioncoecientofthewiregridforthecomponentperpendiculartothe Figure5.13:LimitationofsomeopticalcomponentsintheMichelsonpolarizinginterferometer.<br />
R[E ] wg<br />
data.SincetheFouriertransformofasampledfunctionwithsamplingintervaltisa1=t-periodic FastFourierTransformalgorithm(FFT)[47]becauseofthesamplednatureoftheexperimental interferogram.However,wecannotperformanexactFouriertransform<strong>and</strong>havetousest<strong>and</strong>ard function,thefrequencyresolutionwillbeoftheorderof1=(2Nt)ifNisthenumberofsamplings 5.5.5ExperimentalResultsUsinganAutocorrelationTechnique acquired.Againnotethatt==cisrelatedtotheOPD<strong>and</strong>therefore,foraninterferogram acquiredwithmirrorstepsof,thefrequencyspectrumresolutionis1=(4N). ArstsetofmeasurementswasperformedintheperiodpriorrstlasingoftheIRFEL.Typically,it wasfoundthattheGolaycellcouldeasilydetect,withafairlygoodsignalovernoise,thecoherent transitionradiationpowergeneratedastheelectronbeamwasinlowdutycyclemode(theso-called tune-upmode);signalswithamplitudeoftheorderof1Vwereseeninthiscase.
drivelaserwhereasin(A),suchoperationwasproperlyperformed.Figure.5.14(C)whichreprecorrespondstothelongitudinalphasespacesetupusedfortherstFELlightgeneration.ThedifferencebetweenFigures(A)<strong>and</strong>(B)comesfromtheghostpulsebackground.In(B)theghostpulse wasnotcautiouslyminimizedbyproperlysettingtheelectro-opticscrystalofthephotocathode Figure.5.14(A)<strong>and</strong>(B)depictstwoautocorrelationmeasurementsperformedthesameday,it Autocorrelationmeasurement<br />
metric<strong>and</strong>doesnotvanishwhenthepathlengthinthetwoarmsareidentical.Thisfaultwassentthealgebraicdierencebetweenthetwomeasurements,showsthattheonlydierencebetween thetwoaforementionedmeasurementsresidesinthepresenceofaDCosetforthecasewherethe ghostpulseisnotminimized.Thereforetheautocorrelationitselfisnotatallcontaminatedbythe ghostpulse. initiallythoughttobeduetoanonperfectorthogonalitybetweenthetwomirrorsinthetwo 18002=3600mcorrespondstothedistancebetweentheopticalvacuumwindow<strong>and</strong>theplano- arms;wetriedtocorrectforitbyinstallingpiezoelectricpicomotorsonthemovablemirrorso managedtogettheinterferogramtovanishatthislocation.(2)Therearesecondarybumpslocated atapproximately1800mthatprobablycorrespondstoreectionsinthesystem(thedistance thatitstiltcanbeadjustedremotelywhileperformingameasurement.Unfortunatelywenever Fromtheseinterferogramswecanalsonoticeotherfeatures:(1)TheautocorrelationisnotsymtralpeakoftheinterferogramshowninFig.5.14(A).Fromthispeakwec<strong>and</strong>educethermswidth('110m)<strong>and</strong>alsodistinguishwhetherthecoreofthebeamlongitudinaldistributionisagaussianlikeorsquare-likebunchdistribution.Forsuchapurposewehaveplottedonthesameguretheconvexlens).InFig.5.15(A)wepresentanescan(mirrorstepsizeforthedisplacementis5m)ofthecen-<br />
equivalentgaussian<strong>and</strong>squaredistributionthathavethesameFWHM<strong>and</strong>thesameintegral.<br />
Asecondsetofexperimentsweperformedwastostudyhowsensitivethebunchlengthmeasurement thatconsistsofthesumofthetwopreviousdistributionsbettermatchestheinterferogramcore. DependenceoftheinterferogramonthebeamtransversesizeontheTRscreen Neitherofthesest<strong>and</strong>arddistributionsreallyttheinterferogramcore.Howeveradistribution<br />
waswithrespecttothebeamtransversesizeontheTRradiator.Theprocedureconsistedin<br />
withthecorrespondingtransversebeamsizesaregatheredinTab.5.5.5.<br />
varyinganupstreamquadrupoletriplettovarythebeamspotsizeatthepointofbunchlength prole<strong>and</strong>rmswidtharethencomputed.Theresultsofthemeasurementsforvedierentsettings oftheupstreamopticsarepresentedingure5.16whiletheFWHMoftheinterferogramalong theinterferometeroraCCDcamerabymovingthe\switchermirror".Theimagesrecordedbythe CCDcameraareprocessedaccordinglytothedescriptionofChapter3<strong>and</strong>thebeamtransverse measurement.Foreachsettingofthetripletwemeasured,atthesametime,thebeamtransverse spotsize<strong>and</strong>thenthebunchlength.Forsuchpurposewec<strong>and</strong>irectthetransitionradiationto
CaseInterferogramx (b) (d) (a) (c) (e) FWHM(m)(mm)(mm)(mm) 200 240 320 280 0.42150.34150.3794 0.53160.39870.4604 0.78430.57630.6723 1.25541.11641.1839 1.46671.62401.5433 ypxy<br />
Table5.2:Measuredbunchlength<strong>and</strong>transversebeamdimensionforthecasesreportedinFig.5.16. Computationofthelongitudinalbunchdistribution Usingthemethodologypreviouslyexposed,wehavedevelopedano-lineanalysiscodethatallows thecomputationofthebunchlongitudinaldistribution.<br />
henceforthisthe\average"method. theinterferogram,ortherightpart;wecanalsosymmetrizetheinterferogrambycomputingthe Fouriertransform.Wehaveimplementedthreemethods:wecaneitheruseonlytheleftpartof averageoftheleft<strong>and</strong>rightpartoftheinterferogram.Theautocorrelationsso-obtained<strong>and</strong><strong>their</strong> correspondingFouriertransformsarepresentedrespectivelyinFig.5.17-A<strong>and</strong>Fig.5.17-B.Wecan noticethatthereisnotmuchdierencebetweenthecomputedspectra.Themethodwewilluse form,i.e.theenergyspectrum,willnotbereal.HencetherststepconsistsofsymmetrizingtheBecausetheexperimentallyobtainedinterferogramisnotperfectlysymmetric,itsFouriertransmirrordisplacement.Howevertheinformationcontainedatlargedisplacementmightnotberelevanttocomputetheenergyspectrum.Toverifysuchassumption,wehaveuseddierentlengthof thecentralpartofautocorrelationpresentedingure5.14:weused64,128,256,<strong>and</strong>512points; pointsinthesequencemustbeapowerof2.Fromourpreviousdiscussionwenoticedthatthe stepsizeintheFourierplane(i.e.theenergyspectrum)increasesinverselytothenumberofpoints inthest<strong>and</strong>ardplane(i.e.,theinterferogram).Thereforeanaiveargumentwouldbe,foragiven notethatbecauseweusedanFFTalgorithmthatemploysaradix-2algorithm,thenumberof Moreover,theinterferogram(<strong>and</strong>thereforetheautocorrelation)canbemeasuredforarbitrary<br />
smoothed.Ontheopposite,ifthenumberofpointsistoolarge,becausepointscorrespondingto notprovideanyinformationforthepowerspectrum(i.e.theautocorrelationistheoreticallyzero). Fromgure5.18weseethatindeedthereisacompromiseonthenumberofpoints;ifthisnumber istoosmall,thenestructureofthespectrumislost<strong>and</strong>thereconstructedbunchdistributionis twoargumentsagainstthisfact:(1)themeasurementcantakeuptoonehours(dependingon themirrordisplacementstepsize)<strong>and</strong>(2)forlargevaluesofthedisplacementthetwoTRpulses associatedwiththeelectronbunchdonotoverlapanymore<strong>and</strong>thereforetheinterferogramdoes mirrordisplacementstep,tomeasuretheinterferogramoveralargerange.Practicallythereare<br />
Thislowfrequencypartofthespectrummustsomehowbereconstructed,otherwisetheexperi- frequencycutoinducedbytheniteaperturesizeoftheGolaycelldetectorentrancewindow. pointitseemstovanish.Inourcasetypicalvaluesare1:5mm. mentallycomputedenergyspectrumcannotbeusedtorecoverthebunchlongitudinaldistribution. Forsuchapurposeweextrapolatethislowfrequencyregionofthespectrumusingthefactthat<br />
largedisplacementsconsistsonlyofnoise,thisnoisepropagatesonthespectrum<strong>and</strong>agreatdetail offakestructureappears.Aproperchoiceisto\manually"cutadvisotheautocorrelationatthe Experimentallywecanclearlydistinguish,forawavenumberofapproximately10cm1,thelow
expansionoftheBFFderivedinthischapter,thespectrumcanbeextrapolatedusingtheequation: I()=a0+a22+O(3).Thepointofattachmentofthisparabolaischosentobeintheneighbor- thefrequency;theCTRspectrumhasthesamedependence.Hence,usingthegeneralpolynomial hoodofthelowfrequencycuto.Thecoecienta2<strong>and</strong>a0arecomputedbyusingthecontinuity conditionsatthecutopoint:weassumeboththespectrum<strong>and</strong>itslocalderivativewithrespect towavelengtharecontinuous.Ingure5.19weperformdierentlowfrequencyextrapolationsof forlargefrequency(i.e.lowwavelength),thebunchformfactorhasaquadraticdependenceon<br />
ontherecoveredlongitudinalbunchdistributionisshowningure5.20.Itisinterestingtonote thatonewaytorejectunphysicaldistributionistorejectalltheparabolicextrapolationsthatgive asignicantnumberofnegativevaluesinthebunchlongitudinaldistributions. atthecut-opointtocomputethecoecienta0.Theinuenceofthesedierentextrapolations theenergyspectrumbyvaryingtheparametera2<strong>and</strong>usingonlythecontinuityofthespectrum<br />
5.6ZerophasingTechniqueforBunchLengthMeasurement<br />
awaythatthebunchcentroidcoincideswithazeroacceleratingelectriceld.Hencethecavities investigatedthepossibilityofitsapplicationtomeasurethebunchlengthintheIRFELaccelerator. ThezerophasingmethodusesRFacceleratingcavitiesphased90degreesocresti.e.insuch Ithasbeendemonstratedtoresolvebunchlengthinthesubpicosecondregime[52].Thereforewe Theso-calledzerophasing(orbackphasing)techniquehasproventobeaverypowerfulmethod. 5.6.1BasisoftheMethod<br />
inducealongitudinallydependentenergyrampalongthebunch.Then,bymeansofaspectrometer, theenergydistributionismappedintothetransversedirection,<strong>and</strong>thebeamtransversedensity thedispersioninthechicaneatthebeamprolemeasurementstationisabout2timeslessthan anoptimumchoice:themaximumbeamenergythatcanbedeectedintothedumpisabout readilyavailabletoperformsuchmeasurements:wecanusetheenergyrecoverydumplineorthe thismethodweonlyneedacceleratingcavities<strong>and</strong>spectrometers.Therearetwospectrometers theoneattheOTRprolemonitorlocatedintheenergyrecoverydump. Despitethefactthattheenergyrecoverydumphasbeenchosenasaspectrometer,itisnot ismeasuredwithabeamprolestationlocatedinthedispersiveregion.Thereforetoimplement<br />
24MeV5whichimpliesthatthefourlastcavitiesofthecryomodulemustbeturnedo<strong>and</strong>/or usedaszerophasingcavities.Thenon-zerophasedcavitiesareoperatedunder<strong>their</strong>nominalsettings (acceleratinggradient7.33MV/m,phase-9.6deg)givingabeamenergyof23.74MeV.Atsuchan rst4-bendchicane.Preliminaryconsiderationshaveshownthatthelatterisnoteasilyworkable:<br />
intermediateenergy,thedynamicsof60pCbunchesisnotemittancedominated,requiringastudy ofspacechargeeectsonthemeasurement. IntheFEL,theenergyrecoverydumplineconsistsofaquadrupole<strong>and</strong>anOTRprolemonitor. ThedispersionattheOTRlocationwhenthequadrupoleisturnedoisexpectedtobe=75cm; thislattervaluecanbereduced,ifneeded,usingtheupstreamquadrupole. Itshouldbestressedthatthemeasuredbunchlengthisthebunchlengthattheexitofthefourth cavityi.e.inthemiddleoftheSRF-linac(theparmelapredictedbunchlength<strong>and</strong>phasespace slopeatthislocationare370m<strong>and</strong>-48.74MeV/mrespectively). FollowingnotationofReference[52],wewritethehorizontalpositionxonthebeamprole 5PrivatecommunicationfromR.Legg,January1998
as: measurementstationofoneelectronwithlongitudinalpositionzwithrespecttothebunchcenter<br />
1497MHzitis20:05cm),thedispersionatthebeamprolemeasurementlocation,<strong>and</strong>theaverage thecavitiesusedduringthemeasurement(i.e.operatedatzero-crossing),theRF-wavelength(for beamenergyattheentranceoftherstcavityoperatedatzero-crossing.dE E0arerespectivelythepurebetatroncontributiontotheposition,thetotalacceleratingvoltageof energyspread<strong>and</strong>thespacechargeenergyspreadinducedasthebeamdrifts.x,VRF,RF,<strong>and</strong> whereC0isthecontributionfromRF-inducedenergyspread<strong>and</strong>C1isthesumofinitialintrinsic x=x+2eVRF RF+dE dzE0zdef =x+(C0+C1)z dzisthelongitudinal (5.49)<br />
tor,withthecavitiesrespectivelyturnedo<strong>and</strong>turnedonat<strong>their</strong>90degzero-crossingpoint.Let0x,x,bethebeamsizesmeasuredafterthespectrometerdipoleontheOTRprolemoni- phaseslope;itcanbeexpressedusingthebeammatrixelementasdE Sincethebeamproleontheprolemonitoristheconvolutionofpurebetatroncontributioni.e. theenergypositioncorrelationi.e.56=hzEi. transverse<strong>and</strong>longitudinalphasespace,thesebeamsizescanbeexpressedas: dz=56=(2z)where56is<br />
Becausethesignoftheproduct2C0C1isalternatedasthecavitiesareoperatedat90deg,this quantitycanbeeliminated<strong>and</strong>bycomputingthepuredispersivecontributionduetotheenergy isthehorizontalbetatroncontributiontothebeamspotsize. (x)2=2+(C1C0)22z (0x)2=2+C212z (5.51) (5.50)<br />
spreadinducedbythecavitiesat90degi.e.6(XRMS)2=(x)2(0x)2,itisstraightforwardto deduceananalyticalexpressionforthebunchlength: slope,usingtheformula: Notethatinthecaseofsmallenergyspreadtheseformulaereducetotheonederivedinrefer- FinallywecanalsoestimatethecoecientC1,whichcanprovideinformationonthephasespace C1=jC0j z=h(X+RMS)2+(XRMS)2i1=2 2(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 p2jC0j (5.53) (5.52)<br />
ence[52]. FromEqn.(5.53)thelongitudinalphasespaceslopeis: dE<br />
theenergydistribution.Thermsvalueofafunctiong=fh(istheconvolutionproduct)ishg2i=hf2i+hh2i<br />
Insummary,themeasurementofbunchlength(<strong>and</strong>potentiallyphasespaceslope)reducestothree beamprolemeasurementsforthreedierentsettingsofthezerophasingcavities(90deg,<strong>and</strong> 0deg). 6Thebeamtransversedensityattheprolemeasurementstationisaconvolutionofthebetatrondistributionwith dz=VRF RF(X+RMS)2(XRMS)2 (X+RMS)2+(XRMS)2 (5.54)
Asarstapproximation,wecanestimatetransversespacechargeusingtheK-Venvelopeequations bycalculatingtheratiooftheemittancetermwiththespace-chargeterm(weassumethebeamis cylindrical-symmetric): TransverseSpaceChargeEects<br />
beapproximately0.6atthecavity#5exit.Thereforespacecharge<strong>and</strong>emittancetermsareofthe sameorderindrivingthetransversebeamenvelope. Inequation5.51,onemustinsistthatcontainsthetransversespacechargeeect.Inorder whereIpisthepeakcurrent,I0theAlfvencurrent(17000Aforelectrons),<strong>and</strong>aretheusual relativisticfactors.FortheexpectedvaluesobtainedvianumericalsimulationweestimatedRto R=14Ip I02 ()3x2 (5.55)<br />
tovalidatethederivedequationstocomputethebunchlength<strong>and</strong>phasespaceslope,wemust makesurethat,asitisimplicitlyassumedintheprevioussection,remainsthesameasthe<br />
areturnedwith<strong>their</strong>phasesetat90deg.Hencethetransversespacechargecontributionis inwhichismeasuredwhenthecavitiesareturnedo.Alsoitremainsthesameasthecavities withtheparmelaspacechargeroutineturnedon<strong>and</strong>o.Theeectonthebeamsizebeforethe spectrometer,inallthecases,remainsthesame<strong>and</strong>increasesthebeamrmssizebyapproximately 36m.ThereforethetransversespacechargecontributiontothebeamsizeontheOTRisincluded suchassumptionusingtheparmelacode:thebeamenvelopesalongthebeamlineareplottedin zerophasingcavitiesphasedareturnedon<strong>and</strong>phasedat<strong>their</strong>twozerocrossing7.Wehaveveried<br />
indeeddeconvolvedunambiguouslywhenoneusestheEqn.(5.52)tocomputethebunchlength. gure5.22fordierentcases(dierentnumberofzero-phasingcavitiesused):eachcaseistreated<br />
LongitudinalSpaceChargeEect Thelongitudinalspacechargetendstoinducebunchlengtheningwhichinturnrotatesthelongitu-<br />
wherehdE theslopeatthedipoleentranceisapproximately: dinalphasespace.Henceonewayofassessingtheassociatedeectistostudyhowthephasespace<br />
thespacechargeinducedphasespacerotation. slopeevolvesasthecavitiesarezero-phased.Onecanconceivethatbecauseofthespacecharge<br />
AgainweneedtojustifythatjC1jremainsthesameasthezerophasingcavitiessettingsarechanged: namelywemustmakesurethatthespacechargeinducedslopeisthesameinthedierentcases. dE<br />
Thiscanbeunderstoodsincethezerophasingcavitiesarenotprovidingenergy.Wehavechecked dziinitisthephasespaceslopeupstreamtherstzerophasingcavity,<strong>and</strong>hdE dz=dE dzinit+dE dzSC dziSCrepresents (5.56)<br />
focusingeect.Sucheectisinvestigatedlaterinthisdissertation<strong>and</strong>wasanywayfoundtobeverysmallforthe thisusingparmela:theslopeevolutionforthedierentcasesofzerophasingarepresentedin purposeofthepresentdiscussion;thereforeweignoreitforsakeofsimplicity.<br />
gure5.24.Foreachcasewecomparetheslopecomputedwiththespacechargeroutineturned 7Thereisanothereectthatcansignicantlyaectthetransversebeamsizeontheprolemonitor:thecavity
Slope(Mev/m)36.170633.741231.206727.9008 C1 X+RMS(mm)1.84343.28604.72586.1368 #zero.cav1 C0 XRMS(mm)0.87892.42823.93525.4319 z(mm) 1.16561.08731.00560.8991 3.70287.405711.108514.8114 0.39000.39010.39150.3913 2 3 4<br />
cases<strong>and</strong>wehaveestimatedthisvariationforthenormalizedslopeddzto0.54%/m. Table5.3:rmsbeamhorizontalsizesimulatedwiththeparmelaparticlepushingcodeonthe<br />
5.6.2NumericalSimulationoftheMethod energyrecoverytransferlineprolemeasurementstation.<br />
Wehavenumericallyperformedabunchlengthmeasurementforthenominalsettingsusingthe on<strong>and</strong>o.Itisnoticedthatthevariationoftheslopeduetospacechargeisthesameforallthe<br />
distance).ThebeamsizemeasuredattheOTRlocationareshowningure5.25,forthetwozerophasingvalues90deg,versusthenumberofzero-phasingcavities.Inthegurewealsosimulate bunchlengthof370mthecomputedbunchlengthisalwaysoverestimatedbyabout20m. statementthatwecouldunambiguouslydeconvolvetransversespacechargeeectcontributionto Ontheotherh<strong>and</strong>thecoecientC1isdependentonthenumberofcavities(infactonthedrift themeasurementwiththeparmelaspace-chargeroutineturnedotoverifyagainourprevious checktheconstancyofthemethod.Table1summarizestheresultsweobtained.Forthenominal relationsderivedabove.Wehavedonesuchmeasurementusing1,2,3,<strong>and</strong>4zerophasingcavitiesto<br />
thebeamsize.Ingure5.26,wepresentthebeamdistributioninthetransverseplanealongwith thehorizontalbeamprojection,inthecasewherefourcavitiesareusedaszerophasingcavities. 5.6.3ExperimentalResults Duringtheearlystageofthecommissioningofthelinac,weattemptedabunchlengthmeasurement usingthezerophasingmethod.Wetriedtozerophasedierentnumbersofcavities<strong>and</strong>sincethe bunchlengthwaslargerthanexpectedweneededonlytousetwocavities.<br />
phasespaceslopedE UsingEqn.(5.52)wegetanrmsbunchlengthestimateofz'488112m<strong>and</strong>thelongitudinal thesevaluesyieldXRMS'1:9mm<strong>and</strong>X+RMS'4:7mm. Thermssizeofthehorizontalprojectionofthebeamspots,presenteding.5.27,recordedduring thezerophasingexperimentarerespectively:0x'3:5mm,90 Duringourexperiment,thegradientofthetwozerophasingcavitieswassetto7:33MV=m,the C0denedinEqn.(5.49)isapproximatelyC0'7:19. totalenergyoftheincomingbeamwasestimatedtobe23.75MeV;withsuchvaluetheconstant<br />
optimized.Alsonotethattheerrorbaronthebunchlengthmeasurementisobtainedusingthe<br />
wasperformed:theinjectorbeamdynamicswasnotyetfullyunderstood<strong>and</strong>thesettingsnot predictedwithparmela.Thesediscrepancieswerenotrelevantatthetimethemeasurement dz'82MeV=m.Bothofthesevaluesareindisagreementwiththeparameters x'5:8mm<strong>and</strong>+90 x'4:0mm;
errorpropagation8theoryappliedonEqn.(5.49),assuminganuncertaintyof10%on,VRF,<strong>and</strong><br />
5.7.1Method thebeamsizesmeasurement,<strong>and</strong>arelativeerrorof2%onthebeamenergyinferredfromthedipole magnetstrength.<br />
Theestimationofthebeamenergyspreadisperformedbymeasuringtransversebeamproleina planewherethereissignicantdispersion.InthecaseoftheIRFEL,severallocationscanbeused 5.7IntrinsicEnergySpreadMeasurement<br />
horizontalplaneinourcase,thermsbeamsizeiswritten: <strong>and</strong>variouslocationintherecirculationarc.Intheplanewheredispersionoccurs,i.e.inthe tomeasuretheenergyspread.Typicalhigh-dispersionpointare,symmetrypointsofthechicane<br />
suredorestimatedviamagneticopticscode,thelatterrequiresanemittancemeasurement(ina dispersionfreeregion)<strong>and</strong>thepropagationoftheTwissparameterstothedispersiveregionwhere FromEqn.(5.57)weseethattodeducetheenergyspreadwemustknowthedispersionfunction, Thiscommonlyusedrelationisvalidaslongasnonlinearitiesinthetransportisnegligible.Typically,forthenominalenergy(withoutlasing)spreadintheIRFEL(0.2%RMS)itcanbeused.,butalsothebetatroncontribution~"tothebeamsize.Thoughtheformercanbeeasilymea- x=h()2+~"i1=2 (5.57)<br />
energyspreadistobemeasured.Indeedwecanavoidtheemittancemeasurement9byvaryingthe strengthofanupstreamquadrupolewhileobservingthebeamsizeonthedispersivelocation,until thebeamsizeisminimum.Atthatpointthebetatrontermcontributiontothebeamsizeisthe smallestpossible.Ingure5.28,wepresentthebeamsizevariationfortwoscenariiofenergyspread (i.e.thecasewerethelaserisoi.e.'0:2%<strong>and</strong>oni.e.'2%).Forthelowestenergyspread entranceofamagneticsystem,withazero-energyspread,<strong>and</strong>letx0;,x0;bethesamecoordinates onederivedfromEqn.(5.57).Thisdisagreementcomesfromthenon-negligiblenonlineardispersion atthelocationofthebeamsizemeasurementwhichrendersEqn.(5.57)diculttouse(becauseit onlycontainslineardispersion):Letx0;0,x0;0betheposition<strong>and</strong>divergenceofanelectronatthe theminimumbeamrmssizesimulatedwithdimadiscomparabletothequantity.Howeverfor associatedtoanelectronwithanenergyspread.Insidethebendingsystemthatgeneratesenergy largerenergyspread,weobservediscrepanciesbetweenthevaluecomputedfromdimad<strong>and</strong>the spread,wewillhave: 8Thesystematicerror,z,onthebunchlengthcomputationis: E202RF(X+RMS)2+(XRMS)2 (z)2=2RF(X+RMS)2+(XRMS)2 xf;0=R11x0;0+R12x0;0 xf;=R11x0;+R12x0;+R16+T1662 822V2RF (E0)2+E202RF(X+RMS)2+(XRMS)2 842V2RF ()2+ (5.58)<br />
9SuggestionfromD.R.Douglas<br />
822V4RF (VRF)2+E202RF16(0x)2(0x)2+4(x)2(x)2+4(+x)2(+x)2 3222V2RF(X+RMS)2+(XRMS)2
electronwithnoenergyoset=0,as: Thereforewec<strong>and</strong>eneanorbitosetwithrespecttothe\referenceorbit",i.e.theorbitofthe xdef =xf;xf;0=R12x0;+R16+T1662 =xf;0+R16+T1662<br />
in<strong>and</strong>takingthephysicalsolution: Hencetheenergyspreadcanbeexpressedasafunctionofxbysolvingtheseconddegreeequation =R16 2T166"11+4T166 R216(x)21=2# (5.59)<br />
InthecaseofpracticalvalueinthebendingsystemsoftheIRFEL,athird-orderTaylorexpansion islargelysucient,thereforetheenergyosetofanelectronatpositionxwithrespecttothe referenceorbitis: =x R16+T166(x)2 R3162T166(x)3 R516<br />
(5.60)<br />
monitorlocatedinthecenterofoneoftheby-passchicanes:Usingthesecondordermagneticoptics codedimad,wecomputedtransferthematrixelementstobeR16'42cm<strong>and</strong>T166'45cm. Asanexampleweshallconsideranenergy-spreadmeasurementperformedusingabeamprole (5.61)<br />
Animportantparametertopermitthelasertoturnon,aswewilldiscussindetailinChapter6, 5.8EstimateofLongitudinalEmittanceintheUndulatorVicinity isthelongitudinalemittance.Wedeneitas:<br />
(seeg.5.29).Howeveranenergyspreadmeasurementcanbeperformedonlyinahighdispersion whereisthelongitudinalcoordinateexpressedinunitsofRF-degree<strong>and</strong>isthemomentum spread. energyspread.AttheundulatorlocationwecanmeasurethebunchlengthusingCTRmethods coecienthivanishes<strong>and</strong>thereforetheemittanceissimplytheproductofrmsbunchlength<strong>and</strong> Attheundulatorlocation,thelongitudinalphasespaceisatalongitudinalwaisti.e.thecorrelation ~"=qh2ih2ihi2 (5.62)<br />
thereforeimportanttocheckhowthemeasuredenergyspreadrelatestotheenergyspreadatthe pointi.e.inthemiddleofoneofchicanethankstoanOTRprolemonitor(seeg.5.29).Itis bunchlengthmeasurementstation.Underthevalidityoflineartransfermatrixformalism,itis force<strong>and</strong>wakeeldeects.Theformerhasbeenstudiedbyperformingnumericalsimulationsusing thecodeparmela.Aspicturedingure5.30,wherewecomparethelongitudinalphasespaceat clearthatenergyspreadshouldbethesameoverthewholeregionprovidedtheFEListurnedo. However,wemustbecautiousabouttheapplicabilityoflinearoptics:thereareafeweectsthat chicanemidpoint,theenergydistributionchangeisinsignicant.Thedegradationofenergyspread<br />
thewigglerinsertion<strong>and</strong>theoneatonepotentialpointofmeasurement,namelythedownstream cansignicantlyspoiltheenergyspread.Theprincipaleectsarethelongitudinalspacecharge
inducedbyspacechargeisnotaconcerninthepresentdiscussion.<br />
undulatorvacuumchamberwhichhasarectangularsectionof489mm2. bunchlengthmeasurement<strong>and</strong>theenergyspreadmeasurementstation,thelargestdiscontinuityis introducedbythebeampipesizevariationclosethewigglermagnet:thereisatransitionbetween thest<strong>and</strong>ardbeamvacuumchamber,thathasacircularsectionof50.8mmdiameter,<strong>and</strong>the Thesecondmechanismthatcanpotentiallydeterioratethebeamenergyspreadisduetowakeeld generationasthebeamencountersdiscontinuitiesinthevacuumchamber.Betweenthepointof<br />
functionwrites:!W(x;y;s)=1eZ1 Wakeeldforcesareduetothechangeinboundaryconditionssurroundingaparticle,whichobligate theCoulombeldtoreorganize.Anelectronintheheadofthebunchcancauseanelectriceld atthelocationwheretheboundaryconditionchanges.Anelectronbehindcaninteractwiththis Foranelectrontraversingastructurewithanoset(x;y),withitsvelocityparalleltoz,thewake Ifweassumealltheelectronsarecenteredinthestructure,i.e.x=y=0,theprincipaleect ofwakeeldistointroduceenergyvariationalongthebunchwhichinturncanspoiltherms electriceld<strong>and</strong>therebymodifyitsorbit.Typicallywakeeldsaredescribedbywakefunctions.<br />
energyspread.Thermsenergyspreadinducedbythiseectispurelycomingfromthelongitudinal wakeeld,Wk;itwrites:;wake=e2Z1 1dz!E(x;y;z;t)+cbz^!B(x;y;z;t) (5.63)<br />
where(s)isthebunchdistributionfunction,<strong>and</strong>kkdef energyspreadincreasetothewakeeldeectispresentedingure5.31:itisnoticedthatthe factor. Inthepresentcasethecomputationofthewakeeldisestimatedbyusinga2Dcodetbciwhichalso assumesthebunchisarigidlinechargedistributedalongagaussi<strong>and</strong>istribution.Theexpected 1ds(s)W2k(s)k2k1=2 =1=eR11Wk(s)(s)dsisthetotalloss (5.64)<br />
<strong>and</strong>thecircularvacuumchamberhasbeensmoothedbyintroducinga\trumpet"shapedcopper piece. expectedtohaveasignicantcontributionsincethetotalenergyspreadisthequadraticsumofthe intrinsic<strong>and</strong>wakeeldinducedenergyspread.Moreoverthesteptransitionbetweenthewiggler spread(oftypically50keVat38MeVasachievedinnumericalsimulations).Henceitisnot associatedenergyspreadis,intheworstcase25timeslessthanthebeamintrinsicrmsenergy<br />
Thisismuchlargerthantheexpectedvaluefromtheparmelacode(11:7deg-keV)butstillwithin <strong>and</strong>=0:250:05%whichyieldalongitudinalemittanceofapproximately18:85:5deg-keV. undulatorvicinity<strong>and</strong>istheenergyspreadmeasuredinoneofthehighdispersionlocationsin thechicanes).Typicalbunchlength<strong>and</strong>energyspreadmeasuredduringthecommissioningofthe IRFEL,intheperiodjustpriortorstlaserlightproduction,arerespectivelyz=11030m thelongitudinalemittancereducesto~"z'z(wherezisthebunchlengthmeasuredinthe Henceundertheassumptionthatthereisnomechanismtospoilsignicantlytheenergyspreadof<br />
thespecication33deg-keV.<br />
thebeam,whenthelongitudinalenvelopeisatawaistclosetotheundulatorlocation(.e.hzi'0),
5.8.1Conclusion Inthischapterwehavedevelopedtechniquestomeasurebunchlength.Thesestechniquesincludebothafrequency-basedmethodthatconsistsofmeasuringtheenergyspectrumofcoherent transition(<strong>and</strong>potentiallysynchrotron)radiation,<strong>and</strong>atime-basedmethod.Bothtechniquesare capableofresolvingpicosecond-time-scalebunchlengths.Alongwithbunchlengthmeasurement, someinsightsonthelongitudinalphasespacecanbeobtainedbymeasuringtheenergyspreadfrom which,knowingthebunchlength,thelongitudinalemittancecanbecomputed.
Golay Cell Signal (V)<br />
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totallysuppressedwhilein(B)itwas.(C)givesthedierencebetweenthetwopreviousplots Figure5.14:Completeinterferogramstakenfewminutesapart:In(A),theghost-pulsewasnot<br />
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Figure5.15:FinescanofthecentralpartoftheinterferogrampresentedinFig.5.14(A).The<br />
3<br />
experimentalinterferogram(circle)iscomparedwithaninterferogramgeneratedfromagaussian distribution(solidline)<strong>and</strong>asquaredistribution(dashline)bothofthesedistributionhavethe sameFWHM(A).Theinterferogramiscomparedwithaninterferogramofthesumofasquare<br />
2.5<br />
<strong>and</strong>gaussi<strong>and</strong>istributionbothhavingaFWHMof110m(B).Thebaselineis2.7V.<br />
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Mirror Displacement (microns)<br />
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Figure5.16:InterferogrammeasuredfordierenttransversebeamsizeontheTRradiator.Theleft plotsarethemeasuredinterferogramswhereastherightplotsshowthebeamtransverse(horizontal<br />
2.5<br />
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alongwiththecorrespondingpowerspectrum(B). Figure5.17:Symmetrizationoftheautocorrelationbydierentmethods(A)(seetextfordetail)<br />
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Figure5.18:Energyspectrumobtainedbyconsideringdierentnumbersofdatapointsfromthe<br />
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autocorrelation.Thenumbersareallpowersoftwo,asrequiredbytheFFTalgorithmweused.<br />
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Figure5.19:EnergyspectrumobtainedbyFouriertransforminganautocorrelationwith256data points(solidlinewithsquares)<strong>and</strong>thedierentlowfrequencyextrapolationsconsidered.<br />
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Figure5.20:Recoveredlongitudinalbunchdistributionforthedierentextrapolationofthepower<br />
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driveraccelerator Figure5.21:ExperimentalsetuptomeasurebunchlengthwithzerophasingmethodintheIR-FEL<br />
Cavities tuned off<br />
24 MeV dipole<br />
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To ER dump<br />
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Figure5.22:Horizontalbeamenvelopeevolutionfromtheexitofthefourcavitiesinthecryomodule<br />
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uptothebeamprolestationinthespectrometertransportlinefordierentnumbersofzerophasing<br />
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Figure5.23:picturaleectoflongitudinalspacechargeforceonthephasespacedistribution.<br />
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therstzero-phasingcavityuptotheentranceplaneofthespectrometerdipole.Thenumberof Figure5.24:Longitudinalphasespaceslopeevolutionalongthedriftbetweentheentranceof<br />
0<br />
cavitiesusedforthezerophasingisindicatedbeloweachcurve.Theslopeisnormalizedtothe<br />
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initialenergy.Foreachcasethesimulationisperformedwiththespacechargeroutineinparmela<br />
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turnedon(uppercurve)<strong>and</strong>o(lowercurve).<br />
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rophasingcavitiesFigure5.25:RMShorizontalbeamsizeattheprolemeasurementstationversusnumberofze-<br />
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cavities phased +90<br />
50<br />
40<br />
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0<br />
-10<br />
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-20<br />
-30<br />
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-50<br />
-50 -25 0 25 50<br />
x (mm)<br />
o on-crest<br />
measurement.<br />
Figure5.26:<strong>Beam</strong>spotonthedispersiveOTRmonitorforthethreephasesofthezero-phazing<br />
120<br />
120<br />
120<br />
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x (mm)<br />
x (mm)<br />
x (mm)<br />
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... .. Figure5.27:Typicalzerophasingbasedmeasurement.The2Dfalsecolorimagerepresentsthe<br />
Cavities +90 deg<br />
100<br />
beamspotmeasuredonthedispersiveviewerinthespectrometerlinewhereastherightplotisthe<br />
90<br />
projectionontothehorizontalaxis.Measurementforthecasewhereallthe\zerophasing"cavities<br />
80<br />
areo(topline),arephased-90degw.r.t.themaximumaccelerationphase(middleline)<strong>and</strong>are<br />
70<br />
phased+90degw.r.t.themaximumaccelerationphase(bottomline).<br />
60<br />
50<br />
40<br />
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..<br />
20<br />
-20 -15 -10 -5 0 5 10 15 20<br />
Distance (mm)<br />
Horizontal (energy) axis 40 mm<br />
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-20 -15 -10 -5 0 5 10 15 20<br />
Distance (mm)<br />
. .
σ x (m)<br />
x 10−3<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 5 10 15 20<br />
1<br />
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0.6<br />
0.4<br />
0.2<br />
0 5 10 15 20<br />
Quadrupole Strength (m −2 7<br />
6<br />
5<br />
0 5 10 15 20<br />
1<br />
0.98<br />
0.96<br />
0.94<br />
0.92<br />
0.9<br />
0 5 10 15 20<br />
)<br />
Quadrupole Strength (m −2 <strong>and</strong>=2%(bottomright) beamsizecontributionduetodispersiononlywiththetotalbeamsizefor=0:2%(bottomleft) thebeamsizecontributionduetodispersiononly.Thebottomplotspresenttheratioofthe variationinthecenterofthebunchcompressorversustheexcitationofanupstreamquadrupole Figure5.28:Comparisonfor=0:2%(topleft)<strong>and</strong>for=2%(topright)ofthebeamrmssize usingdimadsimulation(solidlines),usingEqn.(5.57)(crosses).Thedashedlineontheseplotsis<br />
)<br />
Figure5.29:Schematicsoftheavailablediagnosticsintheundulatorregionformeasuringthe longitudinalemittance.<br />
Undulator Magnet<br />
OTR foil<br />
OTR foil<br />
e- beam<br />
from linac<br />
CTR foil<br />
Chicane 1 (Compressor)<br />
Chicane 2 (Decompressor)<br />
σ x (δ)/σ x (no units)<br />
x 10−3<br />
10<br />
9<br />
8
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150<br />
100<br />
.<br />
50<br />
Figure5.30:ComparisonofthelongitudinalphasespaceattheCTRfoil(greydots)<strong>and</strong>the<br />
.<br />
0<br />
bunchlengthattheCTRfoilissmaller(dashedlines)thanatthechicanemid-point(solidlines). decompressormid-point(blackdots).Theenergydistributionareexactlythesamewhilethe<br />
. .<br />
.<br />
-50<br />
-100<br />
-150<br />
-4 -2 0 2 4<br />
(RF-deg)<br />
E (keV)<br />
10<br />
10<br />
0 200 400 600 800 1000<br />
Bunch Length σz (µm)<br />
−2<br />
10 −1<br />
1<br />
Figure5.31:rmsenergyspreadinducedbywakeeldsgeneratedasa60pCbunchofelectrontravels throughthevacuumchambertransitionattheundulatorlocation.<br />
10<br />
Energy Spread ∆E (keV)
<strong>Beam</strong>DynamicsStudies Chapter6<br />
undulatorinthebacklegtransportasitispresentlyenvisionedfortheIRFELUpgrade. ues.Recentlywestartedtostudytransverseemittancegrowthtoassessifwecouldrelocatedthe<strong>and</strong>verifythatthebeamparametersclosetotheundulatorinsertionarewithinthespeciedval- optimize<strong>and</strong>underst<strong>and</strong>thephotoinjector.Theyhavebeenusedtooptimizethebunchingscheme, InthepresentChapterwewouldliketopresentfewapplicationsofthediagnosticswehaveprevi-<br />
6.1StudyofthePhotoinjector ouslydescribed.Essentially,tothepresentdate,theseinstrumentshavebeenhelpfulintryingto<br />
driver-accelerator;everycaremustbetakentopreventanybeamdegradation<strong>and</strong>insurethebeam parametersremainwithinthespeciedvalues.Sincetheinjectortransportslowenergy,highcharge bunches,eectssuchasspacechargehavetobetakenintoaccount. AfewfeaturesoftheDCphotoemission-basedinjector,especiallythebeamgenerationusingthe GaAsphotocathode,havebeendescribedinChapter1.Hereweshallonlyconcentrateonthebeam transportfromthegunexituptothefrontend. Theinjectorbeamlineproispictureding.6.1;itcanbedividedintothreemainregions: Thebeamgeneration<strong>and</strong>lowenergytransportisprobablyoneofthemostcrucialissueforthe<br />
2.Thehighgradientacceleratingstructure,composedoftwoCEBAF-typesuperconducting 1.The350keVtransportlinethatconsistsoftwosolenoidlenses<strong>and</strong>awarmbunchercavity,<br />
Inthissectionwewillstudyeachregion,trytoprovideasimplemodelofbeamevolution,wewill comparewithsimulations,<strong>and</strong>whenpossible,benchmarkwithexperimentalresults<strong>and</strong>provide 3.The10MeVregionthatcanbesubdividedintotwoparts:aquadrupoletelescopethatis chicanethatconsistsofthreebendsarrangedina\staircase"conguration. usedtomatchthetransversephasespaceintothemainlinac<strong>and</strong>anachromaticinjection cavitiescapableofacceleratingthebeamuptoatotalenergyofapproximately10MeV,<br />
tentativeexplanationofdiscrepancies.However,wewillnotdescribethephotoemissionprocess <strong>and</strong>accelerationinthegunchambersinceithasbeentreatedinReference[14]. 137
Solenoidal Lens #2<br />
Accelerating Cavity #4 & #3<br />
Achromatic Chicane<br />
OTR viewer<br />
Figure6.1:SimpliedschematicoftheIRFELphotoinjector(seetextforexplanation).<br />
OTR viewer<br />
Linac Entrance<br />
<strong>High</strong> Voltage<br />
Buncher<br />
Power Supply<br />
GaAs Photocathode<br />
<strong>High</strong> Dispersion<br />
Multislit Mask #1<br />
particlepushingcodeisusedasskeletonforthesimulation[53],theelectrostaticeldinthegun Theinjectorhasbeensubjecttoextensiveintegratedmodelingusingseveralcodes.Theparmela 6.1.1Introduction Thenumericalmodel<br />
Solenoidal lens #2<br />
Multislit Mask #2 OTR viewer<br />
<strong>and</strong>inthesolenoidlensesarecomputedwiththepoissoncode,whereastheelectromagneticeld<br />
pickup cavity<br />
parametershavebeensetaftermanyiterativeoptimizationrunsusingparmela.Theresulting intheRFcavitiesareobtainedfroma3Dmodelusingthemafiafamilycode.Theinjector<br />
Matching Telescope<br />
rmsbeamtransverse<strong>and</strong>longitudinalenvelopesthroughouttheinjectortransport,optimizedfor<br />
betweenthephotocathode<strong>and</strong>theanodecanreach500kV.Ideallyonewouldliketomaintain ThebeamisgeneratedwithaDCphotocathodegunaforementioned.Theacceleratingvoltage agunvoltageof350keVarepresentedingure6.2.<br />
thehighestacceleratingvoltagetominimizethespace-charge-inducedemittancegrowthsincethis forceisproportionalto1=2.Unfortunatelybecauseweencounteredtechnicaldiculties(e.g.eld emissionofthecathodesupport)duringtheguncommissioning,wehadtooperatethegunwith 6.1.2The350keVregion<br />
modelingthatevenwiththisloweracceleratingvoltage,wecouldstillndadequatesettingsto acceleratedinthegun,itsrmstransverse(i.e.radial)beamsizeisstronglydiverging<strong>and</strong>the aloweracceleratingvoltageof350keV(<strong>and</strong>sometime330keV).Itwasassessedvianumerical providetherequiredbeamparametersattheundulatorlocation.Asabunchisemitted<strong>and</strong> bunchiselongatingaspicturedingure6.2.Tocorrectforthestrongdivergence<strong>and</strong>collectall particleofthebunch,asolenoidlenshasbeenlocatedimmediatelydownstreamtheanodeplate.
x,y (mm)<br />
6.0<br />
4.8<br />
3.6<br />
2.4<br />
1.2<br />
x<br />
y<br />
0.0<br />
0 120 240 360 480 600 720 840 960 1080 1200<br />
6.0<br />
5.0<br />
4.0<br />
3.0<br />
2.0<br />
is350keV).<br />
1.0<br />
0 200 400 600 800 1000 1200<br />
Thissolenoid,thatwillbereferredhenceforthasthe\emittancesolenoid",becauseofitsimpact<br />
z (cm)<br />
onthebeamemittance,shouldbeoperatedwiththeoptimummagneticeldtominimizespace- Figure6.2:RMStransverse(x<strong>and</strong>y)<strong>and</strong>longitudinal(z)beamsizesalongtheinjector.The charge-inducedemittancegrowth<strong>and</strong>makesurethereisnobeamlossdownstreamduetoscraping onthevacuumchambers.Typicallytheoptimummagneticelddependsonthechargeperbunch bottomschematicslocatestheopticalelementsalongthebeamline(thegunacceleratingvoltage<br />
GUN BUNCHER CAV1 CAV4 QUAD1 QUAD4 DIP1<br />
DIP3<br />
solenoidarereferredasthebeamgenerationline1.Usually,thisregionisentirelysimulatedwith <strong>and</strong>electronenergy(i.e.acceleratingvoltageofthegun).Generallythegunalongwiththisrst<br />
SOL1 SOL2 CAV2 CAV3 QUAD3 QUAD2<br />
DIP2<br />
numericalmethods:themagnetostaticeld2Dmapinthegun<strong>and</strong>inthesolenoid2arecomputed withmagnetostaticsolvercodesuchaspoisson<strong>and</strong>parmelaisusedtosimulatephotoemission ofelectronmacroparticles<strong>and</strong>trackthemalongthisgenerationline. Inthefollowingwearegoingtoprovideanalyticdescriptionoftheevolutionofthebeamparameters alongthe350keVtransportbeamline.Thisanalyticdescriptionisbasedoncoupleddierential 1thisgenerationlineasbeenthesubjectofaPhDthesisseeReference[14] 2Thegun<strong>and</strong>thesolenoidarecylindricallysymmetricelements<br />
z (mm)
equationthatrequiresomeinitialconditions,thatwewilltaketobeattheemittancesolenoidexit.<br />
Letr<strong>and</strong>zberespectivelythermstransverse<strong>and</strong>longitudinalbeamenvelope.Itiswellknow<br />
(e.g.seereference[54])thatonec<strong>and</strong>escribestheevolutionofthebeamenvelopeviatheso-called<br />
coupledrmsenvelopeequationthatwrite(extendedfromreference[54]):<br />
@2r(s)<br />
@s2+k20r(s)3<br />
10p5Nrc<br />
20301<br />
r(s)z(s) 1g22r<br />
202z!~"2r3r=0 (6.1)<br />
@2z(s)<br />
@s2+k20r(s)3<br />
10p5Nrc<br />
2050g<br />
z(s)2~"z(s)2<br />
z(s)3=0<br />
whereg=g(z=r;b=r)afunctionofthebeamrmssize<strong>and</strong>thevacuumpipediameterb,de-<br />
scribestheeectofthebunchinteractionwithitsimageonthebeamlinevacuumchamber;<br />
rc=e2=(40mc3)istheclassicalradiusofanelectron<strong>and</strong>0isthebunchreducedenergy(from<br />
nowonwewillassumeanenergyof350keV,i.e.0=0.8048<strong>and</strong>0=1.6849.Toconvinceourselves<br />
onthenecessityofusingtheaboveequationsystem,wecanstudythedependenceofthe\space<br />
chargeoveremittanceratio".Forthelongitudinaldirectionwedenethisratioas:<br />
Rrdef<br />
=3<br />
10p5Nrc<br />
30gz<br />
(~"nr)2 (6.2)<br />
Thesamekindoffactorcanbedenedforthetransversedirection:<br />
Rzdef<br />
=3<br />
10p5Nrc<br />
02r<br />
(~"nz)2z 1g22r<br />
202z! (6.3)<br />
Theevolutionoftheseratiosalongthebeamlineusingrmsenvelopenumericallycomputedwith<br />
parmelaareshowningure6.3.Inthe350keVlineitisseenthatspacechargecontributionin<br />
theenvelopeequationcanbeafactor100largerthantheemittancetermcontribution.Evenin<br />
the10MeVregion,thereisstillapredominanceofspacechargetermbyafactor10exceptinthe<br />
bunchingchicanewheredispersionincreasetransversebeamsize<strong>and</strong>thereforelocallyreducespace<br />
chargeforce.Ontheotherh<strong>and</strong>,thelongitudinalratioissignicantlylargerthanunityonlyin<br />
the350keVregion.Itisstronglydampedasthebeamisacceleratedinthe10MeVstructure<strong>and</strong><br />
downstreamthecryounitthelongitudinalenvelopeequationisonlydrivenbytheemittanceterm.<br />
Toapplythermsenvelopeequationtothedierentelementswecanusethefollowingsteps:<br />
foradriftspace,theexternalfocusingparameters,kr<strong>and</strong>kzaresettozero.<br />
thebunchercavityismodeledasa\slopeimpulse":z0buncher<br />
!z0+2RFeV<br />
mc223z<br />
thesolenoidexternalfocusingparameterisestimatedusingtherelationk0=eB0<br />
2mcwhereB0<br />
istheintegratedmagneticeld,whichwehaveestimatedusingapoissongeneratedmagnetic<br />
eldprole.<br />
6.1.3Thehighgradientstructure<br />
Inthissectionwewouldliketodiscussfewinterestingeectsinducedonthetransversebeam<br />
dynamicsbytheCEBAF-typeacceleratingcavities.Thediscussionwillenablethereadertoun-<br />
derst<strong>and</strong>experimentalresultspresentedinthenextsection.
Space Charge over Emittance Ratio (no unit)<br />
10 2<br />
10<br />
0 200 400 600 800 1000 1200<br />
Distance from the Photocathode (cm)<br />
-4<br />
10 -3<br />
10 -2<br />
10 -1<br />
10 0<br />
10 1<br />
Rr tion. EnergyGain Theaccelerationinacceleratingcavitiesisprovidedbythelongitudinalcomponentoftheelectric Figure6.3:\spacechargeoveremittanceratioforthetransverse(Rr)<strong>and</strong>longitudinal(Rz)direc- eldofthefundamentalmode.Sucheldcanbewrittenapproximately:<br />
Rz betweentheparticle<strong>and</strong>theRF-wave.Becauseof<strong>their</strong>energyattherstcavityentrance,350keV, theelectronsarenotrelativistic<strong>and</strong>thereforeoneelectronisnotgoingtokeepthesamerelative phasewithrespecttotheRF-wave,sucheectisnamedphaseslippage.Let'sdenethephase E0isthepeakeld,zisthepositionwithrespecttothecavitycenter,<strong>and</strong>istheosetphase (z)as: Ez=E0cos(kz)cos(!t+)=E0 2(cos(!t+kz)+cos(!t++kz)) (6.4)<br />
TheEqns.(6.5)<strong>and</strong>(6.6)togetherformacoupleddierentialequationsystemthatcanbesolved Moreoverthenormalizedenergygainis: d(z) dz=eE0 (z)def =!tkz=kZz 2mc21(cos((z)+2kz)+cos((z))) 0 p211!dz+ (6.6) (6.5)<br />
numericallyusingst<strong>and</strong>ardtechnique.Figures6.4presentstheenergygaininthetwocavitywith
w.r.t.theRF-wave.ThisfactisaconsequenceofphaseslippagebetweentheRFwave<strong>and</strong>the cavity#4,oneneedstoinjectthebunchwith'40deg. bunchwhichisnotyetrelativistic.Infacttoobtainthemaximumpossibleenergyattheexitof anelectronbeamofinitialenergyof350keV.Itisnotablyseenthatmaximumenergygainprovided bytherstcavity(cavity#4)isnotobtainedbyinjectingthebunchwitharelativephase=0<br />
Reduced Energy Gain (no unit)<br />
2<br />
Cav #4<br />
10<br />
-0.25 -0.125 0.0 0.125 0.25<br />
Distance (m)<br />
0<br />
10<br />
5<br />
2<br />
1<br />
=10 deg.<br />
=-10 deg.<br />
=0 deg.<br />
10<br />
-0.25 -0.125 0.0 0.125 0.25<br />
Distance (m)<br />
0<br />
10<br />
5<br />
2<br />
1<br />
10 0<br />
10<br />
5<br />
2<br />
1<br />
cavity. Radio-FrequencyinducedFocusing Figure6.4:Reducedenergygain,,alongtherst(cavity#4)<strong>and</strong>second(cavity#3)cryounit<br />
=-16 deg.<br />
Thefundamentalmodelongitudinalacceleratingeldalsoinduces,byvirtueofMaxwell'sequations,<br />
=0 deg.<br />
Thisradialeldtogethercombinedwiththeequationofmotion@tp=eEryieldsatransverse equationofmotion,e.g.forthehorizontalplane: aradialelectriccomponentthatwrites: d2x(z) dz2+0(z) Er(z)=r(z) (z)dx(z) dz+d2(z) 2d dzEz dz2x(z)=0 (6.8) (6.7)<br />
Thisequationcanbesolvednumerically,butsomeapproximatedsolutionhavebeenderivedby Chambers[58,57],forpure-modeacceleratingcavities,<strong>and</strong>areinverygoodagreementwiththe<br />
2<br />
Cav #3<br />
2
numericalsolution[56].Thisapproximatesolutioniswrittenintermoftransfermatrix(e.g.in thex-x0phasespace)as: M=0@cos()p2cos()sin() 0icos() p2+1 p8cos()sin()ifcos()+p2cos()sin1A p8i0cos()sin()<br />
elementsattheexitofthecavityofthecavityarerelatedtothebeammatrixelementatthecavity istheaveraged(overtheRFstructure)energygradient:0=eG <strong>and</strong>is,asusual,thephaseoftheinjectionoftheparticlewithrespecttotheon-crestphase.0 estimatedinastraightforwardfashion:usingrstordermatrixformalism,the11beammatrix wherei;faretheinitial<strong>and</strong>nalreducedLorentzfactors,theangleis=1 mc2cos().Thefocallengthcanbe p8cos()ln(f=i) (6.9)<br />
entranceby:<br />
Sincethefocallengthisdenedbythelengthfwherewehaved(ff) Afteradriftalongadistancel,thebeamsizewrites: 11=m211(0) (f) 11,(f) 11=(f) ff12=m11m21(0)<br />
112l(f) 12+l2(f) 11,(f) 22 22=m21(0) 11=dl=0,ityields: 11 (6.11) (6.10)<br />
notthecase:intheCEBAF-typecavities,forinstance,thereareasymmetriesinthevicinityof thehighordermode(HOM)<strong>and</strong>theforwardpower(FP)couplers.Theseasymmetries,inturn, inducedtransverseelectromagneticelds.Thusitrequiresacomplete3Dmodeltoaccurately studytheeectofthesecouplersonthebeamdynamics.Sucha3Dmodelisreadilyavailable Unfortunatelythismodelisderivedassumingperfectaxi-symmetricRFstructurewhichisgenerally fdef =(f) 22=m11 11 (f) m12cosp2cossin 0fsinp2+q181 cossin (6.12)<br />
beamsizethatisequaltothehallowradius,<strong>and</strong>zero-emittance.Aftertheacceleratingcavitiesthe parameterf11<strong>and</strong>f22arecomputed<strong>and</strong>thefocallengthisdeducedusingtheequation6.12.The <strong>and</strong>hasbeenimplementedintheJeersonLabversionofparmelausing3Delectromagneticeld resultscomputedforthetwocavitiesintheinjector,takingintoaccountnon-relativisticeect,are presentedingure6.5. mapgeneratedwiththeeigensolvermafia[64].Inordertocharacterizethefocusingeectofthe cavitywegenerateahallowsheetbeaminthexyspatialcoordinatespacewithzerodivergence<br />
Radio-FrequencyinducedSteering (i.e.x0=y0=0forallmacroparticleinthebeam).Thepropertiesofthiskindofbeamhasa<br />
yieldemittancedilutionviatwoeects:theheadtaileect<strong>and</strong>theskewcoupling.Theformeris<br />
Inasimilarfashionwehavestudied,fortheinjectorcavities,theRF-kickeectonthebeamcentroid.Thekickimpartedduetothepresenceoftransverseeldintheacceleratingstructureversus thephaseoftheelectronbunchwithrespecttotheRF-waveareplottedingure6.6. TheRF-inducedkickduetothepresenceoftheforwardpower<strong>and</strong><strong>High</strong>ordermodecouplerscan
20<br />
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. ..<br />
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15<br />
10<br />
5<br />
0<br />
. . .. .. . .<br />
.<br />
-5<br />
.<br />
-10<br />
.<br />
hor.<br />
-15<br />
ver.<br />
.<br />
.<br />
-20<br />
-100 -80 -60 -40 -20 0 20 40 60 80 100<br />
(RF-deg)<br />
. ..<br />
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.<br />
...<br />
..<br />
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. . . . .<br />
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............ . .. . . . . .<br />
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. .<br />
. ....... ...<br />
. ..<br />
. . . . 9<br />
Energy .<br />
8<br />
.<br />
7<br />
Figure6.5:Focallengthoftherst(cavity#4)<strong>and</strong>second(cavity#3)cryounitcavitiesversus<br />
6<br />
.<br />
5<br />
.<br />
4<br />
Energy<br />
duetothefactthatthekickisdependentontheparticlepositioninsidethebunch,<strong>and</strong>therefore yieldsadierentialmotionbetweenthehead<strong>and</strong>thetailofthebunch.Thisinturnreultsin<br />
3<br />
.<br />
anincreaseofthebunch(projected)emittance.Itisstraightforwardtoestimatetheemittance<br />
hor. 2<br />
growthduetothiseect:itdependsonthebunchlength<strong>and</strong>thebeamparametersatthelocation<br />
.<br />
oftheconsideredcoupler.Thegeneralexpressionfortheemittancegrowthis[62]: ,thephasedierencew.r.t.themaximumenergyphase(so-called\crestphase")<br />
1 ver. .<br />
0<br />
~"x'z<br />
-200 -150 -100 -50 0 50 100 150 200<br />
2~"x0A2x+B02 x (6.13)<br />
(RF-deg)<br />
thecorrespondingeectonemittancegrowthissmallerthanthehead-taileectinduceddilution emittanceistheskewcouplingeectinducedbytheskewquadrupolemomentoftheelectriceldin thecavity:itintroducesacouplingbetweenthevertical<strong>and</strong>horizontaltransverseplane.Generally <strong>and</strong>itcanbereducedwithapropercorrectionscheme[62];thislattereectwillbeignoredinthe forthcomingdiscussion. atthecouplerlocation,<strong>and</strong>A<strong>and</strong>Baccountsforthesteeringeectsofthecavity,theselattertwo termscanbeestimatedusingtrackingcode.Theothercontributionthatcanspoilthetransverse where~"0istheinitialemittance,z,x<strong>and</strong>0xarerespectivelythermsbeamsize<strong>and</strong>divergence<br />
6.1.4The10MeVRegion Aspreviouslymentioned,wehaveinstrumentedthe10MeVregionwithavarietyofdiagnostics devices:beamdensitymonitors,emittancemeasurementdevices,atime-of-ightpickup<strong>and</strong>a<br />
Focal Length (m)<br />
Cav #4<br />
Cav #3<br />
10<br />
E out (MeV)
0.03<br />
.<br />
....<br />
. .. ...<br />
. ..<br />
..... . ..... 0.02<br />
0.01<br />
...<br />
0.0<br />
.<br />
. . -0.01<br />
.<br />
. .<br />
-0.02<br />
hor. . .<br />
ver. . .<br />
-0.03<br />
-150 -100 -50 0 50 100 150<br />
(RF-deg)<br />
.....<br />
.<br />
.<br />
. .<br />
.<br />
.<br />
. . . . . . .<br />
. . . ......<br />
50<br />
Figure6.6:SteeringeectduetoFP<strong>and</strong>HOMcouplerinthecryounitversus,thephase<br />
. . . . ... 0<br />
surementswithnumericalsimulations<strong>and</strong>attempttoexplainpotentialdiscrepancies<strong>and</strong>trytoelectromagneticbeampositionmonitor.Therefore,wecantrytocomparetheexperimentalmea- dierencew.r.t.themaximumenergyphase(so-called\crestphase"). renethemodeltoincorporatetheeectsthatmayaccountforthesediscrepancies.<br />
-50<br />
hor.<br />
ver.<br />
Transverse<strong>Beam</strong>Properties<br />
-100<br />
-150 -100 -50 0 50 100 150<br />
(RF-deg)<br />
themultislitsmaskrevealedemittance70%largerthansimulated.Manyparametricstudieswhere conductedtodetermineifthisemittancegrowthcouldbeminimizedbyuseofavailableparameters: themagneticeldintherst<strong>and</strong>secondsolenoids,<strong>and</strong>thebunchergradient.Mostofoureort Asarstqualitativeobservation,wemeasuredbeamdensityproleontheOTRdensitymonitor beamdensityisshowningure6.7.Thebeamshapearequitesimilar,<strong>and</strong>rmsvalueareinagreementatthe30%level:simu locatedjustdownstreamtheacceleratingstructure.Acomparisonofthemeasured<strong>and</strong>expected<br />
adjustbecauseitalsoaectthebunchingscheme<strong>and</strong>someotheradjustment(acceleratingcavities <strong>and</strong>phasesalongthelinac)arethenneededtopreserveanultrashortbunchlengthinthemain concentratedonthesolenoidlenses<strong>and</strong>thebunchergradient,alsothislatterknobisdicultto rstmeasurementsoftransverseemittancebasedonthephasespacesamplingmethodutilizing x'simu y=1:47mm,<strong>and</strong>exp x=1:77,<strong>and</strong>exp y=1:60mm.However<br />
linacattheundulatorlocation.Anexampleofparametricstudyofthetransverseemittanceversus the\emittance"solenoidmagneticeldispresentedingure6.8.Thisgureclearlyrevealsthe injectoremittancesaredegradedcomparedtotheonepredictedvianumericalmodeling.Onehypotheticexplanationofsuchadisagreementisathemisalignmentofthecryounitwithrespecttothe Angular kick (rad)<br />
Cav #4<br />
Cav #3<br />
100<br />
Equivalent Dipole Strength (G.cm)
15<br />
200 15<br />
180<br />
(A)<br />
(B)<br />
10<br />
10<br />
160<br />
structure(A)withparticlepushingnumericalsimulation(B).<br />
140<br />
5<br />
5<br />
120<br />
beamaxis.Suchhypothesishasbeeninferredfromotherobservationsduringthecommissioning<br />
0<br />
100 0<br />
80<br />
ourdesiretoobtainamorerealisticmodeloftheIRFELelectronbeamtransport,weincludedthe Figure6.7:Comparisonofthetransversebeamdensitymeasuredattheexitoftheaccelerating<br />
−5<br />
−5<br />
60<br />
misalignmentinparmela<strong>and</strong>getemittancevaluesclosertowhatweexperimentallymeasured. misalignmentcurvewasgeneratedbysteeringthebeamby50mradupstreamthecryomodule.In oftheinjectorincludinganoxiousorbitthatcouldnotbestraightened3out.Inthegure6.8,the<br />
40<br />
−10<br />
−10<br />
20<br />
−15<br />
0 −15<br />
EnergySpread<br />
−15 −10 −5 0 5 10 15<br />
−15 −10 −5 0 5 10 15<br />
Horizontal Axis (mm)<br />
Horizontal Axis (mm)<br />
measurementofthermshorizontalbeamsizegivesaccesstothequantity: Theenergyspreadcanbeestimatedbymeasuringthehorizontalbeamproleatthehighdispersion OTR(seegure6.1).Atthislocationthedispersioniscomputedtobe'42cm<strong>and</strong>thereforea<br />
thehighdispersionOTRdensitymonitor.Fromtheseimageswecomputethermsvaluesforthe thislatterstatementisindeedalsoveriedvianumericalsimulationusingthecodeparmela.In Usingtheroutinelymeasuredrmshorizontalemittance(~"x'5-6mm-mrad<strong>and</strong>betafunction gure6.9wepresenttheeectofthebuncherelectriceldonthebeamOTRimagerecordedon x5m)wededucethatthedispersivecontributiontothebeamhorizontalsizeisdominant, x=sx~"x+EE2 (6.14)<br />
horizontal(energyaxis)projection<strong>and</strong>compare<strong>their</strong>valueswiththeoneexpectedfromnumerical simulationingure6.10.Theagreementisseentobereasonable,i.e.within30%,exceptforlarge bunchervoltage,thoughinthisrangewebelievethebeamwasscrapingonthebeamlinevacuum chamberupstreamthelocationofthemeasurement. 3Unfortunatelybecauseoftimeconstraintsthisproblemwasnotaddressed<br />
Vertical Axis (mm)
ε x (mm−mrad)<br />
Nominal<br />
6<br />
4<br />
2<br />
cryounitexitversustheemittancesolenoidmagneticeld.<br />
10<br />
Misaligned<br />
9<br />
Phase-PhaseCorrelationMeasurement Figure6.8:Measured(squares)<strong>and</strong>predictedtransverseemittances(solid<strong>and</strong>dashlines)atthe<br />
8<br />
7<br />
6<br />
5<br />
4<br />
Nominal<br />
3<br />
2<br />
RF-inducedsteeringwhich,inturn,dependsontheinjectionofthebuncheswithrespecttothe<br />
3000 3200 3400 3600 3800<br />
RFacceleratingeldinthecryounit.Henceatthepickupcavitylocation,theTOFofthebunch SimilarlytothebunchcompressioneciencydiagnosticsdescribedintheChapter3,theinjectoris instrumentedwithapickupcavitythatcanbeusedtomeasurethetimeofarrivalofabunch<strong>and</strong> consequentlydeterminephase-phasecorrelationbymodulatingtheRF-phaseofthephotocathode drivelaser.UnfortunatelyinthecaseoftheFELinjector,thepickupcavityislocatedinadispersive regionwhichrendersthetime-of-ight(TOF)dependentonotherparameters4,inparticularto<br />
Solenoid Magnetic Field (Gauss cm)<br />
centroidemittedatthephotocathodesurfacewithinitialphaseiwithrespecttothe\zerocrossing"bunch(i.e.thebunchforwhichtheTOFiszerobydenition)is: whereRFistheTOFcontributionduetotheRF-inducedsteering:RF=Rc!f ;c,i,<strong>and</strong>farerespectivelythelocationofthecryounitexit,thephotocathodesurface,<strong>and</strong>the pickupcavity. TheRFsteeringisdueto(1)theRFtransverseequationofmotioninanacceleratingcavity,<strong>and</strong> (2)theforwardpower<strong>and</strong>highordermodecouplerinducedkicks.Toquantifythecontribution oftheRF-steeringinthephase-phasecorrelationmeasurement,wehavecomparedphase-phase f=RF+Ri!f 55i+Ri!f 56i 51xc+Rc!f 52x0c (6.15)<br />
correlationmapsgeneratedvianumericalsimulationusingparmelawithtwodistinctmodelsof 4ThenecessityofstudyingcarefullythispointwasbroughttomyattentionbyD.R.Douglas<br />
ε y (mm−mrad)<br />
12<br />
10<br />
8<br />
Misaligned
Vertical axis (arbitrary units)<br />
100<br />
150<br />
200<br />
250<br />
300<br />
350<br />
100 200 300 400<br />
100<br />
150<br />
(D)<br />
100<br />
(A) (B) (C)<br />
150<br />
200<br />
250<br />
300<br />
350<br />
100 200 300 400<br />
100<br />
150<br />
100<br />
150<br />
200<br />
250<br />
300<br />
350<br />
100 200 300 400<br />
200<br />
200<br />
200<br />
250<br />
250<br />
250<br />
Figure6.9:<strong>Beam</strong>densitymeasuredonthehighdispersionOTRmonitorforninedierentbunch<br />
300<br />
300<br />
300<br />
350<br />
350<br />
350<br />
100 200 300 400 100 200 300 400 100 200 300 400<br />
100<br />
100<br />
100<br />
(G)<br />
(H)<br />
(I)<br />
150<br />
150<br />
150<br />
200<br />
200<br />
200<br />
250<br />
250<br />
250<br />
thereisnotsignicantdierencebetweenthegeneratedtransfermapsexceptsomebroadeningin fromthelowgradientvalues) theCEBAFcavity:a3Dmafiamodel(whichincludesthecoupler-inducedeects)<strong>and</strong>a2Dcylindricalsymmetricsuperfishmodel.Theresultsarepresentedingure6.11,whichshowsthat gradientsettings(theimages(A)to(I)correspondstothepointspresenteding.6.10starting<br />
300<br />
300<br />
300<br />
350<br />
350<br />
350<br />
100 200 300 400 100 200 300 400 100 200 300 400<br />
thecavitywaslocateddownstreamthecryounitafteradriftofsimilarlengthtoitspresentlocation intheinjectorchicane;sincethecalculationcorrespondstoadispersion-freedrift,thistransfermap givesinsightsontheeectsoftheRFintheaboveequation.Wecanclearlyobservethatthis tocouplers).Inthesamegurewealsocomparethephase-phasecorrelationpatterngeneratedif thecasetheofthemapgeneratedfromthe3Dmafiamodel(whichincorporatestheRF-kickdue<br />
Horizontal axis (arbitrary units)<br />
eectdoesnotwashouttheTOFvariationduetoenergychanges;smalleectsareobservable<br />
Anapplicationofthistypeofmeasurementwastondtheproperoperatingpointofthebuncher<br />
accelerator[22]thatconsistsofcomparingthephase-phasepatternexperimentallymeasuredwith onenumericallygeneratedforanidealsetup. inChapter3).Infact,wecanuseasimilartechniquetotheonealreadyinuseintheCEBAF onlyforlargephotocathodedrivelaserphase.Despitethefactthattheeectissmall,itprevents usfromextractingquantitativeinformationfromthemap(i.e.byperformingnonlineartsas<br />
100<br />
(E) (F)<br />
150
2.2<br />
2<br />
1.8<br />
Figure6.10:Comparisonofthermstransversehorizontalbeamsizemeasuredinthehighdispersion<br />
1.6<br />
OTRmonitorwiththeoneexpectedfromparmela.<br />
1.4<br />
1.2<br />
PARMELA<br />
1<br />
0.8<br />
theinjectorfrontendmatchedtheexperimentallyestimatedkineticenergyof9.56MeV(inferred cavityelectriceld:devisinga\good"bunchingschemeintheinjectorisathreeparametersprob-<br />
0.1 0.2 0.3 0.4 0.5<br />
fromthestrengthoftheinjectionchicanedipoles);duringthesimulationsthecavity#4wasset werenotpreciselycalibrated).Sincethecavity'scontrolelectronics<strong>and</strong>softwareareidentical,<strong>and</strong> becausecavity#4isoperatedatagradient30%higherthancavity#3,weestimatedthegradient withthehelpofparmela:usingthiscodewehavevariedthegradientofthetwocryounitcavities (keepingcavity#4atagradient1.3timeshigherthancavity#3)untilthesimulatedenergyat thebuncher,cavity#4,<strong>and</strong>cavity#3gradients(atthetimeofthemeasurement<strong>their</strong>gradients lemwithonepossiblemeasurement:thephase-phasecorrelationpattern.Thethreeunknownsare<br />
Buncher Gradient (MV/m)<br />
modelishenceforthtermedas\ideal"injector. Firstseriesofmeasurementsperformed pointsatthattime).Thebuncherwasoperatedatagradientof0.32MV/m.Theso-generated tooperateoncrestwhilethecavity#3was-20dego-cresttoreecttheexperimentaloperating<br />
gunchamberwhichinturnwillremovethecesiumfromthephotocathodewafer.<br />
Wehaveinvestigatedthebunchereectonthe320kVgunsetup5aftertheinjectorRF-phases alongwiththepatterngeneratedwithparmelafortheidealinjector.Fromthisgurewedecided wereproperlyreset.TheoptimizedphasearegatheredinTable6.1.4.Thebunchergradientwas initiallysetto0.28MV/m,<strong>and</strong>weinvestigatedtheeectonthe\R55"transfermappatternby systematicallyvaryingthebunchergradient.Therecordedpatternsarepresentedingure6.12 5Atthetimeofthepresentexperimentwewerelimitedtothisvoltage;highervoltagewouldinducearcinginthe<br />
σ x (mm)<br />
MEASUREMENT
4<br />
2<br />
thepickupcavityislocateddownstreamthecryounitafteradriftinadispersion-freeregion. <strong>and</strong>2Dsupersh)atthepickupcavitylocatedintheinjectionchicane<strong>and</strong>(2)forthecasewhere Figure6.11:\R55"transfermapgeneratedwithparmelafor(1)dierentcavitymodel(3Dmaa<br />
0<br />
3D−MAFIA model<br />
−2<br />
2D−Superfish model<br />
MAFIA + Dispersion Free<br />
Table6.1:Nominalinjectorsettingsbeforetherstseriesmeasurement. Buncherzero-crossing0.28MV/m ElementsPhase Laser Cavity#4on-crest Cavity#3-20dego-crest9.10MV/m referencephaseN/A Gradient 11.8MV/m<br />
−4<br />
−40 −20 0 20 40<br />
Drive Laser Phase (RF−Deg)<br />
tooperatethebuncherat0.32MV/msinceitprovidesapatternveryclosetothesimulatedone.<br />
AsecondseriesofmeasurementwasperformedaftertheFELphotonbeamwasoptimizedfor imentalphase-phasecorrelationpatternwiththesimulatedoneoccursforabunchergradientof approximately0.32MV/m;suchagradientdoesnotcorrespondtotheminimumenergyspreadon thedispersiveviewer. Secondseriesofmeasurementsperformed Itisseenfromthegure6.10thatthe\best"bunchergradientdevisedbymatchingtheexper-<br />
60pC/bunch6.Thevalueforthephases<strong>and</strong>gradientsrecordedbeforethemeasurement,after theFELwasoptimized,aregatheredintable6.2.Thephase-phasepatternmeasurediscompared ingure6.13withthe\ideal"oneobtainedbynumericalsimulations.Theagreementbetweenthe measurement<strong>and</strong>simulationisexcellent.Thisagreementvalidatesthemethodforsettingup thelongitudinaldynamicsmanipulationintheIRFELinjector.Basicallythetechniqueconsistsof 6thisoptimizationconsistedofmaximizingtheoutputFELpowerbyvaryingthebunchergradient<br />
T−O−F. at Pickup cavity (RF−Deg)
6<br />
4<br />
2<br />
Figure6.12:\R55"transfermapfordierentexperimentaloperatingpointsofthebunchergradient. Themeasurementsarealsocomparedwiththe\ideal"injectordevisedfromnumericalsimulations. (Firstseriesofmeasurement)<br />
0<br />
−2<br />
gradient=0.28<br />
−4<br />
gradient=0.30<br />
gradient=0.32<br />
Simulation<br />
Table6.2:Nominalinjectorsettingsbeforethesecondseriesofmeasurement. Buncherzero-crossing0.32Mv/m ElementsPhase Laser Cavity#4on-crest Cavity#3-20dego-crest9.10MV/m referencephaseN/A Gradient 11.8MV/m<br />
−6<br />
−40 −20 0 20 40<br />
Drive Laser Phase φIN (RF−Deg)<br />
reproducingthe\optimum"correlationpatternbyonlyplayingwiththebunchergradient. 6.2BunchCompressionStudiesintheLinac Uptonowwehaveonlymentionedthecompressorchicane,withoutmuchdiscussion.Thischicane consistsoffourrectangular-edgedipole-magnetsarrangedsymmetrically.Thoughitspurposeis tobypasstheopticalcavityoftheFEL,italsoservesasabunchcompressorusingthest<strong>and</strong>ard longitudinaltransfermatrixofthechicane<strong>and</strong>letseehowhz2ipropagatefromthechicaneentrance<br />
magneticcompressionscheme.Again,asimplewayofdemonstratingtheprincipleistopropagate thegeneralizedmomentsofthelongitudinaldistributionacrossthechicane.LetRbethe22<br />
T−O−F. at Pickup Cavity φ OUT (RF−Deg)
4<br />
2<br />
0<br />
R55patterngeneratedwithparmela(usingthesecondsetup). (positioni)toitsexit(positionf): Figure6.13:ComparisonofthemeasuredR55patternaftertheFELwasoptimizedwiththe\ideal"<br />
−2<br />
Measurement<br />
Simulation<br />
−4<br />
whichindeedessentiallymodifythecorrelationtermhzii.Hence,undertheassumptionthatthe<br />
−6<br />
bunchisnotsignicantlyvaryingwiththelinacsetpoint,theminimumbunchlengthisgivenby wherewehaveusedthedenitionofrmslongitudinalemittancetowritethefarrightexpression. Inthepresentconguration,theparameterattheentranceofthechicanearesetbythelinac; hz2if=R255hz2ii+2R55R56hzii+R256h2ii=R255hz2ii+2R55R56hzii+R256~"2z+hzi2i hz2ii(6.16)<br />
−40 −20 0 20 40<br />
Drive−Laser Phase φIN (RF−Deg)<br />
caneispresentedingure6.14.Experimentallythetypicalbunchlengthmeasuredintheundulatorcomputedaccordingtoparmelafromthephotocathodesurfaceuptotheexitofthesecondchiditionforachievingminimalbunchlengthafterthecompressorsystemis(dE=dz)i=1=R56.By settingthelinacphase<strong>and</strong>acceleratinggradientsuchthattheaforementionedmatchingcondition isfullled,<strong>and</strong>becauseoftheinitialemittance,theminimumbunchlengthachieved,vianumerical simulations,attheundulatorlocationisabout140m(rms).Theevolutionofthebunchlength introducethelongitudinalphasespaceslope(dE=dz)i=hzii=hz2ii;thereforethe\matching"con- thecondition@hz2if=@hzii=0,whichgivestherelationhzii=hz2ii=1=R56;bydenitionwe<br />
vicinityisapproximately100m(rms),anexampleofreconstructedbunchshape(usingtheinverse HilberttransformtechniquedetailedinChapter5)iscomparedwithabeamprolegeneratedwith parmelaingure6.15(C).Withinthelevelofaccuracy7,wehavegoodagreement.Byvaryingthe phaseofthelinac(therebychangingthequantityhzii)wecanvarythebunchlength;wepresent theresultofsuchanexperimentingure6.15(B)alongwithresultsfromnumericalsimulation(in donotgiveauniquelongitudinaldistribution,<strong>and</strong>therearealotofassumptionsthathavetobeconsidered(e.g. extrapolationoftheCTRspectrumatlowfrequency,...)<br />
7(1)weonlyused500macroparticlesinthenumericalsimulation,(2)theHilberttransformation,aswehaveseen<br />
T−O−F at Pickup Cavity φ OUT (RF−deg)
10 1<br />
5<br />
10<br />
0 500 1000 1500 2000 2500 3000 3500 4000<br />
Distance from photocathode (cm)<br />
-1<br />
10<br />
5<br />
2<br />
0<br />
2<br />
thesecondchicane. Figure6.14:rmsbunchlengthevolutionalongtheIRFELfromthephotocathodeuptotheexitof<br />
Injector Chicane<br />
Chicane<br />
plot(B));forcompletenesswealsoincludeingure6.15(D)thetotalCTRpowerdetected.Wend<br />
Aswementionedintheverybeginningofthisreport,onemotivation,foroperationalpurpose,of tothemeasuredvariation. 6.3<strong>Beam</strong>ParametersMeasurementPriorto\Firstlasing" thatthebunchlengthpredictedvianumericalmodelinghasalesspronouncedvariationcompared<br />
sucientbeamqualitytoenabletheFELtooperatehowevertheachievedparametersaresomewhat largerthantheonethatcouldbetheoreticallyreachedaspredictedfromnumericalmodeling,except theexperimentallyachieved<strong>and</strong>thenumericallyexpectedvalues.Itisseenthatwehaveachieved damagedtheundulatorbecauseofradiationshowersinducedastheelectronbeam"scrapes"on thevacuumchamber.Intable6.3wepresentsomeofthebeamparametersrequiredalongwith magnetwasremoved,themainreasonbeingthatduringthis\tuningperiod"wecouldhave thediagnosticsdevelopedhereinistoverifytheelectronbeamqualityiswithinthespecications<br />
forthebunchlength.Withtheaboveparametersrstlightwasachievedatlowdutycyclewithina toenabletheFELtolase.IntheearlystageofthecommissioningoftheIRFEL,theundulator<br />
betweenthenumericalmodel<strong>and</strong>theachievedparameter.<br />
beam.Onthe<strong>Beam</strong>Physicspointofviewitisinterestingtotrytounderst<strong>and</strong>thediscrepancies coupleofhoursafterweinstalledtheundulatormagnet,<strong>and</strong>twodayslaterwewereabletooperate theFELwithanoutputpowerof150W(cw),therebydemonstratingthequalityoftheelectron<br />
RMS bunch length (mm)<br />
Wiggler Location
σ z (mm)<br />
0.2<br />
0.1<br />
Figure6.15:Variationofthebunchlengthversusthelinacacceleratingphaseperparmelasimu-<br />
0.1<br />
0<br />
−15 −10 −5 0 5 6 7 8 9<br />
lation(A)<strong>and</strong>measured(B)["measured"meansthebunchdistributionwasrecoveredfromthe<br />
∆φ (Deg) (parmela)<br />
∆φ (Deg) (experiment)<br />
CTRautocorrelationusingthetechniquementionedinChapter5].Comparisonbetweentheex-<br />
4<br />
3<br />
(C) Simulation<br />
(D)<br />
3<br />
2<br />
2 Measurement<br />
1<br />
1<br />
extentenergyspread)degradationinthebendingsystemoftheIRFEL.Forsuchapurposeswe perimental<strong>and</strong>simulatedbunchlongitudinaldistribution(C).TotalCTRpowersignalmeasured<br />
0<br />
0<br />
needtoconsiderinthefewfollowingsectionswhatarethemechanismsthatcanleadtoanincrease duringplot(B)experiment(D).<br />
−0.4 −0.2 0 0.2 0.4 5 6 7 8 9<br />
oftheemittance.Itseemswec<strong>and</strong>ividesuchmechanismintotwocategories:therstonearedue Inthislastsectionwewouldliketoreportonanattempttomeasureemittance(<strong>and</strong>toalesser 6.4StudyofPotentialEmittanceGrowth<br />
z (mm)<br />
∆φ (Deg) (experiment)<br />
tothelattice(betatronmismatch,lamentation,chromaticity);thesecondtypeareduetobunch selfinteraction(becauseofCoulombeldorradiationeld). Agrowthofemittancecomesfromanincreaseofoneoftheposition<strong>and</strong>/ordivergence,i.e.xorx0 inthex-x0phasespace.Let'sstartwithaninitialemittanceattheentranceofabeamlinesection Ifweassume,thatduetoperturbationinthebeamlinesection,theposition<strong>and</strong>divergenceofa particlechangeaccordinglyto: ~"0:8<br />
8Wevoluntaryomitthesubscriptxfortheemittance.inthissection<br />
~"0=hx20ihx02ihx0x0i21=2 x0!x0+x (6.17)<br />
Bunch Population (a.u)<br />
0.4<br />
0.3<br />
(A)<br />
σ z (mm)<br />
CTR signal (Volts)<br />
0.4<br />
0.3<br />
0.2<br />
(B)<br />
x0!x0+x0 (6.18)
EnergySpread(%) BunchLength(mm) Parameter Emittances(mm-mrad) LongitudinalEmittance(deg-keV)
0.000125<br />
−0.000125<br />
0.000125<br />
Figure6.16:Phasespacedistortionduetochromaticaberrationatthedecompressorchicaneexit<br />
x−x’<br />
(C) y−y’<br />
(D)<br />
acceleratingphasesincewewillattempttomeasurethetransversehorizontalemittanceversusthe linacgangphase9.Thesimulationswereperformedwithparmela;<strong>and</strong>theresultsshowingthe (topplots)<strong>and</strong>thearc1exit(bottomplots).<br />
dependenceofthebeamparametersversusthelinacphasearepresentedingure6.18;thelinac 6.4.2RF-eects Inthehighenergyregion,wehaveinvestigatedthechangeinparametersduetovariationof<br />
−0.000125<br />
energyduringthesesimulationswassetto38MeV<strong>and</strong>thegangphasethatprovideminimum<br />
Thegeneralclassofeectsthatcanleadtoenergyspreadarebunchselfinteractionviaselfeld atthelinacexitdonotdependontheacceleratingphase. 6.4.3EnergySpreadinducedinaDispersiveregion bunchlengthattheundulatorlocationis'9:5deg.Bothhorizontal<strong>and</strong>verticalemittances<br />
radiation).WehavealreadybrieyconsideredlongitudinalwakeeldattheendofChapter5<strong>and</strong> (spacecharge)orradiationeld(wakeeldduetovacuumchamberirregularities,synchrotron 9AreminderthatinAppendixDweprovideaschematicsoftheRF-controlsystem<br />
−0.000125<br />
(A) (B)<br />
0<br />
x−x’ y−y’<br />
0.000125<br />
−0.00025<br />
0<br />
0.00025
15<br />
Effect of RMS energy Spread<br />
10<br />
Figure6.17:Emittancegrowthduetochromaticaberrationversusthemomentumspread<strong>and</strong><br />
5<br />
bendssystem)cancoupletothetransverseplane<strong>and</strong>yieldemittancedegradation.Thelinearized energyosetofthebeam.<br />
Effect of Energy Centroid Offset<br />
equationofmotionofanelectroninabend,assumingnoexternalfocusing<strong>and</strong>nocouplingbetween thetwotransversephasespaces,forthebendingplane(thex-x'planeinthecaseoftheIRFEL) is[59]: haveshowthatitisverysmallforourbeamparameters.Infactallthebeamlinecomponentshave beenspeciedinsuchawaythatthetotallongitudinallossfactoriswithinsomeimpedancebudget sothatthepotentialbeamdegradationisverysmall[65]. Inthissectionwewouldliketoshowhowenergyspreadgeneratedinadispersiveregion(i.e.a<br />
0<br />
0 0.5 1 1.5<br />
δ (%)<br />
radiusofcurvatureofthetrajectory<strong>and</strong>istherelativeenergyosetw.r.t.thereferenceorbit. wheresisthelongitudinalpositionreferredw.r.t.theentranceofthebendingsystem,xisthe Let'sassume,forsimplicity,thatthemechanismgeneratesanenergyspreadthatisonlydependent onthecurvilinearcoordinates10,i.e.=(0)+(s).Undersuchanassumption,thelatter equationtakestheform: d2x<br />
10generallyspeakingthemechanismisalsodependentontime,foracompletetreatmentsee[69]<br />
d2x ds2+x2x=(0)+(s)<br />
∆ε/ε (%)<br />
ds2+x2x=x (6.22)<br />
x (6.23)
60<br />
0.25<br />
40<br />
20<br />
emittances~"x;y,<strong>and</strong>rmsbeamsizesx;y)versustheoperatingacceleratingphaseofthelinac. Figure6.18:Evolutionofbeamparameters(bunchlengthz,rmsenergyspreadE,transverse<br />
0<br />
0<br />
6<br />
1<br />
x<br />
x<br />
4<br />
y<br />
0.5<br />
y<br />
2<br />
thelatterequationclearlyshowsthat,comparedtotheconstantenergyspreadequation,thereis whichcanbesolvedusingthest<strong>and</strong>ardGreenfunctionperturbativetechnique[59]11Thesolution<br />
0<br />
0<br />
anincrementinangle<strong>and</strong>positionof: oftheaboveequationis: x(s)=cos(s=x)x(0)+xsin(s=x)x0(0)+x(1cos(s=x))(0)+Zs x0(s)=1=xsin(s=x)x(0)+cos(s=x)x0(0)+cos(s=x)(0)+Zs 0xsin(~s=x)(~s)d~s 0cos(~s=x)(~s)d~s(6.24)<br />
−20 −15 −10 −5 0<br />
−20 −15 −10 −5 0<br />
∆φ (RF−Deg)<br />
∆φ (RF−Deg)<br />
achromaticbendingsystem,theachromaticcharacterisbrokenbecauseofEqn.(6.25).Another interestingpointisthatdependingonthebendingsystemdesign,onecanconceiveawayofmaking <strong>and</strong>hxx0i)<strong>and</strong>substitutethemintheeqn.(6.20).Itisinterestingtonotethatinthecaseofan Tocomputetheemittancegrowthweneedtocomputethesecondordermoments(hx2i,hx02i x(s)=Zs x0(s)=Zs 0xsin(~s=x)(~s)d~s<br />
theaboveintegralverysmall(orideallyzero)sothatthenetemittancegrowthisnegligible.Such amethodhasbeendiscussedindetailinreferences[60]<strong>and</strong>[61]. 0cos(~s=x)(~s) (6.25)<br />
principalsolutions(S(t)<strong>and</strong>C(t))accordinglyto: equationwritesx(t)=Rt 11Ifweconsidertherighth<strong>and</strong>sideofthepreviousequationasaperturbationtermp(t;s)thesolutionofthis 0p(~t)d~tG(t;~t)whereG(t;~t)isaGreen'sfunctionthatcanbeconstructedfromthetwo G(t;~t)=S(t)C(~t)C(t)S(~t)<br />
σ z (mm)<br />
ε x,y (mm−mrad)<br />
0.5<br />
∆E (keV)<br />
σ x,y (mm)<br />
100<br />
80
6.4.4BunchSelfInteractionviaCoherentSynchrotronRadiation CSRisalongst<strong>and</strong>ingtopicinseveralsubjects,especiallyinAcceleratorPhysics.Therst comprehensivestudywasperformedbyJ.S.Nodvick<strong>and</strong>D.S.Saxon[4]in1954.Theseauthors accelerator.Itisaconsequenceofthegenerallylongbunchthatarecirculatinginsuchaccelerator: aswewillseeinthischapter,CSRemissionoccursatwavelengthcomparabletothebunchlength. Therefore,forbunchlengthoftheorderofcentimeters(asitiscurrentincircularaccelerator), studiedtheinteractionofchargedparticlemovingonacurvedpathbetweentoperfectlyconducting plane<strong>and</strong>showedhowCSRemissioncouldbepartiallysuppressatagivenwavelengthbythemeans ofthetwoconductingplanethatactasashielding.Indeed,tothebestofourknowledge,CSReect onthe<strong>Beam</strong>Dynamics,<strong>and</strong>CSRemission,haveneverbeenobservedinstorageringorcircular<br />
theTohokuUniversitybyT.Nagazato[66].Thisgroupshowedexperimentallyhowitwaspossible toinferthebunchlength<strong>and</strong>bunchstructureusingthefrequencyspectrumofCSR,usingthe beampipechamber,whichserveasawaveguidefortheCSRpropagation,arealsooftheorderof centimeters<strong>and</strong>sois<strong>their</strong>cutowavelength.ThereforetheCSRemissionis\shielded"bythe theemissionofCSRshouldoccurinthemicrowaveregion:unfortunately,thesizeofthevacuum region,<strong>and</strong>inthefar-infra-redwavelengthhasbeenobservedina100MeVlinearacceleratorof sametechniquewepresentedinChapter4forthetransitionradiation.Theyalsodemonstrate beampipe,i.e.itdoesnotpropagate.Infact,onlyveryrecently,CSRemissioninthefareld thepossibleshieldingofCSRemissionusingtwoparallelconductingplanewithvariablegap[67]. HowevertheanticipatedeectsofCSRonthe<strong>Beam</strong>Dynamics,i.e.transverseemittancedilution, hasneverbeenobserveduptonow. Asimplemodel:steadystateinfreespace WeoutlineinthepresentsectionasimplepictureoftheCSRphenomenon.Forsuchapurposewe startwiththeLienard-Wietchertretardedelectriceld[8]: !E=e"bn!<br />
inthemovingframe,theradiusofcurvature,<strong>and</strong>theanglebetweentheminthelaboratory orequivalentlyby=2sin(=2)withbeingtheanglebetweenthetwoelectrons timet0.Becauseofcausalitytheretardedt0<strong>and</strong>presentttimesarerelatedbyt=t0+R(t0)=c Thesubscriptretmeansthatthequantitiesinsidethebracketsmustbeevaluatedattheretarded !RisavectorfromS0toS,<strong>and</strong>1bn!=1cos(=2)<strong>and</strong>=6(! 2(1bn:!)3R2#ret+ec24bn^(bn!)^!_ (1bn!)3R35ret OS0;! OS)(seegure6.19). (6.26)<br />
frame. Theproblemhasbeentreatedinseveralreferences(e.g.Ref.[69]),itrstconsistsofcalculatingthe electriceldemittedattheretardedtime<strong>and</strong>locationS0atthepresenttime<strong>and</strong>locationS.This electriceldinducesanenergychangeonS,V(ss0),thatdependsontherelativepositions,s<strong>and</strong> alongthebunch.TheenergychangeofareferenceparticleSisgivenbythesuperpositionofthe s0,ofthetwoparticles.InessenceCSRisverysimilartowakeeld:ityieldsanenergyredistribution radiationforceofallthebackparticles: d(ct)=Zs dE1(s0)V(ss0)ds0<br />
(6.27)
InthecaseofarigidlinechargewithaGaussi<strong>and</strong>istribution(s)=N=p22zexps2=(22z),<br />
presentedingure6.20alongwithsimulationresultsusingamodiedversionofparmelathat oneobtainsfortheenergychange[68]: withthefunction12FdenedasF()=R1d0 wakeeldwherethetrailingelectronsinthebunchgenerallyloseenergy,CSReectsyieldanenergy d(ct)= dE(2)1=231=32=34=3<br />
2Ne2<br />
gainforelectronslocatedintheheadofthebunch. includesasimplemodelforCSRbunchselfinteraction(seeAppendixB).Contrarytost<strong>and</strong>ard (0)1=3d zF(s=z) d0e02=2Aplotofthisenergychangeis (6.28)<br />
Bunch at Present time<br />
Bunch<br />
Trajectory<br />
R<br />
Limitationsofthepreviousmodel Figure6.19:SchematicsofCSRselfinteractionofabunch.<br />
S’<br />
Bunch at Retarded time<br />
∆Θ<br />
thestraightsectiontothebendsection.(2)thebunchpropagatesinmetallic(e.g.stainlesssteel) intwoways:(1)itassumesthebunchhasbeenorbitingonacircularpathforever(steadystate assumption)<strong>and</strong>(2)itassumethebunchisinfreespace.Bothofthisassumptionsarenottruein practice:(1)anaccelerator(evencircular)consistsofstraightsectionsjoinedbybendingelements thereforeamorerealisticpictureofCSRshouldincludethetransientCSR,i.e.thepassagefrom ThemodelofCSRbunchselfinteractionbrieyoutlinedintheprevioussectionisoversimplied<br />
O<br />
cut-ofrequencyassociatedwiththegeometricparametersofthevacuumchamber). vacuumchambers<strong>and</strong>thereforeCSRcanbeshielded(i.e.notallowtopropagatebecauseofthe 12sometimetermedas\overtake"function<br />
S
100<br />
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predictionusingasimplenumericalmodelinamodiedversionofthescparmelacodefor200m Figure6.20:AnalyticalcomputationforCSR-inducedenergylossalongagaussianbunch<strong>and</strong><br />
-100<br />
(A)<strong>and</strong>100m(B).Thesystemconsideredisasimpleachromaticchicane(macroparticlewith >0areinthebunchtail).<br />
(B)<br />
TheprimarypurposeoftheexperimentthatwasattemptedintheIRFEListomeasurewhether locatedinthebacklegtransport. 6.5PreliminaryExperimentalResultsonEmittance<strong>and</strong>EnergySpreadMea-<br />
AnothermotivationwastotrytosetuptheIRFELopticssothatwecouldgenerateemittance thetransversehorizontalemittanceissignicantlydegradedaftertherecirculationarc1.ThereasonistoconrmtheviabilityoftheenvisionedUpgradeIRFELinwhichseveralwigglerswillbesurements Theexperimentwasattemptedintwoseriesofruns.Duringtherstrun,wevariedthelinac degradation<strong>and</strong>performsomeparametricstudies. TheexperimentalsetuptomeasureemittancefollowsourdiscussionofChapter3. ofrun,becausetheemittancewasfoundtobelarge(technicalproblemwiththeinjector),we acceleratingphase<strong>and</strong>measuredtheemittancebefore<strong>and</strong>afterthearc1.Inthesecondseries concentratedonmeasuringtheenergyspreadmeasurementonly.<br />
E (keV)<br />
(A)
6.5.1EmittanceMeasurement Arstexperimentconsistedinvaryingthebunchlengthattheundulatorlocation<strong>and</strong>measuring<br />
interaction,ifthosearepresent.Afullyself-consistentcode,writtenbyR.LiofJeersonLab[70], implementedinparmelawefoundthatatsuchpointtheemittanceincreasescomparedtothecase intheundulatorvicinity(asinferredfromthemaximumCTRsignal).Fromasimplisticmodel whereCSRisnotincludedinthecalculationisapproximately10%.Atthetimeofthemeasurement weoperatedthegunwithaverypoorphotocathode<strong>and</strong>couldnotextractmorethat20pCcharge perbunch,atoolowchargetounambiguouslymeasurepotentialeectsfromCSRbunchself ingure6.21.Theemittanceseemstogothroughamaximumforaminimumbunchlength thehorizontalemittanceafterthedecompressorchicane.Atypicalplotobtainedispresented<br />
ranfor60pC<strong>and</strong>anominalemittanceofapproximately7mm-mradyieldeda20%increasein emittance13. Wealsoattemptedtocomparethetransversehorizontalemittancebefore<strong>and</strong>afterthearc1.Inthe casecorrespondingtothenominaloperationofthelinac,whichcorrespondstoaminimumbunch lengthintheundulatorvicinity,noemittancegrowthwasobservedwithintheerrorbars(transverse horizontalemittancesmeasuredwereapproximately18mm-mradnormalizedat38MeV).<br />
22<br />
20<br />
18<br />
Before Arc<br />
16<br />
14<br />
Unfortunatelynoparametricstudyhasbeencompletedtothepresentdate<br />
Figure6.21:TransverseHorizontalemittance<strong>and</strong>totalpowerCTRsignalmeasuredasafunction ofthelinacgangphase. 13R.Li,privatecommunication,thisincreasewasnoticedtobeverydependentontheopticallatticesetup.<br />
12<br />
10<br />
bunch length<br />
CTR signal<br />
8<br />
4 6 8 10 12 14 16 18 20<br />
Linac Phase (RF−deg)<br />
Horizontal Norm. Emittance (mm−mrad)
Inthesecondseriesofrun,wewereabletoextracthighchargeperbunchfromthephotocathode (typically60pC)butwewereneverabletoestablishsucientlylowtransverseemittanceinthe 6.5.2EnergySpreadMeasurement IRFEL14toprovidealowenoughsignal-to-noisetoseedenitivelyapotentialemittancegrowth hadremovedalltransverseeectsoftransportfromtheviewerimages,somethingthatneedstobe donetobesureonlyenergyspreadisbeingobservedontheviewers.However,forafewselectcases atthearcexit.Inthecaseofenergyspread,itwasgenerallydiculttoconvinceourselvesthatwe correspondingtoshortbunchlengthsclosetotheundulator,webelievewehavegooddata,<strong>and</strong> whichindicatestheenergyspreadincreasebetweenthecompressorchicanemidpoint<strong>and</strong>thearc1 exitshouldbeoftheorderof5%15. withintheresolutionofthesemeasurements,theyshownoincreaseinenergyspreadaspicturedin gure6.22.ThisresultisconsistentwithsimulationperformedwiththeJLabselfconsistentcode<br />
−0.01 0 0.01 −0.01 0 0.01 −0.01 0 0.01 −0.01 0 0.01<br />
0.5<br />
energyspread(nounits)].Thebottomplotpresentsthermsrelativeenergyspreadcomputedfrom<br />
0.4<br />
thedistributions. Figure6.22:Energydistributionmeasuredalongthebeamline,atthechicanemidpoint(A)<strong>and</strong> (B)<strong>and</strong>entranceofthearcs(C)<strong>and</strong>(D)[thehorizontalaxisoftheseplotrepresenttherelative<br />
0.3<br />
14thereasonisstillnotunderstoodatthepresenttime 15R.Li,privatecommunication<br />
0.2<br />
0.1<br />
0<br />
0 20 40 60 80<br />
Distance from the cryomodule exit (meters)<br />
δ (%)<br />
Energy Distrib.<br />
(A) (B) (C) (D)
wetriedtoperformedthemeasurements.Howeverapreliminaryconclusionwouldindicatethat CSRisnotasignicanteectintheIRFEL,inthesensethatnotremendousgrowthofthe 6.5.3Conclusion<br />
thefuture,oncethebeamdynamicsinthemachineisfullyoptimized,oneshouldtrytotransport transverseemittanceortheenergyspreadwasobservedevenat60pC.Onlyonemeasurementhas beenperformed<strong>and</strong>theremightbeopticallatticesetupsthatmayprovidealargerenergyspread <strong>and</strong>emittancegrowth.AtminimumwecanconcludethatwiththenominalsetupoftheIRFEL OurexperimentonCSRwasnotverysuccessfulbecauseofbadbeamqualityduringtheperiod<br />
highercharge(e.g.thefull135pCrequiredfortheUpgradeIRFEL),<strong>and</strong>attempttogenerate<strong>and</strong> measurebeamdegradationduetobends.<br />
acceleratorneithersignicantemittancedilutionnorenergyspreadgenerationweremeasured.In
Conclusion Chapter7<br />
Therecirculatortransverseresponsefunctions,thelatticedispersion,<strong>and</strong>thelongitudinaltransfer Theworkpresentedinthepresentreportdescribesindetailtheimplementationofdierenttypesof<br />
functionsR56<strong>and</strong>R55(alongwithnonlinearities)havebeenmeasured<strong>and</strong>comparedwithamag- commissioning,underst<strong>and</strong>ing<strong>and</strong>operatingthersthighaveragepower(kW-level)infra-redfree electronlaseroscillatorthathasbeenbuiltatThomasJeersonNationalAcceleratorFacility. diagnosticsforarelativelyhighbrightnesselectronbeam<strong>and</strong>theapplicationsofthesediagnostics tostudysomebeamdynamicsproblems.Thisworkhadasignicantcontributioninthesuccessof neticopticscodesuchasdimad<strong>and</strong>particlepushingcodesuchasparmela. Wehaveimplementedtwotransversephasespacecharacterizationtechniquestostudythephase theemittancedominatedregime.Theformertechniquebasedontransversephasespacesampling usingamultislitmaskhasallowedsomepreliminaryparametricstudiesofthephasespaceof60pC chargeperbunchbeamproducedatthefrontendofa10MeVphotoinjector<strong>and</strong>hasbeencapable toresolveemittanceaslowas1mm-mradinatestinjectorst<strong>and</strong>. Alongwiththetransversephasespace,afullsix-dimensionalcharacterizationofthephasespacehas spacedensityoftheelectronbeamasitisstillinthespace-chargedominatedregime,butalsoin<br />
coherenttransitionradiationproducedbythebunchedelectronbeam.Thisinterferometerhasalso beenusedtoperformparametricstudyofthebunchlengthevolutionversussomeradio-frequency elementthatplayakeyroleinthebunchingscheme. FinallyafullmodelbasedonmultiparticlesimulationoftheIRFELhasbeenelaborated<strong>and</strong>com- beenattemptedbymeasuringbothbunchlength<strong>and</strong>energyspreadoftheelectronbeam.Because<br />
paredwhenpossiblewithexperiments.Atentativeexperimenttomeasureenergyspreadgeneration oftheultra-shortbeamrequiredtodriveafree-electronlaser(typically
undulatorvicinity.Wehavenotmanaged,uptothepresentdate,tounambiguouslydemonstrate whetherthiseectwasduetocoherentsynchrotronradiation.Asecondseriesofrunswereper-<br />
measurementreportedinthisreport)wouldroughlydoubletheeectprovidedallelsewereequal, formed,whereweconcentratedonthemeasurementofenergyspreadalongthetransportchannel,butthenitstillwouldnotbesignicant.OncethelatticedesignoftheIRFELUpgradehasma-<br />
resultinginnoobservation,withintheprecisionofthemeasurement,ofenergyspreadgeneration (alsothisdataweretakenforoneoperatingpointofthelinaconly). Wecanconclude,ataminimum,thatitshouldbepossiblesetupabeamthatwilltransportaround theBatesarcwithoutsignicantgrowthinenergyspread(whichtranslatestonosignicantgrowth inemittance),<strong>and</strong>thereforefortheFELUpgradeitseemsreasonabletoplanonputtingtheFEL systemsinthebacklegtransferline-abunchchargeof135pC(versusthe60pCusedforthe tured,thenoneshouldbeabletousetheIRDemotosetupabeamattheentrancetotherst ofmeasurements(emittance<strong>and</strong>energyspread)the135pCcase.<br />
BatesbendthatmimicstheIRFELUpgradebeamparameters,<strong>and</strong>trytoperformthesametype
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[33]V.A.Lebedev\Diraction-limitedresolutionoftheopticaltransitionradiationmonitor", [32]P.Piot,J.-C.Denard,P.Adderley,K.Capek<strong>and</strong>E.Feldl,\<strong>High</strong>PowerCW<strong>Beam</strong>Prole [31]S.Leisegang,Z.Phys.132,pp183-187(1952) pp298-305,(1996) MonitoratCEBAF",AIPConferenceProceedings390,A.H.Lumpkin<strong>and</strong>C.E.EybergerEd., Naturforschg.15a,pp1031-1038(1960)<br />
[34]D.Douglas,\Anobservationpointformeasurementofthe45MeVemittanceintheFront [35]P.Piot,\ATcl/TkPackageforCollecting<strong>and</strong>Analyzing<strong>Beam</strong>TranverseDensityMeasure- EndTest",reportTN-90-265,JeersonLab,NewportNewsVA,USA(1990) Nucl.Instr.Meth.,A372,pp.344-348(1996)<br />
[36]P.Piot,G.A.Krat,R.Li,J.Song,\Real-TimeTransverseEmittanceMeasurement",pp. [37]F.J.Sacherer,\rmsenvelopeequationswithspacecharge",reportCERN/SI/Int.DL/70-12, mentsusingOTRintheIRFEL",reportTN-99-031,JeersonLab,NewportNewsVA,USA (1999) 684-689,Proc.InternationalLinearAcceleratorConference,Chicago(1998)<br />
[40]P.Bevington,DataReduction<strong>and</strong>ErrorAnalysisforthePhysicalSciences,McGraw-Hill [39]J.Song,P.Piot,R.Li,etal.\Real-TimePhaseSpaceMonitor",Nucl.Instr.Meth.,A407, [38]M.Crescenti<strong>and</strong>U.Raich,\SecondEuropeanWorkshopon<strong>Beam</strong><strong>Diagnostics</strong><strong>and</strong>Instru- pp.343-349(1998) mentationsforParticleAccelerator"DESYM95-07,pp66-68(1995) CERNGenevaSwizerl<strong>and</strong>(1970)<br />
[43]E.C.Reichenbach,S.K.Park,R.Narayanswamy,"Characterizingdigitalimageacquisition [42]D.E.Groom,\Passageofparticlestroughmatter",Eur.Phys.Jour.C3num(1-4),pp. [41]J.D.Lawson,ThePhysicsofCharged-Particle<strong>Beam</strong>s,2ndEdition,InternationalSeriesof 144-151(1998) MonographsonPhysics75,OxfordSciencePublications(1988) Book(1969)<br />
[44]M.G.Tiefenback,\Space-ChargeLimitsonthetransportofIon<strong>Beam</strong>sinalongalternating [45]B.M.Dunham,D.Douglas,R.Legg,<strong>and</strong>B.Bowling,\AnalysisofEmittanceMeasurement devices",Opt.Eng.30(2),pp170-177(1991) UsingSingle<strong>and</strong>MultipleHarps",reportTN-96-014,JeersonLab,NewportNewsVA,USA gradientsystem",Ph.D.Thesis,LawrenceBerkleyLaboratory,UniversityofCalifornia,report LBL-22465(1986) (1996)
[48]R.Lai,<strong>and</strong>A.J.Sievers,\Phaseproblemassociatedwiththedeterminationofthelongitudinal [47]W.H.Press,S.A.Teukolsky,W.T.Vetterling,<strong>and</strong>B.P.Fannery,NumericalRecipesinC: [46]P.Piot,<strong>and</strong>G.A.Krat,\MonitoringLongitudinalPhaseSpaceofaCharged-Particle<strong>Beam</strong> TheArtofScienticComputing,2ndEdition,CambridgeUniversityPress(1992) withaMultifrequencyCoherentRadiationDevice",reportTN-97-030,JeersonLab,Newport<br />
pp.45764579(1995) shapeofachargedparticlebunchfromitscoherentfar-irspectrum",Phys.Rev.,E52num.2, NewsVA,USA(1997)<br />
[49]R.E.Burge,M.A.Fiddy,A.H.Greenway,<strong>and</strong>G.Ross,\ThePhaseProblem",Proc.R.Soc.<br />
[52]D.X.Wang,G.A.Krat,<strong>and</strong>C.K.Sinclair,\Measurementoffemtosecondelectronbunches [51]T.Larsen,\ASurveyoftheTheoryofWireGrids",IRETrans.Micr.Theo.&Tech.,pp. [50]N.Levinson,<strong>and</strong>R.Redheer,ComplexeVariable,Holden-DayInc.,SanFransisco,Cam- 191-201(1962) bridge,London<strong>and</strong>Amsterdam(1970) Lond.A.350,pp.191-212(1976)<br />
[53]H.Liu,etal.,Nucl.Instr.<strong>and</strong>Meth.A358,pp475-478(1995) [55]P.Piot,G.Biallas,C.L.Bohn,D.R.Douglas,D.Engwall,K.Jordan,D.Kehne,G.A.Krat, [54]M.Reiser,Theory<strong>and</strong>Designofchargedparticlebeams,JohnWiley&Sons,(1994) usingarfzero-phasingmethod",Phys.Rev.,E57num.2,pp.2283-2286(1998)<br />
[56]P.Piot,G.A.Krat,\TransverseRF-focussinginJLABRFCavity",Proc.Euro.Part.Acc. [57]J.B.Rosenzweig,<strong>and</strong>L.Serani,\Transverseparticlemotioninradio-frequencylinearaccel- R.Legg,L.Merminga,J.Preble,T.Siggins,<strong>and</strong>B.C.Byunn,\ExperimentalResultsfroman InjectorforanIRFEL",Proc.Euro.Part.Acc.Conf.'98,pp.1447-1449(1998) Conf.'98,pp.1327-1330(1998)<br />
[60]P.Emma,<strong>and</strong>R.Brikmann,\EmittanceDilutionTroughCoherentEnergySpreadGeneration [59]H.Wiedemann,ParticleAcceleratorPhysicsII,Springer-VerlagBerlinHeidelberg(1995) [58]E.Chambers,unpublishedwork,SLAC,St<strong>and</strong>fordUniversity(1965) inBendingSystem",Proc.Part.Acc.Conf.'97,pp.1679-1681(1997) erator",Phys.Rev.,E49num.2,pp.1599-1602(1994)<br />
[62]Z.Li,\<strong>Beam</strong>DynamicsintheCEBAFSuperconductingCavities",PhDthesis,Collegeof [63]A.W.Chao,PhysicsofCollective<strong>Beam</strong>Instabilitiesin<strong>High</strong>EnergyAccelerators,JohnWiley [61]P.G.O'Shea,\Reversible<strong>and</strong>irreversibleemittancegrowth",Phys.Rev.,E57num.1,pp.<br />
[64]Z.Li,\<strong>Beam</strong>DynamicsintheCEBAFSuperconductingCavities",Proc.Part.Acc.Conf.'93 1081-1087(1998) &Sons,(1993) William&MaryVA-USA(1995) IEEEcatalog#93CH3279-7,pp.179-182,Chicago(1993)
[65]B.C.Yunn,\ImpedancesintheIRFEL",reportTN-96-049,JeersonLab,NewportNews<br />
[68]E.L.Saldin,E.A.Schneidmiller,M.V.Yurkov,reportTESLA-FEL96-14,Deutsches [66]T.Nagazato,etal.,\ObservationofCoherentSynchrotronRadiation",Phys.Rev.Let.,63 [67]R.Kato,etal."Suppression<strong>and</strong>enhancementofcoherentsynchrotronradiationinthepresence VA,USA(1996)<br />
[69]R.Li,C.L.Bohn,<strong>and</strong>J.Bisognano,\Shieldedtransientself-interactionofabunchenteringa Elektronen-Synchrotron-DESY,Hamburg(Deutchl<strong>and</strong>) num.12,pp.1245-1248(1989)<br />
[70]R.Li,\SelfConsistentSimulationofCSR",Proc.Euro.Part.Acc.Conf.'98,pp.1230-1232 circlefromastraightpath",reportTN-97-048,JeersonLab,NewportNewsVA,USA(1997) oftwoparallelconductingplates",Phys.Rev.E57,num3,pp3454-3460(1998).<br />
(1998)
Abbreviations AppendixA<br />
CCD:chargecoupleddevice. BPM:beampositionmonitor. BFF:bunchformfactor.<br />
FWHM:fullwidthhalfmaximum. CW:continuouswave. FFT:fastfouriertransform. CTR:coherenttransitionradiation. CSR:coherentsynchrotronradiation.<br />
ISR:incoherentsynchrotronradiation. IRFEL:infraredfree-electronlaser. HOMcoupler:highordermodecoupler(onacceleratingcavities). HF:high-frequency. FPcoupler:forwardpowercoupler(onacceleratingcavities).<br />
SC:spacecharge. RMS:rootmeansquare. OTR:opticaltransitionradiation. RF:radio-frequency. ODP:opticalpathdierence.<br />
SR:synchrotronradiation. SRF:superconductingradio-frequency.172
TEM:transverseelectricmagnetic.<br />
TOF:timeofight.<br />
TR:transitionradiation.
AppendixB<br />
B.1Linear<strong>and</strong>SecondOrderTransport:Convention <strong>Beam</strong>Dynamics:Notes&Tools<br />
B.1.1TransferMatrix Inthepresentreportweworkinthecoordinatesystem(x;x0;y;y0;;)where: x,y,arethecoordinateinthest<strong>and</strong>ard3Dpositionspace(notethat=2z=RFrepresentsthelongitudinalpositionoftheparticleinunitoftheRFwavelengthoftheacceleratorTopropagateavector!rinalongasectionofbeamline,weuse,providedthesecond-orderapprox- x0<strong>and</strong>y0arethedivergenceinthetransverseplane (withinafactor2))<br />
imationoftheequationofmotionisapplicable: reportcoincidentwiththeenergyaverageofthebunch). istherelativeenergyosetoftheparticlewithareferenceparticle(whichisinthepresent<br />
Rijistherstordermatrix<strong>and</strong>Tijkarethesecondorderterms. B.1.2<strong>Beam</strong>Matrix rout;i=XjRijrin;j+XkXj>kTijkrin;jrin;k+O(r3) (B.1)<br />
Thebeamormatrix,e.g.forthex-x0phasespaceisdenedasfollows: Thesamematrixcanbedenedforthey-y0phasespaces(3j4)or-longitudinalphasespaces (5j6).wehavethefollowingdenitions/properties: xdef =1j2def = 1112 1222!= 174hxx0ihx02i!<br />
hx2ihxx0i (B.2)
B.2Anoteonspacecharge Wedenetheslopeofthephasespaceas:dx=dx0=hxx0i=hx02i thetransversermsemittanceisthedeterminantdet(z)<br />
originatesfromCoulombrepulsionbetweenelectronswithinabunch.Spacechargetendstoinduce emittancegrowth.Inthissection,wewouldliketoshowthattheseforcesareonlyaconcernfor lowenergybeam,inourcase(Q'60pCthespacechargecollectiveeectisonlyimportantin Wehavealreadymentionedthatchargedparticlebeamaresubjecttospacechargeforcethat theinjectorbeamline.Forsuchapurposeweconsideraverysimplemodelofauniformlycharged<br />
Thelongitudinaleldcanalsobecomputedconsideringthebeaminsideapuremetalliccylindrical magneticeldsinsidethebeam,<strong>and</strong>inthebeamreferenceframe,caneasilybecomputedfromthe Maxwellequation<strong>and</strong>are(forr
B.3.1DIMAD TheprogramDIMAD2studiesparticlebehaviorincircularmachines<strong>and</strong>inbeamlines.Thetra- B.3TheSimulationTools<br />
chargedparticlecomputercodes. B.3.2TLIE likeitspredecessorDIMAT,istheresultofmanyyearsofexperimentingwithseveraldierent jectoriesoftherelativisticparticlesarecomputedaccordingtothesecondordermatrixformalism. Itdoesnotprovidesynchrotronmotionanalysisbutcansimulateit.Theprogramprovidestheuser withthepossibilityofdeningarbitraryelementstotailortheprogramtospecicuses.DIMAD,<br />
tosecondorderlikedimad.ThePhysicsbehindthiscodeisbasedontheuseoftheLieAlgebra vectorspacewithaproductverifyingtheproperties(1)(xy)=(x)y=x(y)<strong>and</strong>(2) operatortopropagatetransfertmapalongabeamlinesection.ALiealgebraisanalgebra(i.e.a Tlie3isageneral6DrelativisticdesigncodewithaMADcompatibleinputlanguage.The particularityofTLieisitsabilitytocomputetransfermapatanarbitraryorder<strong>and</strong>notonlyup Pi@f TheLiealgebraoperatorusedin<strong>Beam</strong>DynamicsisthePoissonbracketdenedas:[f;g]= forsuchachoiceisthefactthatwiththehelpofthecanonicalHamiltonequations,wecanwrite y(x1+x2)=yx1+yx2)thatalsoveriestheJocobiidentity:x(yz)+y(zx)+z(xy)=0. forafunctionf(pi;qi): @qi@g @pi@g @qi@f @piwhereg<strong>and</strong>farefunctionsofthegeneralizedvariablespi<strong>and</strong>qi.Thereason<br />
thatfisnotanexplicitfunctionoftimei.e.@f :f:isaLieoperator.ToillustratehowtheTliecodeworks,let'sassume,forthetimebeing, (pi;qi).Inst<strong>and</strong>ardnotation,thePoissonbracketoperatorisoftenwritten:f:g=[f;g]where whereHistheHamiltonianthatgovernstheevolutionofthedistributionfintheconjugatespace df dt=[f;H]=:f:H=:H:f,<strong>and</strong>usingpurelysymbolicequationwehaveintermofoperator: df dt=@f @t+[f;H] @t=0inEqn.(B.8).ThenEqn.(B.8)becomes: (B.8)<br />
e:H:f=f+[H;f]+1=2[H;[H;f]]+:::. ThebeautyofLieAlgebratechniqueresidesinthatthecomputationofthefunctionfatatime dierentialequationise:H:fwheretheexponentiationofLieoperatorisdenedastheseries order,thisistheprincipleonwhichTlieisbased:thehamiltonianHalongwiththecorresponding operatore:H:arecomputedforeachbeamlinepiece<strong>and</strong>areconcatenedusingtheCampbell-Baker- calculationcanbecarriedatanyorderbycomputingthePoissonbracketseriesatthedesired t=t0+,knowingthefunctionfattimet0,justconsistsofcomputingft=e:H:ft0;such dt=:H:sotheLieoperator:H:reducestoasimpletimederivative.Thesolutionofthis<br />
REPORT285SLAC-StanfordUniversityCA-USA(1985) Hausdortheorem,thatstatese:HA!C:=e:HA!B:e:HB!C:,toobtaintheLieoperatorforawhole beamlinesection[A,C]. 3ThecodewaswrittenbyJohannesvanZeijts<strong>and</strong>FilippoNeri<br />
2R.Sevranckx,K.L.Brown,L.Scachinger,<strong>and</strong>D.Douglas,\UsersGuidetotheProgramDIMAD",SLAC
B.3.3PARMELA FeaturesofJeersonLabVersion \Phase<strong>and</strong>RadialMotionin<strong>Electron</strong>LinearAccelerators."Itisaversatilemulti-particlecode thattransformsthebeam,representedbyacollectionofmacroparticles,throughauser-specied linac<strong>and</strong>/ortransportsystem.Itincludesa2-Dspace-chargecalculation<strong>and</strong>anoptional3-D point-to-pointspace-chargecalculation.PARMELAintegratestheparticletrajectoriesthrough theelds.Thisapproachisespeciallyimportantforelectronswheresomeoftheapproximations nothold.PARMELAworksequallywellforeitherelectronsorions.PARMELAcanreadeld usedbyothercodes(e.g.the"drift-kick"methodcommonlyusedforlow-energyprotons)would<br />
AtJeersonLab,amodiedversionofparmelahasbeenproducedbyH.Liu.Itincorporatesa3D distributionsgeneratedbyeitherFISHforrfproblemsorPOISSONformagnetproblems. ModiedSpacechargealgorithm point-by-pointspacechargealgorithmfromK.T.Mc.Donald4.Anoutlineofthealgorithmisas follows.parmelauseatwo-stepmethodtogenerateaspacechargeimpulseoneachmacroparticle: (!)itdeterminesthenetelectromagneticspacechargeeldatthelocationofeachmacroparticle duetoallothermacroparticle,(2)applythespacechargeimpulsetoeachmacroparticle.Then trackthemacroparticlethroughaslice(widthdenedbytheuser)ofthebeamline(forsimple elementthetrackingisperformedusingsecondordertransfermatrix,buttheusercanifdesired eachslice).Thisspacepoint-by-pointalgorithmisverysimplebutbecauseofthe1=r2dependence denethe3Dmapoftheelectromagneticeld.Insuchcasetheequationofmotionisintegratedin (whichcanleadtosingularityornumericalnoise)itmustbeimplementedcarefully.Forinstancein theeventualitytwomacroparticlescomeveryclosetoeachotherthechargeofthemacroparticles inthealgorithmisreduced.Thealgorithmtoreducethemacroparticlechargeisdiscussedindetail elsewhere5 AsimplemodelforCoherentSynchrotronRadiation chargeQi<strong>and</strong>lengthiorbitingonacirculartrajectoryofradiusRidetectedatthepresenttime byanobserverelectronjderived<strong>and</strong>expressedas: TheimplementationoftheCSRinteractionintothePARMELAcodecloselyfollowsthemethod describedbyCarlsten6wheretheelectriceldgeneratedataretardedangle0byalineiofuniform<br />
forshortelectronbunchesincircularmotionusingtheretardedGreen'sfunctiontechnique"Phys.RevE54num1, TN-94-040,JeersonLab,NewportNews,VA-USA(1994) pp838-845(1996)<br />
4K.T.McDonald,IEEETrans.Elect.Dev.35p2052(1988) 6B.E.Carlsten,\Calculationofthenoninertialspace-chargeforce<strong>and</strong>thecoherentsynchrotronradiationforce 5H.Liu,\ConceptofVariableParticleSizeFactorforaPoint-by-PointSpaceChargeAlgorithm",CEBAFreport Ei;j=Qi i"1 r(1!ibnij)"12i2ixj Ri+2i(1cos(0))##fr (B.9)
ofthelinecharge<strong>and</strong>theobserverelectron.Theretardedangle0isrelatedtothepresentangle directionfromthelinecenter<strong>and</strong>theobserverelectron.Thequantityinbracketmustbeevaluated forthepresentanglesrij(resp.fij)thatcorrespondstotheanglebetweentherear(resp.front) wherei,iaretheusualLorentzfactorsfortheorbitinglinei,bnijisthenormedvectoralongthe<br />
wherejisthetotaltransversedisplacementoftheobserverelectronwithrespecttothetrajectory ofthelinecharge. ThedenominatorofEqn.(B.9)canalsobeexpressedasafunctionoftheretardedtimeusinggeo- i<strong>and</strong>theobserverjwehave: byatranscendequationderivedfromgeometricalconsideration.Forinstanceforthelinecharge<br />
metricalconsideration. 2iR2i(0)2=2j+2Ri(Ri+xj)(1cos(0)) (B.10)<br />
linechargecanbereplacedbyamacroparticlewhichcarry,asinPARMELA,auniformcharge. Theextensiontoabunchofelectrondescribedbyamacroparticlemodelisstraightforward:the<br />
whereNisthenumberofmacroparticleinthemodel. Infactthis\pointbypoint"typemacroparticlealgorithmhasalreadybeenimplementedinthe Thereforethetotalelectriceldproducedbythebunchofmacroparticleataretardedtimeonan<br />
JLabPARMELAversiontosimulatemacropaticleinteractionviaspace-chargeforce.Hencewe observermacroparticleatthepresenttimesimplywritesasthesum:<br />
caneasilymodifytheexistingalgorithmtosimulateCSRselfinteraction. TheretardedangleisevaluatedasdescribedbyCarlstenusinganiterativeprocesstosolvethe Etotal j=NXi=1Ei;j (B.11)<br />
transcendentequationEqn.(B.10).ThemodeldescribedintheprevioussectionasbeenimplementedintheJLabversionofPARMELA(bothaHP9000<strong>and</strong>aCrayC90versions).Pratically, whenthebunchentersabendinwhichtheuserwishtoincludeCSRinteraction,theradiusofthe trajectoryofeachmacroparticleisealuated<strong>and</strong>then,basedongeometricalconsideration,allthe parameterinEqn.(B.9)arecomputed<strong>and</strong>theeldduetobunchonamacroparticleisevaluated. intheBENDCSRcardthesubroutinecsriscalled.<br />
Thisoperationisperformedforeachmacroparticle<strong>and</strong>thereforealargenumberofiterationis BENDcard.Thiscardmustbeusedtoindicatethebendingmagnetwheretheuserwishtosimulate CSRinteraction.Secondly,wehavemodiedtheprogramPARMDYNsothatwhentheelectronsare showningrey:rstlywehaveintroducedanewcardBENDCSRwhichfollowthesamesyntaxasthe needed.TypicalCPUtimeneededtorunoneFELchicaneisapproximately1hrwallclock.A blockdiagramofthePARMELAcodeispresentedinFig.B.1.Themodicationperformedare
Initialization MAIN<br />
Stop<br />
INPUT DECK INPUT BEAM START/RESTART<br />
PARMDYM (particle dynamics)<br />
SCHEFF<br />
Space-Charge<br />
Loop on time steps<br />
Loop on particles<br />
SAVE<br />
CSR<br />
Csr Interaction<br />
FigureB.1:Parmelasimpliedalgorithm.<br />
BENDCSR BEND DRIFT QUAD POISSON CELL<br />
SWAP/ OUPUT BRANCHES<br />
Yes<br />
No<br />
LAST<br />
PARTICLE<br />
OUPUT<br />
No End Yes<br />
END
AppendixC<br />
C.0.4Introduction <strong>Beam</strong>Distributions DispersionRelationsforBunched<br />
emissionfromabunchedelectronbeamhavebeenusedbymanyauthors.Howeverwebelieve givethephase,butonlyonetermcontributingtothephase(dependingonthebunchlongitudinal distributionproperties). Dispersionrelationstocomputethephaseassociatedtotheelectriceldgeneratedviacoherent<br />
C.0.5Background themathematicalproofisnotalwaysproperlyderived:thesedispersionrelationsareappliedto aderivationofsuchrelations<strong>and</strong>discussthefactthatthesedispersionrelationdonotafortiori functionswhichcannotbeexpressedasCauchyintegrals.Inthepresentaddendumwepresent<br />
showninthecaseofbunchedelectronbeamthatiftheemittedradiationisobservedatwavelength comparableorlargerthatthebunchlength,theelectronsinthebunchemitcoherently.Insucha Therearemanywayselectronscanemitradiationas<strong>their</strong>environmentismodied.Ithasbeen<br />
where^S(!)istheFouriertransformofthelongitudinaldistributionS(t).Theproblemthathas case,thespectrumoftheradiationwritesas: transformoftheelectriceld,tocomputethebunchformfactorphase. Inthemanypapers,thefunctionlog(^S(!))isdened<strong>and</strong>thecomplexpartofthisfunction(which beenstudiedistousethepowerspectrum,whichisproportionaltothemodulusoftheFourier isthephaseof^S(!))iscalculatedusingthest<strong>and</strong>arddispersionrelations.Howeverweshallsee Etotal(!)=qN(N1)qj^S(!)jE1e(!) (C.1)<br />
belowthatlog^S(!)doesnothavethe\right"properties<strong>and</strong>cannotbegenerallywrittenasa log^S(!)function. Cauchyintegral<strong>and</strong>morecareshouldbeusedwhentryingtorecovertheimaginarypartofthe 180
C.0.6TheDispersionRelationsfor^S(!) WhenS(t),thebunchdistributionhasthe\right"properties,itsFouriertransformhassome interestingproperties:itsimaginary<strong>and</strong>realpartareHilbertconjugate,i.e.theyarerelatedvia Hilberttransformation.The\right"propertiesS(t)mustsatisfyareasfollows:<br />
ThelastiteminsureS(t)isacausalfunction<strong>and</strong>thetwootherareequivalenttoS(t)2L2,the limt!1S(t)=0<strong>and</strong>S(t)!0fasterthan1=t, S(t)=08t
Unfortunatelylog[^S(!)]cannotbeexpressedasaCauchyintegralbecauselog[^S(!)]isnotan dispersionrelationshavebeenappliedtothisfunctiontoretrieveitsimaginarypart(i.e.thephase of^S(!)). analyticinthefullupperhalf-plane.Moreover: Obviouslylog[^SS()]isnota\good"function!Acommonlyusedruseistodenethefunction Ilog[~S()] ()def jj!1 =log^S()log^S() !Z0log[~S()]6=0 (C.5)<br />
whereisanarbitrarypointoftheupperhalf-planewherelog[~S]isanalytic(notethat()isnot Thedispersionrelationsappliedto()yields: singularat=). i()=PZ+1 1(x)dx x (C.7) (C.6)<br />
UsingEqn.(C.6)weget: whichexp<strong>and</strong>sto: ilog[^S()]=ilog[^S()]+()24PZ+1 ilog[^S()]=ilog[^S()]+()24PZ+1 1log[^S(x)]dx 1log[^S(x)]dx (x)(x)PZ+1 (x)(x)log[^S()]PZ+1 1log[^S()]dx (x)(x)35(C.8)<br />
ThetwofarRHStermsarezero,sothatwenallyget: log[^S()]=log[^S()]+iPZ+1+log[^S()]PZ+1<br />
1dx 1dx x(C.9) x<br />
relatelog[j^S()j]tothephaseof^S(),(): Byidentifyingthereal<strong>and</strong>imaginarypartsweobtainthedispersionrelationsforlog[^S()]which log[j^S()j]=log[j^S()j]+PZ+1 1log[^S(x)]dx 1 (x)(x) (x)(x) (x)dx (C.10)<br />
whichaftertakingintoaccountthesymmetryof^S(^S()=^S()yieldsfor=0: ()=()1()PZ+1 ()=(0)2PZ1 1logj^S(x)jlogj^S()j 0logj^S(x)jdx (x)(x)dx x22 (C.12)<br />
(C.11)
have: Howeveritshouldbenotedthatwehaveassumedfromthebeginningthat^S()isanalyticin ^S(0)=12R,sinceS(t)isarealfunctionnormalizedtounity;therefore(0)=0.Sonallywe (0)canbeestimatedbecausetheFouriertransformat!=0istheintegralofS(t);itgives<br />
particularthepointwhere^S()=0willgivesingularityon()whichinturnmustbetakeninto theupperplane.Unfortunatelyitdoesnotimplylog[^S()]hasthesameregionofanalyticity;in accountinthephaseby,whenwriting()asaCauchyintegral,takinginaccountthecontribution totheintegralviatheresiduetheorem. ()=2PZ1 0logj^S(x)jdx x22 (C.13)<br />
Inthegure(gureC.1)belowwepresentresultsforasimplebi-modaldistributionthatconsists ofasumoftwonormaldistributions:wehavegeneratedthreetypesofdistribution(a)(c)<strong>and</strong> (e)<strong>and</strong>foreachofthemwecomparethephasecalculateddirectlyfromthecomputationofthe FouriertransformofthedistributionwiththephaseretrieveusingEqn.(C.13).Onthisverysimple examplewenoticethatthisequationdoesnotafortiorireproducetheveritablephaseoftheFourier transformofthedistribution.
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
2<br />
0<br />
−2<br />
−4<br />
−6<br />
−8<br />
0 1 2 3 4 5 6 7 8 9 10<br />
x 10 6<br />
−10<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
0 1 2 3 4 5 6 7 8 9 10<br />
x 10 6<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
0<br />
−2<br />
−4<br />
−6<br />
−8<br />
−10<br />
0 1 2 3 4 5 6 7 8 9 10<br />
x 10 6 FigureC.1:Exampleofphaseretrievalforabi-gaussian-likebunchdistribution.Threetypeof<br />
(c)<br />
(e)<br />
transform(dashlinesonplot(b),(d),<strong>and</strong>(f))<strong>and</strong>therecoveredphaseusingthedispersionrelation (crosses)onthesameformerplots.<br />
bi-modaldistributions(a),(c)<strong>and</strong>(e)arepresentedalongwiththeexactphaseof<strong>their</strong>Fourier<br />
Distance (a.u.)<br />
Distance (a.u.)<br />
0<br />
(d)<br />
(f)<br />
−5<br />
−10<br />
−15<br />
−20<br />
Frequency (a.u.)<br />
Frequency (a.u.)<br />
Phase (rad) Bunch Population (a.u.)<br />
Phase (rad) Bunch Population (a.u.)<br />
Distance (a.u.)<br />
Frequency (a.u.)<br />
Bunch Population (a.u.)<br />
Phase (rad)<br />
(a)<br />
(b)
FigureD.1:OverviewoftheRF-controlsystemfortheIRFEL. 185<br />
Buncher<br />
Photocathode<br />
Laser<br />
Cryounit<br />
(2 Cavities)<br />
GANG PHASE<br />
SRF- Linac (8 Cavities)<br />
RF Reference (Master Oscillator)<br />
<strong>Beam</strong><br />
- LEGEND -<br />
Phase Shifter<br />
Attenuator<br />
Klystron<br />
AppendixD TheRadio-FrequencyControlSystem
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