High Brightness Electron Beam Diagnostics and their ... - CASA

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powermeasurement,yieldingthelossofphaseinformation. Wehavealsonoticedfromexperimentusingaparticlepushingcodethatthenestructureofthe BFFisaecteddierentlydependingonhowthebunchingprocessisperformed.Forinstance ingure5.6weplottheBFFcomputedfromnumericalsimulationfordierentsettingsofradiofrequencyelementsthatplayakeyroleinthebunchingprocess.Thoughthebunchlengthdoesnot varysignicantly,wecannoticethateachelementaectstheBFFatdierentwavelength.Such parametricstudiesbysystematicallyvaryingeachRF-elements. featurescanbeexperimentallyusedtodeterminewhichelementisnotoperatedatitsnominaloperationpoint(e.g.becauseofdrift,...).Beforeitsapplicationwewillneedtoperformexperimental 2 5 10 -3 2 5 10 -2 10 Wavelength (m) -5 10 -4 10 -3 10 -2 10 -1 10 0 Nominal Buncher +3 deg Cavity 1 +3 deg Laser +3 deg Figure5.6:EectofdierentkeyelementsinthebunchingprocessoftheelectronsontheBFF. 5.3.2RetrievaloftheBunchDistributionbyHilbertTransformingtheBFF photocathodedrivelaser(dottedline)areoperated+3degotheirnominalsettings. TheBFFcorrespondingtothenominalsettingsfortheRFelements(solidline)iscomparedwith thecaseswherethebuncher(dottedline),therstcavityintheinjector(greyline)andthe tiveindexofamaterialfromfromknowledgeoftherealpart.Thistechniquehasbeenrstapplied tothepresentproblembyLaietal.[48];howeverintheliteraturethereisnoclearderivationthat provestheuseofthedispersionrelationforretrievingthephaseoftheBFFislegitimate.We arecommonlyusedinSolidStatePhysicse.g.forreconstructingtheimaginarypartoftherefrac- possibletogetsomeinsightonthephaseinformationusingtheso-calleddispersionrelations3that Aswementionedearlier,theonlyobservablewecanmeasureisthepowerofthecoherentradiation i.e.thesquareamplitudeoftheelectriceld;allthephaseinformationislost.However,itisstill 3AlsoreferredasKramers-Kronig'srelations Bunch Form Factor (no units)
presentanoutlineofthisproofbelow,andadetailedproofinAppendixC.Fromthedenitionof where canbewrittenas: thebunchformfactorwecanwrite:~S()=^S()^S() mathematicstextbooks(seeforinstance[50])andcanbeappliedtoS()tocalculateitsimaginary partknowingitsrealpartbecauseS()isasquareintegrablefunction.Inthepresentcase,the isthewavenumberand^S()istheFouriertransformofthebunchlongitudinaldistribution;it ()isthephaseassociatedtotheFouriertransform.Themethodisdiscussedinstandard ^S()=q~S()exp(i ()) (5.19) (5.18) (=+i0isthecomplexwavenumber): Now,log(^S())isnotsquareintegrableandtheCauchyintegralonlog(^S())doesnotconverge problemisslightlydierent:weknowthemodulusofS()andneedtocomputethephase.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) dispersionrelationsfor^S(seeAppendixCforadetailedderivation),andnallythephaseof^S Let'sintroducethefunction()denedas: ()isnotsingularat=andissquareintegrable.Wecanthenderiveasetof\modied" ()def =log[^S()]log[^S()] (5.22) (5.21) takestheform: Letting0=0andusingthefact^S()=^S()wenallynd: Thislatterequationiswidelyknown,intheliterature,andissometimesreferredasdispersion 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) relation.Oncethephase inverseFouriertransform:S(z)=Z1 ()iscomputedwecanrecovertheinitialdistributionbyusingthe discussionisprovidedinAppendixC). contributiontothephasethatmustbeconsidered.Inthefollowingwewillnotconsidersuchcases byassumingthestandardbunchdistributionisanalyticintheupperhalf-plane(afullydetailed thecomplexplane.Ifithassingularitiesthen,invirtueoftheresiduetheorem,thereareother TwofactsshouldbeemphasedaboutEqn.(5.24)(1)the bezero,and(2)thisequationisapplicableprovidedlog[S()]isanalyticintheupperhalf-partof 0^S()cos(2z (0)termisunknownandisassumedto ())d (5.25)
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HighBrightnessElectronBeamDiagnosti
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eratedupto48MeVpriortothelasingsyst
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IthanktheCEBAFmachineoperators,with
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3.4MeasurementoftheLongitudinalResp
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5.6ZerophasingTechniqueforBunchLeng
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ListofTables 3.2Comparisonofcoecien
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ListofFigures 1.1Comparisonoftheexp
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3.11Momentumcompactionandnonlinearm
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4.18Overviewofthephasespacesampling
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5.16Interferogrammeasuredfordierent
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6.10Comparisonofthermstransversehor
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Chapter1 UVdomainandareplannedtogen
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space-chargeinthelowenergyregime,an
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!Vjitsthevelocity,and!Xjisthepositi
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equation boundarybetweenvacuumandam
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10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.
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1.0 0.8 0.6 0.4 0.2 E=10 MeV E=42 M
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00000000000000000000000000000000000
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harmonicwithproperchoiceoftheKvalue
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Bu(rms)0.28T ParameterValueUnit Nu
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2.7TheJeersonLabIRproject usedasexp
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Study TheFELdriveraccelerator:Latti
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ParameterValue x -0.178 x(m) 8.331
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Thepurposeofmeasuringthetransverser
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computedusingthelatticeset-upuseddu
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etatronexcitationaspicturedingure3.
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∆ x (mm), Corrector 0F00H ∆ x (
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1 0.5 0 spreadoftheparticlewassetto
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2 1 −5 −3 −1 1 3 5 k (m q −
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Experimentallythecalibrationcoecien
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tioned.FromthetransferfunctioninFig
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Pickup Experiment #2 #3 #4 Simulati
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Fromthesebothmeasurementitispossibl
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. . . . . . . . . . . . . . . . . .
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10 5 Sext. ON Figure3.20:Eectofthes
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spaceabeamthatconsistsofNparticles,
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10 1 5 10 0 500 1000 1500 2000 2500
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EnergySpread(%) BunchLength(mm) Par
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15 Effect of RMS energy Spread 10 F
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6.4.4BunchSelfInteractionviaCoheren
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100 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
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Inthesecondseriesofrun,wewereableto
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Conclusion Chapter7 Therecirculator
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[2]Y.Shibata,T.Takahashi,T.Kanai,K.
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[29]J.-C.Denard,C.Bochetta,G.Traomb
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[65]B.C.Yunn,\ImpedancesintheIRFEL"
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TEM:transverseelectricmagnetic. TOF
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B.2Anoteonspacecharge Wedenetheslop
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B.3.3PARMELA FeaturesofJeersonLabVe
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Initialization MAIN Stop INPUT DECK
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C.0.6TheDispersionRelationsfor^S(!)
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have: Howeveritshouldbenotedthatweh
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FigureD.1:OverviewoftheRF-controlsy
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