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

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B.3.1DIMAD TheprogramDIMAD2studiesparticlebehaviorincircularmachinesandinbeamlines.Thetra- B.3TheSimulationTools chargedparticlecomputercodes. B.3.2TLIE likeitspredecessorDIMAT,istheresultofmanyyearsofexperimentingwithseveraldierent jectoriesoftherelativisticparticlesarecomputedaccordingtothesecondordermatrixformalism. Itdoesnotprovidesynchrotronmotionanalysisbutcansimulateit.Theprogramprovidestheuser withthepossibilityofdeningarbitraryelementstotailortheprogramtospecicuses.DIMAD, tosecondorderlikedimad.ThePhysicsbehindthiscodeisbasedontheuseoftheLieAlgebra vectorspacewithaproductverifyingtheproperties(1)(xy)=(x)y=x(y)and(2) operatortopropagatetransfertmapalongabeamlinesection.ALiealgebraisanalgebra(i.e.a Tlie3isageneral6DrelativisticdesigncodewithaMADcompatibleinputlanguage.The particularityofTLieisitsabilitytocomputetransfermapatanarbitraryorderandnotonlyup Pi@f TheLiealgebraoperatorusedinBeamDynamicsisthePoissonbracketdenedas:[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 @piwheregandfarefunctionsofthegeneralizedvariablespiandqi.Thereason thatfisnotanexplicitfunctionoftimei.e.@f :f:isaLieoperator.ToillustratehowtheTliecodeworks,let'sassume,forthetimebeing, (pi;qi).Instandardnotation,thePoissonbracketoperatorisoftenwritten:f:g=[f;g]where whereHistheHamiltonianthatgovernstheevolutionofthedistributionfintheconjugatespace df dt=[f;H]=:f:H=:H:f,andusingpurelysymbolicequationwehaveintermofoperator: df dt=@f @t+[f;H] @t=0inEqn.(B.8).ThenEqn.(B.8)becomes: (B.8) e:H:f=f+[H;f]+1=2[H;[H;f]]+:::. ThebeautyofLieAlgebratechniqueresidesinthatthecomputationofthefunctionfatatime dierentialequationise:H:fwheretheexponentiationofLieoperatorisdenedastheseries order,thisistheprincipleonwhichTlieisbased:thehamiltonianHalongwiththecorresponding operatore:H:arecomputedforeachbeamlinepieceandareconcatenedusingtheCampbell-Baker- calculationcanbecarriedatanyorderbycomputingthePoissonbracketseriesatthedesired t=t0+,knowingthefunctionfattimet0,justconsistsofcomputingft=e:H:ft0;such dt=:H:sotheLieoperator:H:reducestoasimpletimederivative.Thesolutionofthis REPORT285SLAC-StanfordUniversityCA-USA(1985) Hausdortheorem,thatstatese:HA!C:=e:HA!B:e:HB!C:,toobtaintheLieoperatorforawhole beamlinesection[A,C]. 3ThecodewaswrittenbyJohannesvanZeijtsandFilippoNeri 2R.Sevranckx,K.L.Brown,L.Scachinger,andD.Douglas,\UsersGuidetotheProgramDIMAD",SLAC
B.3.3PARMELA FeaturesofJeersonLabVersion \PhaseandRadialMotioninElectronLinearAccelerators."Itisaversatilemulti-particlecode thattransformsthebeam,representedbyacollectionofmacroparticles,throughauser-specied linacand/ortransportsystem.Itincludesa2-Dspace-chargecalculationandanoptional3-D point-to-pointspace-chargecalculation.PARMELAintegratestheparticletrajectoriesthrough theelds.Thisapproachisespeciallyimportantforelectronswheresomeoftheapproximations nothold.PARMELAworksequallywellforeitherelectronsorions.PARMELAcanreadeld usedbyothercodes(e.g.the"drift-kick"methodcommonlyusedforlow-energyprotons)would 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 chargeQiandlengthiorbitingonacirculartrajectoryofradiusRidetectedatthepresenttime byanobserverelectronjderivedandexpressedas: TheimplementationoftheCSRinteractionintothePARMELAcodecloselyfollowsthemethod describedbyCarlsten6wheretheelectriceldgeneratedataretardedangle0byalineiofuniform forshortelectronbunchesincircularmotionusingtheretardedGreen'sfunctiontechnique"Phys.RevE54num1, TN-94-040,JeersonLab,NewportNews,VA-USA(1994) pp838-845(1996) 4K.T.McDonald,IEEETrans.Elect.Dev.35p2052(1988) 6B.E.Carlsten,\Calculationofthenoninertialspace-chargeforceandthecoherentsynchrotronradiationforce 5H.Liu,\ConceptofVariableParticleSizeFactorforaPoint-by-PointSpaceChargeAlgorithm",CEBAFreport Ei;j=Qi i"1 r(1!ibnij)"12i2ixj Ri+2i(1cos(0))##fr (B.9)
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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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Asforthegeometricemittanceonecanden
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(a) ceramic radiator beam x-wire be
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and validundertheassumptionofaunifo
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1000 900 800 700 Figure4.4:Steadyst
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Fraction of Beam enclosed within (%
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5 2 2 3 4 5 6 7 8 9 10 0 10 5 2 Foi
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Actuator Inserts Foil and Mirror CI
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shortterm,touseasbeamdensitymonitor
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Alongwiththeseimplementedoperations
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whereiistheerrorontheithbeamsizemea
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Let'sassumethethinlensapproximation
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obtainedviathestatisticalanalysis.T
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20 18 16 . 14 . 12 . Figure4.14:Rel
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100 10 −3 10 −2 10 −1 10 that
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multislits mask (copper) Aluminum F
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TheReductionoftheSpaceChargeconditi
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. . . . . . . . . . . . . . . . . .
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Table4.7:Typicalsystematicerroronem
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180 8 160 6 Figure4.22:Anexampleof2
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4.0 3.5 3.0 2.5 Figure4.24:Emittanc
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Characterization LongitudinalPhaseS
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concentrateonthebeamparametersinthe
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1.1 1 mrad 0.9 5 mrad mainlyduetoth
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Population Population 100000 90000
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Theequation5.8yields: f()=jZ+1 11Xn
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presentanoutlineofthisproofbelow,an
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contrastintheCEBAFmachine,varyingas
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1.5 1 correspondtothevarianceofveco
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Ε’ 2 B Mirror M 2 Ε2 Polarizer
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FinallyitisinterestingtonotethatFou
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Itsautocorrelationis:S(z)=8>:(1=w2)
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drivelaserwhereasin(A),suchoperatio
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expansionoftheBFFderivedinthischapt
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Asarstapproximation,wecanestimatetr
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errorpropagation8theoryappliedonEqn
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inducedbyspacechargeisnotaconcernin
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