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

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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) whereisanarbitrarypointoftheupperhalf-planewherelog[~S]isanalytic(notethat()isnot Thedispersionrelationsappliedto()yields: singularat=). i()=PZ+1 1(x)dx x (C.7) (C.6) UsingEqn.(C.6)weget: whichexpandsto: 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) ThetwofarRHStermsarezero,sothatwenallyget: log[^S()]=log[^S()]+iPZ+1+log[^S()]PZ+1 1dx 1dx x(C.9) x relatelog[j^S()j]tothephaseof^S(),(): Byidentifyingtherealandimaginarypartsweobtainthedispersionrelationsforlog[^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) 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) (C.11)
have: Howeveritshouldbenotedthatwehaveassumedfromthebeginningthat^S()isanalyticin ^S(0)=12R,sinceS(t)isarealfunctionnormalizedtounity;therefore(0)=0.Sonallywe (0)canbeestimatedbecausetheFouriertransformat!=0istheintegralofS(t);itgives particularthepointwhere^S()=0willgivesingularityon()whichinturnmustbetakeninto theupperplane.Unfortunatelyitdoesnotimplylog[^S()]hasthesameregionofanalyticity;in accountinthephaseby,whenwriting()asaCauchyintegral,takinginaccountthecontribution totheintegralviatheresiduetheorem. ()=2PZ1 0logj^S(x)jdx x22 (C.13) Inthegure(gureC.1)belowwepresentresultsforasimplebi-modaldistributionthatconsists ofasumoftwonormaldistributions:wehavegeneratedthreetypesofdistribution(a)(c)and (e)andforeachofthemwecomparethephasecalculateddirectlyfromthecomputationofthe FouriertransformofthedistributionwiththephaseretrieveusingEqn.(C.13).Onthisverysimple examplewenoticethatthisequationdoesnotafortiorireproducetheveritablephaseoftheFourier transformofthedistribution.
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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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Golay Cell Signal (V) Golay Cell Si
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Golay Cell Ouput (V) Golay Cell Oup
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150 100 5 4 Figure5.19:Energyspectr
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Page 157 and 158: σ x (m) x 10−3 5 4 3 2 1 0 0 5 1
Page 159 and 160: BeamDynamicsStudies Chapter6 undula
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Page 169 and 170: ε x (mm−mrad) Nominal 6 4 2 cryo
Page 171 and 172: 2.2 2 1.8 Figure6.10:Comparisonofth
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Page 177 and 178: EnergySpread(%) BunchLength(mm) Par
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Page 187 and 188: Conclusion Chapter7 Therecirculator
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Page 191 and 192: [29]J.-C.Denard,C.Bochetta,G.Traomb
Page 193 and 194: [65]B.C.Yunn,\ImpedancesintheIRFEL"
Page 195 and 196: TEM:transverseelectricmagnetic. TOF
Page 197 and 198: B.2Anoteonspacecharge Wedenetheslop
Page 199 and 200: B.3.3PARMELA FeaturesofJeersonLabVe
Page 201 and 202: Initialization MAIN Stop INPUT DECK
Page 203: C.0.6TheDispersionRelationsfor^S(!)
Page 207: FigureD.1:OverviewoftheRF-controlsy
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