High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
Let'sassumethethinlensapproximationtobevalid7.Thenthetransfermatrixbetweenthe entranceofthequadrupoleandtheprolemonitoris: Thex;yinthepreviousrelationaretheminimum-functiononewishestoachieveattheprole monitorstation.Thechoiceoftheminimum-functionhastwoimplications. betatronfunctionatthequadrupoleentrance(forinstanceinthehorizontalplane): HencethebetatronfunctionattheOTRlocationcanbeexpressedasafunctionoftheinitial R= x(k)=R212 R21R22! R11R12 f2x;0+R212 1=f1! 10 x;0 (4.38) wherewehaveusethefactthatx;0=x;0=Lwhenonehastakencareofsettingtheupstream opticstosatisfytherelationderivedpreviouslyinEqn.(4.37). Introducingthefocallength(1=f=k1l)andrecallingthatR12=x;0=x(k=0)yields: x(k)=x(0)+R412k21l2 x(0) (4.39) whosederivativewithrespecttothequadrupolestrengthk1is: dx(k) dk=2R212k21l2 x(0) (4.40) latterequationshowsthatthechoiceofx;0,whichwehavesuggestedearliertobeaslargeas possibletoreducetheerroronthebeamsizemeasurement,directlyaectstheslopeofthebeam sizevariationontheprolemonitor:atoolargex;0willgivea\atlooking"variation.Therefore thereisanoptimumbeamvalueforx;0;thisoptimumshouldbedeterminedviaaniterative withthesamekindofrelationintheverticalplane(replacingxindexbyyandR12byR34).The processusingnumericalsimulations. (4.41) 4.4.3Themulti-monitormethod theTwissparametersatthereferencepointbyEqns.(4.24).Indeed,weneedtomakesurethat Anotherwayofvaryingthetransfermatrixbetweenthereferencepointwhereonewishestocompute thesetwoequationsarenotredundant,namelythat: ofthismeasurementisthatnoelementhastobevaried.Howevertogetaprecisemeasurementa dedicatedopticallatticesettinggenerallyneedtobeelaborated. Letanalyzequantitativelythemethod.Thebeamsizeonaprolestationkandlarerelatedto requiresatleastthreemonitorsbutoneshouldusemoreofthemforredundancy.Anadvantage thebeamparametersandthebeamprolemeasurementstationistomeasurethebeamproleat dierentpositionalongthebeamlinewhichareseparatedbynon-dispersiveoptics.Thismethod 20)butitprovideseasieranalyticalresultsanddoesnotchangesignicantlythephysicsofthepresentdiscussion. Thetreatmentofthefullproblemincludingthethicklenstransfermatrixisdonevianumericalmodeling. 7Thisisafalsestatementifweconsiderthewholerangeofthemagneticstrengthforthequadrupoles(20
zero): Thesethreedeterminantsyieldthesameequation(assumingtheR11andR12tobedierentfrom Henceinorderthelatterequationtobeveried,onemusttakecaretosettheopticallatticeso advancebetweenkandl: whichimplies,usingthegeneralformulationofabeamtransfermatrixintermofbetatronphase R11;kR12;lR12;kR11;i6=0 sin()6=0 (4.44) (4.43) Anothercarethathastobetakenistomakesurethatattheprolemeasurementstationthebeam isnotatawaist;thiswillenlargetheerrorbarsonthemeasurement(oneshouldmakethebeam aslargeaspossiblecomparedtotheerroronthebeamsizemeasurement). 4.4.4SimulationofEmittanceMeasurementintheIRFEL (withn2N). thatthebetatronphasebetweentheviewersbeingusedinthemeasurementisdierentfromn Afterthedecompressorchicane,thebeamlineconsistsofaquadrupoletripletandisinstrumented theOTRlocation(suchoptimizationwillbediscussedinmoredetailinChapter5).Afterhaving makesurewecanhavea\right"beamsizevariationoverthequadrupoleexcitationrange.Typically theprolemonitor.Toperformsuchmeasurementoptimallyweneedtosettheupstreamopticsto weusethemagneticopticscodedimadtottheupstreamquadrupolestosatisfyEqn.(4.37)at scanmethod,weusetheOTRmonitorlocatedinthedumpbeamline3.43mdownstreamtheexitof thelastquadrupoleofthistripletquadrupole.Thereforeinthiscasewehaveinvestigatedwhether thisquadrupolecouldbeusetovarythetransfermatrixwhileobservingbeamsizevariationon withtwoOTRviewers.Forthesimulationoftheemittancemeasurementusingthequadrupole minimumbetatronvalueattheOTRlocationminimizestheerrorbarsonthededucedemittance (andontheotherdeducedTwissparameters).Itisseenthat6misareasonablenumberforwhich thesystematicerrorsachievedonemittancemeasurementcanbewellwithinthedesired10%level. plottedinFigure4.14.Fromthisgureonecanobviouslynoticethatthechoiceofthelargest sizevariationispresentedingure4.13.Thededuceduncertaintiesontheemittanceforthesetwo dierentvaluesoftheminimumbetatronfunctionversustheerrorsonbeamsizemeasurementis properlytunedtheseupstreamquadrupoles,wehavenumericallystudiedthevariationofbeam Tofullysimulatethewholemeasurementandbenchmarkourdataanalysisalgorithm,wepropagate sizefortwodierentminimum-functionsatthelocationoftheOTRviewer.Atypicalbeam wesimulatethemeasurement:wevarythequadrupolestrengthandforeachsettingpropagatethe parametersuptothelocationoftheOTRmonitorwherewecomputeandrecordthebeamsize. theentranceofthevaryingquadrupole(i.e.lastquadrupoleofthetripletaforementioned).Then usingthedimadcodetheexpectedparameteratthelinacexit(ascomputedwithparmela)upto Intable4.4wecomparetheresultsobtainedonthecomputedbeamparametersatthequadrupole entrancefacewiththedimadinitialparameters:theresultsareinexcellentagreement.Wealso comparetheerrorbarsobtainedwithourerroranalysiswiththeerrorbarsstatisticallycomputed onasetof200simulationsofthemeasurementinwhichthebeamsizeisrandomlygeneratedalong anormaldensitycenteredonthebeamsizecomputedwiththeopticscodewithavarianceequalto thermsresolution(60m).Theconclusionisthattheerrorpropagationagreeswiththevariances
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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
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