ANALYSISTECHNIQUESANDDIAGNOSTICSOFNON-RELATIVISTICHADRON BEAMS By ChristopherRichard ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics-DoctorofPhilosophy 2020 ABSTRACT ANALYSISTECHNIQUESANDDIAGNOSTICSOF NON-RELATIVISTICHADRONBEAMS By ChristopherRichard Beamdiagnosticsareessentialtotheoperationofhadronparticleaccelerators.They areusedtotunetheaccelerator,verifybeamlinemodes,ensureminimalbeamlosses,and characterizeandmonitorthebeamquality.Byaddingandimprovingthemeasurementsof thebeamproperties,theoperationoftheacceleratorcanbebetterinformedandimproved. Additionandimprovedmeasurementsofthebeampropertiescanberealizedbydeveloping newanalysistechniquesfortheexistingdiagnosticdevices. Thisdissertationpresentsfurtheranalysisofmeasurementsfromtwodevices.Firstly,it discussesconvertingphasespacemeasurementstakenwithanAllisonscannerfromposition- anglecoordinatestoaction-phasecoordinates.Inthiscoordinatesystem,thedistribution isstableunderchangestolinearoptics.Thisallowsfordirectcomparisonofphasespace measurementstakenattlocationsorwithttransversefocusing.Inaddition, thisstabilitycanmakeiteasiertovisualizeandquantifythebeamtails. Secondly,beammeasurementstakenwithBeamPositionMonitors(BPMs)by measuringmultipleharmonicsarepresented.Themeasurementsareprimarilyfocusedon non-relativisticbeamswherethetransverseandlongitudinalcanbetotheBPM signals.Whilethesemeasurementswereunsuccessful,itunderstoodwhytheyfailedand howtoavoidthesameissuesforfuturemeasurements. Lastly,thedesignofateststandtocalibrateBPMsfornon-relativisticispresented. Theteststandreliesonahelicaltransmissionlinecancanpropagatesignalswithphase velocityof0.03c.Itisshown,withtheappropriategeometry,thatthephasevelocity,pulse propagation,andfromthehelicaltransmissionlinecanmatchthethoseofa non-relativisticbunch. Copyrightby CHRISTOPHERRICHARD 2020 ForGrandad v ACKNOWLEDGMENTS Firstandforemost,Iwouldliketothankmyadviser,SteveLidia.Iamgratefulforhis encouragementandpatienceformetopursuemyownideasthatwerefruitfulasoften asnot.IalsomustthankmymentoratFermilab,SashaShemyakin.Ouralmostdaily discussionsduringmytimeworkingwithhimalwaysgavemeseveralideasandpathsforward toconsider.IlearnagreatdealfromSteveandSashaaboutacceleratorphysicsandhow tobeasuccessfulscientist.Inaddition,thankyoutotherestofmyguidancecommittee: SteveLund,PeterOstroumov,ScottBogner,andSelinAviyente. Iamgratefulforeveryonewhohashelpedmewithexperimentsandthroughdiscussions. SpecialthanksmustbegiventoScottCoganwhodevelopedtheTISwaveformsandSasha PlastunfortakingmeasurementsformewhenIcouldnot.Withoutthem,myworkwith theBPMswouldnotbepossible.Inaddition,IamindebtedtotheentirePIP2ITgroup atFermilabLionelProst,Jean-PaulCarneiro,ArunSaini,andBruceHanna.Theywere tremendouslyhelpfulandwelcomingduringmyshortstaywiththem.Also,conversations withGregSaewertweregreatlyhelpfulforthedevelopmentofthehelicaltransmissionline. IcannotthankeveryoneatFermilabwithoutacknowledgingtheAcceleratorScienceand EngineeringTraining(ASET)programatMSU.TheASETprogrammadethisresearch possibleandprovidedmewiththeexperienceofspendingayearworkingatFermilab. Iamimmenselygratefulformyfriendsandfellowgraduatestudents:JamesKakos,Mel Miller,BryanIsherwood,JonathanWong,AndrewLajoie,CrispinContreras,andmany others.Manyofthemhelpeddirectlythroughconversationsandcritiquingmyresearch presentationsandtheyallprovidedmuchneededhumorandemotionalsupport.Lastly, thankyoutomyfamily.IknowmostofthemhavenoideawhatIdo,butregardlessthey vi providedcontinuoussupport. TABLEOFCONTENTS LISTOFTABLES .................................... x LISTOFFIGURES ................................... xi Chapter1Introduction ............................... 1 1.1HadronAccelerators...............................1 1.2Acceleratorfrontends..............................2 1.2.1Hadronbeamsources...........................2 1.2.2Lowenergybeamtransport.......................3 1.2.3Radiofrequencyquadrupole.......................4 1.2.4Mediumenergybeamtransport.....................6 1.2.5Measuredbeamlines...........................6 1.3Transversebeamdynamics............................7 1.3.1Singleparticledynamics.........................7 1.3.1.1Thebetatronfunction.....................10 1.3.1.2Action-phasecoordinates...................11 1.3.2Phasespacedynamics..........................14 1.3.3Spacecharge............................15 1.4Diagnostics....................................17 1.4.1Beampositionmonitoring........................19 1.4.2measurements...........................25 1.4.3Phasespacedistributionmeasurements.................27 1.4.4Beamtailmeasurements.........................29 Chapter2AnalysisofPhasePortraitsusingAction-PhaseCoordinates . 30 2.0.1Allisonscannernoise...........................31 2.1Beamdescriptionusing J ˚ coordinates...................34 2.1.1Coredescription..............................34 2.1.2Discussiononthebeamdistribution...................35 2.1.2.1Centralparameters.......................37 2.1.3Allisonscannerphasedependence....................40 2.1.4Taildescription..............................43 2.2Selectedbeammeasurements...........................46 2.2.1Backgroundnoiseremoval........................46 2.2.2Quadrupolescan.............................50 2.2.3Comparisonofmeasurementsintlocations...........50 2.2.4Distributionattbeamcurrents.................53 2.2.5Scraping..................................54 2.3Futurework....................................60 viii Chapter3Beammeasurementsusingbeampositionmonitors .. 62 3.1BPMsignals....................................63 3.1.1Validityofthepencilbeammodel....................63 3.1.2AmoredetailedBPMmodel.......................64 3.1.3Pickupsignalvariationwithtransversedistribution..........67 3.1.4Buttonsumsignal............................70 3.1.5BPMresponsesimulations........................72 3.2Bunchmeasurements...........................75 3.2.1TISwaveforms..............................75 3.2.2Filtering..............................77 3.2.3Beamlinemeasurements.........................79 3.2.4RFbunchervoltagescan.........................80 3.3Futurework....................................85 Chapter4HelicaltransmissionlineforBPMcalibration .......... 86 4.1Transmissionlinesprimer............................87 4.2Initiallyconsideredteststandgeometries....................92 4.3HelicalRFstructures...............................94 4.4Helicaltransmissionlines-analyticsolution..................95 4.4.1Dispersiondistortioncorrection.....................101 4.4.2Dispersionreductiongeometry......................103 4.4.3Higherorder............................108 4.4.4ImpedanceProperties...........................110 4.4.4.1InductanceandCapacitance..................113 4.5Simulations....................................115 4.5.1Matching.................................116 4.5.2Dispersionmeasurements.........................118 4.5.3Impedancemeasurements........................123 4.5.4Electricscaling...........................124 4.5.5Beamcomparison.............................126 4.5.6helices...............................128 4.6Futurework....................................130 Chapter5Conclusions ................................ 133 BIBLIOGRAPHY .................................... 136 ix LISTOFTABLES Table2.1:ParametersofthePIP2ITMEBTAllisonscanner..........32 Table2.2:PIP2ITMEBTbeamparameters....................32 Table2.3:Averagermsemittancesandcoreandtailparametersforthethree locationsoftheAllisonscanner.....................53 Table3.1:ParametersofthedistributionsusedinFig.3.6............69 Table4.1:Simulatedhelicaltransmissionlinegeometryparameters.......115 x LISTOFFIGURES Figure1.1:Thebeamofhighintensityhadronaccelerators[1].Theyellowlines representbeampowercontours.....................2 Figure1.2:ChargestatesofuraniumproducedbyandECRionsource[2]....3 Figure1.3:SketchofanRFQ[3].Fourvanescreateaquadrupoletofocus thebeamtransversely.Thevanesaremodulatedtocreatealongitu- dinaltoacceleratorthebeam....................5 Figure1.4:PIP2ITfrontend.............................7 Figure1.5:OverviewofthediagnosticsintheFRIBfrontend...........8 Figure1.6:Aninitiallyuniformbeamdistributioninphasespaceevolvingina quadrupolefocusingchannelwithspacecharge[4].Thebeamun- dergoestation,developsan\S"shapeanddilutingtheentire phasespacecausingtheemittancetoincrease.............18 Figure1.7:Asthebeamacceleratestheelectriciscompressedintotothe planeperpendiculartothevelocityresultinginthesame onoppositesidesofthebeampipe[5]..................22 Figure1.8:Simulationresultsforthecalculatedbeampositionusingthelinear = (left)andahigherorderpolynomialcorrection(right)[6]....23 Figure1.9:AsketchofacapacativeBPManditsequivalentcircuitisshownon theleft[7].Duetothecapacitancetoground,BPMssuppressthe lowfrequencycomponentsofthemeasuredsignals(right, R =50 C =3pF)..................................24 Figure1.10:SischematicofanAllisonscanner.Theredlineshowsthe trajectoryofparticlesthroughthedevice................28 Figure2.1:Measuredphaseportraitin x x 0 coordinatesand J ˚ coordinates takenatlocation1............................31 Figure2.2:InitialPIP2ITMEBTation.TheAllisonscannerisinlo- cation1andmeasuresthetransversephasespaceinthehorizontal plane...................................32 xi Figure2.3:FinalofthePIP2ITMEBTwiththeAllisonscannerin location3andorientedtomeasurethetransversephasespaceinthe verticalplane.Transversefocusingisprovidedbyquadrupoleswith twodoubletsandseventriplets.....................32 Figure2.4:Thermsscatterofthepixelamplitudesplottedasafunctionofthe averageamplitudes(blue)phaseportraitsmeasuredatlocationone (left)andlocationthree(right).Theorangelinesshowthefrom Eqs.2.1and2.2..............................35 Figure2.5:Comparisonofthemeasureddistributioninactioninthebeamcore (black)withseveralidealdistributions:Gaussian,KV,UG,andWB. NotethattheUGdistributionisphase-dependent,and,therefore, pixelintensitiesvaryforagivenactionandisrepresentedherebythe areashadedingreen...........................38 Figure2.6:ActiondistributionusingcentralTwissparameters(red)andrms Twissparameters(blue).........................39 Figure2.7:Centralslopeasafunctionoftheportionofthebeamremoved.The curveistoacubicpolynomialtodeterminetheminimum c which isusedtothecentralparamters.................39 Figure2.8:Theshadedarearepresentsthepassedphasespaceareaforagiven positionandvoltagesettingofanAllisonscanner.Thegridisthe displayedpixelsize............................41 Figure2.9:Left:VariationoftheTwissparameterswithquadrupolecurrent. Right:Thecentralslopeisconstantwhenaccountingforof theslitsize................................42 Figure2.10:Theintensitiesarebinnedinactiontodetermine J tr withEq.2.14 toseparatethetailsfromthecore.Theerrorbarsrepresent 3 ˙ ..43 Figure2.11:Phaseportraitinposition-anglephasespace(a)andaction-phase phasespace(c).Thebeamsplitsintotwobranchesseparatedin phaseatlargeactions.Thepixelamplitudeversusaction(b)shows deviationfromthecoredistributionatlargeaction..........45 Figure2.12:Theamplitudeofthe0 th harmonicofthepixelintensitiesasafunc- tionof ˚ foraGaussianmodelisatleastanorderofmagnitude largerthantheamplitudeofthe2 nd harmonic.Forthemeasured beams,theratioofthe0 th and2 nd harmonicamplitudesissimilar tothemodelatlow J .However,thetailscausethe2 nd harmonicto dominateatlargeaction.........................46 xii Figure2.13:VerticalRMSemittancewithhorizontalscraping.Theedata pointscorrespondto1,2,3,4,5mAofbeamcurrentafterscraping.49 Figure2.14:Analysisofphaseprotraitsinaquadrupolescan.(a)phaseportraits in x x 0 coordinatesrecordedatthequadrupolecurrentsincreasing fromlefttorightandfromtoptobottomfrom3.06Ato5.46A.The x and x 0 rangesineachplotare30mmand24mrad,correspondingly. Notvariationoftheslit-correctedcentralslopeandpercent inthecoreareobservedwhilethequadrupolestrengthwasscanned (b).Theaveragebranchphaseagreeswithsmallchangesofthesimu- latedbetatronphase(c).Phaseportraitsinaction-phasecoordinates fortheminimumandmaximumquadrupolecurrentsoverlap(d),(e). Theportionofthebeamoutsideofagivenactionisstableovermost ofthebeam(f)..............................51 Figure2.15:Comparisonoftheamplitudeversusactiondistribution(left)atthe beginningandendoftheMEBTshowsabetweenthehor- izontalandverticalplanes.Thefartailsextendfartheratlocation3 comparedtolocation2(right)whichisasignoftailgrowth.Thepor- tionoutsideagivenactionatlocationonecannotbeusedtosearch fortailgrowthbecauseitrepresentsatplane.Theshaded areasrepresentthermserrorscalculatedbypropagationofthepixel amplitude.........................53 Figure2.16:Thetotalbeamintensity(a),peakpixelamplitude(b),emittance (c),andTwissparameters(d)fortextractionvoltages V extr . ParametersareplottedasfunctionsofthebeamcurrentintheLEBT.55 Figure2.17:Phaseportraitswithscraping.Rows(a)-(d)correspondtomoving intothebeamoneofthescrapersalongthebeamlinepresentedin Fig.2.3;fromtoptobottomM00,M11,M61,M71.Ineachcase, 0.5mAisinterceptedoutoftheinitial5mA.Therow(e)represents thebeamwhentopandbottomscrapersareinsertedinM00and M11stations.Thesolidlinesrepresenttheattemptofpropagating thescrapelinesaccordingto5mAbeamsimulations.Thedashed linesrepresentpropagationwiththephaseadvanceincreasedby10%. Seeotherdetailsinthetext.......................57 Figure2.18:Phaseofthesecondharmonicofthepixelintensitiesasafunctionof phasevarieswithaction.Largervariationsareseenfatherfromthe startoftheMEBTshowingthephaseadvancevariesacrossthebeam.60 Figure3.1:Variationof = whenchanging ˙ x ofaGaussianbeamcenteredat x 0 =2mmand y 0 =1mmfora47mmmapertureBPMwith20mm diameterroundpickups..........................64 xiii Figure3.2:GeometryoftheFRIBMEBTBPMs.Thepiperadiusis23.75mm andthepickupradiusis10mm[6]...................67 Figure3.3:Signalonaandcurved20mmdiameterBPMpickupfroma =0.033beam.Thetwogeometriesgivesimilarresultsupto ˘ 400MHz.68 Figure3.4:Variationinthemeasuredspectraona20mmdiameterpickupina 47.5mmdiameterpipeforanpencilbeam.Thespectraare normalizedtothecenteredcase.....................69 Figure3.5:VariationinthemeasuredspectrumforacenteredroundGaussian beamoftsizes.Thespectraarenormalizedtoacentered pencilbeam................................70 Figure3.6:ComparisonofmeasuredspectrafromaGaussiananddoubleGaus- sianbeamwiththesameandsecondordermomentswith =0.033 (left)and =0.15(right).Atlarge g thetdistributionresults inatmeasuredspectra.....................71 Figure3.7:(Top)Variationofthesummedsignalforanpencilbeam.The variationsinthespectraareafactorof ˘ 7lowerthanthenon- summedsignals.(Bottom)Thesummedsignalsofa1mmo Gaussianbeamnormalizedtothesinglepickupsignals.Thevaria- tionsofthesummedspectraarereducedby20%comparedtothe non-summedsignals...........................73 Figure3.8:CSTsimulationsusingthewaksolvermustbetoauniform squarebeamwithsidelengthgivenbythemeshsize.Forthegiven frequencyrange,thistransversedistributioncanbeignored for > 0.15................................74 Figure3.9:ModeloftheFRIBBPMSinCSTMicrowaveStudio.........75 Figure3.10:Fittingtransmissioncot, S 2 ; 1 ,ofatheCSTbuttonmodelto determinetheimpedance.........................76 Figure3.11:SimulationresultscomparedtoanalyticresultsofBPMpickupsignals fromacenteredpencilbeamusingthewaksolver........76 Figure3.12:ExampleofthesamplingproceduretomeasuretheTISwaveforms. Eachsampletakenbythedigitizerisataerentphasewithrespect tothesignalandcanbeusedtoreconstructtheindividualrepeated pulse(imagecourtesyofS.Cogan)...................77 Figure3.13:CableandboardcalibrationforfourbuttonsonaBPM.......78 xiv Figure3.14:ExampleofameasuredTISwaveformintheFRIBMEBT(left). Thebumpat15nsisapartiallyRFbucket.Thiscausesthe harmonicsof80.5MHztobehigherthantherestoftheharmonicof 40.25MHz(right).............................80 Figure3.15:RawandcalibratedspectraofaBPMintheMEBT.........81 Figure3.16:Fittingthemeasuredspectraandfractionalerrorfromthemeasured values...................................81 Figure3.17:Comparisonofthemeasuredbunchlengthwithsimulationsatthe thirdBPMintheMEBTwhenthebunchercavityvoltageisvaried. TheBPMmeasurement,whileclosetotheexpectedvalues,failto producedtheexpectedtrend.......................82 Figure3.18:ExamplesoftransversemeasurementintheFRIBMEBTmea- suredwithawiremonitor....................83 Figure3.19:Spectraandapencilbeamtomeasurementsat =0 : 185.The droopatlowfrequencyishypothesizedtobecausedbyincorrectly modelingthepickupimpedance.....................84 Figure4.1:Circuitmodelofatransmissionline[8].................87 Figure4.2:CrosssectionofaGoubauline......................93 Figure4.3:NormalizedphasevelocityofaGauboulinewith R i =2mm, R e =20mm, i =10 0 ,andtwotdielectriclayerradii a .Thelowfrequency limitistoolargetoreplicateanon-relativisticbeam..........94 Figure4.4:Crosssectionofhelicaltransmissionlinegeometry.Thegreycircle representsthehelix............................97 Figure4.5:Thedeformedpulseduetodispersionisfoundbyanalyticallypropa- gatingapulseasetdistance.Thiscanbecorrectedbyreversingthe deformedpulseintimeandinputtingitintothetransmissionline. Whenthispulseispropagatedalongthetransmissionline,thedis- persionwillcorrectthepulseatthesetdistance.TheDUTcanbe placedatthislocation..........................102 xv Figure4.6:PropagationofaGaussianpulsealongahelixwithreducingpitch. Thehorizontalaxisisthelongitudinalpositionandtheverticalaxis thetransverseposition.Theverticallinerepresentstheendofthe pitchcompressionsection.Thepulseiscompressedbutmaintainsits formduringthepitchreductionbutdispersiondeformsthepulsein theconstantpitchsection........................104 Figure4.7:Dispersionforttransmissionlinegeometries.Theaddition oftheinnerconductortly v p ( ! ).Helixparameters s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 .........105 Figure4.8:Dispersionscalingwith s (left)and r (right).Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 .........106 Figure4.9:Thelowandhighfrequencylimitsofthephasevelocityscalewith sin( ) : ..................................107 Figure4.10:Despitetoreducedispersionthepulsesnearthehelix aretlydeformedduetodispersion(left).However,the nearthepipewallhavethehighfrequencycomponentssuppressed andthereforemaintaintheirshape.Thissamedistortsthe pulsewhenthehelixisinthepipe.Oupto5mmcanbe achievedwithminimaldeformations(right).Thepulsesshownhere arefromCSTMicrowaveStudiosimulations..............107 Figure4.11:Theshadedareasareforbiddenregionsofatapehelixmodel.Forthe geometry, =0.05, R p =20mm, a =5mm, s =0.5mm,upto8GHz (bluecurve)thehelixoperatesfarfromtheforbiddenregionsandthe sheathhelixwillbeagoodapproximation...............109 Figure4.12:Higherordermodeshavefrequencyat ka ˇ n andpropagate atthespeedoflightatthecutfrequency.Shownisthe threemodesforahelixwithgeometry =0.05, R p =20mm, a =5mm, s =0.5mm, i = 0 ............................109 Figure4.13:Impedancescalingwith s and i .Thesolidlineistheexternal impedanceandthedashedlineistheinternal.Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 .........111 Figure4.14:Thelowfrequencylimitoftheimpedancescaleswith1 = sin( )...111 Figure4.15:Sensitivityofthelowfrequencylimitoftheexternalimpedanceto variationofdtparameters.Theimpedanceismostsusceptible tochangesin s ..............................112 xvi Figure4.16:Scalingoftheinternal C and L with s and i .Thedecreasing s reducingthecapacitanceandincreasestheinductancebyapproxi- matelythesamefactor.Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 .....................114 Figure4.17:MeshingofhelicaltransmissionlineinCSTMicrowaveStudiofor timedomainsimulations.........................115 Figure4.18:Helicaltransmissionlinemodel.Microstripsareusedtomatchthe inputandoutput.............................116 Figure4.19:Lumpedelementsusedforimpedancematching............116 Figure4.20:S 1 ; 1 withandwithoutaresistiveL-networkformatching.......117 Figure4.21:Theradialelectricatthewallshowsminimaldeformationdur- ingpropagationalongthetransmissionline.Theslowpulsesare proceededbyasmallerspeedoflightsignalthatreappearsoncethe slowpulsehasreachedtheendofthetransmissionline........118 Figure4.22:Theelectric0.5mmawayfromthehelixhasastronghar- monicduetothehelixwindings.At1.5mmawayfromthehelixthe theangulardependenceistlyreduced............119 Figure4.23:Probesuiteusedtomeasurethefromthehelix.........120 Figure4.24:Thephasesateachfrequency(topleft)foreachprobeare accordingtoEq.4.75tocalculatethephasevelocity.Thetopleft plotshowstheat500MHz.Theresultingphasevelocityagrees withtheorywithin3%..........................121 Figure4.25:Exactmatchingofthedispersioncanbeachievedbysettingthehelix radiusintheanalyticmodelto a + a usedinthesimulations.The helixradiusinsimulationsis5mmforallpresentedmeasurements with a givenby Da oneachplot.Thestatedhelixradiusisthe radiususedtoanalyticallycalculatethedispersion...........122 Figure4.26:Theanalyticimpedancematchestheresultsfromsimulationswith5%124 Figure4.27:Thecotswerederivedfromtheelectric1.5mm fromthehelix.Theseareusedtocalculatetheanalyticelectric 15mmfromthehelix.Thisagreeswellwithsimulations.125 xvii Figure4.28:Theintheexternalregionissuppressedatlowfrequencycom- paredtotheintheinternalregion.Thismatcheswithsignals measured1.5mmthehelixinsimulations.ForaGaussianinput, theintheexternalregioncanberoughlytotheof twoGaussians...............................127 Figure4.29:Thefromthehelixbestmatcharingbeam. R i =4.5mm, a =5mm, R e =20.65mm, =0.05, i =3 : 5 0 .............129 Figure4.30:Thedispersionandimpedanceseesnotvariationdueto upto10mm.......................130 xviii Chapter1 Introduction 1.1HadronAccelerators Hadronacceleratorsforfundamentalscienceresearchcontinuetopushtohigherbeampower whichisaproductofthebeamenergyandcurrent[1](Fig.1.1).Theenergyfrontieris pushedinprotonacceleratorssuchastheLargeHadronAccelerator(LHC)atCERN.These acceleratorsareusedforhighenergyphysicsresearchtoproduceandstudyexotic,heavy particlessuchastheHiggsboson.Othermachinesincreasethepowerbyincreasingthebeam current.Thesemachines,suchastheFacilityforRareIsotopeBeams(FRIB)currentlybeing builtatMichiganStateUniversity(MSU),requirehighercurrentstoincreaseproduction ratesofrarephenomenatoachievebetterstatisticsfortheusers.Hadronacceleratorsalso haveusesinthemedicale.g.productionofradioactiveisotopesfortumorimaging, andindustry,e.g.processingofsemiconductors. Allofthemachinesinthislargefamilyofacceleratorsrequireminimalbeamlosses.For highpoweraccelerators,thebeamscancausetdamagetotheacceleratorifevena smallportionofthebeamislost.Duetothetlyhighermassofhadronscompared toelectrons,whenhadronsstrikeasurfacetheydeposittheirenergyoverarelativelyshort distantwhichcancausetdamagetothematerial.Forlowerpowermachines,losses needtobeavoidedtoreducemaintenanceandincreaseuptime.Toensureminimallosses, thebeamqualitymustbemeasuredandmonitoredwithadiagnosticsuite. 1 Figure1.1:Thebeamofhighintensityhadronaccelerators[1].Theyellowlinesrepresent beampowercontours. 1.2Acceleratorfrontends Oneoftheimportantsectionsofhadronacceleratorsisthefrontend.Inthefrontend, thebeamiscreated,focused,andinitiallyacceleratedbeforeitentersthemainaccelerator. Thissectionistypicallycomprisedoffourparts:thebeamsource,theLowEnergyBeam Transport(LEBT),theRadioFrequencyQuadrupole(RFQ),andMediumEnergyBeam Transport(MEBT). 1.2.1Hadronbeamsources Hadronbeamsaregeneratedasacontinuous,DCbeamwithenergy ˘ 10skeV/uwhereu isthenumberofnucleons.ProtonacceleratorstypicallyuseH sources[9]becausethese areeasiertoproducethanfullystrippedhydrogen.Theelectronsarestrippedtocreate 2 Figure1.2:ChargestatesofuraniumproducedbyandECRionsource[2]. aprotonbeamfurtherdownthebeamlineoncethebeamasbeenaccelerated.Heavyion acceleratorsrelyonsourcessuchasElectronCyclotronResonance(ERC)ionsourcesto generatehighchargestateions[10].BothH andECRionsourcesproduceavarietyof ions.H sourcescanoutputelectronsandseveralchargestatesofhydrogen.ECRion sourcesproducesawiderangeofchargestates,e.g.theproducedchargestatesofuranium fromanECRionsourceareshowninFig.1.2. 1.2.2Lowenergybeamtransport ThebeamfromtheionsourcegoesintotheLEBT.TheLEBTideallyselectsasinglecharge statecreatedbytheionsourcetoinjectintothemainaccelerator.Acceleratorstypically canonlypropagateonechargestateandtherestwouldbelostbecauseacceleratorsrelyon electricandmagnetictofocusandacceleratethebeam.Toremovethecontaminating 3 particles,thebeamispassedthroughdipolebendingmagnets.Thebendingradius r ofa particlewithmass m ,charge q ,andmomentum p travelingthroughdipolewithstrength B is r = m q p B : (1.1) Therefore,ditchargestatesofagivenisotopewillbebyatamount. Bychoosingthecorrectdipolestrength,thedesiredchargestateandmomentumwillpass throughthebend.However,becausethebeampipehasanon-zeroaperture,itallowsfor somerangeofmomenta p andrangeofchargestates q topassthroughthebend.These contaminatingchargestatesfollowttrajectoriesandcanberemovedbylimitingthe apertureofthebeampipebyinsertingslitstointercepttheunwantedparticles. Inaddition,theLEBTfocusesthebeamtransversely.Transversefocusingreliesonlinear focusingforcesfromeitherelectricormagneticquadruplesorsolenoidmagnets.Thefocusing fromaquadrupoleprovidesuncoupledmotionbetweenthe x and y planes.However, thefromthesemagnetswillfocusthebeaminoneplaneanddefocusintheother. Therefore,atleasttwoquadrupolemagnetsarerequiredtoprovednetfocusinginboth x and y planes.Typicallyquadrupolesaregroupedintostructuresoftwoorthreemagnets, calldoubletsandtripletsrespectively,thatwillfocusinginbothplanes.Solenoidmagnets providefocusingin x and y whichcanmakethemmorecompact.However,theycouplethe motioninthe x and y planesmakingthedynamicsmorecomplex[11]. 1.2.3Radiofrequencyquadrupole TheLEBTtransportsthebeamintotheRFQ.TheRFQprovidestraversefocusingand acceleratesthebeamtoenergiesontheorderof1MeV/u[3].Thetransversefocusingis 4 Figure1.3:SketchofanRFQ[3].Fourvanescreateaquadrupoletofocusthebeam transversely.Thevanesaremodulatedtocreatealongitudinaltoacceleratorthebeam. providedbyatime-varyingelectricmodulatedataradiofrequency.Theisshaped byfourconductingvanes(seeFig.1.3)tocreateaquadrupoleBecausetheelectric ismodulatedintime,thestructurealternatesbetweenfocusinganddefocusingineach plane,resultinginnetfocusinginbothplanes.Toacceleratethebeam,thepoletipsof thevanesareshapedwithasinusoidalmodulationtocreatealongitudinalelectric Becausethisstructurereliesonaradiofrequency(RF)onlyparticlesthatare apositivelongitudinalelectriceldwillbeaccelerated.Particlesthatarealignedwiththe negativewillbedecelerated.ThisresultsintheDCinputbeambecomingseparated intodiscretegroupscalledbunchesseparatedintimebytheRFperiod.Thistimestructure ofthebeamisoftencalledabunchtrain. 5 1.2.4Mediumenergybeamtransport AftertheRFQ,thebeampropagatesthroughtheMEBTwhichtransportsthebeamto themainaccelerator.Transversefocusingisprovidedwithquadrupoleorsolenoidmagnets. Inaddition,theMEBTfocusesthebeamlongitudinallytomaintainthebunchstructure. ThelongitudinalfocusingisprovidedbyanRFbunchercavity.Thesedeviceshavean RFlongitudinalelectricthatisphasedsuchthatparticlewiththeaverageenergywill arrivewhentheelectriciszero.Particleswithhigherenergywillarriveearlyanda negativeelectricwilldeceleratethem.Likewise,lowerenergyparticleswillarrivelate andpositiveelectricwillacceleratethem. TheMEBTalsomustmanipulatethebeamsoitiswithintheacceptanceofthemain acceleratortoavoidlosses.Thisisfurtherdiscussedinsection1.3.2.Inaddition,itmaybe necessarytoremoveparticlesthatarefarfromthecenterofthebeam,typicallycalledhalo ortails,becausetheyarethemostlikelytobelostintheaccelerator.Theseparticlescan beremovedbyinsertingplatescalledscrapersintothebeamlinetointercepttheextraneous particles.ItisoftenpreferabletopurposefullyremovethehaloandtailsintheMEBTwhen theyhavelowenergyanwillnotcausetdamageratherthanriskthembeinglost laterathigherenergieswherethedamagecanbemoredetrimental. 1.2.5Measuredbeamlines Themeasurementspresentedinthisdissertationweretakeninthefrontendsoftwoseparate accelerators.TheistheProtonImprovementPlan2InjectorTest(PIP2IT)[12].This acceleratorisateststandforthefrontendoftheProtonImprovementPlan2(PIP-II)project atFermilab[13].PIP2ITacceleratesH ionsstartingwitha30keVsourceandasolenoid 6 Figure1.4:PIP2ITfrontend. focusingLEBT(Fig.1.4).TheRFQoperatesat162.5MHzandacceleratesthebeamto 2.1MeV.Thetransversefocusinginthe10mMEBTisprovidedbytwoquadrupoledoublets andseventripletsandthelongitudinalfocusingisprovedbythreeRFbunchercavities. MeasurementswerealsotakenwiththefrontendattheFacilityforRareIsotopeBeams (FRIB)atMichiganStateUniversity(MSU)[14].Beamsofheavyionsupto 238 U arecreated withanECRionsourceat7keV/u.TheLEBTfocusesthebeamwithacombinationof solenoidsandquadrupolesstartingatgroundlevelthenverticallydropping30feettothe RFQ(Fig.1.5).TheRFQoperatesat80.5MHzandacceleratestheionsto0.5MeV/u. TheMEBTfocusingthebeamwithfourquadrupoletripletsandtwoRFbunchers. 1.3Transversebeamdynamics 1.3.1Singleparticledynamics Thedynamicsofaparticleofmass m inanelectromagneticbythepotentials ˚ and A canbedeterminedfromtheHamiltonian[11] H t = e˚ + T = e˚ + c q m 2 c 2 +( P e A ) 2 (1.2) where P isthecanonicalmomentum P = p + e A [15].Forsimplicity,thisHamiltonian assumesalinearcoordinatesystemwhichistforthediscussionhereoffrontends 7 Figure1.5:OverviewofthediagnosticsintheFRIBfrontend. becausetheyarepredominatelystraight.Amoregeneralformincurvilinearcoordinatescan beusedwhenbendsareatfeatureoftheoptics[11].Theindependentvariable ofthisHamiltonianistime t .Inaccelerators,itisoftenpreferabletousethelongitudinal coordinate z astheindependentvariableasitdeterminesthelocationinthemachine.With z astheindependentvariableandnormalizingtothetotalmomentum p ,theHamiltonian becomes H z = eA z p s 1 p x p 2 + p y p 2 (1.3) H z = eA z p q 1 x 0 2 + y 0 2 (1.4) where x 0 = dx=dz and y 0 = dy=dz arereferredtoasanglesandmeasuredinradians. 8 Hamilton'sequationsare @x @z = @H z @x 0 (1.5) @x 0 @z = @H z @x (1.6) which,usingtheparaxialapproximation x 0 ˝ 1and y 0 ˝ 1,givestheequationofmotion u 00 = e p z @A z @u : (1.7) Forlinear,uncoupledoptics,e.g.aquadrupolemagnetwith A z = G ( y 2 x 2 ) = 2,theequation ofmotionbecomes u 00 + =0(1.8) where isthefocusingstrengthoftheoptic.ThisisknownasHill'sequation.Ingeneral isafunctionof z asthebeamlineconsistsoffocusingmagnetsseparatedbyregionswithout focusing =0calleddriftspaces.Hill'sequationiswidelyusedinacceleratorphysics becausemostacceleratorsprimarilyrelyonlinearopticstofocusthebeam.However,to satisfyMaxwell'sequations,thefocusingmusthavenon-linearcomponents.Theef- fectsfromthenon-linearcomponentsarelessenedbypurposefullydesigningthemagnetsto minimizethenon-linearitiesandsimulationscanbeperformedifitnecessarytoaccountfor these[16]. ThelineardynamicsoftheparticlescanbedeterminedbyintegratedHill'sequationover thebeamline.However,itisoftenmorepracticaltouseatransfermatrix M thatrelates 9 thecoordinatesoftheparticleatonelocation z 1 toanother z 2 2 6 4 u u 0 3 7 5 z = z 2 = M ( z 2 j z 1 ) 2 6 4 u u 0 3 7 5 z = z 1 : (1.9) Equation1.9holdsforuncoupled,lineartransversemotionasdescribedbyEq.1.8.Itcan alsobegeneralizedtothefull4Dtransversephasespace( x;x 0 ;y;y 0 )andthefull6Ddynam- ics( x;x 0 ;y;y 0 ; z; p z =p z ).Thesematricescanbefoundinliteratureformostbeamline elementssuchasdipoles,quadrupoles,andsolenoids[17]. 1.3.1.1Thebetatronfunction Thebeamlineopticstendtoarrangedinquasi-periodicstructures.Becauseofthis,the motionoftheparticlescanbedescribedinphase-amplitudeform u = p ( z )cos( ˚ ( z ) ˚ 0 )(1.10) where isanoverallamplitudeand ˚ 0 isanarbitraryphaseThedynamicsare determinedbythebetatronfunction ( z )andthebetatronphase ˚ ( z ).Thedynamicsof and ˚ canbedeterminedbytakingthesecondderivativeofEq.1.10,pluggingintoEq.1.8, thenseparatelysettingthecosineandsinetermstozero.Thisgivestwoequations 1 2 00 1 2 0 2 2 ˚ 0 2 + 2 =0(1.11) 0 ˚ 0 + ˚ 00 =0 : (1.12) 10 Equation1.12caneasilybeintegratedtodetermine ( z ) 0 ˚ 0 + ˚ 00 =( ˚ 0 ) 0 =0(1.13) ˚ 0 =constant 1(1.14) ˚ ( z )= ˚ 0 + Z z 0 d z ( z ) : (1.15) Equation1.14canthenbeusedinEq.1.11todeterminethedynamicsofthe ( z ) 1 2 00 1 4 0 + 2 =0 : (1.16) Thebetatronfunctionistypicallyaccompaniedby = 1 2 0 and =(1+ 2 ) .Together, , ,and areknownastheTwissparametersandtheirphysicalmeaningisdiscussedis section1.3.2.Thebetatronphaseadvancehasawellmeaningforperiodicfocusing latticesbecausetheparticletrajectorywillalsobeperiodicwiththeperiodbywhere ˚ ( z )=2 ˇn .Thisisistypicallycharacterizedbythebetatrontunewhichisthenumberof oscillationsperlatticeperiod.Forquasi-periodicfocusingstructures,suchasarefoundin linacs,theexactnatureofthebetatronphaseismorecomplicatedbutthesameintuition holds. 1.3.1.2Action-phasecoordinates TheHamiltonianshowninEq.1.4isusefulbecauseusesthephysicalcoordinates( u;u 0 ;z ). Itcanbesimpleranalyticallytoworkinnormalizedcoordinates( w; _ w;˚ )where ˚ isthe 11 betatronphaseand_ w =d w= d ˚ [11].InthiscoordinatesystemtheHamiltonianis H w = 1 2 w 2 + 1 2 _ w (1.17) Acanonicaltransformationcanbemadetoaction-phasecoordinates( J;˚;˚ )(thatisnota typo,thebetatronphaseisbeingusedforatransverseandlongitudinalcoordinate)using thegeneratorfunction G = 1 2 w 2 tan( ˚ ˚ 0 )(1.18) where ˚ 0 isanarbitraryphase.Thenormalizedcoordinatescanbefoundby @G @w =_ w = w tan( ˚ ˚ 0 )(1.19) @G @ = w 2 2cos 2 ( ˚ ˚ 0 ) J (1.20) andsolvingfor w and_ w w = p 2 J cos( ˚ ˚ 0 )(1.21) _ w = p 2 J sin( ˚ ˚ 0 ) : (1.22) ThisgivestheHamiltonian H w = J (1.23) 12 andtheequationsofmotioninaction-phasecoordinatesare _ J =0(1.24) _ ˚ =1 : (1.25) Theaction J isaconstantofmotionandthetransversedynamicsareentirelydetermined bythebetatronphaseadvance. Torelatethiscoordinatesystemtophysicalcoordinates,Eqs.1.21and1.22canbe writtenintermsof u and u 0 w = u p (1.26) _ w = p u 0 + w: (1.27) Pluggingtheseinto H w = J theactionofaparticlecanbedeterminedintermsofitslocation inphasespace u , u 0 ,andtheTwissparameters J = 1 2 u 0 2 +2 uu 0 + u 2 : (1.28) Thisistheequationofanellipsein u u 0 phasespacewithorientationdeterminedbythe Twissparametersandsizedeterminedby J .Thelocationoftheparticleonthisellipseis determinedby ˚ ˚ = arctan u + u 0 u : (1.29) Thebeamdynamicscanthereforebyinterpretedasparticlestravelingalongtheseellipses ofconstantaction.Inrealspace,theorientationoftheellipsewillevolveaccordingthethe 13 appliedfocusing,buttheactionoftheparticlewillremainconstant. 1.3.2Phasespacedynamics Theabovederivationsconcernthedynamicsofasingleparticle.Unsurprisingly,abeam consistsofacollectionofparticles.Whiletheindividualparticlestravelalongelliptical trajectoriesinphasespaceasbytheactionandbetratronphase,thegeneraldistri- butionisnotnecessarilyelliptical.However,theTwissparametersandemittancearestill welldeforgeneralbeamdistributionsbydescribingtheellipsethatencompassesthe beamdistribution.TheemittanceandTwissparameterscanbefoundfromageneralbeam distributionby = q u 2 u 0 2 h uu 0 i (1.30) = u 2 (1.31) = uu 0 (1.32) = u 0 2 (1.33) where h ::: i denotesaverages.Thesearetypicallytakenfromrmsvalues.Thisbringsphysical meaningtotheparameters:theemittance,aconservedquantity,isthermsareaofthebeam inphasespaceandtheTwissparametersarethesecondordermomentsnormalizedtothe emittance.Thegeometricemittance isoftenscaledbytherelativisticfactorsandreferred toasthenormalizedemittance n = r r .Thisisdonetokeeptheemittanceconstantas thebeamisacceleratred.Equation1.16,determiningtheevolutionofthebetatronfunction throughthebeamline,isadescriptionoftheevolutionofthebeamsize. 14 Liketheordermoments,thesecondordermomentscanalsobepropagatedthrough focusingelementsusingthetransfermatrices.Thesecondordermomentscanbeputin matrixform ˙ = 2 6 4 u 2 uu 0 uu 0 u 0 2 3 7 5 (1.34) whichcanbepropagatedbythetransfermatriceslikeasecondordertensor ˙ ( z = z 2 )= M T ( z 2 j z 1 ) ˙ ( z = z 1 ) M ( z 2 j z 1 ) : (1.35) Notethat j ˙ j = 2 andthereforemustbeconstant.Topreservethedeterminate,thetransfer matricesmustbesymplectic. Thereisamaximumemittance,calledtheacceptance,thatcanbepropagatedthrough theacceleratorwithoutlosses.Fortransversedynamicsthisisdeterminedbytheoptics andtheaperturesofthebeampipeandbeamlineelements.Inaddition,toavoidlosses, theMEBTopticsmustcorrectlyorientthebeaminphasespaceattheendofthefront endto`match'thebeamtotheacceleratoroptics.Thisisachievedbyaligningthebeam distributionwiththeexpectedellipseinphasespacethattheparticletrajectorieswillfollow. Ifthebeamisnotproperlymatched,thebeamwilloscillatearoundthematchedcondition resultinginlargerbeamsizesthatcancauselosses. 1.3.3Spacecharge Thebeamisacollectionofchargedparticlesand,forintensebeams,theelectriccreated bytheseparticleswillthedynamics.Thisresultsinanon-linearadditiontoHill's 15 equation[18] u 00 + = q 3 b 2 b c 2 @˚ @u (1.36) where q and m arethechargeandmassoftheparticles, b istheLorenzfactor, b c isthe beamvelocity,and ˚ isthepotentialoftheThiscanbeapproximatedforauniform beamofradius r b andlinechargedensity as u 00 + = Q r 2 b (1.37) where Q iscalledthenormalizedperveance Q = q 2 ˇ 0 3 b 2 b c 2 : (1.38) Thisofspacechargedecaysrapidlywithincreasingbeamenergyas 3 r 2 r .Nearthe sourcetypicalvaluesof Q areontheorderof10 2 andhigherenergies Q 10 6 .This linearspacechargeforcecausesareductionofthebetatronphaseadvancecalledbetatron tunedepression. Spacechargeforcesalsothebeamsize.Thedynamicsofthesecondordermoments canbedeterminedfromEq.1.37bymultiplyingby u andtakinganaverage uu 00 + D u 2 E Q r b D u 2 E =0(1.39) Thebeamradiuscanbeattwicethermsbeamsize r b =2 u 2 1 = 2 andthisgivesa oftheedgeemittanceof =4 rms .ThesesareexactforaKapchinsky- Vladimirsky(KV)disitrbution[19]andareapproximationsforgeneraldistributions[18]. 16 Thederivativesof r b are r 0 =2 uu 0 u 2 1 = 2 =4 uu 0 r b (1.40) r 00 =4 uu 00 r b + 2 r 3 b : (1.41) PluggingtheseintoEq.1.39givestheenvelopeequationdictatingthedynamicsofthebeam radius r 00 b + b Q r b 2 r 3 b =0(1.42) Thebeamradiusevolvesaccordingtothebeamoptics .Inaddition,thereisathermal defocusingrelatedtotheemittanceandaspacechargedefocusingthatpropor- tionalto Q .Thebeamissaidtobespacechargedominatedif Q=r b ˛ 2 =r 3 b andemittance dominatedif Q=r b ˝ 2 =r 3 b . Forageneralbeamdistribution,thespacechargeforceswillbenon-linear.Theseforces distortthedistributioninphasespacecausingtationresultingintailandhalogrown andincreasingtheemittance(Fig.1.6).Theseneedtobecharacterizedand,if necessary,compensatedforbeforethebeamleavestheMEBTtoensureaqualitybeamis accelerated. 1.4Diagnostics Itiscrucialtoensureahighqualitybeamentersthemainaccelerator.Notonlyisthis importanttominimizebeamlossesbutalsoitchallengingtodeliveraqualitybeamtothe usersiftheinitialconditionsarepoor.Tocharacterizeandmonitorthebeamquality,the frontendshaveextensivediagnosticsuites(e.g.seeFig.1.5)becausethebeamistraveling 17 Figure1.6:Aninitiallyuniformbeamdistributioninphasespaceevolvinginaquadrupole focusingchannelwithspacecharge[4].Thebeamundergoestation,developsan\S" shapeanddilutingtheentirephasespacecausingtheemittancetoincrease. 18 non-relativisticallyandthereforebeamevolvesrapidlyduetoitslowmagneticrigidityand canexperiencetailandhalogrowthcausedbyspacechargeandnon-linearforces.Listed belowisasubsetofthediagnosticdevicesusedinfrontends. 1.4.1Beampositionmonitoring OneofthemostcommondevicesinhadronacceleratorsareBeamPositionMonitors(BPMs). Measuringthebeampositionisimportanttoverifythebeamopticsandalsoensurethebeam isnearthecenterofthebeampipetoavoidlosses. BPMsinhadronacceleratorscommonlyrelyoncapacitivepickupsthatcoupletothe electricgeneratedbythebeam.Thesehavefourpickupslocatedatthetop,bottom, left,andrightsidesofthebeampipe.Tomeasurethebeamposition,theimagecharge generatedbytheelectriceldfromthebeamoneachofthepickupsismeasured.Theradial electricincylindricalcoordinates( r;˚;z )fromapencilbeamlocatedat( r 0 ;˚ 0 )ina pipeofradius R p travelingat v = r c andchargemodulatedwithfrequency ! isgivenby [20] E r ( r;˚;z )= D ! cos ! t + z 0 z r c X n =0 gI n ( gr 0 ) 0 NˇI n ( gR p ) cos( n [ ˚ ˚ 0 ]) I 0 n ( gr ) K n ( gR p ) I n ( gR p ) K 0 n ( gr ) (1.43) where I n and K n arethemoBesselfunctionsoftheandsecondkind[21],primes denotederivativeswithrespecttotheargument,and N = 8 > < > : 2 ; n=0 1 ; else (1.44) 19 isaharmonicfactorand g = ! r r c (1.45) actsasatransversepropagationconstant. g isproportionaltotheratioofthephotonand beammomentaand g issmallwhenthebeammomentumislargeand g islargewhenthe beammomentumissmall.Theimagechargeatapointonthebeampipeis ˙ ( ˚;z )= D ! cos ! t + z 0 z r c X n =0 I n ( gr 0 ) NˇR p I n ( gR p ) cos( n [ ˚ ˚ 0 ])(1.46) Thebeampositioncanbedeterminedbyaremeasuringtheimagechargeattwolocations onoppositesidesofthebeampipeattheazimuthallocations ˚ = ˚ m and ˚ = ˚ m + ˇ .The ofthesetwosignalsdividedbythesumis = P n I n ( gr 0 ) NI n ( gR p ) (cos( n [ ˚ m ˚ 0 ]) cos( n [ ˚ m + ˇ ˚ 0 ])) P n I n ( gr 0 ) NI n ( gR p ) (cos( n [ ˚ m ˚ 0 ])+cos( n [ ˚ m + ˇ ˚ 0 ])) : (1.47) Thiscanbeapproximatedbytakingtermsupton=3 ˇ 2 I 1 ( gr 0 ) I 1 ( gR p ) cos( ˚ 0 ˚ m )+2 I 3 ( gr 0 ) I 3 ( gR p ) cos(3( ˚ 0 ˚ m )) I 0 ( gr 0 ) I 0 ( gR p ) +2 I 2 ( gr 0 ) I 2 ( gR p ) cos(2( ˚ 0 ˚ m )) (1.48) 20 thenexpandingin r 0 upto5 th orderusingMathematica[22] ˇ I 0 ( gR p ) I 1 ( gR p ) cos( ˚ m ˚ 0 ) r 0 +(1.49) I 0 ( gR p ) 24 I 3 ( gR p ) cos(3( ˚ m ˚ 0 )) I 2 0 ( gR p ) 4 I 1 ( gR p ) I 2 ( gR p ) cos( ˚ m ˚ 0 )cos(2( ˚ m ˚ 0 )) I 0 ( gR p ) 8 I 1 ( gR p ) cos( ˚ m ˚ 0 ) r 3 0 + I 0 ( gR p ) 48 I 1 ( gR p ) cos( ˚ m ˚ 0 )+ 7 I 2 0 ( gR p ) 96 I 1 ( gR p ) I 2 ( gR p ) cos( ˚ m ˚ 0 )cos(2( ˚ m ˚ 0 )) I 3 0 ( gR p ) 16 I 1 ( gR p ) I 2 ( gR p ) cos( ˚ m ˚ 0 )cos 2 (2( ˚ m ˚ 0 )) I 2 0 ( gR p ) 96 I 2 ( gR p ) I 3 ( gR p ) cos(2( ˚ m ˚ 0 ))cos(3( ˚ m ˚ 0 )) I 0 ( gR p ) 128 I 3 ( gR p ) cos(3( ˚ m ˚ 0 )) r 5 0 : Thepositionofthebeamcanthenbeestimatedbytakingonlythetermlinearin r 0 ˇ gI 0 ( gR p ) I 1 ( gR p ) r 0 cos( ˚ m ˚ 0 )(1.50) ˇ gI 0 ( gR p ) I 1 ( gR p ) [ x 0 cos( ˚ m )+ y 0 sin( ˚ m )](1.51) where x 0 = r 0 cos( ˚ 0 )and y 0 = r 0 cos( ˚ 0 )isthelocationofthebeamcentroidinCartesian coordinates.Thereforethepositionofthebeamcanbedeterminedusingthe = signal fromahorizontalpairofpickupstomeasure x 0 andaverticalpairtomeasure y 0 .For largeenough gR p theresultisdependenton gR p ,i.e.themeasuredfrequencyandbeam momentum,andthereforemultiplecalibrationsmustbeusedtocorrectlymeasurethebeam positionasthebeamacceleratesand increases.If gR p ˝ 1,generallycorrespondingto 21 Figure1.7:Asthebeamacceleratestheelectriciscompressedintototheplaneper- pendiculartothevelocityresultinginthesameonoppositesidesofthebeam pipe[5]. relativisticbeams r r ˛ 1,thenEq.1.51becomesmomentumandfrequencyindependent ˇ 2 R p [ x 0 cos( ˚ m )+ y 0 sin( ˚ m )] : (1.52) Thedependenceon gR p forlarge gR p isduetotheoftheelectriconthepipe wallgeneratedbyanon-relativisticbeamextendingbeyondthelengthofthebunchandthe aretoneachpickupandvarywithbeamposition.Asthebeamis acceleratedandincreasesinenergy,relativistic`pancake'thedistributioninto theplaneperpendiculartothebeamvelocity(seeFig.1.7)andthelongitudinalof thebecomessimilartothebeamAtthesehigherenergies,varyingthebeam positioncauseslittlevariationintheonthewallandonlychangestheamplitude resultinginnofrequencydependenceinpositionmeasurements[23]. Thelow r ,high gR p aremostimportantinthefrontendsofhadronmachines wherethebeamistravelingnon-relativistically[24].Forexample,atFRIB,with f =161MHz and R p ˇ 20mm,thesewillcausea2%errorinthemeasuredpositionat ˇ 0 : 18 [6].However,intheMEBTthebeamhasvelocity r =0.033correspondingto gR p =2.4and non-relativisticmustbeaccountedfor. 22 Figure1.8:Simulationresultsforthecalculatedbeampositionusingthelinear = (left) andahigherorderpolynomialcorrection(right)[6]. ThelinearresponseofEqs.1.51and1.52holdforsmallbeamfromthecenterof theBPMrelativeto R p .Whenthebeamisfurtheraway,the = signalsvarynon-linearly withbeamposition(seeFig.1.8).Thesenon-linearitiesaremappedonateststandby stringingastraightwirethroughtheBPMandsendingatonedownthewireandrecording theresponseoftheBPMpickups.Thisisrepeatedfortlocationsofthewiretomap theresponsefortbeampositions.Todeterminetheposition,thewirelocationis relatedtothepickupsignalsusingahigherorderpolynomial,e.g.theFRIBBPMsusea orderpolynomial[6]. TheshapeoftheBPMpickupsthemeasuredsignalsandmustbetakeninto account.Thepickupshavesomesizecausingtheirmeasuredsignaltobetheintegrated imagechargeoverthebuttongeometry ˙ pickup ( ! )= Z button ˙ d A but : (1.53) CommonBPMpickupgeometriesareroundbuttonpickups[6],rectangularsplit-platepick- 23 Figure1.9:AsketchofacapacativeBPManditsequivalentcircuitisshownontheleft [7].Duetothecapacitancetoground,BPMssuppressthelowfrequencycomponentsofthe measuredsignals(right, R =50 C =3pF). ups[25],andstriplinepickups[26] ThemeasuredsignalsalsoarebytheimpedanceoftheBPMpickup.TheBPM pickupsarecapacitiveandconnectedtoa50cable(seeFig.1.9).Thissystemactsasa highpasswithcutfrequency ! c =1 =RC where R istheresistanceand C isthe capacitanceofthepickuptoground.ThemeasuredvoltagefromacapacitiveBPMpickup is[7] V meas ( ! ) / !=! c p 1+( !=! c ) 2 ˙ pickup ( ! ) : (1.54) Thissuppressesthelowfrequencycomponentsofthemeasuredsignalcausesthemeasured signalstoappearlikethederivativeoftheelectricatthewall.ForFRIB C =3.3pFand ! c =6.06GHz[6]. TheBPMsignalsare,unsurprisingly,bynoiseandonemainconcernsisthermal noise.Whenmeasuringthevoltageacrossaresistor R attemperature T ,thermsvoltage 24 fromthermalnoiseisrelatedtothemeasuredbandwidth f [27] V T;rms = p 4 k B TR f (1.55) where k B istheBoltzmannconstant.WhiletheBPMpickupshaveabroadbandresponseto thebeam,typicallynarrowbandisusetoreducethethermalnoise.Thenarrowband selectsasingleharmonicofthebunchrepetitionrate.Thisisoftentheharmonic becauseithasthehighestamplitude.However,thesecondharmonicisalsobeusedtoprevent thesignalsfrombeingtedbytheRFacceleratingelementsoperatingattheprinciple harmonic.BecauseBPMsrelyonmeasuringthefrequencycontentofabunchtrain,they canbeusedthetheMEBTbutnottheLEBT.IntheLEBTmonitorscanbeused todeterminethebeamposition. 1.4.2measurements Likepositionmonitoring,measuringthetransverseinrealspaceofthebeamisused toverifythebeamlineoptics.Inaddition,thesemeasurementscanbeusedtocharacterize thebeamqualityandensurethebeamissmallenoughtoavoidlosses.Duetothelowenergy infrontends,devicesthatinterceptthebeamarecommonlyusedmeasurethebeam inthisregion.Forhigherenergybeams,interceptivediagnosticsarelesscommonlyusedand alternativemethodsareemployedsuchasgasjets[28]. Onecommonmonitorisascintillatorscreen[29].Theseareinsertedintothe beamlineandwhenthebeamstrikesthescreen,energyisdepositedthatexcitedthescintil- latormoleculescausingthemtoproducelightwhichisrecordedwithacamera.Themore energythatisdepositedthemorelightthatisproducedsotheareaswithhigherbeamin- 25 tensitywillbebrighter.Assumingtheintensityofthelightislinearwiththebeamintensity, therecordedimageisadirectreplicationofthe2Dtransversedistribution. Scintillatorsmustbechosensuchthattheirresponseislinearwithintherangeofex- pectedbeamintensitiestoavoidsaturation.Anadditionalconsiderationisthescintillating propertieswilldecaywithuse.Ifthebeamisalwaysstrikingthesamespotonthescintilla- tor,theresponseatthehighintensityregionwilldecayfasterwhichwilleventuallydistort themeasureddistributions. Anothercommonlyuseddeviceisawiremonitor.Thesedevicesinsertathin conductingwireintothebeamandwhentheparticlesstrikeit,theyexciteacurrentinthe wirethatcanberecorded.Thecurrentisproportionaltothetotalintensityintercepted bythewire.Bysteppingthewirethroughthebeam,theinonedirectioncanbe mapped[5].Toreducenoise,themeasurementsateachlocationcanbeaveragedintimeand backgroundmeasurementswithoutthebeamcanbetaken.Wiremonitorstypically havethreewirestomeasurethe x and y aswellasoneat45 tomeasure x y coupling. Theresolutionofthesescannersisdeterminedbystepsizeandthesizeofthewire. However,arbitrarilythinwirescannotbeusedbecausethewiremustbeabletowithstand thebeamintensitythatitintercepts.Ifthewireistoothinandoverheatsitcanbedeformed orbreak.Anadditionalconsiderationiswhenthebeamstrikesthewire,thewirecanemit secondaryelectrons.Thiswouldtthemeasuredcurrentaschargeisleavingthewire. Topreventthisthewiresaretypicallyheldatvoltage.Furthermore,ifmultiplewiresare used,thencrosstalkbetweenthewireswilllimittheabilitytomeasurethelowintensity portionsofthebeam. 26 1.4.3Phasespacedistributionmeasurements Thedistributionofthebeamin u u 0 phasespaceisoftenmeasuredinfrontendswithslit scanners.Thesedeviceshaveathinslitthatisinsertedintothebeamlinetoonlyallowa smallsliceofthebeamatasplocationtopass.Theparticlesinthepassedbeamletwill havesomerangeofangleswhichwillcausetheparticlestodiverge.Forexampleaparticle with u 0 =10mradwilldrift1mmtransverselyover100mmwhileaparticlewith u 0 =5mm willonlydrift0.5mm.Therefore,ifthetransverseofthepassedbeamletismeasured downstream,the u 0 oftheparticlesatthelocationoftheslitcanbereconstructed. Theentire u u 0 phasespacedistributioncanbemeasuredbysteppingtheslitthrough thebeamandrecordingthetransverseofthepassedbeamletateachstep.These measurementsofthephasespacedistributionarereferredtohereasphaseportraits. Thereareasmanyformsofthesescannersastherearetypesofmonitors,e.g. pepperpots[30],slit-slitscanners[31]andslit-wireharpscanners[32].Thephaseportraits presentedinchapter2weretakenwithanAllison-typescanner[33]inthePIP2ITMEBT. Allisonscannersconsistofarigidboxwithathinslitoneitherendandavariableelectric dipolebetweentheslits(Fig.1.10).Thebeamisinterceptedbythefrontplateandparticles canonlypassthroughtheslittoselectnarrowpositionrange.Thepassedbeamletis bythedipoleduntilitstrikesrearwall.Iftheparticleshavethecorrectinitial transverseanglethenthewillcausetheparticlestopassthroughthesecondslit andintoaFaradaycuptomeasurethepassedcurrent.ForanAllisonscanneroflength ` andabeamofmass m travelingat v = r c ˝ c thepassedangle x 0 0 foragivenvoltage V is x 0 0 ( V )= qV` 2 2 r c 2 : (1.56) 27 Figure1.10:schematicofanAllisonscanner.Theredlineshowsthetrajectory ofparticlesthroughthedevice. The2Dtransversephasespacecanthereforebemeasuredbysteppingthewholeboxthrough thebeamsothefrontslitcantakeslicesattpositionsandateachpositiontheelectric dipolestrengthisswepttoscanarangeofangles.Ateachposition-dipolesettingthecurrent ontheFaradaycupismeasuredtodeterminetheintensityofthebeaminthesmallphase spaceareathatpassedthroughthescanner. Itisimportanttohandlethebackgroundnoisewhenanalyzingthemeasuredphase portraits.Therearemanymethodsforaccountingforthenoise.Asimpleandinelegant methodistosetacutthresholdinintensitythatishighenoughtoremoveallthenoise[34]. Thislevelcanbesetbyvaryingthecutlevelandcalculatingthermsemittanceforeachcut levelandakneepointwherethenoisestartstothemeasuredvalues.This willinevitablyremoveaportionofthebeamsignal,butifitissmallthentheimpactonthe rmsparameterswillbeminimal.Anothermethodistocalculatethermsemittanceoveran ellipseinphasespacewiththesameTwissparametersasthebeamcore[35].Ifthisellipseis toolargenoisewillthecalculatedrmsemittance.Thereforethenoisecanbeexcluded byvaryingthesizeofthisellipsearangewheretheemittanceisstable.Ref.[36]sets thecutthresholdtoanegativevaluetoincludeallmeasuredpixels,andreliesonperfect cancellationoftherandomnoiseoutsideofthebeamwhencalculatingthermsparameters. 28 1.4.4Beamtailmeasurements Itisgenerallyagreedthatthebeamtailshavedensitiesintherangeof10 1 to10 4 of thepeakdensitywiththehalohavingevenlowerdensity[37]andtypicallyhaveat distributionfromthecore.However,thereareavarietyofoftailsareused basedontheneedsoftaccelerators.Inaddition,beamtailscanbechallengingto quantifybecausethebeamrotatesinphasespaceandthevariationsfrombeamdynamics mustbeisolatedfromvariationscausedbytailgrowth.Thisisparticularlychallengingwhen measuringthetailswithmonitors. Evenwhenmeasuringthedistributionofthebeaminphasespace,itcanbechallenging toquantifythebeamtails.Atypicallypracticeistoonlycalculatethermsparametersof thephaseportraitand`qualify'thebeamtailsvisually. 29 Chapter2 AnalysisofPhasePortraitsusing Action-PhaseCoordinates Amethodthathasbeenusedtoanalyzetailsinsimulationsistoconverttheparticlelocations from u u 0 coordinatestonormalizedcoordinates w _ w [38].Inthiscoordinatesystemthere isawellnotionofradiusbecause w and_ w havethesameunits.Thetailgrowthcan thenbequanedbytheparticledensityasafunctionofradius. Thisideacanbefurtherdevelopedbyusing J ˚ coordinatesinsteadof w _ w .This givesabetterphysicalinterpretationasthe`radius'ofaparticleisitsaction.Theintensity distributionasafunctionof J willbeconstantunderlinearforcesandthereforeitcanbe usedtostudytailgrowthduetonon-linearandcharacterizethebeamquality.In addition,thisanalysiscanbeappliedtobeamlinemeasurementstakenin x x 0 coordinates fromaphasespacescannerbydeterminingtheTwissparametersofthebeamandcalculated J and ˚ ofeachpixelusingEqs.1.28and1.29.Anexampleofaphaseportraitmeasured withanAllisonscannerinthePIP2ITMEBTisshowninFig2.1in x x 0 coordinatesand witheachpixelconvertedinto J ˚ coordinates.Thephaseportraitin J ˚ coordinates shouldbestablemoduloshifts ˚ causedbychangestotheoptics. TheremeasurementspresentedinthischapterweretakenwithanAllisonscannerinthe PIP2ITMEBT.ThePIP2ITMEBTwasassembledandbeammeasurementswereperformed 30 Figure2.1:Measuredphaseportraitin x x 0 coordinatesand J ˚ coordinatestakenat location1. inseveralstagesbetween2016-2018.ItsinitialisshowninFig.2.2and attheendofthe2018runisshowninFig.2.3.Theprimarybeamparameters intheMEBTaresummarizedinTable2.2.TheAllisonscannerwasusedinthreelocations andwasmovedattphasesoftheMEBTconstruction: 1. Location1,downstreamofthesecondquadrupoledoubletasshowninFig.2.2,inthe horizontalposition 2. Location2,inthemiddleofthebeamline,intheverticalposition 3. Location3,towardtheendofthebeamlinedownstreamofallfocusingoptics,asshown inFig.2.3,intheverticalposition. MostofthemeasurementspresentedhereweretakenatthelocationoftheAllison scanner(Fig.2.2).Resultsfromothertwolocationsareexplicitlynoted. 2.0.1Allisonscannernoise MeasurementstakewiththePIP2ITMEBTAllisonscannerunfortunatelyhadsignt electronicnoisecomparedtothemeasuredsignalintensities.Foratypicalscanofa5mA 31 Table2.1:ParametersofthePIP2ITMEBTAllisonscanner ParameterValueUnit Slitsize0.2mm Slitseparation320mm Platevoltage 1000V Platelength300mm Plateseparation5.6mm Maximummeasurableangleat2.1MeV 12mrad Figure2.2:InitialPIP2ITMEBTTheAllisonscannerisinlocation1and measuresthetransversephasespaceinthehorizontalplane Figure2.3:FinalofthePIP2ITMEBTwiththeAllisonscannerinlocation3 andorientedtomeasurethetransversephasespaceintheverticalplane.Transversefocusing isprovidedbyquadrupoleswithtwodoubletsandseventriplets. Table2.2:PIP2ITMEBTbeamparameters ParameterValueUnit Beamenergy2.1MeV Macro-pulserepetitionrate1-20Hz Macro-pulselength0.005-25ms Bunchrepetitionrate162.5MHz PulsebeamcurrentUpto10mA Transverseemittance,rmsnorm. 0.23mmmrad Longitudinalemittance,rmsnorm. 0.34mmmrad 32 beam,itsrmsnoiseis0.2%-0.3%ofthemaximumamplitudefornominaloperation.This limitedthedynamicrangeofthedeviceto ˘ 2ordersofmagnitude.Thesourceofthenoise wasinvestigatedbutnoremedywasfound. Inaddition,tjitterofthebeamcentroidwasmeasuredwiththeBPMsinthe MEBT,butthesourceofthejittercouldnotbelocatednorthejittereliminated[39]. SpectralanalysisoftheBPMreadingsshowsthejitterhasfrequenciesupto ˘ 3Hzwithno dominantharmonics.Therefore,evenindividualangularscansareasittakes ˘ 1s tosweepthevoltageovertheanglerangewhilemeasuringat20Hzandapproximately5 minutestocompleteatypicalphasespacemeasurement[40].IntheMEBT,theamplitude ofthejittervariesalongthebeamlineinaccordancewiththeopticsandreachingupto 0.2mmrmsinamplitude. ThesourceofcentroidjitterwasdeterminedtobeintheLEBT,buttheexactsourcewas notdeterminednorThemotioncanberotatedbetweenthehorizontalandvertical planesbychangingthestrengthofasolenoidfocusingmagnetintheLEBT[41].However, fortheoptimalperformanceoftheRFQ,theLEBTsolenoidsneedtobesetsuchthatthe jitterwasprimarilyintheverticalplane.Thisnoticeablyimpactedthemeasurementstaken atlocationstwoandthree.Themajorityoftheresultspresentedinthefollowingsections usemeasurementstakeninthelocationoftheAllisonscannerwherethescannerwas orientedhorizontallytoreducetheofthejitter. Toestimatetheofthejitteronthepixelamplitude,multiplephaseportraitstaken withthesamefocusingwereusedtoestimatethermsscatterateachpixel ˙ I i [42].At locationone, ˙ I i wasfoundtoincreaseapproximatelylinearlywiththepixelamplitude I i 33 (Fig.2.4left)withthelinear ˙ I i =0 : 0067+0 : 024 I i : (2.1) Theintensitiesvaryby2-3%forpixelsnearthecenterofthebeamandisdominatedby electronicnoiseatlowintensities.Theerrorbarsshowninresultsfrommeasurementstaken atlocationonefollowEq.2.1. Formeasurementstakenatlocationstwoandthree,thejitterintheverticalplanesig- tlyincreasesthevariationinthepixelamplitudeandfollowthegeneraltrend(Fig. 2.4right) ˙ i = 8 > < > : 0 : 01+0 : 3 I i for I i < 0 : 9 0 : 28for I i 0 : 9 (2.2) Thisjitterhasminimalonthemeasuredrmsparametersofthebeamandcausesan errorofonly ˘ 2%[40].However,itconfoundsthedetailedmeasurementsofthedistribution inphasespaceandbeamtailsusing J ˚ coordinates. 2.1Beamdescriptionusing J ˚ coordinates 2.1.1Coredescription WhenthephaseportraitsmeasuredintheMEBTwereconvertedtoaction-phasecoordi- nates,itwasfoundthatinthecentralportionofthebeam,i.e.pixelsatsmallactions,the pixelamplitudeismostlyindependentofthephaseanddecreasesexponentiallywithaction I gauss = I 0 e J c = I 0 e 1 2 c x 2 +2 xx 0 + x 0 2 (2.3) 34 Figure2.4:Thermsscatterofthepixelamplitudesplottedasafunctionoftheaverage amplitudes(blue)phaseportraitsmeasuredatlocationone(left)andlocationthree(right). TheorangelinesshowthefromEqs.2.1and2.2. whichdescribesaGaussiandistributionin x x 0 coordinateswithconstantemittance c . Onsemi-logarithmicscale,Eq.2.3representsastraight-linewithslope 1 c , c isreferred tohereasthecentralslope.SinceEq.2.3describesaperfectGaussiandistribution, c isa measureofhowbroadthecoredistributionisandcanbeinterpretedasthermsemittance ofthebeamiftheGaussiancorewasextendedandthetailsremoved. 2.1.2Discussiononthebeamdistribution ThechoiceofthedescribingthecoreasGaussianwasnotimmediatelyobvious.Oneconsid- erationwas,inthePIP2ITLEBT,thetransversebeamdistributionwasmeasuredtonotbe Gaussian[43].Insteaditwasuniform-Gaussian(UG)causedbythebeambeinginitiallyspa- tiallylimitedbytheionsourceextractionapertureresultinginthebeamdistributioncoming fortheionsourcebeinguniforminpositionandGaussianinangle.TheUGdistribution, projectedontooneplaneinaction-phasecoordinatesis 35 I UG ( J;˚ )= I 0 s 1 J cos 2 ( ˚ ) 2 UG e J sin 2 ( ˚ ) UG H 1 J cos 2 ( ˚ ) 2 UG (2.4) where I 0 isthepeakintensity, UG isthermsemittance,and H istheHeavisidefunction H ( x )= 8 > < > : 0if x< 0 1if x 0 : (2.5) Inaddition,accordingtoRefs.[18,44],afterexperiencingmultiplebetatronoscillations inaperiodicstructure,thetransversedistributionisexpectedtorelaxtowardsanequilibrium distribution.Foraspacechargedominatedbeamtheequilibriumdistributionisuniformand foranemittancedominatedbeamitisaMaxwell-Boltzmanndistributioninactionwhich correspondstoEq.(2.3).ThebeamarrivestotheMEBTafterpassing ˘ 12betatronperiods intheRFQanditwasnotobviousifthiswastfortherelaxation.Also,fornominal 5mAoperationtheperveanceintheMEBTis Q ˘ 10 6 andthermsradiusis r ˇ 3mm andgeometricemittanceis ˇ 3mmmrad.Puttingthesevaluesintotheenvelopeequation (Eq.1.42),wethestrengthofthespacechargeandemittancetermsareonthesame order, Q=r b ˇ 2 =r 3 b ˘ 10 4 ,thereforethebeamwasnotexpectedtobespacechargeor emittancedominated. Todeterminethedistributionofthebeamcoreatlowaction,themeasureddistribution wastoEq.2.3andEq.2.4.Inaddition,thephaseportraitwastotwoothermodels commonlyusedforapproximatingthebeamdistribution:theKapchinskiy-Vladimirskiy 36 (KV)andwaterbag(WB)distributions.Inactiontheyare I KV ( J;˚ )= I 0 H 1 J 2 KV (2.6) I WB ( J;˚ )= I 0 1 J 3 WB H 1 J 3 WB (2.7) where KV and WB arethecorrespondingemittances. InFig.2.5,thehigh-amplitudepixelswithinthenormalizedaction J< 0 : 15mmmrad (containing60%ofthemeasuredbeam),areplottedtogetherwithoftheidealizeddis- tributionsEq.2.3,Eq.2.4,Eq.2.6,andEq.2.7.TheGaussiandistributionEq.2.3is thebestwithreduced ˜ 2 =2 : 07.Forawaterbagdistribution,thereduced ˜ 2 =6 : 55 ( ˜ 2 normalizedtothedegreesoffreedom),however,thismodelabruptlydeviatesfromthe measureddistributionoutsideoftheregion.TheUGandKVdistributionspoorly thedatawiththereduced ˜ 2 of47and687respectively. Thelargevalueof ˜ 2 fortheGaussianindicatesthatEq.2.3doesnotfullycharacterize thedistribution.Moreover,the ˜ 2 valuegrowsquicklywhenincludingintheadditional pixelswithlargeraction.Thegrowthiscausedprimarilybyappearanceatlarge J ofaphase dependenceofthepixelamplitudes,whichisdiscussedbelow. 2.1.2.1Centralparameters TheinitialattemptstocomparethemeasureddatawithEq.2.3showedarelativelylarge scatterofpixelintensitiesforanygivenaction,evenatlowactions(Fig.2.6,blue).Thiswas causedbythechoiceofTwissparametersusedtotheactionwhichcantly thedistributionin J ˚ coordinates.Intially,theTwissparametersusedwerethe rmsparametersoftheentirebeam(referredhereasthermsTwissparameters).Thischoice 37 Figure2.5:Comparisonofthemeasureddistributioninactioninthebeamcore(black)with severalidealdistributions:Gaussian,KV,UG,andWB.NotethattheUGdistributionis phase-dependent,and,therefore,pixelintensitiesvaryforagivenactionandisrepresented herebytheareashadedingreen. ofTwissparametersresultsinalargescatterevenforparticleswithlowactionbecauseit includesthenon-Gaussianbeamtailswhichskewthermsdescriptionofthebeamawaycore distribution. Alternatively,theactioncanbeusingpixelsinthe`central'portionofthebeam. Thecentralportionwasfoundbyremovingthelowerintensitypixelsofthebeamthen Eq.2.13todeterminethe`central'Twissparametersandcentralslope.Thefraction removedwasscannedfrom30-60%ofthetotalintensityin1%steps.Generally,thecentral slopeincreasesatlargeandsmallcuts(Fig.2.7).Theincreaseatsmallcutsisattributed tothetailstheandatlargecutspoorstatisticsincreasesthecentralslope tlywhenthenumberpixelsisbelow ˘ 30.Toavoidbothofthesethe centralslopewastoacubicpolynomialandthecutwaschosentobethepointclosestto theminimumofthecurve.Ifacubicpolynomialcouldnotbethecutwassetto 50%. 38 Figure2.6:ActiondistributionusingcentralTwissparameters(red)andrmsTwissparam- eters(blue). Figure2.7:Centralslopeasafunctionoftheportionofthebeamremoved.Thecurveis toacubicpolynomialtodeterminetheminimum c whichisusedtothecentral paramters. 39 WhenthecentralTwissparametersareusedtoactionandphase,thescatterin thebeam'scentralregionisreduced(Fig.2.6,red).Thisisseeninthereduced ˜ 2 for Eq.2.3topixelswith J< 0 : 15mmmradwhichis ˜ 2 =64whenusingthermsTwiss parametersand ˜ 2 =2whenusingthecentralTwissparameters. 2.1.3Allisonscannerphasedependence Equation1.56tacitlyassumestheslitsintheAllisonscannerareysmallto determinethepassedangleforagivenvoltage.Inrealitytheslitshavesize2 d .This causesthemeasuredphasespaceareaforeveryposition-voltagesettingtobearhomboid (Fig.2.8)withverticesat x 0 + d;x 0 0 (2.8) x 0 d;x 0 0 (2.9) x 0 + d;x 0 0 2 d ` (2.10) x 0 d;x 0 0 + 2 d ` : (2.11) Thisdistortsthemeasureddistributionfromthetruedistribution.Forexample,ifapure 2DGaussianismeasuredwithanAllisonscannerofslittoslitlength ` andslits y 1 and y 2 extendingfrom d to d themeasuredintensitydistributionisgivenbyintegratingoverboth slits[45] I meas ( x;x 0 )= 1 4 d 2 Z d d Z d d exp 1 2 c h ( x + y 1 ) 2 + 2 ( x + y 1 ) x 0 + y 2 y 1 ` + x 0 + y 2 y 1 ` 2 #! d y 1 d y 2 : (2.12) 40 Figure2.8:Theshadedarearepresentsthepassedphasespaceareaforagivenpositionand voltagesettingofanAllisonscanner.Thegridisthedisplayedpixelsize. Theintegrandcanbeexpandedtosecondorderin y 1 and y 2 andtheresultingmeasured distributionuptoorder d 2 is I meas ( x;x 0 )=exp 1 2 c h x 2 +2 xx 0 + x 0 2 i 1+ d 2 6 2 c " c 2 ` 2 ` 2 +2 x + x 0 ` 2 + x 0 + x 2 2 x + x 0 ` x 0 + x : (2.13) Atlarge J ,whentheparametersinTable2.1areused,thisvariationisapproximately2% ofthemeasuredintensityvariationatagivenactionandwasgenerallyignored.However, atlow J ,thisdistortionneedstobeaccountedforwheningactionandthecentral parameters. Theoftheslitscanbeseenbyvaryingthestrengthofaquadrupolemagnetdirectly upstreamoftheAllisonscannertochangetheTwissparametersattheAllisonscanner(Fig. 41 Figure2.9:Left:VariationoftheTwissparameterswithquadrupolecurrent.Right:The centralslopeisconstantwhenaccountingforsoftheslitsize. 2.9left).Becausethisischangingalinearoptic,theactiondistributionand c shouldnot change.IftheslitctisnotaccountedforandthecentralTwissparametersarefound bytakingrmsvaluesoverthehighintensitypixels,thenthecentralslopedecreaseslinearly withthequadrupoleexcitationcurrent(Fig.2.9right).Ifinstead,thecentralslopeand centralTwissparametersarefoundbytoEq.2.13then c isconstantwithin 5%. Therecordedpixelintensityisalsobythethicknessoftheslits[46].This wasneglectedinthesemeasurementsbecausetheisrelatedtotheangleof themeasuredparticleswhichiscomparativelysmallinthePIP2ITMEBTof 12mrad. MostAllisonscannersareusedLEBTswherethelowerenergyresultsinalargerrangeof transverseanglesof ˇ 40mrad.Thethicknessoftheslitsforphaseportraitsrecordedin thePIP2ITMEBT,theslitthicknessisestimatedtohaveatmosta2%onthepixel intensities. 42 Figure2.10:Theintensitiesarebinnedinactiontodetermine J tr withEq.2.14toseparate thetailsfromthecore.Theerrorbarsrepresent 3 ˙ . 2.1.4Taildescription Thedistinctionbetweenthebeamcoreandtailsisbythetransitionaction J tr where thedistributiondeviatesantlyfromGaussianEq.2.3.Thetransitionactionisfound by,determiningthecentralparametersasoutlinedaboveandcalculatingtheaction andphaseofeachpixel.Then,allthepixelsaresortedintonormalizedactionbins J i , typically0.05mmmradinsize,andthemeanamplitude I ( J i )andthestandarddeviation ˙ Int ( J i )oftheamplitudeineachactionbiniscalculated.Thevalueof J tr ischosenasthe actionofthebinwherethemeanamplitudedeviatesfromtheofEq.2.3bymorethan threetimesthestandarddeviationofthemean(seeFig.2.10) I ( J tr ) I 0 e J tr c =3 ˙ Int ( J tr ) : (2.14) Allparticleswithactionlessthanthetransitionactionaretobeinthecore,and particleswithlargeractionareinthetail.Thisgivesameasureoftheextentofthebeam 43 core.Therelativeweightofthecorecanbequanbythefractionofthetotalintensity inthebeamcore.Thepercentageofthebeamintensityinthetailsistypically10-20%of thetotalintensity. Withthesethetransitionactionandthepercentofthebeaminthecoreare constantunderlinearforcesandcanbeusedasametricfortailgrowthduetonon-linear Intheorythemaximumactioncanalsobeused.However,inpractice,becausethe pixelwithmaximumactionhasintensityjustabovethenoiseor,themaximumactionis verynoisyrenderingitanunreliablequantitativemeasure. Atactionsabove J tr thescatterofpixelintensitiesatagivenactionclearlydeviates fromtheGaussiancore(Fig.2.11(b)).Thedominantpartofthisscattercomesfromstrong phasedependencewiththetailbeingsplitintotwo\branches"ofsimilarintensitiesthat areseparatedinphasebyapproximately ˇ radwhichareclearlyevidentwhenthedatais plottedin J ˚ coordinates(Fig.2.11(c)).Thelocationinphaseofthebranches ˚ b is bythephaseofthesecondharmonicofthisdistribution.Thisisfoundbytakingthe Fouriertransformofthepixelintensitiesasafunctionofphasefor J> 1 : 5 J tr .Unfortunately, attemptstoananalyticaldescriptionofthetaildistributiondidnotsucceed. Hence,themeasuredbeamdistributionisdescribedinactionphasecoordinatesbyseven parameters.Thebeamcoreischaracterizedbythecentralslope c andcentralTwisspa- rameters c , c andisbypixelswithactionlessthanthetransitionaction J tr .All particleswithactionlargerthanthetransitionactionareinthetailswhicharecharacterized bythephaseofthebranches ˚ b ,themaximumaction J max ,andthefractionoftheparticles inthecore. Notethatbecausethebeamcentroidjitterisassumedtobefromasinglesourceand thereforepredominatelyalongasinglelineinphasespace,itwouldaddanasymmetry 44 Figure2.11:Phaseportraitinposition-anglephasespace(a)andaction-phasephasespace (c).Thebeamsplitsintotwobranchesseparatedinphaseatlargeactions.Thepixel amplitudeversusaction(b)showsdeviationfromthecoredistributionatlargeaction. foraninitiallysymmetricdistribution.Todetermineifthecentroidjitterwasresponsible ofthemeasuredtails,thiswasmodeledusinga2DGaussiandistributionwithrms parametersequaltothetypicallymeasuredcentralparametersand0.2mmrmsposition jitteradded.Theamplitudesofthezerothandsecondharmonicswerecalculatedforthe givendistributionbytakingtheFouriertransformwithrespecttophaseinnormalizedaction bins J =0.05mmmradwide.Theresultingasymmetry,quandbytheamplitudeofthe secondharmonicinphasecalculatedforpixelswithactions J> 0 : 5mmmrad,wasfoundto beatleastanorderofmagnitudelowerthanobservedinmeasurements(Fig.2.12)atlarge action.Atloweraction,inthecore,theamplitudeofthesecondharmonicissimilarbetween themodelandthemeasurements.ThissupportstheclaimthatthecoreisGaussianand themeasureddeviationsareexpected. 45 Figure2.12:Theamplitudeofthe0 th harmonicofthepixelintensitiesasafunctionof ˚ foraGaussianmodelisatleastanorderofmagnitudelargerthantheamplitudeofthe2 nd harmonic.Forthemeasuredbeams,theratioofthe0 th and2 nd harmonicamplitudesis similartothemodelatlow J .However,thetailscausethe2 nd harmonictodominateat largeaction. 2.2Selectedbeammeasurements 2.2.1Backgroundnoiseremoval Aftertakingascan,thescanneroperatingprogramremovesthebackgroundandcalculates thermsparametersofthephaseportrait.Thebackgroundremovalisperformedbysetting tozeroallpixelswithintensitylessthanathreshold.Bydefault,thethreshold issetto1%ofthepeakintensity[42],whichwasadequatetoremovethenoiseforthe nominal5mAbeam.However,forlowintensitybeamsthisrejectionthresholddoesnot removeallnoise,increasingthereportedemittance.And,forhighintensity,the cutlevelcanbetooaggressive,removingotherwiseobservablebeamtails.Thereforeamore robustmethodisdesired.Section1.4.3describesalternativemethodstoaccountfor theelectronicnoise,however,theyarebasedarounddeterminingthermsparametersofthe 46 distributionandarenotoverlyconcernedwithincludingsomenoiseorexcludingasmall portionofthebeamsignal. Inorderstudybeamtails,wethecutthresholdbasedonthenoiseleveland removeonlythepixelsthatcannotbedistinguishedfromthebeamsignal.Thiscutlevelis establishedby,theareathatismostlikelytocontainonlynoise.Themeasured portraitcellarrayisdividedintofouridenticalquadrants.Inthequadrantwithminimum totalintensity,the6 6pixelsquareintheoutermostcornerofthisrectangleisassumed tocontainonlysignalfromnoise.Thisassumptionwasbytakingmeasurements withoutthebeamandcomparingthesignalsinthecornerstomeasurementswithabeam present.Themeansignalofthissquareiscalculatedandsubtractedfromeachpixelover theentireportrait.Thermsnoiselevel ˙ n inthissquareiscalculatedandthecutthreshold T c issetto T c = A n ˙ n : (2.15) Todeterminethecot A n ,letusconsiderarectangularportraitcontaining N pixels = K M pixelsforwhichamplitudesaredeterminedbyrandomGaussiannoisesothatthe probabilitydensity P p ofapixelwithagivenintensity I p is dP p dI p = 1 p 2 ˇ˙ n e I 2 p 2 ˙ 2 n : (2.16) Theprobability P 0 ofhavingapixelwithamplitude A n timeshigherthanthermsnoise amplitude ˙ n is P 0 =0 : 5erfc A n p 2 (2.17) whereerfcisthestandarderrorfunction[21].Theprobability P 1 ofhavingatleastonepixel 47 abovethethresholdis P 1 =1 (1 P 0 ) N pixels ˇ P 0 N pixels : (2.18) Thiscleaningprocedurecanuse P 1 tosetacutthresholdthatwillremoveallthenoise. Toachievethiswerequire A n ˇ 3 : 3.However,ifafterthecutsinglepixelsremainabovethe thresholdwithallzeroneighborsthenthesecanbeeasilyremoved.Therefore,thecutlevel canbesetlowerandafterwardssinglepixelscanberemoved.Forthis,thecutthreshold needstobesethighenoughsonopairsofneighboringpixels(side-by-sideordiagonally) willbeabovethecutthreshold A n ˙ n .Thetotalnumberofindependentneighboringpairs N pairs is N pairs =4 KM 3( K + M )+2 ; (2.19) whichisapproximatedby N pairs ˇ 4 N pixels for K ˛ 1 ;M ˛ 1 : (2.20) Theprobability P 2 thattwoneighboringpixelsarebothabovethethresholdis P 2 =1 (1 P 2 0 ) N pairs ˇ 4 P 2 0 N pixels : (2.21) Inpractice,itwasacceptedthatonein ˘ 100portraitsmaycontainanun-removednoise pair( P 2 =0 : 01).Thevalueof P 0 iscalculatedfromEq.2.20and2.21andthenthethreshold isdeterminedbyinvertingEq.2.17.Foratypicalnumberofpixelsof3000,themultiplier inEq.2.15is A n ˇ 2 : 2.ForthenominalbeaminPIP2IT,thecutthresholdcalculatedwith 48 Figure2.13:VerticalRMSemittancewithhorizontalscraping.Theedatapointscorre- spondto1,2,3,4,5mAofbeamcurrentafterscraping. thismethodistypically ˘ 0.5%ofthepeakintensity.Thislimitsthedynamicrangeofthe scannertoapproximatelytwoordersofmagnitude. Totesttherobustnessofthismethod,ahorizontalscraperwassteppedthroughthebeam upstreamoftheAllisonscanneratlocationthreeandthephaseportraitsintheverticalplane weremeasuredateachstep.Removalofthebeamhorizontallyresultsinalowerintensityof agivenpixelintheverticalphasespace,sothatthepeakintensitycanbeusedasameasure oftheremainingcurrent.Whenthenoise-basedcutisused,themeasuredemittanceis constantwithin10%,showingthatthebeamellipseisnotx-ycoupled(Fig.2.13).However, whenthesamedataareanalyzedwiththedefault1%cut,theemittanceappearstoincrease whenthepeakintensitygoesbelow ˘ 1.5,correspondingtoabeamcurrentofroughly2mA, duetonoiseodingthephaseportrait. 49 2.2.2Quadrupolescan Totestthatthestabilityofthemeasureddistributionsin J ˚ coordinates,thestrengthofan upstreamquadrupolemagnetclosetotheAllisonscannerwasvaried.Thisisachangetothe linearfocusingopticssotheTwissparameterswillchangebutnosigntchangesinthe distributionoveractionareexpected.Despitethedramaticvisiblechangesoftheportraits in x x 0 coordinates(seeeFig.2.14(a)),thedistributioninaction-phasecoordinatesstays thesame(seeFig.2.14(d,e)),andportionofparticlesoutsideofagivenactionisstablefor morethan99%ofthemeasuredbeam(seeFig.2.14(f)).Theportionofparticlesinthe coreandthecentralslopearefoundtobestable(seeFig.2.14(b))within 3%and 5%, respectively. Thephaseoftheparticlesforscanswithtopticsshiftsbytheinbetatron phaseadvancebetweentheseportraits.Whileaphaseshiftcannotmodifytheappearance ofthephase-independentcore,thephasepositionofthetailsshouldchangeaccordingly. Inthecaseofthepresentedquadrupolescan,thechangeinthephaseadvanceissmall becausethedistancebetweenthevariedquadrupoleandtheAllisonscannerisshort.The observedphasepositionofthebranchesisinagreement,withinmeasurementerrors,with thesimulatedphaseadvance(seeFig.2.14(c)). 2.2.3Comparisonofmeasurementsintlocations Thestabilityofdescriptionofthedistributionsinaction-phasecoordinatesallowsforcom- parisonsofthephaseportraitsofthebeamthathavepassedthroughtlyt optics.Asmentionedabove,phaseportraitswererecordedinthreelocationsandwithtwo scannerorientationsover18months.Theresultsofthesemeasurements,performedwith 50 Figure2.14:Analysisofphaseprotraitsinaquadrupolescan.(a)phaseportraitsin x x 0 coordinatesrecordedatthequadrupolecurrentsincreasingfromlefttorightandfromtop tobottomfrom3.06Ato5.46A.The x and x 0 rangesineachplotare30mmand24mrad, correspondingly.Notvariationoftheslit-correctedcentralslopeandpercentin thecoreareobservedwhilethequadrupolestrengthwasscanned(b).Theaveragebranch phaseagreeswithsmallchangesofthesimulatedbetatronphase(c).Phaseportraitsin action-phasecoordinatesfortheminimumandmaximumquadrupolecurrentsoverlap(d), (e).Theportionofthebeamoutsideofagivenactionisstableovermostofthebeam(f). 51 thesamesettingsfortheionsource,LEBT,andRFQ,aresummarizedinTable2.3.Each resultrepresentsanaverageover10measurementsmadeontdaysinanattempt toseparateday-to-dayvariabilityfrombetweenlocationsandorientation.The presentederrorsarethermsvariationsovereachsetof10measurements. AcrossallthreelocationsoftheAllisonscanner,thermsemittanceisthesamewithin theerrorswhichwasinitiallyinterpretedasnosignitinthedistributions. Howeverin J ˚ phasespace,highervaluesofthecentralslopeandpercentinthecoreatthe location1weremeasured.Thisisattributedtoabetweenthedistributionsinthe horizontalandverticalplanes,sincethesevaluesstayconstantfromlocation2tolocation 3.Thisbetweenthetwoplanesisalsoseeninthelargerspreadinintensitiesat lowactionforthehorizontalplaneatlocation1comparedtotheothertwolocations(Fig. 2.15left).However,directcomparisonbymeasuringbothplanesinsinglelocationwasnot performedtothistheory. Thereisnochange,withinthescatter,in c andthefractionoftheintensityinthe corebetweenlocations2and3inwhichtheAllisonscannermeasuredintheverticalplane. Thisisinterpretedasanabsenceofmeasurablechangesinthebeamcoreparametersin theMEBT.Theextentofthebeamtailscanbemoreeasilyseenbyplottingthetotal intensityofpixelswithactionlargerthansomevalueasafunctionofaction,i.e.integrating theintensity-actiondistribution(Fig.2.15right).Outsideof ˘ 95%ofthemeasuredbeam intensity,thebetweendistributionsislargerthatonewouldexpectfromstatistical andreconstructionerrorsbycomparingwithFig.2.14(f).Theincreaseof particlepopulationoutsideoflargeactionsfromlocation2tolocation3visibleinFig.2.15 (right)maybeasignoftailgrowth.However,duetothelimiteddynamicrangeofthe Allisonscannerandexistenceofbeamjitter,itistomakeaeclaim. 52 Figure2.15:Comparisonoftheamplitudeversusactiondistribution(left)atthebeginning andendoftheMEBTshowsadbetweenthehorizontalandverticalplanes.The fartailsextendfartheratlocation3comparedtolocation2(right)whichisasignoftail growth.Theportionoutsideagivenactionatlocationonecannotbeusedtosearchfortail growthbecauseitrepresentsadtplane.Theshadedareasrepresentthermserrors calculatedbypropagationofthepixelamplitudeations. Table2.3:Averagermsemittancesandcoreandtailparametersforthethreelocationsof theAllisonscanner. Locationrms c %incore 1-horz0 : 20 0 : 0130 : 146 0 : 00388 2 : 5 2-vert0 : 19 0 : 0150 : 117 0 : 01371 11 3-vert0 : 22 0 : 0240 : 123 0 : 01172 10 2.2.4Distributionattbeamcurrents Changingthebeamcurrentisanotherexamplewheredistributionoveractiongavemore detailedinformationaboutthebeamdistributionthanthermsparameters.Figure2.16 showshowthebeamparametersatlocation1varywhenthebeamcurrentisincreasedby increasingtheextractionvoltageoftheionsource.Allothersettings,tunedtooptimize performanceatthenominal5mA,arekeptconstant. Lookingsolelyatthermsparametersthatarewereinitiallyusedtoquantifythedis- 53 tributions,anincreaseinthermsemittanceisseenstartingat ˘ 3mAcoincidentwiththe ofthepeakintensityofthebeam.Thisappearstobeasaturationofthebeam coreresultinginincreasedtails. However,theparametersusedtodescribetheactiondistributiontellatstory. At3mAthefractionofthebeaminthecoreplateausasdoes J tr signifyingminimaltail growthathighercurrents.Thecentralslope,however,continuestoincreaseforallcurrents. Atlowcurrent,thegrowthofthecentralslopeiscompensatedbyareductioninthesizeof thetailsresultinginminimalchangestothermsemittance.Above3mA,thecentralslope continuestobroaden,butthetailsseelessvariation.Itisthebroadeningofthecentral regionthatcausesintheincreaseinthermsemittance,nottailgrowth. 2.2.5Scraping ThePIP2ITMEBTcontainsfoursetsoffourscraperseachsetconsistsofabottom,top, right,andleftscraper.OnegoalofthescrapingsysteminthePIP2ITMEBTistoremove fartailsandinterceptingpartofthebeamwiththescrapingsystemwasforeseenasanormal modeofoperation.Preliminaryestimatesfortailremovalweremadeforaphase-independent GaussianbeaminRef.[47].Inthiscase,itisoptimaltoseparatethescraperby ˇ= 2betatron phaseadvancetominimizethemaximumpassableaction.However,measurementswiththe Allisonscannershowedthesituationtobemorecomplicatedduetothephase-dependent branches.Forillustration,Fig.2.17comparesphaseportraitsrecordedatlocationthree whenremovingbeamwithasinglescraperattlocations.Forthisstudythetop scraperwasmovedintothebeamateachofthescrapingstations,oneatatime,tointercept 10%ofthecurrent(0.5mA)basedonthemeasuredcurrentatthebeamdump.InFig.2.17 phaseportraitswith(red)andwithout(blue)scrapingareoverlappedin x x 0 (left)and 54 Figure2.16:Thetotalbeamintensity(a),peakpixelamplitude(b),emittance(c),andTwiss parameters(d)fortextractionvoltages V extr .Parametersareplottedasfunctions ofthebeamcurrentintheLEBT. 55 J ˚ (right)coordinates.Theactionandphaseofthescrapedbeamsarecalculatedusing thebeamcentroidandthecentralTwissparametersofthenon-scrapedbeamtomaintain thesamenofactionfordirectcomparisons. Figure2.17showsthatscrapingthesamefractionofthebeamcurrentbyerentscrap- ersresultsinrentremovalofthetailsduetoastrongdependenceofthetailintensity onphase.Forexample,insertingthescraperM71(Fig.2.17(d))removesprimarilythetail particles,whilethescraperjustupstream,M61(Fig.2.17(c))missesatportion ofthebranchesanddoesnotreducethemaximumactionofthebeam.Instead,inorderto remove10%ofthecurrent,thescraperremovesparticleswithloweraction.Therefore,in ordertoachievemaximaltailreductionforagivenreductionoftheoutputbeamcurrent, thebeamphasingatthescraperscanbeoptimizedbyadjustingtheopticsand/orscraper locationssuchthatthephasesofthebranchesareat0or ˇ atthescrapers.Alternatively,if suchchangestotheopticsarenotpossible,scrapersthatarenotexpectedtointerceptthe tailscanbepositionedtoremovelessofthetotalbeamcurrentwithminimalincreasetothe passedmaximum J .Thiscanresultinamoretscrapingregimethanwillremoveless ofthetotalcurrentbutstillremovethelargeactionbeamtails. Phaseportraitsforscrapingnearlocation1withscraperM00visiblyshowarounded edgeonthescrapedside.Thisisvisuallydistinctfromscrapingnearlocation3withscraper M71wherephaseportraitshaveastraightscrapededge,Fig.2.17.Thishintsthatfor theupstreamscraperspropagationthroughthebeamlinesmearsthescrapingboundary beyondofwhatisexpectedfromthewidthofthescannerslits.Thiscouldberelated tonon-linearspace-chargeasexpectedfrompreliminarysimulationsinRef.[47].In attemptingtomakeanumericalestimationofthisonecanpropagatethescraper footprintusingthetransportmatrixandcalculatetheportionoftheparticlesbeyondthe 56 Figure2.17:Phaseportraitswithscraping.Rows(a)-(d)correspondtomovingintothe beamoneofthescrapersalongthebeamlinepresentedinFig.2.3;fromtoptobottom M00,M11,M61,M71.Ineachcase,0.5mAisinterceptedoutoftheinitial5mA.Therow (e)representsthebeamwhentopandbottomscrapersareinsertedinM00andM11 stations.Thesolidlinesrepresenttheattemptofpropagatingthescrapelinesaccordingto 5mAbeamsimulations.Thedashedlinesrepresentpropagationwiththephaseadvance increasedby10%.Seeotherdetailsinthetext. 57 cutlineintherecordedportrait.Ascraperwithvertical d fromthebeamcenter producesalineintheAllisonscannerportrait u 0 1 ( u )= u 1 ˚ ) 1 ) d p 0 1 ˚ ) (2.22) wheresubscripts0and1denotethelocationsofthescraperandAllisonscanner,corre- spondingly, ˚ istheverticalbetatronphaseadvancebetweenthetwolocations,and and aretheTwissparameters.Thecenterofthecoordinatesystemisplacedatthecenterof thedistribution.ThermsTwissparametersattheAllisonscanneraremeasureddirectly. Atthescraper,theandthermsbeamsizeand,assumingaconstantemittance, 0 canbereconstructedbysteppingthescraperthroughthebeamandrecordingthepassed currentonthebeamdump.Thedumpcurrentasafunctionofthescraperpositioncan betothestandarderrorfunctiontodeterminetheandsecondordermoments.The phaseadvanceneedstobedeliveredbytheopticsmodel.Thermsparameters,simulatedby TraceWin[48],arefoundinagoodagreementwiththeenvelopemeasurementsperformed withscrapers[49],andthephaseadvancesarecalculatedusingthermsbeamsizesandemit- tancesdeliveredbytheprogram.ThescraperfootprintsdrawnaccordingtoEq.2.22are shownonallplotsofFig.2.17withsolidlines.Theselineswereexpectedtoapproximately coincidewiththescrapededgeofthebeamdistribution.However,thisvisuallyisnotthe case,andnumericalestimationsoftheparticles'overthescraperfootprintscannot bemade.Apossibleinterpretationofthisresultisthattheaccuracyofpredictionwiththe linearmodelwithuniformphaseadvanceforallparticlesbecomesunsatisfactoryforthecase oflongpropagationofthetailswithnon-linearspacechargeandfocusingforces.Simulations performedwiththesameinitialconditionsandthesamemagnetsettingsbutwithazero 58 beamcurrentshowthephaseadvancesthatarelargerthanatnominal5mAby10-20% [47](dependingonthelongitudinalposition).Becausethedensityvariesacrossthebeam, itcanassumedthatthetailparticlesadvanceinthephasewitharatesomewherebetween thezerocurrentandnominalcases.ThedashedlinesinFig.2.17drawnwiththephase advancesincreasedby10%toestimatethereducedbetatrontunedepressionofthetailsfor eachportraitareindeedvisuallyclosertothescraperfootprints. Thisassumptionofnon-uniformphaseadvancesissupportedbyanincreasingphaseshift ofthesecondharmonicasafunctionofactionthroughouttheMEBT.Thesecondharmonic wascalculatedbytakingtheFouriertransformwithrespecttophaseinnormalizedaction bins J =0.05mmmradwide.Thiswasdoneformeasurementsateachofthethreelocations oftheAllisonscannerandthephaseswereshiftedbyaconstanttohavezerophaseathigher J foreasiercomparison(Fig.2.18).Atlocation1,thephaseismostlyconstantwithaction. Atlocationtwo,thephasestartstodecreaseatloweractionsandthisbehaviorbecomes largeratlocationthree.Theshiftinthesecondharmonicisapproximatelyproportionalto theamplitudeofthe0 th harmonicatalllocations,i.e.theamplitudeofthelocaldensity. Thiscanbeinterpretedasthetailshaveatphaseadvancefromthecore resultinginthebeambecoming`S'shaped[18]. Thisincreasing`S'shapecanexplainthatdespitehavingthesame c andfractioninthe core,thedistributionatlocation3extendstohigheractioncomparedtolocation2(seeFig. 2.15).Thetailparticlesareshiftingawayfromtheprimaryaxisoftheellipseby thecentralparametersencompassingthecore.Thiswouldcausethetailstomovetohigher actionswithoutthecentraldistributionorthetotalpopulationoftailparticleschanging. 59 Figure2.18:Phaseofthesecondharmonicofthepixelintensitiesasafunctionofphase varieswithaction.LargervariationsareseenfatherfromthestartoftheMEBTshowing thephaseadvancevariesacrossthebeam. 2.3Futurework Thedevelopmentoftheanalysisofphaseportraitsusing J ˚ coordinateswasintendedfor studyingthebeamtails.Unfortunately,noiseissueswiththeAllisonscannerinthePIP2IT beamlinemadethischallenging.Thethermalnoisewasquitelargeandlimitedthedynamic rangeofthedevicetoroughlytwoordersamagnitude.Toadequatelymeasurethebeam tails,adynamicrangeofatleastthreeorfourordersofmagnitudeisrequired.Whilewe didinvestigatethesourceofthelargenoiseor,nocorrectionwasfound. Inaddition,thebeamjitterdistortsthemeasuredphasespacedistribution.Thejitter appearstoberandom,thereforethemeasureddistributiononaverageshouldbemostly andadequatefordeterminingthermsbeamparameters.However,thejitterdoes thepixelintensitiesontheindividuallevelwhichconfoundsdetailedanalysisofthe phaseportraitsusingaction. Becauseoftheseissues,itischallengingtofurtherthedevelopmentofthisanalysis 60 techniqueusingphaseportraitstakeninthePIP2ITMEBT.Futurestudiesshouldbetaken ofamorestablebeamwithaphasespacescannerwithalargerdynamicrange. Also,theintensitiesoftheindividualpixelsareknowntobebythegeometryof theAllisonscanner.Equation(2.13)providesasimplemodelofthisforaGaussian beamanditisneededtodeterminethecentralparameters.Itwouldbeidealtocorrect theintensityofeachpixelforthisThisprocedureisessentiallyadeconvolutionof themeasuredphaseportraitwiththepassedphasespaceareaforagivenposition-voltage setting(Fig.2.8)andaprocedureforthisisdescribedinRef.[46].However,deconvolution isaninherentlynoisyprocessand,duetheaddednoisefrombeamjitter,attemptstocarry outthiscorrectionfailed.Onceagain,cleanersignalsareneededtoimprovetheanalysis. 61 Chapter3 Beammeasurementsusing beampositionmonitors Measurementsofthebeamdistributioninphasespaceareusefultofullycharacterizethe beam.However,thesedevicescantypicallyonlybeusedinlowenergyregionsofthebeamline becausetheyrelyonthelowermagneticrigidityofbeamtosamplediscreteportionsofthe beam.Inaddition,becausethesedevicesinterceptthebeam,theymustbeabletowithstand thedepositedbeampowerwhichbecomesmorechallengingathigherenergies.Also,because theyareinsertedintothebeamline,thephasespacemeasurementscannotbeperformed duringoperation.Whileslitbasedphasescannersareusefulforcharacterizingthebeam distribution,typically,therewillonlybeacouplephasespacescannersinabeamlineand cannotbeusedtocontinuouslymonitorthebeamproperties. Ontheotherhand,themostprevalentdiagnosticdevicesinalmosteverybeamlineare beampositionmonitors.Thesenon-interceptingdevicesareusedthroughouttheentirety ofacceleratorsasoneoftheprimarytoolstoverifythebeamdynamicsandtuneofthe beamline.StandardanalysisoftheBPMsignalsonlydeterminestheordermoments (centroid)ofthebeam.ExpandingtheanalysisoftheBPMsignalstogivemoreinformation aboutthebeamdistributionwouldallowforpassivemonitoringofmorebeampropertiesat manylocations.Thisadditionalinformationwouldbehighlybfordeterminingthe 62 operationpointsofparticleaccelerators. 3.1BPMsignals 3.1.1Validityofthepencilbeammodel TheelectricgivenbyEq.1.46holdsforasmallbeamtransverselywith chargemodulatedlongitudinallyatasinglefrequency.While,thisisclearlyat abstractionfromanactualbeam,itisanadequatemodeloftheBPMsignalsinmostcases andwidelyused[5].Thesinglefrequencymodelisacceptablebecausetheprocessingofthe BPMpickups'signalstypicallyusenarrowbandtomeasureasingleharmonicofthe bunchrepetitionratetoreducenoise.Inaddition,athigherbeamenergies,the = signals areapproximatelyfrequencyindependent. Thepencilbeamisanacceptablemodelofthetransversedistributionifthebeamis smallcomparedthepiperadius.However,whenthebeamsizecoversanappreciableportion ofthepipeaperture,thesignalsmeasuredbythepickupscanbemodeledbysumming overacollectionofpencilbeamsaterentlocationstogeneratethetransverse Thecomponentsofthisdiscretizedmodelfarfromthecenterwillbebythenon- linearitiesinthepickups'responseandcausethemeasuredpositiontobedistorted[7].This distortionisdependentontheexacttransversedistributionmakingitchallengingtomodel. Becausethisisduesolelytonon-linearitiescausedbythegeometryoftheBPM,itis independentof gR p andnon-relativistic Thisisistypicallyontheorderofafewpercentvariationandtypicallyisnot consideredwhendeterminingthebeamposition.Forexample,consideraGaussianbeam with ˙ y =2mmlocatedat x 0 =2mmand y 0 =1mminapipeofradius R p =23 : 5mm.When 63 Figure3.1:Variationof = whenchanging ˙ x ofaGaussianbeamcenteredat x 0 =2mm and y 0 =1mmfora47mmmapertureBPMwith20mmdiameterroundpickups. ˙ x isvariedfrom1mmto5mm,theresulting = andthereforethepositionmeasurement, variesby2-3%(Fig.3.1).Whilethesebeamarelargerthanideal,theseparameters arerepresentativeofabeamintheFRIBMEBT. 3.1.2AmoredetailedBPMmodel Whilethisthin,singlefrequencymodelofabeamisgenerallyadequatefor positionmeasurements,inrealitythesignalsfromBPMpickupsalsocontaininformationof thelongitudinalbunchprTheBPMsignalsaredirectlyproportionaltotheFourier amplitudeofthelongitudinal D ! atthemeasuredfrequency.Therefore,itispossible, usingbeamlinemodels,tomeasurethelongitudinalsizeofthebunchbyvaryingthelongi- tudinalopticsandmeasuringchangesintheamplitudeofasignalharmonictocalibratefor theseandusetheBPMstomeasurethelongitudinalbunchlength[50,51].Intheory, thebeamlinemodeldependencecanberemovedbyrecordingtheBPMpickups'response atmultiplefrequencies.Inordertoachievethis,amorecompleteanalyticmodelthanEq. 64 1.46isneeded. OneimportantmissingfeatureinEq.1.46isthedependenceonthetransversedistribu- tionwhichismorecomplexthanthelongitudinaldistributiondependence.Thedependence oftheimagechargesonthepipewall ˙ wall foragiventransversedistribution T ( r;˚ )canbe foundbyintegratingEq.1.46for ˙ fromapencilbeamover T ( r;˚ ) ˙ wall ( !;z m ;˚ m )= D ! cos ! t + z 0 z m c ZZ rdrd˚ X n =0 T ( r;˚ ) I n ( gr=R p ) ˇNI n ( gR p ) cos[ n ( ˚ m ˚ )] : (3.1) Theofthetransversedistributionismorediscussedinsection3.1.3. Theimagechangemustnowbeintegratedoverthepickup.Foraroundpickup,which areusedatFRIB,ofradius R b Eq.1.53becomes ˙ pickup ( ! )= Z R b R b d z Z 1 R p q R 2 b z 2 1 R p q R 2 b z 2 R p d ˚ m ˙ wall : (3.2) Forsimplicityalltermsin ˙ wall independentof z m and ˚ m willbeputintoasinglecot 65 F ( g;r;˚ )and ˚ 7! ˚ + ˚ p toaccommodateanyazimuthalpickuplocation ˚ p .Thisgives ˙ meas ( ! )= Z R b R b d z m Z 1 R p q R 2 b z 2 1 R p q R 2 b z 2 R p d ˚ m ZZ r d r d ˚ X n =0 F cos[ n ( ˚ m ˚ p ˚ )]cos ! c ( z m z 0 ) (3.3) ˙ meas ( ! )= ZZ r d r d ˚ X n =0 FR p 2 n cos[ n ( ˚ p + ˚ )] Z R b R b d z m cos ! c ( z m z 0 ) sin n R p q R 2 b z 2 (3.4) ˙ meas ( ! ) ZZ r d r d ˚ X n =0 FR p cos[ n ( ˚ p + ˚ )] P ( ! )(3.5) where P ( ! )= 2 n Z R b R b d z m cos ! c ( z m z 0 ) sin n R p q R 2 b z 2 (3.6) isthetransittimefactorforaroundbuttonpickup.Thereforethecorrectionforthebutton shapecanbeseparatedfromtheofthebeamdistribution. Thisderivationapproximatesthepickupgeometryaswiththeroundpipewall.In practice,buttonpickupsarecommonlyusedbecausetheyaresimpletomanufacture. ShowinFig.3.2istheinsideofaBPMusedatFRIBwhere20mmdiameterpickupsare usedandtheyarerecessed1mmfromtheinnerpipewall.Thisingeometryneeds tobeaccountedforwhenthemeasuredwavelengthisonthesameorderasthepickupsize. Forthe20mmdiameterpickupsusedatFRIB,thefrequencyneedstobeabove ˘ 0.5GHz forabeamtravelingat =0 : 033correspondingtothebeamenergyexitingtheRFQ.This estimateisbyCSTMicrowaveStudio[52]simulationsofthetwoBPMgeometries atthisvelocitywhichshowthebecomestat ˘ 400MHz(Fig.3.3).To limittheofthecurvedapproximation,thebroadbandmeasurementspresentedbelow 66 Figure3.2:GeometryoftheFRIBMEBTBPMs.Thepiperadiusis23.75mmandthe pickupradiusis10mm[6]. withtheFRIBBPMswerelimitedto400MHz. 3.1.3Pickupsignalvariationwithtransversedistribution Thevariationoftheprowithbeamposition,asmentionedinsection1.4.1.,canbe seenfromthemeasuredspectrafromvariousofapencilbeam(Fig.3.4).Because, fornon-relativisticbeams,thesignalsonthepickupsvarywith ! and r ,theyarebest characterizedintermsof gR p withhigher gR p correspondstohigherfrequencyandlower r .Theofthebeamistfor gR p > 1withvariationsontheorder of10sofpercentforvaluesof gR p upto ˘ 10.For gR p < 1thespectraonthebuttonsfrom anbeamarethesameasthespectraforacenteredbeam.Therefore,for gR p > 1 67 Figure3.3:Signalonaandcurved20mmdiameterBPMpickupfroma =0.033beam. Thetwogeometriesgivesimilarresultsupto ˘ 400MHz. Eq.1.51mustbeusedtocorrectlydeterminethepositionandfor gR p < 1Eq.1.52,which independentof g ,cansafelybeused. Toseetheeofthetransversedistribution,theamplitudeoftheintegralinEq.3.1 isplottedinFig.3.5asafunctionof gR p andarenormalizedtotheresultapencilbeam. AGaussianbeamwith ˙ x = ˙ y wasusedandthetransversesizewasvaried.Similarto thebeam,thevariationsduetotransversebeamsizearetfor gR p > 1. For gR p < 1,thetransversesizecanbeneglectedandapencilbeamcanbeassumed. Similarareseenwhenvarying ˙ x or ˙ y individuallywhileleavingtheotherd. Thisdependenceonthetransversedistributionathigh gR p causedbytheelectric distributionandisdistinctfromtheerrorscausedbythetransversebeamdistributiondue tonon-linearitiesdiscussedinsection1.4.1.Becausethetransversedistributioncanthe ontheBPMpickups,thetransversedistributionshouldbeabletobedetermined fromthepickupsignals.Thiswouldrequiremeasurementsatmultiplefrequenciestoseparate thetransversefromlongitudinalSuchwouldmeasurementswould 68 Figure3.4:Variationinthemeasuredspectraona20mmdiameterpickupina47.5mm diameterpipeforanpencilbeam.Thespectraarenormalizedtothecenteredcase. Table3.1:ParametersofthedistributionsusedinFig.3.6. Amp1 x 0 1 y 0 1 ˙ x 1 ˙ y 1Amp2 x 0 2 y 02 ˙ x 2 ˙ y 2 DoubleGaussian10mm1.07mm1.7mm2.49mm SingleGaussian10mm0mm1.7mm1mm0.40mm2mm1.7mm3mm eturnBPMsintotransverseandlongitudinalmonitors. IfaBPMisoperatinginaregionwherewheretransversedistributionsmustbeaccounted for,itisimportanttousetheexactdistribution.Itisnotenttouseamodelwith thesameandsecondordermoments.Forexample,considerabeamwithtransverse thatisthesumoftwoGaussianandabeamwithasingleGaussianwiththe sameandsecondordermoments,e.g.distributionwithparametersgiveninTable3.1. Athigh gR p themeasuredsignalstlybothinamplitudeandresulting inerrorsinthemeasuredposition(Fig.3.6).Thisistocorrectbecauseit requiresanaccuratemodelofthethebeam.Atlower gR p themeasuredspectrabecome nearlyidenticalonallpickupsandthedistributionnolongerneedstobetakenintoaccount. 69 Figure3.5:VariationinthemeasuredspectrumforacenteredroundGaussianbeamof tsizes.Thespectraarenormalizedtoacenteredpencilbeam. 3.1.4Buttonsumsignal Thedependenceonthepickupspectraonthetransversedistributionmakesmeasurements ofthebeamprochallenging.Topartiallyalleviatethissensitivity,thesignalsfromall fourbuttonscanbeaddedtogether.BysummingEq.3.5overthefourpickups,thesummed signalisfoundtobe ˙ sum ( ! )= P ( ! ) ZZ r d r d ˚ X n =0 FR p cos[ n (0+ ˚ )]+cos[ n ( ˇ= 2+ ˚ )]+ cos[ n ( ˇ + ˚ )]+cos[ n (3 ˇ= 2+ ˚ )] (3.7) ˙ sum ( ! )= P ( ! ) Z d A beam X n =0 FR p 4cos( n˚ ) 8 > < > : 1 ;n 0(mod4) 0 ; else (3.8) ForthecircularbuttonpickupsusedforFRIB,onlyazimuthalharmonicsthatarezero modulofourremainaftersumming.Thistlyreducesthedependenceofthesignal 70 Figure3.6:ComparisonofmeasuredspectrafromaGaussiananddoubleGaussianbeam withthesameandsecondordermomentswith =0.033(left)and =0.15(right).At large g thetdistributionresultsinaerentmeasuredspectra. 71 on ˚ aswellasthedependenceonthebeampositionandtransversedistribution.Inthecase ofrectangularpickupsthatcoverthefull2 ˇ solidangle,onlythe n =0componentremains andthesignalisindependentof ˚ . ThescalingswiththetransversedistributiondiscussedabovearerepeatedinFig.3.7 usingthesummedsignalsfortheFRIBBPMgeometry.Thedependenceonbeamfor apencilbeamisreducedbyafactorof ˘ 7andthedependenceonthetransversesizefor aGaussianbeamisreducedby > 20%athigh gR p .Thereductionofdependenceonthe transversebunchshapeisindependentofthesizeofthebeam.Thesereductionsallowfor thesummedsignalthebeamcanbeassumedtobeapencilbeamat gR p ˇ 3whichisan improvementfromthenon-summedsignalswhichrequire gR p ˇ 1toassumeapencilbeam. However,whilethismethodreducesthesensitivitytothetransversedistributionwhich isbfordeterminingthelongitudinalforlowenough r , gR p willbelarge enoughthatthetransversedistributionstillneedstobeaccountedfor.But,duetosymmetry, the x and y cannotbedistinguishedfromeachotherinthesummedsignalandthe positioncanonlybedeterminedmoduloaphaseof ˇ= 4.Thiscanresultintheaggravating situationwherethesamenumberofparametersmustbeusedtodescribethebeamasthe non-summedcase,butlessinformationisobtained.Ingeneral,thesumsignalshouldbe usedwheninformationofthetransversedistributionisnotneededandwhen gR p 2 (1 ; 3). 3.1.5BPMresponsesimulations Therequiredcorrectiontorecoverthebeamparametersfromthemeasuredsignalswere checkedusingCSTMicrowaveStudiosimulations[52].Thesimulationswereperformed usingthewaksolverwithamono-energeticpencilbeamandtheatthepipe wallwasmeasuredatasinglepointtoEq.1.46.Thiseliminatescorrectionsfor 72 Figure3.7:(Top)Variationofthesummedsignalforanpencilbeam.Thevaria- tionsinthespectraareafactorof ˘ 7lowerthanthenon-summedsignals.(Bottom)The summedsignalsofa1mmGaussianbeamnormalizedtothesinglepickupsignals. Thevariationsofthesummedspectraarereducedby20%comparedtothenon-summed signals. 73 Figure3.8:CSTsimulationsusingthewaksolvermustbetoauniformsquare beamwithsidelengthgivenbythemeshsize.Forthegivenfrequencyrange,this transversedistributioncanbeignoredfor > 0.15. thetransversesizeandthepickupgeometryandimpedance;thesefeatureswereplanned beaddedlatertotestEq.3.5.However,simulationsusingthewaksolvershoweda discrepancybetweentheatapointonthewallinsimulationresultsandtheanalytic atthewallfromapencilbeamfor r < 0 : 15(seeFig.3.8). Instead,thesimulationresultsmatchedtheexpectedsignalsfromauniformsquarebeam withsidelengthequaltotwicethemeshcellsizesuggestingthediscrepancyiscausedbyhow CSThandlesthepencilbeaminthewaksolver.Thepencilbeamusedinthewav solvertransverselylaysontheintersectionoffourmeshcells.Thesolverappearstoassume allcellstouchingthethin,pencilbeamarepartofthebeamandmustbeincludedinthe analyticmodel.Thedampingofthetransversedistributionbyreducing gR p isalso bysimulationswith r 0 : 15whichagreewiththeanalyticresultsfromapencil beamanddonotrequireatransversedistribution. Withthisunwittinginclusionofatransversedistributionatlow ,thewaksolver wasusedforsimulationsincludingaBPMmodel.Themodelusedisasimpmodel 74 Figure3.9:ModeloftheFRIBBPMSinCSTMicrowaveStudio. consistingofcylindricalbuttonsthatareconnectedtogroundvia50discreteports (Fig.3.9).Theimpedanceofthesepickupswasmeasuredusingthetransmissioncot S 2 ; 1 betweentwopickupsandfortheresistanceandcapacitance(Fig.3.10)similar totheprocedureinRef.[6].Themeasuredcapacitancetogroundwas4.1pF. TheBPMsignalsmatchtheexpectedanalyticsignalswithin5%for =0 : 033andthe decreasedwhen wasincreased(Fig.3.11).Fromthesesignals,aresonancedue tothebuttonsizecanbeseenaround500MHzfor =0.033.Thisresonanceisnotseenin beamlinemeasurementsduetolowpassering. 3.2Bunchmeasurements 3.2.1TISwaveforms TheBPMsystematFRIBcanmeasurethesignalsfromthebuttonsoverawidebandwidth usingaTimeInterleavedSampling(TIS)methodsimilartothemethoddescribedinRef. [53].Thismethodassumesbeamiscomprisedofaseriesofindividualbunchesthatare 75 Figure3.10:Fittingtransmissioncot, S 2 ; 1 ,ofatheCSTbuttonmodeltodetermine theimpedance. Figure3.11:SimulationresultscomparedtoanalyticresultsofBPMpickupsignalsfroma centeredpencilbeamusingthewaksolver. 76 Figure3.12:ExampleofthesamplingproceduretomeasuretheTISwaveforms.Eachsample takenbythedigitizerisatadtphasewithrespecttothesignalandcanbeusedto reconstructtheindividualrepeatedpulse(imagecourtesyofS.Cogan). assumedtobeidenticalandlongitudinallyspacedatarepetitionrateofeither40.25MHzor 80.5MHz.Thepickupsignalsfromthebunchtrainaresampledbythedigitizerat119MHz. Eachsampleofthedigitizeroccursatatphasewithrespecttothebunches(Fig. 3.12)resultinginanesamplingrateof2.737GHzandcanresolveharmonicsof 40.25MHzupto1.3GHz.However,themeasurementsarelimitedto0.5GHzbyalow-pass ontheboard.ThesemeasuredsignalsarereferredtoherehastheTISwaveforms. 3.2.2Filtering Afterthesignalismeasuredbythepickup,it,passesthroughacableandlowpassto adigitizer.Theresponsetothissystemwascharacterizedbyremovingthecablesfromthe pickupsandinputtingaharmonicof80.5MHzupto483MHzintothecablesandrecording theoutputofthedigitizer.Forthiscalibration,theTISwaveformscouldnotberecorded becausethatsystemrequirestheinputsignaltobephaselockedtothe80.5MHzglobal clockwhilethesignalgeneratorcouldonlylocktoa5MHzor10MHzclock.Insteadthe rawsignalfromthedigitizerwasrecorded.TheFouriertransformsofthesesignalswereused 77 Figure3.13:CableandboardcalibrationforfourbuttonsonaBPM. todeterminetheresponseforeachharmonic.Foranygiveninputfrequency,theharmonics ofthatfrequencywerealsoseen.Thehigherharmonicattwicetheinputfrequencywas atleastthreeordersofmagnitudelowerthantheprimarytoneandignoredintheanalysis. Thiscalibrationisstablewithin 10%fortestedBPMs. ThecalibrationofthefourpickupsononeoftheFRIBBPMsisshowninFig.3.13. Notethattwoofthepickupsare ˘ 1 : 5dBlowerthantheothertwo.Forthesebuttons,the signalprocessingboardincludesaswitchforinjectingsignalswhichcausesthereduction. AllBPMshavetheseswitchesfortwoofthebuttonsandthesemustbecorrectlyaccounted for. Onlyharmonicsof80.5MHzwerecalibratedbecause,atthetimethecalibrationwas performed,theTISwaveformscouldonlymeasureharmonicsof80.5MHz.Thesoftwarewas laterupdatedtomeasureharmonicsof40.25MHz.Theuncalibrated40.25MHzharmonics, exceptfor40.25MHz,arecorrectedusingacubicsplineinterpolationofthe80.5MHz harmonicsmeasurements.The40.25MHzharmonicwasnotusedinmeasurementsbecause 78 itcouldnotbecalibrated.Thesecalibrationswereonlyperformedfortheninewarm BPMsneartheRFQincludingallfourBPMsintheMEBT.ForallotherBPMs,thesignals areapproximatelycorrectedusinganaverageofthesemeasurementsandknowledgeofthe pickupswiththeswitches. 3.2.3Beamlinemeasurements MeasurementsoftheBPMbuttonssignalsweretakenintheFRIBMEBT(Fig.1.5).In thisregionthebeamvelocityis r =0 : 033.TheTISwaveformswererecordedforallfour BPMintheMEBTwhichhavea47mmdiameterapertureandfour,20mmdiameterround pickups.However,mostmeasurementsofinterestweretakenwiththethirdBPMbecause thereisawiremonitordirectlyupstreamofitanditistlydownstreamof abunchercavitytocausevariationsinthebunchlength.TheTISwaveformsrecorded harmonicsof40.25MHzupto483MHzcorrespondingtoarangeof gR p from0.6to7.2. TheTISwaveformweretakentomeasureharmonicsof40.25MHz,howevertheRF frequencyofFRIBis80.5MHz.Thisispossiblebecause,insinglechargestateoperation,a prebuncherbeforetheRFQcausesthebunchescomingoutoftheRFQonlyeveryother RFbucket.However,duetoimperfectbunching,asmallsignalwasseeninthebucketthat wassupposedtobeempty.Thiscausestheevenharmonicsof40.25MHz,i.e.theharmonics of80.5MHz,tobeslightlyhigherthantheoddharmonics(Fig.3.14).Thebeamlinewas alsooperatedwithouttheprebuncher,inthiscaseallRFbucketswereandonlythe harmonicsof80.5MHzarenon-zero. TherawTISwaveformsarecorrectedfortheimpedanceandboard(Fig.3.15) thentoEq.3.5todeterminethetransverseandlongitudinalsizesofthebunch.This assumesthebeamisGaussiantransverselyandlongitudinallywithparameters x 0 , 79 Figure3.14:ExampleofameasuredTISwaveformintheFRIBMEBT(left).Thebump at15nsisapartiallyRFbucket.Thiscausestheharmonicsof80.5MHztobehigher thantherestoftheharmonicof40.25MHz(right). y 0 , ˙ x , ˙ y , ˙ xy , ˙ z ,anamplitude,andantoaccountfornoise.Becausetheamplitude ofeachbuttondependsonthetransversesizeandallfourbuttonspectramustbe simultaneously.ThemeasuredspectraintheMEBTwithin 10%toEq.3.5with primarilyduetothediscrepancybetweentheevenandoddharmonics(Fig.3.16). 3.2.4RFbunchervoltagescan MeasurementsweretakenwiththethirdBPMintheMEBTforarangeofvoltagesof theupstreambunchercavitywithitsettoabunchingphasetochangethebunchlength attheBPM.AlongwiththeTISwaveforms,thetransversewererecordedwitha wirescannerdirectlyupstreamoftheBPM.TheFastFaradayCup(FFC)located downstreamofthethirdBPMcouldnotbeusedtoverifythelongitudinalformost measurementsbecauseaFaradaycupatthesamelocationneededtobeinsertedtoactas abeamstop.Instead,simulationswereusedtocomparetheexpectedlongitudinalsizeto 80 Figure3.15:RawandcalibratedspectraofaBPMintheMEBT Figure3.16:Fittingthemeasuredspectraandfractionalerrorfromthemeasuredvalues. 81 Figure3.17:ComparisonofthemeasuredbunchlengthwithsimulationsatthethirdBPM intheMEBTwhenthebunchercavityvoltageisvaried.TheBPMmeasurement,while closetotheexpectedvalues,failtoproducedtheexpectedtrend. theBPMmeasurements(Fig.3.17).WhiletheBPMmeasurementsgivelongitudinaland transversesizesonthesameorderasthesimulationresults,thelongitudinalmeasurements failtofollowtheexpectedtrendfromchangingthebunchervoltage. Toverifythesimulationresults,separatemeasurementsweretakenwiththeFFCfor thesamerangeofbunchervoltagesandtheyfollowtheexpectedtrendofgoingthrougha minima.InadditiontheFFCmeasurementsthatthelongitudinalbeamis primarilyGaussian. TheabnormalbehavioroftheBPMmeasurementsisbelievedtobecasedbythetrans- versedistributionofthebeam.Measurementsofthetransversewiththewirescanner clearlyshowclearlynon-Gaussian(Fig.3.18),particularlyintheverticalplane,while thetotheBPMmeasurementsassumesaGaussiandistribution.TheTISwaveforms wereforharmonicsof40.25MHzfrom80.5MHzupto402.5MHzcorrespondingtoa rangeof gR p from1.2to6.0andsotheexactformofthetransversedistributionwill 82 Figure3.18:ExamplesoftransversemeasurementintheFRIBMEBTmeasuredwith awiremonitor. themeasuredspectra.Itwasinitiallyhopedthat,whilenotexact,theGaussianmodel wouldbecapableofcorrectlyapproximatingthesecondordermoments.But,thebuncher scanresultsshowedthisisnotthecase. ThereexistsawirepmonitordirectlyupstreamofthethirdBPM.Itispossibleto inputthemeasuredfromthisscannerintoEq.3.5inordertothelongitudinal However,thisprocedurecannotbegenerallyappliedtotheotherBPMalongthe beamlinebecausetherearenomeasurementsofthetransversedistributionelsewhere.Vary- inganupstreamquadrupoletochangethetransversedistributionatthethirdBPMinthe MEBTresultedinavarietyofmeasuredtransversepattheBPM.Becauseofthis,a bettermodelofthetransversedistributiontouseinthewasnotdeveloped. InanattempttoremovethetransversethespectraweremeasuredwithBPMs attheendoftheacceleratinglinacsegementofFRIBwherethebeamistravelingat r =0 : 185correspondingtoamaximum gR p of1.07.Thisistlylowtoignorethe 83 Figure3.19:Spectraandapencilbeamtomeasurementsat =0 : 185.Thedroopat lowfrequencyishypothesizedtobecausedbyincorrectlymodelingthepickupimpedance. transversedistributionandassumeapencilbeam.Thesemeasurementsweretakenwithfour BPMsand,basedonsimulations,thebeamlengthshouldbelinearlyincreasingacrossthe fourlocations.TheBPMmeasurements,onceagain,failtoreproducetheexpectedtrend (Fig.3.19right). Itissuspected,thattheaveragedcableandboardcorrectionsweretand/or theimpedanceoftheBPMisnotproperlybeingcompensated.Themeasuredspectra,after correctionsfortheimpedanceandcableandboarderetainedadroopatlowfrequency (Fig.3.19left)whichisacharacteristicofthepickupimpedance.Inaddition,the longitudinalwasmeasuredtobeprimarilyGaussianwithawirescannerinthisregion ofthebeamlinewhichcontradictstheBPMspectra. 84 3.3Futurework UsingtheBPMsastransverseandlongitudinalmonitorsisacomplicatedmeasure- mentparticularlywiththecomplextransverseintheFRIBMEBT.However,one oftheBPMsintheMEBTisdirectlydownstreamawiremonitoranditmaybe possibletodirectlyinputthosetransversemodelsintoEq.(3.5)andonlyforlongitudinal bunchsize.ThismethodwouldonlyworkforthatsingleBPMandtheinthemodel wouldneedtobecontinuouslyupdatedwhenthemachineisretuned,e.g.foraccelerating atisotope.Anotherpossibilityfortransversemeasurementsathigh gR p ,is toapplythismethodtoatbeamlinethathascleanerdistributions.Forexample,as showninchapter2,thebeamispredominatelyGaussianinthePIP2ITMEBT.Inaddition tofurthermeasurements,theanalyticmodelshouldbefurtherstudiedtounderstandhow closethemodeltobetothebeamdistributionforreasonableerrors. AtFRIB,TISwaveformscanbetakenforBPMsinhigherbeamvelocity,lower gR p regions,wherethesfromthetransversedistributionshouldbedampedoutanda pencilcanbeassumedforforthelongitudinalsize.HoweverBMPsinthissectionof theacceleratorwerenotcalibratedfortheofcableattenuationandlowpass andtheaveragecalibrationwasnott.Oncecalibrationsareperformedonthese BPMs,themeasuredsignalsshouldbeabletobetodeterminethelongitudinalsizeof thebunches. Inaddition,theresponseBPMpickupstoabeamasdescribedinEq.(3.5)makessome assumptionsaboutthebeam.Namely,itassumesthebeaminmono-energeticandthereis nolongitudinal-transversecoupling.Thesetsshouldbeincludedinthemodelandtheir characterized. 85 Chapter4 HelicaltransmissionlineforBPM calibration Asshowninchapter3,measurementswiththeBPMsofnon-relativistic,high gR p beamsare morechallengingthanmeasurementsatlower gR p .Thesemeasurementsarealsochallenging tostudybecausewemustrelyonanalyticandnumericmodelsthat,inpractice,cannot exactlymodelthetrueBPMgeometry.ItwouldbebforBPMmeasurementsat high gR p ,bothbroadbandmeasurementsandpositionmeasurements,tohaveateststand capableofcalibratingandtestingtheBPMs'responseinthehigh gR p regimewheretheywill beoperated.Inordertocalibrateforthefromnon-relativistic,high gR p beams,atest standmustbecapableofreplicatingthemeasuredbeam'sexpectedvelocityandlongitudinal andgeneratethecorrectdistributionontheBPMpickups.Tocalibrateandtest theBPMsignalsexpectedfromthebeamintheFRIBMEBT,thisrequiresreplicatinga 200pslongbunchtravelingat r =0 : 033. Thetypicalteststandforcalibratingfornon-linearitiesinBPMsconsistsofastraight conductingwirethatisstrungthroughtheBPM[5].Atoneatthemeasurementfrequency oftheBPMispasseddownthewireandthesignalsonthepickupsaremeasured.Thisis repeatedwhilemovingthewireoveragridofpositionswithintheBPM.Thewirepositionis thenrelatedtothepickupsignalswithapolynomialtocalibratetheBPMresponse. 86 Figure4.1:Circuitmodelofatransmissionline[8]. Becausetheseteststandsrelyonastraightwire,theypropagatesignalsatthespeed oflightandcannotbeusedtocalibratefornon-relativisticThesedevicesarestill usefulforBPMmeasurementsofhigh gR p beamstocalibrateforthevelocityindependent non-linearandvalidatethemodelsinthespeedoflightlimit.Tocalibrateforthe high gR p anewteststandmustbedevelopedcapableofpropagatingsignalsatthe samevelocityasthebeaminquestion. 4.1Transmissionlinesprimer Beforediscussingpossibleteststandgeometries,thebasicpropertiesoftransmissionlines shouldbeintroduced.Transmissionlinesareusedtotransportpowerandsignalsinthe formofelectromagneticTheytypicallyconsistoftwoconductorsthathaveavoltage betweenthem V ( z;t )andacurrenttravelsalongthem I ( z;t ).Overasmalllength z atransmissionlinecanbeapproximatedbyacircuitwithaseriesresistanceperunit length R ,seriesinductanceperunitlength L ,shuntconductanceperunitlength G ,and shuntcapacitanceperunitlength C asshowninFig.4.1.UsingKirc'slawsthevoltage andcurrentcanbeshowntoobeytheequations[8] 87 V ( z;t ) R zI ( z;t ) L z @I ( z;t ) @t V ( z + z;t )=0(4.1) I ( z;t ) G zV ( z + z;t ) C z @V ( z + z;t ) @t I ( z + z;t )=0 : (4.2) Thecontinuouslimit,knownasthetelegrapherequations,canbefoundbydividingby z andtakingthelimitas z ! 0 @V @z = RI L @I @t (4.3) @I @z = GV C @V @t : (4.4) ForanRF V ( z;t )= V ( z ) e i!t and I ( z;t )= I ( z ) e i!t .Inthiscasethetelegrapher equationsbecome @ V @t = ( R + i!L ) I (4.5) @ I @t = ( C + i!C ) V (4.6) Takingaderivativewithrespectto t givesthewaveequations @ 2 V @t 2 2 V =0(4.7) @ 2 I @t 2 2 I =0(4.8) where isthepropagationconstant = + ih = p ( R + i!L )( G + i!G ) : (4.9) 88 Theelectromagneticwillpropagatealongthetransmissionlinewithphasevelocity v p = ! h : (4.10) Thewaveequationsshowsvoltageandcurrentoscillatewith z i.e. V ( z )= V 0 e z and I ( z )= I 0 e z .Thecurrentandvoltagecanberelatedbypluggingthisformintothe telegrapherequationsresultingin V ( z )= Z I ( z )(4.11) where Z istheimpedanceofthetransmissionline Z = r R + i!L G + i!C : (4.12) Forthecaseofalosslesstransmissionline R = G =0.Inthiscasethethephasevelocity andimpedancebecomefrequencyindependent v p = 1 p LC (4.13) Z = r L C : (4.14) Thisformalizationofthebehaviorofatransmissionlinerequiresknowledgeofthecircuit elements L , C , R ,and G .Foranarbitrarygeometryofconductors,itcanbenon-obvious howtodirectlydeterminethese.Instead,itcanbemorestraightforwardtoderivetheform oftheelectromagneticForageometrywithtwoconductorswithonenestedinside 89 theother,onemethodfordoingthisisusingtheHertzianpotentials[54] r 2 e @ 2 e @t 2 =0(4.15) r 2 m @ 2 m @t 2 =0 : (4.16) If e and m areassumedtoonlyhavea^ z component,thenthearegivenby E = rr e r @ 2 m @t 2 (4.17) H = rr m r @ 2 e @t 2 : (4.18) Oneofthebofthismethodistransverseelectric(TE)andtransversemagnetic(TM) modesnaturallyappear.Forexample,if e =0but m doesnot,then E z =0correspond- ingtoaTEmode.Similarlyif m =0thentheresultingwillbeaTMmode. Withthenestedconductorgeometry,acylindricalcoordinatessystemcanoftenbeused. Incylindricalcoordinateswithadielectricbetweenthetwoconductorswithpermittivity 90 andpermeability ,thegeneralequationsfromHertzianpotentialsare E r = h I 0 n ( r ) A (1) n +K 0 n ( r ) A (2) n (4.19) ! r I n ( r ) B (1) n +K n ( r ) B (2) n e e ihz E = hn r I n ( r ) A (1) n +K n ( r ) A (2) n (4.20) + i! I 0 n ( e r ) B (1) n +K 0 n ( r ) B (2) n e e ihz E z = 2 h I n ( r ) A ( i ) n +K n ( r ) A (2) n i e e ihz (4.21) H r = h ! r I n ( r ) A (1) n +K n ( r ) A (2) n (4.22) I 0 n ( r ) B (1) n +K 0 n ( r ) B (2) n e e ihz H = h i! I 0 n ( r ) A (1) n +K 0 n ( r ) A (2) n (4.23) hn r I n ( r ) B (1) n +K n ( r ) B (2) n e e ihz H z = 2 h I n ( r ) B (1) n +K n ( r ) B (2) n i e e ihz (4.24) where I n and K n arethemoBesselfunctionsoftheandsecondkind[21],primes denotederivativeswithrespecttotheargument, n istheazimuthalharmonicnumber,and and h arethetransverseandlongitudinalpropagationconstantsrespectivelyandarerelated by h 2 = k 2 + 2 (4.25) where k = p isthefreespacepropagationconstant.Thecots A (1) n , A (2) n , B (1) n ,and B (2) n arefoundbyapplyingtheboundaryconditionsattheconductors. Fromthethecurrent I canbedeterminedfromBiot-Savartlawandvoltage V canbedeterminedbyintegratingtheelectricbetweentheconductors.Thesecanbe 91 usedtodeterminetheimpedance.Inadditionthecircuitelements R , L , C ,and G canbe determinedby L = j I j 2 Z S j H j 2 d s (4.26) C = j V j 2 Z S j E j 2 d s (4.27) R = R s j I j 2 Z C 1 + C 2 j H j 2 d ` (4.28) G = ! 00 j I j 2 Z S j E j 2 d s (4.29) where S isthecrosssectionalareaofthetransmissionline, C 1 and C 2 andthecurvesde theboundariesofthetwoconductors, R s isthesurfaceresistance,and = 0 00 [8]. 4.2Initiallyconsideredteststandgeometries Tocreateateststandcapableofreplicatingnon-relativisticbeamsitispossibletousean electronbeamtunedtomatchthevelocityandshapeofthedesiredbeam.However,this reliesonanentirelynewteststandfromthecurrentlyexistingstraightwireonewhich requiresadditionalhardwareandcost.Itwouldbepreferabletomodifytheexistingtest standtoallowforcalibrationfornon-relativistic ApossibleRFstructuretoreplacethestraightwireisaGoubaulinewhichcanpropagate signalsatlessthanthespeedoflightandcanbeusedtoreplicateelectronbeamstocalibrate beamlinedevices[55,56,57].Goubaulinesaresingleconductortransmissionlinescomprised ofaconductingwireofradius R i coveredinadielectricwithradius a andpermittivity i (Fig.4.2).Thisdielectriclayercausesasurfacewavetopropagateatlessthanthespeedof light.ThedispersionrelationforaGoubaulineinadielectric e andcenteredinaconducting 92 Figure4.2:CrosssectionofaGoubauline. pipeofradius R e is i i I 0 ( i R i ) K 0 ( i a ) I 0 ( i a ) K 0 ( i R i ) I 0 ( i R i ) K 1 ( i a )+ I 1 ( i a ) K 0 ( i R i ) = e e I 0 ( e R e ) K 0 ( e a ) I 0 ( e a ) K 0 ( e R e ) I 0 ( e R e ) K 1 ( e a )+ I 1 ( e a ) K 0 ( e R e ) (4.30) where i and e arethetransversepropagationconstantsinthedielectricandoutsidethe dielectricrespectively.Thisisfoundfollowingthesamestepsshowninsection4.4andusing theappropriateboundaryconditions.Thelongitudinalpropagationconstant h isfoundby h 2 = 2 i + k 2 i = 2 e + k 2 e (4.31) Thedispersionrelationisusedtorelatethelongitudinalpropagationconstanttothe frequency.If h islinearwithfrequency,i.e. h = a! ,thenthephasevelocitywillbeconstant andthetransmissionlineissaidtobedispersionfree.Iftherelationisnon-linear,thenthe phasevelocitywillvarywithfrequencyandthetransmissionlineisdisperive. FortheGaubouline,assuming e = 0 ,thehighfrequencylimitofthephasevelocity v p = c= p i whichitreacheswhen ˇ a .Inordertoachieveaphasevelocityof v p =0 : 033 c , amaterialwithadielectricconstantof ˘ 900mustbeusedwhichisimpractical.Inaddition, 93 Figure4.3:NormalizedphasevelocityofaGauboulinewith R i =2mm, R e =20mm, i =10 0 , andtwotdielectriclayerradii a .Thelowfrequencylimitistoolargetoreplicatea non-relativisticbeam. thelowfrequencylimitistlylargerthanthehighfrequencylimit(Fig.4.3).To calibratethebroadbandmeasurementsoftheBPMs,thephasevelocitymustmatchthe beamwithinthemeasuredbandwidth,e.g.upto0.5GHzfortheFRIBBPMs.Thelow frequencylimitcanbereducedbyincreasing a ,however,evenwhenthedielectriclayer atportionofthebeampipe,thereductionisnotenoughtoreachnon-relativistic phasevelocities. 4.3HelicalRFstructures AnRFstructurethatisknowntobecapableofpropagatingatcientlylowphaseveloc- itiesisahelicaltransmissionline.Thelowphasevelocityofhelicesisreliedonfordevices suchasslowtravelingwavetubesforRFpowerrs[58]andthefastkickersforPIP-II [59].TherearemanyapplicationsofhelicalRFstructures,however,manyconcernradiation modesforantennae[60]ortransmissionlinesforsignalpropagationat10sto100sofGHz 94 [61].Generalsolutionshavealsobeenpublished(e.g.[62,63]),however,mostsolutionfocus onderivingimpedance,dispersion,andradiationpropertiesofthesestructures.Forthe desiredteststand,thelargestconcernisthebehavioroftheelectricastheypropagate alongthehelixbecausetheymustreplicatethegeneratedbyabeam. Thegeneralsolutionscanprovideinsightsintothechallengesofusingahelicaltrans- missionlineatlowerfrequencies.Thedispersionrelationforahelixinfreespace[62]shows thehighfrequencylimitofthephasevelocityis v p = c sin( )where isthepitchangle ofthehelix.Thisallowsthesestructurestobecreatedforanydesiredphasevelocityby constructingahelixwiththecorrect ,howeverthelowfrequencylimitofthephasevelocity ofahelixinfreespaceisthespeedofoflight[62].Manyusesofhelicaltransmissionlines canignorethisbecausetheyoperateathighenoughfrequencieswherethephase velocityhasreacheditshighfrequencylimit.However,thisclearlywillnotreplicatethe velocityofanon-relativisticbeamwithinthedesiredfrequencybandand,inaddition,the largediscrepancybetweenthehighandlowfrequencylimitsquicklycausestheinputpulse todeformduetodispersionmakingreplicatingthecorrectbunchshapechallenging.These willthelowerfrequencytransmissionlineandstepsmustbetakentomitigate thisissue. 4.4Helicaltransmissionlines-analyticsolution Thegeometryofahelicalconductormakesexactlysolvingfortheelectromagnetic challenging.However,theboundaryconditionscanbetlyusingthe sheathhelixmodel[62].Thismodelapproximatesahelixasanthincylinder thatconductsonahelicalpathalongthesurfacemakingthestructurea1Dconductorthat 95 islongitudinallyuniform.Sp,theboundaryconditionsatthehelixbecome E i; k = E e; k =0(4.32) E i; ? = E e ; ? (4.33) H i; k = H e; k (4.34) wherethesubscripts e and i denoteintheexternalandinternalregionsofthehelix andthesubscripts k and ? denotethecomponentsparallelandperpendiculartothe directionofconductivityofthesheathhelix.Theseunitvectorsincylindricalcoordinates are ^ k =sin( )^ z +cos( ) ^ (4.35) ^ ? =cos( )^ z +sin( ) ^ (4.36) where isthepitchangleofthehelix.Whilethismodelisanabstractionfromrealhelix, itisshownbelowthatitwellrepresentsthetightlywoundhelicesneededtoreplicatenon- relativisticbeams. Thetransmissionlinefortheteststandconsistsofahelixofradius a andpitch ,a conductingrodofradius R i centeredinsideofthehelix,adielectricwithpermittivity i ofthickness s = a R i thespacebetweentherodandthehelixtosupportthehelix, andthehelixiscenteredinsideaconductingpipeofradius R e .Theanalyticsolutionalso assumestheregionbetweenthehelixandtheouterconductoriswithadielectricwith permittivity e ;however,thisissetto 0 forallstudies(Fig.4.4). ThederivationoftheelectromagneticpresentedherefollowstheworkofS.Sensiper 96 Figure4.4:Crosssectionofhelicaltransmissionlinegeometry.Thegreycirclerepresents thehelix. [62]andothers[63]startingwiththeHertzianpotentials,Eqs.4.15and4.16 Inthisgeometrytheeldsmustbeintheinternalregion,betweentheinner conductorandthehelix,andtheexternalregion,betweenthehelixandouterconductor. Thegeneraldequationsintheinternalregionare E i;r = h i I 0 n ( i r ) A (1) i;n +K 0 n ( i r ) A (2) i;n (4.37) ! r I n ( i r ) B (1) i;n +K n ( i r ) B (2) i;n e e ihz E = hn r I n ( i r ) A (1) i;n +K n ( i r ) A (2) i;n (4.38) + i! i I 0 n ( i r ) B (1) i;n +K 0 n ( i r ) B (2) i;n e e ihz E i;z = 2 i h I n ( i r ) A ( i ) i;n +K n ( i r ) A (2) i;n i e e ihz (4.39) H i;r = h ! i n r I n ( i r ) A (1) i;n +K n ( i r ) A (2) i;n (4.40) i I 0 n ( i r ) B (1) i;n +K 0 n ( i r ) B (2) i;n e e ihz H = h i! i i I 0 n ( i r ) A (1) i;n +K 0 n ( i r ) A (2) i;n (4.41) hn r I n ( i r ) B (1) i;n +K n ( i r ) B (2) i;n e e ihz H i;z = 2 i h I n ( i r ) B (1) i;n +K n ( i r ) B (2) i;n i e e ihz (4.42) 97 where I n and K n arethemoBesselfunctionsoftheandsecondkindand i and h i arethetransverseandlongitudinalpropagationconstantsrespectivelyandarerelatedby h 2 = k 2 i + 2 i = k 2 e + 2 e (4.43) where k i = p i ! isthefreespacepropagationconstant.Theexternalhavethe sameformexceptwithtpropagationconstantsandcotsandaredenotedby exchangingthesubscript i for e . Thecots A (1) i;n , A (2) i;n , A (1) e;n , A (2) e;n , B (1) i;n , B (2) i;n , B (1) e;n ,and B (2) e;n arefoundby applyingtheboundaryconditionsatthehelix(Eqs.4.32,4.33,4.34)andtheinnerandouter conductingsurfaces.Theboundaryconditionsincylindricalcoordinatesattheconductors andthesheathhelixare E i;z R i =0(4.44) E R i =0(4.45) H i;r R i =0(4.46) E e;z R e =0(4.47) E R e =0(4.48) H e;r R e =0(4.49) H i;z a + H a cot( )= H e;z a + H a cot( )(4.50) H i;r a = H e;r a (4.51) E i;z a = E e;z a (4.52) E a = E a (4.53) 98 E e;z a = E a cot( )(4.54) E i;z a = E a cot( ) ; (4.55) Onlyalinearlyindependentsubsetoftheboundaryconditionsareneededtosolveforthe cots.Using A (1) i;n asanoverallamplitudeandEqs.4.44,4.46,4.48,4.49,4.50, 4.52,and4.53thecotsbecome: A (2) i;n = A (1) i;n I n ( i R i ) K n ( i R i ) (4.56) A (2) e;n = A (1) e;n I n ( e R e ) K n ( e R e ) (4.57) A (1) e;n = A (1) i;n 2 i 2 e K n ( e R e ) w 1 ;i K n ( i R i ) w 1 ;e (4.58) B (2) i;n = B (1) i;n I 0 n ( i R i ) K 0 n ( i R i ) (4.59) B (2) e;n = B (1) e;n I 0 n ( e R e ) K 0 n ( e R e ) (4.60) B (1) i;n = iA (1) i;n a 2 i 2 e K 0 n ( i R i ) w 4 ;e i k 2 e w 1 ;i w 2 ;e e k 2 i w 1 ;e w 2 ;i nhw 1 ;i w 1 ;e w 3 ;e K 0 n ( i R i ) nh cot( )+ 2 e 2 i 2 e ˆ 2 e K n ( i R i ) w 1 ;e h i nh cot( )+ 2 e w 3 ;e w 4 ;i e nh cot( )+ 2 i w 3 ;i w 4 ;e i ˙ 1 (4.61) B (1) e;n = B (1) i;n a 2 2 i e K 0 n ( e R e ) w 4 ;i i k 2 e w 1 ;i w 2 ;e e k 2 i w 1 ;e w 2 ;i nhw 1 ;i w 1 ;e w 3 ;i K 0 n ( e R e ) nh cot( )+ 2 i 2 i 2 e a 2 e 2 i K 0 n ( i R i ) w 4 ;e i k 2 e w 1 ;i w 2 ;e e k 2 i w 1 ;e w 2 ;i nhw 1 ;i w 1 ;e w 3 ;e K 0 n ( i R i ) nh cot( )+ 2 e 2 i 2 e 1 (4.62) 99 where w 1 ;i;e =I n i;e a K n i;e R i;e I n i;e R i;e K n i;e a (4.63) w 2 ;i;e =I 0 n i;e a K n i;e R i;e I n i;e R i;e K 0 n i;e a (4.64) w 1 ;i;e =I n i;e a K 0 n i;e R i;e I 0 n i;e R i;e K n i;e a (4.65) w 1 ;i;e =I 0 n i;e a K 0 n i;e R i;e I 0 n i;e R i;e K 0 n i;e a : (4.66) Thedispersionrelationcanbefoundbycreatingamatrixfromtheboundaryconditions usedtodeterminethecotsandsettingthedeterminantto0. 00 I n ( e R e ) K n ( e R e ) 00 e ! R e I n ( e R e ) e ! R e K n ( e R e ) I n ( i R i ) K n ( i R i )00 i ! R i I n ( i R i ) i ! R i K n ( i R i )00 i! cot( ) i i I 0 n ( i a ) i! cot( ) i i K 0 n ( i a ) i! cot( ) e e I 0 n ( e a ) i! cot( ) e e K 0 n ( e a ) 00 2 e + nh a cot( ) I n ( e a ) 2 e + nh a cot( ) K n ( e a ) 2 i I n ( i a ) 2 i K n ( i a ) 2 e I n ( e a ) 2 e K n ( e a ) nh a I n ( i a ) nh a K n ( i a ) nh a I n ( e a ) nh a K n ( e a ) (4.67) 0000 00 e I 0 n ( e R e ) e I 0 n ( e R e ) 0000 i I 0 n ( i R i ) i K 0 n ( i R i )00 2 i + nh a cot( ) I n ( i a ) 2 i + nh a cot( ) K n ( i a ) 2 e + nh a cot( ) I n ( e a ) 2 e + nh a cot( ) K n ( e a ) 00 cot( ) e I 0 n ( e a ) cot( ) e K 0 n ( e a ) 0000 i I n ( i a ) i K n ( i a ) e I n ( e a ) e I n ( e a ) =0 100 Afterthedispersionrelationis 0= 3 i h nh cot( )+ 2 e i 2 w 1 ;i w 1 ;e w 3 ;e w 4 ;i + 3 e h nh cot( )+ 2 i i 2 w 1 ;i w 1 ;e w 3 ;i w 4 ;e + a 2 cot 2 ( ) 2 e 2 i w 4 ;i w 4 ;e e k 2 i w 1 ;e w 2 ;i i k 2 e w 1 ;i w 2 ;e : (4.68) Thisissolvednumericallyfor e asafunctionof k e using 2 e = 2 i + k 2 i k 2 e (4.69) andthen h ( ! )canbedeterminedalongwiththephasevelocity v p ( ! )= k e =h = c . 4.4.1Dispersiondistortioncorrection Inordertoproducethesppulseshapeatthedeviceundertest,itisidealforthe transmissionlinetobedispersionfreesoanypulseinputintothetransmissionlinewillnot bedeformedthroughoutpropagation.However,forahelixinfreespace,thephasevelocity variesfrom =1to =sin( )whichcausessigtdeformationtotheinputpulse.For manyapplicationsthehelixisinsideofpipe(e.g.forRFpowera[58]),andinitial designsoftheteststandaddedaconductingpipearoundthehelix.Forthisgeometry,the lowfrequencylimitofthephasevelocityistlylessthan =1.Forexamplefor theparametersgivenforFig.4.7,thehighfrequencylimitis v p =0 : 05 c andlowfrequency limitis v p =0 : 09 c .However,thedeformationduetodispersionisstillamajorissuewith tdeformationtothepulseshapeseenevenforrelativelyshortpropagationdistances (seeFig.4.5). Onemethodtomitigatethedeteriorationistousedispersionitselftocorrectthepulse 101 Figure4.5:Thedeformedpulseduetodispersionisfoundbyanalyticallypropagatingapulse asetdistance.Thiscanbecorrectedbyreversingthedeformedpulseintimeandinputting itintothetransmissionline.Whenthispulseispropagatedalongthetransmissionline,the dispersionwillcorrectthepulseatthesetdistance.TheDUTcanbeplacedatthislocation. atagivenlocation.Thisisachievedbypropagatingthedesiredpulsealongthehelixand measuringthepulseatthedesiredlocationofthedeviceundertest.Thispulseisthen reversedintimeandinputintothehelix.Thiselysubtractstheaccumulatedphase shiftsateachfrequencyduetodispersionandtheoriginalsignalwillbereconstructedat theplaceofmeasurement(Fig.4.5).Thismethodwasrejectedbecausethedeformedpulses inputintothehelixarerathercomplexandanalyticandnumericalmodelsarerequired togeneratethemwhichmaynottlyreplicatethephysicaldevice.Inaddition, generatingthesesignalswithanarbitrarysignalgeneratorwouldrequireafastrisetimeand wouldhaveprovedchallenging. Anotherconsideredmethodistoslowlydecreasethepitchofthehelixfromaloosely woundhelixattheinputtotherequiredpitchforthedesiredvelocity.Intheregionof decreasing ,thedecreasingpitchcancompressthepulsemorequicklythandispersioncan 102 distortit.Thisresultsinashorterpulseattheendofthecompressionsectionthanwas inputandthepulseattheendwillbewellformedwithminimaldeformation(Fig.4.6). Oncethepulsestartstopropagatealongtheconstantpitchportionofthesystem,the ofdispersiononceagaindeformsthepulseshape.Forthisgeometrythedeviceundertest needsbeplaceddirectlyafterthecompressionsectiontomeasurethedesiredpulseshapeat thecorrectvelocity.However,thisplacementofthedevicemustbedoneverycarefully.If thedeviceistoofartoonesideandthehelixwillhavethewrongpitchandthewill betravelingatthewrongvelocity.Ifthedeviceistoofarintheotherdirection,thenthe willbedeformedduetodispersion. 4.4.2Dispersionreductiongeometry Amorepracticalsolutiontopreventpulsedeformationduetodispersionistoaddaconductor insideofthehelix.Usingtheseparation s =0.5mm,thedispersionbecomestly thanwithouttheinnerconductorwithallothergeometrythesame(seeFig.4.7). Thechangetotalvariationinphasevelocityis v p =0.039withouttheinnerconductorand v p =0.001withtheinnerconductorfortheparametersgiveninFig.4.7.Numerically solvingEq.4.68showsthelowfrequencylimitofthephasevelocitylinearlyvarieswith separation s betweenthehelixandinnerconductorasapproximately lim f ! 0 ( v p ) / 0 : 0025 s: (4.70) for s inmm.Thehighfrequencylimitremainsthesamebutthephasevelocityconverges moreslowlyforsmaller s .Theslowconvergenceresultsinlessvariationin v p overagiven bandwidthdespitethelargervariationbetweenthelowandhighfrequencylimits.For 103 Figure4.6:PropagationofaGaussianpulsealongahelixwithreducingpitch.Thehorizontal axisisthelongitudinalpositionandtheverticalaxisthetransverseposition.Theverticalline representstheendofthepitchcompressionsection.Thepulseiscompressedbutmaintains itsformduringthepitchreductionbutdispersiondeformsthepulseintheconstantpitch section. 104 Figure4.7:Dispersionforttransmissionlinegeometries.Theadditionoftheinner conductortly v p ( ! ).Helixparameters s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 . example,inFig.4.8,forabandwidthupto1GHz, s =1mmhavearangeof v p =0.0028 andfor s =0.1mm v p =0.00057.Therefore,anarrowseparationshouldbeusedwhen constructingahelicaltransmissionlinetolimitthedeformationofpulsesduetodispersion. Thedielectricconstantofthedielectriclayercanalsobevaried.Inthecaseof i = e = 0 ,thehighfrequencylimitofthephasevelocityis v p = c sin( )andthelowfrequencylimit ishigherthanthis.Astheinternaldielectricconstantincreasesforadgeometrythehigh frequencylimitdropsapproximatelyas 1 = 3 i whilethelowfrequencylimitdropsas 1 = 2 i (Fig.4.8right).Thiscausesthelowfrequencylimittodropbelowthehighfrequencylimit. Theinscalingcausesthedisparitybetweenthelowandhighfrequencylimits ofthephasevelocitytoincreasewith i whichisnon-idealformaintainingpulseshapes. However,itscanbemitigatedbyreducing s .Intheory,thedielectricconstantcanbe chosensuchthatthehighandlowfrequencylimitsof v p areequalwhichwouldbeanideal system,however,inpracticethisistorealize. 105 Figure4.8:Dispersionscalingwith s (left)and r (right).Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 . Whenthepitchangleofthehelixischanged,thelowandhighfrequencylimitsofthe phasevelocityscaleassin( )(Fig.4.9).Therefore,thetotalvariationin v p willincrease withthepitchangle.However,withlargerpitchangles,thephasevelocityconvergesmore slowly.Thiscanresultinlessdeformationtothepulseduringpropagation. Itshouldbenotedthatwhiletweremadetothedispersioncurve toreducedeformations,theelectricnearthehelixstillbecomestlydeformed duetodispersion(seeFig.4.10).Thisiscausedbytheshortpulsepropagatingalonghelix of ˘ 200psrmsneededtoreplicatethebunch.Forpulsesthisshortthebandwidthislarge enoughtocausetlytphaseshiftsresultingindistortion.However,the nearthehelixarenotrepresentativeoftheatthepipewallwheretheywillbe measuredbytheBPMbeingcalibrated.Astheradialdistancefromthehelixincreases,the highfrequencycomponentsaresuppressedbecausetheisnon-relativistic.Therefore, theatthepipewallwillpropagatewithminimaldeformationeventhoughthesignal nearthehelixhastdistortions(Fig.4.10). 106 Figure4.9:Thelowandhighfrequencylimitsofthephasevelocityscalewithsin( ) : Figure4.10:Despitetoreducedispersionthepulsesnearthehelixaresignif- icantlydeformedduetodispersion(left).However,thenearthepipewallhavethe highfrequencycomponentssuppressedandthereforemaintaintheirshape.Thissame distortsthepulsewhenthehelixisinthepipe.upto5mmcanbeachieved withminimaldeformations(right).ThepulsesshownherearefromCSTMicrowaveStudio simulations. 107 Fortheteststand,thehelixwillneedtobesetinthepipe.Asshowninsection4.5.6, thisdoesnotchangethedispersionproperties,however,itwillreducetheradialdistance betweenthehelixandthepipewallresultinginashorterpulsebeingmeasuredbyatleast oneoftheBPMpickups.Fortunately,forupto5mm,thehigherfrequenciesare stilltlysuppressedinthemeasuredbandwidthandthemeasuredsignalsarestillnot tlydeformed. 4.4.3Higherorder Thediscussionsabovefocusessolelyonthe n =0azimuthalmode,however,helicaltransmis- sionlinessupporthigherordermodes[62].The n th higherordermodeisexcitedat ka ˇ n . For a =5mmwhichwillbeusedfortheteststand,thiscorrespondsto ! ˇ 10Ghzwhichis welloutsidetherequiredbandwidth.Unlikethe n =0mode,whichhastlyreduced phasevelocityatlowfrequencycomparedtothehelixinfreespace,thehigherordermodes propagateat v p ( ! c )= c atthefrequency(Fig.4.12).Thehighfrequencylimitsof v p ofthehigherordermodesarethesameasthe n =0mode. Thepresentedanalysisreliessolelyonthesheathhelixmodel.Morecomplexmodels canalsobemadeusingatapehelixwithwindingsofwidth[63].Thesemodels introduceforbiddenregionsin k h spaceandneartheboundariesoftheseregions,the dispersiondivergesfromthesheathhelixmodel[62].However,forthegivengeometry,the requiredbandwidthisfarfromaforbiddenregionandthesheathhelixmodelisareasonable approximation(seeFig.4.11).Inaddition,thetapehelixmodelspredictradiatingmodesat lowfrequencies.Itisbelievedthattheseareseeninsimulations,howeverthesetravelnear thespeedoflightandcanbeeasilyseparatedfromtheslowsignals(seeFig.4.21). 108 Figure4.11:Theshadedareasareforbiddenregionsofatapehelixmodel.Forthegeometry, =0.05, R p =20mm, a =5mm, s =0.5mm,upto8GHz(bluecurve)thehelixoperatesfar fromtheforbiddenregionsandthesheathhelixwillbeagoodapproximation. Figure4.12:Higherordermodeshavefrequencyat ka ˇ n andpropagateatthespeed oflightatthecutfrequency.Shownisthethreemodesforahelixwithgeometry =0.05, R p =20mm, a =5mm, s =0.5mm, i = 0 109 4.4.4ImpedanceProperties Oncethecotsandpropagationconstantsarefound,acompletedescriptionof theisknownandtheimpedancecanbecalculated.Forthethreeconductorgeometry describedabove,twoseparateimpedancescanbe Z i whichbetweentheinner conductorandthehelix,and Z e betweenthehelixandpipe.Theimpedanceineachregion wascalculatedby Z i = R a R i E r;i d r a R ˇ ˇ H ;i ( a )d (4.71) Z e = R R e a E r;e d r R e R ˇ ˇ H ;e ( R e )d a R ˇ ˇ H ;i ( a )d : (4.72) Ingeneral,thesearesimilarexceptatlowfrequencies(Fig.4.13). Reducing s maintainsthehighfrequencybehavioroftheimpedancebutreducesthelow frequencylimitandreducesthevariationatlowerfrequencies.Thiscausestheimpedanceto convergemoreslowlytothehighfrequencybehavior(Fig.4.13left).Ideally,asmallenough s shouldbeusedtoachieveanearconstant Z overthedesiredbandwidthsotheinput andoutputofthehelicaltransmissionlinecaneasilymatchedwitharesistivenetwork. Increasingthedielectricconstantreducesthelowfrequencylimitoftheimpedanceand causestheimpedancetostarttodecreaseatalowerfrequency(Fig.4.13right).Whileit ispreferabletousealowerdielectricconstanttoincreasethebandwidthofnearconstant impedanceregion,theofthedielectricisnotastastheof s . Thelowfrequencylimitoftheimpedancescalesas1 =sin ( )whilethehighfrequency limitdoesnotchangewiththepitchangle(Fig.4.14.Thehigherpitchanglesconvergeto thehighfrequencylimitatahigherfrequencywhichcanmakematchingsimpler. 110 Figure4.13:Impedancescalingwith s and i .Thesolidlineistheexternalimpedance andthedashedlineistheinternal.Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 . Figure4.14:Thelowfrequencylimitoftheimpedancescaleswith1 = sin( ) 111 Figure4.15:Sensitivityofthelowfrequencylimitoftheexternalimpedancetovariationof tparameters.Theimpedanceismostsusceptibletochangesin s . Toavoidinternalduetochangestotheimpedancecausedbyvariationsin construction,thegeometricparameterswerevariedintheanalyticmodeltodetermineto whichthesystemismostsensitive.Mostofthegeometryfactors,suchasthepitchand outerpiperadius,haveaminimalimpactontheimpedance.Theimpedanceisprimarily sensitivetothetheinnerconductorradiusandhelixradius,buttheyhaveopposite theimpedance.ForexampleasseeninFig.4.15,a5%increaseinthehelixradiuscauses analmost50%increaseinthelowfrequencylimitoftheimpedancewhileincreasingthe innerconductorradiusby5%reducestheimpedanceby ˘ 50%.Therefore,theseparation s betweenthehelixandinnerconductormustbeconstantalongthetransmissionlinetolimit largevariationsintheimpedanceforevensmallchanges.Theradiusoftheinnerconductor canvaryaslongasthehelixradiusalsochangestokeeptheseparationconstant. 112 4.4.4.1InductanceandCapacitance Theinductance L andcapacitance C perunitlengthofthelosslesshelicaltransmissionline canbedeterminedfromthelongitudinalpropagationconstantandimpedancebasedonthe generalformsderivedfromthetelegraphersequations(Eqs.4.10,4.13,4.14) L i;e = h ! 1 Z i;e (4.73) C i;e = h ! Z i;e : (4.74) Aswiththeimpedance,theseparametersarenedseparatelyfortheinternalandexternal regions. Theconceptofaddingtheinternalconductorwasdevisedasamethodtoincreasethe capacitanceofthetransmissionlinetoreducethephasevelocity(Eq.4.13).Thisis canbeseenbycalculatingtheinternalcapacitancefortvaluesof s (Fig.4.16). Decreasing s doesresultinalargercapacitanceatlowfrequencyandconvergestothesame limitathighfrequencyasexpectedbasedondispersion.However,competingwiththis isadecreaseintheinductanceatlowfrequencywithdecreasing s whichwillincreasethe phasevelocity.Theinductanceisdecreasedbyalmostthesamefactorastheincreasein capacitancewiththelargestdiscrepancyseenatfrequencies < 1GHz.Thissimilarityresults inminimalchangestothephasevelocityandlargechangestotheimpedancewhenchanging s .AsexpectedfromFig.4.13,changing i resultsintchangestothehighandlow frequencylimitsofthecapacitanceandinductance. 113 Figure4.16:Scalingoftheinternal C and L with s and i .Thedecreasing s reducingtheca- pacitanceandincreasestheinductancebyapproximatelythesamefactor.Helixparameters: s =0.5mm, a =5mm, R e =23.75mm, =0.05, i = 0 . 114 Figure4.17:MeshingofhelicaltransmissionlineinCSTMicrowaveStudiofortimedomain simulations. Table4.1:Simulatedhelicaltransmissionlinegeometryparameters. Piperadius, R e 20.65mm Innerconductorradius, R i 4.5mm Helixradius, a 5mm Separation, s 0.5mm Pitchangle, 0.05rad Dielectricconstant, i 3.5 0 Helixwirewidth1mm Helixwirethickness, a 0.1mm 4.5Simulations TimedomainsimulationsofthehelicaltransmissionlinewereperformedinCSTmicrowave studiowithpropertiesgiveninTable4.1unlessotherwisestated.Thisgeometryisexpected tobeclosetowhatwillbeusedforaBPMteststand.Simulationswereperformedupto 1.5GHzwhichiswellbelowthecutofrequencyofthehighermode.Note,whileit isbesttominimize s toimprovepulsepropagation, s =0 : 5mmwasusedbecauseitwas impracticaltocreateameshforsmallervalues.Themeshsizewassettoresolvethedielectric layerandspacingbetweenwindingswithatleasttwomeshcells(Fig.4.17).Thehelixhas thickness a centeredaroundthehelixradius a . 115 Figure4.18:Helicaltransmissionlinemodel.Microstripsareusedtomatchtheinputand output. Figure4.19:Lumpedelementsusedforimpedancematching. 4.5.1Matching Signalswereinputontothehelicaltransmissionlineusingastriplineconnectedtothehelix andinnerconductoroneitherendofthehelix(Fig.4.18).Formostsimulations,thestripline impedancewassetequaltothelowfrequencylimitoftheheliximpedanceandnomatching networkwasused.Alternatively,thestriplineimpedancecanbesetunequaltothethehelix impedancethenmatchedusingaresistiveL-network[64](Fig.4.19).Withthestripline inputgeometry,thesignalisinputintotheinternalregionofthetransmissionlineandthe stripline(withorwithouttheL-network)onlyprovidesanimpedancematchtothisregion. Theexternalregionmustalsomematched.Thisperformedwithanadditionalresistorwith impedanceequaltothelowfrequencylimitof Z e placedbetweentheouterconductorand helix. Withthestriplinematchedtothehelix,agoodmatchisseenwith S 1 ; 1 < -15dBand 116 Figure4.20:S 1 ; 1 withandwithoutaresistiveL-networkformatching. S 2 ; 1 > -1.5dBupto2GHz.Strongresonancesareseencorrespondingtheharmonicsofthe helixlengthforsignalstravelingattheexpectedslowphasevelocity.Theseresonancescanbe dampedbyusingtheresistiveL-networkinthematchingscheme(Fig.4.20).However,the L-networkalsoreducestransmissionby ˘ 11dBduetolossesintheresistors[64].Therefore, tomaximizetheamplitudeofthepropagatedsignal,theL-networkwasonlyusedwhenit wasnecessarytodamptheresonances. Withbothmatchinggeometries,asignaltravelingatthespeedoflightproceedsthe slowpulsethatisafactorof ˘ 10lowerinamplitude(Fig.4.21).Thespeedoflightsignal isnotectedwhenitreachestheendofthetransmissionline,however,itisseenagain whentheslowsignalreachestheendofthehelix.Thissignalmaybeanexcitedhigher ordermode.However,theamplitudeofthefastsignalisindependentof thereforeitisnot an n> 0modeofthesheathhelix.Itcouldalsobearadiatingmode,buttheamplitude doesnotdecreasewithpropagationdistance.Anotherpossibilityistapehelices,whichis usedinsimulations,areknowntosupportadditionalmodesnotpresentinanalyticsolutions usingthesheathhelixapproximation.Itispossibleoneofthesemodesisexcitedandthis 117 Figure4.21:Theradialelectricatthewallshowsminimaldeformationduringpropaga- tionalongthetransmissionline.Theslowpulsesareproceededbyasmallerspeedoflight signalthatreappearsoncetheslowpulsehasreachedtheendofthetransmissionline. geometryshouldbefurtherstudied. Forasinglepulsepropagatingalongthehelix,whichwasusedforallstudiespresented inthisdissertation,thereisenoughseparationbetweenthefastandslowpulsestoeasily distinguishthemandthisabnormalitydoesnottheresults. 4.5.2Dispersionmeasurements ThedispersionwasmeasuredbyinputtingaGaussianpulseintothesystemandmeasuring theradialelectricusingprobesalongthetransmissionline.Nineprobeswereevenly spacedalongthetransmissionline.Theprobeswerebeplacedascloseaspossibletothe helixtomeasurethelargestbandwidth.However,ifaprobeistoocloseitwillmeasurenear fromthewindingofthehelix.Toensureminimalneareightprobes wereplacedaroundthehelixin ˇ= 4increments.Theprobesweremovedfrom0.25mmto 2mmawayfromthehelixtodeterminetheradiuswheretheisnolongertly 118 Figure4.22:Theelectric0.5mmawayfromthehelixhasastrongharmonic duetothehelixwindings.At1.5mmawayfromthehelixthetheangulardependenceis tlyreduced. dependenton becausewithinthesimulatedfrequencyrange,onlythe n =0modeshould beexcited,thereforethereshouldbenoazimuthaldependenceinthe The0.25mmfromthehelixvarieschaoticallywith by 50%.At0.5mmthe helix,thevariationsareslightlyreducedandshowastrongcos( )dependenceduetothe helixwinding(Fig.4.22).Thisbehavioristlyreducedforthe1.5mmradially fromthehelixwithvariationsonthe1-4%level.Whilethe E z stillhasaclearcos( ) dependence,itsamplitudeissmallenoughtoignore. Formeasurementsofthephasevelocity,theprobeswereplaced1.5mmawayfromthe helixtomaximizethemeasuredbandwidthandavoidnearects(Fig.4.23).Thetime signalsfromeachprobeweretransformedintothefrequencydomaintodeterminethephase 119 Figure4.23:Probesuiteusedtomeasurethefromthehelix. shiftsasafunctionoffrequencyandlocation.Alinearofthephasesasafunctionofthe probepositiongivesthephasevelocityforeachfrequency(Fig.4.24) ˚ ( f;z )= f v p z: (4.75) Themeasuredphasevelocityagreeswiththesheathhelixmodelwithin3%upto2GHzat whichpoint,thesignalsaredominatedbynoise. However,nearexactagreementcanbeachievedbyassuming a = a + a =0 : 51mminthe analyticmodel(Fig.4.25).Thisvalueisthesumofthehelixradiusandthehelixthickness inthesimulation.Thisscalingalsoholdsforthickerhelices,however,theagreementwith theoryispoorermostlikelybecausetheassumptionofanthinhelixusedin theanalyticmodelnolongerholds. Thecauseofthisphenomenonisunknownanditisparticularlystrangeconsideringinthe simulationsthehelixconductorisplacedat r 2 [ a a= 2 ;a + a= 2].Thehelixconductor doesnotreach a + a .Inaddition,increasing a willreduce s whichshouldcauset changesintheimpedanceanddispersion,butthesearenotseen.Itisspeculatedthat thesedeviationsarecausedissueswithmeshingthethingeometryofthehelix. 120 Figure4.24:Thephasesateachfrequency(topleft)foreachprobeareaccordingto Eq.4.75tocalculatethephasevelocity.Thetopleftplotshowstheat500MHz. Theresultingphasevelocityagreeswiththeorywithin3%. 121 Figure4.25:Exactmatchingofthedispersioncanbeachievedbysettingthehelixradius intheanalyticmodelto a + a usedinthesimulations.Thehelixradiusinsimulationsis 5mmforallpresentedmeasurementswith a givenby Da oneachplot.Thestatedhelix radiusistheradiususedtoanalyticallycalculatethedispersion. 122 4.5.3Impedancemeasurements TheimpedanceofthehelicallinewasmeasuredusingthecotS 1 ; 1 .Forthis measurement,thestriplineimpedancewaschangedto102.5tobeunequaltothehelix impedanceof Z helix ˇ 64 : 5andmatchedtothehelixusingaresistiveL-networktodamp theresonances.Inaddition,toincreaseS 1 ; 1 theresistorsintheL-networkwereintentionally setincorrectlyforapoorimpedancematch.Withthisnetwork,theheliximpedancecanbe foundby Z helix = R 1 sh + Z 0 1 S 11 1+ S 11 + R 1 ! 1 (4.76) where Z 0 istheimpedanceofthemicrostrip, R istheseriesresistance,and R sh istheshunt resistanceoftheL-network. Therealpartoftheimpedancefromsimulationsagreeswiththeanalyticmodelwithin 5%upto2GHz(seeFig.4.26).Thesimulationalsoshowedasmallreactancethatwas < 15%oftherealimpedancewhichwasnotpresentintheanalyticmodel.Noattemptwas madetomatchthereactancebecausetheandtransmissionwereadequatewiththe currentmatchingnetworks.Theaddedcomplicationsofdevelopingafrequencydependent, reactivenetworktoimprovetheimpedancematchoutweighthebofaslightlybetter match.However,thesourceoftheimaginaryimpedanceshouldbeexploredinthefuture. Inaddition,whenconstructingahelicaltransmissionlinefortheteststand,itwouldbebest touse s< 0 : 5mmwhichwillresultinlessfrequencydependencein Z andabettermatch wouldbeexpected. 123 Figure4.26:Theanalyticimpedancematchestheresultsfromsimulationswith5% 4.5.4Electricscaling Theelectricwascomparedtotheanalyticsolutionbymeasuringtheusing probesatmultipleradiialongthetransmissionline.Themeasured1.5mmthe helixwereusedtodeterminethecotsintheanalyticTheexpected nearthepipewallcanthenbecalculatedusingtheanalyticmodelanditwithin10%with thesimulationresultsupto0.4GHzwherenoisestartstodominatethesimulationresults (Fig.4.27). Anunexpectedfeatureoftheis,whenaGaussianpulseisinput,nearthehelixthe amplitudeoftheradialelectricdecreasesatlowfrequency(Fig.4.28).Thereareno signsoflossesorintheSparametersatlowfrequencytoexplainthisreduction. Itwasfoundthatthisphenomenaiscausedbythecouplingbetweentheinternaland externalregionsofthetransmissionline.Thiscouplingcanbederivedanalyticallybytaking 124 Figure4.27:Thecotswerederivedfromtheelectric1.5mmfrom thehelix.Theseareusedtocalculatetheanalyticelectric15mmfromthehelix.This agreeswellwithsimulations. theratiooftheinternalandexternalradialelectricatthehelix E r;e E r;i r = a = i K 2 0 ( e R e ) w 1 ;i w 2 ;e e K 2 0 ( i R i ) w 1 ;e w 2 ;i : (4.77) ThiscanbeinvertedtodeterminetheappropriateinputpulsetohaveaGaussianle atthehelixintheexternalregionwhichisnecessarytoreplicateagivenbunchshape.This couplingagreeswiththeresultsofsimulations(Fig.4.28).Whileanexactinputcanbe generated,forsimplicity,foraGaussianpulseinputintheinternalregion,thespectranear thehelixcanberoughlytotheoftwoGaussians(Fig.4.28) E r;e r = a ˇ G 1 exp( 2 ˇ 2 ˙ 2 1 f 2 ) G 2 exp( 2 ˇ 2 ˙ 2 2 f 2 ) : (4.78) ThestandarddeviationsandamplitudesofthetwoGaussiancanbedeterminedfromthe standarddeviation ˙ i oftheinputGaussianpulse.FortheparametersgiveninTable4.1, 125 thecotscanbedeterminedby G 1 =0 : 119 e 0 : 653 ˙ i (4.79) G 2 =0 : 066 e 0 : 826 ˙ i (4.80) ˙ 1 =0 : 905 ˙ i 0 : 005(4.81) ˙ 2 =1 : 582 ˙ i +0 : 111(4.82) for ˙ i inns.ThiscanbeinvertedtodeterminetherequiredinputtoachieveaGaussian pulseoflength ˙ i intheexternalregion.However,thequalityofthedoubleGaussian deterioratesforpulseslessthan100psrms. 4.5.5Beamcomparison Theelectricpropagatedbythehelicaltransmissionlineneedstoreplicatethe fromanon-relativistic,high gR p beam.Thelongitudinalcanbesetbytheinput pulsewithproperconsiderationoftheinternal-externalcoupling.Theradialelectric fromalinecharge(Eq.1.43)hasthesamegeneraldependenceoncoordinatesasthe fromthehelix(Eq.4.37).However,thebeamhas g asatransversepropagationconstant whichcorrespondstoaedbeamvelocitywhilethehelixhas e andthephasevelocity varieswithfrequency. Thevariationofthephasevelocityofthehelixmeansachoicemustbemadeforthe velocityofthemodeledbeam.Itwasfoundforcomparingtheupto1GHz,using v p ( f =1GHz)givesthebestagreement.Thevariationinphasevelocitymakesitonly possibletoaccuratelyreplicatethedsfromabeamwithinaspbandwidth.To increasethebandwidththedispersioncurvemustbettedwhichisanadditionreasonto 126 Figure4.28:Theintheexternalregionissuppressedatlowfrequencycomparedto theintheinternalregion.Thismatcheswithsignalsmeasured1.5mmthehelixin simulations.ForaGaussianinput,theintheexternalregioncanberoughlytothe oftwoGaussians. 127 reduce s .Thesmaller s therewillbelessvariationsinthephasevelocity,whichwillresult inbetteragreementoveralargerbandwidth. Thisdisparityinpropagationconstantsalsocausesthetoscaleerentlywith r . Thereforetheofabeamandahelixcannotbethesameovertheentirepipeaperture. InordertoreplicatetheBPMsignalsfromabeam,thefromthehelixatthe pipewallmustmatchtheexpectedfromabeam.Thisexactonthepipewillbe dependentofthechoiceofthetransversedistributionofthemodeledbeam.Todetermine thebesttransversetheradialelectricdofthebeamwastotheofthehelix forapencilbeam,ringbeam,anduniformbeamusingthebeamradiusasaparameter (Fig.4.29).Theringdistributionbestthehelixwithdeviationofatmost10%upto 1GHz.Upto0.5GHz,thebandwidthoftheFRIBBPMs,theringdistributionvariesfrom thehelixbylessthan1%.Thefromauniformbeammatchwellupto ˘ 0.3GHz whichisntforcalibratingpositionmeasurementstakenat161MHz.Thehelix cannotmodelapencilbeam.Thisisprimarilybecausethereisnogeometricparameterof apencil,suchasradius,thatcanbevariedtothe 4.5.6helices Theanalyticandsimulationworkpresentedinthischapterhasallbeenconcernedwith centeredhelices.However,fortheteststandtocalibrateBPMs,thehelixmustbemoved centertoreplicatebeams.Theperformanceandapplicabilityofthehelicaltransmis- sionlineasateststandwouldbelimitedifthedispersionandimpedancepropertiesofthe helicaltransmissionlinevarytlywhenthehelixisinapipe.Toexplorethe ofthehelixinsimulations,ahelixwasmovedcenterinapipein2mm stepsupto10mm.Ateachstepthethedispersionwascalculatedfromprobesmeasuring 128 Figure4.29:Thefromthehelixbestmatcharingbeam. R i =4.5mm, a =5mm, R e =20.65mm, =0.05, i =3 : 5 0 129 Figure4.30:Thedispersionandimpedanceseesnotvariationduetoupto 10mm theelectric1.5mmthehelixasdescribedinsection4.5.2.Notchange inthedispersionwasseenwiththehelixupto2GHz(Fig.4.30).Inaddition, theimpedancewasdeterminedusingthecotfromamismatchedL-network foreachhelixlocation.Once,again,notchangeintheimpedancewasseenwhen thehelixwas(Fig.4.30).Thesearehighlybbehaviorsbecausenoadditional considerationsneedtobetakenwhenthehelixisotoreplicatecenterbeams.The helixcanbetlyinthepipeandtheinputpulsewillnotneedtobechanged becausethereisnochangetodispersionandthematchingnetworkwillnotneedtovaried becausetheimpedanceisalsostable. 4.6Futurework whiletheanalyticandnumericmodelsshowahelicaltransmissionlineiscapableofreplicat- ingtheprandvelocityofanon-relativisticbeam,therearediscrepanciesbetween thetwomodels.Mostnotableisthefastpulsesseeninsimulationsandtheimaginarypart 130 oftheimpedancedonotappearintheanalyticsolution.Thesemaybecausedbythesheath helixapproximationusedanalyticallynotfullydescribingtheldsgeneratedbythetape helixusedinsimulations.Analyticmodelsoftapehelicesexistandpredictadditionalmodes thatcouldexplainthediscrepancies.Whilethesheathhelixapproximationistto showhelicaltransmissionlinescanbeusedforthisteststand,itwouldbebtode- velopananalyticmodelusingatapehelixgeometrytodetermineifthehigherorder willcauseissues. Inaddition,furthertestsofthenumericalmodelshouldbeperformed.Inparticular, furtherstudiesofthediscrepancybetweenthedispersionmeasuredinthesimulationsandthe analyticmodel.Inparticular,thescalingwith a shouldbefurtherexploredtodetermined ifthisisarealorcausedbyissueswiththemeshingofthethingeometryofthehelix anddielectriclayer. Thenextstepforthisprojectistoconstructandtestahelicaltransmissionlinetoverify theresultspresentedinthischapter.Astraightforwardmethodforconstructingthehelix istoinsertaconductingrodintoadielectrictubethencreatingagrovetolaythehelix withathreadcutter.FromdiscussionswiththeattheMSUphysicsmachineshop,it isexpectedthismethodcanatbestachieve s ˘ 0 : 5mm.Thisvalueof s wasusedinmost simulationsanditisttopropagatepulseswithminimaldispersionHowever, asmaller s wouldbepreferableforimpedancematchingandlessvariationsinthephase velocity.Thiswouldrequireatconstructionmethodtobedeveloped. Theinitialconstructionshouldbefocusedonmeasuringthepropertiesofthetransmission lineandnotbeusedasaBPMteststand.Forthistheouterpipeshouldbewith multiplepickupsattpositionsin z tomeasurethedispersion.Thesepickupsshould beassmallasispracticaltolimittheoftheirgeometryonthemeasuredsignalswhich 131 willalloweasiercomparisontotheexpectedstructurefromtheanalyticandnumeric models. 132 Chapter5 Conclusions Itisimportanttocharacterizeandmonitorthebeamqualityinhadronaccelerators.It isparticularlyimportanttodosointhefrontendswherethebeamisnon-relativisticand evolvingrapidly.Thelargediagnosticsuitesinthefrontendsareusedensureaqualitybeam entersthemainacceleratingstructuretoreduceslossesandmaximizetheperformanceofthe accelerator.Theoperationoftheacceleratorcanbeimprovedbygainingmoreinformation aboutthebeamqualitybyaddingmoremeasurementsofthebeampropertiesandtaking moredetailedmeasurementsofthebeamdistribution.Thiscanbeachievedbyfurthering theanalysisofsignalsproducedbyexistingdiagnosticdevices. Oneparticularfeatureofthebeamthatneedscharacterizedisthebeamtailsasthese portionsofthebeamarefarfromthebeamcenterarethemostlikelytobelostinthe accelerator.Toquantifythephasespacedistributionsandbeamtailsmeasuredin x x 0 phasespace,thephaseportraitscanbeconvertedto J ˚ coordinates.Inthiscoordinate systemthedistributionisstableunderlinearopticsandcanbequananddirectly comparedtoothermeasurementstostudytailgrowthcausedbynon-linearforces. Thepresentedmethodforcharacterizingthebeamcoreandtailsreliesontoa modelofthecoredistribution.Forthemeasurementspresentedinthisdisseration,takenin thePIP2ITMEBT,thecorewasGaussian.However,thesamemethodologycanbeused foranydistributionandappliedtootheracceleratorsaslongthereisanadequatemodelof thebeamcoreforng.Withthisanalysistechnique,thebeamtailsareclearlyvisible 133 whenplottingin J ˚ coordinateswhichcouldbeausefultoolinthecontrolroomtohelp visualize,quantify,andreducethetails. Formonitoringthebeamproperties,BPMsareoneofthemostimportantandprevalent devices.Byaddingmeasurementsofmultipleharmonicsofthebunchrepetitionrate,these positionmonitorscanintheoryalsomonitorthebunchsizewhichwouldbehighlyb formonitoringthebeamquality.Unfortunately,itwasshowninthefrontends,wherebeam isnon-relativistic,determiningthebunchsizefromthesemeasurementsischallengingand requiresaadequatemodeloftransverseofthebeam.Whilethemeasurementsin theFRIBMEBTfailedtoreplicatetheexpectedtrends,theresultingmeasurementswere closetotheexpectedvalues.Thisgivessomehopethatforbeamswithsimplertransverse thismethodcanbesuccessfulfornon-relativisticbeam. Itwasalsoshowanalyticallythatthesemulti-harmonicsmeasurementscanbemadeeas- ierbymeasuringrelativisticbeambecausetheofthetransversedistributiondamps outandonlythelongitudinalneedstobeWhile,thisislessinformationabout thebunch,byusingeveryBPMinarelativisticsectiontomeasurethebunchlength,the longitudinaldynamicscanbecontinuouslymonitoredalongthebeamlineevenwhiledeliv- eringthebeamtotheusers.Thisextensivemonitoringiscurrentlynotpossiblewiththe existingdevicesandtechniques. Keytothemulti-harmonicBPMmeasurementsandpositionmeasurementsfornon- relativisticbeamsisunderstandingthesignalsgeneratedbytheBPMpickupsandhow theyrelatetothebeam.Currentstudiesofthesesignalsrelyonanalyticandnumeric models.Tostudyandcalibratethesignalsfromthephysicaldevicesthatwillbeusedinthe beamline,ateststandwasdevelopedusingahelicaltransmissionline.Thiswillallowfurther developmentofthemulti-harmonicanalysisbypropagatingpulseswithknownpropertiesto 134 verifythebehavioroftheBPMsignals. Analyticandnumericmodelsofthisteststandshowitiscapableofreplicatingthe velocityandelectricontheBPMpickupsofanon-relativisticbeam.Withthese propertiesunderstood,allthatremainsconstruct,test,andusethesehelicalstructuresasa teststandforBPMs. Allofthedevicesandtechniquespresentedinthisdissertationpromisetoimproveour abilitytomonitorandcharacterizethebeamquality.Theywarrantfurtherinvestigation andhavethepossibilitytogreatlybthebeamdiagnostic'stoolbox.Thebroadband BPMmeasurementshavethemostobviousadvantageofturningeveryBPMintoanon- interceptivelongitudinalmonitor.Thehelicalteststandshouldaccompanythese BPMmeasurementsformeasurementsatlowenergies.Andtheaction-phaseanalysiscan giveamoredetailedviewofthebeamdistributionwithlessconcernofvariationsbetween separatemeasurements. 135 BIBLIOGRAPHY 136 BIBLIOGRAPHY [1] JieWei.TheVeryHighIntensityFuture.In InternationalParticleAcceleratorConfer- ence(IPAC2014) ,pageMOYBA01,June15-202014. 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