High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
w.r.t.theRF-wave.ThisfactisaconsequenceofphaseslippagebetweentheRFwaveandthe cavity#4,oneneedstoinjectthebunchwith'40deg. bunchwhichisnotyetrelativistic.Infacttoobtainthemaximumpossibleenergyattheexitof anelectronbeamofinitialenergyof350keV.Itisnotablyseenthatmaximumenergygainprovided bytherstcavity(cavity#4)isnotobtainedbyinjectingthebunchwitharelativephase=0 Reduced Energy Gain (no unit) 2 Cav #4 10 -0.25 -0.125 0.0 0.125 0.25 Distance (m) 0 10 5 2 1 =10 deg. =-10 deg. =0 deg. 10 -0.25 -0.125 0.0 0.125 0.25 Distance (m) 0 10 5 2 1 10 0 10 5 2 1 cavity. Radio-FrequencyinducedFocusing Figure6.4:Reducedenergygain,,alongtherst(cavity#4)andsecond(cavity#3)cryounit =-16 deg. Thefundamentalmodelongitudinalacceleratingeldalsoinduces,byvirtueofMaxwell'sequations, =0 deg. Thisradialeldtogethercombinedwiththeequationofmotion@tp=eEryieldsatransverse equationofmotion,e.g.forthehorizontalplane: aradialelectriccomponentthatwrites: d2x(z) dz2+0(z) Er(z)=r(z) (z)dx(z) dz+d2(z) 2d dzEz dz2x(z)=0 (6.8) (6.7) Thisequationcanbesolvednumerically,butsomeapproximatedsolutionhavebeenderivedby Chambers[58,57],forpure-modeacceleratingcavities,andareinverygoodagreementwiththe 2 Cav #3 2
numericalsolution[56].Thisapproximatesolutioniswrittenintermoftransfermatrix(e.g.in thex-x0phasespace)as: M=0@cos()p2cos()sin() 0icos() p2+1 p8cos()sin()ifcos()+p2cos()sin1A p8i0cos()sin() elementsattheexitofthecavityofthecavityarerelatedtothebeammatrixelementatthecavity istheaveraged(overtheRFstructure)energygradient:0=eG andis,asusual,thephaseoftheinjectionoftheparticlewithrespecttotheon-crestphase.0 estimatedinastraightforwardfashion:usingrstordermatrixformalism,the11beammatrix wherei;faretheinitialandnalreducedLorentzfactors,theangleis=1 mc2cos().Thefocallengthcanbe p8cos()ln(f=i) (6.9) entranceby: Sincethefocallengthisdenedbythelengthfwherewehaved(ff) Afteradriftalongadistancel,thebeamsizewrites: 11=m211(0) (f) 11,(f) 11=(f) ff12=m11m21(0) 112l(f) 12+l2(f) 11,(f) 22 22=m21(0) 11=dl=0,ityields: 11 (6.11) (6.10) notthecase:intheCEBAF-typecavities,forinstance,thereareasymmetriesinthevicinityof thehighordermode(HOM)andtheforwardpower(FP)couplers.Theseasymmetries,inturn, inducedtransverseelectromagneticelds.Thusitrequiresacomplete3Dmodeltoaccurately studytheeectofthesecouplersonthebeamdynamics.Sucha3Dmodelisreadilyavailable Unfortunatelythismodelisderivedassumingperfectaxi-symmetricRFstructurewhichisgenerally fdef =(f) 22=m11 11 (f) m12cosp2cossin 0fsinp2+q181 cossin (6.12) beamsizethatisequaltothehallowradius,andzero-emittance.Aftertheacceleratingcavitiesthe parameterf11andf22arecomputedandthefocallengthisdeducedusingtheequation6.12.The andhasbeenimplementedintheJeersonLabversionofparmelausing3Delectromagneticeld resultscomputedforthetwocavitiesintheinjector,takingintoaccountnon-relativisticeect,are presentedingure6.5. mapgeneratedwiththeeigensolvermafia[64].Inordertocharacterizethefocusingeectofthe cavitywegenerateahallowsheetbeaminthexyspatialcoordinatespacewithzerodivergence Radio-FrequencyinducedSteering (i.e.x0=y0=0forallmacroparticleinthebeam).Thepropertiesofthiskindofbeamhasa yieldemittancedilutionviatwoeects:theheadtaileectandtheskewcoupling.Theformeris Inasimilarfashionwehavestudied,fortheinjectorcavities,theRF-kickeectonthebeamcentroid.Thekickimpartedduetothepresenceoftransverseeldintheacceleratingstructureversus thephaseoftheelectronbunchwithrespecttotheRF-waveareplottedingure6.6. TheRF-inducedkickduetothepresenceoftheforwardpowerandHighordermodecouplerscan
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numericalsolution[56].Thisapproximatesolutioniswrittenintermoftransfermatrix(e.g.in thex-x0phasespace)as: M=0@cos()p2cos()sin() 0icos() p2+1 p8cos()sin()ifcos()+p2cos()sin1A p8i0cos()sin()<br />
elementsattheexitofthecavityofthecavityarerelatedtothebeammatrixelementatthecavity istheaveraged(overtheRFstructure)energygradient:0=eG <strong>and</strong>is,asusual,thephaseoftheinjectionoftheparticlewithrespecttotheon-crestphase.0 estimatedinastraightforwardfashion:usingrstordermatrixformalism,the11beammatrix wherei;faretheinitial<strong>and</strong>nalreducedLorentzfactors,theangleis=1 mc2cos().Thefocallengthcanbe p8cos()ln(f=i) (6.9)<br />
entranceby:<br />
Sincethefocallengthisdenedbythelengthfwherewehaved(ff) Afteradriftalongadistancel,thebeamsizewrites: 11=m211(0) (f) 11,(f) 11=(f) ff12=m11m21(0)<br />
112l(f) 12+l2(f) 11,(f) 22 22=m21(0) 11=dl=0,ityields: 11 (6.11) (6.10)<br />
notthecase:intheCEBAF-typecavities,forinstance,thereareasymmetriesinthevicinityof thehighordermode(HOM)<strong>and</strong>theforwardpower(FP)couplers.Theseasymmetries,inturn, inducedtransverseelectromagneticelds.Thusitrequiresacomplete3Dmodeltoaccurately studytheeectofthesecouplersonthebeamdynamics.Sucha3Dmodelisreadilyavailable Unfortunatelythismodelisderivedassumingperfectaxi-symmetricRFstructurewhichisgenerally fdef =(f) 22=m11 11 (f) m12cosp2cossin 0fsinp2+q181 cossin (6.12)<br />
beamsizethatisequaltothehallowradius,<strong>and</strong>zero-emittance.Aftertheacceleratingcavitiesthe parameterf11<strong>and</strong>f22arecomputed<strong>and</strong>thefocallengthisdeducedusingtheequation6.12.The <strong>and</strong>hasbeenimplementedintheJeersonLabversionofparmelausing3Delectromagneticeld resultscomputedforthetwocavitiesintheinjector,takingintoaccountnon-relativisticeect,are presentedingure6.5. mapgeneratedwiththeeigensolvermafia[64].Inordertocharacterizethefocusingeectofthe cavitywegenerateahallowsheetbeaminthexyspatialcoordinatespacewithzerodivergence<br />
Radio-FrequencyinducedSteering (i.e.x0=y0=0forallmacroparticleinthebeam).Thepropertiesofthiskindofbeamhasa<br />
yieldemittancedilutionviatwoeects:theheadtaileect<strong>and</strong>theskewcoupling.Theformeris<br />
Inasimilarfashionwehavestudied,fortheinjectorcavities,theRF-kickeectonthebeamcentroid.Thekickimpartedduetothepresenceoftransverseeldintheacceleratingstructureversus thephaseoftheelectronbunchwithrespecttotheRF-waveareplottedingure6.6. TheRF-inducedkickduetothepresenceoftheforwardpower<strong>and</strong><strong>High</strong>ordermodecouplerscan
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