High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
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TheAngularBFF ThesecondterminEqn.(2.5)inChapter2isthedependenceofthecoherentemissionwithrespect totheangularbeamproperties.Itisinterestingtonotethatthistermiswavelengthindependent <strong>and</strong>thereforeisnotgoingtoinuencethespectrumoftheradiation:itactsasamultiplicative is: factorthatcanreducethetotalpoweremittedbyabunchedbeam.ItsexpressionfromEqn.(1.5) Itisrelativelydiculttoevaluatethisfactorforanarbitrarybunchdistribution.Howeverunderthe assumptionoftransversecylindricallysymmetricbunches,<strong>and</strong>introducingtheangles=6(bn;b), =6(bnyz;bz), =6(bn;dbz),<strong>and</strong>=6(bnxy;dbz)itreducesto[2]: ~A()=jZd ~A(bn)def ZdA( =jZA(!) )cossin (!)d!j2 cossincos sin j2 (5.2) (5.1)<br />
whichinturncanbeexpressedasacompleteellipticintegral(extendedfromRef.[2])ifweassume theangulardistributionwritesasaGaussi<strong>and</strong>istribution:A( ~A()=j202Z=(2) 0x[(1x)K(2x1=2 1+x)+(1+x)E(2x1=2 1+x)]exp[2 )=1=p202exp( 202x2]j2dx 2=(202)<br />
divergenceisoftheorderof0'1mrad,wesatisfytherelation0'1=forthenominal radiation,thespectralpowerhasitsmaximumatanglesoftheorderof'1=<strong>and</strong>sincetheRMS angulardistributioningure5.2.Itisnoticedthattypicallythisintegralisunityinthecasewhere thebeamdivergence0ismuchsmallerthantheangleofobservation.Inthecaseoftransition introduced2.ThenumericalintegrationofEqn.(5.3)ispresentedfordierentRMSwidthofthe wherethecompleteellipticintegraloftherstkind,K(u),<strong>and</strong>secondkind,E(u),havebeen (5.3)<br />
energyof38MeV(i.e.'77).Henceforthwewillassume,exceptwhenexplicitlymentioned,that<br />
equationmoreexplicit.Ifbnyzistheprojectionofthebnunityvectorinthe(y;z)plane,let thisfactorisalwaysunityforourtypicalbeamparameters. TheSpatialBFF TheEqn.(2.5)iswritteninavectorform.WewillworkinCartesiancoordinatestomakethis<br />
wherewehaveassumedwecouldfactorthe3D-spatialbeamdensitydistributionSastheproduct ofthe1DprojectionsSx,Sy<strong>and</strong>Sz. bn!X=(xsinsin+ysincos+zcos),<strong>and</strong>thisequationrewrites: =6(bn;bz)<strong>and</strong>=6(bnyz;bx)thentheargumentoftheexponentialfunctioninEqn.(2.5)writes<br />
Inordertousefrequency-domainanalysistodeduceinformationonthebunchlongitudinaldistribution,itisnecessarythatthe!-dependencecomeonlyfromlongitudinalcoordinatez.Fromthe 2Theellipticintegraloftherst<strong>and</strong>secondkindarerespectivelydenedasK(u)=R=2 ~S(!;bn)def =jZSx(x)Sy(y)Sz(z)expi!c(xsinsin+ysincos+zcos) 0[1u2sin2()]1=2d (5.4)<br />
<strong>and</strong>E(u)=R=2 0[1u2sin2()]1=2d
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