High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Theequation5.8yields: f()=jZ+1 11Xn=0(2iz)n n!(z)dzj2=j1Xn=0(2i)n exp()=1Xn=0n n! n!Z+1 1znS(z)dzj2 (5.12)<br />
Deningthen-ordermomentnasn=Z+1 Eqn.(5.13)becomes: 1znS(z)dz (5.14) (5.13)<br />
thepreviousequationreducesto: Incaseweareathighwecanapproximatetheseriesbyitsrstthreetermsonly.Insuchcase, ~Sz()=j1+2i1+(2i)2 ~Sz()=j1Xn=0(2i)n n!nj2 22+O(3)j2 (5.15)<br />
exp4222whoseTaylorexpansionatsmallfrequencyisalsogivenbyEqn.(5.16).Itis veryinformativetodeveloptheBFFtohigherordertoseewhetherwecanextractinformationon ofthegaussi<strong>and</strong>istributionexpz2=(22)case:forsuchadistributiontheformfactorwrites wherewehaveintroducedthevariance2=221(def thebunchformfactorwithaparabolicfunctionathighfrequency.Thisresultisageneralization FromEqn.(5.16)wenotethatitisstraightforwardtoextractthebunchlength,,bytting =14222+O(3) =z). (5.16)<br />
thehighermomentsofthebunchlongitudinaldistribution.Performingsuchderivationyieldsthe generalformoftheBFF[46](with=2): ~Sz()=j1Xn=0(1)n2n = =1Xn=04n 1Xn=0(1)n2n2n (2n)!2n+i1Xn=0(1)n2n+1 (2n)!!2+ 1Xn=0(1)n2n+12n+1 (2n+1)!2n+1j2<br />
+1Xn=04n+2 ((2n)!)22n+21Xn=0Xm
Change language