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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presentanoutlineofthisproofbelow,<strong>and</strong>adetailedproofinAppendixC.Fromthedenitionof<br />
where canbewrittenas: thebunchformfactorwecanwrite:~S()=^S()^S() mathematicstextbooks(seeforinstance[50])<strong>and</strong>canbeappliedtoS()tocalculateitsimaginary partknowingitsrealpartbecauseS()isasquareintegrablefunction.Inthepresentcase,the isthewavenumber<strong>and</strong>^S()istheFouriertransformofthebunchlongitudinaldistribution;it ()isthephaseassociatedtotheFouriertransform.Themethodisdiscussedinst<strong>and</strong>ard ^S()=q~S()exp(i ()) (5.19) (5.18)<br />
(=+i0isthecomplexwavenumber): Now,log(^S())isnotsquareintegrable<strong>and</strong>theCauchyintegralonlog(^S())doesnotconverge problemisslightlydierent:weknowthemodulusofS()<strong>and</strong>needtocomputethephase.By takingthelogarithmofthelatterequation,wecomebacktothedeterminationoftheimaginary partofthefunctionlog[S()]fromtheknowledgeofitsrealpartlog[jS()j]: log(^S())=log(q~S())+i Ilog(^S())log(^S()) ()=1=2log(j~S()j)+i jj!1 !Z0log(^S())!1 () (5.20)<br />
dispersionrelationsfor^S(seeAppendixCforadetailedderivation),<strong>and</strong>nallythephaseof^S Let'sintroducethefunction()denedas: ()isnotsingularat=<strong>and</strong>issquareintegrable.Wecanthenderiveasetof\modied" ()def =log[^S()]log[^S()] (5.22) (5.21)<br />
takestheform: Letting0=0<strong>and</strong>usingthefact^S()=^S()wenallynd: Thislatterequationiswidelyknown,intheliterature,<strong>and</strong>issometimesreferredasdispersion wherePdesignatestheCauchyprincipalvaluefortheintegral. ()= (0)1(0)PZ+1 ()= (0)2PZ+1 1log[j^S()j]log[j^S()j] 0log[j^S()j] ()(0)d 22d (5.24) (5.23)<br />
relation.Oncethephase inverseFouriertransform:S(z)=Z1 ()iscomputedwecanrecovertheinitialdistributionbyusingthe<br />
discussionisprovidedinAppendixC).<br />
contributiontothephasethatmustbeconsidered.Inthefollowingwewillnotconsidersuchcases byassumingthest<strong>and</strong>ardbunchdistributionisanalyticintheupperhalf-plane(afullydetailed thecomplexplane.Ifithassingularitiesthen,invirtueoftheresiduetheorem,thereareother TwofactsshouldbeemphasedaboutEqn.(5.24)(1)the bezero,<strong>and</strong>(2)thisequationisapplicableprovidedlog[S()]isanalyticintheupperhalf-partof 0^S()cos(2z (0)termisunknown<strong>and</strong>isassumedto ())d (5.25)
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