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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AppendixC<br />
C.0.4Introduction <strong>Beam</strong>Distributions DispersionRelationsforBunched<br />
emissionfromabunchedelectronbeamhavebeenusedbymanyauthors.Howeverwebelieve givethephase,butonlyonetermcontributingtothephase(dependingonthebunchlongitudinal distributionproperties). Dispersionrelationstocomputethephaseassociatedtotheelectriceldgeneratedviacoherent<br />
C.0.5Background themathematicalproofisnotalwaysproperlyderived:thesedispersionrelationsareappliedto aderivationofsuchrelations<strong>and</strong>discussthefactthatthesedispersionrelationdonotafortiori functionswhichcannotbeexpressedasCauchyintegrals.Inthepresentaddendumwepresent<br />
showninthecaseofbunchedelectronbeamthatiftheemittedradiationisobservedatwavelength comparableorlargerthatthebunchlength,theelectronsinthebunchemitcoherently.Insucha Therearemanywayselectronscanemitradiationas<strong>their</strong>environmentismodied.Ithasbeen<br />
where^S(!)istheFouriertransformofthelongitudinaldistributionS(t).Theproblemthathas case,thespectrumoftheradiationwritesas: transformoftheelectriceld,tocomputethebunchformfactorphase. Inthemanypapers,thefunctionlog(^S(!))isdened<strong>and</strong>thecomplexpartofthisfunction(which beenstudiedistousethepowerspectrum,whichisproportionaltothemodulusoftheFourier isthephaseof^S(!))iscalculatedusingthest<strong>and</strong>arddispersionrelations.Howeverweshallsee Etotal(!)=qN(N1)qj^S(!)jE1e(!) (C.1)<br />
belowthatlog^S(!)doesnothavethe\right"properties<strong>and</strong>cannotbegenerallywrittenasa log^S(!)function. Cauchyintegral<strong>and</strong>morecareshouldbeusedwhentryingtorecovertheimaginarypartofthe 180
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