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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adiationpulseoflengthNu0u.centeredonthewavelength0s'0u.Inotherterms,theelectron asanNucounter-propagatingradiationeldwithaLorentz-contractedwavelength0u=uu.Thus itoscillatesNutimesalongaverticallineperpendiculartothewiggleraxis,therebyemittinga actsasarelativisticmirror<strong>and</strong>reecttheincomingradiationviaComptonback-scattering.Infact providedweobservewavelengththatarelargerthantheComptonwavelengthCompton=hc 0sisalsoshiftedbytheComptonwavelength,butthisshiftisnegligibleforrelativisticelectrons<br />
Usingtheaveragez-velocityhcosi=(1K2 whereistheangleofobservationreferencedtotheaxisoftheundulator. agoodassumptioninthecaseofIRFEL.Thereforethefundamentalwavelengthoftheundulator radiationis: 1=u(1hicos) 42+O(K4))(isthetrajectorydeectionangle), (2.19) mc2,<br />
onendsthatthefundamentalwavelengthis:<br />
widthwillhavethelimit=!0SincetheelectrononlymakesNuoscillationsintheundulator thegeneratedradiationcontainsthesamenumberofwavelengths<strong>and</strong>thereforethedurationof wavelengthsaresuppressed.Iftheundulatorwouldhaveainnitenumberofperiod,theline wavelengthrepresentsthewavelengthoftheeldcomponentthatinterfereconstructively.Other Infactalltheharmonicarealsopresenti.e.thewavelengthn=1=nwithn2N.The 1=u 22(1+K2 2+22) (2.20)<br />
thepulseisT=Nu=c.TheFouriertransformofaplanewavetruncatedafterNuoscillations frequencydependence: issinc-function4,hencethefrequencyspectrumofthespontaneousundulatorradiationhasthe<br />
to: on-axis(=0)componentisofinterested,i.e.isamplied,thefundamentalwawelengthreduces Whichmeanstheradiationispeakedatthefrequency!n=2c=n.Thewidthofthespectrum isabout! !=1 Nu.ItisimportanttonotethatinthecaseoftheFEL-oscillator,sinceonlythe d!d/sinc2Nu!!n d2W !n (2.21)<br />
reference[9])<strong>and</strong>areworthmentioninginthepresentdiscussion.Theopticalwavegeneratedfrom anundulatorcanbewellapproximated,ifNuislargeenough,byapureTEwave.Insuchcase, theform[9]: theon-axisradiationonlycontainsoddharmonic.Thepowerspectralangulardistributionisof Finallyweneedtoelaboratethepowerdensityspectrum.Wehavequalitativelyexplainedthe sincdependencebuttherearemanyotherpropertiesthathavebeenderived(seeforinstance 1=u 22(1+K2 2) (2.22)<br />
Thelatterequationisplottedingure2.9forthetwodierentvaluesofKthatareconsidered withKdef 4Thecardinalsinusfunctionisdenedas:sinc(x)=sin(x) =K=p1K2=2. d!d/mK(1K2=2)hJ(m+1)=2(mK2=4)J(m1)=2(mK2=4)i2def d2W x<br />
=Q (2.23)
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