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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equation boundarybetweenvacuum<strong>and</strong>amediumofrelativedielectricpermittivity=absolute=0(where 0isthevacuumelectricpermittivity),theproblemconsistsofsolvingthescalar<strong>and</strong>vectorpotential<br />
atthemediainterface:thefollowingcomponentsoftheelectromagneticeldmusthavecontinuity: Ek,B?,Hk,<strong>and</strong>D?(\k"<strong>and</strong>\?"correspondstothecomponentsparallel<strong>and</strong>perpendicularto theinterfacesurface).Moreovertheelectriceldsolutionofthehomogeneousequation(i.e.the electromagneticeldsolutionofEqn.(2.8)mustbematchedwiththeproperboundarycondition inthetwomediai.e.vacuum(byletting=1)<strong>and</strong>inthemediawithpermittivity.Thehomogeneoussolutionofthelatterequationgivestheradiationeld(photon;!Aphoton).Theobtained r2"!A#1c2@t"!A#=10e"(!r;t) !(!r;t)# (2.8)<br />
typesofradiationarefound:aforwardradiationwhichisemittedinthedirectioncenteredaround thedirectionofmotionoftheelectron,<strong>and</strong>abackwardradiationemittedaroundthespecularaxis ofreectionoftheinterface.Themostgeneralexpressionforthetransitionradiationemittedin is[7]: radiationpotentials)mustsatisfyr:!Eradiation=0everywhere.Whensolvingthisproblemtwo anangleofincidence thebackwarddirectionbyanelectronmovingfromvacuumtoamediumofpermittivitywith d!d= d2Wk j12z+zqsin2()2zxcos(x)sin2()xzcos(x)qsin2() 43[(12xcos2(x))22xcos2()]2sin() (denedintheplanexz)withrespecttotheinterfacenormaldirection Z0e22zcos2()j1j2<br />
d2W? d!d= 43[(1xcos(x))22zcos2()]sin2() 1+zqsin2()xcos(x)cos()+qsin2() e22x4zcos2(y)cos2()j1j2 1 (2.9) j2<br />
Z0=120isthevacuumfreespaceimpedance,<strong>and</strong>thedierentanglesarepresenteding- thedirectionofobservation<strong>and</strong>thexoryaxis.Theseanglesaredenedbycos(x)=sin()cos() Aprioritransitionradiationspectrumhasnodirectdependenceonthefrequency!ofobservation; ure2.2.Thedependenceon referencedw.r.t.thezaxis. <strong>and</strong>cos(y)=sin()sin(),istheazimuthalangleinthex-yplane<strong>and</strong> j1zqsin2()xcos(x)qsin2()+cos()j2(2.10)<br />
inrealitythisdependenceiscomingfromtheelectricpermittivity=(!). isinx=sin( )<strong>and</strong>z=cos( )<strong>and</strong>x;yaretheanglesbetween<br />
spectralenergydistributionemittedinthebackwarddirectionviatransitionradiationreducesto: Undernormalincidence,i.e. =0(x=y=0),onlythe"k"componentremains,<strong>and</strong>the istheincidenceangle<br />
d!d=e22sin2()cos2() d2W2c(12cos2())2j<br />
1+qsin2()cos()+qsin2()j2(2.11)<br />
(1)12+qsin2()
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