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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y<br />
electron<br />
ψ<br />
O<br />
z<br />
densitymeasurementwhereweusedcarbontoproducetransitionradiation.<br />
θ<br />
Finally,anotherimportantcaseistheoneofaperfectconductor(i.e.(!)!1,8!).Forsuch<br />
φ<br />
classofmaterial,<strong>and</strong>undernormalincidence,thelatterEqn.(2.11)reducestothewellknown ThiscaseisofimportanceforourdiscussionontheelaborationofanoninterceptiveTR-based Figure2.2:Denitionoftheanglesusedinequations(2.9)<strong>and</strong>(2.10).<br />
x<br />
relation3:<br />
Observation<br />
d!d=Z0e22sin2() d2W<br />
Point<br />
referencedwithrespecttothespecularaxis. wheretheelectronincomesontheinterfacewitha45degincidence;theangle,inthiscase,being TheEqn.(2.12),inthelimitofanultra-relativisticelectron(i.e.!1)isalsovalidinthecase intercepttheelectronbeamwithverythinfoil.Inourcase,thefoilismadeofaluminumorcar- Thecongurationgenerallyusedtogeneratetransitionradiationinaparticleacceleratoristo 44c(12cos2())2(1;1) !Z0e2 43(2+2)2 2 (2.12)<br />
atsmalleranglesinceitis1=()2.Intheextremecasewhere=p2themaximumoccursat angleof90degw.r.t.thespecularaxis.Ingure2.3(B),wecomparetherenormalized(compared toitsmaximumvalue)TRangulardistributionemittedbyintheforwarddirectionbyanelectron bon.Thistypeofcongurationallowstogeneratebothbackward(atthevacuum-to-aluminum<br />
normallyincidentonacarbon<strong>and</strong>aluminumfoil.Typicalradiationpatternarepresentedinthe gulardistributionofbackwardTR,forthecaseofanaluminuminterface,generatedbyanelectron undernormalincidenceispresentedingure2.3(A)forthreedierentvaluesoftheLorenzfactor interface)<strong>and</strong>forward(atthealuminum-to-vacuuminterface)transitionradiation.Atypicalan-<br />
antennadiagramingure2.4forthecaseofnormal<strong>and</strong>45degincidenceoftheelectronbeamon thefoil.Inthecaseofnormalincidence,thepatternissymmetricwithrespecttotheelectronaxis. .Astheelectronenergyincrease,themaximumoftheangulardistributiongetlarger<strong>and</strong>occur<br />
chargeusuallyusetorendereasierthetreatmentofboundaryvaluesproblem.Inthepresentcase,theproblemofan electronmovingtowardaninniteperfectlyconductingplanecanbereducetoanelectron<strong>and</strong>itselectromagnetic imagetravelingtowardeachother.Thepassagefromtheelectronintotheperfectconductoristhenequivalenttothe collisionoftheelectronwithitsimage,formalismtotreatsuch\collapsingdipole"isreadilyavailable(seereference[8] Chap.(15)).<br />
3Infactthisrelationcanbederiveddirectly,withoutsolvingthewaveequation,byusingthemethodofimage
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