High Brightness Electron Beam Diagnostics and their ... - CASA
High Brightness Electron Beam Diagnostics and their ... - CASA High Brightness Electron Beam Diagnostics and their ... - CASA
y electron ψ O z densitymeasurementwhereweusedcarbontoproducetransitionradiation. θ Finally,anotherimportantcaseistheoneofaperfectconductor(i.e.(!)!1,8!).Forsuch φ classofmaterial,andundernormalincidence,thelatterEqn.(2.11)reducestothewellknown ThiscaseisofimportanceforourdiscussionontheelaborationofanoninterceptiveTR-based Figure2.2:Denitionoftheanglesusedinequations(2.9)and(2.10). x relation3: Observation d!d=Z0e22sin2() d2W Point 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) atsmalleranglesinceitis1=()2.Intheextremecasewhere=p2themaximumoccursat angleof90degw.r.t.thespecularaxis.Ingure2.3(B),wecomparetherenormalized(compared toitsmaximumvalue)TRangulardistributionemittedbyintheforwarddirectionbyanelectron bon.Thistypeofcongurationallowstogeneratebothbackward(atthevacuum-to-aluminum normallyincidentonacarbonandaluminumfoil.Typicalradiationpatternarepresentedinthe gulardistributionofbackwardTR,forthecaseofanaluminuminterface,generatedbyanelectron undernormalincidenceispresentedingure2.3(A)forthreedierentvaluesoftheLorenzfactor interface)andforward(atthealuminum-to-vacuuminterface)transitionradiation.Atypicalan- antennadiagramingure2.4forthecaseofnormaland45degincidenceoftheelectronbeamon thefoil.Inthecaseofnormalincidence,thepatternissymmetricwithrespecttotheelectronaxis. .Astheelectronenergyincrease,themaximumoftheangulardistributiongetlargerandoccur chargeusuallyusetorendereasierthetreatmentofboundaryvaluesproblem.Inthepresentcase,theproblemofan electronmovingtowardaninniteperfectlyconductingplanecanbereducetoanelectronanditselectromagnetic imagetravelingtowardeachother.Thepassagefromtheelectronintotheperfectconductoristhenequivalenttothe collisionoftheelectronwithitsimage,formalismtotreatsuch\collapsingdipole"isreadilyavailable(seereference[8] Chap.(15)). 3Infactthisrelationcanbederiveddirectly,withoutsolvingthewaveequation,byusingthemethodofimage
10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −4 10 −3 10 −2 10 −1 10 0 (A) (B) Aluminum 100 therenormalizedTRforwardangulardistributionemittedbyanelectronpassingthroughavacuumaluminum(solidline)andvacuum-carbon(dashedline)interface(B).Forcarbonthepermittivity appropriatecurve)asanelectronpassesfromtheinterfacevacuum-aluminum(A).Comparisonof 10 twodierentcrystaldirection.The5.7valueisthesmallestpermittivity.Privatecommunication isassumedtobe5.7.(Carbonormoreexactlygraphitehastwodierentelectricpermittivityforits fromGoodfellowInc.,London,U.K.). Figure2.3:DistributionofforwardTRradiationfordierentvalueof(mentionedclosetothe 2 Carbon Howeverinthecaseofnon-normalincidencethereisadis-symmetryinthelobesamplitude.This inthecaseof45degincidence,forultra-relativisticelectrons.Let'sstudyhowtheradiation,in θ (rad) termofenergy,isdistributedarounditsmaximum.Forsuchapurposeweneedtoevaluatethe dis-symmetrytendstobereducedastheelectronenergyisincreased,andbecomesinsignicant, θ (rad) θ (rad) integrals:dW aboveangularintegralto==2: Thereforethetotalenergyradiatedinthehemisphereisobtainedsettingtheupperlimitofthe d!=Zdd2W =+(12)argtanh() 2(21)cos()(1+2)argtanh(cos())(2+22+2cos(2)) d!d=Z2 2 0dZ0dd2W 2(2+2+cos(2)) + d!d dW d!tot=+(1+2)2log1+ 1 (2.13) Firstlywenotethatforultra-relativisticelectronthetotalenergyemittedinthehemispherehasa logarithmicdependenceontheenergyinlog(42). 1=-coneversustheenergyoftheincidentelectron.Wenotethatforultra-relativisticelectrons, Ingure2.5(B)wepresentthedependenceofthefractionofthetotalenergyencompassedinthe mostoftheenergyislocatedoutsidethis1=-cone.Despitetransitionradiationhasasharp maximumlocatedatthe1=-cone,itspowerisnot,likeforinstanceforsynchrotronradiation, TR Spectral Angular Intensity (a.u) TR Spectral Angular Intensity (a.u) OTR Distribution (a.u.) 2 (2.14)
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10<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
−3<br />
10 −2<br />
10 −1<br />
10 0<br />
10 1<br />
10 2<br />
10 3<br />
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10<br />
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−4<br />
10 −3<br />
10 −2<br />
10 −1<br />
10 0<br />
(A) (B)<br />
Aluminum<br />
100<br />
therenormalizedTRforwardangulardistributionemittedbyanelectronpassingthroughavacuumaluminum(solidline)<strong>and</strong>vacuum-carbon(dashedline)interface(B).Forcarbonthepermittivity appropriatecurve)asanelectronpassesfromtheinterfacevacuum-aluminum(A).Comparisonof<br />
10<br />
twodierentcrystaldirection.The5.7valueisthesmallestpermittivity.Privatecommunication isassumedtobe5.7.(Carbonormoreexactlygraphitehastwodierentelectricpermittivityforits fromGoodfellowInc.,London,U.K.). Figure2.3:DistributionofforwardTRradiationfordierentvalueof(mentionedclosetothe<br />
2<br />
Carbon<br />
Howeverinthecaseofnon-normalincidencethereisadis-symmetryinthelobesamplitude.This inthecaseof45degincidence,forultra-relativisticelectrons.Let'sstudyhowtheradiation,in<br />
θ (rad)<br />
termofenergy,isdistributedarounditsmaximum.Forsuchapurposeweneedtoevaluatethe dis-symmetrytendstobereducedastheelectronenergyisincreased,<strong>and</strong>becomesinsignicant,<br />
θ (rad)<br />
θ (rad)<br />
integrals:dW<br />
aboveangularintegralto==2: Thereforethetotalenergyradiatedinthehemisphereisobtainedsettingtheupperlimitofthe d!=Zdd2W =+(12)argtanh() 2(21)cos()(1+2)argtanh(cos())(2+22+2cos(2)) d!d=Z2 2 0dZ0dd2W 2(2+2+cos(2)) + d!d<br />
dW d!tot=+(1+2)2log1+ 1 (2.13)<br />
Firstlywenotethatforultra-relativisticelectronthetotalenergyemittedinthehemispherehasa logarithmicdependenceontheenergyinlog(42). 1=-coneversustheenergyoftheincidentelectron.Wenotethatforultra-relativisticelectrons, Ingure2.5(B)wepresentthedependenceofthefractionofthetotalenergyencompassedinthe mostoftheenergyislocatedoutsidethis1=-cone.Despitetransitionradiationhasasharp maximumlocatedatthe1=-cone,itspowerisnot,likeforinstanceforsynchrotronradiation,<br />
TR Spectral Angular Intensity (a.u)<br />
TR Spectral Angular Intensity (a.u)<br />
OTR Distribution (a.u.)<br />
2 (2.14)
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