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Lecture10:
Coherent Synchrotron Radiation
YunhaiCaiSLACNa4onalAcceleratorLaboratory
June15,2017
USPASJune2017,Lisle,IL,USA
W (z) = 231/3ρ2/3
∂∂zz−1/3
Wakefield due to CSR is given by
For z>0. It vanishes when z<0 (force is on the particle ahead). where ρ is the bending radius.
1D CSR Wakefield in Free Space
• Simplicity• Universal• Informofderiva4ve
5/31/17 YunhaiCai,SLAC 2
CSR Microbunching in Bunch Compressors
ElegantCSRenergylossa.er
BC1measuredwithBPM
opera;ngpoint
ElegantMeas-1Meas-2
HorizontalemiKanceaLerBC1vs.RFphase
Elegantopera;ngpoint
• CSRlimitsfurtherimprovementoflongitudinalemiKanceandlimitspeakbeamcurrentbelow3kA
• 1Dmodelresultsareingoodagreementwithdata,asshowninthefollowingBC1examples
• 3Dmodelmaybenecessaryatmuchhigherpeakcurrent
CourtesyofYuantaoDing
CSR Instability in Electron Storage Rings
Measuredburs4ngthresholdatANKASeeM.Kleinetal.PAC09,p4761(2009)
),cos/()(8
3/12
02
3/7revrfsrf
thbthz ffVIZc
ϕρχξπ
σ =
Scalinglawforbunchedbeam:
,34.05.0)( χχξ +=th χ=σzρ1/2/h3/2and
K.Bane,Y.Cai,andG.Stupakov,PRSTAB13,104402(2010)
G.Wustefeldatal.PAC10,p2508(2010)
4YunhaiCai,SLAC5/31/17
TransverseForceinCurvedGeometry
x ''+ xρ2
=δρ+
ecp0βs
[Ex +βyBs −βs (1+xρ)By ],
y '' = ecp0βs
[Ey +βs (1+xρ)Bx −βxBs ]
ρ
!"#"$"
Thecurvilinearcoordinate
Equa4onofmo4on:
Curvatureterms
• Curvaturetermsareconceptuallyimportant• Ex,Ey,Bx,By,andBsaretheself-fields• Noexplicitdependenceonthepoten4als• Equa4onsarederivedfromtheHamiltonianbyCourant-Synder
5/31/17 YunhaiCai,SLAC 5
!"
#"
$"
%
ρ
&"
θ
% '"("
Lienard-Wiechert Formula
!E = e[
!n −!β
γ 2 (1− !n ⋅!β)3R2
]ret + (ec)[!n × (!n −
!β)×!"β
(1− !n ⋅!β)3R
]ret,!B = !n ×
!E
• Spacechargeissuppressedby1/γ2• Iden4fyradiatedfieldwithCSR• Subjecttoretardedcondi4on:
SpaceChargeRadiatedField
t ' = t − Rc
5/31/17 YunhaiCai,SLAC 6
κ =Rρ= χ 2 + 4(1+ χ )sin2α ,
α =θ / 2,χ = x / ρ
Electrical and Magnetic Fields
where
Es =eβ 2[cos2α − (1+ χ )][(1+ χ )sin2α −βκ ]
ρ2[κ −β(1+ χ )sin2α]3
Ex =eβ 2 sin2α[(1+ χ )sin2α −βκ ]
ρ2[κ −β(1+ χ )sin2α]3
By =eβ 2κ[(1+ χ )sin2α −βκ ]ρ2[κ −β(1+ χ )sin2α]3
• Theyaresimplestexpressions,especiallyinthedenominatorandchosentosuppressthenumericalnoisenearthesingularity.
5/31/17 YunhaiCai,SLAC 7
cRtt −='
ℓ = v(t − t ')− (s− s ')
RetardedTime:
Timeofflightatposi4ons:
ξ =α −β2
χ 2 + 4(1+ χ )sin2α
Itisthevariableforthewake.Thearcdistancetothesourcepar4cleatthe4met.Wederiveitsrela4ontoα.
Retarded Time and Longitudinal Position
s’sℓ
whereξ =-/2ρ and.ℓ
A
BC
ℓ = z '− z
5/31/17 YunhaiCai,SLAC 8
Solutions of the Retarded Condition
Numericalsolu4onisonmesh:512x512usingMathema4catakingseveralhours.Thedifferencesbetweenthenumericandanaly4csolu4onsareatanorderof10-6.Herewehaveused γ=500.
NumericalAnaly4cal
α 4 +3(1−β 2 −β 2χ )β 2 (1+ χ )
α 2 −6ξ
β 2 (1+ χ )α +
3(4ξ 2 −β 2χ 2 )4β 2 (1+ χ )
= 0
Expandinguptothefourth-orderofαoftheretardedcondi4on,wehave
9YunhaiCai,SLAC5/31/17
Analytical Solution of the Retarded Condition
024 =+++ ζηαυαα
Ithasanaly4calsolu4ondiscoveredbyFerrari(1522-1565)byaddingandsubtrac4ngatermtomakeadifferenceoftwoperfectsquares.Tofindtheterm,weneedtofirstfindtherootsofathird-orderequa4on.Arootmisgivenby,
Ingeneral,wewanttofindtherootsofthedepressedquar4cequa4on:
3/13/12
)363
(3
Ω+Ω++−= −υζυm
⎪⎪⎩
⎪⎪⎨
⎧
++−+−
−+−+=
)22)(22(
21
)22)(22(
21
mmm
mmm
ηυ
ηυ
α
Thesolu4onofα:ξ>=0
ξ<0
32
23232
)363
()216616
(216616
υζυζυηυζυη+−+−++−=Ω
where
5/31/17 YunhaiCai,SLAC 10
• Thescalingwithrespecttoγisdifferent.Herewehaveusedγ=500.• Thecentrifugalforceismuchhardtocomputenumericallybecauseofthecancella4onbetweentheelectricandmagne4cforces.
LongitudinalFieldandCentrifugalForce ρ2Es/eγ4ρ2Fx/e2γ3
11YunhaiCai,SLAC5/31/17
• Thescalingwithrespecttoγisdifferent.Herewehaveusedγ=500.• Thecentrifugalforceismuchhardtocomputenumericallybecauseofthecancella4onbetweentheelectricandmagne4cforces.
LongitudinalFieldandCentrifugalForce
-2 -1 0 1 2-2
-1
0
1
2
3γ3ξ
γ2χ
ρ2
eγ4Es
-2 -1 0 1 2-2
-1
0
1
2
3γ3ξ
γ2χ
ρ2
e2 γ3Fx
5/31/17 YunhaiCai,SLAC 12
A Longitudinal Potential Ψs Differen4atetheretardedcondi4on,
Es =∂ψs
∂ξ
where
Esdξ =eβ 2[cos2α − (1+ χ )][(1+ χ )sin2α −βκ ]
ρ2κ[κ −β(1+ χ )sin2α]2 dα
= d(eβ 2 ( cos2α − 1
1+ χ)
2ρ2[κ −β(1+ χ )sin2α] )
ξ =α −β2
χ 2 + 4(1+ χ )sin2α
Wehave,dξ = (1− β(1+ χ )sin2α
χ 2 + 4(1+ χ )sin2α)dα
CombiningitwiththelongitudinalelectricfieldEs,wefind
ψs (ξ,χ ) =eβ 2 ( cos2α − 1
1+ χ)
2ρ2[κ −β(1+ χ )sin2α]
or
13YunhaiCai,SLAC5/31/17
Transverse Force and Potential Ψx
ψx (ξ, χ ) =e2β 2
2ρ2{ 1χ (1+ χ )
[(2+ 2χ + χ 2 )F(α, −4(1+ χ )χ 2 )− χ 2E(α, −4(1+ χ )
χ 2 )]
+κ 2 − 2β 2 (1+ χ )2 +β 2 (1+ χ )(2+ 2χ + χ 2 )cos2α −κβ(1+ χ )sin2α[1−β 2 (1+ χ )cos2α]
[κ 2 −β 2 (1+ χ )2 sin2 2α],
Similarly,
where,Fx =
∂ψx
∂ξ
Fx =eβ 2[sin2α − (1+ χ )βκ ][(1+ χ )sin2α −βκ ]
ρ2[κ −β(1+ χ )sin2α]3and,
• TheTransverseforceistheLorentzforceandplusthecurvatureterm• Thecurvaturetermisnecessaryfortheanaly4calexpression• F(α,k)andE(α,k)aretheincompleteellip4cintegralsofthefirstandsecondkind• Fy=0,sothepar4clesstayintheplaneiftheyareini4allyinthehorizontalplane
Curvatureterm
5/31/17 YunhaiCai,SLAC 14
• Thescalingwithrespecttoγisdifferent.Herewehaveusedγ=500.• The“logarithmic”singularityisclearlyseeninthetransversepoten4alalongthelineofχ=0.
Longitudinal and Transverse Potentials ρ2Ψs/eγρ2Ψx/e2
5/31/17 YunhaiCai,SLAC 15
Wakefields Fromtheequa4onsofthemo4on,thechangesofthemomentumdevia4onandkickaregivenby,
δ ' = reNb
γWs (z, χ ),
x '' = reNb
γWx (z, χ )
wherereistheclassicalelectronradius,Nbthebunchpopula4on,andthewakes,
Ws (z, χ ) = Ys∫∫ (z− z '2ρ
, χ − χ ')∂λb(z ', χ ')∂z '
dz 'dχ ',
Wx (z, χ ) = Yx∫∫ (z− z '2ρ
, χ − χ ')∂λb(z ', χ ')∂z '
dz 'dχ ',
withYs=2ρΨs/(eβ2),Yx=2ρΨx/(eβ)2andλbisthenormalizeddistribu4on.
• Theseareaddi4onalchangeswhenintegra4ngthroughthebend.
16YunhaiCai,SLAC5/31/17
Gaussian Bunch Wakes
-6 -4 -2 0 2 4 6-3×106
-2×106
-1×106
0
1×106
q=z
σz
Ws[m-2
]
-6 -4 -2 0 2 4 6
-2.5×106
-2.0×106
-1.5×106
-1.0×106
-500000
0
χ
σχ
Ws[m-2
]
-6 -4 -2 0 2 4 6
0
50000
100000
150000
200000
250000
300000
q=z
σz
Wx[m-2
]
-6 -4 -2 0 2 4 6
50000
100000
150000
200000
250000
χ
σχ
Wx[m-2
]
x=-σx
x=σx
ρ=1m,γ=500,σx=σz=10µm,
z=σz z=-σz
Λ
2πρσ z
e−q2
2
Λ =1124γE − 4+ ln(
2ρ2
σ x2 )+
1324ln( σ z
2
2ρ2)5/31/17 17
Estimate of Emittance Growth
ΔεN =12γβx < (Δx '− < Δx ' >)
2 >,
IncreaseoftheprojectedemiKance:
ΔεN = 7.5×10−3 βx
γ( NbreLB
2
ρ5/3σ z4/3 )
2,
Fromthelongitudinalcontribu4onabendingmagnet:
ΔεN =(−3+ 2 3)24π
βx
γ(ΛNbreLB
ρσ z
)2,
Itleadsto38%increaseoftheemiKanceforthelastdipole.Fromthecentrifugalforce,wehave
Thisgives29%increaseoftheemiKance.
Symbol γ εN σz Nb βx ρ LBValue 10,000 0.5µm 10µm 109 5m 5m 0.5m
TheparametersforthelastbendofBC2inLCLS
18YunhaiCai,SLAC5/31/17
Summary • Thetransverseforceinthecurvetedcoordinateisessen4allytheLorentzforcebutwithasubs4tu4onofthetransversemagne4cfield,Bx,y->(1+x/ρ)Bx,y
• Thecurvaturetermplayakeyroleforderivingthepoint-chargewakefieldexplicitlyintermsoftheincompleteellip4cintegralsofthefirstandsecondkind
• EmiKancegrowthduetothecentrifugalforceisatthesamelevelofthecontribu4onthroughtheenergychanges
• Asteady-statetheoryofthecoherentsynchrotronradia4onintwo-dimensionalfreespaceisdeveloped
5/31/17 YunhaiCai,SLAC 19
References 1Dtheory:1) J.B.Murphy,S.Krinsky,andR.L.Gluckstern,“LongitudinalWakefieldforanElectronMovingona
CircularOrbit,”Par4cleAccelerator,Vol.57,pp.9-64,19972) M.DohlusandT.Limberg,“EmiKancegrowthduetowakefieldsoncurvedbunchtrajectories,”
Nucl.Instr.andMeth.inPhys.Res.A393(1997)494-4993) E.L.Saldin,E.A.Schneidmiller,andM.V.Yurkov,“Onthecoherentradia4onofanelectronbunch
movinginarcofacircle,”Nucl.Instr.andMeth.InPhys.Res.A398(1997)373-3944) M.Borland,“Simplemethodforpar4cletrackingwithcoherentsynchrotronradia4on,”Phys.
Rev.STAccel.Beams,4,070701(2001)2Dandbeyond:1) R.Talman,“Novelrela4vis4ceffectimportantinaccelerators,”Phys.Rev.LeK.56,1429,19872) Y.S.DerbenevandV.D.Shiltsev,“Transverseeffectsofmicrobunchradia4veinterac4on,”SLAC-
PUB-7181,June19963) G.V.Stupakov,“Effectofcentrifugaltransversewakefieldformicrobunchinbend,”SLAC-
PUB-8028,RevisedMarch20064) G.V.Stupakov,“Synchrotronradia4onwakeinfreespace,”SLAC-PEPRINT-2011-034,Proc.Of
PAC97,Vancouver,Bri4shColumbia,Canada(1997)5) ChengkunHuang,ThomasJ.T.Kwan,andBruceE.Carlsten,“Twodimensionalmodelfor
coherentsynchrotronradia4on,”Phys.Rev.STAccel.Beams.16,010701,(2013)6) Ohmi’stalkintheoryclub,SLAC2016
20YunhaiCai,SLAC5/31/17
Acknowledgements
• ManydiscussionswithK.OhmiwhovisitedSLACrecently
• Helpfullydiscussionswithmycolleagues:KarlBane,RobertWarnock,GennadyStupakov
• BenefitedfromseveraltalksinthetheoryclubbyGennadyStupakov
• YuantaoDingforprovidingLCLSparameters
5/31/17 YunhaiCai,SLAC 21