Lecture 10 - USPAS · Lecture 10: Coherent Synchrotron Radiation Yunhai Cai SLAC Naonal Accelerator...

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


!"

#"

$"

%

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


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.


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


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


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


•  Thescalingwithrespecttoγisdifferent.Herewehaveusedγ=500.•  Thecentrifugalforceismuchhardtocomputenumericallybecauseofthecancella4onbetweentheelectricandmagne4cforces.

LongitudinalFieldandCentrifugalForce ρ2Es/eγ4ρ2Fx/e2γ3


•  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


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


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


•  Thescalingwithrespecttoγisdifferent.Herewehaveusedγ=500.•  The“logarithmic”singularityisclearlyseeninthetransversepoten4alalongthelineofχ=0.

Longitudinal and Transverse Potentials ρ2Ψs/eγρ2Ψx/e2


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.


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


Summary •  Thetransverseforceinthecurvetedcoordinateisessen4allytheLorentzforcebutwithasubs4tu4onofthetransversemagne4cfield,Bx,y->(1+x/ρ)Bx,y

•  Thecurvaturetermplayakeyroleforderivingthepoint-chargewakefieldexplicitlyintermsoftheincompleteellip4cintegralsofthefirstandsecondkind

•  EmiKancegrowthduetothecentrifugalforceisatthesamelevelofthecontribu4onthroughtheenergychanges

•  Asteady-statetheoryofthecoherentsynchrotronradia4onintwo-dimensionalfreespaceisdeveloped


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


Acknowledgements

•  ManydiscussionswithK.OhmiwhovisitedSLACrecently

•  Helpfullydiscussionswithmycolleagues:KarlBane,RobertWarnock,GennadyStupakov

•  BenefitedfromseveraltalksinthetheoryclubbyGennadyStupakov

•  YuantaoDingforprovidingLCLSparameters


Lecture 10 - USPAS · Lecture 10: Coherent Synchrotron Radiation Yunhai Cai SLAC Naonal Accelerator...

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