csr impedance from a wiggler and the impact on the beam instability in damping rings

KEK-SLAC ISG9 Meeting

CSR Impedance From a WigglerCSR Impedance From a Wiggler and the Impact on the Beam Instability and the Impact on the Beam Instability

in Damping Ringsin Damping Rings

Tor Raubenheimer,Tor Raubenheimer,andand

Juhao Wu, Gennady StupakovJuhao Wu, Gennady Stupakov

December 10 - 13, 2002December 10 - 13, 2002

Presented at the Presented at the 9th KEK-SLAC9th KEK-SLAC International Linear Collider Study International Linear Collider Study GroupGroup

(ISG) Meeting(ISG) Meeting , KEK, Japan, December 10 - 13, 2002 , KEK, Japan, December 10 - 13, 2002


MotivationMotivation• Damping rings for future Linear Colliders, e.g., Damping rings for future Linear Colliders, e.g.,

NLC, requires very fast damping NLC, requires very fast damping long long wigglers;wigglers;

• Extracted beam must be very stableExtracted beam must be very stable sensitive sensitive to bursting instabilities;to bursting instabilities;

• Microwave instabilities from vacuum system Microwave instabilities from vacuum system are real concerns;are real concerns;

• Recent CSR studies indicate additional Recent CSR studies indicate additional potential problem.potential problem.

NLC/JLC ATF ALS TESLA LEP circumference / km 0.3 0.14 0.2 17 26 E / GeV 1.98 1.3 1.9 5 46

R / m 5.5 3.94 80 / e-4 9.09 6.0 9 / e-4 2.95 19.0 1.2 wiggler length / m 46 21 6? 432 wiggler K 54 70 56

particle / bunch/e+10 0.75 1.0 2.0

bunch length /mm 3.6 5.0 6.0

Damping RingDamping Ring Nominal ParametersNominal Parameters

-140

-120

-100

-80

-60

-40

-20

0

-180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100

Injection111 m

Extraction111 m

PMDR298 m

PPDR231 m

Transport161 m

NLCNLC TESLATESLA


CSR instabilityCSR instability• Stupakov-Heifets theory Stupakov-Heifets theory [[PRST-AB, PRST-AB,

5(2002)0544025(2002)054402]] indicates a potential indicates a potential instability due to the CSR in instability due to the CSR in dipolesdipoles;;

• Experimental observations Experimental observations [John Byrd, [John Byrd, et al.et al. , , PRL 89(2002)224801PRL 89(2002)224801; NSLS/BNL; BESSY-II]; NSLS/BNL; BESSY-II]

20

15

10

5

0

Bursti

ng t

hreshold

(m

A)

2.01.81.61.41.2

Energy (GeV)

2.0 mm threshold 3.2 mm threshold

Model predictions

---- Courtesy of J. Byrd---- Courtesy of J. Byrd



CSR ImpedanceCSR Impedance• Theory is developed Theory is developed ((knownknown,, [Stupakov-Heifets, [Stupakov-Heifets, 20022002]] ) )

• We need impedanceWe need impedance– Dipoles (Dipoles ( > 0 > 0) () (knownknown, , [Murphy, [Murphy, et al.et al., , 19951995]]))

• ScalingScaling

• longest wavelength determines thresholdlongest wavelength determines threshold

– Wigglers (Wigglers ( < 0 < 0) () (We need to solveWe need to solve))

• Could we approximate the wiggler as a Could we approximate the wiggler as a sequence of dipoles with an effective radius?sequence of dipoles with an effective radius?

3/23/1 / )( RkAikZ )1 3)(3/2(3 3/1 iA

C

L

R

k

C

L

R

kiAkZ W

W

D

D3/2

3/1

3/2

3/1

)()()(

CSR Wiggler WakeCSR Wiggler Wake• Early studies Early studies [Yong-Ho Chin,[Yong-Ho Chin, LBL-29981 (1990); LBL-29981 (1990); Saldin Saldin et et

al.al., , NIMA 417 (1998) 158NIMA 417 (1998) 158]]

• We introduce Wake asWe introduce Wake as

therefore, the wake Green functiontherefore, the wake Green function

• Our results are Our results are universaluniversal by expressing G with by expressing G with only only oneone parameter , as long as K is parameter , as long as K is large.large.

sdsd

)sG(s-sdksWs

w

)( )(

ds

sdGksw w

)()(

sK

kw2

2


Numerical G-functionNumerical G-function

Start over from retarded potentialStart over from retarded potential

Wake has the correct dependences:Wake has the correct dependences:Zeros and peaks at nZeros and peaks at nFELFEL/2 and scaling/2 and scaling

as sas s-1/3-1/3 for small s and s for small s and s-1-1 for large s for large s


Simple Physics ModelSimple Physics Model (ala Derbenev)(ala Derbenev)

• Dipole:Dipole:

• Wiggler:Wiggler:

sin|| eEF

•WW|| || = 0, for s = n = 0, for s = n FEL FEL ;;

•WW|| || = max, for s = = max, for s = FELFEL/2;/2;

•WW||||~ s ~ s - 1/3- 1/3 , for s , for s 0; 0;

•WW||||~ s ~ s - 1- 1 , for s >> , for s >> FELFEL..

WW||||~ s ~ s - 1/3- 1/3

ss-1/3-1/3 comes from comes from retarded positionretarded position


Real Part of the ImpedanceReal Part of the Impedance

Real part of the Real part of the impedance with impedance with peaks at npeaks at nFELFEL which which

continues to continues to increase towards increase towards cc. .

Check with the Check with the wiggler radiation wiggler radiation spectrumspectrum[Alferov, [Alferov, et al.et al. , , 19741974; Krinsky, ; Krinsky, et al.et al. , ,

19831983]], perfect , perfect agreement. agreement.

r wavenumbe FEL4

2

2

K

kk w


Imaginary Part of the ImpedanceImaginary Part of the Impedance

r wavenumbe FEL4

2

2

K

kk w


Complete Numerical ResultsComplete Numerical ResultsNLC damping ringNLC damping ring

• Growth rateGrowth rate

WigglersWigglers

DipolesDipoles

Dipoles and Dipoles and wigglerswigglers

Why dips Why dips not not peaks???peaks???


Inside Wiggler Inside Wiggler

NLCNLC

TESLATESLA

Yeah, right Yeah, right peaks!!!peaks!!!


Estimate at FEL frequencyEstimate at FEL frequency

• At FEL and harmonics, Z = R – i L, with At FEL and harmonics, Z = R – i L, with L>>1, R>0, but R<<L.L>>1, R>0, but R<<L.

• Gaussian in energy spreadGaussian in energy spread

• Dispersion relation isDispersion relation is

• 3D FEL theory give instability growth 3D FEL theory give instability growth length about length about 5050 m for the NLC wiggler. m for the NLC wiggler.

)2/(

0

00

20

2

2

e

n

k

L

1 2

0for

0)(Im


ATF Damping RingATF Damping RingThreshold vs. wavelengthThreshold vs. wavelength

• We hope to get experimental verification!We hope to get experimental verification!


Pipe cut-off Pipe cut-off wavelength wavelength (Assuming 3 (Assuming 3 cm height )cm height )

Wiggler onWiggler onWiggler offWiggler off

Nominal Nominal currentcurrent

DiscussionDiscussion• The fact that the impedance of the dipole and The fact that the impedance of the dipole and

that of the wiggler have different scaling at that of the wiggler have different scaling at low-frequency, supports a wigger ring design low-frequency, supports a wigger ring design together with other study results. together with other study results.

• For practical CSR purpose, (< 0.1 fFor practical CSR purpose, (< 0.1 fFELFEL), the ), the

wiggler impedance iswiggler impedance is

• What about waveguide cut-off and resonances?What about waveguide cut-off and resonances?• How to compare with experiment?How to compare with experiment?

k

kkikKkZW Log

2

4

)( 2

2


References:References:• G. Stupakov & S. HeifetsG. Stupakov & S. Heifets, , PRST-AB 5(2002) 054402PRST-AB 5(2002) 054402; ; • J. Byrd, J. Byrd, et al.et al. , , PRL 89(2002) 224801PRL 89(2002) 224801;;• G. Carr, G. Carr, et al.et al. , NIMA 463( , NIMA 463(2001)3872001)387; ; B. Podobedov, B. Podobedov, et al.et al. , ,

PAC2001, P. 1921PAC2001, P. 1921; ; S. Kramer, S. Kramer, et al.et al. , E , EPAC2002, P. 1523PAC2002, P. 1523;;• M. Abo-Bakr, M. Abo-Bakr, et al.et al. , , PRL 88(2002) 254801PRL 88(2002) 254801;; • Yong-Ho ChinYong-Ho Chin, , LBL-29981 (1990);LBL-29981 (1990); E.L. Saldin, E.L. Saldin, et al.et al. , , NIMA NIMA

417(1998)158;417(1998)158;• J.B. Murphy, J.B. Murphy, et al.et al. , , PAC95, P. 2980;PAC95, P. 2980; Part. Accel. 57(1997)9;Part. Accel. 57(1997)9; • J.B. Murphy, J.B. Murphy, et al.et al. , , NIMA 346(1994)571;NIMA 346(1994)571; J.B. Murphy, J.B. Murphy, et et

al.al. , , PAC95, P. 284;PAC95, P. 284; • Derbenev, Derbenev, et al.et al. , , DESY-TESLA-FEL95-05; DESY-TESLA-FEL95-05; R.L. Warnock & R.L. Warnock &

P. MortonP. Morton, , Part. Accel. 25(1990)113;Part. Accel. 25(1990)113; A. Faltens, L.J. LaslettA. Faltens, L.J. Laslett, , Part. Accel. 4(1973)151;BNL-20550(1975);Part. Accel. 4(1973)151;BNL-20550(1975);

• D.F. Alferov, D.F. Alferov, et al.et al. , , Sov. Phys. Tech. Phys. 18(1974)1336;Sov. Phys. Tech. Phys. 18(1974)1336; S. S. Krinsky, Krinsky, et al.et al. , , Handbook on Syn. Rad. Vol. 1(1983) 65.Handbook on Syn. Rad. Vol. 1(1983) 65.



Calculation detailsCalculation details

• The rate of energy lossThe rate of energy loss

s

w sd

sdzKssDsdke

cdt

d )()ˆ,,ˆˆ(2

)ˆ,(

]ˆsin)ˆcos1(ˆcos)[sinˆ,(2

ˆ

1)ˆ,,ˆˆ( 222

2

zBK

zzzzBK

szKssD

zzzB ˆsin)sincos(ˆcos)sincos1()ˆ,(

)}cos1(2)ˆ2sinsinˆ2cos(cos

]sin)cos1(2{[42

12

ˆ22

zz

KKs

wherewhere

and and is determined by is determined by

0

)ˆ,,ˆ( ˆ1

)( zKsDzdsGandand

sks w2ˆ

zkz wˆ


Analytical checkAnalytical check

• Estimate of the extreme pointsEstimate of the extreme points

They are plotted as “+”s in the plot for They are plotted as “+”s in the plot for G.G.

,1,0 with)12(

1

4

)12(for ;

])12[(4

)12(4

,2,1 with 2

for ; 0

)(

2 nn

n

n

n

nn

G


Analytical checkAnalytical check

• Short-range limit Short-range limit [[perturbative approachperturbative approach]]

• Simple approachSimple approach

3/1

3/2

34

5

611

3 4)(

G

3/13/12 )3(

2)(

RG

2

222

2

ˆcos

)ˆ(

1

zKk

zRw

0

3/2 ˆcos ˆ1

zzdDipolesDipoles

Effective Effective radiusradius

zkz wˆ


Undulator (X-ray FEL)Undulator (X-ray FEL)

Yeah, right Yeah, right peaks!!!peaks!!!

LCLSLCLS

TESLA XFELTESLA XFEL


How to treat wiggler How to treat wiggler (An estimate)(An estimate) • Treat the wiggler as 2N pieces of DipolesTreat the wiggler as 2N pieces of Dipoles

• ImpedanceImpedance

• Threshold in NLC Damping Ring drops Threshold in NLC Damping Ring drops from 1.75×10from 1.75×101010 with dipoles only to 7.25 with dipoles only to 7.25 ×10×1099. .

)cos(

)( 1 zkKkzR ww

Kk

R w21 averageaverage

C

L

R

k

C

L

R

kiAkZ W

W

D

D3/2

3/1

3/2

3/1

)()()(

It is serious!!! Recall the design value is 7.5 ×10It is serious!!! Recall the design value is 7.5 ×1099;;

But is bending magnet approach for wiggler valid?But is bending magnet approach for wiggler valid?

AcknowledgementsAcknowledgements

• Thank A.W. Chao, S.A. Heifets, Z. Thank A.W. Chao, S.A. Heifets, Z. Huang of SLAC;Huang of SLAC;

• Thank J.B. Murphy, J.-M. Wang , S. Thank J.B. Murphy, J.-M. Wang , S. Krinsky, B. Podobedov, S. Kramer of Krinsky, B. Podobedov, S. Kramer of NSLS / BNL.NSLS / BNL.

• Work supported by contract DE-AC03-Work supported by contract DE-AC03-76SF00515 with US DOE.76SF00515 with US DOE.


An estimate on the FEL instabilityAn estimate on the FEL instability• 1-D theory predicts very short-gain length1-D theory predicts very short-gain length

• However, minimal beam-radiation overlapHowever, minimal beam-radiation overlap

• 3-D TDA simulation gives, for NLC 3-D TDA simulation gives, for NLC wigglers,wigglers,

but the wiggler is only about 50 m.but the wiggler is only about 50 m.

m 31wigglerGL

Single pass isn’t dangerous, but how about Single pass isn’t dangerous, but how about multi-turn? Need numerical simulations multi-turn? Need numerical simulations with momentum compaction for ring.with momentum compaction for ring.


Dipoles and wigglersDipoles and wigglers

DipolesDipoles

WigglersWigglers Why Why peaks peaks not not dips???dips???


Complete Numerical ResultsComplete Numerical ResultsNLC damping ring thresholdNLC damping ring threshold

• Low-frequency (up to )Low-frequency (up to )

with the FEL wavenumber.with the FEL wavenumber.

• High-frequency ( )High-frequency ( )

Analytical Asymptotic ExpressionAnalytical Asymptotic Expression

)] Log[ 24

()(FEL

2

2

k

kk

ik

KkZ

2

2

FEL4

K

kk w

1.0 FELkk

FELkk 3/1

22

23/1

34

5

611

6)(

wk

KkAikZ

Average over one periodAverage over one period

Effective dipole Effective dipole radiusradius

)1 3)(3/2(3 3/1 iA

Real part of the Real part of the impedance is impedance is linear in k; linear in k; Imaginary part Imaginary part is k Log[k].is k Log[k].


Length ScalesLength Scales• Cooperative/formation length for a radiation Cooperative/formation length for a radiation

wavelength wavelength ff

– Dipole:Dipole:– Wiggler: Periodicity leads Wiggler: Periodicity leads

to interferenceto interference

• Pipe cut-off wavelength: Pipe cut-off wavelength: 1 mm1 mm;;

• Long-range wake applicable for: Long-range wake applicable for:

3 224~ ff RL m13~~ FEL f

For For < < FELFEL ““maymay”” approximate as 2N dipoles ; approximate as 2N dipoles ;

but for longer, has to include interferencebut for longer, has to include interference; need ; need look at FEL frequency and harmonics look at FEL frequency and harmonics

FEL


Low-frequency ImpedanceLow-frequency ImpedanceAnalyticalAnalytical

• Wiggler impedance:Wiggler impedance:

• Total impedance:Total impedance:

FEL2

2

Log2

4

)(kkkikK

kZW

CL

kZCL

kZkZ WW

DD )()()(

Nominal (10Nominal (101010)) Threshold from full-Threshold from full-impedance (10impedance (101010))

Threshold from low- Threshold from low- frequency (10frequency (101010))

NLCNLC 0.750.75 1.31.3 1.31.3

TESLATESLA 2.02.0 78.878.8

We may use the low-frequency impedance for We may use the low-frequency impedance for practical purpose!practical purpose!


csr impedance from a wiggler and the impact on the beam instability in damping rings

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