1. KEK-SLAC ISG9 Meeting CSR Impedance From a Wiggler and the Impact on the Beam Instability in Damping Rings Tor Raubenheimer, and Juhao Wu, Gennady Stupakov December 10 - 13, 2002 Presented at the 9th KEK-SLAC International Linear Collider Study Group (ISG) Meeting, KEK, Japan, December 10 - 13, 2002

2. KEK-SLAC ISG9 MeetingMotivation Damping rings for future Linear Colliders, e.g., NLC, requires very fast damping  long wigglers;Damping rings for future Linear Colliders, e.g., NLC, requires very fast damping  long wigglers; Extracted beam must be very stable  sensitive to bursting instabilities;Extracted beam must be very stable  sensitive to bursting instabilities; Microwave instabilities from vacuum system are real concerns;Microwave instabilities from vacuum system are real concerns; Recent CSR studies indicate additional potential problem.Recent CSR studies indicate additional potential problem.

3. Damping RingNominal Parameters Damping Ring Nominal Parameters NLCTESLA KEK-SLAC ISG9 Meeting

4. CSR instability Stupakov-Heifets theory [PRST-AB, 5(2002)054402] indicates a potential instability due to the CSR in dipoles;Stupakov-Heifets theory [PRST-AB, 5(2002)054402] indicates a potential instability due to the CSR in dipoles; Experimental observations [John Byrd, et al., PRL 89(2002)224801; NSLS/BNL; BESSY-II]Experimental observations [John Byrd, et al., PRL 89(2002)224801; NSLS/BNL; BESSY-II] Model predictions ---- Courtesy of J. Byrd KEK-SLAC ISG9 Meeting

5. CSR Impedance Theory is developed (known, [Stupakov-Heifets, 2002] )Theory is developed (known, [Stupakov-Heifets, 2002] ) We need impedanceWe need impedance –Dipoles (  > 0) (known, [Murphy, et al., 1995] ) ScalingScaling longest wavelength determines thresholdlongest wavelength determines threshold –Wigglers (  < 0) (We need to solve) Could we approximate the wiggler as a sequence of dipoles with an effective radius?Could we approximate the wiggler as a sequence of dipoles with an effective radius?

6. CSR Wiggler Wake Early studies [Yong-Ho Chin, LBL-29981 (1990); Saldin et al., NIMA 417 (1998) 158]Early studies [Yong-Ho Chin, LBL-29981 (1990); Saldin et al., NIMA 417 (1998) 158] We introduce Wake asWe introduce Wake as therefore, the wake Green function Our results are universal by expressing G with only one parameter, as long as K is large.Our results are universal by expressing G with only one parameter, as long as K is large. KEK-SLAC ISG9 Meeting

7. Numerical G-function Start over from retarded potential Wake has the correct dependences: Zeros and peaks at n FEL /2 and scaling as s -1/3 for small s and s -1 for large s KEK-SLAC ISG9 Meeting

8. Simple Physics Model (ala Derbenev) Dipole:Dipole: Wiggler:Wiggler: W || = 0, for s = n FEL ;W || = 0, for s = n FEL ; W || = max, for s = FEL /2;W || = max, for s = FEL /2; W || ~ s - 1/3, for s  0;W || ~ s - 1/3, for s  0; W || ~ s - 1, for s >> FEL.W || ~ s - 1, for s >> FEL. W || ~ s - 1/3 s -1/3 comes from retarded position KEK-SLAC ISG9 Meeting

9. Real Part of the Impedance Real part of the impedance with peaks at n FEL which continues to increase towards c. Check with the wiggler radiation spectrum [Alferov, et al., 1974; Krinsky, et al., 1983], perfect agreement. KEK-SLAC ISG9 Meeting

10. Imaginary Part of the Impedance KEK-SLAC ISG9 Meeting

11. Complete Numerical Results NLC damping ring Growth rateGrowth rate Wigglers Dipoles Dipoles and wigglers Why dips not peaks??? KEK-SLAC ISG9 Meeting

12. Inside Wiggler NLC TESLA Yeah, right peaks!!! KEK-SLAC ISG9 Meeting

13. Estimate at FEL frequency At FEL and harmonics, Z = R – i L, with L>>1, R>0, but R >1, R>0, but R<<L. Gaussian in energy spreadGaussian in energy spread Dispersion relation isDispersion relation is 3D FEL theory give instability growth length about 50 m for the NLC wiggler.3D FEL theory give instability growth length about 50 m for the NLC wiggler. KEK-SLAC ISG9 Meeting

14. ATF Damping Ring Threshold vs. wavelength We hope to get experimental verification!We hope to get experimental verification! KEK-SLAC ISG9 Meeting Pipe cut-off wavelength (Assuming 3 cm height ) Wiggler on Wiggler off Nominal current

15. Discussion The fact that the impedance of the dipole and that of the wiggler have different scaling at low-frequency, supports a wigger ring design together with other study results.The fact that the impedance of the dipole and that of the wiggler have different scaling at low-frequency, supports a wigger ring design together with other study results. For practical CSR purpose, (< 0.1 f FEL ), the wiggler impedance isFor practical CSR purpose, (< 0.1 f FEL ), the wiggler 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? KEK-SLAC ISG9 Meeting

16. References: G. Stupakov & S. Heifets, PRST-AB 5(2002) 054402;G. Stupakov & S. Heifets, PRST-AB 5(2002) 054402; J. Byrd, et al., PRL 89(2002) 224801;J. Byrd, et al., PRL 89(2002) 224801; G. Carr, et al., NIMA 463(2001)387; B. Podobedov, et al., PAC2001, P. 1921; S. Kramer, et al., EPAC2002, P. 1523;G. Carr, et al., NIMA 463(2001)387; B. Podobedov, et al., PAC2001, P. 1921; S. Kramer, et al., EPAC2002, P. 1523; M. Abo-Bakr, et al., PRL 88(2002) 254801;M. Abo-Bakr, et al., PRL 88(2002) 254801; Yong-Ho Chin, LBL-29981 (1990); E.L. Saldin, et al., NIMA 417(1998)158;Yong-Ho Chin, LBL-29981 (1990); E.L. Saldin, et al., NIMA 417(1998)158; J.B. Murphy, et al., PAC95, P. 2980; Part. Accel. 57(1997)9;J.B. Murphy, et al., PAC95, P. 2980; Part. Accel. 57(1997)9; J.B. Murphy, et al., NIMA 346(1994)571; J.B. Murphy, et al., PAC95, P. 284;J.B. Murphy, et al., NIMA 346(1994)571; J.B. Murphy, et al., PAC95, P. 284; Derbenev, et al., DESY-TESLA-FEL95-05; R.L. Warnock & P. Morton, Part. Accel. 25(1990)113; A. Faltens, L.J. Laslett, Part. Accel. 4(1973)151;BNL-20550(1975);Derbenev, et al., DESY-TESLA-FEL95-05; R.L. Warnock & P. Morton, Part. Accel. 25(1990)113; A. Faltens, L.J. Laslett, Part. Accel. 4(1973)151;BNL-20550(1975); D.F. Alferov, et al., Sov. Phys. Tech. Phys. 18(1974)1336; S. Krinsky, et al., Handbook on Syn. Rad. Vol. 1(1983) 65.D.F. Alferov, et al., Sov. Phys. Tech. Phys. 18(1974)1336; S. Krinsky, et al., Handbook on Syn. Rad. Vol. 1(1983) 65. KEK-SLAC ISG9 Meeting

17. Calculation details The rate of energy lossThe rate of energy loss where and  is determined by and

18. KEK-SLAC ISG9 Meeting Analytical check Estimate of the extreme pointsEstimate of the extreme points They are plotted as “+”s in the plot for G.

19. KEK-SLAC ISG9 Meeting Analytical check Short-range limit [perturbative approach]Short-range limit [perturbative approach] Simple approachSimple approach Dipoles Effective radius

20. KEK-SLAC ISG9 Meeting Undulator (X-ray FEL) Yeah, right peaks!!! LCLS TESLA XFEL

21. KEK-SLAC ISG9 Meeting How to treat wiggler (An estimate) Treat the wiggler as 2N pieces of DipolesTreat the wiggler as 2N pieces of Dipoles ImpedanceImpedance Threshold in NLC Damping Ring drops from 1.75×10 10 with dipoles only to 7.25 ×10 9.Threshold in NLC Damping Ring drops from 1.75×10 10 with dipoles only to 7.25 ×10 9. average It is serious!!! Recall the design value is 7.5 ×10 9 ; But is bending magnet approach for wiggler valid?

22. Acknowledgements Thank A.W. Chao, S.A. Heifets, Z. Huang of SLAC;Thank A.W. Chao, S.A. Heifets, Z. Huang of SLAC; Thank J.B. Murphy, J.-M. Wang, S. Krinsky, B. Podobedov, S. Kramer of NSLS / BNL.Thank J.B. Murphy, J.-M. Wang, S. Krinsky, B. Podobedov, S. Kramer of NSLS / BNL. Work supported by contract DE-AC03- 76SF00515 with US DOE.Work supported by contract DE-AC03- 76SF00515 with US DOE. KEK-SLAC ISG9 Meeting

23. An 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 wigglers,3-D TDA simulation gives, for NLC wigglers, but the wiggler is only about 50 m. Single pass isn’t dangerous, but how about multi-turn? Need numerical simulations with momentum compaction for ring. KEK-SLAC ISG9 Meeting

24. Dipoles and wigglers Dipoles Wigglers Why peaks not dips??? KEK-SLAC ISG9 Meeting Complete Numerical Results NLC damping ring threshold

25. Low-frequency (up to )Low-frequency (up to ) with the FEL wavenumber. High-frequency ( )High-frequency ( ) Analytical Asymptotic Expression Average over one period Effective dipole radius Real part of the impedance is linear in k; Imaginary part is k Log[k]. KEK-SLAC ISG9 Meeting

26. Length Scales Cooperative/formation length for a radiation wavelength fCooperative/formation length for a radiation wavelength f –Dipole: –Wiggler: Periodicity leads to interference Pipe cut-off wavelength: 1 mm;Pipe cut-off wavelength: 1 mm; Long-range wake applicable for:Long-range wake applicable for: For < FEL  “may” approximate as 2N dipoles ; but for longer, has to include interference ; need look at FEL frequency and harmonics KEK-SLAC ISG9 Meeting

27. Low-frequency Impedance Analytical Wiggler impedance:Wiggler impedance: Total impedance:Total impedance: Nominal (10 10 ) Threshold from full- impedance (10 10 ) Threshold from low- frequency (10 10 ) NLC0.751.31.3 TESLA2.078.8 We may use the low-frequency impedance for practical purpose! KEK-SLAC ISG9 Meeting