LCLS Multi-bunch Wake Effects

1. LCLS Multi-bunch Wake Effects K. Bane 16 February 2011 Thanks to Gennady, Yuantao, Franz-Josef

2. Outline • Introduction • Two bunches • -- Longitudinal wake effect—fundamental mode component • -- Longitudinal—estimate of the effect of higher modes • -- Transverse—two modes • Multiple bunches, transverse • Conclusion

3. Introduction • Study multi-bunch wake effects in linacs of LCLS-II • Focus on two bunches • Longitudinal wake will change energy and chirp at tail bunch • Transverse wake will increase sensitivity to injection jitter; static effect due to misalignments is not important • Assume wake does not affect internal structure of trailing bunch • eN= 250 pC; tmax= 100 ns Schematic of two-bunch operation Und. 1 Und. 2 Accelerator

4. 1 nC Machine and beam parameters in the three linacs (250 pC) (Yuantao)

5. Longitudinal • SLAC linac is disk-loaded travelling wave structure; operating frequency is f0= 2.856 GHz; 2π/3 phase advance per cell; average: iris radius a= 1.2 cm, cavity radius b= 4 cm, iris thickness t= 0.6 cm, period p= 3.5 cm • 84 cells in 3 m structure; constant : a becomes smaller as one moves down the structure; b is adjusted to keep f0 unchanged (=> transverse mode is detuned) • Longitudinal wake wz(t)= 2zn cos(2πfnt) exp(-πfnt/Qn) • Fundamental mode dominates, others will detune • Q~ 10,000 => 1/e time ~ 1s => we can ignore the damping • For SLAC linac z= 20 V/pC/m; since all bunches are on crest we get a negative (linear) bunch-to-bunch energy variation.

6. energy gain Im V eVrf= (Ef  E0)/cos(), Vw= 2eNLz, tan(+)= Vrfsin()/[Vrfcos() Vw] = tan()/[1  eVw/(Ef  E0)] or |Vtot/Vrf| -cos2() eVw/(Ef  E0)  sin()cos() eVw/(Ef  E0) Re V   Vrf Vtot Vwake Phasor diagram of voltage at second bunch Longitudinal wake effect at second bunch due to first (fundamental) mode • In LCLS jitter: Vrf~ .035%--.1%, ~ .05 deg --Franz-Josef • For M bunches, effect at Mth bunch is factor (M 1) stronger

7. Longitudinal HOMs • Longitudinal HOMs can change the loss • Accurate calculation, with coupling from cell-to-cell, down the beam pipe, and into the couplers is difficult • To get a pessimistic estimate, assume no damping. Begin with the first 9 HOMs only. Note TM cut-off is 12.1 GHz Largest longitudinal modes in SLAC linac

8. Longitudinal HOMs (cont’d) Wake of fundamental and HOMs 1-9. Red dots show value at intervals of rf Wake of HOMs 1-9, compared to the fundamental, at intervals of rf • This suggests that the inaccuracy of ignoring HOMs <~ 20%

9. Longitudinal HOMs (cont’d) • Time domain calculations: I. Zagorodnov’s program Echo (2D) • Begin with a 10-cell model of the SLAC structure; Gaussian driving bunch, • z= 2 mm; calculate wake to 16 m behind driving bunch • Cylindrically symmetric => no couplers, also no wall losses => pessimistic Note TM cut-off is 12.1 GHz Fourier transform of wake: agrees with frequency domain results

10. Longitudinal HOMs (cont’d) Wake as function of distance behind driving bunch, as obtained by time-domain calculation. Red dots give values spaced by rf behind driving bunch. • Red dots have an rms value of ~30% =>  30% variation of wake at bunch positions

11. Transverse Wake Effects • Main dipole mode is at f= 4.2 GHz with kick factor = 1.2 V/pC/mm/m • To increase detuning of this mode, 1/3rd of structures have been dimpled by +2MHz, and 1/3rd by +4MHz • Have calculated long-range wake of first two dipole bands using a double-band circuit model (see K. Bane, R. Gluckstern, SLAC-PUB-5783, 1992). Model used is disk-loaded structure with non-rounded irises. • Freq [GHz] Blue Book: Calc Meas • 4.140 4.1394 4.13964 • 4.149 4.1478 4.14750 • 4.156 4.1545 4.15400 • 4.162 4.1605 4.15972 • 4.168 4.1657 4.16482 • Blue Book: total in a structure, (R/Q)y= 460 Ω (calc), 400 Ω (meas) • y= (/4)(R/Q)y*(/c)/L, with L= 84*.035 • y: .082 (here), .077 (Blue Book measured), .088 (BB calculated)

12. Coupled Dipole Modes of First Two Bands

13. Long Range Dipole Wakefields (Blue nom, red +2MHz, yel + 4MHz) • Wake envelope damps because of frequency • spread: 1/(200 MHz)= 5 ns • Note that peak of red dots is given by the beating of two frequencies: • f1n0/f0= (n1 +/ 1/4) with • f0= 2.856 GHz, f1= 4.15 GHz, n0= 6, n1= 9 • (8.72  8.75) Red dots give w(h0)

14. Beam Break-Up (BBU) Calculation • Equation of motion is given by: • Chao et al have solved this equation for the single bunch instability using a perturbation approach (see Chao’s book); the strength of the effect is characterized by the strength parameter, ; for small  the first term suffices • For multi-bunch problem the same approach can be used (see e.g. K. Bane, Z. Li, 2000). Again, for small  the first term suffices • with the sum wake Sm= wj

15. Quantifying Effect of Wake y’ y’ r2 r1 y y Initial phase space Final phase space Wake sensitivity parameter: = ||r2-r1||/||r1||

16. Linac 2 Upsilon of second bunch, as function of harmonic number h. Note that rf wavelength is 10.5 cm.

17. Solve equation of motion for two bunches (smooth focusing), for different bunch spacings. • Initially offset both bunches by same amount • If acceptable offset of beam from design orbit is say 0.1y (for FEL considerations), then results mean that in worst situation (h 6), second bunch will be offset at end of Linac 2, with respect to the first bunch, by 0.095 y as function of harmonic number h. Note that rf wavelength is 10.5 cm.

18. 2nd bunch for different spacings (blue dots) first bunch (red) orbit of first bunch Final phase space of 2nd bunch • In linear regime, wake kick is in quadrature to oscillation: if y0~ cos(ks), then yw~ s sin(ks) • Injection jitter tolerance reduced by factor (1+2)1/2

19. Linac3 Linac 1

20. Transverse Multibunch • Consider M equally spaced bunches • Wake sensitivity parameter: = Max(||rm-r1||)/||r1|| 10 8 6 h= 2 4 4 6 h= 2 8 10 (Left) Sum wake as function of number of bunches for different values of bunch spacing (Top right) max (Bottom right)  • Simulations gives much larger  than before • Although max and  correlate, they are no longer equal

21. Transverse Multibunch (Cont’d) h  M 3D and contour plots of  obtained from simulations. Note, there is only data at integer values of M and h

22. Conclusion In multi-bunch operation the longitudinal wake will change final energy and chirp of trailing bunches; transverse wake will change their final position in transverse phase space, due to injection jitter (or structure misalignments) In two-bunch operation, with 250 pC per bunch, longitudinal and transverse wake effects are relatively mild Transverse effect is greatly reduced by keeping the two bunches >= 30 buckets (10 ns) apart Higher longitudinal modes will change the longitudinal effect by <~ 30% (rms); the exact values at all bunch spacings can in principle be found by 3D time-domain calculations (e.g. Kwok Ko’s group) With two bunches the parameter ~  gives the machine, beam parameter scaling With many bunches longitudinal effect increases at bunch M by factor (M-1); transverse effect becomes much larger, and , although it still correlates with  is no longer equal to it Higher mode transverse effect can be calculated: it should be small