Power efficiency vs instability (or, emittance vs beam loading) Sergei Nagaitsev, Valeri Lebedev, and Alexey Burov Fermilab/UChicago Oct 18, 2017

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1 Power efficiency vs instability (or, emittance vs beam loading) Sergei Nagaitsev, Valeri Lebedev, and Alexey Burov Fermilab/UChicago Oct 18, 2017

2 Acknowledgements We would like to thank our UCLA colleagues for discussions and computer simulations and G. Stupakov for fruitful discussions. 2

3 Our motivation for this study came from: 3

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6 Is drive-to-beam 50% efficiency possible??? 6

7 Why is power efficiency important? Because power = cost 7

8 Acceleration in ILC cavities The ILC cavity: ~1 m long, 30 MeV energy gain; f 0 = 1.3 GHz, wave length 23 cm The ILC beam: 3.2 nc (2x10 10 ), 0.3 mm long (rms); bunches are spaced ~300 ns (90 m) apart Each bunch lowers the cavity gradient by ~15 kv/m (beam loading 0.05%); this voltage is restored by an external rf power source (Klystron) between bunches; (~0.5% CLIC) Such operation of a conventional cavity is only possible because the Q-factor is >> 1; the RF energy is mostly transferred to the beam NOT to cavity walls. 8

9 Acceleration in a blow-out regime The Q-factor is very low (~1) must accelerate the trailing bunch within the same bubble as the driver! Cannot add energy between bunches, thus a single bunch must absorb as much energy as possible from the wake field. 9 M. Tzoufras et al., PRL 101, (2008) To achieve L ~10 34, bunches should have ~10 10 particles (similar to ILC and CLIC). In principle, we can envision a scheme with fewer particles/bunch and a higher rep rate, but the beam loading still needs to be high for efficiency reasons.

10 Transverse beam break-up (head-tail instability) Transverse wakes act as deflecting force on bunch tail beam position jitter is exponentially amplified Short-range transverse wake W " z = 8z a ' a 35 mm (ILC) a 3.5 mm (CLIC) a ~ 0.1 mm (PWFA) 10

11 Case I: ~50% power efficiency Courtesy of UCLA 11

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13 Case II: ~25% power efficiency Courtesy of UCLA 13

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15 Beam breakup in various collider concepts ILC Not important; bunch rf phase is selected to compensate for long wake and to minimize the momentum spread CLIC Important; bunch rf phase is selected to introduce an energy chirp along the bunch for BNS damping (~0.5% rms). May need to be de-chirped after acceleration to meet final-focus energy acceptance requirements PWFA the subject of our study Critical; BNS damping requires a large energy chirp (see below). De-chirping and beam transport is very challenging because of plasma stages (small beta-function in plasma ~1 cm). In essence, requires a final-focus optics between every stage. 15

16 CLIC strategy: BNS damping + < µm alignment of cavities 16

17 Strategy was also used at the SLC 17

18 We start with the Lu plasma bubble equation We assume the driving bunch intense enough to produce an 1 electron-free plasma bubble with radius R. According b >> k - p to Lu et al. : 2 2 d rb ædrb ö 2 dnd rb d 2 ç + = d n 2 0 r b d x è x ø p x R b = L d N d πn 0 L d 3 8N d πn 0 L d E! = 2πn 0 er b dr b dξ R b N d π 3 3 L d n 0 1/8, N d n 0 L d 3 >> 1 Example: N = 10 ; n = 4 10 cm ; L= 25 µm d Rk» b p

19 Power transfer from drive to trailing bunches Following M. Tzoufras et al., PRL 101, (2008) Trapezoidal line density distribution à constant electric field p P= en E c= e n cr d d 0 b p en0 c ( 2 2 ær ) 2ö b Pt = ecntet = rt2 - rt1 ç + r 2 t1 4 èrt 2 ø h P Pt rt2 - r æ t1 Rb r ö t1 = = 2 ç P Rb èrt2 Rb ø 19

20 Example The power transfer efficiency of 50% and the transformer ratio of 2. For n 0 =10 17 cm -3 the drive bunch parameters are chosen to be R b k p =5, L d k p =2.5 yielding the decelerating field of E d = 50 GV/m and N d = The trailing bunch parameters are: r t2 =0.518R b, r t1 =0.373R b, E t = 100 GV/m, N t =

21 Instability of the trailing bunch The Beam Break-up (BBU) instability is characterized by the ratio of the wake deflection force to the focusing force. F r = 2πn 0 e 2 r Need to find W! x for the bubble regime. First, in a quasilinear regime, where σ^ is the rms size of plasma channel For a hollow channel 1 Focusing force x1 2 dnt 1 ^ 1 dx x -L Ft º F( x ) = e r ò W ( x, x) dx ^ ( ) t p 2 ^ è øe è min ø Defocusing force (varies along bunch) k ædn ö ær ö w max p W^ = 2 ç sin( kp( s- s )) ln ç, kp = s n r c D n! 1 n ( ( )) ( ) W» 2k sin k s- s ln 2, s» k ^ 3-1 p p ^ p 21

22 Wakes in the bubble regime Longitudinal (from the Lu equation): W = 4 r b 2 ; (Δz << r,k 1 ) b p (similar to a dielectric channel and periodic array of cavities) For reference, see: A. V. Fedotov, R. L. Gluckstern, and M. Venturini (PRST-AB (1999)) Transverse : 2 8Dz -1 W^» òw 2 dz = ; ( D z << r, ) 4 b k p rb rb -1 b( ) >> p -- local bubble radius at bunch location, r z k z (This is true for a dielectric channel, array of cavities and resistive wall) For reference, see also: Karl Bane, SLAC-PUB-9663 and S. S. Baturin and A. D. Kanareykin, PRL 113, (2014). Recent findings: r! ( z) r ( z) + k - 1 b b p to account for bubble wall thickness 22

23 Our estimate for the transverse wake 8x W ( xx, )»! qx (!),! x = x - x ^ 3 ( x) r ( x ) 2 2 rb b 2 ( x ) q ( x) is the Heaviside step function. We believe this estimate is on the low side. The actual wake is likely to be greater. Now, let s find the ratio of the defocusing (wake) force to the focusing force: L 4 4 F t t rt2 Lt -x é ær ö æ æ b R ö öù b ht =- = ò dx êr 3 t2ç r 4 - x ç ú t2 Fr rt1 r ( ) 0 b x ê r ç t2 r t2 ú ë è ø è è ø øû Recall that Pt rt2 - r æ t1 Rb r ö t1 hp = = 2 ç P Rb èrt2 Rb ø r b >> k -1 p 23

24 The efficiency-instability relation 2 hp r ht» 41-h R ( ) This formula does not include any details of beams and plasma, being amazingly universal! Note: this formula is an estimate from a low side. On a high 2 side, we estimate it as: h» h 2 / 4 1-h Example: à P t 2, 0.7 h = 25% à 0.021< < b ( ( ) ) t P P h = 50% < < 0.25 P P h t h t 24

25 Instability development 2 d X X 2h t + = 2 2 ( ) ( ) ò X( x ) x -x dx. dµ 1 +D p/ p 1 +Dp/ p Lt 0 x p - 1 X = ; b = kp 2g d = dz/ b p0 For and D p p= it was solved in: h! 1 / 0 t C. B. Schroeder, D. H. Whittum, and J. S. Wurtele, Multimode Analysis of the Hollow Plasma Channel Wakefield Accelerator, Phys. Rev. Lett. 82, n.6, 1999, pp Approximate solutions (it s a very good fit, <10% deviation): 2 A Ê ( µh ) ˆ µht 100 t = exp ; 1.57 A Á ( µh ) Ë + h t x µ b t A A 0 2 Ê 2 ( µh ) ˆ Á t µht 1.57 Á Ë ( µht ) ht = exp Á ;

26 Note that A is a normalized particle amplitude. For a constant plasma density and without instability A would stay constant, while the initial physical amplitude x should decrease as 1 4 1/g 26

27 Examples (FACET-II) Plasma: n 0 = cm, 60 cm long channel p i =10 GeV/c for both the drive and the trailing bunches, and the final momentum of trailing bunch p f =21 GeV/c, N d =1x10 10 and N t =4.3x10 9 A hp = 50%, h t» 0.12, µht» 11.5» 5.8 A If one reduces the power efficiency: A hp = 25%, h t» 0.021, µht» 2» 1.3 A Of course, the final momentum is now p f =15.5 GeV/c (for the same number of particles) 2 2 d x æ A ö de n = g i, 2 bi = 2b ç i A è 0 ø 2g k p i

28 BNS damping Assume a constant long. density trailing bunch. Chromatic detuning of tail particles allows to keep amplitudes constant x 1 2h t - ( x - x ) dx = 1 Dp ò 1+ æ Dp ö L t p ç p è ø Dp( x) p =- 2 x ht L We believe that the collider final focus optics and transitions between stages can not tolerate D >, so h 0.01 This limits the power transfer efficiency to < 18% 2 t p p 1% t 28

29 Conclusions We have found a universal efficiency-instability relation for plasma acceleration. Should allow for tolerance and instability analysis without detailed computer simulations. We considered only the ideal trapezoidal distributions. Real-life distributions are worse (from the efficiency perspective). In a blowout regime, plasma focusing is just strong enough to keep the instability in check for low power efficiencies (<25%) Even for such efficiencies, external focusing and hollow channels are not viable concepts because of transverse instability. Presents obvious difficulties for positrons BNS damping is possible but external optical systems limit the momentum spread to ~1% max. Thus, the power efficiency (drive to trailing) can not exceed ~18%. 29

30 Summary We wish FACET-II success and would like to be part of its science program. Our conclusions require confirmation by computer simulations and by experiments, especially in regimes not covered by the Lu equations (small bubble size). 30