Calculation of wakefields for plasma-wakefield accelerators

Transcription

1 1 Calculation of wakefields for plasma-wakefield accelerators G. Stupakov, SLAC ICFA mini-workshop on Impedances and Beam Instabilities in Particle Accelerators September 2017

2 2 Introduction to PWFA Plasma wake excited by relativistic particle bunch Blow-out regime when nb /ne > 1 Acceleration and focusing by plasma Accelerating field scales as ne1/2 Typical: ne 1017 cm-3, kp-1 = 17 µm, E & 10 GV/m, G & MT/m C. Joshi, W.B. Mori, (2006) 1 m, 20 MV/m 100 µm, 20 GV/m

3 3 Hosing instability in PWFA Courtesy of Weiming An from UCLA.

4 4 Plasma wakefields The terminology of wakefields in plasma can be confusing. The original meaning of the wake in plasma is the field generated by the driver that accelerates the witness beam. The driver is a beam of charged particles (PWFA) or a laser beam (LWFA). In this presentation, by wakefields I mean the fields (longitudinal and transverse) with which the witness bunch acts on itself. They are generated by the leading charges and act on the trailing charges of the witness bunch. In linear approximation, validforn b n p, one can assume that the perturbation of the plasma density is small, n e n e.thewakefield problem can be solved analytically for arbitrary charge distribution of the driver and witness bunches 1. This approach, unfortunately, does not work in the blowout regime. 1 T. Katsouleas et al., Particle Accelerators, 22, 81(1987).

5 5 Wakefields in the blowout regime z r b In the absence of theory some researchers 2 use for the short-range wakefields formulas that work for a round pipe with resistive wall, corrugated pipe, dielectric pipe, etc. They replace the pipe radius a in these formulas by the bubble radius r b at the location of the source charge, w`(z) = 4 r 2 b h(z) w t (z) = 8z r 4 b h(z) h(z) is the step function (in SI system of units multiply by Z 0 c/4 ). Our goal is to calculate the wakes by solving Maxwell equations with correct plasma responce. 2 V. Lebedev, A. Burov, S. Nagaitsev, arxiv: (2017).

6 6 Relativistic point charge moving in free space = In wakefield theory for relativistic beams we assume v = c. When a point charge q is moving in vacuum, its field is E r = B = 2q r (z - ct) What happens if the point charge is moving in uniform, cold plasma of density n 0?

7 7 Point charge moving through plasma The remarkable result of Ref. 3 is the existence of the electromagnetic shock wave (EMSW) E r = B = 2qk p K 1 (k p r) (z - ct) = where k p =! p /c = p 4 n 0 e 2 /mc 2 and K 1 is the modified Bessel function. For r kp -1 we recover E r, B 2q (z - ct)/r; forr kp -1 the field decays exponentially, E r, B / e -kpr / p k p r.remarkably,thefieldsin EMSW are linear functions of charge. The only external dimensionless parameter in the problem is = q e r ek p = N d r e k p q p n 0 For n 0 = cm -3, q = 1 nc we have k -1 p = 53 µm, = N. Barov et al., PRAB 7, (2004).

8 Plasma equations This is the system of equations (in dimensionless units) that governs the plasma dynamics in axisymmetric geometry. We assume a steady state with everything depending on = t - z and r = p x 2 + y 2.Introduce = - A z, Eq. for E z @r E r = -@ r = n e (1 - v z ) - 1! kp -1 E! Emc! p /e Eq. for B rb n ev n ev z Eqs. of motion for plasma electrons dp r d = 1 + The continuity r - B, dr d = p r 1 [n e (1 - v z rn ev r = v z = 1 (1 + ) 8 Remarkably, for a given plasma flow, n e, v r and v z, the fields are found through an integration over r in each slice.

9 9 Numerical solution of PWFA equations We 6 developed a matlab code that solves an axisymmetric plasma bubble generated by a Gaussian driver and witness bunches. Illustrations: the driver with z = 13 µm, r = 5 µm, plasma density cm -3 (k -1 p = 26 µm). = = - - ξ - ξ Plots of the longitudinal electric field. One unit of electric field is 19.2 GV/m. 6 G. Stupakov, P. Baxevanis, V. Khudik, to be published.

10 10 Longitudinal wake in the bubble = E z (0, ) - = = - ξ I developed theory that calculates a jump in E z immediately behind the witness charge, E z (r, ). Remarkably,thetheorypredictsthatthis jump is proportional to the (dimensionless) witness charge w (the charge has not to be small). So we can introduce the longitudinal wake is w` = E z (0, )/ w.

11 11 Calculation of the longitudinal wake First, one needs to calculate the strength of the EMSW, D(r, ), atthe location of the witness charge: E r (r, ) =D(r, 0 ) ( - 0 ) (here 0 is the position of the source charge in the bubble). It satisfies the rd = n e0(r, ) 0(r, ) D Here n e0 and charge. Then 0 are the quantities in the bubble without rd E z = - 1 r This result can be benchmarked against the wakefields in a hollow plasma channel.

12 12 Wakefields in a hollow plasma channel Wakefields for a hollow plasma channel were calculated in 7 in linear approximation (small charge limit). Longitudinal wake w`(z) =2apple cos c z apple = 2 a 2 K 0 (ak p ) K 2 (ak p ) In my analysis I use n e0 (r) =n 0 h(r - a) and 0 (r) =1 and obtain w`(0) = 4 a 2 K 0 (ak p ) K 2 (ak p ) The wake w`(0) is valid not only in the linear, but in nonlinear regime as well. 7 C. Schroeder, D. Whittum, J. Wurtele. PRL, 82, 1177(1999).

13 13 Longitudinal wake as a function of (//) =/ ξ (μ) This wake is in good agreement with the simulated jump in witness charge on the axis of the bubble. E z of a

14 14 Transverse wake in the bubble 1 The source charge is now o axis, the o set is assumed small. The shock wave is not axisymmetric, E r / ^D(r, ) cos, E / ^D(r, ) sin. Behindthewave E z (r,, ) = ^E z (r, ) cos. The fields satisfy the following rr ^D + 1 r ^D - 4 ^D r 2 = n e0(r, ) 0(r, ) ^D ^E z = -@ r ^D - 2 ^D r The transverse wake is w t is found from the Panofsky-Wenzel relation and it is a linear function of the distance between the source and the witness, w t = wt 0 ( 1 - ). Our result agrees with the linear approximation of the transverse wake in a plasma channel calculated by Schroeder et al. w 0 t = 8 a 4 K 1 (ak p ) K 3 (ak p )

15 15 Transverse wake as a function of / (/// ) =/ ξ (μ)

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18 16 BBU instability of the witness bunch With the model for the wakefields in the plasma bubble, we apply them to the beam-breakup instability of the witness bunch. z X (s, z) X (s, z) is the transverse o set of the slice, z is the coordinate in the bunch, s is the distance along the @s + (s)k2 (s) X (s, z) =N b r e Z 1 f w (z 0 )w t (z 0 - z)x (s, z 0 )dz 0 Here (s) is the energy increase with distance due to acceleration, k (s) is the focusing, f w is the longitudinal distribution in the bunch. p Assume (s) = 0 + gs, k (s) =k 0 0/ (s). Ifthefocusingisduetoplasma ions, then k 0 = k p / p 2 0. We can solve the BBU equation numerically for an arbitrary distribution function.

19 17 Numerical solution for a Gaussian bunch Parameters of Weiming An simulations: the driver has z = µm, r = 3.65 µm, Q = 1.6 nc, (I peak = 15 ka); the witness has z = 6.38 µm, r = 3.65 µm, Q = 0.69 nc, (I peak = 13 ka). Plasma density cm -3. The distance between the bunches is a) 108 µm and b) 150 µm. 150 µm 150 µm (/) µm / (/// ) 108 µm - ξ (μ) ξ (μ)

20 18 Numerical solution for a Gaussian witness bunch 108 µm 150 µm One way to characterize BBU is to calculate the projected emittance: 2proj (s) = h(x - X )2 ih(x 0 - X 0 )2 i - h(x - X )(X 0 - X 0 )i where the averaging means h...i = Z dz(...)fw (z)

21 19 BBU instability projected emittance 108 µm 150 µm ϵ/ ϵ/ () () For a particular application this result can be translated into the jitter tolerance for the witness bunch.

22 20 Summary A method is developed to calculate longitudinal and transverse short-range wakes in the PWFA blowout regime. The calculation requires the knowledge of the energy-density radial distribution in the bubble, which can be taken from 2D simulations of PWFA. We developed a matlab code that solves axisymmetric plasma bubble excited by a driver with arbitrary longitudinal current distribution (run a few minutes on a desktop computer). The calculated transverse wakefield is then used for the study of BBU instability. The strength of the instability critically depends on the position of the witness bunch in the bubble.

23 21 Acknowledgments I thank X. Xu for benchmarking numerical calculations of the matlab code and P. Baxevanis for the code development.