K.V. Lotov- Fine wakefield structure in the blowout regime of plasma wakefield accelerators

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Fine wakefield structure in the blowout regime of plasma wakefield accelerators

K.V. Lotov

Budker Institute of Nuclear Physics, 630090, Novosibirsk, Russia(Received 19 December 2002; published 6 June 2003)

For simulations of plasma wakefield acceleration (PWFA) and similar problems, we developed two-dimensional fully electromagnetic fully relativistic hybrid code LCODE. The code is very fast due to

explicit use of several simplifying assumptions (quasistatic approximation, ultrarelativistic beam, andthe symmetry).With LCODE, we make high-resolution simulations of the blowout regime of PWFA andstudy the temperature effect on the amplitude of the accelerating field spike.

DOI: 10.1103/PhysRevSTAB.6.061301 PACS numbers: 52.40.Mj, 52.35.Mw, 52.65.Ww

I. INTRODUCTION

Acceleration and focusing of particles with plasmas isof a great interest because the field strength in the plasmacan be made several orders of magnitude higher than thatin conventional accelerating structures or focusing lenses[14]. The accelerating field in the plasma can be excitedeither by a powerful laser pulse (or several pulses) [57]

or by a charged par ticle beam (usually elect ron beam) [8].The latter variant is usually termed PWFA (plasma wake-field accelerator). Here we consider the wave excitation byelectron beams.

To compete with conventional accelerators, the plasmaaccelerator must have high efficiency, high final energy,and a good quality of the accelerated beam. Various con-figurations of the drive beam were proposed in order tomeet these requirements [812]. One option is to use asingle high-density drive bunch (so-called blowout re-gime) [12]. In this regime, the head of the driver ejectsall of t he plasma electrons from its propagation channel,and the most part of the beam propagates in the electron-

free region. The blowout regime was extensively studiedboth theoretically [1218] and experimentally [4,1924].It has a number of good features. The linear focusingforce is good for preservation of the accelerated beamemittance. The accelerating field, which does not dependon radius inside the ion channel, helps to minimize theenergy spread. The maximum value of the acceleratingfield is much greater than the decelerating field within thedriver, which gives hope to get an energy gain far inexcess of the driver particle energy.

In this paper we study the blowout regime of PWFA.With the two-dimensional code LCODE [25], we calculatethe wakefield structure near the point of the maximumaccelerating field and separate numerical and physicaleffects. We show that the enormously high spike of theaccelerating field is an artifact of adopted models andcannot be used for acceleration of a reasonably largenumber of particles.

The paper is organized as follows. In Sec. II we de-scribe the code and compare it with other codes [15,17,18]by simulating the SLAC E-157 experiment [4,23]. InSec. III we present high-resolution simulations of the first

wakefield period and discuss the temperature effect on theamplitude of the accelerating field spike.

II. THE CODE

Two-dimensional fully electromagnetic fully relativis-tic code LCODE [25] supports both Cartesian and cylin-drical geometries. For definiteness, we write out all

equations for the cylindrical case.We use the cylindric coordinates r;;z and the co-moving simulation window (Fig. 1). Since the lengt h scaleof beam evolution is typically much longer than t hebunch length, we work in the so-called quasistatic ap-proximation [26,27]. Namely, when calculating theplasma response we consider the beam as rigid andfind the fields as functions ofr and z ct, where c isthe speed of light. Then we use these fields to modify thebeam, etc.

The beam is modeled by macroparticles. The plasmacan be modeled either as a relativistic electron fluid or bymacropar ticles. The beam model and electron fluid model

are described in detail in [27]. Briefly, each beam macro-particle is characterized by r and coordinates (rb andb), r and z momenta (pbr and pbz), and angular momen-tum (Mb). They are changed every time step:

drbdt

vbr;dbdt

vbz c;dMb

dt 0; (1)

dpbrdt

qbEr B;dpbz

dt qbEz; (2)

~vv b c~ppb

m2bc2 p2br p2bz M2b=r2b

q ; (3)

FIG. 1. (Color) Geometry of the problem.

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where charges qb and masses mb are equal for all macro-particles. The time step t for beam par ticles can be madelarge enough for highly relativistic beams (Fig. 1) tospeed up simulations.

The macroparticle model of the plasma was speciallydeveloped for simulations of the nonlinear PWFA andsimilar problems. T his is a Lagrangian model ratherthan a conventional particle-in-cell model. Namely,

each plasma macropar ticle is characterized by six quan-tities: transverse coordinate (r), th ree components of t hemomentum (pr, p, and pz), mass M, and charge q. Thelongitudinal coordinate () is not a parameter, but anargument. Thus, a plasma macroparticle presents a par-ticle tube, i.e., a group of real particles started from agiven radius (Fig. 2). Since, in the quasistatic approxima-tion, the beam is unchanged while we calculate theplasma response, all particles started from r0 copy themotion of each other, and their parameters (r, pr, p, andpz) can be found as functions of.

Parameters of plasma macroparticles are initializedahead of the beam (at

0) and then calculated slice

by slice:

d~pp

d d~pp

dt

dt

d q

vz c

~EE 1c~vv ~BB

; (4)

dr

d vr

vz c; ~vv ~pp

M2 p2=c2p : (5)

Plasma current and charge density are obtained by sum-mation over plasma macroparticles lying within a givenradial i nterval:

~jj AXi

qi~vvic vz;i

; AXi

qic vz;i

; (6)

where A is a normalization factor. The denominator in (6)appears since the contribution of a particle tube to thedensity and current depends on the macroparticle speedin the simulation window.

Knowing ~jj and at some , we obtain the fields in thislayer from Maxwell equations which, under the assump-tion

@=@ @=@z @=@ct; (7)

take the form

1

r

@

@rrEr 4 b

@Ez@

; (8)

1

r

@

@rrBr

@Bz@

; (9)

@

Er

B

@ @Ez

@r 4

cj

r; (10)

@Bz@r

4c

j; E Br: (11)

Here b is the beam charge density, and we neglect thecomponents jbr and jb of t he beam current.

To provide stability of the algorithm, we solve in finitedifferences, instead of (8) and (9), the following equa-tions:

@

@r

1

r

@

@rrEr

!2p

c2Er 4

@ b

@r 1

c

@jr@

!

2p

c2~EEr;

(12)

@

@r

1

r

@

@rrBr

!2p

c2Br

4

c

@j@

!2p

c2~BBr; (13)

where ~EEr and ~BBr are the fields at the previous layer, and

!p 4n0e

2=mp

is the unperturbed plasma frequency.These equations are obtained by differentiation of (8) and(9), and substitution of (10) and (11) into the result.Addition of the fields (with or without the t ildes) toboth sides of the equalities does not change the equa-tions and has the effect only when we proceed to final

differences.The boundary conditions for Eqs. (10)(13) are that ofa perfectly conducting tube of the radius rmax:

Er0 Br0 B0 0; Ezrmax 0; (14)

Zrmax0

2rBz dr r2maxB0; Brrmax 0; (15)

where B0 is an external longitudinal magnetic field, if any(the presence of this field does not change t he symmetryof the system).

The plasma response is calculated layer by layer to-wards the decreasing [from rig ht to lef t in (Fig. 1)]. Weuse (4)(6) to predict ~jj and at the next layer, calculatethe fields at this layer, move plasma macroparticles usingthe average fields, then correct ~jj, , and fields at the newlayer. The algorithm allows easy shortening of the stepin the regions of a fine field structure. The shortening ismade automatically when the current density jjzj exceedssome th reshold value. The shift of beam part icles accord-ing to Eqs. (1)(3) is made in parallel with the calculationof the plasma response, so t hat every beam time step

FIG. 2. (Color) Trajectory of a plasma macropar ticle in thesimulation window.

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requires only one computing passage of the simulationwindow.

The code allows an arbitrary initial distribution ofbeam particles over the simulation window, but we usu-ally take the initial beam density of the form

nb nb0

2er2=22r

1 cos

2

L

; L < < 0:

(16)

The cosine distribution over is more convenient thanthe Gaussian one because it smoothly vanishes outside aninterval of a finite length. For

L 22

pz; (17)

the distribution (16) is close to the Gaussian one

nb nb0 exp r

2

22r L=2

2

22z

(18)

(Fig. 3) and contains t he same number of particles.Because of the explicit use of the simplifying assump-

tions (quasistatic approximation, ultrarelativistic beam,and the symmetry), the code is very fast. To demonstratethis, we si mulate the SLAC E-157 experiment [4,23]. Wetake the beam a nd plasma para meters close to those usedin earlier simulations [15,17,18] (Table I). To simulate theaccelerated particles correctly, we take the half-Gaussiantrailing edge of the beam, while the front edge is the half-cosine. Similarly to [15,17], we use a square grid of thesize 0:05c=!p, nine macroparticles per cell for theplasma, and 25 macroparticles for the beam (near theaxis). The time step for the beam corresponds to 1.4 cm.

The simulation results are shown in Fig. 4. They are ingood agreement wit h [15,17]. Even the nar row field spike

just behind the maximum accelerating field observed inPEGASUS and OSIRIS simulations [15] is reproduced.The only drastic difference is that the full beam dynamicsrun [Fig. 4(b)] takes about 5 min on a desktop Pentium-IIrather than days on a multiprocessor Cray [15].

III. FINE STRUCTURE OF THE WAKEFIELDSimulations of the blowout PWFA with various codes

are in perfect agreement everywhere except the region ofthe sharp accelerating field spike just behind the beam.The dependence of the spike amplitude on the grid size[15] and a noisy electric field after the spike suggest thatthe resolution of earlier runs is not sufficient for correctdescription of this region. At the same time, it is thisspike that promised ultrahigh accelerating gradients a ndgreatly contributed to attractiveness of the blowout re-gime. With this motivation, we make high-resolutionsimulations of the near-spike region.

We take the same beam parameters (Table I), but amuch denser gr id. T he simulation window, in units ofc=!p, is 20 (in ) 5 (in r). Both r and steps are0:0025c=!p, and near the spike the step is automati-cally decreased down to 2:5 105c=!p. The beam ismodeled by 3:4 105 macropar ticles, and 4 104plasma macroparticles are used (20 macroparticles percell).

The simulation results are shown in Fig. 5. The graphsfor zero plasma temperature (Te 0) are shown in the left

FIG. 3. (Color) Cosine and Gaussian distributions.

TABLE I. Beam and plasma parameters.

Number of beam particles, N 3:7 1010Beam radius, r 70 mBeam half-length, z 0.63 mmBeam energy, Wb 30 GeVPlasma density, n0 2:1 1014 cm3Plasma length, L 1.4 m

FIG. 4. (Color) (a) Longitudinal electric field, (b) energy andnumber of beam particles after 1.4 m plasma for the low-resolution simulation of the E-157 experiment.

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FIG. 5. (Color) Plasma and field behavior in the vicinity of the highest accelerating field: (a) on-axis values of Ez for different runsand t he location of the magnification window; (b),(c) on-axis values of Ez in the region of the field spike; (d),(e) plasma electrondensity (multiplied by r for better visibility); (f ),(g) longitudinal electric field; a nd (h),(i) focusing force with zero (b),(d),(f),(h)and nonzero (c),(e),(g),(i) plasma temperatures.


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column; the right column shows Te 0:1 eV accordingto E-157 data published in [23]. The location of the de-tailed window is indicated in Fig. 5(a). The wakefieldcalculated in Sec. II with a coarse grid is also shown inFig. 5(a) for comparison.

It is seen from Fig. 5(a) that the initial plasma tem-perature has a visible effect on the wakefield only in theregion of the field spike. The spike is suppressed by the

temperature [Figs. 5(b) and 5(c)]. The reason why thishappens is explained by maps of the plasma electrondensity [Figs. 5(d) and 5(e)], where the product nr isshown; the factor ofr compensates the geometry-inducedsingular ity of the density. It is seen that the field growth isstopped when some plasma electrons arrive at the axis.For the initially cold plasma, the boundary of the blowoutregion is sharp. T his boundary, when approaching theaxis, produces a high current density

jr envr / 1=r (19)(because of the singular density growth at a nearly con-stant radial speed) a nd, according to (10), the singularityof Ez. For the warm plasma, the edge of the blowoutregion is diffuse, currents of inward and outward movingelectrons almost cancel each other, and no field singular-ity appears.

In the transverse direction, the mai n field spike is alsovery narrow [Figs. 5(f) and 5(g), notice the scale]. In theregions of a nonzero plasma electron density, a strongdefocusing force acts on t he beam [Figs. 5(h) a nd 5(i)]. Asa result, the region favorable for particle acceleration isvery narrow in the vicinity of the spike (the triangulararea of positive values in Figs. 5(h) a nd 5(i)], and we havea trade-off between the width of the favorable cross

section and the accelerating field in it. Note that behindthe spike the negative focusing force is almost constant upto very small radii, which means huge defocusing gra-dients near the axis.

The above picture of spike formation suggests a simpleanalytical dependence of the spike amplitude on theplasma temperature. Assuming the scaling (19) is true,we can find, for zero plasma temperature, the on-axis fieldat the leading edge of the spike:

Ez 4

c

Z10

jrdr 4e

c

Zr1r0

n1vrr1r

dr

A1 A2 lnr0;A2 > 0;

(20)

where r0 is the radius of the electron-free region[Fig. 5(d)], r1 is a typical radius of the perturbed plasma,n1 is the electron density there, and Ai are some con-stants. For straight electron trajectories [which is true inour case, as follows from Fig. 5(d)], we have

r0 vr0

c vz0s ; (21)

where vr0 < 0 and vz0 are electron velocities near thespike, and s is the coordinate of the field extremum.Whence

Ez A3 A2 ln s; (22)and we observe a logarithmic singularity of the longitu-dinal electric field.

When the plasma has a nonzero initial temperature,

electrons slightly deflect from their zero-temperature tra-jectories, and some of them ar rive at the axis ea rlier thanin the zero-temperature case. Typical deviation of anelectron at the distance jsj from the beam head is

r0 jsj

c

Tem

r(23)

[Fig. 5(e)]. At r0 r0, the logarithmic increase of jEzjstops, and we observe the maximum field

Ez;max A1 A2 lnr0 A4 A5 lnTe; A5 > 0:(24)

Coefficients Ai here depend on the driver shape, current,and radius.

This nonrigorously obtained scaling is surprisinglywell confirmed by simulations (Fig. 6). The thin line inFig. 6 is determined by the formula

Ez;maxGV=m 0:98 0:14 logTeeV: (25)At high plasma temperatures, we observe a systematicdeviation of the field amplitude from this scaling; sincethe ratio r1=r0 is not very high there, wide plasmaregions contribute to the i ntegrals (20), a nd the logarith-mic precision of our estimates becomes worse. At lowtemperatures, r

0

approaches the gr id step, and numeri-cal noises come into play.

As follows from the above, the main field spike can beused for acceleration only in idealized conditions of aperfectly symmetric and cold plasma and a very narrowbeam. In real experiments the conditions are rather farfrom the ideal ones. The plasma could have transversedensity gradients, and t he beam is not perfectly cylindri-cal a nd straight. These effects cannot be analyzed with atwo-dimensional code, but we can expect that account of

FIG. 6. (Color) The temperature dependence of the spike am-plitude: simulation (crosses) and approximation (line).

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three-dimensional effects wil l make t he situation worse,no additional length scale or field scale will appear, andthe maximum accelerating field behind the beam will notbe an order of magnitude higher than the deceleratingfield within the beam.

ACKNOWLEDGMENTS

This work was supported by a Science support founda-

tion grant, an SB RAS grant for young researchers, andthe Russian Foundation for Basic Research, GrantsNo. 00-15-96815 and No. 03-02-16160a.

[1] E. Esarey, P. Sprangle, J. Kral l, and A. Ting, IEEE Trans.Plasma Sci. 24, 252 (1996).

[2] W. P. Leemans et al., Phys. Plasmas 5, 1615 (1998).[3] D. Umstadter, Phys. Plasmas 8, 1774 (2001).[4] C. Joshi et al., Phys. Plasmas 9, 1845 (2002).[5] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267

(1979).[6] P. Sprangle, E. Esarey, J. Kral l, a nd G. Joyce, Phys. Rev.

Lett. 69, 2200 (1992).[7] N. E. A ndreev et al., JETP Lett. 55, 571 (1992).[8] P. Chen, J. M. Dawson, R.W. Huff, and T. Katsouleas,

Phys. Rev. Lett. 54, 693 (1985); 55, 1537(E) (1985).[9] P. Chen et al., Phys. Rev. Lett. 56, 1252 (1986).

[10] R. D. Ruth , A.W. Chao, P. L. Morton, and P. B. Wilson,Part. Accel. 17, 171 (1985).

[11] K.V. L otov, Nucl. Inst rum. Methods Phys. Res., Sect. A410, 461 (1998).

[12] J. B. Rosenzweig, B. Breizman, T. Katsouleas, and J. J.Su, Phys. Rev. A 44, 6189 (1991).

[13] N. Barov and J. B. Rosenzweig, Phys. Rev. E 49, 4407(1994).

[14] J. K rall and G. Joyce, Phys. Plasmas 2, 1326 (1995).

[15] S. Lee, T. Katsouleas, R. Hemkel, and W. Mori, Phys.Rev. E 61, 7014 (2000).

[16] E. S. Dodd et al., Phys. Rev. Lett. 88, 125001 (2002).[17] R. Hemkel, W. Mori, S. Lee, and T. Katsouleas, Phys.

Rev. ST Accel. Beams 3, 061301 (2000).[18] D. L. Bruhwiler et al., Phys. Rev. ST Accel. Beams 4,

101302 (2001).[19] J. B. Rozenzweig et al., Phys. Rev. Lett. 61, 98 (1988).[20] J. B. Rozenzweig et al., Phys. Rev. A 39, 1586 (1989).[21] N. Barov, M. E. Conde, W. Gai, and J. B. Rosenzweig,

Phys. Rev. Lett. 80, 81 (1998).[22] N. Barov et al., Phys. Rev. ST Accel. Beams 3, 011301

(2000).[23] M. J. Hogan et al., Phys. Plasmas 7, 2241 (2000).

[24] C. E. Clayton et al., Phys. Rev. Lett. 88, 154801 (2002).[25] LCODE is available free at www.inp.nsk.su/~lotov/lcode.[26] P. Sprangle, E. Esarey, and A. Ting, Phys. Rev. Lett. 64,

2011 (1990).[27] K.V. Lotov, Phys. Plasmas 5, 785 (1998).


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K.V. Lotov- Fine wakefield structure in the blowout regime of plasma wakefield accelerators