Principles of self-modulated proton driven plasma wake field acceleration

Principles of self-modulated proton driven plasma wake field accelerationAlexander Pukhov, Tobias Tuckmantel, N. Kumar, A. Upadhyay, K. Lotov et al. Citation: AIP Conf. Proc. 1507, 103 (2012); doi: 10.1063/1.4773682 View online: http://dx.doi.org/10.1063/1.4773682 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1507&Issue=1 Published by the American Institute of Physics. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

Principles of Self-Modulated Proton Driven Plasma Wake Field Acceleration

Alexander Pukhova, Tobias Tuckmantela, N. Kumara, A. Upadhyaya, K.Lotovb,c, V. Khudikd, C. Siemond, G. Shvetsd, P. Mugglie, and A. Caldwelle

aInstitute for Theoretical Physics I, University of Dusseldorf, 40225 Germany bBudker Institute of Nuclear Physics, 630090 Novosibirsk, Russia

cNovosibirsk State University, 630090 Novosibirsk, Russia dThe University of Texas at Austin, Department of Physics and

Institute for Fusion Studies, Austin TX 78712, U.S.A. eMax-Plank-Institut fur Physik, 80805 Munchen , Germany

PACS: 52.40.Mj, 52.65.-y, 52.59.-f

BASICS OF SELF-MODULATIONAL INSTABILITY

Plasma is a promising medium for high gradient acceleration of charged particles. It can sustain fields orders of magnitude higher than the breakdown fields of conventional accelerators [1]. One can excite strong plasma waves either with lasers or with charged particle beams [2-4]. One of the very attractive approaches is to use already existing TeV proton beams as a driver to generate plasma wake fields. Due to the limitation set by the transformer ratio, the energy gain of the witness beam cannot be much larger than the driver energy [5]. Employing a TeV proton driver allows in principle to accelerate an electron bunch to TeV energies in one single stage thus alleviating the technical burden of multi-staging. To excite a high amplitude plasma wave, the driver must be shorter than the plasma wavelength. The existing proton bunches are some 10 cm long, or longer. A plasma that maintains Langmuire waves of 10 cm length is very dilute, with the electron density less than 311cm10 31en The wave breaking limiting field of plasma is

GV/mcm10 314 3e

WBn

E (1)

As we are interested in a high gradient acceleration, the plasma density must be higher than 314 cm10 3 . The plasma wavelength

Advanced Accelerator ConceptsAIP Conf. Proc. 1507, 103-110 (2012); doi: 10.1063/1.4773682

© 2012 American Institute of Physics 978-0-7354-1125-8/$30.00

103

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

mmcm10 315

ep n

3

p (2)

is then much shorter than the available proton bunches. A long proton bunch with the standard Gaussian profile sent through dense plasma excites no wake. However, the bunch is subject to instabilities as it propagates through theplasma. The well-known one is the hosing instability: the centroid of the bunch starts oscillating around the propagation direction at the plasma wavelength [6]. Although this will lead to plasma wave excitation, this kind of wake is useless for acceleration and should be avoided.

FIGURE 1. Proton bunch self-modulation in plasma and the wake field generation (schematics).

Another kind of instability is the beam self-modulation as illustrated in Fig. 1. Let us assume the head of the proton bunch excites a small amplitude wake. The transverse fields of this wave will modify the bunch radius downstream. These ripples of the bunch density excite their own wake that tends to amplify the initial perturbation: the positive feedback is established [7,8].

and linear plasma response, the equation on the bunch radius evolution is

03

2

2

2

'sin)'(ˆ)'(

ˆˆˆ

drI

rr

b

b (3)

Here )(,,/ˆ 0 ctzkzkrrr pbb rr are the normalized variables and 22 /2 mcnek bb b22 2 is the

proton betatron wavenumber [8]. The positive feedback is due to the bunch radius term present in the integrand

In the limit of small beam emittance , equation (3) has an exponentially growing solution with growth rate [8].

3/1

2433

/1

2223

zmnmn

bpe

bp

bp (4)

p bk r 1

104

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

Thus, an initially smooth proton bunch undergoes a process of self-modulation, where it is split automatically in a sequence of micro-bunches each one plasma wavelength short. This structured proton bunch excites a highamplitude plasma wave suitable for particle acceleration.

SIMULATIONS OF SELF-MODULATION INSTABILITY

protons at 450 GeV/c with normalized emittance n = 3.5μm and length σz =12cm. There is also an option tooptimize the SPS bunch and increase the number of protons in the bunch to 3.5·1011. We use the standard and the optimized SPS bunch parameters in our 3D PIC simulations with the newly developed hybrid code H-VLPL3D [9]. This new code simulates the background plasma hydrodynamically while high energy bunches are treated with a fullkinetic algorithm. The hydrodynamic part of the code introduces much less numerical dispersion into the plasmawaves than a PIC approach with the same resolution. The bunch focused to σr = 0.19 mm is sent through plasmawith the free electron density ne = 7.76 · 1014 cm −3. The maximum bunch density on axis is nb = 1.5 · 1012 cm−3 forthe standard SPS bunch and nb = 4.5 · 1012 cm −3 for the optimized case. To avoid beam hosing and to seed the self-modulation, we assumed the bunch is hard-cut in the middle [10]. The maximum field evolution is shown in Fig. 2

FIGURE 2. Maximum amplitude of the wake field generated by the standard and optimized SPS bunch.

A wake field strength of GV/m can be maintained over several meters with the SPS proton bunch. This shouldallow for the demonstration of GeV energy gains by injected electrons in the pilot experiments planned at CERN.

105

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

However, it is not only the accelerating field amplitude that is important for the high energy acceleration. The wake phase velocity is another important parameter. The modulation mode not only has an imaginary frequency part responsible for the exponential growth. It also has a real part that modifies the phase velocity of the growing wave:

3/1

2211

zmnmn

vvbpe

bbph

b

(6)

Thus, during the linear growth stage of the modulation instability, the wake field is slower than the drive bunch.Figure 3 shows the phase velocity of a plasma wake field generated by the standard SPS bunch. We see that at thevery head of the proton bunch, the wake has the same phase velocity as the driver. Further downstream, the phase velocity deviates significantly. The largest velocity drop is observed during the development of the instability. Later,at the nonlinear saturation stage, the wake reaches the velocity of the proton bunch. This has a significant influenceon the particle acceleration.

FIGURE 3. Phase velocity of the wake field generated by the standard SPS bunch.

Z, m

(vph – c )/ c, x10-4

05

10

0

-1

-2

-3

-4

200100

0

106

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

SIDE INJECTION OF ELECTRONS

Because of the low phase velocity of the wake at the stage of linear instability growth, one should avoid early electron injection into the plasma wave. At the same time the low phase velocity provides an opportunity for trapping of low energy particles. Simulations show that a few MeV electrons can be successfully trapped in the wake generated by the proton bunch. Figure 3 demonstrates that there is an optimal injection point just at the transition from the linear instability to the saturation regime, somewhere at the 5 meters of propagation distance in this case. The wake still has a low phase velocity here, but is fast accelerating. Thus, if one injects electrons here, they are accelerated together with the wake and stay in phase for a long distance. It appears that one of the possibilities is side injection of electrons in the wake. In the simulation, we send a bunch of electrons at a small angle with respect to the proton bunch axis. Propagating together with the plasma wave over a relatively long distance, the electrons are sucked in by the transverse fields of the wake. Apparently, the structure of the wake allows for an automatically correct phasing of electrons in the accelerating field.

FIGURE 4. Test electron bunch (dark green discs) and the accelerating field (the red-blue wave) at z = 5 m. The electrons are captured and split into micro bunches located exactly in the accelerating and focusing phases of the wake.

Fig. 4 shows an electron bunch with the 10 MeV energy initially side-injected at an angle of 0.005 radian with respect to the proton bunch propagation direction. The electron bunch trajectory is designed to cross the driver axis at z = 6 m, and ξ = 20 cm behind the bunch head. Due to the small injection angle, however, the electrons are sucked in into the wake much earlier, at the position z ≈ 5m. The wake transverse fields put most of the bunch electrons into the focusing and accelerating phase. The electron beam is split into micro-bunches located exactly in the accelerating and focusing phases of the wake. Due to this configuration, the side injected beam results in a maximum energy gain of 1.2 GeV and a rather narrow energy spectrum as shown in Fig. 5. For comparison, we also show the spectrum of electrons injected on-axis at the very beginning of the interaction. Clearly, the low phase velocity of the wake did not allow the electrons to gain much energy and the spectrum has a large energy spread.

When electrons are side-injected, a significant fraction of the injected bunch charge is trapped. When an electron enters the wake field from side, it not only experiences transverse oscillation, but also gains energy due to the longitudinal component of the wake – if it enters at the right accelerating phase. Thus, at the swing out of the wake, the relativistic -factor of the electron increases. The heavier electron moves slower in the transverse direction and has a smaller excursion radius. It stays in the wake. In our simulations, up to 50% of the side injected electrons are trapped in the wake.

107

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

FIGURE 5. �lectron energy spectra at z = 10 m. There were two groups of electrons. �ne was injected on-axis at the plasma entrance� another was side-injected at the angle of 0.005 radian. In the latter case, the electron bunch trajectory should cross the driver axis at z = 6 m, and ξ = 20 cm behind the bunch head. The on-axis injection lead to a broad spectrum and low energy gain. The side injection results in a �uasi-monoenergetic beam at 1.2 GeV energy and about 1% energy spread. The initial electron energy is 20 MeV for on-axis injection and 10 MeV for side injection.

PHASE VELOCITY CONTROL BY PLASMA DENSITY GRADIENTS

As we have shown, the phase velocity of plasma wake generated by a long proton bunch in the self-modulated regime may be lower than the driver velocity. This influences the acceleration process significantly. �owever, one can control the wake phase velocity by a plasma density gradient. Indeed, when the plasma density – and correspondingly the plasma fre�uency – is changing along the propagation distance, the relative phase of the wake at particular position behind the head of the bunch is changing as well. The longer the protons bunch, the stronger is the phase dependence on the plasma density gradient.

To elucidate the effect, we have performed an additional simulation with the same beam parameters, but introducing a positive linear plasma density gradient� ne (z) = ne0 (1 � z/d) with characteristic length d = 200m. The phase velocity obtained in this simulation is shown in Fig. 6. The phase velocity at the head of the beam takes a dive as defined by the growing mode dispersion. �owever, the positive plasma density gradient compensates for the mode dispersion and at the tail of the beam the wake phase velocity becomes e�ual to the speed of light and even superluminous.

FIGURE 6. Phase velocity control by positive plasma density gradient. a) Phase velocity at z=2.5 m. b) Full dynamics picture� the transparent plane marks the speed of light.driver axis at z = 6 m, ξ = 20 cm behind the bunch head.

108

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

TRANSVERSE COHERENT TRANSITION RADIATION

Coherent transition radiation (CT�) is one of the most common techni�ues used for diagnosis of a longitudinal structure of charged particles bunches �11-15�. The method particularly demonstrated its power to characterize accelerated electron bunches in laser-plasma experiments �16-1��. An elementary charge propagating through a medium with a particular dielectric permittivity is dressed by a field matched to that medium. When the charge traverses a sharp boundary between two media with different permittivities, its field must be adjusted. The unmatched field can be radiated. The strongest radiation is observed when a charge passes a boundary between a conductor and vacuum. A point-like relativistic charge with the relativistic factor γ emits a radiation burst that is collimated within a cone with the opening angle θ ≈ 1/γ around the axis, although the emission is exactly zero in the propagation direction itself. The radiation is broadband. A bunch of particles can emit this radiation coherently at the wavelength comparable with its longitudinal structure. In the concept of proton bunch-driven plasma wake field accelerator [1�–20�, a long proton bunch is sent through plasma undergoes self-modulation at the plasma wave period and excites a strong resonant wake field. We show thatCT� is the method of choice for diagnosis of the proton bunch modulation after it exits the plasma cell. The nature of the proton bunch modulation is such that the proton bunch radius is modulated, but the total bunch current remains the same in each cross-section. For this reason, there will be no signatures of the proton bunch modulation in the forward coherent transition radiation. The classic forward CT� is cast useless in this case. Moreover, it is important in the experiment to distinguish between the axisymmetric modulation mode when the radius of the proton bunch is changing periodically ��� and the possible hosing mode when the proton bunch centroid oscillates periodically in the transverse plane ���. It appears that the transverse coherent transition radiation (TCT�) does contain the signature of the bunch modulation and allows for distinguishing between the axisymmetric modulation mode and the hosing. The TCT� is emitted perpendicularly to the particle bunch propagation direction and its amplitude does not depend on the particles γ−factor as soon as it is large enough. Because the radius of the proton bunch is comparable with the modulation wavelength, the bunch does radiate when it crosses a sharp boundary of two media with different electromagnetic properties, optimally, the boundary between vacuum and a conducting plate. An axially symmetric proton bunch modulation leads to a radially symmetric TCT�, see Fig. �a. The hosing leads to an asymmetric TCT�, see Fig. �b. The main lobes of the TCT� are located in the hosing plane.

FIGURE 7. TCT� emitted by a proton bunch exiting from a metallic foil. A) �nly the radius of the proton bunch is modulated,not the current. The colored surfaces show the electric field component normal to the metal foil. B) A hosed proton bunchshakes off the radiation fields anti-symmetrically in the plane of hosing.

a) b)

109

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

The radiation field magnitude reaches several 100kV/m values at 1mm distance from the bunch axis and decays as 1�� with the distance from the bunch axis.

SUMMARY

We have shown that the self-modulational instability of a charged beam in plasma corresponds to a growing mode with a slow phase velocity. The wake velocity is much lower than that of the driver. The wake slowdown is due to the real part of the fre�uency of the unstable mode. Although this effect limits electron energy gain at the stage of the linear instability growth, the low phase velocity can be harnessed to inject low energy electrons into the wake of a highly relativistic driver. We also have shown that the side injection of electrons at a small angle with respect to the driver axis may drastically improve the �uality of acceleration. The transverse field of the wake sucks in the injected electrons and automatically puts them into the right acceleration phase. Finally, we show that the wake phase velocity can be controlled by a longitudinal plasma density gradient. It has been also shown that a particle bunch that is modulated in radius or hosed does radiate TCT� even when the total current of the bunch remains unmodulated. The radiation is emitted due to the retardation effect� the observer located in the transverse plane sees signals of particles traversing the conductor plate at retarded times. An axisymmetric particle bunch with a periodically modulated radius emits axisymmetric waves. �n the contrary, a hosed bunch whose centroid oscillates in a transverse plane emits in a direction perpendicular to this plane

ACKNOWLEDGMENTS

This work was supported in parts by DFG and BMBF (Germany). V.�., C. S., and G. S. acknowledge funding support from the �S Department of �nergy under the grant D�-FG02-04��41321.

REFERENCES

1. T. Tajima and �. M. Dawson, �aser �lectron Accelerator, Phys. �ev. �ett. 43, 26�-2�0 (1���).2. Chan �oshi and Victor Malka �Focus on �aser- and Beam-Driven Plasma Accelerators�. �ew �ournal of

Physics. (2010)3. C. �oshi et al., Phys. Plasmas 9, 1�45-1�55 (2002).4. C. �oshi, �Plasma Accelerators,� Scientific American (February 2006), 294, 40-4�3.5. �. �. Bane, P. Chen, and P. B. Wilson, I��� Trans. �ucl. Sci. 32, 3524 (1��5).6. D. �. Whittum, Phys. Fluids 5B, 4432 (1��3).�. �. �umar, A. Pukhov, �. �otov. Phys. �ev. �ett. 104, 255003 (2010).�. A. Pukhov, �. �umar, T. Tuckmantel, A. �padhyay, �. �otov, P. Muggli, V. �hudik, C. Siemon, and G. Shvets,

Phys. �ev. �ett. 107 , 145003 (2011)�. C. B. Schroeder, C. Benedetti, E. Esarey, F. J. Grüner, and W. P. Leemans, Phys. �ev.�ett. 107, 145002 (2011).

�. T. Tuckmantel, A. Pukhov, �. �iljo, M. �ochbruck, I��� TPS, 38, 23�3 (2010)10. P. Muggli et al., Phys. �ev. �ett. 101, 054�01 (200�), P. Muggli et al., Phys. �ev. ST Accel. Beams 13, 052�03 (2010).11. �. �osenzweig, G. Travish, A. Tremaine � Coherent transition radiation diagnosis of electron beam microbunching �. �uclear

Instruments and Methods in Physics �esearch Section A� 365 , 255–25� (1��5).12. � �. �appek, A.�. Sievers and �.B. Blum, Phys. �ev. �ett. 67, 2�62 (1��1).13. �.-C. �ihn, P. �ung, C. Settakorn, and �.Weidemann, Phys. �ev. � 53 , 6413 (1��6).14. �.�ai and A.�. Sievers, Phys. �ev. � 52, 45�6 (1��5).15. �. Shibata, et al., Phys. �ev. � 50, 14�� (1��4)16. W. P. �eemans, C. G. �. Geddes, �. Faure, Cs. Toth, �. van Tilborg, C. B. Schroeder, �. �sarey, G. Fubiani, D. Auerbach,

B. Marcelis, et al., Phys. �ev. �ett. 91(�), 0�4�02 (2003).1�. C. B. Schroeder, �. �sarey, �. van Tilborg, and W. P. �eemans, Phys. �ev. � 69(1), 016501 (2004).1�. �. van Tilborg, C. B. Schroeder, C. V. Filip, Cs. Toth, C. G. �. Geddes, G. Fubiani, �. �uber, �. A. �aindl, �. �sarey, and

W. P. �eemans, Phys. �ev. �ett. 96 , 014�01 (2006).1�. A. Caldwell, �. �otov, A. Pukhov, and F. Simon, �at. Phys, 5 , 363 (200�)�20. A. Caldwell, �. �otov, A. Pukhov and G. �ia, Plasma, Phys. Controlled Fusion 53 , 014003 (2011).

110

Downloaded 28 Jan 2013 to 210.212.157.162. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions