# Plasma Beam Dumps for the EuPRAXIA Facility

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## Abstract

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## 1. Introduction

^{12}Watt) or petawatt (10

^{15}Watt) laser drivers [7,8,9].

## 2. Plasma Beam Dumps

## 3. Simulation Results of Plasma Beam Dumps for EuPRAXIA Beams

^{18}cm

^{−3}. The Fourier–Bessel particle-in-cell (FBPIC) code [16] is used to perform simulations of beam–plasma interaction. Although particles in FBPIC have 3D cartesian coordinates, this code adopts a spectral solver, which uses a set of 2D radial grids, each of them representing an azimuthal mode, m ($m=0,1,\dots $). The first mode, $m=0$, represents axisymmetric fields, i.e., fields with cylindrical symmetry, with no dependence on the azimuthal angle, $\theta $. Additional modes can be used to represent departures from the cylindrical symmetry (for example, a linearly polarised laser can be represented by the second azimuthal mode, $m=1$). Among the many interesting features implemented in FBPIC, it is worth highlighting the mitigation of spurious numerical dispersion by the spectral solver algorithm, including the zero-order numerical Cherenkov effect [17], and the adoption of the openPMD meta data standard [18]. The simulations presented in this document were performed using the first azimuthal mode ($m=0$). Hence, a 2D axisymmetric geometry is adopted. The longitudinal simulation domain is $-2.5\pi /{k}_{p}\le \xi \le 5{\sigma}_{\xi}$, where $\xi \equiv z-ct$ is the co-moving coordinate and ${\sigma}_{\xi}$ is the longitudinal bunch root mean square (RMS) size. Transversally, the domain is $-20{\sigma}_{r}\le x,y\le 20{\sigma}_{r}$, where x and y are the transverse coordinates, and ${\sigma}_{r}$ is the transverse RMS bunch length. This domain is enough to evaluate the relevant dynamics, such as the formation of re-acceleration peaks and the behaviour of defocused particles. The longitudinal and transverse resolutions are ${\sigma}_{\xi}/20$ and ${\sigma}_{r}/20$, respectively, and the total number of particles per cell is 4, being 2 along each coordinate z and r.

#### 3.1. Plasma Beam Dumps for the 1 GeV Beam

^{17}cm

^{−3}. For this plasma density and considering the beam charge and dimensions presented in Table 1, a beam–plasma density ratio of ${n}_{b}/{n}_{0}\simeq 3.1$ is obtained. The magnitude of this ratio defines the regime of the wakefield excited by the beam propagating in the plasma. If ${n}_{b}/{n}_{0}\ll 1$, the wakefield presents a perfectly harmonic aspect, and is said to be in the linear regime. As this ratio increases, the wakefield changes from a sinusoidal to a sawtooth-like shape. Although no longer harmonic, under certain conditions, some analytical estimates for the wakefield still hold for $1\lesssim {n}_{b}/{n}_{0}\lesssim 10$ [19]. Therefore, the propagation regime within the aforementioned interval of ${n}_{b}/{n}_{0}$ can be classified as ranging from the quasi-linear, or quasi-non-linear [20], to the non-linear regime. This is the case for the parameters chosen for this work. Besides the wakefield propagation regime, the choice of plasma density must also take into account the electron bunch size. While the longitudinal wakefield oscillates from decelerating (positive electric field) to accelerating (negative electric field) phases, the transverse wakefield alternates from focusing to defocusing phases. For a plasma beam dump, a density is chosen that can contain the whole bunch within the first wakefield phase, which is longitudinally decelerating and transversely focusing. This ensures that the bunch will be simultaneously decelerated and focused as it propagates in the plasma. With all the parameters defined, a particle-in-cell (PIC) simulation is performed.

_{0}= 9.9 × 10

^{17}cm

^{−3}, at 6 cm, to 10 n

_{0}= 9.9 × 10

^{18}cm

^{−3}, at approximately 17 cm. This particular plasma density profile can be obtained by imposing a constant rate of change for the plasma wavelength $\lambda $ with respect to the propagation distance s, i.e., $d\lambda /ds=\mathrm{constant}$ [14].

#### 3.2. Plasma Beam Dumps for the 5 GeV Beam

^{17}cm

^{−3}, i.e., the same value adopted in the previous case. In panel (b), the corresponding energy spectrum for this propagation distance is also shown. Qualitatively, the phase space of Figure 5a is equivalent to that shown in Figure 1a for the 1 GeV beam; particles at the middle and tail of the bunch lose their energies, and some particles at the tail already started picking up energy from the wakefield. However, since the beam energy is 5-fold higher in this case, the propagation distance to reach this point is 26 cm, which is approximately 4.3-fold longer with the 6 cm observed in Figure 1a. Figure 5b demonstrates that the 5 GeV beam also shows a large number of particles decelerated to small energies. As in the 1 GeV beam, a smaller fraction of beam particles (the fraction within the bunch head vicinities in panel a) maintain their initial energies.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Li, X.; Mosnier, A.; Nghiem, P.A.P. Slice energy spread optimization for a 5 GeV laser-plasma accelerator. Nucl. Instrum. Meth. Phys. Res. A
**2018**, 909, 49. [Google Scholar] [CrossRef] - Tajima, T.; Dawson, J.M. Laser electron accelerator. Phys. Rev. Lett.
**1979**, 43, 267. [Google Scholar] [CrossRef] [Green Version] - Mangles, S.; Murphy, C.D.; Najmudin, Z.; Thomas, A.G.R.; Collier, J.L.; Dangor, A.E.; Divall, E.J.; Foster, P.S.; Gallacher, J.G.; Hooker, C.J.; et al. Monoenergetic beams of relativistic electrons from intense laser-plasma interactions. Nature
**2004**, 431, 535. [Google Scholar] [CrossRef] [PubMed] - Geddes, C.G.R.; Toth, C.; van Tilborg, J.; Esarey, E.; Schroeder, C.B.; Bruhwiler, D.; Nieter, C.; Cary, J.; Leemans, W.P. High-quality electron beams from a laser wakefield accelerator using plasma-channel guiding. Nature
**2004**, 431, 538. [Google Scholar] [CrossRef] [PubMed] - Faure, J.; Glinec, Y.; Pukhov, A.; Kiselev, S.; Gordienko, S.; Lefebvre, E.; Rousseau, J.-P.; Burgy, F.; Malka, V. A laser-plasma accelerator producing monoenergetic electron beams. Nature
**2004**, 431, 541. [Google Scholar] [CrossRef] [PubMed] - Esarey, E.; Schroeder, C.B.; Leemans, W.P. Physics of laser-driven plasma-based electron accelerators. Rev. Mod. Phys.
**2009**, 81, 1229. [Google Scholar] [CrossRef] - Leemans, W.P.; Nagler, B.; Gonsalves, A.J.; Tóth, C.; Nakamura, K.; Geddes, C.G.R.; Esarey, E.; Schroeder, C.B.; Hooker, S.M. GeV electron beams from a centimetre-scale accelerator. Nat. Phys.
**2006**, 2, 696. [Google Scholar] [CrossRef] - Geddes, C.G.R.; Nakamura, K.; Plateau, G.R.; Toth, C.; Cormier, M.E.; Esarey, E.; Schroeder, C.B.; Cary, J.R.; Leemans, W.P. Plasma-density-gradient injection of low absolute-momentum-spread electron bunches. Phys. Rev. Lett.
**2008**, 100, 215004. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gonsalves, A.J.; Nakamura, K.; Daniels, J.; Benedetti, C.; Pieronek, C.; de Raadt, T.C.H.; Steinke, S.; Bin, J.H.; Bulanov, S.S.; van Tilborg, J.; et al. Petawatt laser guiding and electron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide. Phys. Rev. Lett.
**2019**, 122, 084801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, H.C.; Tajima, T.; Habs, D.; Chao, A.W.; Meyer-ter-Vehn, J. Collective deceleration: Toward a compact beam dump. Phys. Rev. Accel. Beams
**2010**, 13, 101303. [Google Scholar] [CrossRef] - Bonatto, A.; Schroeder, C.B.; Vay, J.-L.; Geddes, C.G.R.; Benedetti, C.; Esarey, E.; Leemans, W.P. Passive and active plasma deceleration for the compact disposal of electron beams. Phys. Plasmas
**2015**, 22, 083106. [Google Scholar] [CrossRef] - Bonatto, A.; Schroeder, C.B.; Vay, J.-L.; Geddes, C.R.; Benedetti, C.; Esarey, E.; Leemans, W.P. Compact Disposal of High-energy Electron Beams Using Passive or Laser-driven Plasma Decelerating Stage. In Proceedings of the AIP Conference Proceedings, San Jose, CA, USA, 13–18 July 2012; Volume 1777, p. 040002. [Google Scholar]
- Hanahoe, K.; Xia, G.; Islam, M.; Li, Y.; Mete-Apsimon, Ö.; Hidding, B.; Smith, J. Simulation study of a passive plasma beam dump using varying plasma density. Phys. Plasmas
**2017**, 24, 023120. [Google Scholar] [CrossRef] [Green Version] - Jakobsson, O.; Bonatto, A.; Li, Y.; Zhao, Y.; Nunes, R.P.; Williamson, B.; Xia, G.; Tajima, T. Tailored plasma-density profiles for enhanced energy extraction in passive plasma beam dumps. Plasma Phys. Control. Fusion
**2019**, 61, 124002. [Google Scholar] [CrossRef] - Nunes, R.; Bonatto, A.; Xia, G. A Passive Plasma Beam Dump Study with Application to EuPRAXIA. In Proceedings of the IPAC’19, Melbourne, Australia, 19–24 May 2019; pp. 3–3581. [Google Scholar]
- Lehe, R.; Kirchenb, M.; Igor, A.; Andriyashc, B.; Godfreyad, B.; Vaya, J.-L. A spectral, quasi-cylindrical and dispersion-free particle-in-cell algorithm. Comput. Phys. Commun.
**2016**, 203, 66–82. [Google Scholar] [CrossRef] [Green Version] - Godfrey, B.B. Numerical Cherenkov instabilities in electromagnetic particle codes. J. Comput. Phys.
**1974**, 15, 504–521. [Google Scholar] [CrossRef] - Huebl, A.; Rehwald, M.; Obst-Huebl, L.; Ziegler, T.; Garten, M.; Widera, R.; Zeil, K.; Cowan, T.E.; Bussmann, M.; Schramm, U.; et al. Spectral control via multi-species effects in pw-class laser-ion acceleration. arXiv
**2018**, arXiv:1903.06428. [Google Scholar] [CrossRef] - Lu, W.; Huang, C.; Zhou, M.M.; Mori, W.B.; Katsouleas, T. Limits of linear plasma wakefield theory for electron or positron beams. Phys. Plasmas
**2005**, 12, 063101. [Google Scholar] [CrossRef] - Londrillo, P.; Gatti, C.; Ferrario, M. Nuclear instruments and methods in physics research A. Nucl. Inst. Method Phys. Res. A
**2014**, 740, 236–241. [Google Scholar] [CrossRef] - Cowley, J.; Thornton, C.; Arran, C.; Shalloo, R.J.; Corner, L.; Cheung, G.; Gregory, C.D.; Mangles, S.P.D.; Matlis, N.H.; Symes, D.R.; et al. Excitation and Control of Plasma Wakefields by Multiple Laser Pulses. Phys. Rev. Lett.
**2017**, 119, 044802. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Adli, E.; Ahuja, A.; Apsimon, O.; Apsimon, R.; Bachmann, A.-M.; Barrientos, D.; Batsch, F.; Bauche, J.; Bohl, T.; Bernardini, M.; et al. Acceleration of electrons in the plasma wakefield of a proton bunch. Nature
**2018**, 561, 363–367. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) The EuPRAXIA 1 GeV beam longitudinal phase space and (

**b**) energy spectrum, both plotted after 6 cm propagation in plasma.

**Figure 2.**A tailored plasma density profile designed to eliminate the re-acceleration of particles in the bunch tail in the EuPRAXIA 1 GeV beam. The plasma density exhibits uniform behaviour until s = 6 cm, followed by a non-linear growth that reaches a 10 times higher density at s = 16.8 cm compared to the former uniform plasma density.

**Figure 3.**(

**a**) The EuPRAXIA 1 GeV beam longitudinal phase space and (

**b**) energy spectrum, after 16.8 cm propagation in plasma. At this propagation distance, particles with lower energies, which can be clearly seen in Figure 1a, are not present because they were ejected by the defocusing phase of the transverse wakefield. The energy spectrum for the remaining beam (7 pC, at this propagation distance) is represented by the blue columns in Figure 3b. In addition, the light-red shaded region shows the energy and charge distribution for the whole beam, including the transversely ejected particles.

**Figure 4.**EuPRAXIA 1 GeV beam energy plots, as a function of propagation distance in plasma, for the density profile shown in Figure 2.

**Figure 5.**(

**a**) The beam longitudinal phase space and (

**b**) energy spectrum, after 26 cm propagation in plasma, for the EuPRAXIA 5 GeV beam.

**Figure 6.**A tailored plasma density profile designed to eliminate the re-acceleration of particles in the bunch tail in the EuPRAXIA 5 GeV beam. The plasma density exhibits uniform behaviour until s = 26 cm, followed by a non-linear growth that reaches a 15-fold higher density at s = 36 cm, compared to the former uniform plasma density.

**Figure 7.**(

**a**) The EuPRAXIA 5 GeV beam longitudinal phase space and (

**b**) energy spectrum, after 36 cm propagation in plasma, for the EuPRAXIA 5 GeV beam. Compared to Figure 5a, this figure shows that the density profile from Figure 6 is effective in eliminating the lower energy particles reaching the accelerating wakefield phase, thus preventing the formation of a re-acceleration peak. The energy spectrum for the remaining beam (18 pC, at this propagation distance) is represented by the blue columns in panel (

**b**). In addition, the light-red shaded region shows the energy and charge distribution for the whole beam, including the transversely ejected particles.

**Figure 8.**The energy plots as a function of propagation distance in plasma, for the plasma density profile shown in Figure 6 for the EuPRAXIA 5 GeV beam.

Beam Energy | 1 GeV | 5 GeV |
---|---|---|

Bunch charge | 30 pC | 30 pC |

Transverse bunch size (${\sigma}_{r}$) | 1.4 μm | 1.4 μm |

Longitudinal bunch length (${\sigma}_{\xi}$) | 2.0 μm | 2.0 μm |

Energy spread | 1.0% | 1.0% |

Angular divergence (rad) | 1.0 × 10^{−5} | 1.0 × 10^{−5} |

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**MDPI and ACS Style**

Xia, G.; Bonatto, A.; Pizzato Nunes, R.; Liang, L.; Jakobsson, O.; Zhao, Y.; Williamson, B.; Davut, C.; Wang, X.
Plasma Beam Dumps for the EuPRAXIA Facility. *Instruments* **2020**, *4*, 10.
https://doi.org/10.3390/instruments4020010

**AMA Style**

Xia G, Bonatto A, Pizzato Nunes R, Liang L, Jakobsson O, Zhao Y, Williamson B, Davut C, Wang X.
Plasma Beam Dumps for the EuPRAXIA Facility. *Instruments*. 2020; 4(2):10.
https://doi.org/10.3390/instruments4020010

**Chicago/Turabian Style**

Xia, Guoxing, Alexandre Bonatto, Roger Pizzato Nunes, Linbo Liang, Oscar Jakobsson, Yuan Zhao, Barney Williamson, Can Davut, and Xueying Wang.
2020. "Plasma Beam Dumps for the EuPRAXIA Facility" *Instruments* 4, no. 2: 10.
https://doi.org/10.3390/instruments4020010