Laser plasma accelerators are able to produce high quality electron beams from 1 MeV to 1 GeV. The next generation of plasma accelerator experiments will likely use a multi-stage approach where a high quality electron bunch is first produced and then injected into an accelerating structure. In this paper we present scaled particle-in-cell simulations of a 10 GeV stage in the quasi-linear regime. We show that physical parameters can be scaled to be able to perform these simulations at reasonable computational cost. Beam loading properties and electron bunch energy gain are calculated. A range of parameter regimes are studied to optimize the quality of the electron bunch at the output of the stage

Bruhwiler, D. L.

Cormier-Michel, Estelle

Cowan, B.

Esarey, E.

Geddes, C. G. R.

Leemans, W. P.

Paul, K.

Schroeder, C. B.

UNT Digital Library

Scaled simulations of a 10 GeV acceleratorEstelle Cormier-Michel∗, C.G.R. Geddes∗, E. Esarey∗,†, C.B. Schroeder∗,D. L. Bruhwiler∗∗, K. Paul∗∗, B. Cowan∗∗ and W.P. Leemans∗,†∗LOASIS program, Lawrence Berkeley National Laboratory, Berkeley, CA 94720†University of Nevada Reno, Reno, NV 89557∗∗Tech-X corporation, Boulder, CO 80303Abstract. Laser plasma accelerators are able to produce high quality electron beams from 1 MeV to1 GeV. The next generation of plasma accelerator experiments will likely use a multi-stage approachwhere a high quality electron bunch is first produced and then injected into an accelerating structure.In this paper we present scaled particle-in-cell simulations of a 10 GeV stage in the quasi-linearregime. We show that physical parameters can be scaled to be able to perform these simulationsat reasonable computational cost. Beam loading properties and electron bunch energy gain arecalculated. A range of parameter regimes are studied to optimize the quality of the electron bunchat the output of the stage.Keywords: Laser plasma accelerator, 10 GeV stage, PIC simulationPACS: 52.38.Kd, 41.75.Jv, 52.65.RrINTRODUCTIONLaser plasma accelerators are able to produce very high gradient electric fields whichmake them suitable to accelerate particles to high energies in a very short distance.Production of high quality electron beams has been demonstrated, with energies up to1GeV reached in a few centimeters [1]. Until recently, acceleration of these electronbeams relied on the self-trapping of background plasma electrons. Experiments haveshown that it is now possible to control the quality of the electron beam by controllingthe injection process and then further accelerating the beam [2, 3, 4]. In this paper weexamine the design of a 10 GeV accelerator stage where we assume that a high quality,mono-energetic electron beam is externally injected in a “dark current free” (i.e., wherethere is no self-trapping) plasma accelerating structure. Parameters of the acceleratingstage are studied in order to optimize charge, energy and energy spread of the bunch.SCALING LAWSThe acceleration length, and hence the energy gain, in a laser plasma based accelera-tor, is limited by the dephasing of the electrons, i.e., the electrons stop being acceleratedwhen they outrun the wake [5]. To increase the dephasing length and increase the energygain up to 10 GeV, the density must be decreased to the order of 1017 cm−3 giving an ac-celeration length of about a meter [6]. Using a fully self-consistent Particle-In-Cell (PIC)algorithm to simulate such a stage is still impractical with state of the art computing fa-cilities. The use of reduced models [7, 8, 9] is therefore required to model accelerationof particle beams to these high energies as it can greatly reduce the required numberof simulation hours. Here we use the fully self-consistent PIC algorithm, implementedin the code VORPAL [10], to simulate the evolution of an externally injected electronbunch over the accelerating stage by scaling the physical quantities with the plasma den-sity. In this way, it is possible to use short scaled simulations at high density and deducethe properties of the accelerated electron bunch by verifying and using scaling laws. Thisalso allows us to fully simulate the laser pulse evolution up to depletion.In this paper we study a laser plasma accelerator stage in the standard regime [5]. Thenormalized laser intensity profile is of the form a20 exp(−2x2/L2)exp[−2(y2 + z2)/w20]where a20 = 7.32× 10−19(λ0[µm])2I0[W/cm2] with λ0 the laser wavelength and I0 thelaser intensity, x is the longitudinal direction, L the laser pulse length, and w0 the laserspot size. The laser is guided by a transverse parabolic plasma channel whose depth isadjusted, when the power of the laser is close to the critical power, to compensate for theself-focusing such that the laser beam radius undergoes minimal variation. The injectedelectron bunch profile is nb ∼ exp(−x2/2σ2L)exp[−(y2 + z2)/2σ2r ], with σL the bunchlength and σr the bunch radius.In the scaled simulations the dimensions of the problem are kept constant comparedto the plasma wavelength λp, i.e, kpL, kpw0, kpσL and kpσr are constant while thedensity is varied, where kp = 2π/λp ∝√n0 is the plasma wave number and n0 theplasma density. The normalized laser strength a0 is also fixed, fixing the ratio of thelaser power over the critical power. Using the fixed parameters kpL = 2, kpw0 = 5.3and a0 = 1, and n0 = 1019 and 1018 cm−3 we verify that the peak accelerating electricfield scales as√n0 as expected [11]. The laser evolution is also consistent. Althoughthe laser betatron oscillation and Rayleigh length are not constant compared to λp, thisissue is overcome by the use of the plasma channel and indeed we verify that the spotsize stays constant along the guiding structure and that the same amount of energy isdepleted after a dephasing length. Because of this, the wakefield structure also scaleswith the density, i.e., contours of the accelerating field overlay within a few percent,when normalizing the lengths by λp; this has also been confirmed using n0 = 1017 cm−3and reduced models. We also verify that the dephasing length scales as λ 3p/λ20 ∝ n−3/20[6]. Note that the wake phase velocity depends on the plasma density for a constant laserwavelength. This confirms the scaling laws predicted in the linear regime and extendsthem to multi-dimensions and to the quasi-linear regime.The energy gain of a stage is given by LaccEx ∝ 1/n0, where Lacc is the acceleratinglength which is equal to the dephasing length in the linear regime. The linear theorypredicts that, for the parameters given above, an energy gain of 100 MeV is obtained atn0 = 7× 1018 cm−3 and 10 GeV is achieved at n0 = 7× 1016 cm−3. A gain of 8 GeVhas been demonstrated with test particles using a simulation in the boosted frame in1D at 1017 cm−3 [12, 13], consistent with the linear theory which predicts a gain of 7GeV in this case. By using test particles in the PIC simulations, we find that becausethe wake is slightly non-linear, 97 MeV and 1120 MeV can be achieved respectivelyat 1019 and 1018 cm−3. We can then predict that we will obtain 10 GeV with n0 = 1017cm−3 over Lacc = 1 m. We next perform simulations with the PIC algorithm at n0 = 1019cm−3 where the beam is accelerated to 100 MeV in 1 mm to determine beam loadingperformance and electron beam quality. Using the scaling laws given here we can deducethe properties of the bunch in a 1017 cm−3 stage.BEAM LOADINGA charged electron beam, when propagating into the plasma, creates its own wake whichdamps the wakefield created by the leading laser pulse. The maximum amount of chargethat can be loaded is that for which the wakes created by the electron beam and the laserpulse are equal. For a short bunch (kpσL < 1) in the linear regime (Ex/E0 < 1) it is givenby [14]:Nmax =12kpreExE0k2pσ2rHR≈ 9.3×109 Ex/E0√n0/1016cm−3k2pσ2rHR(1)withHR =k2pσ2r2exp(k2pσ2r /2)Γ(0,k2pσ2r2)(3D) (2)HR =√π2kpσr exp(k2pσ2r /2)Erfc(kpσr√2)(2D) (3)where E0 = c2mkp/e is the cold non-relativistic wavebreaking limit, re is the classicalelectron radius, and Γ(a,x) =∫∞x e−tta−1dt.Simulations verify that Eq. (1) is accurate for a mildly nonlinear laser-driven wake-field (a0 ∼ 1). Figure 1 shows the percentage of beam loading in terms of peak ac-FIGURE 1. Beam loading, in terms of peak accelerating electric field, as a function of charge, normal-ized by√n0(cm−3)/1017×(HR/k2pσ2r)/(Ex/E0). The dashed line is theory, the points are for 2D (+)and 3D (∗) simulations and for the following cases from the lighter gray to the darker: kpL = 2, a0 = 1,n0 = 1019 cm−3, kpσL = 0.05 and beam radii kpσr = 0.3 (orange), kpσr = 1 (magenta), kpσr = 1.8,(green); scaling with a0 is shown at kpL = 1, a0 =√2, n0 = 1019 cm−3, kpσr = 0.3 (blue); scaling withdensity is shown at kpL = 2, a0 = 1, n0 = 1018 cm−3, kpσr = 1.8 (black).celerating electric field as a function of the electron bunch charge, Q, normalized by√n0(cm−3)/1017×(HR/k2pσ2r)/(Ex/E0). The dashed line is the coefficient given byEq. (1) and the points represent different simulated cases. The scaling is verified for dif-ferent values of σr, n0, and Ex/E0, with kpσL = 0.05, for both 2D and 3D simulations.Simulations in 2D at n0 = 1017 cm−3 also agree with the scaling. This data implies thatwe can load 60 pC for kpσr = 0.3 and 200 pC for kpσr = 1 with n0 = 1017 cm−3 anda0 = 1, kpL = 2 and kpw0 = 5.3.ACCELERATING STAGEWe now consider the design of an accelerating stage with beam loading. Since theelectron bunch creates its own wake, strong electric field gradients can be introducedinside the bunch which induce energy spread as the electrons are accelerated. It ispossible to shape the bunch such that Ex stays longitudinally constant inside the bunch[14]. This can be achieved by using a triangular bunch, or by increasing the bunch lengthσL while keeping the charge constant. Ideally, minimizing the energy spread by thismethod requires that the bunch stays at the same phase over the acceleration length.This can be achieved by axially tapering the plasma density profile. In principle, someoptimum tapering can be found such that the electron bunch does not dephase [15].Here we simply consider a linear plasma taper and a gaussian electron bunch loadedin the second bucket after the laser pulse. Because the beam is loaded further behind thedriver a milder taper is needed. Although the tapering is not optimum and the electronbunch still dephases, energy gain is increased and momentum spread is low. Figure 2(a)shows the electron momentum distribution (solid line) with and without plasma densityFIGURE 2. (a) Electron (solid line) and positron (dashed line) longitudinal momentum distributionwithout (peak centered around px ∼= 25 MeV/c) and with (peak centered around px ∼= 90 MeV/c) plasmataper for a0 = 1, kpL = 2, kpw0 = 5.3, n0 = 1019 cm−3, kpσL = 0.5, kpσr = 1, Q = 22.8 pC (correspondingto Q = 228 pC at n0 = 1017 cm−3). (b) Electron spectra with plasma taper for stages at n0 = 1019 cm−3,kpw0 = 5.3, kpσL = 0.5, kpσr = 1 and with approximately equivalent beam loading at a0 = 1, kpL = 2,Q = 22.8 pC (black) and a0 =√2, kpL = 1, Q = 31.4 pC [magenta (gray)].taper with kpσL = 0.5 (so the accelerating field is constant longitudinally inside thebunch at the loading phase) and kpσr = 1, and for the laser parameters a0 = 1, kpL = 2and kpw0 = 5.3. The charge in the bunch is 22.8 pC, the wakefield being approximately50% beam loaded. The bunch is loaded with an initial momentum of 10 MeV/c and aninitial momentum spread of 0.9 MeV/c (9%) full width at half maximum (FWHM). Theplasma taper is of the form n(x) = n0(13.2 x[cm] + 1) where n0 = 1019 cm−3 on axis,i.e., the plasma density varies by 40% in 0.3 mm. Without plasma taper the electronbunch is accelerated to ∼= 24.5 MeV/c (i.e., 14.5 MeV/c energy gain) over 0.3 mm with∼= 12% momentum spread FWHM. With plasma taper the electron bunch is acceleratedto ∼= 90 MeV/c (i.e., 80 MeV/c energy gain) over ∼= 0.7 mm with ∼= 3% momentumspread FWHM. The plasma taper has allowed the electron bunch to gain four timesthe energy and reduced the energy spread by allowing phase locking. Scaling thesequantities to a stage at n0 = 1017 cm−3, we can achieve 8 GeV energy gain for a 228 pCcharge electron bunch in 0.7 m, and with few percent energy spread.Because the stage is in the quasi-linear regime (a0 ∼ 1) the wake is nearly symmetricand a positron bunch can be accelerated in the same manner. Figure 2(a) shows thepositron momentum distribution (dashed line) for the same parameters as for the electronbunch described above. The axial plasma taper is of the form n(x) = n0(6.8 x[cm]+ 1)with n0 = 1019 cm−3 on axis. Without taper, the positron bunch is accelerated to ∼= 25.5MeV/c with 7% momentum spread FWHM in 0.2 mm; with taper, the final energy is∼= 92 MeV/c with 4% momentum spread FWHM in 0.7 mm.In the previous cases the laser pulse energy is about 30% depleted at the end of theacceleration stage. For a given laser energy, it is possible to increase the efficiency ofthe stage by reducing the laser pulse length and hence increasing the laser strengtha0. Simulations show that for kpL = 1 and a0 =√2, the laser depletion length is ofthe order of the dephasing length, i.e. more of the laser energy is transferred into theplasma wake. The higher a0 in this case results in a 40% higher accelerating field Ex/E0and hence more charge can be loaded into the wake. Figure 2(b) shows the electronmomentum distribution in the previous case (kpL = 2 and a0 = 1) and in the case kpL = 1and a0 =√2, corresponding to the same laser energy, and with the bunch parameterskpσL = 0.5, kpσr = 1 and Q = 31.4 pC. The wake is about 50% beam loaded, as in theprevious case. The same axial plasma taper is used. With about 40% more charge thanin the previous case, the electron bunch is accelerated to 96 MeV/c (i.e., ∼= 86 MeV/cenergy gain) with∼= 4% momentum spread FWHM. Scaling these quantities to a stage atn0 = 1017 cm−3, we can achieve almost 9 GeV energy gain for a 300 pC charge electronbunch in 0.7 m, and with few percent energy spread.CONCLUSIONPIC simulations have been used to model the evolution of an electron beam in a laserdriven plasma accelerating structure. By using and extending the scalings of the physicalconstants with the plasma density, we are able to deduce the properties of the electronbeam for longer stages at lower density and with higher energy gain. We have verifiedthe scaling for the guiding, the wake structure and for the beam loading. Using thescaled simulations, we have shown that it is possible to accelerate a few hundreds ofpC of charge to about 10 GeV at a density of 1017 cm−3 using a 110 J, 60–120 fs laserand an axial plasma taper which allows increase of the energy gain and reduction of themomentum spread. In the quasi-linear regime, it is also possible to accelerate positronsquasi-symmetrically. Further work will be done to study the evolution of the emittance ofthe electron bunch in the accelerating stage and to optimize the laser driver parametersto increase the stage efficiency. Furthermore, the effect of the electron beam betatronoscillation, which is not constant compared to λp, on emittance and energy spread willalso be studied with scaled simulation at different densities and with reduced models.ACKNOWLEDGMENTSThe authors thank B.A. Shadwick for insightful discussions. We also acknowledgethe assistance of the VORPAL development team. This work was supported by theDirector, Office of Science, Office of High Energy Physics, of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231 and DE-FG02-04ER84097 andby SciDAC grant No. DE-FC02-07ER41499. The simulations were performed at theNational Energy Research Scientific Computing Center (NERSC).REFERENCES1. W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tóth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B.Schroeder, and S. M. Hooker, Nature Phys. 2, 696–699 (2006).2. J. Faure, C. Rechatin, A. Norlin, A. Lifschitz, Y. Glinec, and V. Malka, Nature 444, 737–739 (2006).3. C. G. R. Geddes, K. Nakamura, G. R. Plateau, C. Tóth, E. Cormier-Michel, E. Esarey, C. B.Schroeder, J. R. Cary, and W. P. Leemans, Phys. Rev. Let. 100, 215004 (2008).4. A. J. Gonsalves, D. Panasenko, K. Nakamura, C. Lin, E. Monaghan, C. Toth, C. G. R. Geddes, C. B.Schroeder, E. Esarey, and W. P. Leemans, “Injection in a capillary-discharge waveguide using anembedded gas jet,” in this proceedings, AIP, 2008.5. E. Esarey, P. Sprangle, J. Krall, and A. Ting, IEEE Trans. on Plasma Sci. 24, 252–288 (1996).6. E. Esarey, B. A. Shadwick, C. B. Schroeder, and W. P. Leemans, “Nonlinear Pump Depletion andElectron Dephasing in Laser Wakefield Accelerators,” in Eleventh Advanced Accelerator Concepts,edited by V. Yakimenko, AIP, New York, 2004, vol. 737, pp. 578–584.7. C. Huang, V. Decyk, C. Ren, M. Zhou, W. Lu, W. Mori, J. Cooley, T. A. Jr., and T. Katsouleas, J.Comp. Phys. 217, 658–679 (2006).8. J.-L. Vay, Phys. Rev. Let. 98, 130405 (2007).9. B. Cowan, D. L. Bruhwiler, P. Messmer, K. Paul, C. G. R. Geddes, E. Esarey, and E. Cormier-Michel,“Laser wakefield simulation using a speed-of-light frame envelope model,” in this proceedings, AIP,2008.10. C. Nieter, and J. R. Cary, J. Comp. Phys. 196, 448–473 (2004).11. L. Gorbunov, and V. Kirsanov, Zh. Eksp. Teor. Fiz. 93, 509–518 (1987).12. D. L. Bruhwiler, J. R. Cary, B. Cowan, K. Paul, C. G. R. Geddes, P. J. Mullowney, P. Messmer,E. Esarey, E. Cormier-Michel, W. P. Leemans, and J. L. Vay, “New Developments in the Simulationof Advanced Accelerator Concepts,” in this proceedings, AIP, 2008.13. C. G. R. Geddes et al., Journal of Physics: Conference Series 125, 012002 (2008).14. T. Katsouleas, S. Wilks, P. Chen, J. Dawson, and J. Su, Part. Acc. 22, 81–99 (1987).15. P. Sprangle, B. Hafizi, J. R. Peñano, R. F. Hubbard, A. Ting, C. I. Moore, D. F. Gordon, A. Zigler,D. Kaganovich, and T. M. Antonsen, Phys. Rev. E 63, 056405 (2001).

Scaled simulations of a 10 GeV accelerator

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Scaled simulations of a 10 GeV accelerator

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