ARTICLESPUBLISHED ONLINE: 7 MARCH 2010 | DOI: 10.1038/NPHYS1538

Exploring laser-wakefield-accelerator regimes fornear-term lasers using particle-in-cell simulationin Lorentz-boosted framesS. F. Martins1, R. A. Fonseca1, W. Lu2, W. B. Mori2,3 and L. O. Silva1*Plasma-based acceleration offers compact accelerators with potential applications for high-energy physics and photon sources.The past five years have seen an explosion of experimental results with monoenergetic electron beams up to 1 GeV on acentimetre-scale, using plasma waves driven by intense lasers. The next decade will see tremendous increases in laser powerand energy, permitting beam energies beyond 10 GeV. Leveraging on the Lorentz transformations to bring the laser and plasmaspatial scales together, we have reduced the computational time for modelling laser–plasma accelerators by several orders ofmagnitude, including all the relevant physics. This scheme enables the first one-to-one particle-in-cell simulations of the nextgeneration of accelerators at the energy frontier. Our results demonstrate that, for a given laser energy, choices in laser andplasma parameters strongly affect the output electron beam energy, charge and quality, and that all of these parameters canbe optimized.

In laser-wakefield acceleration1–5 (LWFA), an intense and shortlaser pulse is sent through many Rayleigh lengths of tenuousplasma. The radiation pressure of the laser creates a plasma-

wave wake on which electrons can surf to very high energy. Inrecent experiments, the wake has been a fully nonlinear and three-dimensional (3D) structure that is now often referred to as a bubble.The wakefields (both accelerating and focusing) within this bubblehave ideal properties for accelerating electrons. The transverse, thatis, focusing, fields increase linearly with the radial distance andthe accelerating field is independent of the radial coordinate6,7.Although it is possible to excite quasi-linear wakes that are alsowide (width of the order of the wake wavelength or larger), muchof the recent research has been on creating nonlinear and weaklynonlinear wakes in which most of the plasma electrons are expelledradially. In addition to the recent advances in experimental resultsthere has also beenmuch progress on theory and simulation7–11.

The next generation of large-scale laser projects with powersabove a petawatt and energies above hundreds of joules12,13 willprovide new avenues for designing compact particle accelerators.It will also permit, for the first time, accessing the fully nonlinearregimes described in the recent theories8–10. Here, we address howto choose the plasma density and length, the lasermatching spot sizeand the laser intensity to obtain the desired electron beam energygain and output beam charge. Specifically, we identify the optimalparameters for using a 250 J laser to produce monoenergetic+-10GeV electron beams with nanocoulombs of charge in asingle LWFA stage. By using full-scale 3D particle-in-cell (PIC)simulations in a Lorentz-boosted frame, we show that self-injected11-GeV and externally injected 40-GeV electron beams can beachieved for such laser energies, provided that the pulse durationand spot size are appropriately chosen. These simulations representa revolutionary leap in numerical modelling capability.

A laser–plasma system can be configured with different physicalparameters to generate distinct accelerating structures. From energy

1GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal, 2Department of Electrical Engineering, University ofCalifornia, Los Angeles, California 90095, USA, 3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA.*e-mail: luis.silva@ist.utl.pt.

balance, the total energy in the accelerated electron beam will beless than the incident laser energy. For a monoenergetic beam, thetotal output energy is a product of the charge times the energy perelectron. Therefore, there is an important trade-off between the two,which can be explored in different physical regimes.

The physical parameters for three possible options with a250 J laser are shown in Table 1. These are identified from thecondition for self-guiding of the laser pulse and the forma-tion of the optimal accelerating structure (a0 & 4), and the dis-tinct regimes identified for 2 . a0 . 2ω0/ωp (henceforth theblowout regime)10, and for a0 & 2ω0/ωp (henceforth the bub-ble regime). The laser spot size, w0, and full-width at half-maximum pulse length, τFWHM, are defined such that the laserenergy is given by energy = I (π/2)w2

0 τFWHM√π/8ln2, where I

is the laser intensity. The normalized vector potential of alinearly polarized laser eA/mc2 ≡ a0 can be obtained througha0= 0.853×10−9λ(µm)

√

I (W cm−2).The first option is to compress the laser to as short a pulse

length as possible, τ . 30 fs, and then choose a plasma density,ne ' 1019 cm−3, so that the pulse length roughly matches halfof a plasma period. This is the idea of the bubble regime8,9,where the spot size is also roughly matched to the bubble radius,w0' cτ '

√a0. For these conditions, the laser intensity of the 250 J

laser would be 6× 1021 Wcm−2. In this regime, the electrons arecontinuously injected and a quasi-monoenergetic beam is formedas the beam dephases in the wake and rotates in phase space. Thereis tremendous beam loading and the loaded wake is noisy, with asignificantly reduced amplitude. The acceleration length is typicallysmall owing to the quick depletion of the laser. This regime leads toa lower energy output beam, typically maximizing the beam charge,and it can be simulated using standard PIC techniques.

A second regime was proposed in ref. 10, where it was shownthat a laser pulse with a moderate intensity, I . 1019 Wcm−2,propagating in a plasma with a density ne . 1018 cm−3 leads to a

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1538

Table 1 | Laser/plasma parameters for the different LWFAregimes of a 250 J laser.

Self-guiding External-guiding

Bubble-SI* Blowout-SI Blowout-EI†

Lasera0 53.0 5.8 2.0Spot (µm) 10 50 100Duration (fs) 33 110 160Power (PW) 9.4 2.8 1.4

PlasmaDensity (1016 cm−3) 1,500 27 2.2Length (cm) 0.25 22 528

Electron beam(simulation)

Energy (GeV) 3 (3.4) 13 (5–13) 53 (40)Charge (nC) 14 (25) 2.0 (0.6–2.2) 1.5 (0.3)

Parameters are obtained from refs 9 and 10, for the bubble and blowout regimes, respectively.Ref. 9 uses a circularly polarized laser. Here, for consistency with the other cases, we use linearlypolarized lasers and modify their formulae appropriately. The decrease in laser intensity from thestrongly nonlinear regime (a0=53) to the weakly nonlinear regime (a0=2) is accompanied by asteep increase in the acceleration length, and thus on the computational modelling requirements,because the laser wavelength (0.8 µm) must always be resolved in a full PIC simulation.

*Self-injected (SI) electrons.†

Externally injected (EI) electrons.

more controlled and stable blowout of the electrons when propermatching of the laser spot size and pulse length is achieved; thelength over which the pump is depleted can nearly match thedephasing length and self-guiding is possible. Electrons can beself-injected or externally injected into the rear of the bubble whilenot radically modifying the accelerating structure. The injectedelectrons can clamp further injection through beam loading, leadingto beams with a smaller energy spread and a better quality14. Fora 250 J laser, this regime can be achieved for a pulse length of110 fs and a matched spot size of 50 µm, with a peak intensityof 7.8× 1019 Wcm−2. The matched conditions ensure maximumacceleration of the injected particles, and therefore the maximumenergy of the output beam. If self-guiding is desirable, then a0 stillneeds to be sufficiently high7 (a0' 5.8), and this is accompanied byself-injection, which typically occurs at lower a0 (a0 ' 3–4; refs 7,14, 15) for matched conditions (electron beams can be externallyinjected and this could load the wake to prevent any self-injection).However, the numerical study of this regime becomes a grandcomputational challenge owing to the need to simulate accelerationdistances over 0.1–1m, while still resolving the laser wavelength(that is, more than 106 laser wavelengths of a 0.8 µm laser).

To further increase the final output energy (with less charge),a third option for the blowout regime was also proposed inref. 10, where lower laser intensities are used and the laser isexternally guided in a plasma channel. For a 250 J laser, thisregime can be achieved for a pulse length of 160 fs and amatched spot size of 100 µm, corresponding to a peak intensity of8.6×1018 Wcm−2; no self-trapping occurs and externally injectedelectrons are required. The most effective injection mechanismis still under intensive research. Conventional radiofrequencyaccelerators can provide 10–100-fs pulses with nanocoulombs ofcharge at gigaelectronvolt energies; however, such an injector wouldbe expensive, large and synchronization could be challenging.All optical alternatives have been proposed with colliding pulseinjection16,17 and density gradient injection18,19. In addition, theacceleration length is between 1 and 10m, which implies the needfor more advanced plasma sources to generate long and stableplasma channels. As a consequence of the longer propagationdistances involved, the computational requirements increase by

another order of magnitude as compared with the second regime,andmodelling this option requires 104 timesmore processing hoursthan modelling the first option.

In this article, wemodel all of these three options for a fixed laserenergy. Full 3D PIC simulations demonstrating a +10GeV energystage in the regime of ref. 10 are computationally challenging andhave never been carried out: a standard simulation in the laboratoryreference frame can takemore than one year on 512 processors, with>109 particles simulated for >108 time iterations. However, it wasrecently suggested that it is possible to minimize the time and spacescales to be numerically resolved, by carrying out the simulations inan optimal Lorentz frame (different from the laboratory frame)20.This concept can be generalized to laser–plasma interactions as longas backscattered radiation is not important. In a LWFA simulation,the longitudinal grid resolution is set by the laser wavelength, whichis usually the smallest relevant physical structure. The numberof time steps is set by the longest length, which is the plasmalength, that is, the acceleration length. The disparity in plasmalength to the laser wavelength can be eliminated if the simulationis carried out in a frame moving with the laser pulse in the plasma.The number of cells remains the same because the number oflaser wavelengths within the laser is an invariant. However, theplasma length contracts and the laser wavelength (hence cell size)increases, leading to a shorter acceleration length and a larger timestep. Therefore, when the plasma length is long compared withthe laser pulse duration, the computational gains compared withthe standard PIC method in a moving window are proportionalto γ 2(1+ β)2, where γ = (1− β2)−1/2 is the relativistic factorof the boosted frame20.

We begin by presenting results from a standard full PICsimulation of the bubble regime in the laboratory frame using theparameters from Table 1. The numerical parameters are detailed inthe Methods section. Figure 1 shows a time sequence of the bubble,the laser envelope, the wakefield, the longitudinal phase space andthe energy spectrum. The final spectrum, Fig. 1h, shows a final peakenergy of 3.4 GeV (close to predictions in ref. 9), with∼11% energyspread and 16.7 nC of charge FWHM, confirming the high chargeand relatively low final electron energy associated with this range ofparameters. However, the emittance of the beam (electrons between3.0 and 3.7GeV) is close to 0.5×103 mmmrad, owing to the hightransverse momentum acquired by the injected electrons. In fact,the physics involved in this regime is very nonlinear and complex.As can be seen in Fig. 1a–d, the maximum bubble radius variesconsiderably as the laser propagates through the plasma (we discussthis in more detail later). In addition, electrons are continuouslyinjected, leading to very strong beam loading. This can be seen inthe lineouts of the accelerating field (E1) and in the plasma densityslices, where the wakefield is noisy and the sheath in the bubbleis not well defined. The front part of the self-injected electronshas dephased (reached decelerating fields) within only 1mm ofpropagation (Fig. 1b). These electrons then begin to deceleratewhile the rest of the bunch continues to accelerate. The peak energysaturates after the laser is nearly pump depleted. Interestingly, thepoint where the E1 changes sign moves forward faster than thelaser pulse, which slips backward at a rate between c − vg andthe etching rate in ref. 10, where vg is the linear group velocityof light in a plasma.

The total amount of charge between 3.0 and 3.7GeV is 25 nC,which is above the value predicted in ref. 9 (14 nC). The highercharge can be understood from ref. 11, where it was shown thatthe product of charge times loaded wakefield depends on themaximum blowout radius. Therefore, the same wakefield can beloaded with more charge with the trade-off being that the loadedgradient is lower. If in equation (6) in ref. 11 one uses a maximumblowout radius of Rb ≈ 2.5w0, and assumes the loaded wakefieldis half the theoretical peak value (eE1/mcωp)≈ kpRb/4 then one

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NATURE PHYSICS DOI: 10.1038/NPHYS1538 ARTICLESx 2

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Figure 1 | Simulation results for a LWFA in the bubble regime with a 250 J laser in the laboratory frame. The simulation was carried out in the laboratoryframe, that is, the rest frame of the plasma. a–d, 2D central slices of electron charge density in the x1x2 plane (black to red colour table) and laser envelope(red to white) after 0.3, 1.0, 2.1 and 2.6 mm into the plasma; the laser propagates to the right. White lineouts represent the longitudinal electric field (E1)down the centre of the simulation box. One tick mark corresponds to 5mcωp/e. e–h, Longitudinal electron phase spaces; the red lines on the right axiscorrespond to the energy spectra of the plasma electrons. The injected charges FWHM for the different temporal snapshots are, respectively, 4.6 nC,26.0 nC, 17.3 nC and 16.7 nC. The energy spreads are 22, 25, 11 and 11%. At the end of the simulation, the bunch duration is 30 µm FWHM. The phasevelocity of the bubble vφ (the velocity of the spike in E1), measured in terms of the parameter α, such that vφ = 1−αω2

p/ω20, varies between 0.42 and 0.80

for different propagation distances. Linear theory corresponds to α= 1/2 and the nonlinear theory for moderate a0 in ref. 10, α= 3/2.

gets 29 nC of charge. A detailed comparison to the beam loadingtheory is difficult as the wakefield is not flat and it evolves overthe propagation length. Furthermore, charge is continually injectedso the injected beam becomes longer. Its length eventually exceedsthe initial diameter of the bubble; therefore, the wakefield becomeselongated and deviates from its initial spherical shape. Furthermore,as the front of the beam dephases, its own space charge cancontribute to the wake (not load it) and this can further elongate itsshape14. Nevertheless, despite the complicated and highly nonlinearevolution, the energy spectrum has a surprisingly ‘monoenergetic’bump near the peak energy. Although this regime does not favourhigh electron acceleration or good beam quality, it might provide avery compact accelerator for other applications21.

Next, we present results from a simulation (in a boosted frame)of the self-guiding/self-injection blowout regime (parameters inTable 1). This simulation confirms the predictions in ref. 10 thatenergies in excess of 10GeV and bunches with a few nanocoulombs(Fig. 2) are possible in this regime with 250-J lasers. As can be seenin both the lineout of E1 and the 2D slice of the plasma densityin Fig. 2b, the wakefield and sheath are much cleaner than for theprevious case. In addition, the loaded wakefield is also relativelyflat (see the deviation from the dashed line in Fig. 2c) and theblowout radius remains relatively constant and roughly equal theinitial laser spot size for the entire acceleration length. Althoughnot pronounced (see below), the evolution in time of the laserspot size and vector potential combined with the beam loadingfrom previously injected electrons leads to more than one injectionperiod in the first blowout region (bucket). The end result is thatthere are three distinct bunches in space and in energy (Fig. 2c),with energies between 5 and 13GeV, with energy spreads between8 and 20% and with charges FWHM between 0.6 and 2.2 nC. Thetheory in ref. 11 predicts 4 nC of charge if the loaded wakefieldis half of the peak value, which agrees well with the total charge(all injected electrons) in all three bunches (4.8 nC), correspondingto an overall efficiency of 16%. The obtained emittances range

from 5 to 25mmmrad, where the higher values are in the laserpolarization plane.

The phenomenological theory and matching conditions inref. 10 are meant to be a guide. There is much room for tuningthe laser spot size, pulse length and axial profile to optimize thebeam energy and charge (we note that one could externally injectparticles in this regime as well), energy spread and efficiency. Forexample, we have carried out several extra simulations in whichthe laser spot size (we kept the laser energy and axial profile thesame) was increased and decreased by up to 20%. For larger initialspot sizes, the blowout radius is closer to the spot size, but in eachcase three bunches were injected. To illustrate the potential for finetuning, when the spot size was increased to 55 µm, themicro-bunchat 11GeV had 0.3 nC of FWHMcharge, energy spread of 3.7% and anormalized emittance of 3.4 and 9.2mmmrad in the two transverseplanes.More detail is given in captions of Figs 2 and 4.

Finally, we present results from a LWFA simulation in theweakly nonlinear blowout regime using external guiding of thelaser using a plasma channel. The use of a guiding structure relaxesthe requirements for the laser intensity, and leads to higher beamenergies (but less charge).Moderate intensities (a0'2) can be used,such that the optimum accelerating structure associated with theblowout region is excited without the occurrence of self-injection10.Recently, in ref. 11, a theory was presented for how to optimallyplace an external beam. The analytical results are, however, strictlyvalid for large blowout radius, so in this regime simulations can bevery useful to optimize the amount and placement of the charge.

This weakly nonlinear regime is characterized by longer laserpulses and longer propagation distances (5.28-m-long plasma, seeTable 1). Therefore, the full PIC simulation of this scenario is notpossible at present with standard numerical techniques, because itneeds to cover more than seven orders of magnitude in space andtime. This regime has also been studied using the quasi-static PICcode, QuickPIC (refs 22, 23), which allows for faster computations,but cannot be used to study laser propagation near pump depletion

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1538

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Figure 2 | Simulation results for a LWFA in the self-guiding/self-injection blowout regime of a 250 J laser. The simulation is carried out in a boostedframe, where the plasma moves to the left with γ = 10. The longitudinal distances presented correspond to the position of the simulation window in theboosted frame. As the plasma is contracted and moving to the left at nearly the speed of light, the corresponding laboratory position of the simulation boxrelative to the plasma is given by xlab= (1+β)γ xb, where γ = 10 is the relativistic factor of the boost. a, 3D physical shape of the wakefield in the boostedframe. Green and yellow iso-surfaces represent plasma density, orange iso-surfaces represent the electric field in the laser polarization plane (x3) and thedark blue dots constitute the injected electrons inside the formed wakefield structure. b, 2D central slice of the plasma density, with injected electronscoloured by energy (blue: low, red: high). The black line represents the longitudinal electric field lineout at the centre of the simulation box. The injectedpeak energies are 5.3 GeV, 7.9 GeV and 11.3 GeV, with energy spreads of 19%, 20% and 9% and FWHM charges of 0.7 nC, 2.2 nC and 0.6 nC, respectively.c, Electron energy as a function of longitudinal position at the end of the plasma jet. The red line on the right axis corresponds to the energy spectrum ofthe injected electrons.

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Figure 3 | Simulation results for a LWFA in the external-guiding/external-injection blowout regime of a 250 J laser. As in the self-injection case, thesimulation is carried out in a boosted frame, where the plasma moves with γ = 10. a, Illustration of the plasma channel that guides the pulse over 5.28 m,providing stable accelerating conditions for an externally injected beam. b, Peak energy evolution with propagation distance for the trailing bunch placed atthe back of the laser wakefield. Vertical line segments represent the energy spread at a given propagation distance, for the beam loaded case (black) and atest particle case that does not affect the accelerating structure (grey and there are fewer points). The inset represents the longitudinal electric fieldcreated by the laser pulse together with the tailored injected electrons—solid line. The profile of the trapezoidal trailing beam is defined to generate a flataccelerating structure, and thus to minimize the energy spread of the accelerated beam11. The dashed grey line corresponds to the accelerating fieldgenerated by the laser pulse alone.

distances. The boosted frame bridges the gap between the differentscale lengths, and opens the possibility to carry out these full-lengthfully kinetic simulations.

The plasma channel provides the external guiding of the laserpulse, and thus ensures that a stable wakefield is generated. Such aquasi-static structure is required for the controlled acceleration ofan externally injected beam to very high energies. The numericalexperiment was done with a properly tailored electron beam of80 pC that initially flattened the wake11 (Fig. 3b, inset). As the beamphase slips forward the wake does not remain flat, the gradientdecreases and the energy spread increases. In addition, in thisregime the phase velocity decreases as the laser pump depletes.This is shown in Fig. 3b, where the energy spread of the beam is

shown as a function of propagation distance. The FWHM energyspread at 30GeV is ∼10%, increasing to 20% at the final 40GeV.(For the non-shaped bunch, the value is already 25% at 20GeV.)From the simulation data we determine an average acceleratingfield that is smaller than the Emax/2 assumed for the scaling lawsin ref. 10. The beam loading decreases the maximum electric fieldsampled by the electrons, and thus lowers the output energy belowthe estimated 53GeV (C. Huang, in preparation). We have scannedthe size (length and radius) of the injected beam using simulationsfor short propagation distances to study the initial wake formationand have found that the maximum charge that could be loadedin this configuration is ∼0.3 nC. Assuming that a bunch withthis charge would be accelerated to 40GeV gives an efficiency

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NATURE PHYSICS DOI: 10.1038/NPHYS1538 ARTICLES

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Figure 4 | Comparison of laser/plasma matching conditions for the three regimes studied. a, Plots of the plasma bubble radius in the transverse directionnormalized to the initial laser spot size as a function of the laser propagation distance normalized to the total acceleration length, Lacc. The bubble regime ischaracterized by a substantial deviation of the radius compared with the spot size (w0= 10 µm) and by a significant variation with propagation distance.The matched parameters of the blowout regime (Table 1) lead to a more stable ion channel and laser guiding (w0= 50 µm). Improved guiding in thisself-injection regime may be obtained with minimal adjustment of the initial laser parameters, as suggested by the results obtained with 10% variation ofthe initial laser spot size (dashed lines). Extremely stable wakefields and laser guiding can be obtained with a lower intensity (w0= 100 µm), for whichexternal guiding of the laser is provided by a pre-formed plasma channel. b, Energy spectra for the blowout regime with w0= 50±5 µm. The small laserspot variations influence the guiding and injection process, with consequences on the final electron energy, energy spread and charge.

of∼5%. However, there is much room to optimize the efficiency.For example, the laser pulse could be shortened (and even shaped)and an axial density profile could be used to more closely matchthe dephasing to pump depletion lengths. This is indicated by thefinal energy of the laser after the 5.28m, which is nearly 50%of the initial value.

The stability of the three different options can also be assessedby examining the evolution of the radius of the wake (Fig. 4a).The bubble regime is accompanied by strong variations in thebubble radius: it is initially 1.75 of the laser spot size, and thenrapidly increases to values nearly three times the initial spotsize. Eventually, it begins to decrease as the laser pump depletes(Lpd≈ cτω2

0/ω2p= 1.6mm). For the blowout regime, the maximum

blowout radius remains relatively stable: it starts out at within 20%of the initial spot size, indicating thematching condition in ref. 10 isa good guide. Figure 4 shows that the stability of the blowout radiusis slightly better for the larger initial spot size. For a channel-guidedlaser, the blowout radius (and laser spot size) changes very little.The matched conditions guarantee that the maximum energy isdetermined by the dephasing length, and thus themaximum energyof the output beams and the efficiency are optimized. In Fig. 4b,the final electron spectra is plotted for the three different spot sizesin the self-guiding regime of the blowout case. This reinforces thepoint that it is possible to tune and optimize the charge, energyand energy spread, with potential to obtain high-quality beamsat the energy frontier.

MethodsThe simulations were carried out with OSIRIS (ref. 24), a 3D fully relativistic PICcode, which has been widely used to simulate various LWFA scenarios (see, forinstance, refs 4, 14, 25). In the PICmethod, the trajectories of particles are advancedusing the relativistic equations of motion. Currents and charge densities from theparticles are deposited onto a grid, and the electric andmagnetic fields are advancedusing finite-difference equations on the grid. The simulations were carried out inthe laboratory frame and in Lorentz-boosted frames. In this section, we describe theboosted-frame technique, give the numerical parameters for each simulation andthen define how the charge and emittances were calculated.

When carrying out a PIC simulation in a different Lorentz frame, the electricfield, magnetic field and particle quantities must be transformed and initializedproperly. The relativistic PIC algorithm, which is covariant, then evolves the systemregardless of the Lorentz frame. Density and momentum transformations for theplasma are straightforward because we typically initialize particles at rest. The initialelectromagnetic fields (except the laser) are initially null. To initialize the laser

pulse, a Lorentz transformation of a Gaussian beam must be carried out includingthe fact that different parts of the laser pulse now are focused to different positionsand, as expected from length contraction, the Rayleigh length is reduced by a factorof γ (where γ is the relativistic factor of themoving frame).

The boosted-frame technique was implemented in OSIRIS, and the completedetails can be found in a separate manuscript26. Several analyses were carried outfor benchmarking purposes. These include direct laboratory/boost comparisons,benchmarks with QuickPIC (ref. 22) in the laboratory frame for several LWFAregimes and direct comparison with experimental results27. The benchmarksinclude cases where self-injection and external injection occurs. The self-injectioncases are more challenging. Results show a strong quantitative agreement of theinjected charge and of the final bunch energy. Overall qualitative agreement is alsoobserved for injection thresholds, accelerating gradients and bunch shape. A setof distinct numerical algorithms and parameters was also tested, namely a scanof boost speeds with distinct resolutions, higher order field solvers28, linear andquadratic particle shapes29 and modified Boris pusher30 was carried out. Resultsshow consistent convergence.

Similarly to the laboratory frame, simulations in a boosted frame are alsocarried out in the moving-window configuration (that is, the simulation boxtravels at the speed of light) to further reduce the computational requirements.The moving window also ensures backward-propagating radiation exits morerapidly, avoiding noise build-up. For large boost velocities (γ & 10), however, themoving window might be insufficient to remove the backward radiation, and fieldsmoothing must be used. For the boosted simulations presented here, we have useda binomial filter in the longitudinal direction, applied to the electric and magneticfields every five iterations. To minimize the impact on the well-resolved andimportant wavelengths, the filter is compensated to ensure minimal attenuation ofthe central laser frequency for the given numerical grid26.

The simulation of the first scenario (Table 1, first column) was carried out inthe laboratory frame with a moving window of size (98 µm)3. The computationalwindow has 4,000×300×300 cells with periodic boundary conditions in thetransverse directions. A total of 1.4×109 particles with second-order shapes29 werepushed for 105 iterations, corresponding to 0.25 cm of pre-formed plasma (densityis n0=1.5×1019 cm−3). Current smoothing and compensation are also used. Giventhe ultrahigh laser intensity, mobile hydrogen ions were used.

The self-guiding/self-injection in the blowout regime (Table 1, secondcolumn) was carried out in a boosted frame (γ = 10), with a moving windowof size 0.31× 0.05× 0.05 cm3, divided into 7,000× 250× 250 cells. A totalof 3.5× 109 particles with second-order shapes were pushed for 3× 104iterations, corresponding to 2.17 cm of pre-formed plasma (n0 = 2.7×1017 cm−3)in the boosted frame (21.7 cm in the laboratory). Current smoothing andcompensation are also used.

The external-guiding/external-injection in the blowout regime (Table 1, thirdcolumn) was also carried out in a boosted frame (γ = 10), with a moving windowof size 0.65×0.11×0.11 cm3 divided into 10,000×128×128 cells. A total of1.3×109 particles with second-order shapes were pushed for 5×105 iterations,corresponding to 5.28m of pre-formed plasma in the laboratory frame. Currentsmoothing and compensation was used. A plasma channel was used with a plasmadensity on axis of n0 = 2.2×1016 cm−3 and a transverse plasma profile given by

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1538

n= n0(1+0.0488r2), with r being the transverse distance to the channel centre,in c/ωp units. An externally injected beam is simulated as a 80 pC Gaussian beamplaced at the position of higher acceleration, with an initial energy of 250 MeVin the laboratory frame.

The number of particles in the boosted-frame simulations refers to electronsand ions, although ion velocity is not updated to reduce computational cost.We emphasize that the boosted-frame scheme strongly reduces the number ofiterations required, but that the number of particles to process at each timestep does not change from the laboratory case, unless more particles per cell areused to increase statistics.

When calculating the charge in a main or micro-bunch, the peak energyis determined and then the charge within the FWHM of this energy is used.In other cases, mentioned explicitly, the total charge within a minimum andmaximum is used. The normalized beam emittance in the xi direction, expressedas εN ,xi =

√〈x2

i 〉〈p2xi 〉−〈xipxi 〉2/mc , is estimated with an integration over all the

numerical particles in the respective bunch.Finally, the laser beam is simulated with a linearly polarized, diffraction-limited

pulse focused at the entrance of the plasma. The longitudinal profile of the laserelectric field is symmetric and given by 10τ 3−15τ 4+6τ 5, with τ =

√2t/τFWHM,

and τFWHM being the pulse duration at FWHM. The transverse profile of the laser isGaussian with a spot defined at FWHM in intensity.

We have verified that the energy loss of the accelerated beam resulting fromradiation emission is negligible when compared with the energy gain. Using theestimates of ref. 10 for the parameters of the externally injected beam at 40GeV,the ratio of energy loss resulting from synchrotron radiation to the energy gain inthe accelerating field is of the order of 2×10−3. For the self-injected electrons, thisratio is 2×10−4 for the 11GeV bunch.

Received 18 August 2009; accepted 26 January 2010;published online 7 March 2010

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AcknowledgementsThis work was partially supported by Fundação Calouste Gulbenkian, by Fundação paraa Ciência e a Tecnologia, under grants PTDC/FIS/66823/2006 and SFRH/BD/35749/2007(Portugal), Laserlab-Europe/LAPTECH, EC FP7 Contract No. 228334, by theUS Department of Energy (DOE) under grant numbers DE-FC02-07ER41500,DE-FG02-92ER40727 and DE-FG52-09NA29552, by the NSF under grant numbersPHY-0904039 and PHY 0936266 and by the University of California Lab ResearchProgram. S.F.M. and L.O.S. would like to thank KITP, where a part of this research wasconcluded, partially supported by the National Science Foundation under Grant numberPHY05-51164. We thank the DEISA Consortium (www.deisa.eu), co-funded throughthe EU FP6 project RI-031513 and the FP7 project RI-222919, for support within theDEISA Extreme Computing Initiative. The simulations presented here were producedusing IST Cluster (IST/Portugal), Dawson cluster (UCLA), Jügene (Jülich, Germany)and NERSC supercomputers.

Author contributionsS.F.M., code development, simulations, data analysis and manuscript preparation;R.A.F., code development; W.L., data analysis; W.B.M. and L.O.S., result analysisand manuscript writing.

Additional informationThe authors declare no competing financial interests. Reprints and permissionsinformation is available online at http://npg.nature.com/reprintsandpermissions.Correspondence and requests formaterials should be addressed to L.O.S.

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