Towards the Optimisation of Direct Laser Acceleration
A.E. Hussein, A.V. Arefiev, T. Batson, H. Chen, R.S. Craxton, A.S. Davies, D.H. Froula, Z.Gong, D. Haberberger, Y. Ma, P.M. Nilson, W. Theobald, T. Wang, K. Weichman, G.J. Williams, L.Willingale
TTowards the Optimisation of Direct LaserAcceleration
A. E. Hussein , ∗ , A. V. Arefiev , T. Batson , H. Chen ,R. S. Craxton , A. S. Davies , D. H. Froula , Z. Gong ,D. Haberberger , Y. Ma , P. M. Nilson , W. Theobald ,T. Wang , K. Weichman , G. J. Williams , L. Willingale Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, MI 48109,USA University of California San Diego, San Diego, CA 92093 USA Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Laboratory for Laser Energetics, University of Rochester, Rochester, NY 14623,USA Center for High Energy Density Science, The University of Texas, Austin, TX78712, USA ∗ Present address: Department of Electrical and Computer Engineering, University ofAlberta, Edmonton, AB, T6G 2R3, CanadaE-mail: [email protected]
Abstract.
Experimental measurements using the OMEGA EP laser facilitydemonstrated direct laser acceleration (DLA) of electron beams to (505 ±
75) MeVwith (140 ±
30) nC of charge from a low-density plasma target using a 400 J, picosecondduration pulse. Similar trends of electron energy with target density are also observedin self-consistent two-dimensional particle-in-cell simulations. The intensity of the laserpulse is sufficiently large that the electrons are rapidly expelled from along the laserpulse propagation axis to form a channel. The dominant acceleration mechanismis confirmed to be DLA and the effect of quasi-static channel fields on energeticelectron dynamics is examined. A strong channel magnetic field, self-generated bythe accelerated electrons, is found to play a comparable role to the transverse electricchannel field in defining the boundary of electron motion. a r X i v : . [ phy s i c s . p l a s m - ph ] J a n owards the Optimisation of Direct Laser Acceleration
1. Introduction
Modern laser technology and the realization of high-intensity, short-pulse lasersystems using chirped-pulse amplification [1] has expanded the frontiers of physics forfundamental research and novel technological applications including laser-based schemesfor charged particle acceleration. In all laser-plasma interactions, the pivotal step andbasis of all subsequent phenomena is governed by the transfer of energy between thelaser fields and plasma electrons. The generation of copious, high-energy electrons iskey for driving secondary particle and radiation sources, such as energetic ions [2, 3, 4],hard X-rays [5, 6, 7] neutrons [8, 9] and electron-positron beams [10, 11]. Elucidatingand optimizing the dynamics of electron heating and acceleration for different regimes ofplasma density and laser pulse duration is central to the development of these sources.A laser field can propagate through a plasma if the electron density n e is belowthe critical density, n crit ≡ m e ω / (4 πe ), where m e is the electron mass, ω is thelaser frequency and e is the electron charge. In this regime, the mechanism for laser-driven electron acceleration is highly dependent on laser pulse duration ( τ L ). Thedominant electron acceleration mechanism can be inferred from the relationship between τ L and the plasma frequency, ω pe = (cid:112) πn e e /m e . When τ L (cid:39) /ω pe , as is typicalfor femtosecond-duration pulses and low-density targets, laser wakefield acceleration(LWFA) dominates, and electrons can be accelerated up to many-GeV energies by thelongitudinal electric field of electron plasma waves [12]. The wakefield structure formsbecause electrons within the focal region of the laser pulse experience the ponderomotiveforce, expelling them from regions of high intensity to form a cavity containing theheavier ions. Once the laser pulse passes, the electrons return towards the axis, and thewakefield structure is formed. LWFA electron beams can be high-energy (many GeV)[13, 14, 15, 16], mono-energetic [17, 18, 19] and low divergence (on the order of a fewmilliradians [20]), however, the total charge of the electron beam is typically low, on theorder of tens of picocoulombs [21]. Higher charge electron beams are preferable for thegeneration of secondary sources.For picosecond (ps) duration pulses, the laser pulse duration is typically muchgreater than the plasma period. At low intensities, ps pulses can accelerate electrons via self-modulated laser wakefield acceleration (SM-LWFA), which has been shownto produce electron beams with charge on the order of tens of nanocoulombs (nC)[22, 23, 24]. However, as the laser intensity is increased (typically above 10 W/cm for a λ = 1 µ m laser), the sustained ponderomotive force means the electrons areunable to return into the electron depleted region, therefore a wakefield is unable toform (except perhaps at the rising intensity of the leading edge of the laser pulse), andinstead an ion channel is established. Within this channel, strong radial space chargefields can be present and direct laser acceleration (DLA) mechanisms become dominant.Eventually, the radial electric field leads to a “Coulomb explosion” of the ions [25, 26],reducing the strength of the radial electric fields of the ion channel.The basis of DLA is the transfer of energy directly from the laser to plasma owards the Optimisation of Direct Laser Acceleration v × B force [27, 28].In vacuum, the maximum energy an electron initially at rest can gain directly from aplane electromagnetic wave, with normalized amplitude increasing from zero to a isgiven by: γ vac = (1 + a /
2) [29], where a = | e | E / ( m e cω ) is the normalized amplitudeof a laser pulse with electric field E and γ is the relativistic factor. In this vacuum case,a λ = 1 µ m wavelength pulse with a = 7 should yield γ max = 25 . (cid:15) max (cid:39)
13 MeV).However, experimental measurements of DLA in a plasma have demonstrated electronenergies vastly exceeding this limit [30, 31, 32].Enhanced electron energy is largely attributed to the formation of the ion channelthrough the ponderomotive expulsion of electrons in the transverse direction. Thischannel evolves on the ion timescale and is associated with transverse and longitudinalelectric fields that are quasi-static relative to the timescale of electron motion. Whilethese fields are much weaker than the laser field, they can have a profound impact on thedynamics of electrons injected into the channel [33, 28, 34, 35, 29, 36], in particular bymitigating electron dephasing from the laser pulse. Under specific conditions in whichthe electron oscillation frequency matches the laser frequency, a resonance effect hasbeen postulated to occur, increasing the transverse momentum of the electron, which isthen transformed into longitudinal momentum through the v × B force. [28]. However,it has been previously shown [45] that the dynamics of an electron irradiated by a planewave within a static ion channel are non-linear, with strongly modulated eigenfrequencyleading to a threshold process rather than a linear resonance. Strong quasi-staticazimuthal magnetic fields are also generated through the driving of longitudinal electroncurrents by the intense laser pulse [7, 37, 38]. A sufficiently strong azimuthal magneticfield may play a role in reinjecting an escaping electron into the beam volume to undergofurther acceleration [27]. The impact of these fields has been largely neglected in favorof quasi-static electric fields in the context of DLA.While DLA electron beams are typically broadband and of lower peak energy thanLWFA beams [30, 31], this mechanism can produce high-charge electron beams ( ∼ owards the Optimisation of Direct Laser Acceleration ±
75) MeV and up to (140 ±
30) nanocoulomb charge, was experimentallyobserved. 2D PIC simulations demonstrate similar trends of electron energy gain asa function of target density, providing insight into key phenomena governing electronacceleration in this regime. Further, 2D PIC simulations demonstrate that the magneticfield of the plasma channel plays an important role in the confinement and subsequentacceleration of plasma electrons by the laser field. owards the Optimisation of Direct Laser Acceleration Figure 1. a)
Schematic of the experimental configuration showing the generation of anunderdense plasma plume using a long-pulse heater beam, a 1.0 ps beam for electronacceleration, a 263 nm optical probe beam, and the location of beam diagnostics. b) Layout of radiochromic film and the magnetic spectrometer for diagnosing theaccelerated electron beam (not to scale). c) An angular filter refractometry (AFR)image used to extract the plasma density profile.
2. Experimental setup
Experiments were performed at the OMEGA EP laser system at the University ofRochester Laboratory for Laser Energetics. A schematic of the setup is given in Fig. 1a).An underdense plasma target was produced using a single long-pulse UV heater beam(2.5 ± ± λ = 351 nm) with an 800 µ mdiameter super-Gaussian spatial profile focal spot incident on a flat CH foil (125 µ mthickness). Electron acceleration was driven using a (1.0 ± λ = 1.053 µ m owards the Optimisation of Direct Laser Acceleration ± f / a (cid:39) .
0. The electron beam pointing and divergence were recorded usinga stack of radiochromic film (RCF), positioned along the axis of the ps laser pulse ata distance of 8 cm behind the focal plane, as shown in Fig. 1b). The approximately16 mm thick stack consisted of 10 sheets of HD-V2 film followed by two sheets of MD-V2-55 film interleaved with aluminium filters, with a 100 µ m aluminium filter at thefront. The RCF stack was tilted 12 ◦ from normal to prevent back reflection of the laser.A hole in the center of the RCF stack allowed a direct line of sight to an absolutelycalibrated magnetic spectrometer (EPPS [46]) 48.6 cm away from the focal plane formeasurements of the electron spectrum along the axis of the main interaction beam.The plasma density was varied by changing the interaction height of the 1.0 pslaser pulse above the plane of the CH foil, within a range of (1.5 – 2.0) mm. The timingbetween the ns and ps beams was 1.7 ns for the lowest density presented here and 2.5 nsfor all others. The plasma density was measured by angular filter refractometry (AFR)[47], with example data shown in Fig. 1c). A fit to the data was found such that theplasma density profile can be approximated as a Gaussian function in two dimensions.In these experiments, peak plasma densities, n , ranging between (0.0095 - 0.11) n crit were investigated, where n crit = 1.0 × cm − for λ = 1.053 µ m. The quoted densityvalues refer to the peak density along the axis of the short-pulse laser in Fig. 1b). Giventhe interaction with the plasma plume at least 1 mm from the target and at late times inits evolution ( >
3. Particle-in-cell simulations
Two-dimensional PIC simulations using the
EPOCH code [49] (version 4.17.9) wereperformed to examine a laser pulse at relativistic intensity interacting with a plasmaof sub-critical density. The simulations were designed to match the conditions of theOMEGA EP laser system. The 1.053 µ m wavelength pulse was linearly polarized in y , and propagated in x . The time profile of the laser intensity was sin ( πt/τ ) with a τ L = 1 . τ = 2 . τ L ). Two co-incident pulses and focal spots wereused to approximate the experimental energy distribution in the focal plane: spot sizesof 3.4 µ m and 17 µ m, with laser intensities I = 3.78 × W/cm and I = 2.81 × W/cm , respectively, corresponding to vacuum normalized vector potentials a of5.5 and 1.5.The simulation box was (2200 × µ m, spanning x = [-900,1300] µ m and y =[-100,100] µ m, with 30 cells per λ in x , and 6 cells per λ in y and three macroparticles owards the Optimisation of Direct Laser Acceleration Figure 2. a)
Snapshot of the normalized laser intensity in vacuum, at t = − . b) Simulated plasma density profile, informed by AFRmeasurements, for a plasma characterized by n = 0 . n crit . per cell for both electrons and ions. As shown in Fig. 2, the laser entered the box at y =0, propagating from left to right, and traveled through vacuum before coming to focusin the plasma at x = 490 µ m. The peak plasma density, n , was scaled from the profileextracted from AFR measurements (Fig. 1c)) to yield peak densities of (0.005 - 0.1) n crit along the laser trajectory. Here we assume the same plasma length at each density. Ina vacuum simulation, the laser reached peak intensity at a distance of 410 µ m into thesimulation box and a time referenced as t = 0 ps. Fully ionized carbon ions were treatedas mobile and open boundary conditions were employed. Simulations were run at leastuntil the accelerated electron beam exited the simulation box. Up to 10 ps of interactiontime was simulated. A vertical probe plane placed at x = 1295 µ m in the simulationbox recorded the positions, momenta and weight of all electrons with energy exceeding10 MeV passing through the plane in the laser propagation direction (i.e. moving right).The electron and carbon densities, electromagnetic fields, current and particle locationswere recorded every 250 fs, and time-averaged over five laser periods. Subsequently,particle tracking was conducted for time intervals from (-0.25 to 6.25) ps, with outputsof fields, density, electron position and momentum every 25 fs.The aim of these 2D simulations is to investigate experimental trends and toillustrate the physics of electron acceleration using picosecond duration laser pulsesin underdense plasma, rather than for direct comparison with experimental results.While effects like diffraction and self-focusing may be underestimated in 2D simulations,recent work has shown that 2D simulations in this regime are qualitatively similar to 3Dsimulations [59], and therefore reasonably capture the key physical phenomena relevantfor interpretation of our experiments. owards the Optimisation of Direct Laser Acceleration Figure 3. a)
Experimental spectra of escaped electrons for different peak plasmadensities. b) Shot-to-shot variation in measured electron spectrum over five shotsobtained at an estimated peak plasma density of n = 0 . n crit . c) Total electronspectrum collected outside of the plasma at a probe 5 µ m from the end of thesimulation box, representing the beam exiting the plasma. d) Comparison betweenexperimental and simulated average electron energies at the probe, showing goodqualitative agreement (the left axis corresponds to experimental data).
4. Results and Analysis
Experimental electron energy spectra from five different plasma densities are shownin Fig. 3a). Significant acceleration of electron beams with a Maxwellian distributionextending to (505 ±
75) MeV is observed at a plasma density of 0.028 n crit , indicatingthe existence of an optimal density for the generation of energetic electron beams.The electron spectra are shown to be reproducible at nominally identical experimentalconditions ( n = 0.055 n crit ) with the average over five shots plotted in Fig. 3b),where the shaded region represents the standard deviation. In 2D PIC simulations,the escaping electron beam for electron energies >
10 MeV was diagnosed outside ofthe plasma, as the electrons passed through the probe at x = 1295 µ m (Fig. 3c), forcomparison with the experimentally measured beam. While the existence of an optimal owards the Optimisation of Direct Laser Acceleration Figure 4. a-c)
Radiochromic film (RCF) images, along the axis of the laser beamfor three plasma densities, shown in deposited dose, serving as a diagnostic of beampointing and divergence. The hole in the center of the RCF stack is aligned withthe 1.0 ps main interaction beam. The signal around the hole is due to line-of-sightradiation and is therefore ignored in calculations of total charge. d-f )
Electron angularenergy distribution from 2D simulations at t = 1.75 ps for electrons with energy > density in the simulations is not as dramatic as that observed in experiments, an optimaldensity for electron acceleration is also observed in simulations, with the highest energybeams produced at 0 . n crit .The average electron energy, evaluated from (10 - 300) MeV, is plotted for bothsimulations and experiments in Fig. 3d). According to Ref. [47], the total error inthe plasma density calculation using AFR is about ± ±
3) MeV at 0.028 n crit . For0.055 n crit , the electron spectrum is averaged over data from the five repeated shots ofFig. 3b), yielding an average energy of (33 ±
3) MeV, where the quoted error reflectsthe standard deviation from five repeated experiments. The average electron energyappears to plateau for the highest densities in the experiments, however this trend isnot reproduced in simulations, potentially owing to the underestimation of self-focusingand filamentation effects in 2D. owards the Optimisation of Direct Laser Acceleration A stack of RCF positioned along the laser axis of the ps pulse provided informationabout the pointing, divergence and charge of the resultant electron beam (see Fig. 1b).Scans of the final layer of MD-v2-55 film at the rear of the stack are shown from threedifferent densities in Figs. 4(a-c), in which the raw RCF signal was converted to dose,following Ref. [50]). The assumed center of the electron beam is indicated by ellipses inFigs. 4(a-c). Similar behavior has previously been attributed to space-charge-inducedion motion that can seed hosing-type instabilities [51]. Here, no such hosing is observedat low density in simulations. At low density, the formation of beamlets is reminiscentof forking in the electron beam at high energies, which has previously demonstratedas a characteristic of DLA [52, 59]. Such forking is also observed in our simulations,presented in Fig. 4(d-f), where the angular energy distribution is plotted for electronswith energy greater than 20 MeV.In all cases, the centroid of the electron beam or beamlets in Fig. 4(a-c) lies abovethe original laser axis (centered approximately on the RCF hole) by about (1.75 -2.5) ± ±
30 mrad for the lowest to highest density. These resultsindicate that the highest energy electrons may not be directed towards, or measured by,the magnetic spectrometer, which has a line-of-sight through the hole in the RCF film.The perturbation of the electron beam from the laser axis may be due to refraction ofthe laser pulse in the plasma gradient of the plume towards regions of lower density.The upward refraction of the laser beam is also present in simulations, evidencedby the angular distribution of the most energetic electrons in Figs. 4(d-e) above thelaser axis ( θ = 0). At the highest density (Fig. 4f), the angular distribution ofenergetic electrons appears to be nearly centered on the axis of the laser pulse; however,propagation instabilities such as filamentation are most severe at high density, and cansignificantly impact beam pointing. Simulations indicate that deflection of the electronbeam from the laser axis may also be due to the formation of sheath fields [4, 53] as thebeam exits the plasma.The beam divergence as a function of plasma density from experiments wascalculated by applying a Gaussian fits to the electron beam profiles on the RCF inFigs. 4(a-c), ranging from about (300-400) mrad FWHM with increasing with plasmadensity. Similar trends were reproduced in simulations. Multiple beamlets at low densitywere considered as a single beam for comparison. The total charge within the electronbeam was estimated by determining the total number of electrons with energy > n crit ), there was insufficient signal on the RCF above background tomake estimates, so no estimates for this density are provided. The number of electronsmeasured on the electron spectrometer for this density was three orders of magnitudelower than observed at 0.028 n crit . owards the Optimisation of Direct Laser Acceleration Figure 5.
Series of 2D PIC snapshots in the ( x, y )-plane at t = 0.75 ps. The laserenters the box at height y = 0 and propagates from left to right. a-c) The laserintensity, with E y normalized to E = 5.63 × statV/cm. d-f ) The electron density,averaged over 5 laser periods. g-i)
The electron phase space density N e for electronswith energy exceeding 20 MeV, where p x is the longitudinal momentum in arbitraryunits. Note that these snapshots have been cropped from the full simulation window. The highest estimated charge beam, reaching (140 ±
30) nC, was obtained at theoptimal plasma density (0.028 n crit ). The total charge in the beam for [0.11, 0.055,0.038] n crit were [111, 64, 70] nC, respectively, with a standard error of ±
30 nC definedby the variation from five repeated shots at 0.055 n crit . These charge estimates areconsidered as an upper bound on the total charge in the electron beam, as they do nottake into account spatial variation along the beam profile. However, the measured beamcharge is lower than previous results [32], which may be due to the presence of the RCFstack along the axis of the accelerated electron beam. Using the beam charge estimatesand average energy in the electron beam, the conversion efficiency into electrons withenergy greater than 10 MeV was estimated to reach a maximum of (0.48 ± n crit . Given the variation in electron beam pointing observed in Figs. 4(a-c), the highest energy electrons may not be measured on the electron spectrometer,thereby reducing the average electron energy and resulting in an underestimation of theconversion efficiency. Two-dimensional simulations provide insight into laser propagation effects and the roleof the quasi-static plasma channel on electron acceleration at different plasma densities.Snapshots of the laser intensity, electron density, and phase space density are given inFig. 5 at a simulation time of 0.75 ps, as the laser self-focuses in the plasma plume.It is clear that the plasma density plays an important role in laser self-focusing andinstability growth. At the lowest plasma densities, 0 . n crit , a clear channel is formed,but is associated with moderate electron acceleration (c.f. Fig. 3c). Additionally, at owards the Optimisation of Direct Laser Acceleration . n crit , with moderate filamentation of the laser pulse occurring withenhanced self-focusing relative to the lower density, but not impacting the ultimateformation of a channel propagation through the plasma at later times. At the highestdensity, 0 . n crit , electrons are stochastically accelerated in the first few picosecondsof the interaction (c.f. Fig. 6c)). Subsequently, the propagation becomes unstable,resulting in filamentation and transverse break-up of the laser pulse. When a plasmachannel cannot be formed, due to high levels of filamentation as observed at high density,there is no guiding of electron beams for enhanced electron energy gain from the laserfield. However, instability growth at the beginning of the interaction can stochasticallyaccelerate electrons, potentially impacting electron injection and pre-acceleration.The phase space density of electrons with energy exceeding 20 MeV with respectto the longitudinal position and momentum is shown in Fig. 5(g-i), indicating thatelectron acceleration occurs along the length of the laser pulse at all densities. While abubble structure can be observed at the leading edge of the laser pulse in Fig. 5(d,e),the sustained strength of the ponderomotive force prevents the formation of a plasmawave or wakefield, and the electron channel density becomes almost completely depletedalong the laser axis. Further, as is evident in Fig. 5(g-i), the majority of electrons arebeing accelerated within and along the cavitated channel rather then at the leadingbubble structure. The temporal and spatial dynamics of individual electrons provide further details on theacceleration process. The electron energy distribution in Fig. 6(a-c), sampled at timeintervals of 250 fs from (-2.25 to 7.25) ps, demonstrates electron energy gain from 20MeV up to >
200 MeV over 2 ps. For all densities, the electron energy saturates andthe energetic electron beam exits the box at approximately 6 ps.To investigate the behavior of energetic electrons throughout this process,individual electron tracking was performed. Electrons with energy greater than 20MeV, and a maximum energy
E > [160 , , n = [0.01, 0.02, 0.1] n crit ,respectively, at t = 2.75 ps were tracked from (-0.25 - 2.75) ps, with outputs every 25fs, to investigate differences in their trajectories close to the maximum acceleration.The momentum gain of examples of these electrons is shown in Fig. 6(d-e). For n = 0 . n crit , the electron undergoes clear periodic oscillations under the action of thelaser and quasi-static channel fields, gaining energy with each cycle. At lower density, theelectron is subject to weaker quasi-static channel fields, and undergoes fewer oscillations,here achieving a lower electron energy over the same period of time. At high density, thetrajectory of energetic electrons is chaotic and unstable, indicating likely energy gain bystochastic processes [31] associated with self-focusing and growth of the filamentation owards the Optimisation of Direct Laser Acceleration Figure 6.
Electron energy distribution and particle tracking from 2D PIC simulations. a-c)
The energy distribution function of all electrons as a function of time. d-f )
Electron trajectories in momentum space ( p x , p y ) for randomly selected high energyelectrons, from t =(-0.25 - 2.75) ps, with outputs every 25 fs. g) Work done by thetransverse and longitudinal electric fields on an electron achieving maximum energy(
E >
180 MeV) at t = 4.5 ps and n = 0.02 n crit . h) The components of energy gainin ( W x , W y ) space at t = 4 ps for the optimal density (0.02 n crit ). The red dashedline divides the space into two regions: DLA-dominated region in the upper left andaccelerated by longitudinal fields (associated with plasma waves) in the lower right. instability.From the position, momentum and fields sampled by individual electrons at eachtime step of the simulation, the relative contributions to the total energy gain of eachelectron due to the transverse electric field ( E y ) and the longitudinal electric field ( E x )can be calculated. The work done by E x is given by W x = −| e | (cid:82) t E x · v x dt (cid:48) , and isassociated with plasma waves, while the work by the transverse field, W y is given by W y = −| e | (cid:82) t E y · v y dt (cid:48) , and is characteristic of DLA [54, 40]. At the optimal density,the temporal evolution in energy gain for an electron with energy >
180 MeV at t =4.5 ps is found to be dominated by W y (Fig. 6g). owards the Optimisation of Direct Laser Acceleration >
10 MeV, the electron distribution in energy gain space,( W x , W y ), is plotted in Fig. 6h) at t =4 ps for n = 0 . n crit . The majority of electronspopulate the region where W y > W x , confirming DLA as the dominant accelerationmechanism, consistent with the oscillatory behavior of high energy electrons in Fig. 6e).Additionally, the considerable acceleration and deceleration of electrons indicates thatthis process could be an efficient X-ray source. Indeed, previous work has suggested thatDLA produces higher-amplitude betatron oscillations than achieved in the wakefieldregime, enabling X-ray sources with much higher energies [41, 42]. Examination of the relative strengths of the quasi-static channel fields gives insight intotheir effect on electron confinement and acceleration. In Figs. 7(a-e), snapshots of thelaser field, plasma density, and time-averaged electric and magnetic fields are shownfor n = 0 . n crit at t = 1 .
75 ps. All fields are normalized to the vacuum maximumamplitude of the laser field, denoted (
E, B ) , where E = 5.63 × statV/cm and B = 5.65 × G. The location and time-history of 4 tracked electrons at t = 1.75 psare shown in Fig. 7(f), where the shaded region has a width of 12 µ m, providing areference for the amplitude of transverse electron oscillations. These electrons, whichare representative of similar high-energy electrons investigated during particle tracking,undergo clear oscillations within a confined boundary.The fields in Figs. 7(c-e) are time-averaged over five laser cycles and represent thequasi-static channel fields (the averaged values are denoted using angular brackets).By visual inspection, it is clear that the longitudinal channel field, (cid:104) E x (cid:105) , in Fig. 7c) issignificantly weaker than the transverse field (Fig. 7d). This is expected during DLA,since the ponderomotive force prevents a plasma wave from forming. However, fieldscoinciding with density perturbations from x = [-100, 0] µ m in Fig. 7b) may be indicativeof electron injection into the plasma channel by surface wave structures, and likely playa role in electron injection during DLA [55, 56, 32].The quasi-static transverse electric field, (cid:104) E y (cid:105) , and magnetic field, (cid:104) B z (cid:105) , resultfrom a collective plasma response to the laser pulse. Transverse electron expulsionleads to charge separation that we characterize using a charge density ρ . Thecorresponding electric field (see Fig. 7d) reaches a maximum value of | (cid:104) E y (cid:105) | /E ≈ . x = − µ m. The laser pulse also drives a longitudinal electron current by pushingthe plasma electrons in the forward direction. We use a current density j < x = − µ m, its maximum relative magnitude, (cid:104) B z (cid:105) /B ≈ . | (cid:104) E y (cid:105) | /E .To understand the impact of the quasi-static channel fields on the dynamics ofa laser-accelerated electron, we use a standard test-particle approach where a single owards the Optimisation of Direct Laser Acceleration Figure 7.
PIC simulation snapshots of channel fields for n = 0 . n crit at t = 1 . a) The normalized laser intensity, with E y normalized to E . The laser pulseextends beyond the window, spanning approximately x = − µ m to x = 150 µ m. b) Electron plasma density. c) The longitudinal electric field, (cid:104) E x (cid:105) , normalized to E . d) The transverse electric field, (cid:104) E y (cid:105) , normalized to E . e) The out-of-plane magneticfield, (cid:104) B z (cid:105) , representing the quasi-static channel field B chan , normalized to B . f ) Location and time-history of selected high energy electrons at t = 1 .
75 ps, where theshaded region has a width of 12 µ m. g,h) Line-outs of (cid:104) E y (cid:105) and (cid:104) B z (cid:105) , respectively, at x = − µ m, denoted by the vertical dashed line in (d-f). The slope of these fields isestimated by a linear fit at the center of the channel ( y = 5 µ m), denoted by the dottedline in d), yielding ∆ (cid:104) E y (cid:105) / ∆ y = 2 . × statV/cm , and ∆ (cid:104) B z (cid:105) / ∆ y = − . × G/cm. Note that these snapshots have been cropped from the full simulation windowto investigate the region of the pulse and channel where the highest energy electronsare located. The brackets (cid:104)·(cid:105) denote time-averaging over five laser periods. owards the Optimisation of Direct Laser Acceleration d p dt = −| e | E − eγm e c [ p × B ] , (1) d x dt = cγ p m e c , (2)where E and B are the electric and magnetic fields acting on the considered electron, x and p are the electron position and momentum, t is the time, and γ = (cid:112) p / ( m e c )is the relativistic γ -factor. This simplified model can be reasonably applied when thetransverse displacement of electrons is less than the transverse size of the laser pulse, asshown in Fig. 7(a,f).We approximate the laser pulse by a plane electromagnetic wave propagating alongthe x -axis with a superluminal phase velocity, v ph > c . The superluminosity accountsfor the presence of the plasma and the finite size of the channel that effectively actsas a wave-guide. The plane-wave approximation neglects the longitudinal laser electricfield. This field is smaller than the transverse component roughly by a factor of λ /R ,where R is the channel radius and λ is the laser wavelength in vacuum. For simplicity,we neglect the temporal change of the laser amplitude and the laser deflection observedin simulations. Then, the linearly polarized laser electric and magnetic fields can bewritten as [57]: E wave = E cos( ξ ) ˆ y, (3) B wave = B cos( ξ ) ˆ z, (4)where ξ = ω ( t − x/v ph ) is the phase variable, ω = 2 πc/λ is the laser frequency, and B = ( c/v ph ) E .In order to find the quasi-static electric and magnetic fields of the channel, weassume that ρ and j are constant in the channel cross-section. We also neglect theirvariation along x . We then readily find from Maxwell’s equations that: E ychan = 4 πρ ( y − y ) , (5) B zchan = 4 πj ( y − y ) /c, (6)where the axis of the channel is located at y = y . Therefore, the total electric andmagnetic fields acting on the considered electron are: E = E wave + E chan = [0 , E wave + E chan , , (7) B = B wave + B chan = [0 , , B wave + B chan ] . (8)It can be verified from the equations of motion that the following quantity isconserved as the electron moves along the channel under the action of these fields: γ − up x m e c + ( y − y ) λ ( uκ B + κ E ) = C , (9) owards the Optimisation of Direct Laser Acceleration C is a constant, u ≡ v ph /c is a normalized phase velocity, and κ B and κ E are twodimensionless parameters defined in terms of j and ρ as, κ E ≡ π | e | ρ λ m e c , (10) κ B ≡ − π | e | j λ m e c . (11)The obtained conservation law is helpful in determining the amplitude of transverseelectron displacements.Typically, electrons are injected into the laser pulse from the channel walls beforebeing accelerated. These electrons will reach the axis of the channel with an appreciabletransverse momentum. It is thus appropriate to consider an electron with the followinginitial momentum on the axis of the channel: p x = 0 and p y = p i . The constant ofmotion for this electron is its initial γ -factor, such that C = γ i . Since the longitudinalmomentum and the γ -factor increase subject to the condition that γ − p x / ( m e c ) > γ − p x / ( m e c ) →
0. Itfollows from Eq. (9) that: | y − y | max = λ (cid:20) γ i + ( u − γuκ B + κ E (cid:21) / . (12)In order to find κ B and κ E from our simulations, we note that: κ E = | e | λ m e c ∂E ychan ∂y , (13) κ B = − | e | λ m e c ∂B zchan ∂y . (14)In the 2D PIC simulations, E ychan and B zchan are represented by (cid:104) E y (cid:105) and (cid:104) B z (cid:105) ,respectively. Therefore, for each field, the rate of change in y can be approximatedby ∆( (cid:104) E y , B z (cid:105) ) / ∆ y . Line-outs from (cid:104) E y (cid:105) and (cid:104) B z (cid:105) at x = − µ m and t = 1 .
75 ps areshown in Figs. 7(g,h), from which the slope of these fields is estimated to be linear nearthe channel axis at y (cid:39) µ m (denoted by the dotted line in Figs. 7(d,g,h)), yielding:∆ (cid:104) E y (cid:105) / ∆ y = 2 . × statV/cm , and ∆ (cid:104) B z (cid:105) / ∆ y = − . × G/cm. Using thesevalues in Eqs. (13) and (14), we obtain κ E = 0 .
088 and κ B = 0 . E ychan and B zchan both play a role in transverse electron dynamics. Given that thesevalues are comparable, it is clear that the impact of the channel magnetic field onelectron dynamics is important in the considered regime of the direct laser acceleration.The smallest amplitude of the transverse oscillations in the considered quasi-staticelectric and magnetic fields is found by setting u = 1 and γ i = 1 in Eq. (12), whichyields: | y − y | ∗ = λ √ κ B + κ E = 2 . µ m (15)for the obtained values of κ E = 0 .
088 and κ B = 0 . µ m ( | y − y | = 6 µ m). owards the Optimisation of Direct Laser Acceleration γ -factor using Equation12, as it is likely comparable to a = 5 due to the transverse injection, assuming u = 1.
5. Conclusions
Experiments and 2D PIC simulations demonstrate an optimal electron density for DLA,resulting in measurements of electron beams with energies up to (505 ±
75) MeV andup to (140 ±
30) nC of charge. Good agreement between experimental trends and fullyself-consistent 2D PIC simulations enabled investigation and diagnosis of the underlyingmechanisms of DLA. The channel magnetic field was found to play an important role indefining the transverse extent of the energetic electrons, forming a boundary for electronmotion with the transverse electric channel field. These observations are supported bytheoretical work highlighting the profound role of a quasi-static azimuthal magneticfield on electron energy gain via
DLA [28, 58, 59, 57], where much of previous workhas primarily focused on channel electric fields. This result is particularly compellingfor electron acceleration using longer pulse duration and higher laser intensities becausemagnetic fields are robust to ion motion, while electric channel fields have been shownto undergo field reversal following ion acceleration [60].This demonstration of high energy, high charge electron beams using picosecondpetawatt-class laser systems could enable new applications such as positron productionthrough the interaction of energetic electrons with a high-intensity laser pulse [61],or experimental verification of the two-photon Breit-Wheeler process [62]. Moreover,investigations into the motion of energetic electrons suggest that DLA can be usedto drive bright X-ray sources with ultrashort duration [41] and the capability to beaccurately synchronized to short pulse laser-initiated events. Such sources could beused to image and diagnose high-energy-density physics experiments [6, 63].
Acknowledgements
This work was supported by the National Laser Users’ Facility under grant numberDE-NA0002723. Simulations for this work were performed using the EPOCHcode (developed under UK EPSRC grants EP/G054940/1, EP/G055165/1 andEP/G056803/1) and HPC resources provided by the Texas Advanced ComputingCenter at The University of Texas. A.E. Hussein acknowledges funding from theNational Science and Engineering Research Council of Canada and the University ofCalifornia President’s Postdoctoral Fellowship program. H. Chen and G. J. Williamswere supported under the auspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract DE-AC52-07NA27344. The authorsthank Jens Von Der Linden at Lawrence Livermore National Laboratory for assistancewith the calibration of the electron positron spectrometer. owards the Optimisation of Direct Laser Acceleration References [1] Strickland, D and Mourou, G 1985,
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