This is the accepted manuscript made available via CHORUS. The article has been
published as:

Production of relativistic electrons at subrelativistic laser
intensities
G. J. Williams et al.
Phys. Rev. E 101, 031201 — Published 4 March 2020
DOI: 10.1103/PhysRevE.101.031201

Production of Relativistic Electrons at Sub-Relativistic Laser Intensities
G. J. Williams,1 A. Link,1 M. Sherlock,1 D. A. Alessi,1 M. Bowers,1 A. Conder,1, ∗ P. Di Nicola,1 G. Fiksel,2
F. Fiuza,3 M. Hamamoto,1 M. R. Hermann,1 S. Herriot,1 D. Homoelle,1 W. Hsing,1 E. d’Humières,4 D.
Kalantar,1 A. Kemp,1 S. Kerr,1 J. Kim,5 K. N. LaFortune,1, ∗ J. Lawson,1, † R. Lowe-Webb,1 T. Ma,1 D.
A. Mariscal,1 D. Martinez,1 M. J.-E. Manuel,6 M. Nakai,7 L. Pelz,1 M. Prantil,1 B. Remington,1 R.
Sigurdsson,1 C. Widmayer,1 W. Williams,1 L. Willingale,2 R. Zacharias,1 K. Youngblood,6 and Hui Chen1
1

Lawrence Livermore National Laboratory, Livermore, California 94550, USA
Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, Michigan 48109, USA
3High Energy Density Science Division, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
4
CELIA, Universit de Bordeaux, Talence, France
5
Center for Energy Research, University of California, San Diego, California 92093, USA
6
General Atomics, San Diego, California 92186, USA
7
Institute of Laser Engineering, Osaka University, 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan(Dated: February 13, 2020)2

Relativistic electron temperatures were measured from kilojoule, sub-relativistic laser-plasma interactions. Experiments show an order of magnitude higher temperatures than expected from aponderomotive scaling, where temperatures of up to 2.2 MeV were generated using an intensity of
1 × 1018 W/cm2 . 2D particle-in-cell simulations suggest that electrons gain superponderomotiveenergies by stochastic acceleration as they sample a large area of rapidly changing laser phase. We
demonstrate that such high temperatures are possible from sub-relativistic intensities by using lasers
with long pulse durations and large spatial scales.

Experimental and theoretical studies in the past
decades on laser-solid interactions at relativistic intensities (IL > 1018 W/cm2 for λ ∼ 1 µm) have established that electrons can be generated with energies of
10s of MeV at high efficiencies [1–6]. These electrons provide the energy for the production of secondary sources
such as x-rays [7, 8], ions [9, 10], neutrons [11, 12] andpositrons [13]. Understanding electron acceleration in all
laser regimes is critical to applications based on laserdriven sources for scaled experiments that study astrophysical phenomenon [14], compressed heated material[15, 16], and high energy density conditions [17].
The mechanisms responsible for laser fields accelerating electrons to relativistic energies are highly sensitive
to plasma conditions and laser intensity. For interactions
with steep plasma density profiles, the dominant energy
transfer mechanism is j×B heating [1, 2]. In the presenceof plasma densities less than critical (nc ≈ 10−21 cm−3 ),simulations predict that electrons can gain energy by dephasing from a relativistic-intensity laser field [18–25].
The processes by which electrons dephase from the laser,
however, are not well understood at sub-relativistic intensities.
In recent years, large scale short pulse laser facilities
have become available [26–29]. Of particular interest is
the newly commissioned Advanced Radiographic Capability (ARC) laser [29] at the National Ignition Facility(NIF) [30], which provides a novel diagnostic to NIF experimental platforms. The ARC is unique as the high-∗ Present address: LCLS, SLAC National Accelerator Laboratory,

Menlo Park, California 94025 USA
Special Technologies Laboratory, Santa Barbara, California 93111, USA† Present address:

est energy short pulse laser and is a source for highflux, moderate-energy x-ray radiography [31, 32]; however peak intensities are only marginally relativistic dueto the large spot size and long-focal length of the final optics. Previous experiments have hinted that electron temperatures exceeding ponderomotive scaling [33–35] may
be possible at relatively low intensities. However, questions remain as to the exact mechanism responsible forthese observations and whether one can take advantageof the high energies at ARC to generate spectra whichhave applications normally associated with relativistic
intensity lasers. For example, proton radiography or isochoric heating applications conservatively require electron temperatures of ∼2 MeV and 10s of joules of protons[36]. Positron-electron pair creation becomes efficient forelectron temperatures of ∼4 MeV with high-energy laser
platforms capable of generating a charge neutral plasma
with pair yields above 1014 [37, 38].
In this letter, we report on the first science experiments performed using the ARC laser in which the electron spectra, temperature scaling, and conversion efficiencies were measured. We find that sub-relativistic
laser intensities can be used to generate electrons withrelativistic temperatures >10× higher than ponderomotive scaling predictions suggest. 2D particle-in-cell (PIC)
simulations highlight the importance of numerous filamentations in intensity and density that initiate stochastic acceleration responsible for such high temperatures.
Previous simulation work has suggested efficient superponderomotive electron acceleration is possible at relativistic intensities; however, we demonstrate this is also
possible at lower intensities when the pulse length is longand the spot size is sufficiently large. Two experiments
were performed that varied laser intensity to observe a

2

MeV

1.3

=

MeVkT

0.83

=

(MeV)109101010111012Electrons/sr/MeV(b)kT

EnergyIntensityN170517024681012Electronchange in accelerated electron temperature, where MeVtemperatures were measured at nominal on-target intensities of IL ≤ 1018 W/cm2 . The results presented here
provide insight into electron acceleration mechanisms atlarge spatial and temporal scales.
A model of the target and diagnostic orientation isshown in Fig. 1, where the laser angle of incidence onto
an Au target was between 10-15◦ for each of the four
ARC beamlets. See Supplemental Materials for additional details of the experimental geometry and laser parameters [39]. The target rear surfaces are directed towards the equatorial diagnostic instrumentation manipulator (DIM) lines of sight at (90◦ , 78◦ ) and (90◦ , 315◦ ),which hold the primary particle diagnostics. This created a prism-shaped target where charged particles weredirected towards the DIMs by target normal sheath acceleration (TNSA). Four ARC beamlets, synchronously
timed, were spatially separated by 1 mm for each experiment in order to avoid uncertain overlap between beamlet
spots. As a result, the overall signal was interpreted as
equivalent to the sum of four single beamlets.
Two experiments (NIF shot IDs N170514-002 andN170517-003) were performed with a nominal energy
per beamlet of 150 J and 600 J at 10 ps pulse duration, which provided a relative low (4 × 1017 W/cm2 )and high (1 × 1018 W/cm2 ) intensity interaction, respectively. ARC is a fundamental wavelength laser operating
at λ = 1053 nm. Measurements of the on-shot energy,
spectrum, prepulse, and pulse duration were recorded forone of the beamlets. Beamlet foci are elliptically shaped
spots with minor and major axes of ∼15 µm and ∼38 µm
full width at half maximum (FWHM), respectively andcontain 30% of the total beam energy. The 1D equivalentis a 26 µm spot. Here, we reference the peak intensity forsimplicity and as a comparison to previous works. Propagated error in the laser parameter calculations conservatively suggest an uncertainty in peak intensity of 50%.
The prepulse energy contrast was measured on-shot tobe greater than 10−8 up to 0.5 ns before the main pulse.
Two NIF electron-positron-proton spectrometers [40](NEPPS) measured energetic electrons and protons fromtwo equatorial DIMs at a standoff of 56 cm each. The
high intensity experiment used a copper step filter in the

IntensityN170514High

FIG. 1. (a) Diagram of the experimental setup with ARC
beamlets incident on a gold prism target. (b) Polar view of
the target.

MeVLow

(b)

2.2

DIM
(90°,315°)(a)=

Target
Stalk

m
m

mm

ARC
Beams

B354B

MeVkT

4 mm

1.9

B354A

0.89

2.1 m
m

B353A

=

B353B

(0◦ , 0◦ ) direction to measure the spectrum of x-rays emitted from the target.
Electron acceleration was quantified by measuring thespectral slope to determine a characteristic temperature for each experiment. Electron spectra collected byeach NEPPS are shown in Fig. 2. An exponential fit
to the high energy region of each spectrum was performed between 1.5 MeV and the detection threshold of∼5 × 108 electrons/MeV/sr. The propagated uncertainty
in the measurement is shown as a shaded area aroundthe spectra (∼20%). The measured electron temperatures are consistent between the two lines of sight for
the low intensity experiment with 0.89 ± 0.10 MeV and
0.83 ± 0.07 MeV for (90◦ , 78◦ ) and (90◦ , 315◦ ) respectively. An asymmetry in flux is observed with ∼3× moresignal towards (90◦ , 315◦ ). This is hypothesized to be
due to geometric asymmetries with respect to the laser
and the dynamic sheath field evolution that influences
the quantity of refluxing electrons. As the collisionalmean free path is much less than the target thicknessfor electrons with energy <10 MeV, most are stopped
in the target and the higher energy electrons experience
multiple deflections leading to a wide angular distribution prior to exiting the target. If the electron beam
divergence is assumed to be Gaussian with a 2-4 steradians (65◦ -130◦ ) FWHM, the total number of electrons
between 1.5 MeV to the detection threshold is estimatedto be 0.76-2.1 × 1012 with a total energy of 0.26-0.76 J.
This corresponds to a conversion efficiency from laser toescaping high-energy electrons of 0.04-0.11%.
The high intensity experiment showed an increase inelectron temperatures while retaining the observed asymmetric flux with preference towards the (90◦ , 315◦ ) direction. The electron temperatures differed between lines ofsight where (90◦ , 78◦ ) observed 2.25 ± 0.35 MeV and
109101010111012Electrons/sr/MeV(a)kT

DIM(90°,78°)

DIM(90°,78°)

Electrons/sr/MeV Electrons/sr/MeV

Polar
View

DIM
(90°,315°)1.
9Alignment
Fiducial

10 12
Low Intensity
N170514High Intensity
N170517kT = 2.2
MeV

10 11
10 10
10 9

kT = 0.89
MeV

(a)

10 12

kT = 1.3
MeV

10 11
10 10
10 9

(b)
0

kT = 0.83
MeV

2

4

6

8

10

12

Electron Energy (MeV)

FIG. 2. Electron measurements from (90◦ , 78◦ ) (a) and (90◦ ,315◦ ) (b) viewing angles. 1σ error bounds are shown inshaded regions. Dashed lines are single-exponential temperature fits between 1.5 MeV and the detection limit.

3(m)00.020.040.06Count

lengthpath(d)0200400600Transverse(m)00.020.040.06Count

lengthpath(d)0200400600Transverse(m)00.020.040.06Count

lengthpath(d)0200400600Transverse

Davies collisions [45] for electrons greater than 25 keV.
Hard x-ray emission was included inside the solid gold
target through an application of the Integrated Tiger Series code [46] which runs inline with the PIC simulation.
The simulation box was 400 µm in the transverse dimension and 350 µm along the laser direction with 10 cells
per wavelength and 15 steps per optical cycle. The radiation hydrodynamic code HYDRA [47] initialized the
preformed plasma profile using measured laser prepulsethat provided the initial condition to the PIC simulation.
This initial plasma had a scale length of 15.2 µm below
critical density, 7.5 µm between nc and 7nc , and 1.3 µm
from 7nc to solid density.
The PIC simulations show the development of multiple∼3 µm plasma filaments (Fig. 3(a)) close to the quarter
critical surface. The location, magnitude and size of filaments in our simulations are in agreement with the priorwork of Brady et al. [48], and as a result, they are attributed to the filamentation instability seeded by Raman
scattering. Filaments are found in both plasma density
and laser intensity, where both cases show peak intensity was enhanced by a factor of ∼4 inside the filaments(see Fig. 3(b) and Fig. 4(a)). However, this intensity increase is not sufficient to explain the energetic tail in theelectron spectrum by ponderomotive arguments because
the measured temperatures are significantly greater thanponderomotive at four times the nominal intensity (0.26-

(d)
0.06

Count

1.34 ± 0.12 MeV in the (90◦ , 315◦ ) direction. Assuming
the same angular divergence, an estimated 3.8-11 × 1012
electrons escaped the target with integrated energies between 1.9-5.8 J. The conversion efficiency into MeV electrons increased to 0.08-0.24%. Protons with maximum
cutoff energies of 3.5 MeV were also observed on thisexperiment. X-ray bremsstrahlung emission was fit toa single-component exponential with a temperature of
155 ± 30 keV and indicates a strong multi-temperatureelectron profile: a high-flux, colder spectrum at the ponderomotive temperature that stops within the target anda low-flux, hotter spectrum that escapes [41].
The data set obtained at ARC is supplemented byexperiments performed at Omega EP [28] using ARCequivalent laser conditions. Here, the f /2 optic was defocused to approximate the ARC spot with 30% of theenergy contained in a 25 µm spot. A 5 µm Au foil was
irradiated at normal incidence with 550 J of incident
laser energy. Pulse duration was scanned between 30,
10, and 3 ps to change the on-target intensity from 0.3,
0.8, and 2.9×1018 W/cm2 , respectively. Electron spectra
were measured normal to the foil rear surface and analyzed using the same method outlined above. The temperatures derived from the data were found to increase
with intensity as 0.73 ± 0.14 MeV, 1.76 ± 0.24 MeV and3.10 ± 0.58 MeV.
Electron temperatures reported here are significantly
higher than those expected from acceleration by the laser
ponderomotive potential. When a laser interacts with asteep density interface, such that electrons observe only
a few cycles of the laser field, PIC calculations [1] predict a Maxwellian electron distribution with a temperature of kTPond = 0.511[(1 + I18 λ2µm /1.37)1/2 − 1] MeV,
where I18 is the intensity in units of 1018 W/cm2 andλµm is the wavelength in units of µm. For the laser
intensities considered here, 0.4 and 1.0 × 1018 W/cm2 ,kTPond = 0.08 MeV and 0.18 MeV, respectively. The observed electrons have superponderomotive temperaturesthat are between 7× and 12× higher than the ponderomotive scaling, depending on line of sight and intensity.
Two dimensional PIC simulations were performed togain understanding of the acceleration mechanisms responsible for producing the observed electron spectra.
The hybrid PIC code Chicago [42] was used to model
the laser-plasma interaction of a single ARC beamlet on
a surrogate gold target for both the low and high intensity cases. These simulations are novel due to the large
spatial and temporal scales, with runs of ∼0.5 MCPUhours to simulate 40 ps of the interaction. The simulation
employed a radial approximation to the ARC laser spot
as a superposition of two Gaussian spots of 26 µm and120 µm FWHM containing 33% and 67% of the incident
energy, respectively, that was delivered in a sin2 temporal profile with a 10 ps FWHM. Peak vacuum intensities were 0.3 and 1 × 1018 W/cm2 . Simulations were 2D
cartesian and collisions were handled using a cell average
Lee-More-Desjarlais collision model [43, 44] for thermal
electrons (electrons below 25 keV) and Atzeni-Schiavi-

0.04

0.02

0

0

200

400

600

Transverse path length ( m)FIG. 3. (a) Electron density at τ = −3 ps with nc /4 andnc contours drawn in red and black, respectively. (b) Laser
intensity at the same time step (c) Example trajectory of electron with a 2× ponderomotive final energy and nc /10 contour
shown in addition to those in (a). Electron motion is incoherent and rapid energy gains occur as it moves transversely
across a changing laser phase. 100 fs time steps are labeled.
(d) Histogram of transverse path length for electrons withenergies above 180 keV.

10

3

2

1

-10

10(a)(b)psTotal

ps9

ps6

ps3

ps0

ps-3

ps-6

(electrons/MeV)-9(MeV)108101010121014dN/dE

Plasma0510152025E(W/cm2)1018VacuumWith

10 14
VacuumWith Plasma(a)0(ps)01234Ipeak

18

-5

0

(ps)

5

10

dN/dE (electrons/MeV)

4

Ipeak (W/cm 2)

0.55 MeV for the low and high intensity cases, respectively). The filaments grow rapidly upon being seeded,reaching nonlinear saturation within 100 fs and will bepresent within any multi-picosecond interaction of sufficient intensity.
Simulations reveal that a majority of the electron energy gain occurs in the coronal region (z < −30 µm)
away from the strong density filaments and that workon the electrons appears to be done by the laser, rather
than self-generated, electric fields. The existence of laser
filaments is important because their random nature leads
to a reflected wave which itself has significant transverseand longitudinal variations in phase. The stochastic acceleration that is enabled by the presence of filaments
has been studied by Nakamura et al. [49] and, while
an analytic scaling was not possible, simplified FokkerPlanck simulations predicted the generation of an energetic tail at energies well above ponderomotive scaling.
There is no evidence of ion channel formation in our conditions that could trap electrons within a filament, as has
been observed for significantly higher intensities [3]. Instead, electrons experience rapid, stochastic accelerations
as they execute highly non-uniform trajectories in thecoronal plasma, crossing regions of significantly different laser phase. A typical superponderomotive electron
trajectory, color-coded by energy, is shown in Fig. 3(c).
In this way, electrons are able to gain energy over hundreds of microns in the coronal plasma and can easily
reach relativistic energies so long as they undergo significant phase change. They are therefore not constrainedby one-dimensional energy scaling [49].
This is in comparison to small-spot laser interactions,which typically allow for electrons to travel only ∼10 µm
in the transverse direction before being ejected from locally higher intensity regions, which ultimately limits thenumber of stochastic accelerations possible. We find thatthe vast majority of superponderomotive electrons (defined as those with energy >180 keV) sample many laser
filaments while in the corona as indicated by
R their significant (3 µm) transverse path lengths, |vx |dt (seeFig. 3(d)). A large number of filaments can only be generated when the laser spot size is many times larger than
the filament size and the plasma scale-length is sufficient
to sustain these filaments. This suggests that lasers with
sub-relativistic intensities that have large focal spots andlong pulse durations can more easily generate relativistic
electrons. Plasma turbulence could also be responsible
for the dephasing, as shown in Ref. [50], but the electromagnetic fields associated with the filaments are typically much greater in magnitude than the electrostatic
response and the energy jumps visible in Fig. 3(c) are
due primarily to the laser fields.
Simulated electron spectra from the laser-plasma interaction were characterized at an extraction plane 10 µm
inside the solid density region to capture the particles
which would enter the target. This distribution was revolved assuming axisymmetry to get a 3D equivalent electron distribution to compare to the experiment. The-10-50510

10(a)(b)psTotal

ps9

ps6

ps3

ps0

ps-3

ps-6

(electrons/MeV)-9(MeV)108101010121014dN/dE

Plasma0510152025E(W/cm2)1018VacuumWith(ps)01234Ipeak-10-50510

4

-9 ps
-6 ps
-3 ps
0 ps
3 ps
6 ps
9 ps
Total

10 12

10 10
10 8
0

10

(b)

5

10

15

20

25

E (MeV)

FIG. 4. (a) Input laser intensity (red) for the high intensity case and cycle-averaged laser intensity on the simulation
grid (black). An increase is observed due to self-focusing inside density filaments. Self-focusing occurs in both intensity
simulations. (b) Simulated time-integrated and time-resolved(± 0.1 ps) electron spectra for the high intensity case.

time-integrated and time-resolved electron distributions
are shown in Fig. 4(b) for the high intensity case and indicate a maximum hot electron temperature of 2.5 MeV
for the energy region of 8-20 MeV. Simulations of the low
intensity case showed a high energy electron tail with atemperature of 0.56 MeV.
We note that the measured electron temperatures appear to agree with the scaling relationship for superponderomotive acceleration described in Pukhov et al. [3].
This is despite the fact that the attributed acceleration
mechanisms are unique and that we do not observe confining ion channels in our simulations of the kind discussed in Ref [3]. The ponderomotive and Pukhov scalings are shown in Fig. 5 along with the experimental andsimulated electron temperatures as a function of laser
intensity.
The simulations show the high intensity case had a conversion efficiency of 7.8% from laser to electrons above
1.5 MeV at the extraction plane. The conversion efficiency to observable electrons is estimated by injecting
the simulated spectrum into a 3D model of the target
using the Monte Carlo code Geant4 [51]. Modeling the
dynamic TNSA fields using PIC code was not tractable
due to the large spatial scale and asymmetry of the target that would necessitate a 3D simulation. However, anelectric field that develops on the target can be describedby a target charging model in the large surface area limit(here, ∼1 cm2 ) [52]. The protons serve as a measurement of the capacitor charge that downshifts the escaping
electrons while the spectrum temperature is maintained.
Electrons escaping in the target have their energy reduced by 3.5 MeV and show a conversion efficiency fromlaser energy to electrons above 1.5 MeV of 0.02%. Angular emission of the scattered electrons created a forward
going cone with FWHM of 130◦ . This agrees with experimental estimates within a factor of four.
Proton acceleration expected at these laser conditions
is estimated using the simulated time-resolved electron

Electron Temperature (MeV)

5spectrum and a 10 µm Au target. 1D PIC simulations of
the generated TNSA fields show a maximum proton cutoff energy of ∼20 MeV with 30 J of protons produced, suf-

ficient for radiographic requirements and isochoric heating of materials to 10s of eV. Monte Carlo simulations
showed the high intensity experiment generated an emitted positron yield of ∼1×109 /sr, which was not observed
but would have appeared at the detection threshold of thediagnostics. A robust positron signal could be measurable with a doubling of the electron temperature resulting
in a yield of 5 × 1010 pairs/sr emitted from the target.
In summary, the first ARC science campaign was completed to measure energetic electrons generated from a
sub-relativistic laser target interaction. Electrons wereobserved to have a temperature exceeding 2 MeV, an order of magnitude above the ponderomotive scaling. 2D
PIC simulations demonstrated that electrons gain theirenergy through stochastic accelerations in the coronalplasma as they traverse a large area of rapidly changing laser phase. The results described here establish thathighly relativistic electrons can be generated with subrelativistic lasers by using long pulse durations and large
spatial scales.
We thank the NIF Experimental Operations team for
implementing these experiments. We gratefully acknowledge the NIF Discovery Science Program for the experimental allocation. This work was performed under
the auspices of the U.S. Department of Energy by theLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and funded by the LLNLLDRD program under tracking code 17-ERD-010.

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ARC Data
Omega EP Data

2
N1705171
N170514

0
10 17

10 18

10 19

Laser Intensity (W/cm 2)
FIG. 5. Measured and simulated electron temperatures.
Square and circle markers differentiate ARC data at (90◦ , 78◦ )and (90◦ , 315◦ ) lines of sight, respectively, and are placed at
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