Abstract DLA to the energy gain of the electrons


direct laser acceleration (DLA) of electrons in a laser wakefield accelerator
(LWFA) operating in the forced or quasi-blowout regimes has been investigated
through experiment and simulation.  When
there is a significant overlap between the trapped electrons and the drive
laser in a LWFA cavity, the resulting electrons can gain energy from both the
LWFA and the DLA mechanisms. 
Experimental work investigates the properties of the electron beams
produced in a LWFA with ionization injection by dispersing those beams in the direction
perpendicular to the laser polarization. 
These electron beams show certain spectral features that are
characteristic of DLA.  These
characteristic spectral features are reproduced in particle-in-cell
simulations, where particle tracking was used to elucidate the roles of LWFA
and DLA to the energy gain of the electrons in this experimental regime and to demonstrate
that such spectral features are definitive signatures of the presence of DLA in

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the field of laser wakefield acceleration (LWFA) 1 matures, emphasis is
shifting toward utilizing LWFA as a source of electron beams and x-rays for
applications.  There is an increasing
emphasis on producing electron beams from LWFAs that can meet the stringent
beam requirements (narrow divergence, small emittance, narrow energy spread)
necessary for use in staged plasma accelerators 2 and free electron lasers. Simultaneously,
betatron x-rays from LWFA are being utilized for applications 3-7, which
places an emphasis on optimizing LWFA to produce these x-rays.  Even though these applications require
optimization of different electron beam properties, all applications benefit
from a more-complete understanding of the dynamics of electron energy gain in
LWFA and how those dynamics affect properties such as electron beam energy, divergence,
source size, shape, and energy spread.

For the
range of plasma densities (mid-1018 to a few 1019 cm-3)
and laser pulse durations (35-45 femtoseconds full width at half maximum) that
are typically used in many current LWFA experiments in the forced or
quasi-blowout regimes, the laser pulse length is on the order of the wake wavelength;
therefore it may occupy the entire first bucket of the wake.  In such experiments, the wakefield structure
has a desirable transverse and longitudinal field structure for generating a self-injected
electron bunch, but it also has the conditions needed for direct laser
acceleration (DLA) 8, 9 if there is an overlap between the accelerating
electrons and the transverse electric field of the laser pulse 10-16.  It is therefore important to understand the
role that not only the longitudinal electric field of the wake, but also the
other fields—namely, the transverse fields of the ion column and of the laser
itself—play in determining the ultimate energy gained by the electrons.  In this paper, we show through experiments
direct, observable signatures in the produced electron beams that indicate that
DLA makes a significant contribution to the electrons’ energy in LWFAs operated
in such a configuration. Three-dimensional (3D) particle-in-cell (PIC) simulations are used
to elucidate the energy dynamics that lead to this contribution.


the matched, self-guided 17 blowout regime of LWFA 18, an ultrashort,
intense laser pulse propagates through either an underdense plasma or a neutral
gas.  In the latter case, the leading
edge of the laser pulse ionizes the neutral gas, and the pondermotive force of
the laser then expels the plasma electrons out and around the main body of the
pulse. On the femtosecond (fs) timescale of the laser, the more-massive ions
remain relatively immobile, so an ion column forms behind the drive laser. The
expelled plasma electrons are drawn back to the laser axis by the Coulomb force
of the ion column, where they overshoot and oscillate about the axis and
thereby set up a wake structure. The charge separation generated by this wake
structure produces a longitudinal electric field that is capable of
accelerating electrons trapped in the wake at gradients > 1 GeV/cm. Those
electrons that are injected off-axis will undergo betatron oscillations in
response to the linear transverse focusing force of the ions 19, 20.

can become trapped in a LWFA by a variety of methods 21-31, but in the experiments
and simulations presented here, the ionization injection 32-34 technique is
used. In this technique, the plasma is produced by the laser ionization of a
neutral gas mixture comprised of a gas with a low ionization potential
(commonly He or H2) doped with a gas with high ionization potential
(commonly N2 or Ar).  The lower-intensity front edge of the laser
pulse ionizes the outer (typically L) shell electrons of the dopant gas along
with all the electrons in the gas with a low ionization potential.  Because the inner (typically K) shell
electrons of the higher-Z atoms have a much higher ionization potential, they
are ionized only near the peak of the laser pulse within a fully formed wake
and are subsequently trapped without slipping all the way to the back of the
wake.  Compared to
self-trapping, this method of ionization injection permits trapping in a LWFA
at reduced plasma densities and laser powers. 

a LWFA operating in the forced or quasi-blowout regime, the ion column acts as
a very strong wiggler.  Trapped electrons
that are being accelerated by the wake undergo betatron oscillations in
response to the transverse electric field of the ion column.  Therefore, if a LWFA is configured such that
some of the trapped electrons undergo betatron oscillations in the plane of
polarization of the laser’s electric field, the transverse field of the drive
laser can give the electrons additional transverse momentum.  This transverse momentum can then be
converted to longitudinal momentum via the v
x B force.  Thus, the DLA mechanism 8, 9 can accelerate
electrons by this coupling of the transverse field of the laser through the
betatron motion of the electrons.  As a
result, there is a potential for those electrons to be accelerated by the DLA
mechanisms in addition to the LWFA mechanism in a LWFA where the drive laser
overlaps the trapped electrons 10-16.

It has been noted that DLA is the inverse of
the ion channel laser mechanism 35.  DLA in LWFA is also similar to inverse free
electron laser (IFEL) acceleration 36, 37, except that the static magnetic
undulator used in an IFEL is replaced by the transverse electric field of the
ions in DLA and the resonance condition need not be strictly obeyed as in the
IFEL 11, 13-14.  In principle, the
resonance condition for DLA is similar to that for an IFEL 38; i.e., in an
ideal situation, the laser pulse overtakes the electrons by one wavelength per
betatron oscillation once the electrons come into resonance with the
fundamental (N=1) harmonic, where the electrons are bunched on a
laser-wavelength scale 8, 39-42.  However,
unlike in an IFEL, sustained resonance for DLA is more difficult to design
because in the latter case, the normalized undulator strength K >> 1 and
the energy and betatron frequency of the electrons as well as the laser
properties are continuously and rapidly changing 11, 13, 14. 

The condition for energy gain from the DLA
mechanism is typically expressed using the one-dimensional resonance condition
for a single electron 8, 9


N is an integer indicating the harmonic of the betatron frequency

??  =                                                                         (2)

 is the velocity of the electron in the
longitudinal direction, and v? and ?0 are the phase
velocity and frequency, respectively, of the electromagnetic wave (i.e.,
laser).  Essentially, this resonance
condition means that in order for an electron to gain energy from DLA, a
harmonic of the betatron frequency  must equal the Doppler-shifted laser frequency
 witnessed by the electron 8, 9, 11, 13, 14.  It is well known that in LWFAs, especially
those not in the ideal blowout regime 18, the properties of the drive laser,
including  and , evolve throughout the acceleration
distance.  Furthermore,
as electrons are accelerated in a LWFA, their longitudinal momentum, and
therefore , increases, and their betatron
frequency is expected to fall as ?-1/2 as seen in Equation 2.  Despite these evolving quantities, electrons
that are being accelerated in a LWFA are able to gain significant energy from
DLA because the quantities evolve together such that a quasi-resonance is set
up and the electrons are in a phase where they gain energy from the DLA
mechanism for more than one-half of each betatron cycle 11, 13, 14.

determine if a LWFA is operating in a regime where DLA is expected to contribute
to the energy gain of the electrons, the LWFA can be characterized using the
ratio of the laser pulse length ?laser relative to the nonlinear
plasma wavelength ?wake.  This ratio can be represented by the
dimensionless pulse length parameter 13, 14

= c?laser/?wake = ?p?laser/(2?a01/2)                                  (3)

If the laser pulse length c?laser
is equal to the a0-dependent length of the first bucket 18 ?wake ?  then Tp
= 1.  Here, kp = ?p/c,
and a0 is the normalized vector potential a0 = eE0/mc?0
? 8.6 x 10-10 ??m, where I0 is the laser intensity and ?
is the wavelength of the laser. In the case where Tp is 0.5 or
less, the laser does not overlap the trapped electrons in the LWFA while they
are being accelerated; those electrons gain energy purely from the longitudinal
wakefield 10, 13, 14. When Tp reaches 0.6 or more, the laser pulse
will overlap the trapped electrons, and DLA can play a role in the acceleration
of those electrons 10, 13, 14. A Tp
> 1 indicates a significant overlap between the transverse laser field and
the trapped electrons 13, 14.

Experimental Methods and Results

this section, we show definitive experimental evidence of the presence of DLA
in nonlinear LWFAs where the laser pulse overlaps the trapped electrons.  We first demonstrate that the electron beams
are indeed interacting with the drive laser when there is an overlap between
the laser and trapped electrons, as might be expected in a DLA-assisted LWFA
experiment.  We then show that the
transverse structure of the dispersed electron beams exhibits characteristic
features that are indicative of DLA as an additional acceleration

experiments presented in this paper were conducted at UCLA using an 815-nm
Ti:Sapphire laser with a fixed pulse length ?laser of 45 ± 5 fs full
width at half maximum of intensity and a spot size w0 of 6.7
?m.  The laser was run with powers P up
to 10 TW, which correspond to an a­0 up to 2.6.  An f/6 off-axis parabola (OAP) system focused
the main laser pulse at the entrance of a variable-length (0.1-2 mm) gas cell 43,
44 as shown in Figure 1.  The
gas cell was filled with a 95% He/5% N2 neutral gas mixture using a
pulsed solenoid valve.  The gas mixture
was used so that ionization injection 32 could be used to both inject charge
early into the wake and increase the amount of trapped charge.  The
plasma density was measured on every shot using a Michelson interferometer and
was varied by changing the gas pressure 43, 44.  The produced electron beams were dispersed in
energy with a 0.92 tesla (T) dipole magnet onto a plastic scintillator or a LANEX
screen and recorded using a PI-MAX intensified CCD camera.  This electron spectrometer could be rotated
by 900 so that the electron beam could be dispersed parallel to or
orthogonal to the linear laser polarization 13, 14.



Figure 1: Experimental
setup.  The thick red line shows the main laser pulse
being focused by the f/6 OAP system at the entrance of the gas cell.  The laser is linearly polarized in the plane
of the page. The thin red line shows the probe for the Michelson
interferometer.  A typical interferogram
is shown.  The electrons are dispersed by
the 0.92 T dipole magnet onto a scintillator or a LANEX and imaged by a PI-MAX
3 camera.  The dipole magnet and screen
could be rotated by 900 so that the electron beams could be
dispersed parallel or orthogonal to the laser polarization.  The dipole magnet typically was located 3.2
cm downstream from the gas cell, and the distance from the end of the magnet to
the screen was 7.0 cm.  A typical measured
electron spectrum is shown.

Because the energy gain from DLA relies on
the coupling between the transverse laser field and the betatron motion of the
electrons, the first observable signature of an interaction
between the laser and the trapped electrons in a LWFA is that the undispersed
electron beam should be elliptical in the direction of the laser polarization 45.
The white ellipses in Figure
are fits to the 50% contour of the undispersed electron beams from 10
consecutive shots where the laser had horizontal, linear polarization and a
vacuum a0 of ~1.5.  The plasma
density was ~1.7 x 1019 cm-3, which yields a Tp
value of ~1.3, and the gas cell length was 900 ?m.  The fits show a strong ellipticity in the
direction of the laser polarization with an average measured half-width at
half-maximum (HWHM) divergence of 12.2 mrad. 
In contrast, the average measured HWHM divergence in the perpendicular
direction was 5.6 mrad.  The direction of
the linear polarization of the drive laser was then rotated 900
using a thin (1 mm) quartz half wave plate for high-laser-energy
applications.  The ellipticity of the
undispersed electron beams rotated with the laser polarization, as shown in Figure
which indicates that the trapped electrons’ transverse momentum is being
enhanced in the polarization plane.  With
the vertical laser polarization, the average measured HWHM divergence in the
direction of the laser polarization was 13.0 mrad, and the average measured
HWHM divergence in the perpendicular direction was 6.5 mrad.  Therefore, under the laser-plasma parameters
described above, the measured divergence of undispersed electron beams
emanating from the LWFA shows ellipticity that is correlated to the
polarization of the laser pulse.  This correlation
demonstrates that the electrons are indeed interacting with the drive laser. Although
DLA is expected to preferentially increase the divergence of the electron beam
in the plane of the laser polarization, the observed ellipticity in the
divergence of the undispersed electron beams in Figure
2 in
itself is not definitive proof that DLA is present in the LWFA 13.  Rather, because the energy gain from DLA relies on the coupling
between the transverse laser field and the betatron motion of the electrons, a
signature of this transverse coupling must be present in the energy gain of the
electrons to demonstrate the presence of DLA in LWFA

Figure 2: (a, b)  Fits (white ellipses) to the 50% contour of
undispersed electron beams from a series of 9 and 10, respectively, consecutive
laser shots when using horizontal and vertical, respectively, linear laser polarization.  (Inset) 
Typical undispersed electron beam from data shown in (a) with 50%
contour points marked by the black crosses and the fit to that point marked by
the white ellipse.


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