11institutetext: Laboratoire d’Optique Appliquée, ENSTA-ParisTech, CNRS, Ecole
Polytechnique, Université Paris-Saclay, Palaiseau, France

# Plasma Wake Accelerators: Introduction and Historical Overview

V. Malka

###### Abstract

Fundamental questions on the nature of matter and energy have found answers
thanks to the use of particle accelerators. Societal applications, such as
cancer treatment or cancer imaging, illustrate the impact of accelerators in
our current life. Today, accelerators use metallic cavities that sustain
electric fields with values limited to about 100 ${\,\mathrm{MV/m}}$. Because
of their ability to support extreme accelerating gradients, the plasma medium
has recently been proposed for future cavity-like accelerating structures.
This contribution highlights the tremendous evolution of plasma accelerators
driven by either laser or particle beams that allow the production of high
quality particle beams with a degree of tunability and a set of parameters
that make them very pertinent for many applications.

Keywords
Accelerator; laser; plasma; laser plasma accelerator; laser wakefield.

##  0.1 Introduction

This article corresponds to the introductory lecture given at the first CAS-
CERN Accelerator School on Plasma Wake Acceleration on 21–28 November, 2014.
Having this school dedicated to Plasma Accelerators at CERN, where an
important part of the story of high energy physics has been written and where
the worlds larger accelerators and the brighter and more energetic particle
beams are produced, represents in itself the realization of a dream that shows
the maturity and the vitality of the field. Having an unexpected level of
participation shows also the dynamism of this field of research with an
impressive growth of groups in Europe and all over the world. Accelerator
physics started almost 130 years ago with the discovery of the cathodic tube.
Since then, accelerators have gained in efficiency and in performance
delivering energetic particle beams with record energy and luminosity values.
During the last century, they have been developed for fundamental research,
for example, for producing intense picosecond X-ray pulses in synchrotron
machines, or more recently even shorter, few femtosecond X-ray pulses in free
electron laser machines. Such short X-ray pulses are crucial for the study of
ultra-fast phenomena, for example in biology, to follow the DNA structure
evolution, or in material science to follow the evolution of crystals. Higher
energy accelerators are crucial to answering important questions regarding the
origins of the universe, of dark energy, of the number of space dimensions,
etc. The largest one available, the Large Hadron Collider, has for example
confirmed two years ago the existence of the Higgs boson. Figure 1 illustrates
few of the many fundamental discoveries that have been made this last century
and have permitted matter from $10^{-10}$ to $10^{-20}$ m spatial resolution
to be probed.

Figure 1:  Evolution of accelerators and their main related discoveries

Moreover, as illustrated in Fig.2, with an annual market of more than a few US
billions of dollars, accelerators are used today in many fields such as cancer
therapy, ion implantation, electron cutting and melting, and non-destructive
inspection, etc.

The accelerating field in superconducting radio-frequency cavities is, due to
electrical breakdown of the metallic cavity, limited to about 100
$\,\mathrm{MV/m}$. It is for this reason that an increase of the particle
energy requires an increase of the acceleration length. In the 1950s, Budker
and Veksler [2] proposed using plasma collective fields to accelerate charged
particles more compactly. In the pioneering theoretical work performed in
1979, Tajima and Dawson [3] showed how an intense laser pulse can excite a
wake of plasma oscillations through the non-linear ponderomotive force
associated to the laser pulse. In their proposed scheme, relativistic
electrons were injected externally and were accelerated in the very high
${\,\mathrm{GV/m}}$ electric field sustained by relativistic plasma waves. In
this early article [3], the authors proposed two schemes: the laser beat wave
and the laser wakefield. Several experiments were performed at the beginning
of the 1990s following on from their ideas, and injected few
${\,\mathrm{MV/m}}$ electrons have gained energy in ${\,\mathrm{GV/m}}$
accelerating gradients using either the beat wave or the laser wakefield
scheme. In 1994, at the Rutherford Appleton Laboratory, using the 40
${\,\mathrm{TW}}$ powerful Vulcan Laser, hundreds of ${\,\mathrm{GV/m}}$
accelerating gradients have been generated and used to trap electrons from the
plasma itself, and to accelerate them [4] to few tens of ${\,\mathrm{MV/m}}$
over only 1 mm distance. ${\,\mathrm{TV/m}}$ accelerating gradients have since
been demonstrated in the non-linear regime in the forced laser wakefield
scheme. Figure 3 illustrates the compactness of a plasma accelerating cavity.

Figure 2:  Market of industrial accelerators and their main societal
applications

In 1985, Chen and Dawson [5] proposed to use a bunched electron beam to drive
plasma wakes with, again, ${\,\mathrm{GV/m}}$ accelerating gradients. Soon
after, the first experiments on Particle WakeField Acceleration (PWFA) were
achieved using low energy electron beam drivers. In 1996, T. Katsouleas and C.
Joshi proposed to use an ultra-relativistic electron beam delivered by the
SLAC linac to drive ${\,\mathrm{GV/m}}$ accelerating fields. In 2009, the
possibility of driving plasma-wakefield acceleration with a proton bunch was
proposed [6], and the authors demonstrated through numerical simulations that
${\,\mathrm{TeV}}$ energy levels could be reached in a single accelerating
stage driven by a ${\,\mathrm{TeV}}$ proton bunch.

Figure 3: Compactness of plasma ‘cavity’. Left: Radiofrequency cavity. Right:
Non-linear laser plasma wakefield. The laser pulse in yellow propagates from
left to right, the iso-electronic density is shown in blue and the electron
bunch in red.

In both cases, the accelerating gradient results from the rapid electron
plasma oscillation that follows the electronic perturbation. If, in an
initially uniform and non-collisional plasma a slab of electrons are displaced
from their equilibrium position, the restoring force which is applied to this
electron slab drives them towards the equilibrium position. For the time scale
corresponding to the electrons motion, the motion of the ions can be neglected
because of inertia. The typical frequency of electron oscillations around the
equilibrium position is called the electron plasma frequency
$\omega_{\mathrm{pe}}$:

$\omega_{\mathrm{pe}}=\sqrt{\frac{n_{\mathrm{e}}e^{2}}{m_{\mathrm{e}}\varepsilon_{0}}},$
(1)

where $n_{\mathrm{e}}$ is the unperturbed electron density.

If $\omega_{\mathrm{pe}}<\omega_{0}$ (where $\omega_{0}$ is the laser
frequency) then the characteristic time scale of the plasma is longer than the
optical period of the incoming radiation. The medium cannot stop the
propagation of the electromagnetic wave. The medium is then transparent and it
is called ‘under-dense’. When $\omega_{\mathrm{pe}}>\omega_{0}$ then the
characteristic time scale of the electrons is fast enough to adapt to the
incoming wave and to reflect totally or partially the radiation, and the
medium is called ‘over-dense’.

These two domains are separated at frequency $\omega_{0}$, which corresponds
to the critical density, $n_{\mathrm{c}}=\leavevmode\nobreak\
\omega_{0}^{2}m_{\mathrm{e}}\varepsilon_{0}/e^{2}$. For a wavelength
$\lambda_{0}=1{\,\mu\mathrm{m}}$, one obtains $n_{\mathrm{c}}=1.1\times
10^{21}{\,\mathrm{cm^{-3}}}$. The typical range of electron densities of laser
plasma accelerators with current laser technology, is $[10^{17}$
cm${}^{-3}-10^{20}{\,\mathrm{cm^{-3}}}]$.

In a uniform ion layer, the density change $\delta n$ for a periodic
sinusoidal perturbation of the electron plasma density is written

$\delta n=\delta n_{\mathrm{e}}\sin(k_{\mathrm{p}}z-\omega_{\mathrm{p}}t),$
(2)

where $\omega_{\mathrm{p}}$ and $k_{\mathrm{p}}$ are the angular frequency and
the wave number of the plasma wave.

This density change leads to a perturbation of the electric field
$\delta\vec{E}$ via the Poisson equation:

$\vec{\nabla}\delta\vec{E}=-\frac{\delta n\;e}{\varepsilon_{0}}.$ (3)

This gives

$\delta\vec{E}(z,t)=\frac{\delta
n_{\mathrm{e}}\;e}{k_{\mathrm{p}}\varepsilon_{0}}\cos(k_{\mathrm{p}}z-\omega_{\mathrm{p}}t)\vec{e_{z}}.$
(4)

The electric field associated to the relativistic plasma wave, i.e. with a
phase velocity close to the speed of light
$v_{\mathrm{p}}=\omega_{\mathrm{p}}/k_{\mathrm{p}}\sim c$ can be described by

$\delta\vec{E}(z,t)=E_{0}\frac{\delta
n_{\mathrm{e}}}{n_{\mathrm{e}}}\cos(k_{\mathrm{p}}z-\omega_{\mathrm{p}}t)\vec{e_{z}},$
(5)

where $E_{0}=m_{\mathrm{e}}c\omega_{\mathrm{pe}}/e$.

In the linear case, as shown in Fig. 4, the relative density perturbation is
much smaller than one, and the density perturbation with the electric field
has a sinusoidal profile. Note that the electric field is dephased by $-\pi/4$
with respect to the electron density. A $1\%$ density perturbation at a plasma
density of $10^{19}{\,\mathrm{cm^{-3}}}$ corresponds to 3 ${\,\mathrm{GV/m}}$.
In the non-linear case, for a $100\%$ density perturbation at a plasma density
of $10^{19}{\,\mathrm{cm^{-3}}}$ the accelerating field reaches 300
${\,\mathrm{GV/m}}$.

Figure 4: Density perturbation with the corresponding electric field

We now examine the electron motions in this oscillating electric field in the
simplified case of a one-dimensional plasma wave. Figure 5 represents an
example of an electron trajectory in a plasma wave. In this phase space, the
closed orbits correspond to trapped particles. Open orbits represent untrapped
electrons, either because the initial velocity is too low or to high. The
curve which separates these two regions is called the separatrix. This
separatrix gives the minimum and maximum energies for trapped particles. This
is comparable to the hydrodynamic case, where a surfer has to crawl to gain
velocity and to catch the wave.

Figure 5: Upper panel: Potential in phase space. Lower panel: Trajectory of an
electron injected ion the potential of the plasma wave in the frame of the
wave with the fluid orbit (dashed line), the trapped orbit and in between in
red the separatrix.

For an electron density much lower than the critical density
$n_{\mathrm{e}}\ll n_{\mathrm{c}}$, we find
$\gamma_{\mathrm{p}}=\omega_{0}/\omega_{\mathrm{p}}\gg 1$ and

$\Delta W_{\mathrm{max}}=4\gamma_{\mathrm{p}}^{2}\frac{\delta
n_{\mathrm{e}}}{n_{\mathrm{e}}}mc^{2}.$ (6)

For an electron travelling along the separatrix, the time necessary to reach
maximal energy is infinite because there exists a stationary point at energy
$\gamma_{\mathrm{p}}$. for other closed orbits, the electron successively
gains and loses energy during its rotation in the phase space. In order to
design an experiment, one needs an estimation of the distance an electron
travels before reaching maximal energy gain. This length, which is called the
dephasing length $L_{\mathrm{deph}}$, corresponds to a
$\lambda_{\mathrm{p}}/2$ rotation in the phase space. In order to have a
simple analytical estimation, one can assume that the energy gain is small
compared to the initial energy of the particle and that the plasma wave is
relativistic $\gamma_{\mathrm{p}}\gg 1$, then the dephasing length is written

$L_{\mathrm{deph}}\sim\gamma_{\mathrm{p}}^{2}\lambda_{\mathrm{p}}.$ (7)

In these formulas, we have considered a unique test electron, which has no
influence on the plasma wave. In reality, a massive trapping of particles
modifies the electric fields and distorts the plasma wave. This is called the
space-charge or beam loading effect (which results from the Coulomb repulsion
force). Finally, this linear theory is difficult to apply to non-linear
regimes which are explored experimentally.

##  0.2 Laser wakefield accceleration

###  0.2.1 Laser wakefield: the linear regime

The ponderomotive force of the laser excites a longitudinal electron plasma
wave with a phase equal to the group velocity of the laser close to the speed
of light. Two regimes have been proposed to excite a relativistic electron
plasma wave.

In the standard laser wakefield acceleration (LWFA) approach, a single short
laser pulse excites the relativistic electron plasma wave. As the
ponderomotive force associated with the longitudinal gradient of the laser
intensity exerts two successive pushes in opposite directions on the plasma
electrons, the excitation of the electron plasma wave is maximum when the
laser pulse duration is of the order of $1/\omega_{\mathrm{p}}$. For a
linearly polarized laser pulse with full width at half maximum (FWHM)
$\sqrt{2\ln 2}\,L$ (in intensity), the normalized vector potential, also
called the force parameter of the laser beam, is written

$a(z,t)=a_{0}\exp\left[-\left(\frac{k_{0}z-\omega_{0}t}{\sqrt{2}k_{\mathrm{p}}L}\right)^{2}\right].$
(8)

In the linear regime, $a_{0}\ll 1$, the electronic response obtained behind a
Gaussian laser pulse can be easily calculated [7]. In this case, the
longitudinal electric field is given by

$\vec{E}(z,t)=E_{0}\frac{\sqrt{\pi}a_{0}^{2}}{4}k_{\mathrm{p}}L\exp(-k_{\mathrm{p}}^{2}L^{2}/4)\cos(k_{0}z-\omega_{0}t)\vec{e_{z}}.$
(9)

Equation (9) explicitly shows the dependence of the amplitude of the wave on
the length of the exciting pulse. In particular, the maximal value for the
amplitude is obtained for a length $L=\sqrt{2}/k_{\mathrm{p}}$ as shown in
Fig. 6 for a laser with a normalized vector potential $a_{0}=0.3$.

Figure 6: Amplitude of the electric field as function of the length of a
Gaussian laser pulse for a normalized vector potential $a_{0}=0.3$.

One can note that in the linear regime, the electric field has a sinusoidal
shape and reaches maximal values of a few ${\,\mathrm{GV/m}}$. For example,
for an electron density $n_{\mathrm{e}}=10^{19}$ cm-3, the optimal pulse
duration equals $L=2.4$ $\mu$m (equivalent to a pulse duration $\tau=8$ fs).
For $a_{0}=0.3$, the maximal electric field is in the ${\,\mathrm{GV/m}}$
range. Figure 7 illustrates the density perturbation and the corresponding
longitudinal electric field produced at resonance by a low intensity,
$I_{\mathrm{laser}}=3\times 10^{17}$ W$/$cm2, laser pulse of 30 s duration.

Figure 7: Density perturbation (top) and electric field (bottom) produced in
the linear regime

In experiments carried out at LULI, relativistic plasma waves with $1\%$
amplitude have been demonstrated. As indicated in Fig. 8, 3 ${\,\mathrm{MeV}}$
electrons have been injected into a relativistic plasma wave driven by a 300
fs laser pulse, some of which were accelerated up to 4.6
${\,\mathrm{MeV}}$[8]. The electron spectra has a broad energy distribution
with a Maxwellian like shape, as expected when injecting an electron beam with
a duration much longer that the plasma period, and in this case with a
duration much longer than the plasma wave live-time.

Figure 8: Electrons spectra obtained at LULI in the laser wakefield scheme

###  0.2.2 Laser beatwave

Before the advent of short and intense laser pulses, relativistic plasma waves
were driven by the beatwave of two long laser pulses of a few tens of
picoseconds (i.e. with duration much greater than the plasma period). In this
case, the plasma frequency $\omega_{\mathrm{p}}$ has to satisfy exactly the
matching condition, $\omega_{\mathrm{p}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \omega_{1}-\leavevmode\nobreak\ \omega_{2}$, with
$\omega_{1}$ and $\omega_{2}$ the frequencies of the two laser pulses. The
first observation of relativistic plasma waves was performed using the Thomson
scattering technique by the group of C. Joshi at UCLA [9]. Acceleration of 2
${\,\mathrm{MeV}}$ injected electrons up to 9 ${\,\mathrm{MeV}}$ [10] and
later on, up to 30 ${\,\mathrm{MeV}}$ [11], were demonstrated by the same
group using a CO2 laser of about 10 $\mu$m wavelength. At LULI, 3
${\,\mathrm{MeV}}$ electrons were accelerated up to 3.7 ${\,\mathrm{MeV}}$ in
beat wave experiments with Nd:Glass lasers of about 1 $\mu$m wavelengths by a
longitudinal electric field of 0.6 ${\,\mathrm{GV/m}}$ [12]. Similar works
were also performed in Japan at University of Osaka [13], in the UK at the
Rutherford Appleton Laboratory [14], and in Canada at the Chalk River
Laboratory [15]. Electron spectra obtained at LULI in the laser beat wave
scheme are shown on Fig. 9.

Figure 9: Electrons spectra obtained at LULI in the laser beat wave scheme

In order to reduce the coupling between electron waves and ion waves which was
a limiting factor of previous experiments performed with 100 ps Nd lasers
[16], experiments done at the Rutherford Appleton Laboratory with a 3 ps laser
pulse have shown excitations of higher amplitude relativistic plasma waves
[17].

###  0.2.3 Self-modulated laser wakefield

In all of these experiments, because of the duration of the injected electron
bunch, which is much longer than the plasma period and even longer than the
life time of the plasma, only a very small fraction of injected electrons were
accelerated and the output beam had a very poor quality with a Maxwellian-like
energy distribution.

Thanks to the development of powerful laser systems with short pulse duration
(500 fs), a new regime that allows self-injection of electrons in very intense
accelerating gradients with values exceeding 100${\,\mathrm{GV/m}}$ has been
discovered. The cumulative effects of the self-focusing and the self-
modulation of the laser envelope by the initial perturbation of the electron
plasma density generates a train of laser pulses which become resonant with
the plasma wave. These effects are described in Fig. 10. The self-modulated
laser wakefield regime occurs when the laser pulse duration exceeds the plasma
period and when the laser power exceeds the critical power for self-focusing
[18, 19, 20]. The initial Gaussian laser pulse becomes modulated at the plasma
wavelength during its propagation. This mechanism, which is close to a Forward
Raman Scattering Instability [21], can be described as the decomposition of an
electromagnetic wave into a plasma wave at a frequency shifted by the plasma
frequency.

In an experiment done at the Rutherford Appleton Laboratory, a relativistic
plasma wave was excited by an intense laser (${>}5\times 10^{18}$ W$/$cm2),
for a short duration (${<}1$ ps), by a 1.054 $\mu$m wavelength laser pulse in
the self-modulated laser wakefield regime. This is the decay (induced by a
noise level plasma wave) of the strong electromagnetic pump wave
($\omega_{0},k_{0}$) into the plasma wave
($\omega_{\mathrm{p}},k_{\mathrm{p}}$) and two forward propagating
electromagnetic cascades at the Stokes ($\omega_{0}-n\omega_{\mathrm{p}}$) and
anti-Stokes ($\omega_{0}+n\omega_{\mathrm{p}}$) frequencies, $n$ being a
positive integer, and $\omega$ and $k$ being the angular frequency and the
wavenumber, respectively, of the indicated waves. The spatial and temporal
interference of these sidebands with the laser produces an electromagnetic
beat pattern propagating synchronously with the plasma wave. The
electromagnetic beat exerts a force on the plasma electrons, reinforcing the
original noise level plasma wave which scatters more sidebands, thus closing
the feedback loop for the instability.

The solid curve in Fig. 11 shows the electromagnetic frequency spectrum
emerging form the plasma with a density of ${>}5\times 10^{18}$ cm-3, where
the abscissa is the shift in frequency of the forward scattered light from the
laser frequency in units of $\omega_{\mathrm{p}}$. The upshifted anti-Stokes
and downshifted Stokes signals at $\Delta\omega/\omega_{\mathrm{p}}=\pm 1$ are
clearly visible as is the transmitted pump at
$\Delta\omega/\omega_{\mathrm{p}}=0$ and the second and third anti-Stokes
sidebands. These signals are sharply peaked, and their widths indicate that
the plasma wave which generated these signals must have a coherence time of
the order of the laser pulse. The dashed curve shows the spectrum when the
density is increased to $1.5\times 10^{19}$ cm-3. The most startling feature
is the tremendous broadening of the individual anti-Stokes peaks at this
higher density. This broadening corresponds to wave-breaking and is mainly
caused by the loss of coherence due to severe amplitude and phase modulation
as the wave breaks. As wave-breaking evolves, the laser light no longer
scatters off a collective mode of the plasma but instead scatters off the
trapped electrons which are still periodically deployed in space but have a
range of momenta producing, therefore, a range of scatter frequencies.

Figure 10: Evolution of the laser pulse and plasma density in the self-
modulated laser wakefield regime

During experiments carried out in the UK in 1994 [4], the amplitude of the
plasma waves reached the wave-breaking limit, where electrons initially
belonging to the plasma wave are self-trapped and accelerated to high
energies. The fact that the external injection of electrons in the wave is no
longer necessary is a major improvement. Electron spectrums extending up to 44
${\,\mathrm{MeV}}$ have been measured during this first campaign, and up to
104 ${\,\mathrm{MeV}}$ in the second campaign. This regime has also been
reached for instance in the United States at CUOS [22], and at NRL [23].
However, because of the heating of the plasma by these relatively ‘long’
pulses, the wave-breaking occurred well before reaching the cold wave-breaking
limit, which limited the maximum electric field to a few 100
${\,\mathrm{GV/m}}$. The maximum amplitude of the plasma wave has also been
measured to be in the range 20–60% [24].

Figure 11: Frequency and electron spectrum the self-modulated laser wakefield
regime for two different electron plasma densities: $0.54\times 10^{18}$ cm-3
(in red) and $1.5\times 10^{19}$ cm-3 (in blue).

Experiments performed at LOA since 1999 have shown that an electron beam can
also be produced using a compact 10 Hz laser system [25]. Fig. 12 shows two
typical electron spectra obtained at $5\times 10^{19}$ cm-3 and $1.5\times
10^{20}$ cm-3. The 0.6 J, 35 fs laser beam was focused tightly to a 6 $\mu$m
focal spot leading to a peak laser intensity of $2\times 10^{19}$ W$/$cm2.
Electron distributions with electron energy greater than 4 ${\,\mathrm{MeV}}$
are well fitted by an exponential function, characteristic of an effective
temperature for the electron beam. These effective temperatures are 8.1
${\,\mathrm{MeV}}$ (2.6 ${\,\mathrm{MeV}}$) for electron density of $5\times
10^{19}$ cm-3 ($1.5\times 10^{20}$ cm-3), to which correspond typical values
of 54 ${\,\mathrm{MeV}}$ (20 ${\,\mathrm{MeV}}$) for the maximum electron
energy. This maximum energy is defined by the intersection between the
exponential fit and the detection threshold. One can observe an important
decrease in the effective temperature and in the maximum electron energy for
increasing electron densities.

This point is summarized in Fig. 12 where we present the maximum electron
energy as a function of the electron density. It decreases from 70
${\,\mathrm{MeV}}$ to 15 ${\,\mathrm{MeV}}$ when the electron density
increases from $1.5\times 10^{19}$ cm-3 to $5\times 10^{20}$ cm-3. Also
presented in Fig. 12 is the theoretical value [26]

$W_{\mathrm{max}}\approx
4\gamma_{\mathrm{p}}^{2}(E_{z}/E_{0})mc^{2}F_{\mathrm{NL}}.$ (10)

Figure 12: Left: Typical electron spectra obtained at $5\times 10^{19}$ cm-3
(squares) and $1.5\times 10^{20}$ cm-3 (circles). The corresponding effective
temperatures are 8.1 ${\,\mathrm{MeV}}$ (2.6 ${\,\mathrm{MeV}}$) for electron
density of $5\times\leavevmode\nobreak\ 10^{19}$ cm-3 ($5\times 10^{20}$
cm-3). Right: Maximum electron energy as a function of the plasma electron
density. Experimental data: squares. Theoretical data: line.

Here, the maximum electron energy is greater than the conventional one given
by the simple formula $W_{\mathrm{max}}\approx
2\gamma_{\mathrm{p}}^{2}(E_{z}/E_{0})mc^{2}$, where $\gamma_{\mathrm{p}}$ is
the plasma wave Lorentz factor (which is equal to the critical density to
electron density ratio $n_{\mathrm{c}}/n_{\mathrm{e}}$) and $E_{z}/E_{0}$ is
the electrostatic field normalized to $E_{0}$
($E_{0}=cm\omega_{\mathrm{p}}/e$). The factor of two is due to self-
channelling induced by the space-charge field which focuses accelerated
electrons for all phases. The correction factor $F_{NL}\approx(\gamma_{\perp
0}n_{0}/n)^{3/2}$ corresponds to a non-linear correction due to the
relativistic pump effect and to self-channelling. In this formula, $n_{0}$ is
the initial electron density, $n$ the effective one and $\gamma_{\perp 0}$ is
the Lorentz factor associated to the laser intensity: $\gamma_{\perp
0}=(1+a_{0}^{2}/2)^{1/2}$. The electron density depression is estimated by
balancing the space-charge force and laser ponderomotive force, and evaluated
by $\delta
n/n=(a_{0}^{2}/2\pi^{2})(1+a_{0}^{2}/2)^{-1/2}(\lambda_{p}/w_{0})^{2}$.

In the lower electron density case, the depression correction will introduce
an important increase of the maximum energy gain which is multiplied by a
factor of 2 at $1.5\times 10^{19}$ cm-3. For densities greater than $1.0\times
10^{20}$ cm-3, the main contribution is due to the relativistic pump effect,
as outlined on the plot in Fig. 12. It is also crucial to note that the fact
that the electron maximum energy increases when the electron density decreases
demonstrates that electrons are mainly accelerated by relativistic plasma
waves. The maximum electron energy calculated at lower density overestimated
the experimental ones, indicating that the dephasing length becomes shorter
than the Rayleigh length. In order to solve this problem, experiments were
performed at LOA using a longer off-axis parabola, more energetic electrons
have been measured, with a peak laser intensity ten times smaller than in this
first experiment.

Electron beams with Maxwellian spectral distributions, generated by compact
high repetition rate ultra-short laser pulses, have been also at this time
been produced in many laboratories around the world: at LBNL [27], at NERL
[28] and in Germany [29] for instance, and are now currently produced in more
than 20 laboratories worldwide.

#### Forced laser wakefield

The forced laser wakefield regime [30] is reached when the laser pulse
duration is approximately equal to the plasma period and when the laser waist
is about the plasma wavelength. This regime allows a reduction in heating
effects that are produced when the laser pulse interacts with trapped
electrons. In this regime, highly non-linear plasma waves can be reached as
can be seen in Fig. 13.

Figure 13: Density perturbation (top) and electric field (bottom) produced in
the non-linear regime

The laser power needs also to be greater that the critical power for
relativistic self-focusing in order for the laser beam to shrink in time and
in space. Due to self-focusing, pulse erosion can take place, which can allow
efficient wake generation. Since the very front of the pulse is not self-
focused, the erosion will be more severe. The wake then is mostly formed by
this fast rising edge, and the back of the pulse has little interaction with
the relativistic longitudinal oscillation of the plasma wave electrons.
Indeed, the increase of plasma wavelength due to relativistic effects means
that the breaking and accelerating peak of the plasma wave sits behind most,
if not all, of the laser pulse. Hence its interaction, and that of the
accelerated electrons with the laser pulse, is minimized, thus reducing
emittance growth due to direct laser acceleration. Thanks to short laser
pulses, plasma heating in the forced laser wakefield regime is significantly
lower than in the self-modulated wakefield regime. This allows much higher
plasma wave amplitudes to be reached, as well as higher electron energies.
Thanks to a limited interaction between the laser and the accelerated
electrons, the quality of the electron beam is also improved. Indeed, the
normalized transverse emittance measured using the pepper pot technique has
given values comparable to those obtained with conventional accelerators with
an equivalent energy (normalized r.m.s. emittance $\varepsilon_{n}=3\pi$ mm
mrad for electrons at $55\pm 2$ ${\,\mathrm{MeV}}$) [31].

The three-dimensional simulations realized for this experiment showed that the
radial plasma wave oscillations interact coherently with the longitudinal
field, so enhancing the peak amplitude of the plasma wave. This, coupled with
the aforementioned strong self-focusing, are ingredients absent from one-
dimensional treatments of this interaction. Even in two-dimensional
simulations, it was not possible to observe electrons beyond 200
${\,\mathrm{MeV}}$, as measured in this experiment, since except in three-
dimensional simulations, both the radial plasma wave enhancement and self-
focusing effects are underestimated. Hence it is only in three-dimensional
simulations that $E_{\mathrm{max}}\sim E_{\mathrm{wb}}$ can be reached. That
such large electric fields are generated demonstrates another important
difference between FLW and SMWF regimes, since in the latter, plasma heating
by instabilities limits the accelerating electric field to an order of
magnitude below the cold wave-breaking limit. It should be noted that the peak
electric field inferred for these FLW experiments is in excess of 1
${\,\mathrm{TV/m}}$, considerably larger than any other coherent accelerating
structure created in the laboratory.

Figure 14: Typical experimental (blue squares) and calculated (black curve)
electron spectrum obtained at $n_{\mathrm{e}}=\leavevmode\nobreak\
2.5\leavevmode\nobreak\ \times 10^{19}$ cm-3 with a 1 J, 30 fs laser pulse
focused down to a waist of ${w}_{0}=18\mu$m.

#### Bubble regime

In 2002, theoretical work based on three-dimensional particle-in-cell (PIC)
simulations have shown the existence of a robust acceleration mechanism called
the bubble regime [32]. In this regime, the dimensions of the focused laser
are shorter than the plasma wavelength in longitudinal and also transverse
directions, the laser shape appearing like a ball of light. If the laser
energy contained in this spherical volume is large enough, the ponderomotive
force of the laser expels radially and efficiently electrons from the plasma,
which forms a cavity free from electrons behind the laser surrounded by a
dense region of electrons. Behind the bubble, electron trajectories intersect
each other. Electrons are injected into the cavity and accelerated along the
laser axis, thus creating an electron beam with radial and longitudinal
dimensions smaller than those of the laser (see Fig. 15).

Figure 15: Left: acceleration principle in the bubble regime. Right: typical
quasi-monoenergetic electron spectra measured at LOA.

The signature of this regime is a quasi-monoenergetic electron distribution
that results from the localization of injection at the back of the cavity,
which gives similar initial properties in the phase space to injected
electrons. Since electrons are trapped behind the laser pulse, this reduces or
even suppresses interaction with the electric field of the laser. The trapping
process stops when the charge contained in the cavity compensates the ionic
charge, and the rotation in the phase space also leads to a shortening of the
spectral width of the electron beam [33].

Several laboratories have obtained quasi-monoenergetic spectra: in France [34]
with a laser pulse shorter than the plasma period, but also with pulses
slightly longer than the plasma period in the U.K. [35], in the United States
[36], then in Japan [37] and in Germany [38]. The interest in such a beam is a
result of its importance for a number of applications: it is now possible to
transport and to refocus this beam by magnetic fields. With a Maxwellian-like
spectrum, it would have been necessary to select an energy range for the
transport, which would have decreased significantly the electron flux.
Electrons in the ${\,\mathrm{GeV}}$ level were also observed in this regime
using a uniform plasma [39] or in plasma discharge, i.e. a plasma with a
parabolic density profile [40] with a more powerful laser which propagates at
high intensity over a longer distance. With the development of PW class
lasers, a few ${\,\mathrm{GeV}}$ electron beam has been reported [41, 42, 43].

In all the experiments performed so far, the laser plasma parameters were not
sufficient to fully enter the bubble/blowout regimes. Yet, with the increase
of laser system power, this regime will be reached, and significant
improvement of the reproducibility of the electron beam is expected.
Nevertheless, since self-injection occurs through transverse wave-breaking, it
is hardly appropriate for a fine tuning and control of the injected electron
bunch. Figure 16 shows electron distributions obtained for different
densities. It illustrates the transition from a Maxwellian-like spectrum
obtained in high density cases in the self-modulated laser wakefield, to the
forced laser wakefield regime with an emerging monoenergetic component at
moderate density, to a spectrum containing a very well defined monoenergetic
component. This transition occurs for densities around $n_{\mathrm{e}}=$1–3
$\times 10^{19}$ cm-3. The best coupling for obtaining a high charge and a
quasi-monoenergetic electron beam is at $n_{\mathrm{e}}=6\times 10^{18}$ cm-3.
For this density, the image shows a narrow peak around 170 ${\,\mathrm{MeV}}$,
indicating efficient monoenergetic acceleration with a 24% energy spread
corresponding to the spectrometer resolution.

Figure 16:  Electron beam distribution for different plasma densities showing
the transition from the self-modulated laser wakefield and the forced laser
wakefield to the bubble/blow-out regime. From top to bottom, the plasma
density values are $6\times 10^{18}\,\mathrm{cm^{-3}}$, $1\times
10^{19}\,\mathrm{cm^{-3}}$, $2\times 10^{19}\,\mathrm{cm^{-3}}$ and $5\times
10^{19}\,\mathrm{cm^{-3}}$.

###  0.2.4 Injection in a density gradient

One solution to control electron injection with current laser technology was
proposed by S. Bulanov $et$ $al.$ [44]. It involves a downward density ramp
with a density gradient scale length $L_{\mathrm{grad}}$ smaller than the
plasma wavelength $\lambda_{\mathrm{p}}$. Injection in a downward density ramp
relies on the slowing down of the plasma wave velocity at the density ramp.
This decrease of the plasma wave phase velocity lowers the threshold for
trapping the plasma background electrons and causes wave-breaking of the
wakefield in the density ramp. This method can therefore trigger wave-breaking
in a localized spatial region of the plasma. Geddes _et al._ [45] showed the
injection and acceleration of high charge (${>}300$ pC) and stable quality
beams of ${\simeq}0.4$${\,\mathrm{MeV}}$ in the downward density ramp at the
exit of a gas jet ($L_{\mathrm{grad}}\simeq
100\,\mu\mathrm{m}\gg\lambda_{\mathrm{p}}$). These results, although very
promising, have the disadvantage that the low energy beam blows up very
quickly out of the plasma, due to the space-charge effect. To circumvent this
issue, one should use a density gradient located early enough along the laser
pulse propagation so that electrons can be accelerated to relativistic
energies [46]. This can be achieved by using, for instance, a secondary laser
pulse to generate a plasma channel transverse to the main pulse propagation
axis [47]. In this case, the electron beam energy could be tuned by changing
the position of the density gradient. In this pioneering experiment, the
electron beam had a large divergence and a Maxwellian energy distribution
because of a too low laser energy. However, two-dimensional PIC simulations
showed that this method can result in high quality quasi-monoenergetic
electron beams [48].

At LOA a density gradient across a laser created plasma channel was used to
stabilize the injection [49]. The experiment was performed at an electron
density close to the resonant density for the laser wakefield
($c\tau\sim\lambda_{\mathrm{p}}$) to guaranty a post acceleration that
delivered high quality electron beams with narrow divergences (4 mrad) and
quasi-monoenergetic electron distributions with 50 to 100 pC charge and 10%
relative energy spread.

The use of density gradients at the edges of a plasma channel showed an
improvement of the beam quality and of the reproducibility with respect to
those produced in the bubble/blowout regime with the same laser system and
with similar laser parameters. However, the electron energy distribution was
still found to fluctuate from shot to shot. The performance of the experiments
could be further improved and could potentially lead to more stable and
controllable high quality electron beams. In particular, sharper gradients
with $L_{\mathrm{grad}}\simeq\lambda_{\mathrm{p}}$ coupled with a long plasma
can lead to better beam quality [50].

Figure 17:  Top: target schematic representation with embedded supersonic gas
jet into a capillary that is filled with hydrogen gas. Bottom: the charge
(squares), energy (circles) and energy spread (triangles) as a function of the
peak jet density. From A. J. Gonsalves et al. [51].

For example, at LBNL, as shown in Fig. 17, electrons at 30 ${\,\mathrm{MeV}}$
were produced in a density ramp and accelerated up to 400 ${\,\mathrm{MeV}}$
in a second stage 4 cm parabolic plasma channel formed with a plasma discharge
[51]. Here also, the density gradient injection led to an improvement of the
stability and of the electron beam quality. The electron energy, divergence,
charge and relative energy spread were found to be respectively 400
${\,\mathrm{MeV}}$, 2 mrad, 10 pC and 11%. It was shown that steeper density
transitions, with $L_{\mathrm{grad}}\ll\lambda_{\mathrm{p}}$, can also cause
trapping [52]. Such injection was successfully demonstrated experimentally
using the shock-front created by a knife-edge inserted in a gas jet [53, 54].

Figure 18 illustrates the improvement of injection in a sharp density
gradient, with a characteristic length of the order of the plasma wavelength
and a peak electron density of about $5\times 10^{19}\,\mathrm{cm^{-3}}$. The
experiment was performed at the Max-Planck-Institut fur Quantenoptik using a
multi-TW sub-10 fs laser system that delivered for this experiment pulses with
65 mJ energy on target and with a duration of 8 fs FWHM. The laser pulse was
focused down to a spot diameter of $12\,\mu\mathrm{m}$ FWHM into the gas
target yielding a peak intensity of $2.5\times 10^{18}\,\mathrm{W/cm}^{2}$.
The comparison between the self-injection and density transition injection
shows a reduction of the relative energy spread and of the charge of a about a
factor of 2.

Figure 18:  A few representative shots of the 10% of all the shots with the
lowest energy spread for self-injection (top) and injection at a density
transition (bottom). The horizontal axis in each image corresponds to the
transverse electron beam size; the vertical axis shows electron energy. From
K. Schmid et al.[54].

###  0.2.5 Injection with colliding laser pulses

In 2006, stable and tunable quasi-monoenergetic electron beams were measured
by using two counterpropagating laser beams in the colliding scheme. The use
of two laser beams instead of one offers more flexibility and enables one to
separate the injection from the acceleration process [55]. The first laser
pulse, the pump pulse, is used to excite the wakefield while the second pulse,
the injection pulse, is used to heat electrons during the collision with the
pump pulse. After the collision has occurred, electrons are trapped and
further accelerated in the wakefield, as shown in Fig. 19.

To trap electrons in a regime where self-trapping does not occur, one has
either to inject electrons with energies greater that the trapping energy or
dephase electrons with respect to the plasma wave. As mentioned earlier,
electrons need to be injected in a very short time
(${\ll}\lambda_{\mathrm{p}}/c$) in order to produce a monoenergetic beam. This
can be achieved using additional ultra-short laser pulses whose only purpose
is to trigger electron injection.

Umstadter $et$ $al.$ [22] first proposed to use a second laser pulse
propagating perpendicular to the pump laser pulse. The idea was to use the
radial ponderomotive kick of the second pulse to inject electrons. Esarey $et$
$al.$ [56] proposed a counter-propagating geometry based on the use of three
laser pulses. This idea was further developed by considering the use of two
laser pulses [57]. In this scheme, a main pulse (pump pulse) creates a high
amplitude plasma wave and collides with a secondary pulse of lower intensity.
The interference of the two beams creates a beatwave pattern, with a zero
phase velocity, that heats some electrons from the plasma background. The
force associated with this ponderomotive beatwave is proportional to the laser
frequency. It is therefore many times greater than the ponderomotive force
associated with the pump laser, that is inversely proportional to the pulse
duration at resonance. As a result, the mechanism is still efficient even for
modest laser intensities. Upon interacting with this field pattern, some
background electrons gain enough momentum to be trapped in the main plasma
wave and then accelerated to high energies. As the overlapping of the lasers
is short in time, the electrons are injected over a very short distance and
can be accelerated to an almost monoenergetic beam.

This concept was validated in an experiment [55], using two counter-
propagating pulses. Each pulse had a duration of 30 fs FWHM, with $a_{0}=1.3$
and $a_{1}=0.4$. They were propagated in a plasma with electron density
$n_{\mathrm{e}}=7\times 10^{18}$ cm-3 corresponding to
$\gamma_{\mathrm{p}}=k_{0}/k_{\mathrm{p}}=15$. It was shown that the collision
of the two lasers could lead to the generation of stable quasi-monoenergetic
electron beams. The beam energy could be tuned by changing the collision
position in the plasma.

Figure 19:  Scheme of the principle of the injection with colliding laser
pulses: (a) the two laser pulses propagate in opposite direction; (b) during
the collision, some electrons get enough longitudinal momentum to be trapped
by the relativistic plasma wave driven by the pump beam and (c) trapped
electrons are then accelerated in the wake of the pump laser pulse.

PIC simulations in one dimension have been used to model electron injection in
the plasma wave at the collision of the two lasers, and their subsequent
acceleration. In particular, the PIC simulations were compared to existing
fluid models [56] with prescribed electric field. They showed significant
differences, such as changes in the behaviour of plasma fields and in the
amount of injected charge. The fluid approach fails to describe qualitatively
and quantitatively many of the physical mechanisms that occur during and after
the laser beams collision [58]. In this approach, the electron beam charge was
found to be one order of magnitude greater than in the PIC simulations. For a
correct description of injection, one has to describe properly (i) the heating
process, e.g. kinetic effects and their consequences on the dynamics of the
plasma wave during the beating of the two laser pulses and (ii) the laser
pulse evolution which governs the dynamics of the relativistic plasma waves
[59]. Unexpectedly, it was shown that efficient stochastic heating can be
achieved when the two laser pulses are crossed polarized. The stochastic
heating can be explained by the fact that for high laser intensities, the
electron motion becomes relativistic which introduces a longitudinal component
through the $\mathbf{v}\times\mathbf{B}$ force. This relativistic coupling
makes it possible to heat electrons even in the case of crossed polarized
laser pulses [60]. Thus the two perpendicular laser fields couple through the
relativistic longitudinal motion of electrons. The heating level is modified
by tuning the intensity of the injection laser beam or by changing the
relative polarization of the two laser pulses [61]. This consequently changes
the volume in the phase space of the injected electrons and therefore the
charge and the energy spread of the electron beam.

Figure 20 shows, at a given time (42 fs), the longitudinal electric field
during and after collision for parallel and crossed polarization. The solid
line corresponds to PIC simulation results whereas the dotted line corresponds
to fluid calculations. The laser fields are represented by the thin dotted
line. When the pulses have the same polarization, electrons are trapped
spatially in the beatwave and cannot sustain the collective plasma
oscillation, inducing a strong inhibition of the plasma wave which persists
after the collision. When the polarizations are crossed, the electron motion
is only slightly disturbed compared to their fluid motion, and the plasma wave
is almost unaffected during the collision, which tends to facilitate trapping.

Figure 20:  Longitudinal electric field computed at $t=43$ fs in one-
dimensional PIC simulation (solid red line), and in fluid simulations (dotted
blue line). The transverse electric field is also represented (thin dotted
line). The laser pulse duration is 30 fs FWHM, the wavelength is 0.8 $\mu$m
with $a_{0}=2$ and $a_{1}=0.4$. The laser pulses propagate in a plasma with
electron density $n_{\mathrm{e}}=7\times 10^{18}$ cm-3. In (a) the case of
parallel polarization and in (b) the case of crossed polarization.

Importantly, it was shown that the colliding pulse approach allows control of
the electron beam energy which is done simply by changing the delay between
the two laser pulses [55]. The robustness of this scheme permitted also very
accurate studies of the dynamics of the electric field in presence of a high
current electron beam to be carried out. Indeed, in addition to the wakefield
produced by the laser pulse, a high current electron beam can also drive its
own wakefield as shown in Fig. 21.

Figure 21:  In red, normalized longitudinal electric field. (a) The laser (in
pink) wakefield. (b) The electron bunch (in blue) wakefield. (c) Field
resulting from the superposition of the laser and electron beam wakefields.
The normalized vector potential is $a_{0}=1$, the laser pulse duration is 30
fs, $n_{\mathrm{e}}=7\times 10^{18}$ cm-3, $n_{\mathrm{beam}}=0.11\times
n_{\mathrm{e}}$, the bunch duration is 10 fs and its diameter is 4 $\mu$m.
From C. Rechatin, Ph.D. thesis.

The beam loading effect contributes to the reduction of the relative energy
spread of the electron beam. It was demonstrated that there is an optimal load
which flattened the electric field, leading to the acceleration of all the
electrons with the same value of the field, and producing consequently an
electron beam with a very small, 1%, relative energy spread [62]. Thanks to
the beam loading effect, the most energetic electrons can be slightly slowed
down and accelerated to the same energy as the slowest ones. In cases of low
charge beam, this effect does not play any role and the energy spread depends
mainly on the heated volume. For a very high current, the load is too high and
the most energetic electrons slow down so much that they eventually obtain
energies smaller than the slowest electrons [62], increasing the relative
energy spread. The existence of an optimal load was observed experimentally
and supported by full three-dimensional PIC simulations. It corresponds to a
peak current in the 20–40 kA range. The decelerating electric field due to the
electron beam was found to be in the ${\mathrm{GV/m/pC}}$ range.

###  0.2.6 Injection triggered by ionization

Another scheme was proposed recently to control the injection by using a
high-$Z$ gas and/or a high-$Z$/low-$Z$ gas mixture. Thanks to the large
differences in ionization potentials between successive ionization states of
the atoms, a single laser pulse can ionize the low energy level electrons in
its leading edge, drive relativistic plasma waves, and inject in the wakefield
the inner level electrons which are ionized when the laser intensity is close
to its maximum.

Such an ionization trapping mechanism was first demonstrated in electron beam
driven plasma wave experiments on the Stanford Linear Collider (SLAC) [63].
Electron trapping from ionization of high-$Z$ ions from capillary walls was
also inferred in experiments on laser wakefield acceleration [64]. In the case
of a self-guided laser driven wakefield, a mixture of helium and trace amounts
of different gases was used [65, 66]. In one of these experiments, electrons
from the K shell of nitrogen were tunnel ionized near the peak of the laser
pulse and were injected into and trapped by the wake created by electrons from
majority helium atoms and the L shell of nitrogen. Because of the relativistic
self-focusing effect, the laser propagates over a long distance with peak
intensity variations that can trigger the injection over a long distance and
in an inhomogeneous way, which leads to the production of a high relative
energy spread electron beam. Importantly, the energy required to trap
electrons is reduced, making this approach of great interest to produce
electron beams with a large charge at moderate laser energy. To reduce the
distance over which electrons are injected, experiments using two gas cells
were performed at LLNL [67], as shown in Fig. 22. By restricting electron
injection to a small region, in a first short cell filled with a gas mixture
(the injector stage), energetic electron beams (of the order of 100
${\,\mathrm{MeV}}$) with a relatively large energy spread were generated. Some
of these electrons were then further accelerated in a second, larger,
accelerator stage, consisting of a long cell filled with low-Z gas, which
increases their energy up to 0.5 ${\,\mathrm{GeV}}$ while reducing the
relative energy spread to ${<}5\%$ FWHM.

Figure 22:  (a) Schematic of the experimental set-up showing the laser beam,
the two-stages gas cell, on the left the injector part and on the right the
accelerator part. (b) Magnetically dispersed electron beam images from a 4 mm
injector-only gas cell (top) and the 8 mm two-stages cell (bottom). From B. B.
Pollock et al. [67].

###  0.2.7 Longitudinal injection

As has been shown, electron trapping is generally achieved by the wave-
breaking of the plasma wake, a process that is by nature uncontrollable and
leads generally to poor quality electrons. The presented controlled injection
techniques, such as colliding pulse injection, ionization-induced injection
and density gradient injection, have been developed to overcome this
shortcoming. These methods offer an improved control on the acceleration and
lead to better electron features, but they imply generally complex set-ups.
For this reason, self-injection remains the most common method for injecting
electrons in the plasma wake. Two distinct physical mechanisms can be
distinguished: longitudinal and transverse self-injection. In longitudinal
self-injection, the trajectory of injected electrons is mainly longitudinal,
with a negligible transverse motion. As shown by the schematic in Fig. 23, the
injected electrons pass through the laser pulse and gain energy while crossing
the plasma wave. When they reach the rear of the first plasma period, their
velocity exceeds the wake phase velocity and the electrons are eventually
injected. The only electrons that are trapped are those that were initially
close to the axis where the laser intensity and the wakefield amplitude are
the highest and where the ponderomotive force is small. The longitudinal self-
injection mechanism is analogous to one-dimensional longitudinal wave-
breaking. In contrast, transverse self-injection occurs in the bubble regime,
where the laser ponderomotive force expels electrons from the propagation axis
and forms an electron-free cavity in its wake. As shown in Fig. 23 and in Fig.
15, the injected electrons are initially located at approximately one laser
waist from the axis. They circulate around the laser pulse and the bubble, and
attain a velocity larger than the wake phase velocity when reaching the axis
at the rear of the bubble.

Figure 23:  Schematic for longitudinal and transverse self-injections. (a)
Typical trajectory of an injected electron in the longitudinal self-injection
mechanism. (b) Typical trajectory of an injected electron in the transverse
self-injection mechanism. The blue colour scale represents the electron
density. The red to yellow colour scale indicates the laser intensity. The
trajectories are given by the green lines.

During its propagation, the laser pulse evolves, the self-phase-modulation
modifies its duration and the relativistic self-focusing modifies its initial
transverse shape. As a consequence, the generated wakefield is not uniform
along the laser propagation axis and electrons can be self-injected at
different positions of the plasma accelerator. Electrons in the second bunch
originate from positions close to the laser waist, as expected in the case of
trapping by transverse self-injection. In contrast, electrons in the first
bunch come from regions close to the axis. When these electrons are injected,
the laser spot radius is large and the normalized laser amplitude is still
low; hence, the radial ponderomotive force close to the axis is small.

Figure 24:  Left: Evolution of the normalized laser amplitude. The coloured
areas indicate the injection regions. Right: Electron spectra for a density of
$1.1\times 10^{19}\,\mathrm{cm^{-3}}$. Mean $5\%$ cut-off energy as a function
of the cell length. The solid line is a parabolic fit, the dashed line is a
visual guide. The peak accelerating electric field is
$E_{\mathrm{max}}^{z}=340\pm 65{\,\mathrm{GV/m}}$ for the first bunch and
$E_{\mathrm{max}}^{z}=185{\pm}40{\,\mathrm{GV/m}}$ for the second bunch.

Thus, on-axis electrons are only weakly deviated when crossing the laser
pulse, and they remain in the region of largest accelerating field $E_{z}$.
Moreover, the laser amplitude increases steeply in the region of first
injection because of laser self-focusing (see Fig. 24). This reduces the wake
phase velocity via the relativistic shift of the plasma wavelength
$\lambda_{\mathrm{p}}$ [68]. The strongly reduced wake phase velocity lowers
the threshold for trapping, such that electrons can catch up the plasma wave
and be injected despite a low $a_{0}$, similarly to density gradient
injection.

##  0.3 Particle beam wakefield accceleration

###  0.3.1 Electron driven plasma wakefield

Wakefields in a plasma can be also driven by an electron bunch that has, at
resonance, a length of about half the plasma wavelength. Whereas in the laser
wakefield case the radiation pressure, known as the ponderomotive force,
pushes away the plasma electrons, here the force is due the space-charge of
the electron beam. The plasma electrons are strongly blown out radially, but
because of the space-charge attraction of the plasma ions, they are attracted
back towards the rear of the beam where they overshoot the beam axis and set
up a wakefield oscillation. Here again, charged particles injected in an
appropriately phased trailing pulse can then extract energy from the
wakefield. Because of the lack of accelerators that deliver suitable electron
beams, there are fewer particle-beam-driven plasma acceleration experiments
compared with laser accelerator experiments. The first beam-driven plasma
wakefield experiments were carried out at the Argonne Wakefield Accelerator
Facility in the 1980s [69]. Since then, important experiments done at SLAC by
the UCLA/USC/SLAC collaboration have mapped the physics of electron and
positron beam-driven wakes and has shown acceleration gradients of
$40{\,\mathrm{GV/m}}$ using electron beams with metre-scale plasmas [70]. In
the first important SLAC experiments only one electron bunch was used to
excite the wakefield. Since the energy of the drive pulse was 42
${\,\mathrm{GeV}}$, both the electrons and the wake are moving at a velocity
close to c, so there is no relative motion between the electrons and the
wakefield. Because the electron bunch was also longer that the plasma period,
most of the electrons in the drive bunch lose energy in exciting the wake, but
some electrons in the back have gained energy from the wakefield as the
wakefield changes its sign. Thanks to the high quality, low emittance of the
electron bunch, its intensity was so high that the 42 ${\,\mathrm{GeV}}$
electron beam passed through a column of lithium vapour 85 cm long, the head
of the beam created a fully ionized plasma and the remainder of the beam
excited a strong wakefield. Figure 25 shows the energy spectrum of the beam
measured after the plasma. The electrons in the bulk of the pulse that lost
energy in driving the wake are mostly dispersed out of the field of view of
the spectrometer camera and so are not seen in the spectrum. However,
electrons in the back of the same pulse are accelerated and reach energies up
to 85 ${\,\mathrm{GeV}}$. The measured spectrum of the accelerated particles
was in good agreement with the spectrum obtained from computer simulations of
the experiment, as Fig. 25 shows. As said Prof. C. Joshi, ‘This is a
remarkable result when one realizes that while it takes the full 3 km length
of the SLAC linac to accelerate electrons to 42 ${\,\mathrm{GeV}}$, some of
these electrons can be made to double their energy in less than a metre’.

Figure 25:  Energy spectrum of the electrons in the 35–100 ${\,\mathrm{GeV}}$
range. The dispersion (shown on the top axis) is inversely proportional to the
particle energy (shown on the bottom axis). The head of the pulse, which is
unaffected by the plasma, is at 43 ${\,\mathrm{GeV}}$. The core of the pulse,
which has lost energy driving the plasma wake, is dispersed partly out of the
field of view of the camera. Particles in the back of the bunch, which have
reached energies up to 85 ${\,\mathrm{GeV}}$, are visible to the right. Figure
26:  (a) The dispersed electron beam profile without plasma interaction, where
the spectrometer is set to image 22.35 ${\,\mathrm{GeV}}$. (b) and (c) The
dispersed beam profile after the electron bunches have interacted with the
plasma, where the spectrometer is set to image 20.35 ${\,\mathrm{GeV}}$ and
22.35 ${\,\mathrm{GeV}}$. (d) The spatially integrated spectrum (in $x$) or
the linear charge density of the bunches shown in (c) (solid blue line) along
with the final spectrum obtained from the simulation (solid green line in
(d)). The core of the accelerated trailing beam is shown for the data (dashed
red line).

In this former experiment, a small fraction of electrons of the beam was
injected and accelerated. As a consequence, the quality of the accelerated
electrons was poor with a long Maxwellian-like tail and therefore with also a
poor energy transfer efficiency. For future high energy physics purposes, a
high efficiency is mandatory to achieve an affordable and compact high-energy
collider. To improve this, a second important experiment has been performed at
SLAC. In this plasma wakefield acceleration experiment, a charge-density wake
with high accelerating fields has been driven by an ultra-relativistic bunch
of charged particles (the drive bunch) through a plasma followed by a second
bunch of relativistic electrons (the trailing bunch) in the wake of the drive
bunch at an appropriate distance that has been efficiently accelerated to
higher energy. Whereas in the previous experiment, the total charge of
accelerated electrons was insufficient to extract a substantial amount of
energy from the wake, here high efficiency acceleration of the trailing bunch
of electrons has been demonstrated. Accelerations of approximately 70–80 pC of
the trailing bunch have been achieved in an accelerating gradient of about 4.4
${\,\mathrm{GV/m}}$. As presented in Fig. 26, these particles have gained
approximately 1.6 ${\,\mathrm{GeV}}$ of energy per particle, with a final
energy spread as low as $1\%$ and an energy-transfer efficiency from the wake
to the bunch that exceeded $30\%$. This acceleration of a distinct bunch of
electrons containing a substantial charge and having a small energy spread
with both a high accelerating gradient and a high energy-transfer efficiency
represents a milestone in the development of plasma wakefield acceleration
into a compact and affordable accelerator technology. $6\%$ of the initial
electron beam energy (36 J) was transferred to the trailing bunch. This value
is comparable to the laser to electron beam energy transfer efficiency from
LPAW. The main advantage here being that the driver is more efficient that the
laser driver. Accelerators have indeed today a wall-plug efficiency more that
10 times larger than lasers.

###  0.3.2 Proton driven plasma wakefield

As it has been shown in all these former experiments, the energy gain was
limited by the energy carried by the driver (about 40 J for an e-beam driver
and about 100 J for a laser driver) and by the propagation length of the
driver in the plasma (few tens of centimetres for the e-driver and few
centimetres for the laser driver). The laser pulse and electron bunch driver
schemes therefore require the use of many acceleration stages in the tens of
${\,\mathrm{GeV}}$ each in order to gain ${\,\mathrm{TeV}}$ energy levels. A
10 ${\,\mathrm{GeV}}$ stage that delivers an nC of charge corresponds to an
energy of 10 J, and it will correspond to 10 kJ for a 10 ${\,\mathrm{TeV}}$
stage. If one assumes $10\%$ energy transfer efficiency from the driver to the
trail bunch, this indicates that the driver energy must contain about 100 kJ.
In 2009, for the first time, plasma-wake excitation by a relativistic proton
beam has been considered [6]. In this ideal case, the proton driver beam has
to be resonant with the plasma and it was predicted on the basis of numerical
simulations that 10 ${\,\mathrm{GeV}}$ electrons injected could be accelerated
to 0.5 ${\,\mathrm{TeV}}$ in a 450 m proton wakefield. Unfortunately such a
short proton bunch does not exist, and therefore it has been proposed to use
the CERN SPS 19 kJ, 400 ${\,\mathrm{GeV}}$ proton beam that is produced
routinely.

Figure 27:  Design of the layout of the AWAKE experiment

Because the length of the driver (about 10 cm) is much longer than the plasma
wavelength (about 1 mm) at a density large enough (of $10^{14}$ cm-3) to reach
a ${\,\mathrm{GV/m}}$ accelerating field, the interaction has to occur in the
self-modulated regime. In this regime, also called the self-modulation
instability (SMI) [71], the proton bunch is split during its propagation in
several micro-bunches that excite resonantly a strong plasma wave. A first
experiment, called AWAKE [72], will be performed at CERN in the next two years
to demonstrate the possibility of accelerating injected electrons in proton
driven plasma wakefield. For this, an extremely uniform long plasma has been
developed with a control precision for the density that must be within $0.5\%$
over metres. To optimize the coupling efficiency, electrons will be injected
at an angle after the self-modulated instability has reached saturation. The
conceptual design of the proposed AWAKE experiment is shown in Fig. 27 where
the laser and proton bunches are made co-linear. The laser that will ionize
the metal vapour is required to seed the self-modulation instability. The
self-modulated proton bunch (shown on the left hand side) enters a second
plasma section where it drives the plasma wakefield structure (shown on the
right side). The electrons are injected in the wakefields and the 2
${\,\mathrm{GeV}}$ accelerated electrons will be measured with an electron
spectrometer. The AWAKE experiment will be installed in the CERN Neutrinos to
Gran Sasso (CNGS) facility. Approximately $5\%$ of electrons are supposed to
be trapped and accelerated to the end of the 10 m plasma with accelerating
gradients in the few ${\,\mathrm{GV/m}}$. In addition to the electron
spectrometer, several other diagnostics will be used to characterize the
proton beams to better understand the physics of self-modulation. Coherent
Transition Radiation (in the visible and in the infra-red) produced when the
proton beam passes through a thin foil, will be measured using a streak
camera. Additionally, transverse coherent transition radiation will be
produced and detected using electro-optical sensors; this will be the first
experimental use of this recent concept [73]. First protons to the experiment
are expected at the end of 2016 and this will be followed by an initial 3–4
year experimental program of four periods of two weeks of data taking.

##  0.4 Future of the laser plasma accelerators

The tremendous progress that has been made in plasma acceleration [74, 75,
76], from the first acceleration of externally injected electrons in a
${\,\mathrm{GV/m}}$ laser wakefield, self-injection in a
$100{\,\mathrm{GV/m}}$ laser wakefield with first a 100 ${\,\mathrm{MeV}}$
broad spectra to the series of experiments with the production of a quasi-
monoenergetic electron beam in a laser wakefield with a compact 10 Hz laser
system have contributed to boosting this field of research in which tens of
laboratories/teams are playing important roles in a competitive and friendly
approach. The evolution of short-pulse laser technology with diode pump lasers
or fibre lasers, a field in rapid progress, will eventually contribute to the
improvement of laser plasma acceleration and their societal applications, in
material science for example for high resolution gamma radiography [77, 78],
in medicine for cancer treatment[79, 80], in chemistry [81, 82] and in
radiobiology [83, 84, 85]. In the near future, the development of compact free
electron lasers could open the way to the production of intense X-ray beams,
in a compact way, by coupling the electron beam with undulators. Thanks to the
very high peak current of a few kiloamperes [86], comparable to the current
used at LCLS, the use of laser plasma accelerators for free electron lasers,
the so-called fifth generation light source, is clearly identified by the
scientific community as a major development. Alternative schemes to produce
ultra-short X-ray beams, such as Compton, betatron or Bremsstrahlung X-ray
sources, have also been considered. Tremendous progress has been made
regarding the study of betatron radiation in a laser plasma accelerator. Since
its first observation in 2004 [87] and the first monitoring of electron
betatronic motion in 2008[88], a number of articles have reported in more
detail this new source, including measures of a sub ps duration[89] and of a
transverse size in the micrometre range [90]. Betatron radiation was used
recently to perform with high spatial resolution, of about 10 microns, X-ray
contrast phase images in a single shot mode operation[91, 92]. In parallel,
similar huge progress has been performed in accelerating electrons and
positrons using electron or positron bunches, with here, a gain of a few tens
of ${\,\mathrm{GeV}}$ in a few tens of centimetres accelerating gradient.
Wakefields driven by electron beams are good candidates to boost electron
energy in a metre long plasma device. The requirement for the driver being
very close to the one for FEL purposes, shorter radiation wavelength could be
produced by doubling, for example, the electron energy delivered from SLAC or
from DESY. Acceleration of electrons and positrons with these drivers are also
very relevant for a staging approach for high energy physics purposes. The
AWAKE experiment will certainly contribute to defining the roadmap for future
larger-scale R&D projects on laser, electron or proton driven plasma wakefield
acceleration for future high energy colliders for particle physics. The
success of plasma wakefield accelerators will open a pathway towards many
exciting societal application, a compact FEL radiation source and a
revolutionary plasma-based ${\,\mathrm{TeV}}$ lepton collider. This revolution
could then enable ground-breaking discoveries in many domains, including
particle physics.

## Acknowledgements

I acknowledge the support of the European Research Council for funding the
PARIS and X-Five ERC projects (contract number 226424 and 339128), from CARE,
EUCARD and UCARD2 and from LASERLAB1,2 and 3. I warmly acknowledge my many
excellent collaborators from LOA and all over the world, my new friends R.
Bailey and B. Holzer from CERN for initiating and organizing with success this
first CAS-CERN Accelerator School on Plasma Wake Acceleration.

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