Journal of Modern Physics, 2012, 3, 1983-1990
http://dx.doi.org/10.4236/jmp.2012.312248 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Influence of a Magnetic Guide Field on Self-Injection in
Wakefield Acceleration
Alain Bourdier, Guillaume Girard, Sébastien Rassou, Xavier Davoine, Mathieu Drouin
CEA, DAM, DIF, Arpajon, France
Received September 28, 2012; revised October 28, 2012; accepted November 7, 2012
ABSTRACT
The influence of an external static field applied in the direction of propagation of a high intensity driving laser pulse on
the electron trapping in laser wakefield acceleration is explored. It is shown that, in the case of self-injection, the elec-
tric charge accelerated can be enhanced in some physical situations.
Keywords: Laser Wakefield Acceleration; Magnetic Guide Field; Electron Injection; Self-Trapping
1. Introduction
Particle accelerators have important applications in many
fields, from medicine to high energy physics [1-6]. La-
ser-plasma accelerator have a decisive advantage over
conventional accelerators: plasma supports electric fields
orders of magnitude higher than the breakdown-limited
field in radio-frequency cavities of conventional linacs.
The acceleration gradients in conventional accelerators
are limited to a few tens of MV/m, while they can reach
and even exceed 100 GV/m in plasma. Ultra short light
pulses that can be generated with a plasma accelerator
are powerful tools for time resolved studies of molecular
and atomic dynamics [7].
Among the various laser-plasma accelerator concepts,
the laser-wakefield accelerator (LWFA) [1,8] is the most
promising [2]. One of the key applications of laser-
plasma interaction is particle acceleration based on the
excitation of a strong plasma wakefield by a laser. As
low density plasma is always considered, the phase ve-
locity of the plasma wake is close to the speed of light, so
the charged particles which are loaded in the wake can be
accelerated to very high energy.
One of the most critical issues in the LWFA process
may be how to control and enhance the beam energy and
the charge. It is shown in this article that a static longitu-
dinal magnetic field can enhance particle trapping.
When a laser pulse propagates in plasma, the pon-
deromotive force which is proportional to the intensity
gradient of the pulse pushes electrons forwards and side-
ways. Because the ions are much heavier than the elec-
trons and so do not respond to the ponderomotive force,
the electrons are dragged back towards their original po-
sition by the space charge field and start to oscillate, cre-
ating a plasma wave and the so-called wakefield. When
the intensity of the pulse is high enough the “bubble”
regime is reached [9,10]. Hence, a cavity structure gen-
erally called bubble is driven in low density plasma by
the laser pulse through the ponderomotive force. The
laser pulse can excite a plasma wave in different ways.
The excitation is most effective when the laser pulse is
shorter than the plasma wavelength 2π
p
p
c

  where
p

is the plasma frequency. Usually, electrons oscillat-
ing in the plasma wave under the wave-breaking limit
cannot be accelerated by the wakefield since they are out
of phase with it. If the laser intensity reaches a threshold
value, some electrons can oscillate so fast in the plasma
wave that they can reach the wave velocity. In this case,
these electrons overtake the wave and wave breaking
occurs. Then, they stop to oscillate in the wave and are
injected in the wakefield and start to propagate with it.
As the electron reaches the highest velocity at the back of
the bubble, they are injected at this position in the bubble.
Wave breaking turns out to be important as it leads to
abundant self-trapping of electrons in the wakefield.
However, the electron beams created this way do not
usually have the stability and reproducibility that are
required for applications [11]. This is because the mech-
anism responsible for injecting electrons into the wake-
field is based on highly nonlinear phenomena, and is
therefore hard to control. It is well known that trapping
of the background electrons begins much below the lon-
gitudinal wave-breaking limit [12-14]. The transverse
wave-breaking regime which has to be studied with a two
dimensional approach is the situation where a static
magnetic field should play an important role. It was al-
ready shown by Hur, Gupta, and Suk [14] and by Vieira
C
opyright © 2012 SciRes. JMP

A. BOURDIER ET AL. 1984
et al. [15] in the case of a transverse constant magnetic
field that a static magnetic field can play an important
part in order to control the electron injection in the ac-
celerating cavity, it should play a role similar to the one
of a perturbing counter propagating wave. As a conse-
quence, we started a deeper exploration of the positive
effect of a static magnetic field which has been chosen,
in this work, to be applied in parallel with the propaga-
tion direction of the driving laser pulse. Following a pa-
per previously published by Hur, Gupta, and Suk [14],
the effect of such a field on the trapped beam is studied
again in this article. The idea was to verify if the mag-
netic field can be considered as an important controlling
knob for the trapped charge in the case of self-injection.
2. Electron Injection Enhanced by a Static
Magnetic Field—PIC Code Simulation
Results
The influence of a constant homogeneous magnetic guide
field on LWFA is studied here with a code. Numerical
simulations were conducted using the two-dimensional
PIC code CALDER [16]. Their results were compared to
those obtained by Hur, Gupta, and Suk with code XOOPIC
[17]. In their paper Hur, Gupta, and Suk claim that a
static magnetic field applied in parallel with the direction
of propagation of the driving laser pulse enhances the
particle trapping in the first bubble. They show from
two-dimensional PIC code simulations that the total
charge of the trapped beam and its maximum energy in-
crease with the magnitude of the guide field.
In order to test our code, the simulation domain con-
sidered in this work was chosen to be as close as possible
to the one considered by Hur, Gupta, and Suk. The wave-
length of the laser was assumed to be λ = 1 μm in all the
simulations. A moving window was employed to reduce
the computational time. It defines the simulation domain
which was divided in nearly the same number of cells as
in the simulations performed by Hur, Gupta, and Suk.
The simulation box considered here, which drifts with
the moving window, is 80 μm in the laser propagation
direction (z) and 120 μm in the transverse direction (x). A
trapezoidal electronic density profile in the longitudinal
direction is assumed, with one homogeneous slab sur-
rounded by two density gradients, Lg is the density gra-
dient length.
In order to compare our results to their results, the same
parameters were considered: the plasma density was as-
sumed to be: , and the normalized
vector potential of the laser pulse:
18 3
410 cm 3.n
23.5Amcae .
The pulse duration was Δt = 38 fs and the spot size of the
laser was assumed to be 12 μm large. When using
CALDER, the 2D1/2 version was used.
As some parameters of the simulations performed by
Hur, Gupta, and Suk are not specified in their paper, dif-
ferent situations were considered. No significant charge
trapped in the first bubble was found considering the
physical parameters chosen by them. Electrons are found
to be mainly trapped in the second and third bubble
(Figure 1). The simulation results shown just below cor-
respond to a time which is normalized to the laser fre-
quency 0
ˆ
tt . Normalized space variables: 




0
ˆ,ici ixz
 
and a normalized momentum: ˆi
i
ppmc are also in-
troduced.
This result is confirmed when calculating the electron
energy distribution in the three first bubbles (Figure 2).
In this article, the number of particles, N, expressed in
106 particles per MeV is calculated for a 8 μm thick
plasma in the direction (y). This assumption had to be
done as 2D simulations have been performed.
The influence of the plasma density gradient Lg was
Figure 1. Electron density as a function of the longitudinal
position from 2D PIC simulations. a = 3.5, 3
18
3.4 10cmn

 .
Figure 2. Electron energy distribution in the three first bub-
bles. a = 3.5, 3, Lg = 200 μm. (a) First bub-
ble; (b) Second bubble; (c) Third bubble.
18
3.4 10cmn 

A. BOURDIER ET AL. 1985
explored first. Simulations show that the plasma density
gradient has a strong influence on the electron energy
distribution in the second bubble. Still no one of the dif-
ferent values of Lg considered allows particles to be
trapped in the first bubble.
The influence of the magnetic field on the distribution
in the second bubble was also explored. It has been shown
that the initial constant homogeneous guide magnetic field,
when inferior to 250 T, has almost no influence on the
electron energy distribution in the second bubble.
The time step used in our simulations is just below the
limit given by the Courant condition
 
22
1y

 

11tx .
This upper limit is in our case:


1
00.16 fs
0.310
L
t   ,
and the time step used in our simulations is


10.13 fs 
0
0.25t  
while the size of meshes is given by:

2
5 10μm


0
0.313zc  
and


00.4 μm
 2.5xc .
In order to check if the results published by Hur, Gupta,
and Suk, who observed electrons trapped in the first bub-
ble, were due to a bad optimization of their time step,
simulations with smaller time steps were performed as a
slower group velocity might have led to some trapping in
the first bubble. No electron trapped in the first bubble
was observed even when using a time step ten times
smaller than the one usually used in the PIC simulations,
that is to say the one chosen just below the Courant con-
dition.
Figure 3. Electron longitudinal momentum distribution as a
function of the longitudinal position. a = 7, 3
n
Figure 4. Electron energy distribution in the first bubble. a =
7, 3
18
3.4 10cmn


319
0 cm



.
For this value of the intensity of the laser pulse (a = 3.5)
higher electron densities up to 11 were con-
sidered in order to diminish the group velocity of the wave.
No electron trapping in the first bubble was observed.
Then, the laser intensity was increased in order to try to
have some self injection in the first bubble. Figures 3 and
4 show that some electrons are trapped in the first bubble
when a = 7.
Figure 4 shows that a very strong magnetic field along
the direction of propagation of the wave allows the
wakefield to trap more electrons in the first bubble.
When B0 = 0 the charge trapped in the bubble is close to
δq = 0.56 picocoulombs, when B0 = 50 T the charge is δq
= 0.86 picocoulombs and when B0 = 120 T one has δq =
2.4 picocoulombs. Between B0 = 0 and B0 = 120 T the
trapped charge is multiplied by 4.3. In this case, the
magnetic field has no visible effect on the dimension of
the bubble. Although these charges are very weak they
show a trend: the charge increases with the magnitude of
the static magnetic field. The non-relativistic plasma
frequency in the presence of a magnetic field can be ap-
proximated by 12
22
 
mp where ωm and ωp rep-
resent the plasma frequencies of the magnetized plasma
and the unmagnetized plasma, respectively. The cyclotron
frequency is defined by
18
3.4 10cm

 ,
B0 = 0.
0.eB m120 TB When 0


,
one has

22 2
122 10 


p
which is too small to
give any significant modification in the plasma frequency,
as a consequence, the plasma itself behaves as if there
were no magnetic field.
Figure 5 displays the magnetic field density along the
direction of propagation of the wave for an initial constant
magnetic field B0 = 120 T. It shows that some magnetic
field along the z-axis is built up close to the back of the
cavity.
No magnetic field is created when the initial static
magnetic field is zero.
Figure 6 shows the distributions in momentum along

A. BOURDIER ET AL. 1986
Figure 5. Map of the longitudinal magnetic field. a = 7,
3, B0 = 120 T. 18
3.4 10cm n
(a)
(b)
Figure 6. Electron momentum along the y axis distribution
in the first bubble. a = 7, 3. (a) B0 = 0; (b) B0
= 120 T.
18
3.4 10cm n
the y axis versus z when B0 = 0 and B0 = 120 T.
Due to the static magnetic field, electrons which drift
on the rim of the cavity turn around the back of the bubble.
A transverse current is built up (Figure 6(b)). Conse-
quently, a strong static magnetic field is created along the
z axis. Thus, a relatively small amplitude initial magnetic
field is enough to build up a strong one. When a = 7 and a
= 7.5, Figures 4 and 7 show that the total charge of the
trapped electrons in the first bubble increases with the
initial magnitude of the magnetic field. This result is quite
similar to the one obtained by Hur, Gupta, and Suk [14].
In this case (a = 7.5), when B0 = 0 the trapped charge in
the high energy peak is δq = 2.85 picocoulombs, when B0
= 120 T, one has δqB = 7.7 picocoulombs, the significant
parameter is 2.7rqq



B. When this strong static
magnetic field is applied the accelerated charge is multi-
plied by almost three.
Then, a higher wave intensity was considered: a = 8.
Figure 8(a) shows that a high energy peak close to 250
MeV is created. Figures 8(b) and (c) show that the static
magnetic field has no significant influence on the bubble
size. The correction to the plasma frequency due to the
initial magnetic field is still

22 2
122 10 

p

ˆ4860t
. At
, when the initial static magnetic field has a
magnitude of 120 T, the longitudinal component of the
magnetic field reaches about 350 T (Figure 9), then


22
1 20.35  
p. This ratio is made smaller when
taking into account the relativistic mass increase of the
electrons. Thus, no significant deformation of the cavity
due to the magnetic field is expected and no deformation
is observed.
In this case, many more electrons turn around the mag-
netized cavity, then a more intense transverse current is
produced.
Figure 9 shows the evolution of the very strong z
component of the magnetic field built up by the electrons
turning around the neck of the bubble versus time for two
initial values of B0.
Figure 7. Electron energy distribution in the first bubble in a
neighborhood of the high energy peak. for different values of
the initial static magnetic field a = 7.5, 3
18
3.4 10cmn

 .

A. BOURDIER ET AL. 1987
(a)
(b)
(c)
Figure 8. (a) Electron energy distribution in the first bubble;
(b) Electron density when B0 = 0; (c) Electron density when
B0 = 120 T. a = 8, 3.
18
3.4 10cm  n
A transverse magnetic field is also created. Figure 10
shows the Bx component of the magnetic field close to the
rear of the bubble.
Figure 11 shows the electron energy distribution for
the trapped bunch close to the high energy peak displayed
in Figure 8(a) for different values of B0.
−− B
0
= 120 T
−− B
0
= 250 T
18 3
10 cm
Figure 9. Longitudinal component of the magnetic field close
to the rear of the bubble versus time. a = 8, n = 3.4 ×

.
Figure 10. Evolution of the x component of the magnetic
field close to the rear of the bubble a = 8, 3
18
3.4 10cmn

 .
Figure 11. Electron energy distribution in the first bubble,
for different values of the initial static magnetic field in a
neighborhood of the high energy peak. a = 8, n = 3.4 ×
18 3
10 cm

.

A. BOURDIER ET AL. 1988
When B0 =0, one has δq = 0.5 picocoulombs and when
B0 = 120 T one has δq = 4.6 picocoulombs, the trapped
charge is multiplied by 9.2rqq
B

 .
The evolution of the electron energy distribution is
similar to the one shown in Figure 3 in the article pub-
lished by Hur, Gupta, and Suk. Many trajectories are bent
by the magnetic field, keeping particles closer to the rear
of the bubble. Then, the electron trapping is more likely to
occur [18].
In conclusion, we have found again, for a = 7 and a = 8,
results similar to those previously published by Hur,
Gupta, and Suk for their electron density but for different
values of the laser intensity.
More high energy particles are trapped in the high en-
ergy peak in the first bubble due to a strong magnetic field
and to values of the pulse normalized vector potential
close to a = 9 (Figure 12).
When the intensity reaches a = 10, many electrons are
trapped in the first bubble even when there is no magnetic
field (Figure 13).
The electron energy distribution is not very much af-
fected by the magnetic field (Figure 14).
The number of trapped particles is close enough to the
beam loading limit [19,20], as a consequence the accel-
erated beam has a large energy spread with or without a
strong magnetic field [21].
In this case, a strong longitudinal magnetic field is still
built up.
For this very high intensity, electron trapping is no
longer enhanced by a static longitudinal magnetic field
and the static magnetic stops having a positive effect on
the quality of the beam trapped in the first bubble.
3. Conclusions
The goal of this work was mainly to verify the very in-
Figure 12. Electron energy distribution in the first bubble,
for different values of the initial static magnetic field in a
neighborhood of the high energy peak. a = 9, n = 3.4 ×
.
(a)
(b)
Figure 13. Electron density. a = 10, 3 n
18
3.4 10cm

 . (a)
When B0 = 0; (b) When B0 = 120 T.
Figure 14. Electron energy distribution in the first bubble. a
= 10, 3
18
3.4 10cmn

 .
teresting results previously published by Hur, Gupta, and
Suk [14] concerning the influence of a constant magnetic
field in the enhancement of the charge trapped by the
18 3
10 cm

A. BOURDIER ET AL. 1989
first bubble in the case of auto-injection. Although their
results are not confirmed for the physical parameters they
have considered, similar results are found in slightly dif-
ferent situations. Unfortunately, the enhanced trapping
by the static magnetic field is only observed in a small
range of the various values of the physical parameters. It
seems that, close to the saturation level, when self-injec-
tion leads to the injection of a charge close to the maxi-
mal charge allowed by the beam-loading limit, adding a
static magnetic field does not enhance the particle trap-
ping.
It is shown that a current is created in a plane perpen-
dicular to the direction of propagation of the wave which
intensifies the initial static field which is applied.
Finally, it is confirmed in this paper that a constant
magnetic field is a very important controlling knob for
improving the charge of the beam in the LWFA process
in the case of self-injection.
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