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Laser Physics
Related content
PAPER
Combined influence of azimuthal and axial
magnetic fields on resonant electron acceleration
in plasma
To cite this article: Arvinder Singh et al 2017 Laser Phys. 27 110001
- Multi-GeV electron acceleration by a
periodic frequency chirped radially
polarized laser pulse in vacuum
Harjit Singh Ghotra and Niti Kant
- Electron energy enhancement by
frequency chirp of a radially polarized laser
pulse during ionization of low-density
gases
Kunwar Pal Singh, Rashmi Arya, Anil K
Malik et al.
- Topical Review
R Bingham, J T Mendonça and P K
Shukla
View the article online for updates and enhancements.
This content was downloaded from IP address 129.8.242.67 on 27/10/2017 at 09:47
Laser Physics
Astro Ltd
Laser Phys. 27 (2017) 110001 (6pp)
https://doi.org/10.1088/1555-6611/aa8759
Combined influence of azimuthal and
axial magnetic fields on resonant electron
acceleration in plasma
Arvinder Singh1, Jyoti Rajput1,2 and Niti Kant2
1
Department of Physics, National Institute of Technology Jalandhar, Jalandhar, Punjab, India
Department of Physics, Lovely Professional University, G.T. Road, Phagwara 144411, Punjab, India
2
E-mail: nitikant@yahoo.com
Received 8 April 2017
Accepted for publication 20 August 2017
Published 26 October 2017
Abstract
Resonant enhancement in electron acceleration due to a circularly polarized laser pulse in
plasma, under the combined influence of external azimuthal and axial magnetic fields, is
studied. We have investigated direct electron acceleration in plasma by employing a relativistic
single particle simulation. The plasma is magnetized with an azimuthal magnetic field applied
in the perpendicular plane and an axial magnetic field applied along the direction of laser
beam propagation. Resonance takes place between electron and electric field of the laser pulse
for the optimum value of the combined magnetic field, which supports electron acceleration
to higher energies, up to the betatron resonance point. The optimum value of these magnetic
fields is highly sensitive to laser initial intensity and laser initial spot size. The effects of laser
intensity, initial spot size, and laser pulse duration are taken into consideration in optimizing
the magnetic field for efficient electron acceleration. Higher electron energy gain, of the order
of GeV, is observed by employing terawatt circularly polarized laser pulses in plasma under
the influence of combined magnetic field of about 10 MG.
Keywords: electron acceleration, circularly polarized laser, magnetic field, plasma
(Some figures may appear in colour only in the online journal)
1. Introduction
ultrashort femtosecond lasers can generate magnetic fields of
the orders of several hundred MG—nearly 350 MG [11]. The
azimuthal magnetic field is generated by both linearly polarized and circularly polarized lasers, whereas an axial magn­
etic field is produced only in circularly polarized lasers [12].
In case of circularly polarized lasers, electrons spiral about
the propagation direction, and hence produce axial magnetic
field via the azimuthal current [13]. The self-matching resonance acceleration (SMRA) regime has been proposed with
the aid of analytical modelling and 3D-PIC simulations. With
the combination of the self-generated magnetic fields (in both
axial and azimuthal directions) and the laser pulse, relativistic electrons undergo two processes: trapping and resonance
[14]. Researchers [15–17] have shown that suitable external magnetic fields also have impressive effects on electron
acceleration, due to resonance. Direct electron acceleration in
vacuum has been analysed in the presence of axial magnetic
Recent advancement in short pulse laser technology has discovered new scopes of laser-plasma interaction, which have
proved successful in explaining many exciting facts about
plasma—e.g. laser confinement fusion, laser induced particle acceleration [1, 2, 5] etc. The first idea for accelerating particles in plasma with the help of various fields was
given more than six decades ago [3]. However, Tajima and
Dawson [4] proposed an idea that electrons can be accelerated to very high energies by short pulse lasers. Accordingly,
an additional acceleration can be obtained from the interaction of these electron beams with external or self-generated
magnetic fields. Such an electron gains additional energy due
to spontaneous magnetic fields [6–9]. These resultant generated magnetic fields can be of the order of multiple gigagauss [10]. Recently, it has been experimentally observed that
1555-6611/17/110001+6$33.00
1
© 2017 Astro Ltd Printed in the UK
A Singh et al
Laser Phys. 27 (2017) 110001
where z0 is the initial position of the pulse peak, wave vec1/2
tor k = (ω/c) × (1 − ωp2 /ω 2 ) , ω is the angular frequency
of laser pulse, r2 is given by (x2 + y2 ), r0 is the initial spot
size of the laser beam, τ is laser pulse duration, group veloc1/2
ity vg = c(1 − ωp2 /ω 2 ) , c is the velocity of light in vacuum, ωp2 = 4πn0 e2 /m is relativistic plasma frequency, e and
m = m0 γ are the charge and relativistic mass of the electron
respectively (here m0 is the rest mass of the electron), and
1/2
γ = 1 + P2x + P2y + P2z
is the Lorentz factor, where Px , Py
and Pz are the x, y and z components of electron momentum.
The electromagnetic fields related to the vector potential of
the laser pulse are
field using symmetry of radially polarized laser beams, which
improves trapping and acceleration of electrons to the level
of GeV [18]. Recently, the effect of axial magnetic fields on
electron acceleration due to axicon focused Gaussian radially
polarized laser has been investigated, and it was observed that
electron energy can be enhanced effectively without the use
of petawatt lasers [19]. The combined influence of frequency
chirp and azimuthal magnetic field has been studied, employing linearly and circularly polarized lasers, and it is reported
that the combined effect of chirping and azimuthal magnetic
field not only enhances the electron energy gain, but retains
this electron energy gain for longer distances [20, 23]. Results
have shown that small energy spread of the order δε/ε 10−2
is obtained by laser induced electron acceleration in vacuum
in magnetic field [21].
The purpose of the present study is to analyse the effect
of external magnetic fields of the order of a few megagauss
on direct electron acceleration in plasma, by optimizing the
circularly polarized laser parameters to achieve resonance,
so as to get higher electron energy gain and minimum values
of radiation losses. Betatron oscillations are more prominent
in magnetized plasma as compared to unmagnetized plasma.
Energy transfer to an electron due to circularly polarized laser
is much more efficient, as compared to linearly polarized laser
in vacuum [22]. In circularly polarised laser, there is production of axial and azimuthal magnetic field; so, in this paper,
we place emphasis on the electron acceleration process due to
interaction between circularly polarized laser field and external axial and azimuthal magnetic fields in plasma. We study
the dependence of the maximum electron energy on the laser
intensity, spot size and external combined magnetic field. The
results of a relativistic, single particle simulation for direct
acceleration of electron are presented. It is observed that there
exists an optimum value of these magnetic fields to establish
resonance with the laser fields. Above this value, deceleration
of the electron takes place. We have performed computer simulation for a single particle code to get the optimum values of
magnetic fields. The value of magnetic fields employed in the
paper is of the order of a few MG, which is experimentally
available [26], hence favouring our model. The structure of the
paper is as follows: The equations governing the motion of the
electron for circularly polarized laser beam in plasma in the
presence of azimuthal and axial magnetic fields are form­ulated
in section 2. Numerical results for maximum electron energy
gain γm with the optimum value of external magnetic fields
parameter b0 , laser initial intensity parameter a0 , laser initial
spot size parameter r0 and laser pulse duration param­eter τ are
discussed in section 3, and a conclusion is drawn in section 4.
∂A
EL = −
and BL = ∇ × A.
(2)
∂t
The external azimuthal magnetic field [23, 24] can be written
as follows:
B0
(yx̂ − xŷ).
Bθ =
(3a)
r0
Further, the external axial magnetic field is given by
Bz = ẑB0 .
(3b)
Here B0 is the magnitude of both azimuthal and axial magn­
etic field. So, the combined external magnetic field comp­
onent, which includes the azimuthal and axial magnetic fields,
is given by the equation,
B0
(yx̂ − xŷ) + ẑB0 .
BT =
(3)
r0
When a charged particle moves through the electromagnetic
fields, it experiences a force, which is known as the Lorentz
force. The relativistic Lorentz force equations are:
dPx
∂Ax
∂Ax
B0
− e xvz − eB0 vy ,
=e
+ evz
(4)
dt
∂t
∂z
r0
∂Ay
∂Ay
dPy
B0
− e yvz − eB0 vx ,
=e
+ evz
(5)
dt
∂t
∂z
r0
∂Ay
dPz
∂Ax
B0
B0
− evy
= −evx
+ e xvx + e yvy .
(6)
dt
∂z
∂z
r0
r0
The equation for rate of change of the electron energy is given
by
∂Ay
d(γm0 c2 )
∂Ax
(7)
.
= evx
+ evy
dt
∂t
∂t
If the electron is at origin in the beginning with Px = Px0 and
Py = Py0 , then the integration of the equations (4) and (5)
yields
2. Relativistic analysis
Consider the propagation of a circularly polarized laser pulse
in plasma with a vector potential [27]
A=
B0
xz − eB0 y,
r0
(8)
Px = 2eAx + Px0 − 2eA0 exp(−z20 /v2g τ 2 ) − e
2
[t − (z − z0 )/vg ]
r2
− 2 ,
A0 [x̂ cos(ωt − kz) + ŷ sin(ωt − kz) exp −
τ2
r0
B0
Py = 2eAy + Py0 − e yz + eB0 x.
(9)
r0
(1)
From equations (6) and (7)
2
A Singh et al
Laser Phys. 27 (2017) 110001
3. Results and discussions
2
2
0
Pz − γm0 c − eB
2r0 (x + y ) = 0,
(10)
2
2
0
Pz − γm0 c − eB
2r0 (x + y ) = C3 .
d
dt
The combined laser fields and the external magnetic fields
are employed to generate betatron oscillations in the elecIf the electron has momentum Pz = Pz0 in the beginning,
trons. The intense laser field’s ponderomotive force facilitates
1/2
C3 = Pz0 − (P2x + P2y + P2z + m20 c2 ) . Therefore, from equa- electron motion in the forward direction in the accelerating
tion (10), we may write
platform, and the applied magnetic fields play an interesting
role in enhancing its energy. So, here, we consider the comeB0 2
1/2
(x + y2 ).
Pz = Pz0 + γm0 c − (P2x0 + P2y0 + P2z0 + m20 c2 ) +
bined effect of these external magnetic fields on the electron
2r0
(11) energy. We observe the resonant enhancement in electron
energy due to these magnetic fields. Throughout this paper,
Using the dimensionless variables:
we have chosen the normalized initial position of pulse peak
a0 → eA0 /m0 c, ax → eAx /m0 c, ay → eAy /m0 c, b0 → eB0 /m0 ω,
z0 = −100, normalized initial electron momentum Pz0 = 1
t → ωt, τ → ωτ , r0 → ωr0 /c, x → ωx/c, y → ωy/c,
and nor­malized laser propagation constant k = 0.99.
z → ωz/cz0 → ωz0 /c, Px0 → Px0 /m0 c, Py0 → Py0 /m0 c,
Figure 1(a) shows the variation of electron maximum
energy γm as a function of normalized magnetic field b0 . In
Pz0 → Pz0 /m0 c, k → kc/ω, vx → vx /c, vy → vy /c, vz → vz /c.
this figure, electron energy has been analyzed at different valIn terms of the above dimensionless variables, equations ues of normalized laser intensity a0 . Figure 1(b) shows the
maximum electron energy gain calibration curve. The fig(7)–(11) can be written as follows:
ure represents the variation of a0 and optimized magnetic field
b0 dx
1 z2
0
parameter b0 as a function of the electron’s maximum energy
=
P + 2ax − 2a0 exp − 2 2 − x z − b0 y
dt
γ x0
r0
k τ gain γm . For each value of laser intensity there is a corre­
(12) sponding optimum value of magnetic field at which resonance
is established which leads to maximum energy gain. It can be
dy
1 b0 y
z
+
b
x
=
P
+
2a
−
(13)
y
0
seen from figure 1(a) that for laser intensity parameter a0 = 10
y0
dt
γ
r0
(corresponding to a value of I0 ~ 1.38 × 1020 W cm−2), ini
tial spot size r0 = 80 (corresponding value is 12.7 µm) and
dz
1 b0 2
2
2
2 1/2
2
(x
=
P
+
[γ
−
(1
+
P
+
P
+
P
)
]
+
+
y
)
magnetic field parameter b0 = 0.09 (corresponding value of
x0
y0
z0
dt
γ z0
2r0
B0 ~ 9.6 MG), which is feasible, we can achieve a maximum
(14)
electron
energy gain of 0.68 GeV. Similarly, for laser intensity
dγ
dx ∂ax
dy ∂ay
parameter a0 = 20 (corresponding value of I0 ~ 5.52 × 1020
= + (15)
W cm−2) and the optimized magnetic field parameter b0 = 0.2
dt
dt
∂t
dt
∂t
(corresponding value of B0 ~ 21 MG), the maximum energy
2
2
where, ax = a0 cos(t − k z ) exp[−(t − (z − z0 )/k ) /τ −
gain
is raised to 1.35 GeV. The practically achieved values
2
2
(r /r02 )] and ay = a0 sin(t − k z ) exp[−(t − (z − z0 )/k ) /
of
these
magnetic fields are up to 100 MG [7]. Figure 1(b)
2
2
τ − (r /r02 )]. Equations (12)–(15) constitute a set of coupled shows that if we increase laser intensity parameter a0 , then the
ordinary differential equations. The initial position of the elec- value of external magnetic field parameter b0 also increases,
tron is assumed at origin with initial momenta (Px0 ,Py0 ,Pz0 ). to achieve higher values of maximum electron energy gain
To examine the validity of our model, we have solved the coupled γm . Thus, for the higher values of laser intensity parameter
differential equations (12)–(15) numer­ically by using computer a0 , optimum values of magnetic field parameter b0 to achieve
simulation to find the conditions for maximum electron energy resonance condition, in order to get maximum energy gain,
gain in the presence of external magnetic fields in plasma.
appear to be higher. At resonance, the electron cyclotron freThe electron may lose a small amount of energy during quency and the Doppler-shifted frequency become comparadeceleration by means of radiations, which can be calculated ble, so that maximum energy is transferred to the electron.
from Lenard’s result in terms of radiative power:
Even after passing the laser pulse, electron energy gain is
retained due to the effect of these magnetic fields [25]. Thus,
2
2 6
2 2
2e γ
dv
dv
(16)
.
P(t) =
− v ×
these magnetic fields help in retaining the resonance condition
3c
dt
dt
for longer, and deflect the electron to keep it in accelerating
In case of circular accelerators, such as synchrotrons and phase, so that effective increase in electron energy takes place.
Figure 2(a) shows the variation of electron energy γ as a
betatrons, momentum of the charged particle changes
drastically in the direction of rotation of particle, but function of normalized time t for different values of laser iniin energy per revolution is very small as tial spot size r0 ; figure 2(b) shows the variation of nor­malized
the change
p
initial laser initial intensity a0 and laser initial spot size r0 as
d /dτ (= γ ω |p|) (1/c)dE/dτ . Here ω = cβ/ρ , nor­
a function of electron maximum energy gain γm . The magnimalized velocity β = v/c and ρ is the orbit radius. For high tude of axial and azimuthal magnetic fields is taken here as
energies (β ≈ 1), the radiative loss/revolution can be esti- 9.6 MG. The other parameters are the same as taken in fig4
mated as δ E(MeV) = 8.85 × 10−2 [E(GeV)] /ρ(m). In a 10 ure 1. It is shown that the maximum electron energy increases
GeV electron synchrotron, the loss/revolution is 8.85 MeV.
with the laser intensity parameter a0 , because the longitudinal
3
A Singh et al
Laser Phys. 27 (2017) 110001
(a)
(a)
(b)
Figure 2. (a) Electron energy gain γ as a function of normalized
time t for different values of normalized laser initial spot size r0
at b0 = 0.09. (b) Maximum energy gain γm calibration curve for
normalized intensity a0 and normalized laser initial spot size r0 at
b0 = 0.09. Other parameters are the same as in figure 1.
(b)
Figure 1. (a) Maximum energy gain γm as a function of normalized
magnetic field parameter b0 for various values of normalized
intensity a0 (b) Maximum energy gain γm calibration curve for a0
and b0 . Other parameters are Pz0 = 1 and k = 0.98, z0 = −100
and τ = 300 .
Figure 3 shows the variation of electron energy gain γ
as a function of normalized distance z for normalized laser
pulse duration τ = 100 (corresponding laser pulse duration
53 fs), 300 (corresponding laser pulse duration 160 fs) and
500 (corresponding laser pulse duration 266 fs). The other
parameters are optimized from figures 1 and 2. We observe
that electron gains more energy due to 266 fs laser pulse duration as compared to 160 fs and 53 fs, for a set of optimized
parameters. Energy exchange between electron and laser field
in the applied magnetic fields is maximum for longer laser
pulse durations. For the chosen optimized laser and magnetic
field parameters, energy gain of 0.133 GeV is achieved for 53
fs laser pulse duration, 0.67 GeV for 160 fs and 1.01 GeV for
266 fs pulse duration. So, electron energy is sensitive to laser
pulse duration τ in presence of magnetic fields.
Figure 4 shows the variation of acceleration gradient dγ/dz
and energy gain γ as a function of accelerating distance z .
The acceleration gradient dγ/dz = −ev · E/vz is calculated,
and is plotted as a function of accelerating distance z with
a0 = 10 and b0 = 0.09. The maximum acceleration gradient
of 60 GeV m−1 is observed with laser peak intensity a0 = 10.
The maximum energy gain of 0.67 GeV is observed with peak
momentum of electron depends on the square of laser intensity amplitude. So, energy gain increases with increasing
amplitude of laser intensity. The maximum energy gain also
depends on the laser initial spot size parameter r0 . It is clear
from the figure 2(b) that maximum energy gain increases for
smaller spot size. For higher values of laser intensity param­
eters, optimum values of laser spot size to achieve higher
electron energy gain appear to be smaller. So, the electron
will be accelerated more effectively for smaller spot size and
higher intensity. At a0 = 10, under the influence of magnetic
fields b0 = 0.09 and initial spot size parameter r0 = 80, we
obtain energy gain of 0.68 GeV, which can be increased up
to 1.3 9 at a0 = 20(corresponding value of I0 ~ 5.52 × 1020
W cm−2) and initial spot size parameter r0 = 40 (corre­
sponding value is 6.4 µm). So, the net electron energy gain
during acceleration is sensitive to both the laser intensity and
the laser spot size.
4
A Singh et al
Laser Phys. 27 (2017) 110001
Figure 3. Electron energy gain γ as a function of z for a0 = 10,
b0 = 0.09, and r0 = 80 for laser pulse duration τ = 100, 300 and
500. Other parameters are the same as in figure 1.
Figure 6. Acceleration gradient as a function of distance z for
tightly focused chirped laser pulse with intensity parameter a0 = 5,
10, and 15. Other parameters are the same as taken in figure 1.
intensity 1.38 × 1020 W cm−2. With the higher energy laser
pulse, a steeper acceleration gradient is observed.
Figure 5 shows electron energy γ as a function of nor­
malized time t for different laser intensities under the influence of axial magnetic field and combined axial and azimuthal
magn­etic field. It has been observed that electron energy gain
is higher in the presence of combined magnetic field, as compared to that in the presence of axial magnetic field only. In
circularly polarised laser, there is production of self generated
axial and azimuthal magnetic fields. Hence, the application of
combined axial and magnetic fields acts as a booster in order
to accelerate the electron up to GeV energy. For a0 = 10, in
the presence of axial magnetic field of ~21 MG, energy gain
of about 1.4 GeV is observed, whereas in the presence of
combined magnetic field of ~9.6 MG, energy gain is about
2.4 GeV.
Electron dynamics may be analyzed by the study of acceleration gradient. Figure 6 shows the variation of acceleration
gradient as a function of distance z. Acceleration gradient
may be calculated from the relation dγ/dz = −e (β · E) /βz ,
which can be easily derived from the Lorentz equation itself.
Here, energy gradient is calculated for different values of nor­
malized laser intensity parameter a0 = 5, 10 and 20, for optimum values of magnetic field as taken from figure 1(a). Other
parameters are also the same as in figure 1. We have observed
acceleration gradients of about 32 GeV m−1 due to tightly
focused chirped laser for a0 = 5. For a0 = 10 and 20, values
of acceleration gradient rise up to 52 GeV m−1 and 83 GeV
m−1 respectively. Thus, for higher values of the laser intensity
parameter, a steeper acceleration gradient is observed.
Figure 4. Acceleration gradient dγ/dz and energy gain γ as a
function of accelerating distance z with a0 = 10 and b0 = 0.09.
Other parameters are the same as in figure 1.
4. Conclusion
Resonant enhancement in the acceleration of electrons by circularly polarized laser pulses in the presence of azimuthal and
axial magnetic fields has been investigated. Effective exchange
Figure 5. Electron energy gain γ as a function of t for a0 = 10 and
20 for b0 = 0.09 and bz = 0.2. Other parameters are the same as in
figure 1.
5
A Singh et al
Laser Phys. 27 (2017) 110001
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magnetic field on axicon laser-induced electron acceleration
High En. Den. Phys. 18 20
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32 205
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laser pulse on magnetic field influenced electron acceleration in vacuum Opt. Commun. 365 231
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effects on electron acceleration by two lasers of different
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laser in a plasma Phys. Plasmas 11 3992
[28] Sung J H, Lee S K, Yu T J, Jeong T M and Lee J 2010 0.1 Hz
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of energy between the electron and laser electric field takes
place in the resonance region, leading to enhanced electron
energy gain. This resonance region is found more interesting
than other regions of lower energy gain for various applications. In order to achieve higher maximum energy gain (~GeV),
laser and plasma parameters, along with the combined magn­
etic fields, are optimized. It is also observed that maximum
electron energy gain increases for smaller laser spot size for
the optimum value of magnetic fields. It is possible to achieve
higher electron energies under the combined effects of laser and
magnetic fields. The azimuthal and axial magn­etic fields confine and collimate the electron beam along the z-axis effectively
by rotation of electrons around the z-axis in a small x–y plane
area, so as to get a highly energetic electron beam. The specifications required for our scheme are available at the Center
for Relativistic Laser Science (CoReLS) of Institute for Basic
Science, Korea [28]. Results of the present investigation may be
useful in the development of laser driven particle accelerators.
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