1. Analysis of radial and longitudinal field of plasma wakefield generated by a Laguerre-
Gauss laser pulse
Ali Shekari Firouzjaei and Babak Shokri
Citation: Physics of Plasmas 23, 063102 (2016); doi: 10.1063/1.4953052
View online: http://dx.doi.org/10.1063/1.4953052
View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/23/6?ver=pdfcov
Published by the AIP Publishing
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2. Analysis of radial and longitudinal field of plasma wakefield generated
by a Laguerre-Gauss laser pulse
Ali Shekari Firouzjaei and Babak Shokri
Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839-63113, Iran
(Received 16 February 2016; accepted 17 May 2016; published online 3 June 2016)
In the present paper, we study the wakes known as the donut wake which is generated by Laguerre-
Gauss (LG) laser pulses. Effects of the special spatial profile of a LG pulse on the radial and longi-
tudinal wakefields are presented via an analytical model in a weakly non-linear regime in two
dimensions. Different aspects of the donut-shaped wakefields have been analyzed and compared
with Gaussian-driven wakes. There is also some discussion about the accelerating-focusing phase
of the donut wake. Variations of longitudinal and radial wakes with laser amplitude, pulse length,
and pulse spot size have been presented and discussed. Finally, we present the optimum pulse dura-
tion for such wakes. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953052]
I. INTRODUCTION
Laser plasma interaction is a central problem in plasma
physics. It is essential to understand its different aspects
which are related to laser plasma acceleration schemes,
advanced fusion concepts, and novel radiation sources.1–8
When a laser pulse propagates through underdense plasma, a
running plasma wave is produced by the ponderomotive
force of the laser pulse. This wave oscillates at the frequency
xp=
ffiffiffi
c
p
, where xp ¼ ð4pn0e2
=mÞ1=2
is the non-relativistic
electron plasma frequency. e, m, and, n0 denote charge,
mass, and density of electrons, respectively, and finally, c is
the electron relativistic factor. The produced accelerating
fields can be three orders of magnitude greater than those in
conventional accelerators.9,10
Many different aspects of
intense laser plasma interactions have been studied at length
in the last decades. In the present work, we concentrate on
the problem of laser wakefield generation when the laser
pulse is not purely Gaussian. In recent years, there has been
an increasing interest on lasers with Laguerre-Gauss (LG)
shape because of their two special properties: special spatial
profile and orbital angular momentum (OAM) states.11–15
The usage of LG laser pulses to control the transverse focus-
ing fields in laser wakefield acceleration was explored,16
and
the influence of higher order Laguerre–Gaussian laser pulses
in electron acceleration was also examined.17
In another
work,18
the angular momentum of particles in the longitudi-
nal direction produced by the LG laser was investigated, and
the enhancement was compared with those produced by the
usual laser pulses.
Recent papers have introduced a new type of wakes
named donut-shaped wake which is produced by LG laser
pulses.19,20
In the present paper, we model the wake gener-
ated by the LG laser pulse in a weakly non-linear regime in
two dimensions. Furthermore, the excited longitudinal and
transverse electrostatic wakefields are obtained via numeri-
cal study of the non-linear formula. We consider several
aspects of the donut-shaped wakes and discuss the advan-
tages and challenges of these types of wakes and finally com-
pare the results with the Gaussian-driven wake case. This
paper has the following structure. In Section II, we drive the
basic equations of our model. In Section III, we present
some conclusions and discuss on the possible implications of
spatial profile effect of laser pulses on longitudinal accelerat-
ing and radial focusing of donut wakes. Finally in Section
IV, the results are summarized.
II. ANALYTICAL INVESTIGATION
To investigate wakefield excitation, we adopt a com-
moving frame ðx ¼ x; y ¼ y; n ¼ z À ct; t ¼ tÞ where (x; y)
are the transverse coordinates and t and z are the time and
propagation distance, repectively.10
We further use the
quasi-static approximation @=@t ! 0.9
When a laser pulse
propagates in low-density plasma in the z direction, a wake-
field will be excited behind the laser pulse, such that the
wakefield satisfies the well-known wave equation9,10
kÀ2
p
@2
/
@2n
¼
1 þ a r; zð Þ2
2 1 þ /ð Þ2
À
1
2
; (1)
where kp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4pe2n0=mc2
p
is the plasma wave number. This
equation can be solved in two dimensions for the scalar poten-
tial / which is normalized to mc2
=e, and the electric field
which could be inferred through Ezðn; rÞ ¼ ÀE0@/=@n and
Erðn; rÞ ¼ ÀE0@/=@r, where the parameter E0 ¼ mcxp=e
refers to the cold non-relativistic wave breaking limit.9
We use cylindrical coordinates ðr; h; zÞ where r ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
x2 þy2
p
is the transverse distance to the axis and h is the azimuthal
angle. The normalized vector potential (aL ¼ eAL
mc2) of the LG
laser pulse is given by aLðn;rÞ¼a0ajjðnÞarðrÞ, where a0 is
the peak laser vector potential; ajjðnÞ is the longitudinal
profile of intensity given by ajjðnÞ¼expðÀðtÀz=cÞ2
=2sl
2
Þ
where sl, the laser pulse duration, is the root-mean-square
(RMS) of length which is related to the full width at half
maximum (FWHM) through sFWHM ¼2
ffiffiffiffiffiffiffiffiffi
2ln2
p
sl; arðrÞ is the
transverse laser profile given by arðrÞ¼cl;pðr=w0Þjlj
expðÀr2
=w0
2
þilhÞLjlj
p ð2r2
=w2
0Þ, where w0 is the laser spot
1070-664X/2016/23(6)/063102/5/$30.00 Published by AIP Publishing.23, 063102-1
PHYSICS OF PLASMAS 23, 063102 (2016)
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3. size, Ljlj
p is a Laguerre polynomial with radial index p and az-
imuthal index l, and finally, cl;p are normalizing factors.18,20
As we are interested in the donut type wake, we introduce a
laser pulse with ðl;pÞ¼ð1;0Þ. For convenience, we have
introduced the dimensionless time s¼xpsFWHM. The spatial
coordinates are normalized to 1=kp. Solving the above differ-
ential equation numerically and using the LG laser profile,
we can present and discuss the results for the transverse fo-
cusing and longitudinal accelerating donut wakefield.
III. NUMERICAL RESULTS
In this section, simulation results of the LG laser pro-
duced wake are compared to the results of the Gaussian laser
pulse. Under the baseline simulation conditions, we use a
laser pulse with normalized peak vector potential a0 ¼ 0:3,
normalized FWHM duration sFWHM ¼ kp=2c, where kp is
plasma wavelength, and normalized spot size of w ¼ 20.
Before analyzing the results, it should be noticed that all pa-
rameters are considered the same for both of the mentioned
laser pulses.
First, we compare Gaussian laser produced wakes with
the one generated by a LG laser pulse. Figure 1 shows the
simulated longitudinal accelerating field Ezðn; rÞ and the
transverse focusing field Erðn; rÞ in a weakly nonlinear re-
gime for both types of wake. Longitudinal and transverse
fields of the donut and Gaussian pulse generated wakes are
presented in Figures 1(a), 1(b) and 1(c), 1(d), respectively.
The accelerating phase of positive and negative charged par-
ticles in a donut wake consists of two slices in both sides of
the propagation axis n ¼ 0. The radius of each slice of the
accelerating phase in the donut wake is about a quarter of the
size of the Gaussian-driven wake.
Regarding these figures and considering the same laser
pulse amplitude, spot size, and pulse duration, there will be a
larger wake field for a donut wake (about ten times more).
According to the previous works on Gaussian laser produced
wakes,9
the most interesting and useful situation for charged
particle acceleration is when particles are injected near the
axis. This is different for a donut wake. Particles should be
injected off axis in donut wakes because the longitudinal
accelerating field vanishes or has a minimum on the axis.
According to Figure 1, for the LG case, the radiation source
should be larger than the normal pulse case.
Another important fact which is considered in this paper
is the overlap region for accelerating-focusing phase (AFP).
A particle of charge q propagating in the positive z direction
in the plasma wakefield is accelerated if qEz 0 and is
focused if q@rEr 0. These inequalities define the AFP and
thus determine the volume of space which is useful for elec-
tron trapping and acceleration. Because of the curvature of
the phase surface in the donut wake, the regions of radial fo-
cusing are shifted toward longitudinal accelerating regions,
and consequently, the overlap between the AFP increases
respect to the Gaussian-driven case. In addition, as seen
from Figure 1, the AFP of a donut wake is larger for positive
charged particles than that for the negative ones. The phase
difference between the accelerating and focusing fields is
generally well known in the quasi-linear regime. There exists
a kp=4 region that is both focusing and accelerating for either
electrons or positrons. However, in the quasi-linear case, the
formation of the AFP for positrons is mainly due to the varia-
tion of the transverse shape and intensity of the driver pulse.
It can be seen from Figure 1(b) that the transverse field
phase is shifted backward about p compared to the Gaussian
case. Due to this phase shift, as presented in Figures 1(a) and
FIG. 1. Accelerating and focusing
wakefield generated by, (a) and (b) LG
laser pulse, (c) and (d) Gaussian laser
pulse. Accessible phase regions of fo-
cusing and accelerating for both elec-
trons and positrons are present.
063102-2 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016)
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4. 1(b), the AFP for the positive charged particles is situated in
the first half of the bucket, so they could be injected into the
first bucket. This result is in contrast to Figures 1(c) and 1(d)
which show that the first bucket has no AFP for the positive
charges.
This phase shift also leads to a shift in the AFP of the
negative charged particles from the second quarter to the first
quarter in the first bucket. This finding is important in the self
injection process of a highly nonlinear regime and also other
methods of injection in which charged particles are injected
at the rear point of the first plasma wake. In the Gaussian pro-
duced wake, as the background plasma electrons are trapped
in the first half of the wake region (self-injection), negative
charges will experience a radial expulsion force. Therefore,
this will increase the radial emittance, resulting in the produc-
tion of broad energy spectra of charges before being acceler-
ated in the AFP region. In fact, this feature of the donut wake
will be favorable for laser wakefield accelerators (LWFA).
After the above general comparison, we start to investi-
gate parameter sensibility of donut wakes. Plots of Figure 2
show details of variation of the longitudinal and radial fields
versus laser spot size. It is obvious that laser spot size has a
strong effect on both the longitudinal accelerating and radial
focusing field amplitude. Furthermore, the nonlinear effects
of wave steepening and period lengthening are clearly evi-
dent. This will cause an enhancement in nonlinear plasma
wavelength.10
Thus, one can change the wakefield properties
(length and amplitude) by varying the radial profile of the
laser pulse.
To understand more the latest result, Figure 3 is plotted
for varying pulse amplitude and pulse spot size simultane-
ously. It is found out that the longitudinal accelerating field
could be greater than the cold wave breaking limit even for
the quasi-linear laser amplitude case. So, it can be concluded
that there is a transition from the quasi-linear to the non-
linear wake regime. This finding is really important. In previ-
ous works, it was shown that the maximum wake amplitude
for a Gaussian laser pulse in the linear laser amplitude re-
gime is Ez ¼ 0:76a2
0E0, which is really smaller than the cold
wave breaking limit.9,10
In Figure 4, the effect of the pulse duration on the donut
wake is presented while keeping pulse amplitude fixed. At a
fixed electron density, we examine three different pulse
lengths csFWHM ¼ kp=2 ; kp=4 ; kp=6. It is obvious that the
pulse duration influences the wakefields. One could find
FIG. 2. Variation of (a) longitudinal and (b) transverse, component of donut
wakefield with spot size as a function of kpn and kpr for spot size changed to
w ¼ 25.
FIG. 3. Variation of (a) longitudinal (b) transverse donut wakefield as a
function of kpn and kpr. Normalized spot size changed to w ¼ 25 and nor-
malized amplitude changed to a ¼ 0:6 simultaneously.
063102-3 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016)
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5. from Figure 4 that the maximum wake amplitude will be
generated by the pulse length csFWHM ¼ kp=2. To find out
the pulse length at which the laser pulse will most efficiently
drives the plasma wave, we plot the maximum normalized
wake amplitude versus the pulse length in Figure 5. It is evi-
dent that the optimum pulse duration for a donut wake occurs
at s ¼ 2:6. This is different from the results reported for the
Gaussian-driven wakes.9,10
IV. SUMMARY AND CONCLUSION
In this paper, we have studied the case of the laser gener-
ated wakefield by an intense LG pulse in a weakly non-linear
regime. An analytical formula was introduced and numerical
results have been demonstrated. We have explained some
general features of the donut wakes and then compared them
with those of the Gaussian produced wakes. This regime
could be well characterized, and the acceleration process
could be optimized for maximum electron and positron ener-
gies. We showed that the donut wake has a larger focusing-
accelerating phase for positive charged particles compared to
the Gaussian wakes. Our results also show that unlike
Gaussian-driven wakes, there is no possibility to inject
charged particles on axis in donut wakes. Finally, variation of
the laser spot size and laser amplitude were presented and
their effects were discussed. In addition, effects of laser pulse
duration were examined, and the optimum pulse duration to
resonantly drive the wakefield was obtained.
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FIG. 4. Longitudinal wakefield gener-
ated by different pulse durations, (a)
csFWHM ¼ kp=2, (b) csFWHM ¼ kp=4,
and (c) csFWHM ¼ kp=6.
FIG. 5. Behaviour of normalized maximum donut wake amplitude versus
pulse length.
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063102-5 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016)
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