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- 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

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2016 13:32:01 - PHYSICS OF PLASMAS 23, 063102 (2016)

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

paper has the following structure. In Section II, we drive the

basic equations of our model. In Section III, we present

Laser plasma interaction is a central problem in plasma

some conclusions and discuss on the possible implications of

physics. It is essential to understand its different aspects

spatial profile effect of laser pulses on longitudinal accelerat-

which are related to laser plasma acceleration schemes,

ing and radial focusing of donut wakes. Finally in Section

advanced fusion concepts, and novel radiation sources.1–8

IV, the results are summarized.

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

II. ANALYTICAL INVESTIGATION

pﬃﬃﬃ

xp= c, where xp ¼ ð4pn0e2=mÞ1=2 is the non-relativistic

To investigate wakefield excitation, we adopt a com-

electron plasma frequency. e, m, and, n0 denote charge,

moving frame ðx ¼ x; y ¼ y; n ¼ z À ct; t ¼ tÞ where (x; y)

mass, and density of electrons, respectively, and finally, c is

are the transverse coordinates and t and z are the time and

the electron relativistic factor. The produced accelerating

propagation distance, repectively.10 We further use the

fields can be three orders of magnitude greater than those in

quasi-static approximation @=@t ! 0.9 When a laser pulse

conventional accelerators.9,10 Many different aspects of

propagates in low-density plasma in the z direction, a wake-

intense laser plasma interactions have been studied at length

field will be excited behind the laser pulse, such that the

in the last decades. In the present work, we concentrate on

wakefield satisfies the well-known wave equation9,10

the problem of laser wakefield generation when the laser

pulse is not purely Gaussian. In recent years, there has been

1 þ aðr; zÞ2

an increasing interest on lasers with Laguerre-Gauss (LG)

@2/

1

kÀ2

¼

À ;

(1)

shape because of their two special properties: special spatial

p

@2n

2ð1 þ /Þ2

2

profile and orbital angular momentum (OAM) states.11–15

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

p

ﬃﬃﬃﬃﬃﬃﬃﬃ

The usage of LG laser pulses to control the transverse focus-

where kp ¼

4pe2n0=mc2 is the plasma wave number. This

ing fields in laser wakefield acceleration was explored,16 and

equation can be solved in two dimensions for the scalar poten-

the influence of higher order Laguerre–Gaussian laser pulses

tial / which is normalized to mc2=e, and the electric field

in electron acceleration was also examined.17 In another

which could be inferred through Ezðn; rÞ ¼ ÀE0@/=@n and

work,18 the angular momentum of particles in the longitudi-

Erðn; rÞ ¼ ÀE0@/=@r, where the parameter E0 ¼ mcxp=e

nal direction produced by the LG laser was investigated, and

refers to the cold non-relativistic wave breaking limit.9

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

the enhancement was compared with those produced by the

We use cylindrical coordinates ðr; h; zÞ where r ¼

x2 þ y2

usual laser pulses.

is the transverse distance to the axis and h is the azimuthal

Recent papers have introduced a new type of wakes

angle. The normalized vector potential (aL ¼ eAL) of the LG

mc2

named donut-shaped wake which is produced by LG laser

laser pulse is given by aLðn;rÞ ¼ a0ajjðnÞarðrÞ, where a0 is

pulses.19,20 In the present paper, we model the wake gener-

the peak laser vector potential; ajjðnÞ is the longitudinal

ated by the LG laser pulse in a weakly non-linear regime in

profile of intensity given by a

2

jjðnÞ ¼ expðÀðt À z=cÞ2=2sl Þ

two dimensions. Furthermore, the excited longitudinal and

where s

transverse electrostatic wakefields are obtained via numeri-

l, the laser pulse duration, is the root-mean-square

(RMS) of length which is related to the full width at half

cal study of the non-linear formula. We consider several

pﬃﬃﬃﬃﬃﬃﬃﬃﬃs

aspects of the donut-shaped wakes and discuss the advan-

maximum (FWHM) through sFWHM ¼ 2 2ln2 l; arðrÞ is the

tages and challenges of these types of wakes and finally com-

transverse

laser

profile

given

by

arðrÞ ¼ cl;pðr=w0Þjlj

pare the results with the Gaussian-driven wake case. This

expðÀr2=w 2

0 þ ilhÞLjljð2r2=w2Þ, where w

p

0

0 is the laser spot

1070-664X/2016/23(6)/063102/5/$30.00

23, 063102-1

Published by AIP Publishing. - 063102-2

A. S. Firouzjaei and B. Shokri

Phys. Plasmas 23, 063102 (2016)

size, Ljlj is a Laguerre polynomial with radial index p and az-

Regarding these figures and considering the same laser

p

imuthal index l, and finally, c

pulse amplitude, spot size, and pulse duration, there will be a

l;p are normalizing factors.18,20

As we are interested in the donut type wake, we introduce a

larger wake field for a donut wake (about ten times more).

laser pulse with ðl;pÞ ¼ ð1;0Þ. For convenience, we have

According to the previous works on Gaussian laser produced

introduced the dimensionless time s ¼ x

wakes,9 the most interesting and useful situation for charged

psFWHM. The spatial

coordinates are normalized to 1=k

particle acceleration is when particles are injected near the

p. Solving the above differ-

axis. This is different for a donut wake. Particles should be

ential equation numerically and using the LG laser profile,

injected off axis in donut wakes because the longitudinal

we can present and discuss the results for the transverse fo-

accelerating field vanishes or has a minimum on the axis.

cusing and longitudinal accelerating donut wakefield.

According to Figure 1, for the LG case, the radiation source

III. NUMERICAL RESULTS

should be larger than the normal pulse case.

Another important fact which is considered in this paper

In this section, simulation results of the LG laser pro-

is the overlap region for accelerating-focusing phase (AFP).

duced wake are compared to the results of the Gaussian laser

A particle of charge q propagating in the positive z direction

pulse. Under the baseline simulation conditions, we use a

in the plasma wakefield is accelerated if qEz > 0 and is

laser pulse with normalized peak vector potential a0 ¼ 0:3,

focused if q@rEr < 0. These inequalities define the AFP and

normalized FWHM duration sFWHM ¼ kp=2c, where kp is

thus determine the volume of space which is useful for elec-

plasma wavelength, and normalized spot size of w ¼ 20.

tron trapping and acceleration. Because of the curvature of

Before analyzing the results, it should be noticed that all pa-

the phase surface in the donut wake, the regions of radial fo-

rameters are considered the same for both of the mentioned

cusing are shifted toward longitudinal accelerating regions,

laser pulses.

and consequently, the overlap between the AFP increases

First, we compare Gaussian laser produced wakes with

respect to the Gaussian-driven case. In addition, as seen

the one generated by a LG laser pulse. Figure 1 shows the

from Figure 1, the AFP of a donut wake is larger for positive

simulated longitudinal accelerating field Ezðn; rÞ and the

charged particles than that for the negative ones. The phase

transverse focusing field Erðn; rÞ in a weakly nonlinear re-

difference between the accelerating and focusing fields is

gime for both types of wake. Longitudinal and transverse

generally well known in the quasi-linear regime. There exists

fields of the donut and Gaussian pulse generated wakes are

a kp=4 region that is both focusing and accelerating for either

presented in Figures 1(a), 1(b) and 1(c), 1(d), respectively.

electrons or positrons. However, in the quasi-linear case, the

The accelerating phase of positive and negative charged par-

formation of the AFP for positrons is mainly due to the varia-

ticles in a donut wake consists of two slices in both sides of

tion of the transverse shape and intensity of the driver pulse.

the propagation axis n ¼ 0. The radius of each slice of the

It can be seen from Figure 1(b) that the transverse field

accelerating phase in the donut wake is about a quarter of the

phase is shifted backward about p compared to the Gaussian

size of the Gaussian-driven wake.

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. - 063102-3

A. S. Firouzjaei and B. Shokri

Phys. Plasmas 23, 063102 (2016)

1(b), the AFP for the positive charged particles is situated in

focusing field amplitude. Furthermore, the nonlinear effects

the first half of the bucket, so they could be injected into the

of wave steepening and period lengthening are clearly evi-

first bucket. This result is in contrast to Figures 1(c) and 1(d)

dent. This will cause an enhancement in nonlinear plasma

which show that the first bucket has no AFP for the positive

wavelength.10 Thus, one can change the wakefield properties

charges.

(length and amplitude) by varying the radial profile of the

This phase shift also leads to a shift in the AFP of the

laser pulse.

negative charged particles from the second quarter to the first

To understand more the latest result, Figure 3 is plotted

quarter in the first bucket. This finding is important in the self

for varying pulse amplitude and pulse spot size simultane-

injection process of a highly nonlinear regime and also other

ously. It is found out that the longitudinal accelerating field

methods of injection in which charged particles are injected

could be greater than the cold wave breaking limit even for

at the rear point of the first plasma wake. In the Gaussian pro-

the quasi-linear laser amplitude case. So, it can be concluded

duced wake, as the background plasma electrons are trapped

that there is a transition from the quasi-linear to the non-

in the first half of the wake region (self-injection), negative

linear wake regime. This finding is really important. In previ-

charges will experience a radial expulsion force. Therefore,

ous works, it was shown that the maximum wake amplitude

this will increase the radial emittance, resulting in the produc-

for a Gaussian laser pulse in the linear laser amplitude re-

tion of broad energy spectra of charges before being acceler-

gime is Ez ¼ 0:76a2E

0

0, which is really smaller than the cold

ated in the AFP region. In fact, this feature of the donut wake

wave breaking limit.9,10

will be favorable for laser wakefield accelerators (LWFA).

In Figure 4, the effect of the pulse duration on the donut

After the above general comparison, we start to investi-

wake is presented while keeping pulse amplitude fixed. At a

gate parameter sensibility of donut wakes. Plots of Figure 2

fixed electron density, we examine three different pulse

show details of variation of the longitudinal and radial fields

lengths csFWHM ¼ kp=2 ; kp=4 ; kp=6. It is obvious that the

versus laser spot size. It is obvious that laser spot size has a

pulse duration influences the wakefields. One could find

strong effect on both the longitudinal accelerating and radial

FIG. 2. Variation of (a) longitudinal and (b) transverse, component of donut

FIG. 3. Variation of (a) longitudinal (b) transverse donut wakefield as a

wakefield with spot size as a function of kpn and kpr for spot size changed to

function of kpn and kpr. Normalized spot size changed to w ¼ 25 and nor-

w ¼ 25.

malized amplitude changed to a ¼ 0:6 simultaneously. - 063102-4

A. S. Firouzjaei and B. Shokri

Phys. Plasmas 23, 063102 (2016)

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.

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.

FIG. 5. Behaviour of normalized maximum donut wake amplitude versus

pulse length.

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