Influence of properties of the Gaussian laser pulse and magnetic field on the electron acceleration in laser–plasma interactions

Optics & Laser Technology 45 (2013) 613–619

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Influence of properties of the Gaussian laser pulse and magnetic field on the electron acceleration in laser–plasma interactions H.R. Askari n, A. Shahidani Department of Physics, Vali-e-Asr University, Vallayat Blvd., Rafsanjan, Iran

a r t i c l e i n f o

abstract

Article history: Received 25 April 2012 Received in revised form 18 May 2012 Accepted 18 May 2012 Available online 27 June 2012

In this paper, by using the hydrodynamics fluid equations in underdense plasma and Maxwell’s equations, the general equation for the wake potential, which can be directly used for different envelopes of optical laser pulse, is obtained. The analytical solution of the equation for the Gaussian short-intense laser pulse is also acquired. It is shown that the wake field, which leads to electron acceleration, depends on intensity, length, frequency and energy of pulse and the electron number density. We also investigate the generation of wake field and the energy gain acquired by an electron in the plasma in the presence or absence of external magnetic field. It is shown that the efficiency of wake field is increased in the presence of an external magnetic field. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Laser pulse Electron acceleration Plasma

1. Introduction The study of the charged particle acceleration has had great importance in different subject of physics such as, nuclear physics, particle physics and astrophysics. It has been the subject of great interest for the experimental as well as the theoretical researches and the important applications of industrial and medicinal [1,2]. The appreciable attempts have been made to achieve compact ultrahigh gradient accelerators using ultra-short and ultra-intense lasers in the plasmas. In accelerators based laser–plasma interaction, ultra-short and ultra-intense lasers generate a large longitudinal accelerating electric field, which is called the wake field. The wake field is excited via the ponderomotive force of an ultrashort and ultra-intense laser. The ponderomotive force is proportional to the gradient of the laser intensity which pushes electrons outward and separates them from the ions, thus creating a travelling electric field. The wake field accelerates particles to high energies at distances much shorter than those in conventional accelerators. The wake electric field can have a magnitude more than 100 GV/m. The characteristic scale length of the accelerating structure is of the order of the plasma wavelength lp E10–30 mm, for typical plasma densities of ne E1024– 1026 m  3. Several experiments have shown that laser–plasma accelerators can produce high quality electron beams, with quasi

n

Corresponding author. Tel.: þ98 913 1917717; fax: þ 98 391 3226800. E-mail addresses: [email protected], [email protected] (H.R. Askari), [email protected] (A. Shahidani). 0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.05.023

monoenergetic energy distributions more than 100 MeV level [3–5]. There are four different accelerators based on laser–plasma interactions. These are the laser wake field accelerator, the selfmodulated laser wake field accelerator, the plasma beat wave accelerator, and the plasma wake field accelerator [6]. In laser wake field acceleration, a large amplitude wake field is excited by a high intensity short laser pulse when its length matches with half of the plasma wavelength. The ponderomotive force is proportional to the negative gradient of pulse intensity. Envelope pulse-dependent ponderomotive force separates the electrons from the laser pulse area and a strong electric wake field due to charge separation is created in the plasma. Therefore, plasma medium can be the appropriate medium for the acceleration of charged particles. In order to achieve higher acceleration, the laser–plasma interaction length as well as the amplitude of the wake field should be maintained at higher magnitudes. Kingham et al. [7] have shown that the wake fields can be enhanced by the nonlinearities in response of the plasma to the ponderomotive force. Andreev et al. [8] have proven that the amplitude of the laser wake field has also been increased by the ionization processes. Sprangle et al. [9] have shown that Tapered plasma channels have been proposed to achieve greater electron acceleration in the laser wake field mechanism. Lin et al. [3] have shown that the compression of the low intensity pulse by the plasmas might be a possible way to excite large amplitude wake field. By choosing Gaussian-like (GL) pulse, rectangular-triangular (RT) pulse and rectangular-Gaussian (RG) pulse, Hitendra et al. [10] have calculated the maximum energy gain acquired by an electron for all these three types.

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H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

In this paper, we use the continuity equations in underdense plasma and Maxwell’s equations and obtain the general equation for the wake potential. The equation depends on envelopes of optical laser pulse and it is solved for the Gaussian short-intense laser pulse. The analytical solution of the equation shows that the wake field, which leads to electron acceleration, depends on intensity, wide and energy of pulse. We show that wake field amplitude reaches its maximum value when the pulse length is of the order of the plasma wavelength and electron energy gain is increased in the presence of an external magnetic field as well.

@E @B ¼ , @z @t

ð12Þ

@B 1 @E ¼ m3 en3 v?  2 , @z c @t

ð13Þ

@2 fw en0 ¼ e: e0 @z2

ð14Þ

By considering the system to be nonevolving, i.e., all the quantities depend only on x ¼z  ngt, plasma is placed in a weakly relativistic case and the presence of constant magnetic field B3 e^ ?  e^ J , and the above equations reduce to

2. Wake potential and electric field The hydrodynamics fluid equations in weakly collisional cold electron plasma and Maxwell’s equations are given by [11,12] @n ! ! þ r Uðn n Þ ¼ 0, @t ! ! ! ! dp ¼ eð E þ n  B Þ, dt

! ! r U B ¼ 0,

ð4Þ ð5Þ

It is assumed that the ion motion is neglectable, and the plasma in absence of the applied optical fields is at equilibrium state. Under equilibrium conditions, electron number density n3 is ! constant and drift velocity u 0 is zero. It is also assumed that a laser pulse with high intensity I0, angular frequency o ¼2pf and time pulse duration t is propagating in the þz-axis direction. By considering a weakly relativistic case, where v? cvJ , we approximate ð7Þ

ð16Þ

ng

@v? @n? e e e þ vJ ¼  E þ vJ B þ vJ B0 , m m m @x @x

ð17Þ

@E @B ¼ ng , @x @x

ð18Þ

@B ng @E ¼ m3 en3 n? þ m3 en0e n? þ 2 , @x c @x

ð19Þ

¼

2

@x

e

e3

n0e :

ð20Þ

Integration of the above equations under the condition that n0e ,

nll and n? vanish as 9x9-N, the following equation is obtained for the wake potential:  2 op @2 jW e þ jW  2 ng 2mn2g @x

! c2 eB0 1 E2  mvg v2g

! c2 1 E ¼ 0, v2g

ð21Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where op ¼ ðe2 n3 Þ=ðme3 Þ is plasma frequency and vg ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1ðo2p =o2 Þ is the group velocity of laser. Eq. (21) is the general equation for the wake potential which depends on envelope of electric field of laser, E. The above equation can be rewritten as @2 jw 2

@x

2

þ kp jw ¼

2 X

b2i f i ðxÞ,

ð22Þ

i¼1

where kp ¼ op/vg ¼2p/lp and ! eE23 c2 eE3 B3 2 b21 ¼ 1 , b2 ¼ mvg 2mn2g v2g

! c2 1 , v2g

f 1 ðxÞ ¼ E2 , f 2 ðxÞ ¼ E, ð8Þ

where indexes ‘‘99’’ and ‘‘?’’ show components of vector quantities along and perpendicular to the direction of the pulse propagation, respectively. It is better to solve equations by means of the perturbation method. The electron density is taken as n3 þ n0e where n0e is the electron number density variation. By using Eqs. (1)–(5), the following equations are obtained for the quantities of system: @ne @ðne vJ Þ ¼ 0, þ @z @t

@vJ @vJ e @ jW e e þ nll ¼  n? B n? B0 , m @x m m @x @x

ð23Þ

and

and pJ  mvJ ,

ng

@2 jW

! ! ! where B , E , c, v , n, m and e are magnetic and electric fields, the light velocity in vacuum, the average velocity, number density, and mass and electrical charge of electron, respectively; ! ! ! ! J ¼ n e v is the current density, and p ¼ gm v is relativistic momentum where the relativistic factor g is defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2 ð6Þ g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1þ 2 2 : 2 2 m c 1ðv =c Þ

mv? mv? p? ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  mv? , 2 2 2 1ððvJ þ v? Þ=c Þ 1ðv2? =c2 Þ

ð15Þ

ð2Þ

ð3Þ

e0

@n0e @vJ @n0 @vJ þ n0 þ vJ e þ n0e ¼ 0, @x @x @x @x

ð1Þ

! ! 1 @E ! ! , r  B ¼ m0 j þ 2 c @t

! ! eðni ne Þ rUE ¼ ,

ng

ð9Þ

@v? @v? e e þvJ ¼  Eþ vJ B, m m @t @z

ð10Þ

@vJ @vJ e @j e þvJ ¼  v? B, m @z m @t @z

ð11Þ

ð24Þ

It can be shown that special solution of Eq. (20) is as

jw ðxÞ ¼

2 X

ji ðxÞ,

ð25Þ

i¼1

where

ji ðxÞ ¼

b2i kp

Z

x a

0

0

0

sinkp ðxx Þf i ðx Þdx ,

ð26Þ

By choosing an appropriate shape for the laser pulse, the wake potential is obtained. Shape of laser envelope is chosen Gaussian so that its field profile is obtained as ! 2 E ¼ e^ x E3 egðxðL=2ÞÞ , ð27Þ pffiffiffi where E3 is the maximum amplitude of Gaussian electric field, 1= g pffiffiffi is defined as the beam waist radius, i.e., 1= g ¼ xE ¼ E3 xE ¼ E3 =e and

H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

L is the width of pulse. By substituting the Gaussian pulse profile (Eq. (27)) in Eq. (26), we obtain the following solutions !   rffiffiffiffi 2 kp b21 p L sin kp x , ð28Þ j1G ðxÞ ¼ exp  2 kp g 4g

j2G ðxÞ ¼

b22

sffiffiffiffiffiffi 2p

g

kp

2

exp 

kp

!

2g

sin kp



 L x : 2

ð29Þ

The following expressions are obtained for the wake potential in two cases, in absence and presence of magnetic field in plasma   1 L ¼3 , ð30Þ jBWG ðxÞ ¼  A1WF sin kp x kp 2 a3 jBWG ðxÞ ¼ 

1 ðA1WF þ A2WF Þsin kp kp



 L x , 2

ð31Þ

A2WF ¼

ð32Þ

! ! pffiffiffiffiffiffi 2 kp eB3 E3 2p c2 : 1 exp  pffiffiffi mng g 4g v2g

ð33Þ

! ! Using the relation E w ¼  r jw , the wake fields are obtained as:   !B ¼ 3 L ¼3 e^ , cos kp x E WG ðxÞ ¼ ABWF 2 x

ð34Þ

  !B a 3 L a3 e^ , cos kp x E WG ðxÞ ¼ ABWF 2 x

ð35Þ

¼3 a3 ¼ A1WF and ABWF ¼ A1WF þ A2WF . The above equaso that ABWF tions show that A1 and A1 þA2 are amplitudes of the wake fields in absence and presence of magnetic field in plasma, respectively. We can obtain A1 and A2 in terms of intensity I3 of laser as ! pffiffiffiffi e pLI3 c2 2 1 exp ð0:01kp L2 Þ, ð36Þ A1WF ¼ 5me3 vg c2 v2g

A2WF ¼

2 eB3 L 5 mc

sffiffiffiffiffiffiffiffiffi ! pI 3 c 2 2 1 exp ð0:02kp L2 Þ, e3 ng v2g

B¼0 is of the order of the The above equation shows that Lmax plasma wavelength and depends on n3 and o. We can also calculate the maximum wake field in the presence of magnetic a0 field at LBmax such that it is obtained from the following equation 2

rffiffiffiffiffiffiffiffiffi

e3 ng I3

a0 2 a0 2 Þ Þ exp ð0:01kp ðLBmax Þ Þ ¼ 0, ð10:02kp ðLBmax 2

2

ð39Þ We can calculate A1 which is maximum at 0 ¼0 nB3max

me3 o2 B ¼ @1 e2

A2WF

2eB3 ¼ mc

¼0 nB3max

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi ffi 10 2Ep c2 2 pffiffiffiffi 1 exp ð0:02kp L2 Þ: e3 L p v2g

so that 1

0:04o2 L2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A, 2 2 2 ð0:02o L þ 3c Þþ ð0:02o2 L2 þ 3c2 Þ2 0:08o2 L2 c2

ð40Þ

ð42Þ

3. The electron energy gain In order to calculate the electron acceleration, the interaction between electrons and Gaussian wake fields could be studied by solving the following Lorentz motion equation ð43Þ

where p is relativistic momentum. For simplicity of calculations, we define the new variable Z ¼ kp(x L/2), which is related to the phase of the wake electric field. By using the definitions of Z and x, we obtain the following relations dg peEw ðZÞ ¼ , dt gm2 c2

ð44Þ

and " sffiffiffiffiffiffiffiffiffiffiffiffiffi # dZ 1 ¼ kp c 1 2 ng : dt g

ð45Þ

the resultant By dividing dg=dt with dZ=dt andpintegrating ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equation and by taking dx=dt ¼ nx ¼ c 11=g2 and b ¼ ng/c, the following relation is obtained: Z qffiffiffiffiffiffiffiffiffiffiffiffi e gb g2 1 ¼  2 ð46Þ EðZÞdZ: mc kp This equation is a general equation for the electron acceleration. By assuming the initial value g as g3 at x ¼ 3 and integrating Eq. (46), Dg is calculated. Then by using DW¼mc2Dg, the electron energy gain is obtained as    kp L eA1WF , ð47Þ DW BG ¼ 3 ¼ sin Z þ sin 2 ð1bÞkp

ð37Þ

For Gaussian pulse, we assume that L is Dx between the points pffiffiffi that their electric fields are 0:06E3 , i.e., L ¼ 5= g. By using Eq. (36), we can prove that the maximum wake field in the absence of B¼0 magnetic field is at Lmax where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼0 ¼ 1:1254lp ¼ 2:25pc ðme3 Þ=ðe2 n3 Þ1=o2 , LBmax ð38Þ

a0 2 ð10:02kp ðLBmax Þ Þþ 2cB3

¼0 nB3max depends on L and o. We can also express A1WF and A2WF in terms of the pulse energy Ep of laser as ! pffiffiffi 50 2eEp c2 2 1 exp ð0:01kp L2 Þ, ð41Þ A1WF ¼ mpe3 c2 L2 v2g

dp ¼ eEw ðxÞ, dt

where we introduce the quantities A1 and A2 as ! ! pffiffiffiffi 2 kp eE23 p c2 A1WF ¼ 1 exp  , p ffiffiffi 4g 2mv2g g v2g

615

DW BG a 3 ¼

   kp L e : ðA1WF þ A2WF Þ sin Z þ sin 2 kp ð1bÞ

ð48Þ

4. Results and discussions By using hydrodynamics fluid and Maxwell’s equations, the general equation for the wake potential has been solved for the Gaussian short-intense laser pulse. The analytical solutions of this system, Eqs. (36) and (37), show that the amplitude of wake field depends on intensity, width, frequency and energy of pulse, magnetic field and the electron number density. We could consider the effects of these parameters on the wake field. We use the data and the results of the experiments that were performed by Faure et al. [13], I3 ¼ 3:4  1022 W=m2 , l ¼ 820 nm, n3 ¼ 7:5  1024 m3 , EWF Z 100 GV=m and DW G Z100 MeV: 4.1. Effect of the electron number density depends on L and o. In order to According to Eq. (41), nmax 3 study the L effect, the amplitude of wake field AWF is plotted as a

616

H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

250

200

A wf (Gv/m)

Awf (Gv/m)

300

L =21 μm 150

L =22 μm

n o =7.5×1024 m − 3 n o =8.5×1024 m − 3

100

L =23 μm

n o =9.5×1024 m − 3

L =24 μm

100 2

4

6

8

10

12

14

n o ×1024 (m − 3 )

700

0 0

5

10

L (µm)

15

20

25

λ=0.82 μm λ=1.053 μm

600

A wf (Gv/ m)

n o =6.5×1024 m − 3

200

λ=1.2 μm λ=1.4 μm

500 400 300 200 100 2

4

6

8

10

12

14

n o ×1024 (m − 3 ) Fig. 1. Diagrams of amplitude of wake field AWF in terms of n3 for I3 ¼ 3:4  1022 W=m2 , (a) l ¼820 nm and the different values of L¼21 mm, 22 mm, 23 mm and 24 mm; (b) L¼23 mm and different values of l ¼820 nm, 1.053 mm, 1.2 mm and 1.4 mm that are denoted by large to small dashed curves, respectively.

function of the electron number density n3 for a special frequency and four different L (Fig. 1a). The values of laser wavelength and intensity are chosen as 820 nm and 3:4  1022 W=m2 , respectively. At reso3 ¼ 0 nance values nB3max ¼ 6:40557  1024 m3 , 5.83748  1024 m  3, 5.34172  1024 m  3 and 4.90649  1024 m  3, the wake field has the maximum values Amax 265:392 GV=m, WF ¼ 278:716 GV=m, 253:285 GV=m and 242:236 GV=m for the different values of L, 21 mm, 22 mm, 23 mm and 24 mm, respectively. It is concluded that with increasing L, magnitude of Amax WF reduces and its resonance state 3 ¼ 0 shifts to the lower value of nB3max . Fig. 1b presents plots of amplitude of wake field AWF in terms of the electron number density n3 for four different values of laser wavelength and special L. It is shown that the wake field has the maximum values Amax WF ¼ 242:236 GV=m, 3 ¼ 0 399.872 GV/m, 519.721 GV/m and 708.275 GV/m at nB3max ¼ 5:34172  1024 m-3 , 5.3361  1024 m  3, 5.33183  1024 m  3 and 5.32512  1024 m  3, for I3 ¼ 3:4  1022 W=m2 , L¼23 mm (t ¼ 30.6871 fs) and the different values of l, 820 nm, 1.053 mm, 1.2 mm and 1.4 mm, respectively. It is shown that with increasing l, magnitude of amplitude of wake field AWF increases and its resonance state 3 ¼ 0 shifts to the lower value of nB3max . 4.2. Effect of length width L and time duration t ¼0 Based on Eq. (41), LBmax depends on n3 and o. Fig. 2 presents the amplitude of wake field AWF in terms of L for l ¼820 nm, I3 ¼ 3:4  1022 W=m2 and different values of n3 . It is shown that the wake field has the maximum values Amax WF ¼ 311:99 GV=m,

Fig. 2. (a) Diagrams of amplitude of wake field AWF in terms of L for I3 ¼ 3:4  1022 W=m2 , l ¼ 820 nm and the different values n3 ¼ 6:5  1024 m3 , 7.5  1024 m  3, 8.5  1024 m  3 and 9.5  1024 m  3 that are denoted by large to small dashed curves, respectively. (b) Diagrams of amplitude of wake field AWF in terms of in terms of L and n3 for l ¼ 820 nm and I3 ¼ 3:4  1022 W=m2 .

¼0 335.334 GV/m, 357.206 GV/m and 377.862 GV/m at LBmax ¼ 14:7264 mm, 13.7054 mm, 12.8701 mm and 12.1702 mm, for the different values of n3 , 6.5  1024 m  3, 7.5  1024m3, 8.5  1024 m  3 and 9.5  1024 m  3, respectively. It is observed that with increasing n3 , value of Amax WF increases and its resonance ¼0 position shifts to the lower value of LBmax . In Fig. 2b, a three dimensional diagram of AWF is plotted in terms of L and n3 for the same values of l and I3 . The results are such as results of the diagrams of Figs. 1 and 3a. By using the relation between L and time duration t of laser pulse, we obtain the wake field EWF as a function of time duration t. In Fig. 3a, amplitude of wake field AWF is plotted in terms of t for l ¼820 nm and different values n3 . It is observed that the wake field has the maximum values Amax WF ¼ 469:567 GV=m, 453.255 GV/m, 3 ¼ 0 438.042 GV/m and 423.821 GV/m at nB3max ¼ 6:4516  1024 m3 , 6.01274  1024 m  3, 5.61723  1024 m  3 and 5.25953  1024 m  3, for the different values of tmax ¼ 28 fs, 29 fs, 30 fs and 31 fs, respectively. The results are such as results of L. Fig. 3b presents the amplitude of wake field AWF in terms of t for the same above values l and I3 and different values of n3 . It is shown that the wake field has the maximum values Emax WF ¼ 550:571 GV=m, 591.765 GV/m, ¼0 630.363 GV/m and 666.816 GV/m at tBmax ¼ 19:6868 fs, 18.3274 fs, 17.2156 fs and 16.2843 fs, for the different values n3 , 6.5  1024 m  3,

H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

617

25000 L=21µm

450

L=22µm

20000

L=23µm A wf (Gv/m)

A wf (Gv/m)

400 τ=28fs

350

L=24µm 15000 10000

τ=29fs

300

5000

τ=30fs τ=31fs

250

0 0.0

2

4

6

8

10

0.2

0.4

0.6

n o ×1024 (m − 3 )

0.8

1.0

p

12

Fig. 4. Diagram of amplitude of wake field AWF in terms of op =o for I3 ¼ 3:4  1022 W=m2 , n3 ¼ 1:5  1025 m3 , and different L¼24 mm, 25 mm, 26 mm and 26 mm.

0.030 500 n o = 6.5×1024 m − 3

0.025

n o = 7.5×1024 m − 3

400

0.020

n o = 8.5×1024 m − 3

A wf (Gv/m)

A wf (Gv/ m)

600

n o = 9.5×1024 m − 3

300

0.015

Io = 3.4×1018 W / m 2 B o = 0T

0.010 10

15

20

25

30

B o = 10T

τ(fs)

0.005

Fig. 3. (a) Diagrams of amplitude of wake field AWF versus n3 for different values 28 fs, 29 fs, 30 fs and 31 fs (b) Diagrams of amplitude of wake field AWF in terms of t for different values n3 ¼ 6:5  1024 m3 , 7.5  1024 m  3, 8.5  1024 m  3 and 9.5  1024 m  3 that are denoted by large to small dashed curves, respectively, and I3 ¼ 3:4  1022 W=m2 , l ¼820 nm.

0

2

4

6

n o ×10

24

8

10

12

8

10

12

(m − 3 )

250

200

A wf (Gv/m)

depends on o and L. Fig. 4 shows the diagram of wake nmax 3 field in terms of op =o by assuming I3 ¼ 3:4  1022 W=m2 , n3 ¼ 7:5  1024 m-3 and different L, 21 mm, 22 mm, 23 mm and 24 mm. It is observed that the amplitude of wake field AWF has the maximum values Amax WF ¼ 2485:84 GV=m, 1821.01 GV/m, 1323.45 GV/m and 953.998 GV/m at the resonant positions op/o9max ¼0.520705, 0.50384, 0.487856 and 0.472707, respectively. It is observed that with increasing o, magnitude of Emax WF increases and its resonant position shifts to the lower value of op/o9max.

B o = 30T

0.000

7.5  1024 m  3, 8.5  1024 m  3 and 9.5  1024 m  3, respectively. It is observed that with increasing n3 , value of Emax WF reduces and its ¼0 resonance position shifts to the lower value of tBmax . 4.3. Effect of the frequency of laser pulse

B o = 20T

150 Io =3.4 ×1022 W /m 2

100

Bo =0T Bo =10T

50

Bo =20T Bo =30T

0 0

2

4

6

n o ×10

24

(m − 3 )

4.4. Effect of the magnetic field

Fig. 5. Diagrams of amplitude of wake field AWF versus n3 for l ¼ 0.820 mm, L¼ 23 mm and different values B3 and two values of intensities, (a): I3 ¼ 3:4  1019 W=m2 and (b):6  1022 W=m2 .

In Fig. 5a and b, amplitude of wake field AWF is plotted in terms of n3 for l ¼0.820 mm, L¼23 mm and different values B3 and two values of intensities, I3 ¼ 3:4  1019 W=m2 and 6  1022 W/m2, respectively. It is calculated that the wake field has the maximum B3 ¼ 10T 3 ¼ 0 3 ¼ 20T values ABWFmax ¼ 0:0242 GV=m, AWFmax ¼ 0:0257 GV=m, ABWFmax ¼ B3 ¼ 30T B3 ¼ 0 0:0274 GV=m and AWFmax ¼ 0:0291 GV=m at n3max ¼ 5:34 B3 ¼ 10T B3 ¼ 20T 1024 m-3 , n3max ¼ 5:034  1024 m-3 , n3max ¼ 4:77  1024 m3 24 3 B3 ¼ 30T and n3max ¼ 4:54068  10 m , respectively for I3 ¼ 3:4  1019

W=m2 (Fig. 5a). When it is chosen I3 ¼ 6  1019 W=m2 , the wake field B3 ¼ 0 3 ¼ 10T has the maximum values AWFmax ¼ 242:236 GV=m, ABWFmax ¼ B3 ¼ 20T B3 ¼ 30T 242:385 GV=m, AWFmax ¼ 242:534 GV=m, AWFmax ¼ 242:683 GV=m B3 ¼ 10T B3 ¼ 20T 3 ¼ 0 at nB3max ¼ 5:341  1024 m3 , n3max ¼ 5:338  1024 m3 , n3max ¼ 24 3 24 3 B3 ¼ 30T 5:33516  10 m and n3max ¼ 5:332  10 m , respectively. It is seen that the wake field the effect of magnetic field is neglectable when intensity increases and the maximum amplitude of walk field is at lower L and n3 Fig. 6.

618

H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

Io =2×1022 W /m 2 Io =3×1022 W /m 2

200

Io =4×1022 W /m 2

E wf (Gv/m)

100

0

−100

−200

0

10

20

30

40

50

(µm)

Ep=220mj

300

Ep=270mj Ep=320mj

200

E wf (Gv/m)

100 0 −100 −200 −300 0

10

20

30

40

50

60

(µm) Fig. 6. Diagrams of (a) wake field and (b) electron energy gain in terms of coordinate x for l ¼ 0.820 mm, L¼ 23 mm, n3 ¼ 7:5  1024 m3 and for three different: (a) intensities I3 ¼ 3  1018 W=m2 , 4  1018 W/m2and 5  1018 W/m2(b) pulse energies: Ep ¼ 220 MJ, Ep ¼270 MJ and Ep ¼320 MJ that are denoted by solid, dashed and dotted curves, respectively.

4.5. Effect of intensity and energy of laser pulse Fig. 4 show diagram the amplitude of wake field AWF in terms of x at l ¼0.82 mm, L¼23 mm and n3 ¼ 4:34349  1025 m3 by assuming that the intensity has the three different values: I3 ¼ 1  1022 W=m2 , 2  1022 W=m2 and 3  1022 W=m2 . When it is compared three graphs, we realize that the wake field is being larger, whenever intensity and consequently the energy of laser pulse increases. The Eqs. (36) and (37) also show that the A1 is proportional with pffiffiffiffiffi I3 and Ep, respectively and A2 are proportional pffiffiffiffi with I3 and Ep , respectively. 4.6. The electron energy gain The electron energy gain is obtained by the Eq. (48). Fig. 4a shows diagram electron energy gain in terms of n3 for l ¼0.82 mm, L¼23 mm and I3 ¼ 3:4  1022 W=m2 . It is shown that the electron energy gain is maximum for special n3 . In Fig. 4b, the electron energy gain is plotted in terms of I3 for l ¼0.82 mm, L¼23 mm and n3 ¼ 7:5  1024 m3 . It is observed that the electron energy gain increases with increase of I3 . The Eqs. (36) and (37) also show that

pffiffiffiffi the A1 is proportional with I3 and A2 is proportional with I3 , and Eqs. (47) and (48) also show the electron energy gain is proportional A1 and A2. It is observed that the results are consistent with the results of the experiments that are performed by Faure et al. [13], EWF Z 100 GV=m and DW G Z100 MeV for I3 ¼ 3:4  1022 W=m2 , l ¼820 nm and n3 ¼ 7:5  1024 m3 (Fig. 7).

5. Conclusion In this paper, the general equation for the wake potential, which can directly be used for different envelopes of optical laser pulse, is obtained. The analytical solutions of the equation for the Gaussian short-intense laser pulse have shown that the wake field depends on intensity, length, frequency and energy of pulse, the electron number density of plasma and the external magnetic field. It is shown that for special values of the electron number density of plasma, frequency, length L and time duration t of laser pulse, the electric wake field is maximum. It is shown that the results are consistent with the results of the experiments that are performed.

H.R. Askari, A. Shahidani / Optics & Laser Technology 45 (2013) 613–619

a

References

250

Δ WG (Mev)

200 150 100 50 0 0

5

10

15 n ×10

24

b

(m

20 −3

25

30

)

200

150

Δ WG (Mev )

619

100

50

0 0

500

1000

1500

2000

2500

3000

I0 ×1019 (m− 3) Fig. 7. (a) Diagrams of the electron energy gain in terms of coordinate n3 for l ¼0.820 mm, L¼ 23 mm and I3 ¼ 3:4  1022 W=m2 (b) Diagrams of the electron energy gain in terms of coordinate I3 for l ¼ 0.820 mm, L¼ 23 mm and n3 ¼ 7:5  1024 m3 .

[1] Katsouleas T, Joshi C, Mori WB. Plasma physics with GeV electron beams, comments on plasma physics and controlled fusion. Comments on Modern Physics 1999;1:99. [2] Lin Hai, Xu Zhizhan, Chen Li-Ming, Usami JCKieffer S, Ohsawa Y. Laser wakefield and self-modulation of driving pulse. Phys Plasmas 2003;10:3371. [3] Mangles S, et al. Mono – energetic beams of relativistic electrons from intense laser – plasma interactions. Nature 2004;431:535–8. [4] Geddes CGR. High – quality electron beems from a laser wakefield accelerator using plasma – channal guiding. Nature 2004;431:538–41. [5] Faure J, et al. A laser – plasma accelerator producing monoenergitic electron beams. Nature 2004;431:541–4. [6] Huang C, et al. QUICKPIC: A highly efficient particle – in – cell code for modeling wakefield acceleration in plasmas. Journal of Computational Physics 2006;217:657–79. [7] Kingham RJ, Bell AR. Enhanced wakefields for the 1D laser wakefield accelerator. Physics Review Letter 1997;79:4810–3. [8] Andreev NE, Chegotov MV, Veisman ME. Wakefield generation by elliptically polarized femtosecond laser pulse in ionizing gases. IEEE Transactions on Plasma Science 2000;28:1098–105. ˜ ano JR, Hubbard RF, Ting A, Moore CI, et al. Wakefield [9] Sprangle P, Hafizi B, Pen generation and GeV acceleration in tapered plasma channels. Physical Review E 2001;63:056405. [10] Malik Hitendra K, Kumar Sandeep, Nishida Yasushi. Electron acceleration by laser produced wake field: Pulse shape effect. Optics Communications 2007;280:417–23. [11] Bittencourt JA. Fandementals of Plasma Physics. Suo Paulo, Brazil: Research Scintist Professor, Institute for Space Research (INPE); 1986. [12] Jakson JD. Classical Electrodynamics. Reprint of the 3rd, John Wiley & Sons; 1999 p. 320. [13] Faure J, et al. Controlled injection and acceleration of electrons in plasma wakefields by colliding laser pulses. Nature 2006;444:737–9.