Eur. Phys. J. D (2014) 68: 57DOI: 10.1140/epjd/e2014-40643-4

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Effects of relativistic and ponderomotive nonlineartieson the beat wave generation of electron plasma waveand particle acceleration in non-paraxial region

Priyanka Rawat1, Ram Kishor Singh2, Ram Pal Sharma2, and Gunjan Purohit1,a

1 Department of Physics, DAV (PG) College, Dehradun, 248001 Uttarakhand, India2 Centre for Energy Studies, Indian Institute of Technology, 110016 New Delhi, India

Received 22 October 2013 / Received in final form 5 December 2013Published online 21 March 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. In this communication, the combined effect of relativistic and ponderomotive nonlinearitieson the generation of electron plasma wave by cross focusing of two intense laser beams at differencefrequency (Δω ≈ ω1 − ω2 ≈ ωp) and acceleration of electrons in laser produced homogeneous plasmais analysed in the non-paraxial region. On account of these nonlinearities, two laser beams affect thedynamics of each other, and cross focusing takes place. It is observed that the focusing of laser beamsbecomes fast in the non-paraxial region by expanding the eikonal and other relevant quantities up to thefourth power of the radial distance (r). Modified coupled equations for the beam width of laser beams,electric field amplitude of the excited electron plasma wave and energy gain at beat wave frequency arederived, when relativistic and ponderomotive nonlinearities are operative. These coupled equations aresolved analytically and numerically to study the cross focusing of two intense laser beams in plasma andits effect on the variation of the amplitude of the electron plasma wave and energy gain. It is observedfrom the results that both nonlinearities significantly affect the amplitude of plasma wave excitation andparticle acceleration in the non-paraxial region in comparison to the paraxial region.

1 Introduction

In laser plasma interaction, the generation of large am-plitude electron plasma wave by high intensity lasers(1019–1022 W/cm2) is a subject of interest because ofits potential use for ultrahigh gradient electron acceler-ation [1,2]. At such intensities, motion of electrons be-comes fully relativistic and accelerating field is very high(>1 GV/cm). High intensity lasers can be used to drivelarge amplitude relativistic plasma waves, which can ac-celerate charged particle to high energies. Various schemesare operative for particle acceleration by lasers such aslaser beat wave accelerator (LBWA), laser wake field ac-celerator (LWFA), etc. In LBWA scheme, a large ampli-tude electron plasma wave (with phase velocity close tothe speed of light) is resonantly excited by beating twointense lasers with a frequency difference equal to theplasma frequency [3,4]. The charge separation associatedwith the excited electron plasma wave produces an elec-tric field up to several GV/cm due to the resonance effect,if the plasma frequency is close to the frequency differ-ence between the two laser beams. The energy from theexcited plasma wave is transferred to the plasma parti-cle by Landau damping if the plasma is collisionless andhence the acceleration of particles takes place [5]. In beat

a e-mail: purohit gunjan@yahoo.com

wave scheme, electron plasma wave can attain large am-plitude under the resonance condition, however, in thisprocess there is a problem of detuning of resonance condi-tion, which is attributed to the modified plasma frequency(ωp0 ∝ √

1/me) due to the change in electron mass be-cause of their relativistic speeds in very large amplitudeof the wakefield. In this process, focusing of the lasersbeams is essential to avoid an unacceptable loss of laserenergy from the acceleration region. The behaviour of theone laser beam is affected by the presence of second laserbeam in the plasma (i.e. cross focusing of the laser beams)and one can control the focusing/defocusing of laser beamsby choosing the parameters of another beam [6,7]. An im-portant consideration in the beat wave scheme is to havesufficiently intense lasers such that the time to reach sat-uration is short compared to the ion plasma period [8].The efficiency of beat wave accelerator is strongly depen-dent on the saturation amplitude level reached by the ex-cited plasma wave as well as effects of numerous plasmanonlinearities.

There has been lot of experimental and theoretical re-search work reported by various researchers on the accel-eration of charged particles in plasmas [9–12]. The pio-neering work on laser plasma accelerators based on largeamplitude relativistic plasma waves generated by lasers,first proposed by Tajima and Dawson in 1979 [1]. Ghizzo

Page 2 of 9 Eur. Phys. J. D (2014) 68: 57

et al. [13] developed a Hilbert-Vlasov code to study of theplasma beat wave accelerator with high radius of driverfrequency to plasma frequency. Plasma wave excitationby a resonant laser beat wave has been first time ob-served by Clayton et al. [4] and this was extended inthe form of beat wave accelerated electrons by Ebrahimet al. [14]. Kitagawa et al. [15] observed beat wave excitedplasma wave and energetic electrons with the energy ofmore than 10 MeV. The electric field of the excited plasmawave was more than 1 GV/m. Clayton et al. [10] experi-mentally demonstrated that high gradient acceleration ofexternally injected 2.1 MeV electrons by a laser beat wavedriven relativistic plasma wave. The corresponding gradi-ent was 0.7 GV/cm over a length of 1 cm. Laser-plasmaaccelerators are capable of producing beams of 1 GeV elec-tron energy over 3 cm of acceleration length [16]. Parti-cle acceleration by relativistic electron plasma waves hasbeen demonstrated in a number of experiments produc-ing more than 100 MeV electron beams in distances ofabout 1 mm [16–19]. The resulting accelerating field ashigh as 1 GV/cm has been achieved in these experiments.Recently, different theoretical processes involve the devel-opment of experimental techniques in laser plasma accel-erators is reported in the literature [20–23].

The generation of electron plasma wave by beat waveprocess and particle acceleration has been theoreticallyand experimentally studied in great detail by earlier re-searchers [15,24–28], by taking relativistic/ponderomotivenonlinearity separately. However, not much work has beenreported when both relativistic, as well as ponderomotivenonlinearities are present simultaneously and such stud-ies have also been carried out within the framework ofnear axis approximation. Some theoretical studies on non-linear theory of nonparaxial laser propagation have beenreported in the plasma channel in the adiabatic, as wellas the non-adiabatic limit [29,30]. In laser plasma inter-action, both relativistic and ponderomotive nonlinearitiesare simultaneously arise depending on the time scale ofthe pulse at high laser power flux. Hora [31] developed thetheory of relativistic self focusing of laser beam in plasmasand reported that self focusing of laser beam in a plasmais stronger due to the combined contribution of both rel-ativistic and ponderomotive nonlinearities. Sun et al. [32]analysed self focusing of intense laser beam in plasma inthe presence of relativistic and ponderomotive nonlineari-ties. Mori et al. [33] presented the evaluation of self focus-ing of intense electromagnetic waves in plasmas using twodimensional particles in cell periodic simulations in rela-tivistic ponderomotive regime. Ming-Ping et al. [34] anal-ysed the effect of relativistic and ponderomotive nonlin-earities on an intense laser pulse propagation in a plasmachannel and found that the laser focusing is released bythe coupling of relativistic and ponderomotive nonlinear-ities. Xiongping et al. [35] developed non-paraxial theoryof relativistic filamentation of intense laser beam in inho-mogeneous plasma and observed that relativistic nonlin-earity has a decisive effect on the formation of beam fil-amentation in intense laser-plasma interaction comparedwith the ponderomotive effects. Recently, Patil et al. [36]

analysed the combined effect of relativistic and pondero-motive self focusing of laser beam propagation in a plasma.It is observed that the ponderomotive self focusing signif-icantly contributes in the relativistic self-focusing of thelaser beam. Gupta et al. [37] reported the combined ef-fect of both relativistic and ponderomotive nonlinearitieson the excitation of electron plasma wave by cross focus-ing of two laser beams and particle acceleration processin paraxial region. Sharma and Chauhan [38] investigatedthe effect of cross focusing of two laser beams on plasmawave excitation and particle acceleration in non-paraxialregion, taking into account only the relativistic nonlinear-ity. Excitation of an upper hybrid wave by two intenselaser beams and particle acceleration in paraxial regionwere analysed by Purohit et al. [39] in which both rela-tivistic and ponderomotive nonlinearities are simultane-ously operative.

It may be of interest to analyse the generation of elec-tron plasma wave by cross focusing of two intense laserbeams and particle acceleration in non-paraxial region,when relativistic and ponderomotive nonlinearities are op-erative. It is in this context, this study presents non-paraxial theory of the propagation of two intense laserbeams (in short duration τpe < τ < τpi, where τ is thelaser pulse duration, τpi is the ion plasma period, andτpe is the electron time period) in laser produced plasmaand the generation of electron plasma wave at the beatfrequency (Δω = ω1 − ω2). In this case electrons are ex-pelled from the high intensity region by a ponderomotiveforce whereas ions are not expelled due to their inertia andthe nonlinearity in the dielectric constant of the plasmacomes by electron mass variation due to laser intensitiesand due to changes in electron density on account of theponderomotive force [40–42]. It is due to the limitations ofthe paraxial theory [43], we have considered non-paraxialregion in which the higher order terms in the expansionof dielectric constant and eikonal were taken into accounti.e. the laser intensity profile and other relevant quanti-ties of plasma were expended up to the 4th power of ra-dial distance (r). The electric field of the excited electronplasma wave at difference frequency and the energy of theelectron accelerated by the electron plasma wave throughbeat wave process is calculated. The nonlinear evaluationof dielectric constant, beamwidth and intensity of laserbeams, beat wave generation of electron plasma wave andthe acceleration of electrons are studied by using the ex-tended paraxial ray approximation given by Akamanovet al. [43] and developed by Sodha et al. [44], as well asWKB approximation.

In Section 2, we have developed an expression for theeffective dielectric constant of the plasma and derived thedifferential equation governing the nature of the dimen-sionless beam width parameters of two co-propagatinglaser beams in non-paraxial region when relativistic andponderomotive nonlinearities are operative. We have de-rived the electron plasma wave excitation equation at beatwave frequency and electron acceleration in Section 3. Theimportant results and conclusions based on the current in-vestigation are presented in Sections 4 and 5, respectively.

Eur. Phys. J. D (2014) 68: 57 Page 3 of 9

2 Analytical formulation

2.1 Nonlinear dielectric constant of plasma

Consider the propagation of two co-axial Gaussian laserbeams of frequencies ω1 and ω2 along the z-direction inhomogeneous collisionless unmagnetized plasma. The ini-tial intensity distribution of the beams are given by:

E1E∗1

∣∣∣z=0

= E210 exp

(

− r2

r21f

21

)

,

E2E∗2

∣∣∣z=0

= E220 exp

(

− r2

r22f

22

)

, (1)

where r is the radial coordinate of the cylindrical coordi-nate system, r1 and r2 are their initial beam widths andf1 and f2 are dimensionless beam width parameters at z.

The dielectric constant of the plasma is given by:

ε01,02 = 1 − ω2p0

ω21,2

(2)

where ωp0 is the electron plasma frequency given by ω2p0 =

4πn0e2/m0γ (where e is the charge of an electron m0 is its

rest mass and n0 is the density of plasma electron in theabsence of laser beam) and relativistic factor is given by:

γ = (1 + α1E1E∗1 + α2E2E

∗2 )

12 ,

where

α1,2 =e2

m20c

2ω21,2

. (3)

Equation (3) is valid when there is no change in plasmadensity. The relativistic ponderomotive force [40–42] isgiven by:

Fp = −m0c2∇ (γ − 1) . (4)

Using the electron continuity equation and the currentdensity equation for second order correction in the electrondensity equation [40,41], the modified electron density (n)because of relativistic-ponderomotive force is

n = n0 + n2 = n0 +c2n0

ω2p0

(

∇2γ − (∇γ)2

γ

)

,

where n2 is the second order electron density and

n

n0= 1 +

2c2

ω2p0

[

− 1γ

(a1

r21f

41

+a2

r22f

42

)

− r2

γ3

(a1

r41f

81

+a2

r42f

82

)

× e−r2

(

1r21f2

1+ 1

r22f2

2

)

+r2

γ

(a1

r41f

61

+a2

r42f

62

)]

× e−r2

(

1r21f2

1+ 1

r22f2

2

)

.

The intensity dependent effective dielectric constant of theplasma at frequencies ω1 and ω2 (denoted by ε1 and ε2,respectively) is given by:

ε1,2 = ε01, 02 + φ1,2 (E1E∗1 , E2E

∗2 ) , (5)

where the nonlinear part of the dielectric constant isgiven by:

φ1,2 (E1E∗1 , E2E

∗2 ) =

ω2P0

ω21,2

(

1 − n

n0γ

)

.

To investigate the cross focusing of the two laser beams,we expand the dielectric constant in equation (5) aroundr = 0 in non-paraxial region by Taylor’s expansion

ε1,2 = εf1,2 + γ1,2r2 + η1,2

r4

2,

where

εf1,2 = ε01,02 +ω2

p0

ω21,2

[

1 +(

−1 +a1

γ f41 r2

1k2p

+a2

γ f42 r2

2k2p

)

×(

1 +a1

f21

+a2

f22

)−1/2]

,

γ1,2 =ω2

p0

ω21,2

{X

2γ3− 2

(a1

r21f

41

+a2

r22f

42

)

×[

1γ2k2

p

(1

r21f

21

+1

r22f

22

)

+X

γ4k2p

]

+2

γ4k2p

(a1

r41f

81

+a2

r42f

82

)

− 2γ2k2

p

(a1

r41f

61

+a2

r42f

62

)}

,

η1,2 = − ω2p0

ω21,2

{Y

2γ3− 3X2

4γ5+ 2

(a1

r21f

41

+a2

r22f

42

)

×[

1γ4k2

p

(1

r21f

21

+1

r22f

22

)2

− Y

γ4k2P

+2X2

γ6k2p

]

−(

a1

r41f8

1

+a2

r42f

82

)

×[

4γ4k2

p

(1

r21f

21

+1

r22f

22

)

+4X

γ6k2p

]

+(

a1

r21f6

1

+a2

r62f

62

)

×[

2γ2k2

p

(1

r21f

21

+1

r22f

22

)

+2X

γ4k2P

]}

,

where

X =[a1 (α01 − 1)

r21f

21

+a2 (α02 − 1)

r22f

22

]

and

Y =[

a1

f21

(2α12 − 2α01 + 1

r41f

41

)

+a2

f22

(2α22 − 2α02 + 1

r42f

42

)]

where a1 = α1E210 and a2 = α2E

220 or a1,2 = αA2

10,20 aresquare of dimensionless vector potential, α = e2/m2

0c4 and

k2p = ω2

p0/c2.

2.2 Cross focusing of two laser beams

The wave equation governing the electric vectors of thetwo laser beams in plasma can be written as:

∂2E1,2

∂Z2+

1r

∂E1,2

∂r+

∂2E1,2

∂r2+

ω21,2

c2ε1,2E1,2 = 0. (6)

Page 4 of 9 Eur. Phys. J. D (2014) 68: 57

In writing equation (6), we have neglected the ∇(∇.E)term which is justified as long as (ω2

p0/ω21,2) (1/ε1,2). In

ε1,2 � 1 [44]. Assuming the variation of the electric fieldsto be equal to

E1,2 = A1,2(x, y, z)e−ik1, 2z.

The wave equation becomes

− k21,2A1,2 − 2ik1,2A1,2 +

(1r

∂

∂r+

∂

∂r2

)

A1,2

+ω2

12

c2ε1,2A1,2 = 0 (7)

A1,2 is a complex function of space. Further, it is assumedthat the variation of A1,2 may be presented by [43]:

A1,2 = A01,2(r, z)e−ik1,2S1,2(r,z), (8)

where A01,2 and S1,2 are the real function of space.Substituting equation (8) into equation (7) and sepa-

rating real and imaginary part of resulting equation, thefollowing set of equation is obtained:

The real part from equation (7)

1 + 2∂S12

∂z+

(∂S12

∂r

)2

=1

k21,2A01,2

×(

∂2A012

∂r2

+1r

∂A012

∂r

)

+ω2

12ε1,2

c2k21,2

, (9)

where

A201,2 =

(

1 +α01,2r

2

r21,2f

21,2

+α21,2r

4

r41,2f

41,2

) (

E21,2

f21,2

)

e

(

− r2

r21,2f2

1,2

)

(10)are the laser beam intensities, f1,2 are the dimensionlessbeam width parameter for beam 1 and 2, respectively, and

S1,2 =r2

2f1,2

df1,2

dz+

r4

r41,2

S21,2. (11)

Equation (10) gives the intensity profile of the laser beamsin the plasma along the radial direction. The intensityprofile of both laser beams depends on the beam widthparameters f1,2 and the coefficient (α01,2 and α21,2) of r2

and r4 in the non-paraxial region.By substituting equations (10), (11) and (5) into equa-

tion (9) and equating the coefficient of r2 on both sides ofthe resulting equating, we obtained the governing equa-tion of beam width parameter

d2f1,2

dz2=

1k21,2f

41,2r

41,2

+ω2

p0f1,2γ1,2

c2k21,2

+8α21,2 − 3α2

01,2 − 2α01,2

k21,2f

41,2r

41,2

. (12)

In similar way, by equating the coefficient of r4 on bothsides of the resulting equation we obtained the following

equation:

∂S21,2

∂z=

ω2p0r

41,2η1,2

4c2k21,2

− (7α01,2α21,2 − 2α201,2 − α3

01,2)k21,2r

21,2f

61,2

− 4S21,2

f1,2

df1,2

dz.

(13)

The imaginary part of equation (7) is given by:

∂A201,2

∂z+ A2

01,2

(∂2S1,2

∂z+

1r

∂S1,2

∂r

)

+∂S1,2

∂r

∂A201,2

∂r= 0.

(14)By substituting equations (10) and (11) into equation (14)and equating the coefficient of r2 on both sides of the re-sulting equation, we obtained the equations for the coef-ficient α01,2,

∂α01,2

∂z= −16S21,2f

21,2

r21,2

. (15)

In similar way, by equating the coefficient of r4 gives theequation for the coefficient α21,2,

∂α21,2

∂z= 8 (1 − 3α01,2)

S21,2f21,2

r21,2

. (16)

Equation (12) determines the focusing/defocusing of thelaser beams, along with the distance of propagation in theplasma when relativistic and ponderomotive nonlinearitiesare operative. In order to have a numerical appreciationof the cross focusing in the non-paraxial region and theeffect of the change of parameters of the plasma and laserbeams, we have performed the numerical computation ofequations (12), (13), (15) and (16). We have also solvedthe coupled equations and obtained the numerical resultsfor typical laser-plasma parameters.

3 Generation of beat plasma waveat difference frequency

To study the effect of relativistic and ponderomotive non-linearities on the generation of the plasma wave by thebeat wave process in non-paraxial region, we start withthe following equation:

(i) The continuity equation

∂

∂tN + ∇ · (N.V ) = 0 (17)

(ii) The momentum equation

m

[∂

∂tV + (V.∇) V

]

= −eE − e

cV × B − 2ΓemV

− 3KB

NTe∇N (18)

Eur. Phys. J. D (2014) 68: 57 Page 5 of 9

(iii) Poisson’s equation

∇E = − 4πeN (19)

where N is the total electron density, E is the sum ofthe electric field vectors of the electromagnetic waves, andthe self-consistent field V is the sum of drift velocities ofthe electron in the electromagnetic field and self consistentfield; other symbols have their usual meanings.

Using equations (17)–(19), we obtained the fol-lowing equation governing the plasma wave in hotplasma [25,26,44]

∂2N

∂t2+ 2Γe

∂N

∂t− V 2

th∇2N − e

m∇ · (NE)

= ∇ ·[N

2∇(V.V ) − V

∂N

∂t

]

, (20)

where 2Γe is the Landau damping factor given byKrall [45], and Vth is the thermal speed of the electron.Therefore, the equation for the electron wave at the dif-ference frequency (Δω = ω1 − ω2) reduce to:

− (ω1 − ω2)2N1 + 2iΓ (ω1 − ω2)N1 − V 2th∇2N1

+

(

ω2p0

γ

n

n0

)

N1∼= 1

4N0∇2V1 · V ∗

2 , (21)

where N1 is the component of electron density oscillatingat frequency Δω. The drift velocities of electron in thepump field at the frequency ω1,2 are

V1,2 =eE1,2

m0iω1γ.

Therefore,

V1.V∗2 = H1

G121 G

122

γ2e−H2r2

e−i[(k1−k2)z+(k1S1−k2S2)],

H1 =e2

m20ω1ω2

E10E20

f1f2, H2 =

12r2

1f21

+1

2r22f

22

,

G1 = 1+α01r

2

r21f

21

+α21r

4

r21f

21

, and G2 = 1+α02r

2

r22f

22

+α22r

4

r42f

42

.

(22)

Equation (21) contains two plasma waves (both at differ-ent frequency). The first one is supported by hot plasmaand the second by the source term at the different fre-quency. If ωp0/Δω � 1, the phase velocity is almost equalto the thermal velocity of the electron and Landau damp-ing occurs. But as ωp0/Δω ≈ 1, the phase velocity isvery large as compared to the electron thermal velocityand Landau damping is negligible [24–26]. The solutionof equation (21) with in the WKB approximation can beexpressed as:

N1 = N10(r, z)e−i(kz+S) + N20(r, z)

× e−i[(k1−k2)z+(k1s1−k2s2)], (23)

where

k2 =

[

(ω1 − ω2)2 − ω2

p

]

V 2th

,

and N10 and N20 are the slowly varying real function ofthe space coordinate. Substituting N1 from equation (23)into equation (21), the equation for N10 and N20 canbe obtained by equating the coefficient of exp−i(kz + s)and exp−i[(k1 − k2)z + (k1s1 − k2s2)] in the resultingequation.

The equation for N20 is given by:

N20

N0= − H1e

−H2r2(G1G2)

12

4γ2(

Δω2 − Δk2V 2th − ω2

p0γ

nn0

)

×{[

Δk2 + 4H2 − 4r2H22

−(

G′1

2G1− G′

2

2G2

)2

+4H2r

2 − 1rγ2

∂γ2

∂r

]

+1

G1

[G′′

1 + G′1/r

2− 2rG′

1K2 − G′1

γ2

∂γ2

∂r

]

+1

G2

[G′′

2 + G′2/r

2− 2rG′

2H2 − G′2

γ2

∂γ2

∂r

]}

. (24)

Using Poisson’s equation one can obtain the electric vec-tor E (Δω) of the plasma wave excited at the differencefrequency.

E(Δω)

=im0ω

2p0H1e

−H2r2(G1G2)1/2e−i[(k1−k2)z+(k1S1−k2S)2]

4eΔkγ2(Δω2 − Δk2V 2th − ω2

p0γ

(nn0

)

×{[

Δk2 + 4H2 − 4r2H22 −

(G′

1

2G1− G′

2

2G2

)2

+4H2r

2 − 1rγ2

∂γ2

∂r

]

+1

G1

[G′′

1 + G′1/r

2− 2rG′

1H2 − G′1

γ2

∂γ2

∂r

]

+1

G2

[G′′

2 + G′2/r

2− 2rG′

2H2 − G′2

γ2

∂γ2

∂r

]}

, (25)

Page 6 of 9 Eur. Phys. J. D (2014) 68: 57

where

G′1 =

2α01r

r21f

21

+4α21r

3

r41f

41

, G′′1 =

2α01

r21f

21

+12α21r

2

r41f

41

,

G′2 =

2α02r

r22f

22

+4α22r

3

r42f

42

and G′′2 =

2α02

r22f

22

+12α22r

2

r42f

42

∂γ2

∂r=

a1

f21

[2(α01 − 1)r

r21f

21

+2(2α12 − α01)r3

r41f

41

− 2α12r5

r61f

61

]

× e− r2

r21f2

1 +a2

f22

[2(α02 − 1)r

r22f

22

+2(2α22 − α02)r3

r42f

42

−2α22r5

r62f

62

]

e− r2

r22f2

2 .

Equation (25) gives the expression for the electric vector ofthe excited plasma wave at the difference frequency (Δω)by the two laser beams in the non-paraxial region, whenthe self focusing of two laser beams and focusing of theplasma wave (supported by hot plasma) due to the modi-fication of the background electron concentration by pon-deromotive force and relativistic effects are taken intoaccount.

3.1 Charged particle acceleration

The excited electron plasma wave transfers its energy toaccelerate the electrons at the difference frequency of laserbeams. The equation governing electron momentum andenergy are [8]:

dp

dt= −eE(Δω)

anddγ

dt= −eE(Δω)V

m0c2,

where γ =√

1 + P 2/m20c2 is the energy gain by the elec-

tron or electron energy.This equation can be written as:

dγ

dz= −eE(Δω)

m0c2, (26)

where E (Δω) is given by equations (25) and (26) givesthe electron energy. Equation (26) has been solved nu-merically, where we have used f1 and f2 by equation (12).

4 Numerical results and discussion

In this study, we have investigated first the propagation oftwo intense laser beams at difference frequency in plasmain the non-paraxial region (α01,2 �= α21,2 �= 0) whenrelativistic and ponderomotive nonlinearities are opera-tive. When two laser beams simultaneously propagatingthrough the plasma, the density of plasma will be varyingdue to ponderomotive force and relativistic mass varia-tion. Cross focusing of laser beams takes place in plasma,

(a)

(b)

Fig. 1. (a) Variation of the beam width parameters (f1 and f2)with the normalized distance of propagation ξ(ξ = zc/ω1r

210)

for the fixed power of the first laser beam and different powersof second laser beam at ωp0 = 0.025ω1, when both relativisticand ponderomotive nonlinearities are operative. (b) Variationof the beam width parameters (f1 and f2) with the normalizeddistance of propagation ξ(ξ = zc/ω1r

210) for different plasma

frequency at (a1 = 2 and a2 = 3), when both relativistic andponderomotive nonlinearities are operative.

when the self focusing of one beam is affected by the pres-ence of another beam or the behaviour of f1 is governedby f2. The behaviour of cross focusing is influenced bythe coupling terms i.e. γ1,2 term in the powers of r2 andη1,2 in the powers of r4. We solve equation (12) for var-ious laser intensities, which are greater than the criticalvalue. The following set of the parameters has been usedin the numerical calculation: r1 = 15 μm, r2 = 20 μm,ω1 = 1.778 × 1015 rad/S, ω2 = 1.740 × 1015 rad/S, andωp0 = 0.02ω1, 0.025ω1 and 0.03ω1. For initial plane wavefronts of the beams, the initial conditions for f1,2 aref1,2 = 1 and df1,2/dz = 0, α01,2 = α21,2 = 0 and S21,2 = 0at z = 0. Laser intensities (I0) corresponding to normal-ized intensities (a1 = 2, a2 = 3, 4, 5) used in numeri-cal solution are 3.3 × 1018 W/cm2, 4.98 × 1018 W/cm2,6.65× 1018 W/cm2 and 8.31 × 1019 W/cm2, respectively.

Figure 1a depicts the variation of the beamwidth pa-rameter (f1,2) of the two laser beams for a fixed powerof first laser beam (a1 = 2) and different powers of thesecond laser beam (a2 = 3, 4 and 5). At z = 0 in equa-tion (12), dS21,2/dz is positive and α01,2 starts decreas-ing, however, α21,2 increases with the distance in z. Thecombined effect of these terms directly affects the focus-ing behaviour of two laser beams. It is observed that inthe non-paraxial region the focusing of the laser beamsalong the distance of propagation becomes faster in com-parison to paraxial region (α01,2 = α21,2 = 0), (resultsnot shown here) [38] due to the participation of off-axis

Eur. Phys. J. D (2014) 68: 57 Page 7 of 9

parts (α01,2 �= α21,2 �= 0). In paraxial region, it is notedthat the intensity of laser beams is maximum at r = 0along the distance of propagation as α01,2 = α21,2 = 0.While in non-paraxial region the laser intensity becomesminimum at r = 0, it assumes a ring structure or a split-ted beam profile due to the contribution of higher orderterms [46,47]. It is seen that the rate of focusing of secondlaser beam becomes slower and the beamwidth of laserbeam also decreases with the increase of intensity of thefirst laser beam. This is due to the contribution of thefirst laser beam in equation (12), which governs the beamwidth profile f2. Here, we can make a comparison betweenonly relativistic/ponderomotive case and combined effectof relativistic and ponderomotive nonlinearities on the fo-cusing of laser beams in paraxial and non-paraxial regions.It is observed that relativistic nonlinearity plays a mainrole in intense laser beam propagation properties and thefocusing of laser beam in a plasma is stronger due to thecombined contribution of both relativistic and pondero-motive nonlinearities in paraxial as well as non-paraxialregion [31]. The effect of an ultra intense laser pulse onthe propagation of an electron plasma wave is analysed byKumar et al. [48] in the relativistic ponderomotive regimeand observed that the intensity of beam self-focusing andelectron plasma wave are strengthened in comparison toonly relativistic or ponderomotive nonlinearity. A com-parison of laser beam propagation in paraxial and non-paraxial region in relativistic ponderomotive regime is alsoreported by Sharma [49], the nature of the results weresame as in our case. Gupta et al. [37] investigated plasmawave excitation by cross focusing of two laser beams inparaxial region, when both relativistic and ponderomotivenonlinearities are simultaneously operative and concludedthat cross-focusing as well as plasma wave generation atthe difference frequency significantly changed in contrastto only relativistic nonlinearity. The ranges of laser powerswere approximately same (1018 W/cm2) as in the presentstudy. Figure 1b illustrates the effect of plasma densityon the cross focusing process of two laser beams in theplasma. Both of the beams show oscillatory self focusingat a1 = 2 and a2 = 3. It is obvious that focusing of bothlaser beams is faster and the beamwidth of laser beams ishigher at higher plasma density. These two laser beams in-teract with each other and generate electron plasma wavein plasma.

In order to understand the dynamics of generation ofthe electron plasma wave by the beat wave process in thenon-paraxial region, we solve equation (25) numerically byusing same set of parameters used in Figures 1a and 1b.Equation (25) gives the behaviour of the electric field ofthe excited electron plasma wave at the plasma frequency(ωp0 = ω1 − ω2). The electric field of the excited electronplasma wave depends on the parameters of laser beamsand their beam width parameters and the coefficients of r2

and r4 (α01,2, α21,2). Figure 2a presents the variation ofthe normalized intensity of electron plasma wave againstthe normalized distance of propagation for a fixed powerof first laser beam and different powers of the second laserbeam. It is obvious from the figure that the intensity of

(a)

(b)

Fig. 2. (a) Variation of the normalized intensity of electronplasma wave (EPW) with the normalized distance of propaga-tion ξ(ξ = zc/ω1r

210) for the fixed power of the first laser beam

and different powers of second laser beam at ωp0 = 0.025ω1,when both relativistic and ponderomotive nonlinearities areoperative. (b) Variation of the normalized intensity of electronplasma wave (EPW) with the normalized distance of propaga-tion ξ(ξ = zc/ω1r

210) for different plasma frequency at (a1 = 2

and a2 = 3), when both relativistic and ponderomotive non-linearities are operative.

the electron plasma wave decreases in an oscillatory man-ner when the normalized distance increases. Similar be-haviour has been observed when we increase the powerof the second laser beam in relativistic and ponderomo-tive regime. This is due to the dependence of E (Δω) onthe coefficients of r2 and r4 (α01,2, α21,2) in non-paraxialregion. The overall effect is that the intensity of the ex-cited electron plasma wave decrease with the increase ofthe power of the second laser beam. This is on account ofthe fact that the ponderomotive force on electrons reducesthe background electron density; therefore the amplitudeof the excited electron plasma wave gets reduced [37]. Fig-ure 2b presents the effect of the variation in the plasmafrequency on the excited electron plasma wave for differ-ent power of two laser beams. The intensity of the excitedelectron plasma wave again shows oscillatory behaviourand decrease with the normalized distance of propaga-tion. It is observed that the intensity as well as focusingof the electron plasma wave increases with the increase ofplasma electron density.

Figures 3a and 3b, respectively presents the effect ofchange in the laser power and plasma frequency on the en-ergy gain of electrons (γ) at the beat frequency with thenormalized distance of propagation. From the results, itis clear that the energy gain increases with the increase ofpower of the second laser beam, as well as plasma electrondensity. Equation (26) indicate that the energy gain of the

Page 8 of 9 Eur. Phys. J. D (2014) 68: 57

(a)

(b)

Fig. 3. (a) Variation of the energy gain (γ) of the electronwith the normalized distance of propagation ξ(ξ = zc/ω1r

210)

for the fixed power of the first laser beam and different powersof second laser beam at ωp0 = 0.025ω1, when both relativis-tic and ponderomotive nonlinearities are operative. (b) Varia-tion of the energy gain (γ) of the electron with the normalizeddistance of propagation ξ (ξ = zc/ω1r

210) for different plasma

frequency at (a1 = 2 and a2 = 3), when both relativistic andponderomotive nonlinearities are operative.

electrons depend on the electric field of the excited elec-tron plasma wave, as well as coefficients of r2 and r4 (α01,2,α21,2) in non-paraxial region. Due to the contribution ofterms α01,2 and α21,2, the maximum energy gain increasesfrom on-axis to off-axis position.

5 Conclusions

In conclusion, we develop a theoretical model for beatwave generation of electron plasma wave by two intenselaser beams and the acceleration of electrons in non-paraxial region when relativistic and ponderomotive non-linearities are operative. It is observed that the focusingprocess becomes faster when the off-axial contribution istaken into account. This also results in a significant changein the generation of electron plasma wave at the differ-ence frequency as well as particle acceleration process.The amplitude of electron plasma wave is reduced, whenwe increase the intensity of second laser beam due to thecontribution of ponderomotive force. The energy gain bythe electrons is modified at different laser intensities andplasma frequency. The analytical model developed in thisstudy is more realistic as both relativistic and ponderomo-tive nonlinearities and off-axis contributions are taken intoaccount at high laser power flux. The results of presenttheory may be useful in understanding the saturation levelof plasma beat wave, which is essential for more energy

gain from the longitudinal electric field of the saturatedplasma beat wave that may be used in laser beat wavebased particle accelerators.

This work was supported by Uttarakhand Council of Sci-ence and Technology (UCOST), Government of Uttarakhand,Dehradun, India and the University Grant Commission(UGC), Government of India, New Delhi, India.

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