Optics & Laser Technology 44 (2012) 781–787

Contents lists available at SciVerse ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlastec

Stimulated Brillouin scattering of elliptical laser beam in collisionless plasma

Arvinder Singh n, Keshav Walia

Department of Physics, National Institute of Technology, Jalandhar, India

a r t i c l e i n f o

Article history:

Received 5 September 2011

Received in revised form

20 October 2011

Accepted 20 October 2011Available online 3 December 2011

Keywords:

Elliptical laser beam

Self-focusing

Back-reflectivity

92/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlastec.2011.10.028

esponding author. Tel.: þ91 9914142123; fa

ail address: arvinder6@lycos.com (A. Singh).

a b s t r a c t

This paper presents an investigation of self-focusing of elliptical laser beam in collisionless plasma and

its effect on stimulated Brillouin scattering. The pump beam interacts with a pre-excited ion-acoustic

wave leading to Brillouin back-scattered process. The transverse intensity gradient of a pump beam

generates a ponderomotive force, which modifies the background plasma density profile in a direction

transverse to pump beam axis. This modification in density effects the incident laser beam, ion-acoustic

wave and back-scattered beam. Non-linear differential equations for the beam width parameters of

pump laser beam, ion-acoustic wave and back-scattered beam are set up and solved numerically. It is

observed from the analysis that the focusing of waves enhances the SBS back-reflectivity.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The interaction of intense laser beams with plasma is a activefield of research due to its importance in laser driven fusion [1–4].It is well established that various laser-plasma instabilities likeself-focusing, harmonic generation, stimulated Raman scattering(SRS) and stimulated Brillouin scattering (SBS), etc. [5–12], comeinto existence during the interaction of laser pulses with plasmasand play significant role as far as the transfer of energy from laserto the plasma is concerned. The presence of these instabilitiesresults in the significant loss in the incident laser energy, whichleads to poor laser plasma coupling. These instabilities can alsomodify the intensity distribution, affecting the uniformity ofenergy deposition. In particular, SBS governs the amount of laserenergy that can be propagated over long distances throughplasma. Control of SBS is crucial in the context of laser-driveninertial confinement fusion (ICF). In SBS, the incident electro-magnetic wave (EM) resonantly decays into scattered EM waveand an ion-acoustic wave (IAW). The beating between theincident and Scattered EM waves reinforces the density perturba-tion, thus leading to unstable loop. As a result, SBS can lead toscattering of a significant portion of the incoming laser light andthus prevent an efficient coupling of the laser light with thetarget. Since, many laser matter interaction applications such asadvanced radiation sources, laser fusion, and relativistic non-linear optics depend critically on the amount of transmitted laserenergy through the plasma. Consequently, SBS is studied bothexperimentally and theoretically. In solid target experiments,energy reflectivities attributed to SBS have varied from 0 to 50%

ll rights reserved.

x: þ0181 2690320.

[13–16]. There is a vast difference between the reported results oftheory and experiments in spite of intensive research work doneon studies of SBS during the last two decades [17–20]. Thismismatch between the results of theory and experiment may bedue to the idealized theoretical assumptions made in the theory.Theoretical explanation of low reflectivity observed in large scalefusion experiments [21–25] is one of the main challenge fortheoretical researcher.

In most of the theoretical investigations on non-linear phe-nomena self-focusing and stimulated back scattering have beencarried out separately by ignoring the interplay among them. Inorder to understand the interplay among various instabilities, it isvery important to investigate the evolution of these instabilitiesin the nonlinear regime, where they co-exist and effect eachother. In light of considerable current interest in self-focusing andBrillouin scattering, a lot of work has already been done in thepast [26–30]. The fundamental works on the nonlinear propaga-tion of laser beam in different media have been carried out bytaking cylindrical gaussian beams [31,32]. Since many lasersystems produce a beam, which is more nearly elliptical thancircular in cross-section and it has been observed that the focusedelliptically polarized light for the semiconductors is crucial forcreation of photoinduced nonlinear optical effects [33]. It istherefore worthwhile to study this practical situation. So ourmotivation of present work is to study the effect of self-focusingof elliptical laser beam on the Brillouin scattering process incollisionless plasma.

In the present paper, Brillouin Scattering of a elliptical laserbeam from a collisionless plasma has been investigated. Thepump wave (o0, k0) interacts with pre-excited ion-acoustic waveðo,kÞ to generate a scattered wave ðo0�o, k0�kÞ. As a specificcase, back scattering for which kC2k0 has been discussed. Thepump beam exerts a ponderomotive force on the electrons, leading

A. Singh, K. Walia / Optics & Laser Technology 44 (2012) 781–787782

to redistribution of carriers and consequently, the pump beambecomes self-focused. The dispersion relation for ion-acoustic waveis also significantly modified. The phase velocity of the ion-acousticwave becomes minimum on the axis and increases away from it.Therefore, if appropriate conditions are satisfied, the ion-acousticwave may also get focused. Since the scattered intensity is propor-tional to the intensities of the pump and ion-acoustic wave, it istherefore expected that the self-focusing should lead to enhancedback-scattering.

In Section 2, solution of the wave equation for the pump waveis derived in the paraxial ray approximations. Differential equa-tions for the beam width parameters of the pump wave are alsoderived. In Section 3, the wave equation for the ion-acoustic waveis solved in the paraxial ray approximation and differentialequations for the beam width parameters of the ion-acousticwave are also derived. In Section 4, the wave equation for theback-scattered wave is solved in the paraxial ray approximationand differential equations for the beam width parameters of theback-scattered wave are also derived. Expression for the reflec-tivity ‘R’ of the back-scattered beam is also derived. Finally, adetailed discussion of the results is presented in Section 5.

2. Solution of wave equation for pump wave

Consider the propagation of high power elliptical gaussianlaser beam of frequency o0 and wave number k0 in a collisionlessplasma along z-axis. The transverse intensity distribution of anelliptical laser beam along the wave-front at z¼ 0 is given by

EiE%

i 9z ¼ 0 ¼ E200 exp �

x2

a20

�y2

b20

" #ð1Þ

where a0 and b0 are the initial dimensions of the laser beam atz¼ 0 in x and y directions respectively, Ei is the electric fieldvector of pump beam and E00 is the axial amplitude of the beam.

The transverse intensity gradient generates a ponderomotiveforce, which leads to modification in the background electrondensity (N0). Following Sodha et al. [32] the electron density Noe

in the presence of laser beam may be written as

N0e ¼N0 exp �3

4am

MEiE

%

i

� �ð2Þ

where the non-linearity parameter a is given by

a¼ e2M

6kBT0gm2o20

ð3Þ

where Noe is the modified electron density in the presence of laserbeam, N0 is the electron density in the absence of beam, e and m

are the charge and mass of plasma electrons. kB is Boltzmann’sconstant, g¼ 3 is the ratio of specific heats for electron gas, and T0

is the equilibrium plasma temperature, M is the mass of ion.The wave equation governing the electric field Ei of the pump

laser in plasma can be written as

r2Eiþo2

o

c21�

o2pN0e

o2oN0

" #Ei ¼ 0 ð4Þ

Now, following Akhmanov et al. [31] and Sodha et al. [32], thesolution of Ei can be written as

Ei ¼ E0 exp½iðo0t�k0ðS0þzÞÞ� ð5Þ

E2o ¼

E2oo

f 01f 02

� exp�x2

a2of 2

01

" #exp

�y2

b2of 2

02

" #ð6Þ

So ¼1

2x2 1

f 01

df 01

dzþ

1

2y2 1

f 02

df 02

dzþFoðzÞ ð7Þ

k0 ¼o0

cE1=2

0 ð8Þ

where E0 is the real function of x, y and z. S0 is the eikonal for themain beam, F0ðzÞ is a constant whose value will not be requiredexplicitly in further analysis. E0 is the linear part of the dielectricconstant, c is the speed of light, op is the plasma frequency, f 01

and f 02 corresponds to dimensionless beam width parameters ofpump beam and satisfy the following differential equations:

d22f 01

dz2¼

1

k20a4

0f 301

�o2

p

o20e0

�3

4am

ME2

00

� �exp �

3

4am

M

E200

f 01f 02

!1

a20f 2

01f 02

ð9Þ

d2f 02

dz2¼

1

k20b4

0f 302

�o2

p

o20e0

�3

4am

ME2

00

� �exp �

3

4am

M

E200

f 01f 02

!1

b20f 2

02f 01

ð10Þ

where f 01 ¼ f 02 ¼ 1 and df 01=dz¼ df 02=dz¼ 0 at z¼0. Eqs. (9)and (10) describe the change in the dimensionless beam widthparameters f 01 and f 02 of pump beam on account of the competi-tion between diffraction divergence terms and nonlinear refrac-tive terms as the beam propagates in the collisionless plasma.

3. Solution of wave equation for ion-acoustic wave

The laser beam interacts with the ion-acoustic wave and leadsto its excitation. To analyze the excitation process of ion-acousticwave in the presence of ponderomotive non-linearity. We startwith the following set of fluid equations [34].

Continuity equation:

@nis

@tþr � ðN0VisÞ ¼ 0 ð11Þ

Momentum equation:

@Vis

@tþgiv

2th

N0rnisþ2GiV is�

e

MEsi ¼ 0 ð12Þ

where nis is the perturbation in the ion density, Vis is the velocityof ion-fluid, vth is the ion-thermal velocity, gi is the ratio ofspecific heat of ion-gas, Gi is the landau damping factor of the ionwave, Esi is the electric field associated with the generated ion-acoustic wave, satisfying the poisson’s equation

r � Esi ¼�4peðnes�nisÞ ð13Þ

Where nes and nis corresponds to perturbations in the electronand ion densities, and are related to each other by followingequation:

nes ¼ nis 1þk2l2

d

N0e

N0

26643775�1

ð14Þ

where k is the propagation constant for ion-acoustic wave,

ld ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT0=4pN0e2

pis Debye length. The landau damping

coefficient Gi for IAW is given by [35] 2Gi ¼ k=ð1þk2l2dÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipkBTe=8M

p½ffiffiffiffiffiffiffiffiffiffiffim=M

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiTe=Ti

3p

expð�ðTe=TiÞ=ð1þk2l2dÞÞ�, where Te and Ti are the

electron and ion temperatures.Following standard techniques, equation for the space time

evolution of perturbation in the ion density can be obtained as

@2nis

@t2þ2Gi

@nis

@t�gv2

thr2nisþo2

pi

N0e

N0

k2l2d

1þk2l2d

nis ¼ 0 ð15Þ

A. Singh, K. Walia / Optics & Laser Technology 44 (2012) 781–787 783

where vth ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBTi=M

pis the ion thermal velocity. In order to solve

Eq. (15), we follow the approach developed by Akhmanov et al.[31] and Sodha et al. [32] and express

nis ¼ n0ðx,y,zÞ exp½iðot�kðzþSðx,y,zÞÞ� ð16Þ

where n0 is the slowly varying real function of x, y and z. o and k

are the frequency and propagation constant for ion-acousticwave, S is the eikonal for the ion-acoustic wave. Substituting‘nis’ from Eq. (16) in Eq. (15) and separating the real andimaginary parts, one obtains

2@S

@zþ

@S

@x

� �2

þ@S

@y

� �2

¼1

k2n0

@2no

@x2þ@2no

@y2

� �þ

o2p

k2v2th

1�N0e

N0

� �k2l2

d

1þk2l2d

ð17Þ

@n2o

@zþ@S

@x�@n2

o

@xþ@S

@y�@n2

o

@yþn2

o

@2S

@x2þ@2S

@y2

� �þ

2Gi

gv2th

�on2

o

k¼ 0 ð18Þ

For solving the coupled Eqs. (17) and (18), we assume theinitial variation of the density perturbation in the ion acousticwave to be

n209z ¼ 0 ¼ n2

00 expð�x2=a2Þ expð�y2=b2Þ ð19Þ

where a and b are the initial dimensions of the ion acoustic waveat z¼ 0 in x and y directions.

Following Akhmanov et al. [31] and Sodha et al. [32], thesolution of Eqs. (17) and (18) can be written as

no ¼nooffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f ia1f ia2

p exp �x2

2a2f 2ia1

!exp �

y2

2b2f 2ia2

!expð�kizÞ ð20Þ

where ki is damping factor

Sðx,y,zÞ ¼1

2x2 1

f ia1

df ia1

dzþ

1

2y2 1

f ia2

df ia2

dzþFðzÞ ð21Þ

where n00 is the axial amplitude of density perturbation of ion-acoustic wave, ‘S’ is the eikonal for the ion-acoustic wave, FðzÞ is aconstant whose value will not be required explicitly in furtheranalysis. f ia1 and f ia2 are the dimensionless beam width para-meters of ion acoustic wave. To obtain an equations for the beamwidth parameters, we employ the paraxial ray approximation andthen equating the coefficients of x2 and y2 on both sides, weobtain from Eq. (17), the following equations for f ia1 and f ia2:

d2f ia1

dz2¼

1

k2a4f 3ia1

�o2

p

gik2v2

th

�3

4am

ME2

00

� �

�exp �3

4am

M

E200

f 01f 02

!f ia1

a20f 3

01f 02

k2l2d

1þk2l2d

ð22Þ

d2f ia2

dz2¼

1

k2b4f 3ia2

�o2

p

gik2v2

th

�3

4am

ME2

00

� �

�exp �3

4am

M

E200

f 01f 02

!f ia2

b20f 3

02f 01

k2l2d

1þk2l2d

ð23Þ

Eqs. (22) and (23) describe the variation in the dimensionlessbeam width parameters f ia1 and f ia2 of ion acoustic wave onaccount of the competition between diffraction divergence termsand nonlinear refractive terms with the distance of propagation in thecollisionless plasma with f ia1 ¼ f ia2 ¼ 1 and df ia1=dz¼ df ia2=dz¼ 0at z¼ 0.

4. Solution of wave equation for back-scattered wave

The high frequency electric field EH may be written as asum of the electric field Ei of the incident beam and Es of the

scattered wave, i.e.

EH ¼ Ei expðiootÞþEs expðiostÞ ð24Þ

where Es is due to scattering of the pump beam from the ion-acoustic wave (i.e Brillouin Scattering), os represents scatteredfrequency. The vector EH satisfies the wave equation

r2EH�rðr � EHÞ ¼

1

c2

@2EH

@t2þ

4pc2

@JH

@tð25Þ

where JH is the total current density vector in the presence of highfrequency electric field EH . Equating the terms at scatteredfrequency os, one gets

r2Esþo2

s

c21�

o2pN0e

o2s N0

" #Es ¼

o2posnn

2c2ooNo

" #Ei�rðr � EiÞ ð26Þ

In order to solve Eq. (26), second term on right hand side hasbeen neglected by assuming that the scale length of variation ofthe dielectric constant in the radial direction is much larger thanthe wavelength of pump. The solution of Eq. (26) may be obtainedin the form

Es ¼ Eso expðþ iksozÞþEs1 expð�iks1zÞ ð27Þ

where

k2so ¼

o2s

c21�

o2p

o2s

" #¼o2

s

c2eso ð28Þ

ks1 and os satisfy phase matching conditions [35], whereos ¼oo�o and ks1 ¼ ko�k.

Here Eso and Es1 are the slowly varying real functions of x, y

and z. kso and ks1 are the propagation constants of scattered wave.Using Eq. (27) in Eq. (26) and separating terms with differentphases we obtain

�k2s0E2

s0þ2iks0@Es0

@zþr

2?Es0þ

o2s

c2es0þ

o2p

o2s

� 1�N0e

N0

� �" #Es0 ¼ 0

ð29Þ

�k2so1E2

s1þ2iks1@Es1

@zþr

2?Es1þ

o2s

c2esoþ

o2p

o2s

� 1�N0e

N0

� �" #Es1

¼1

2

o2p

c2

nn

No

os

ooEo expð�ikoSoÞ ð30Þ

The solution of Eq. (30) may be written as

Es1 ¼ E0s1 expð�ikoSoÞ ð31Þ

Substituting Eq. (31) into Eq. (30) and neglecting terms containingspace derivatives by assuming ðrob2p=koÞ, one obtains thefollowing equation:

E0s1 ¼�1

2

o2p

c2

n%

N0

os

o0

bEE0

k2s1�k2

s0�o2

p

c2 1�N0e

N0

h ih i ð32Þ

where bE is a unit vector along E.By putting Eso ¼ Esoo expðþ iks0SsÞ in Eq. (29) and separating the

real and imaginary parts one can obtain

2 �@Ss

@zþ

@Ss

@x

� �2

þ@Ss

@y

� �2

¼o2

P

esoo2s

1�N0e

N0

� �þ

1

k2soEsoo

r2?Esoo ð33Þ

@E2soo

@zþ@Ss

@x

@E2soo

@xþ@Ss

@y

@E2soo

@yþE2

soor2?Ss ¼ 0 ð34Þ

A. Singh, K. Walia / Optics & Laser Technology 44 (2012) 781–787784

Here, Esoo is the real function of x, y and z, Ss is the Eikonal for thescattered wave. Solutions of Eqs. (33) and (34) can be written as[31,32]

E2soo ¼

B21

f s1f s2exp

�x2

a21f 2

s1

" #exp

�y2

b21f 2

s2

" #ð35Þ

Ss ¼1

2x2 1

f s1

df s1

dzþ

1

2y2 1

f s2

df s2

dzþFsðzÞ ð36Þ

Here, a1 and b1 are initial dimensions of scattered beam at z¼0 inx and y directions. B1 is the amplitude of the scattered beam,whose value is to be determined later by applying boundarycondition. f s1 and f s2 are the dimensionless beam width para-meters of the scattered beam and satisfies the following differ-ential equations:

d2f s1

dz2¼

1

k2s0a4

1f 3s1

�o2

p

o2s es0

�3

4am

ME2

00

� �exp �

3

4am

M

E200

f 01f 02

!f s1

a20f 3

01f 02

ð37Þ

and

d2f s2

dz2¼

1

k2s0b4

1f 3s2

�o2

p

o2s es0

�3

4am

ME2

00

� �exp �

3

4am

M

E200

f 01f 02

!f s2

b20f 3

02f 01

ð38Þ

where f s1 ¼ f s2 ¼ 1 and df s1=dz¼ df s2=dz¼ 0 at z¼0. Eqs. (37) and(38) describe the change in the dimensionless beam width para-meters f s1 and f s2 of scattered beam on account of the competitionbetween diffraction divergence terms and nonlinear refractiveterms as the beam propagates in the collisionless plasma.

Fig. 1. Variation of beam width parameter f 01 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

5. Expression for back-reflectivity

It is clear from Eq. (20) that, the ion-acoustic wave is dampedas it propagates along z-axis and hence the scattered waveamplitude should also decrease with increasing z. Therefore, theappropriate boundary condition would be

Es ¼ Eso expðþ iksozÞþEs1 expð�iks1zÞ ¼ 0 ð39Þ

at z¼ zc . Here, zc is the distance at which amplitude of thescattered wave is zero. Therefore, at z¼ zc , one can obtain

B1 ¼1

2

o2p

c2

os

o0

n00

N0

E00 expð�kizcÞ

k2s1�k2

s0�o2

p

c2 1�N0eN0

h ih i�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif s1ðzcÞf s2ðzcÞ

f 01ðzcÞf 02ðzcÞf ia1ðzcÞf ia2ðzcÞ

sexpð�iðk0S0þks1zcÞÞ

expðþ iðks0ScþksozcÞÞð40Þ

with the conditions 1=a21f 2

s1ðzcÞ ¼ 1=a2f 2ia1ðzcÞþ1=a2

0f 201ðzcÞ and

1=b21f 2

s2ðzcÞ ¼ 1=b2f 2ia2ðzcÞþ1=b2

0f 202ðzcÞ. Here f 01ðzcÞ, f 02ðzcÞ, f ia1ðzcÞ,

f ia2ðzcÞ, f s1ðzcÞ, f s2ðzcÞ are the values of dimensionless beam widthparameters of pump beam, ion-acoustic beam and scattered beamat z¼ zc .

Reflectivity ‘R’ is defined as the ratio of scattered power toincident power and is given by

R¼1

4

1

a0b0

o4p

c4

o2s

o20

n200

N20

expð�2kizÞ

k2s1�k2

s0�o2

p

c2 1�exp �34a

mM

E200

f 01 f 02

� �h ih i2

�f s1ðzcÞf s2ðzcÞ

f 01ðzcÞf 02ðzcÞf ia1ðzcÞf ia2ðzcÞ

�

2664a1b1�2 expð�kiðz�zcÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 01f 02f ia1f ia2f s1f s2

p�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 01ðzcÞf 02ðzcÞf ia1ðzcÞf ia2ðzcÞ

f s1ðzcÞf s2ðzcÞ

scosðks1þks0Þðz�zcÞ

�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12a2

1f 2

s1

þ 12a2f 2

ia1

þ 12a2

0f 2

01

r 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2b21 f 2

s2

þ 12b2f 2

ia2

þ 12b2

0 f 202

qþ

expð�2kiðz�zcÞÞ

f 01f 02f ia1f ia2

�f 01ðzcÞf 02ðzcÞf ia1ðzcÞf ia2ðzcÞ

f s1ðzcÞf s2ðzcÞ

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

a2 f 2ia1

þ 1a2

0f 2

01

r 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

b2f 2ia2

þ 1b2

0 f 202

q3775 ð41Þ

6. Discussion

Eqs. (9), (10), (22), (23), (37) and (38) for the dimensionless beamwidth parameters f 01 and f 02 of the pump beam, f ia1 and f ia2 of theion-acoustic beam, f s1 and f s2 of the scattered beam respectively aresolved numerically for the following set of parameters

o0 ¼ 1:778� 1014 rad s�1 ðCO2 laserÞ

a0 ¼ a¼ 20 mm, b0 ¼ b¼ 12:5 mm, a1 ¼ 40 mm, b1 ¼ 25 mm

o2p

o20

¼ne

ncr¼ 0:25,0:45, aE2

00 ¼ 1;2,3

Vth ¼ 3� 107 cm=s,n00

N0¼ 0:01

The first term on the right hand side of Eqs. (9), (10), (22), (23),(37) and (38) represents the diffraction phenomenon and the secondterm that arises due to the collisionless non-linearity, represents thenon-linear refraction. The relative magnitude of these terms deter-mines the focusing/defocusing behavior of the beams.

Figs. 1 and 2 describe the variation f 01 and f 02 as a function ofdimensionless distance of propagation x for different values ofintensities aE2

00 ¼ 1:0,2:0,3:0 at a fixed value of plasma density

Fig. 2. Variation of beam width parameter f 02 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

Fig. 3. Variation of beam width parameter f ia1 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

Fig. 4. Variation of beam width parameter f ia2 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

Fig. 5. Variation of beam width parameter f s1 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

Fig. 6. Variation of beam width parameter f s2 against the normalized distance of

propagation x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,2:0,3:0.

A. Singh, K. Walia / Optics & Laser Technology 44 (2012) 781–787 785

o2p=o2

0 ¼ 0:25. It is observed from these figures that with increasein the intensity of laser beam, there is a decrease in self-focusing.This is due to the fact that the non-linear refractive terms inEqs. (9) and (10) are very sensitive to the intensity of laser beam.Therefore, as the intensity of the laser beam is increased,refractive terms become relatively weaker than diffractive terms.

Figs. 3 and 4 describe the variation of f ia1 and f ia2 against thenormalized distance of propagation x for different values of inten-sities aE2

00 ¼ 1:0,2:0,3:0 at a fixed value of plasma densityo2

p=o20 ¼ 0:25. From Eqs. (22) and (23) of beam width parameters

f ia1 and f ia2 of ion-acoustic wave, it is observed that non-linearrefractive terms are sensitive to the intensity of the laser beam as wellas on the beam width parameters of the main beam. Therefore due tothe increase in the intensity of the laser beam, non-linear refractiveterms become weaker as compared to the diffractive terms. In fact inthe refractive term both the exponential term involving intensity ofthe laser beam and beam width parameters of the main beam,strongly, reduce the second non-linear term as compared to the firstdiffractive term as the intensity of the laser beam is increased. Due tothis defocusing of ion-acoustic wave is observed with the increase inthe intensity of the main beam as evident from Figs. 3 and 4.

Figs. 5 and 6 describe the variation of fs1 and fs2 of back-scatteredbeam against the normalized distance of propagation x for different

values of intensities aE200 ¼ 1:0,2:0,3:0 at a fixed value of plasma

density o2p=o2

0 ¼ 0:25. It is observed from these figures that withincrease in intensity parameter focusing length of scattered beamincreases. This is due to the weakening of non-linear refractive

Fig. 7. Variation of reflectivity R against the normalized distance of propagation

x¼ Z=Rd for o2p=o2

0 ¼ 0:25 and for intensity aE200 ¼ 1:0,3:0.

Fig. 8. Variation of reflectivity R against the normalized distance of propagation

x¼ Z=Rd for o2p=o2

0 ¼ 0:25,0:45 and for intensity aE200 ¼ 3:0.

A. Singh, K. Walia / Optics & Laser Technology 44 (2012) 781–787786

terms as compared to the diffractive terms at higher values ofintensity. It is further observed that the extent of self-focusing ofscattered beam decreases with increase in the intensity parameter.

Fig. 7 describes the variation of reflectivity R against thenormalized distance of propagation x for different values of pumpbeam intensity aE2

00 ¼ 1:0,3:0 for a fixed value of plasma densityo2

p=o20 ¼ 0:25. It is observed from the figure that reflectivity of the

scattered wave is larger for aE200 ¼ 1:0 than for aE2

00 ¼ 3:0. This isdue to the fact that self-focusing is larger for aE2

00 ¼ 1:0 ascompared to aE2

00 ¼ 3:0. It is thus observed that back-scatteredpower and hence reflectivity increases due to self-focusing.

Fig. 8 describes the variation of reflectivity R against thenormalized distance of propagation x for different values ofplasma density o2

p=o20 ¼ 0:25,0:45 for a fixed value of intensity

aE200 ¼ 3:0. It is clear from the figure that the reflectivity of the

scattered wave increases with increase of plasma density. This isdue to the reason that with increase in plasma density, kld

decreases, which leads to decrease in ion-acoustic wave damping.

7. Conclusion

In the present work, effect of intensity of the pump beam andplasma density on stimulated Brillouin scattering process is

studied by following paraxial ray approximation, when collision-less non-linearity is operative. Following important observationsare made from present analysis:

(1)

The effect of increase of pump beam intensity is to decreasethe extent of self-focusing of the pump and scattered beam.

(2)

Increase in pump beam intensity leads to increase in defocus-ing of the ion acoustic wave.

(3)

There is decrease/increase in SBS reflectivity with increase inpump beam intensity/plasma density.

SBS plays an important role as it scatters a significant fractionof laser energy and thus inhibits its transfer to the plasma.Therefore, SBS reflectivity parameter plays very important roleto understand the physics of laser induced fusion.

Acknowledgments

Authors would like to thank Department of Science andTechnology (DST), Government of India for providing financialassistance for carrying out this work.

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