Self-focusing of Gaussian laser beam in collisionless plasma and its effect on stimulated Brillouin scattering process Arvinder Singh n , Keshav Walia Department of Physics, National Institute of Technology Jalandhar, India article info Article history: Received 16 November 2011 Received in revised form 13 August 2012 Accepted 24 October 2012 Available online 8 November 2012 Keywords: Self-focusing Ponderomotive force Ion-acoustic wave Back-reflectivity abstract This paper presents an investigation of self-focusing of Gaussian laser beam in collisionless plasma and its effect on stimulated Brillouin scattering process. The pump beam interacts with a pre-excited ion- acoustic wave thereby generating a back-scattered wave. On account of Gaussian intensity distribution of laser beam, the time independent component of the ponderomotive force along a direction perpendicular to the beam propagation becomes finite, which modifies the background plasma density profile in a direction transverse to pump beam axis. This modification in density affects the incident laser beam, ion-acoustic wave and back-scattered beam. We have set up non-linear differential equations for the beam width parameters of the main beam, ion-acoustic wave, back-scattered wave and SBS-reflectivity with the help of moment and paraxial theory approach. Results of the moment theory approach have been compared with that of paraxial theory approach. It has been observed from the analysis that reflectivity of the back-scattered wave is less in moment theory approach as compared to paraxial theory approach. & 2012 Elsevier B.V. All rights reserved. 1. Introduction There has been considerable interest in the non-linear inter- action of intense laser beams with plasmas on account of its relevance to laser induced fusion and charged particle accelera- tion [1–8]. During the interaction of laser pulses with plasmas, various laser-plasma instabilities like self-focusing, harmonic generation, stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), etc. [9–14], come into picture and play important role as far as the transfer of energy from laser to the plasma is concerned. These instabilities result in the significant loss in the incident laser energy and hence lead to poor laser plasma coupling. Therefore, these instabilities are being studied theoretically and experimentally. SBS is a parametric instability in plasma in which an electromagnetic wave interacts with an ion acoustic wave to produce a scattered electromagnetic wave. SBS has been a concern in inertial confinement fusion (ICF) applica- tion, because it occurs up to the critical density layer of the plasma and affects the laser plasma coupling efficiency. SBS produces a significant level of back-scattered light; therefore, it is important to devise techniques to suppress SBS. There is a vast difference between the reported results of theory and experiments in spite of intensive research work done on studies of SBS during the last two decades [15–18]. This mismatch between the results of theory and experiment may be due to the idealized theoretical assumptions made in the theory. Theo- retical explanation of low reflectivity observed in large scale fusion experiments [19–23] is one of the main challenge for theoretical researchers. Most of the work on scattering instabilities is done with the assumption of a uniform laser pump. Since most of the electro- magnetic beams have non-uniform distribution of irradiance along the wavefront, there was a need to take into account this non-uniformity in the theory of scattering instabilities. It is well known that such beams exhibit the phenomenon of self-focusing/ self-defocusing. The non-uniformity in the intensity distribution of the laser also affects the scattering of a high power laser beam. In light of considerable current interest in self-focusing and Brillouin scattering, lot of work has already been done in the past [24–28]. In most of the above mentioned works, investigations have been carried out in the paraxial approximation due to small divergence angles of the laser beams involved. In some experi- ments, where solid state lasers are used, wide angle beams are generated for which the paraxial approximation is not applicable. Also, if the beam width of laser beam used is comparable to the wavelength of the laser beam, paraxial approximation is not valid. Paraxial theory approach [29–31] takes into account only paraxial region of the beam, which in turn leads to large error in the analysis. In this theory non-linear part of the dielectric constant is Taylor expanded up to second order term and higher order terms Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/optcom Optics Communications 0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.10.063 n Corresponding author. Tel.: þ91 9914142123; fax: þ91 181 2690320. E-mail address: [email protected] (A. Singh). Optics Communications 290 (2013) 175–182
Transcript
Optics Communications 290 (2013) 175–182
Contents lists available at SciVerse ScienceDirect
Optics Communications
0030-40
http://d
n Corr
E-m
journal homepage: www.elsevier.com/locate/optcom
Self-focusing of Gaussian laser beam in collisionless plasma and its effecton stimulated Brillouin scattering process
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 16 November 2011
Received in revised form
13 August 2012
Accepted 24 October 2012Available online 8 November 2012
This paper presents an investigation of self-focusing of Gaussian laser beam in collisionless plasma and
its effect on stimulated Brillouin scattering process. The pump beam interacts with a pre-excited ion-
acoustic wave thereby generating a back-scattered wave. On account of Gaussian intensity distribution
of laser beam, the time independent component of the ponderomotive force along a direction
perpendicular to the beam propagation becomes finite, which modifies the background plasma density
profile in a direction transverse to pump beam axis. This modification in density affects the incident
laser beam, ion-acoustic wave and back-scattered beam. We have set up non-linear differential
equations for the beam width parameters of the main beam, ion-acoustic wave, back-scattered wave
and SBS-reflectivity with the help of moment and paraxial theory approach. Results of the moment
theory approach have been compared with that of paraxial theory approach. It has been observed from
the analysis that reflectivity of the back-scattered wave is less in moment theory approach as compared
to paraxial theory approach.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
There has been considerable interest in the non-linear inter-action of intense laser beams with plasmas on account of itsrelevance to laser induced fusion and charged particle accelera-tion [1–8]. During the interaction of laser pulses with plasmas,various laser-plasma instabilities like self-focusing, harmonicgeneration, stimulated Raman scattering (SRS), stimulatedBrillouin scattering (SBS), etc. [9–14], come into picture and playimportant role as far as the transfer of energy from laser to theplasma is concerned. These instabilities result in the significantloss in the incident laser energy and hence lead to poor laserplasma coupling. Therefore, these instabilities are being studiedtheoretically and experimentally. SBS is a parametric instability inplasma in which an electromagnetic wave interacts with an ionacoustic wave to produce a scattered electromagnetic wave. SBShas been a concern in inertial confinement fusion (ICF) applica-tion, because it occurs up to the critical density layer of theplasma and affects the laser plasma coupling efficiency. SBSproduces a significant level of back-scattered light; therefore, itis important to devise techniques to suppress SBS. There isa vast difference between the reported results of theory andexperiments in spite of intensive research work done on studies
ll rights reserved.
x: þ91 181 2690320.
of SBS during the last two decades [15–18]. This mismatchbetween the results of theory and experiment may be due tothe idealized theoretical assumptions made in the theory. Theo-retical explanation of low reflectivity observed in large scalefusion experiments [19–23] is one of the main challenge fortheoretical researchers.
Most of the work on scattering instabilities is done with theassumption of a uniform laser pump. Since most of the electro-magnetic beams have non-uniform distribution of irradiancealong the wavefront, there was a need to take into account thisnon-uniformity in the theory of scattering instabilities. It is wellknown that such beams exhibit the phenomenon of self-focusing/self-defocusing. The non-uniformity in the intensity distributionof the laser also affects the scattering of a high power laser beam.In light of considerable current interest in self-focusing andBrillouin scattering, lot of work has already been done in the past[24–28]. In most of the above mentioned works, investigationshave been carried out in the paraxial approximation due to smalldivergence angles of the laser beams involved. In some experi-ments, where solid state lasers are used, wide angle beams aregenerated for which the paraxial approximation is not applicable.Also, if the beam width of laser beam used is comparable to thewavelength of the laser beam, paraxial approximation is not valid.Paraxial theory approach [29–31] takes into account only paraxialregion of the beam, which in turn leads to large error in theanalysis. In this theory non-linear part of the dielectric constant isTaylor expanded up to second order term and higher order terms
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182176
are neglected. Recently, Sharma et al. [32] used the higher orderparaxial theory in which the off-axial part of the laser beam istaken into consideration up to certain extent by including higherorder terms and compared their results with that of paraxialtheory approach [31]. However, the moment theory [33,34] isbased on the calculation of moments and does not suffer from thisdefect. In moment theory approach, full non-linear part of thedielectric constant is taken as a whole in calculations. This theoryhas recently been used in the past for studying the effect of self-focusing of Gaussian laser beam on self-channeling, Secondharmonic generation [35–37]. So, our motivation of the presentwork is to study the effect of self-focusing of Gaussian laser beamon stimulated Brillouin scattering (SBS) process in collisionlessplasma with the help of moment theory approach. Momenttheory is difficult to apply wherever the propagation of morethan one wave is involved and therefore one always prefer toapply paraxial theory and higher order paraxial theory, in whichthe mathematical calculations become simpler as compared tomoment theory approach. To the best of our knowledge, so far noone has used moment theory approach to study the SBS process.So, the novelty of the present work is that, we have consideredthe full non-linear part of the dielectric constant in the presentinvestigation and compared the results with that of paraxialtheory approach.
The pump wave (o0, k0) interacts with pre-excited ion-acoustic wave (o,k) to generate a scattered wave (o0�o, k0�k).As a specific case, back scattering for which kC2k0 has beendiscussed. The pump beam exerts a ponderomotive force on theelectrons, leading to redistribution of carriers and consequently,the pump beam becomes self-focused. The dispersion relation forion-acoustic wave is also significantly modified. The phase velo-city of the ion-acoustic wave becomes minimum on the axis andincreases away from it. Therefore, if appropriate conditions aresatisfied, the ion-acoustic wave may also get focused.
In Section 2, the wave equation for the laser beam has been setup and differential equations for the beam width parameter of thelaser beam have been derived with the help of moment theoryand paraxial theory approach. In Section 3, we have set up thewave equation for the ion-acoustic wave and used the momenttheory as well as paraxial theory approach to obtain the differ-ential equations for the beam width parameter of the ion-acousticwave. In Section 4, the wave equation for the back-scattered wavehas been set up and differential equation for the beam widthparameter of the back-scattered wave have been derived. InSection 5, expression for the reflectivity ‘R’ of the back-scatteredbeam has been derived. Finally, a detailed discussion of theresults has been presented in Section 6.
2. Solution of wave equation for pump beam
Consider the propagation of Gaussian laser beam of frequencyo0 and wave vector k0 in hot collisionless and homogeneousplasma along z-axis. When a laser beam propagates throughplasma, the transverse intensity gradient generates a pondero-motive force, which modifies the plasma density profile in thetransverse direction as
N0e ¼N00 exp �3
4am
ME0 � E
n
0
� �ð1Þ
where a¼ e2M=6KBT0gm2o20. e and m are the electronic charge
and mass, M and T0 are mass of ion and equilibrium temperatureof plasma respectively. N0e is electron concentration in thepresence of laser beam, N00 is the electron concentration in theabsence of laser beam, KB is Boltzman’s constant and g is ratio oftwo specific heats.
The initial intensity distribution of beam along the wavefrontat z¼0 is given by
E0:E%0 9z ¼ 0 ¼ E2
00 exp½�r2=r20� ð2Þ
where r2 ¼ x2þy2 and r0 is the initial width of the pump beamand r is radial co-ordinate of the cylindrical co-ordinate system.E0 is the electric field vector of pump beam and E00 is the axialamplitude of the beam.
Slowly varying electric field E0 of the pump beam satisfies thefollowing wave equation:
r2E0�rðr:E0Þþ
o20
c2EE0 ¼ 0 ð3Þ
In the Wentzal–Kramers–Brillouin (WKB) approximation, thesecond term rðr � E0Þ of Eq. (3) can be neglected, which isjustified when
c2
o20
1
Er2 ln E
���� ����51
r2E0þo2
0
c2EE0 ¼ 0 ð4Þ
where E¼ E0þFðE0E%0 Þ , E0 and FðE0E%
0 Þ are the linear and non-linear parts of the dielectric constant respectively. Here
qis the electron plasma frequency. Further taking E0 in Eq. (4) as
E0 ¼ Aðr,zÞexp½ifo0t�k0zg� ð5Þ
where Aðr,zÞ is a complex function of its argument. The behaviorof the complex amplitude Aðr,zÞ is governed by the parabolicequation obtained from the wave equation (4) in the WKBapproximation by assuming variations in the z direction beingslower than those in the radial direction
idA
dz¼
1
2k0r2?AþwðE0 � E
%0 ÞA ð6Þ
where
wðE0 � E%0 Þ ¼
k0
2E0ðE�E0Þ and E¼ E0þFð9E09
2Þ
2.1. Moment theory approach
Now from the definition of the second order moment, themean square radius of the beam is given by
/a2S¼
RRðx2þy2ÞAA% dx dy
I0ð7Þ
From here one can obtain the following equation:
d2/a2S
dz2¼
4I2
I0�
4
I0
ZZQ ð9A92
Þ dx dy ð8Þ
where I0 and I2 are the invariants of Eq. (6) [33]
I0 ¼
ZZ9A92
dx dy ð9Þ
I2 ¼
ZZ1
2k20
9r?9A92�F
� �dx dy ð10Þ
With [34]
Fð9A92Þ ¼
1
k0
Zwð9A92
Þdð9A92Þ ð11Þ
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182 177
and
Q ð9A92Þ ¼
9A92wð9A92Þ
k0�2Fð9A92
Þ
" #ð12Þ
For z40, we assume an energy conserving Gaussian ansatz forthe laser intensity [29,30]
AA%¼
E200
f 20
exp �r2
r20f 2
0
( )ð13Þ
From Eqs. (7), (9) and (13) it can be shown that
I0 ¼ pr20E2
00 ð14Þ
/a2S¼ r20f 2
0 ð15Þ
where f 0 is dimensionless beam width parameter and r0 is beamwidth at z¼ 0. Now, from Eqs. (8)–(15) we get
d2f 0
dx2þ
1
f 0
df 0
dx
� 2
¼2k2
0
pE200f 0
I2�
ZZQ ð9E09
2Þ dx dy
� �ð16Þ
where x¼ ðz=k0r20Þ is the dimensionless propagation distance.
Eq. (16) is a basic equation for studying the self-focusing of aGaussian laser beam in a non-linear, non-absorptive medium.Now, by making use of (1), (10)–(13) and (16) we get
d2f 0
dx2þ
1
f 0
df
dx
� 2
¼1
f 30
�opr0
c
� �2
�1
f 0
f 20
a1E200
exp�a1E2
00
f 20
!�1�
Zexpð�btÞ�1
t
" #" #ð17Þ
Initial conditions of plane wavefront are df 0=dx¼ 0 and f 0¼1 atx¼ 0. Eq. (17) describes the change in the beam width parameterof a Gaussian beam on account of the competition betweendiffraction divergence and non-linear refractive terms as thebeam propagates in the collisionless plasma.
2.2. Paraxial ray approximation
Further assuming the variation of Aðr,zÞ as
Aðr,zÞ ¼ A0ðr,zÞexp½�ik0S0ðr,zÞ� ð18Þ
where A0ðr,zÞ and S0 are real functions of r and z (S0 being theekional). On substituting A in Eq. (6) and separating the real andimaginary parts of the resulting equation, the following set ofequations is obtained:
2@S0
@z
� þ
@S0
@r
� 2
¼1
k20A0
r2?A0þ
o2p
o20E0
1�N0e
N0
� �ð19Þ
and
@A20
@zþ
@S0
@r
� @A2
0
@rþA2
0r2?S0 ¼ 0 ð20Þ
Following [29,30], the solutions for Eqs. (19) and (20) can bewritten as
A20 ¼
E200
f 20
exp �r2
r20f 2
0
" #ð21Þ
S0 ¼r2
2b0ðzÞþF0ðzÞ ð22Þ
where
b0ðzÞ ¼1
f 0
df 0
dz
and
k0 ¼o0E
1=20
c
The parameter b�10 may be interpreted as the radius of the
curvature of the main beam and F0ðzÞ is the phase shift, whichwe do not require for the further analysis as we are interested inthe intensity of the laser beam rather than its phase. Onsubstituting Eqs. (21) and (22) in Eq. (19) and on equating thecoefficients of r2 on both sides, we get the following differentialequation for the beam width parameter f 0 of the laser beam:
d2f 0
dx2¼
1
f 30
�opr0
c
� �2
�3
4am
ME2
00
� exp �
3
4am
M
E200
f 20
!1
f 30
ð23Þ
where f 0 ¼ 1 and df 0=dz¼ 0 at z¼0. Eq. (23) describes the changein the dimensionless beam width parameter f 0 of pump beam onaccount of the competition between diffraction divergence termand non-linear refractive term as the beam propagates in thecollisionless plasma.
3. Solution of wave equation for ion-acoustic wave
The laser beam interacts with the ion-acoustic wave (IAW) andleads to its excitation. To analyze the excitation process of ion-acoustic wave in the presence of ponderomotive non-linearity.We start with the following set of fluid equations [38].
Continuity equation:
@nis
@tþr � ðN0VisÞ ¼ 0 ð24Þ
Momentum equation:
@Vis
@tþgiv
2th
N0rnisþ2GiV is�
e
MEsi ¼ 0 ð25Þ
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 generatedion-acoustic wave, satisfying Poisson’s equation:
r � Esi ¼�4peðnes�nisÞ ð26Þ
where nes and nis corresponds to perturbations in the electron andion densities, and are related to each other by following equation:
nes ¼ nis 1þk2l2
d
N0e
N0
26643775�1
ð27Þ
where k is the propagation constant for ion-acoustic wave,ld ¼
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 ð28Þ
where vth ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBTi=mi
pis the ion thermal velocity.
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182178
3.1. Moment theory approach
Further taking nis as
nis ¼ n1ðr,zÞexpðiðot�kzÞÞ ð29Þ
where n1 is the slowly varying real function of r and z. o and k arethe frequency and propagation constant for ion-acoustic wave.Substituting the value of nis from Eq. (29) in Eq. (28), one obtainsin the WKB approximation
@n1
@z¼�
i
2kr2?n1�iPn1�
Gon1
kgv2th
ð30Þ
where P can be written as
P¼o2
pi
2kgiv2th
k2s l
2d
1þk2s l
2d
1�Noe
N0
� By the definition of second order moment
/a2S¼1
I0
ZZðx2þy2Þn1nn
1 dx dy ð31Þ
where I0 is zeroth order moment and can be written as
I0 ¼
ZZn1nn
1 dx dy ð32Þ
Now, solution to Eq. (30) is of the form
n1 ¼n00
fexp
r2
2a2f 2�kiz
" #ð33Þ
From Eqs. (31)–(33), it can be shown that
I0 ¼ pn200a2 expð�2kizÞ ð34Þ
and
/a2S¼ a2f 2 expð�2kizÞ ð35Þ
Now, with the help of Eqs. (31) and (35), it can be shown that
d2f
dx2þ
1
f
df
dx
� 2
¼ expð2kizÞ1
4f 3�
1
4gopro
vth
� 2 k2s l
2d
1þk2s l
2d
"1
f
f 20
f 2I1þ
f 40
f 4I2
" ##þ
G2o2fr40
2g2v2th expð�2kiðzÞÞ
ð36Þ
where
I1 ¼
Zta1�1 1�exp �
a1E200 � t
f 20
!" #dt
and
I2 ¼
ZlogðtÞ 1�exp �
a1E200t
f 20�
!" #ta1�1 dt
Eq. (36) describes the variation in the dimensionless beamwidth parameter f of ion acoustic wave on account of thecompetition between diffraction divergence and non-linearrefractive terms with the distance of propagation in the collision-less plasma with f¼1 and df=dz¼ 0 at z¼0.
3.2. Paraxial ray approximation
Assuming the variation of nisðr,zÞ as
nisðr,zÞ ¼ n1ðr,zÞexp½iðot�kðzþSðx,y,zÞÞ� ð37Þ
where n1 is the slowly varying real function of r and z. o and k arethe frequency and propagation constant for ion-acoustic wave, S
is the eikonal for the ion-acoustic wave. Substituting for ‘nis’ fromEq. (37) in Eq. (28) and separating the real and imaginary parts,
one obtains
2@S
@zþ
@S
@r
� 2
¼1
k2n1
r2?n1þ
o2p
k2v2th
1�N0e
N0
� �k2l2
d
1þk2l2d
ð38Þ
@n21
@zþ@S
@r�@n2
1
@rþn2
1r2?Sþ
2Gi
gv2th
�on2
1
k¼ 0 ð39Þ
The solution to Eqs. (38) and (39) can be written as
n1 ¼n00
fexp
r2
2a2f 2�kiz
" #ð40Þ
where ki is damping factor
Sðr,zÞ ¼1
2r2 1
f
df
dzþFðzÞ ð41Þ
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 is the dimensionless beam width parameters of ion-acoustic wave. To obtain an equations for the beam widthparameter, we employ the paraxial ray approximation and thenequating the coefficients of r2 on both sides, we obtain from Eq.(38), the following equation for f:
d2f
dz2¼
k20r4
0
k2a4f 3�
o2p
gik2v2
th
�3
4am
ME2
00
�
�exp �3
4am
M
E200
f 20
!f
f 40
k20r2
0
k2l2d
1þk2l2d
ð42Þ
Eq. (42) describes the variation in the dimensionless beam widthparameters f of ion-acoustic wave on account of the competitionbetween diffraction divergence term and non-linear refractiveterm with the distance of propagation in the collisionless plasmawith f¼1 and df=dz¼ 0 at z¼0.
4. Solution of the wave equation for back-scattered beam
The high frequency electric field EH may be written as a sum ofthe electric field E0 of the incident beam and Es of the scatteredwave, i.e.
EH ¼ E0 expðiootÞþEs expðiostÞ ð43Þ
where, Es is due to the scattering of the pump beam from the ionacoustic wave(i.e. Brillouin scattering), 0os
0 represents scatteredfrequency. The vector EH satisfies the wave equation
r2EH�rðr � EHÞ ¼1
c2
@2EH
@t2þ
4pc2
@JH
@tð44Þ
where, JH is the total current density vector in the presence ofhigh frequency electric field EH . Equating the terms at scatteredfrequency 0os
0, we get
r2Esþ
o2s
c21�
o2pNoe
o2oNo
" #Es ¼
o2posnn
2c2ooNo
" #E0�rðr � E0Þ ð45Þ
In order to solve Eq. (45), 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 to Eq. (45) may be obtainedin the form
Es ¼ Eso expðþ iksozÞþEs1 expð�iks1zÞ ð46Þ
where
k2so ¼
o2s
c21�
o2p
o2s
" #¼o2
s
c2eso ð47Þ
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182 179
ks1 and os satisfy phase matching conditions [39], where os ¼
oo�o and ks1 ¼ ko�k. Here Eso and Es1 are the slowly varying realfunctions of r and z. kso and ks1 are the propagation constants ofscattered wave. Using Eq. (46) in (45) and separating terms withdifferent phases we obtain
�k2s0E2
s0þ2iks0@Es0
@zþr
2?Es0þ
o2s
c2es0þ
o2p
o2s
� 1�N0e
N0
� " #Es0 ¼ 0
ð48Þ
�k2so1E2
s1þ2iks1@Es1
@zþr
2?Es1þ
o2s
c2esoþ
o2p
o2s
� 1�Noe
No
� " #
Es1 ¼1
2
o2p
c2
nn
No
os
ooEo ð49Þ
Now, from Eq. (49), neglecting terms containing space deriva-tives by assuming ðrob2p=koÞ, one obtains the followingequation:
Es10¼ �
1
2
o2p
c2
n%
N0
os
o0
bEE0
k2s1�k2
s0�o2
p
c21�
N0e
N0
� �" # ð50Þ
where bE is a unit vector along E.From Eq. (48), again as considered earlier, one obtains in the
WKB approximation
@Es0
@z¼�
i2ks0r
2?Es0�iPEs0 ð51Þ
4.1. Moment theory approach
Now, from definition of second order moment
/a2S¼1
I0
Z Zðx2þy2ÞEs00En
s00 dx dy ð52Þ
where I0 is zeroth order moment and can be written as
I0 ¼
ZZEs00En
s00 dx dy ð53Þ
The solution to Eq. (51) is of the form
E2s00 ¼
B21
f 2s
exp�r2
b2f 2s
" #ð54Þ
Here, Esoo is the real function of r and z. b is the initial dimensionof scattered beam at z¼0, B1 is the amplitude of the scatteredbeam, whose value is to be determined later by applying bound-ary condition. f s is the dimensionless beam width parameter ofthe scattered beam.
Now, from Eqs. (52)–(54), it can be shown that
I0 ¼ pB21b2
ð55Þ
and
/a22S¼ b2f 2
s ð56Þ
With the help of Eqs. (52) and (56), one can get
d2f s
dx2þ
1
f s
df s
dx
� 2
¼k2
0r40
k2s0b4f 3
s
�k2
0r40f 2
0
b4k2s0f 3
s
opro
c
� �2
I3þr2
0f 20
b2f 2s
I4
" #ð57Þ
where
I3 ¼
Zta2�1 1�exp �
a1E200 � t
f 20
!" #dt
and
I4 ¼
ZlogðtÞ 1�exp �
a1E200 � t
f 20
!" #ta2�1 dt
where f s¼1 and df s=dz¼0 at z¼0. Eq. (57) describes the changein the dimensionless beam width parameter f s of scattered beamon account of the competition between diffraction divergenceand non-linear refractive terms as the beam propagates in thecollisionless plasma.
4.2. Paraxial ray approximation
By putting Eso ¼ Esoo expðþ iks0SsÞ in Eq. (51) and separating thereal and imaginary parts one can obtain
2 �@Ss
@zþ
@Ss
@r
� 2
¼o2
P
esoo2s
1�N0e
N0
� �þ
1
k2soEsoo
r2?Esoo ð58Þ
@E2soo
@zþ@Ss
@r
@E2soo
@rþE2
soor2?Ss ¼ 0 ð59Þ
Here, Esoo is the real function of r and z, Ss is the eikonal for thescattered wave. Solutions to Eqs. (58) and (59) can be written as
E2s00 ¼
B21
f 2s
exp�r2
b2f 2s
" #ð60Þ
Ss ¼1
2r2 1
f s
df s
dzþFsðzÞ ð61Þ
Here, B1 is the amplitude of the scattered beam, whose value is tobe determined later by applying boundary condition. f s is thedimensionless beam width parameters of the scattered beam andsatisfies the following differential equation:
d2f s
dx2¼
k20r4
0
k2s0b4f 3
s
�o2
pk20r2
0
o2s es0
�3
4am
ME2
00
� exp �
3
4am
M
E200
f 20
!f s
f 40
ð62Þ
where f s¼1 and df s=dz¼0 at z¼0. Eq. (62) describes the changein the dimensionless beam width parameters f s of scattered beamon account of the competition between diffraction divergenceterm and non-linear refractive term as the beam propagates inthe collisionless plasma.
5. Expression for back-reflectivity
Now, the value of B1 is calculated with the boundary conditionthat Es¼0 at z¼ zc .
Es ¼ Eso expðþ iksozÞþEs1 expð�iks1zÞ ¼ 0 ð63Þ
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
c21�
N0e
N0
� �" # f sðzcÞ
f 0ðzcÞf ðzcÞ
expð�iðks1zcÞ
expðiks0zcÞ
ð64Þ
with the condition
1
b2f 2s
¼1
a2f 2þ
1
r20f 2
0
Here f 0ðzcÞ, f ðzcÞ, f sðzcÞ are the values of dimensionless beamwidth parameters of pump beam, ion-acoustic beam andscattered beam at z¼ zc .
1.5 2.5
f
1.0
0.8
0.6
0.4
0.2
0.00.0 0.5 1.0 2.0 3.0
ξ
αE002 = 1.0
E002 = 2.0
Fig. 2. Variation of beam width parameter f against the normalized distance of
propagation x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182180
Now, reflectivity R is defined as ratio of scattered flux toincident flux and is given by
R¼1
4
o2p
c4
o2s
o20
n20
N200
1
k2s1�k2
s0�o2
p
c21�exp �
3
4am
M
E200
f 20
expð�1:0Þ
!" #" #2I1�I2�I3½ �
ð65Þ
where
I1 ¼f 2
s ðzcÞ
f 20ðzcÞf
2ðzcÞ
1
f 2s
exp �2kizc�r2
b2f 2s
!ð66Þ
I2 ¼�2f sðzcÞ
f 0ðzcÞf ðzcÞ
1
ff 0f s
exp �r2
2b2f 2s
�r2
2a2f 2�
r2
2r20f 2
0
!expð�kiðzþzcÞÞcosðks1þks0Þðz�zcÞ ð67Þ
I3 ¼1
f 2f 20
exp �r2
a2f 2�
r2
r20f 2
0
�2kizc
!ð68Þ
0.5 1.5 2.5
0.3
f s
1.11.00.90.80.70.60.50.4
0.20.1
0.0 1.0 2.0 3.0ξ
αE002 = 1.0
αE002 = 2.0
Fig. 3. Variation of beam width parameter f s against the normalized distance of
propagation x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.
6. Discussion
The differential equations (17), (23), (36), (42), (57) and (62)for the beam width parameters f 0 of the pump beam, f of theion-acoustic beam, f s of the scattered beam respectively havebeen solved numerically for the following set of parameters;o0 ¼ 1:778� 1014 rad s�1, o2
p=o20 ¼ 0:6. The first term on the
right hand side of Eqs. (17), (23), (36), (42), (57) and (62)represents the diffraction phenomenon and the second term thatarises due to the colisionless non-linearity, represents the non-linear refraction. The relative magnitude of these terms deter-mines the focusing/defocusing behavior of the beams.
Fig. 1 describes the variation of beam width parameter f 0 ofthe pump beam as a function of dimensionless distance ofpropagation x for different values of intensities aE2
00 ¼ 1:0,2:0. Itis observed from the figure that with increase in the intensity oflaser beam, there is an increase in self-focusing. This is due to thefact that the non-linear refractive term in Eq. (17) is sensitive tothe intensity of laser beam. Therefore, as we increase the intensityof the laser beam, refractive term becomes relatively strongerthan diffractive term.
0.5 1.5 2.5
1.0
αE002 = 2.0
f 0
ξ
0.8
0.6
0.4
0.2
0.00.0 1.0 2.0 3.0
αE002 = 1.0
Fig. 1. Variation of beam width parameter f0 against the normalized distance of
propagation x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.
Fig. 2 describes the variation of beam width parameter f of theion acoustic wave against the normalized distance of propagationx for different values of intensities aE2
00 ¼ 1:0,2:0. It is observedfrom the figure that the extent of self-focusing of the ion-acousticwave increases with increase in intensity parameter. This isbecause, as we increase the intensity of the laser beam, non-linear refractive term dominate the diffractive term and hencethere is an increase in focusing of the beam at higher intensity.
Fig. 3 describes the variation of beam width parameter fs ofback-scattered beam against the normalized distance of propaga-tion x for different values of intensities aE2
00 ¼ 1:0,2:0. It isobserved from the figure that with increase in intensity para-meter the extent of focusing of the scattered beam increases. Thisis due to the weakening of diffractive term as compared to non-linear refractive term at higher value of intensity.
Fig. 4 describes the variation of reflectivity R against thenormalized distance of propagation x for different values of pumpbeam intensity aE2
00 ¼ 1:0,2:0. It is observed from the figure thatreflectivity of the scattered wave is larger for aE2
00 ¼ 2:0 than foraE2
00 ¼ 1:0, which is due to the fact that self-focusing is appreci-ably larger in the former case. Thus, self-focusing of beams leadsto increase in back-scattered flux and hence reflectivity.
Fig. 5 shows the variation of the beam width parameter f 0
of the pump beam due to the paraxial theory and the momenttheory with dimensionless distance of propagation x for
R
ξ
0.000010
0.000008
0.000006
0.000004
0.000002
0.000000
0.002 0.004 0.006 0.008 0.010
αE002 = 2.0
αE002 = 1.0
Fig. 4. Variation of reflectivity R against the normalized distance of propagation
x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.
0.5 1.5 2.5
f 0
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.10.0 1.0 2.0 3.0
ξ
αE002 = 2.0
Fig. 5. Variation of the beam width parameter f0 against the normalized distance
of propagation x¼ Z=Rd for aE200 ¼ 2:0 and o2
p=o20 ¼ 0:6. (Solid line corresponds to
result of the moment theory and dotted line corresponds to that of the paraxial
theory approach.)
0.5 1.5 2.5
f
1.11.00.90.80.70.60.50.40.30.20.1
0.0 1.0 2.0 3.0ξ
αE002 = 2.0
Fig. 6. Variation of beam width parameter f against the normalized distance of
propagation x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 2:0. (Solid line
corresponds to result of the moment theory and dotted line corresponds to that of
the paraxial theory approach.)
0.5 1.5 2.5
f s
1.0
0.8
0.6
0.4
0.2
0.00.0 1.0 2.0 3.0
ξ
αE002 = 2.0
Fig. 7. Variation of beam width parameter f s against the normalized distance of
propagation x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 2:0. (Solid line
corresponds to result of the moment theory and dotted line corresponds to that of
the paraxial theory approach.)
0.008
R
0.000010
0.000008
0.000006
0.000004
0.000002
0.000000
0.002
0.004
0.006
0.010
0.012
0.014
0.016
0.018
0.020
ξ
αE002 = 2.0
Fig. 8. Variation of reflectivity R against the normalized distance of propagation
x¼ Z=Rd for o2p=o2
0 ¼ 0:6 and for intensity aE200 ¼ 2:0. (Solid line corresponds to
result of the moment theory and dotted line corresponds to that of the paraxial
theory approach.)
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182 181
aE200 ¼ 2:0. Solid line corresponds to the moment theory and
dotted line corresponds to the paraxial theory approach. It isobserved from figure that there is strong focusing of the beam in
the moment theory as compared to the paraxial theory onaccount of participation of off-axis parts.
Fig. 6 shows the variation of the beam width parameter f of theion-acoustic wave with dimensionless distance of propagation xfor aE2
00 ¼ 2:0 for the paraxial theory and the moment theoryapproach. Solid line corresponds to the moment theory anddotted line corresponds to the paraxial theory approach. It isagain observed from figure that there is strong focusing of thebeam in the moment theory as compared to the paraxial theory.
Fig. 7 shows the variation of the beam width parameter f s ofthe back-scattered beam against the normalized distance ofpropagation x for aE2
00 ¼ 2:0 for the paraxial theory and themoment theory approach. Solid line corresponds to the momenttheory and dotted line corresponds to the paraxial theoryapproach. It is observed from the figure that the self-focusinglength is less in the moment theory as compared to the paraxialtheory.
Fig. 8 describes the variation of reflectivity R against thenormalized distance of propagation x for pump beam intensity
A. Singh, K. Walia / Optics Communications 290 (2013) 175–182182
aE200 ¼ 2:0 for the paraxial theory and the moment theory
approach. Solid line corresponds to the moment theory anddotted line corresponds to the paraxial theory approach. It isobserved from the figure that reflectivity of the scattered wave islarger for paraxial ray approximation as compared to that formoment theory approach. This observation is similar to that ofSharma et al. [32].
Results of present analysis are similar to that Sharma et al. [32]in which the authors compared the results of higher orderparaxial approach with that of paraxial theory [31]. In higherorder paraxial approach, one additional term in the Taylor seriesexpansion is included in the analysis as compared to the paraxialapproach and thus off-axial part of the laser beam is taken intoconsideration up to certain extent. The beauty of the momenttheory approach is that it takes into consideration the completelaser profile instead of only certain part as considered in thehigher order paraxial approach and is therefore more realistic.
7. Conclusion
In the present investigation, moment theory has been used tostudy the stimulated Brillouin scattering (SBS) of laser beam incollisionless plasma. Results are compared with the paraxial rayapproximation. Following important observations are made frompresent analysis.
(1)
Self-focusing of pump wave and ion-acoustic wave is strongerin the moment theory approach as compared to that ofparaxial theory.
(2)
Self-focusing length of back-scattered wave is less in themoment theory as compared to the paraxial theory.
(3)
Reflectivity of the back-scattered wave is larger for paraxialtheory as compared to that for moment theory approach.
Results of the present investigation are useful for understand-ing physics of laser-induced fusion in which SBS plays a majorrole, as it produces a significant level of back-scattered light andthus leads to poor laser plasma coupling.
Acknowledgment
The authors would like to thank Department of Science andTechnology (DST), Government of India, for the support of this work.
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