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j o ur nal hom epage: www.elsev ier .de / i j leo
timulated Raman scattering of gaussian laser beam in relativistic plasma
rvinder Singh ∗, Keshav Waliaepartment of Physics, National Institute of Technology Jalandhar, India
a r t i c l e i n f o
rticle history:eceived 30 May 2012ccepted 24 October 2012
a b s t r a c t
This paper presents an investigation of Stimulated Raman Scattering of gaussian laser beam in relativis-tic Plasma. The pump beam interacts with a pre-excited electron plasma wave and thereby generate aback-scattered wave. Due to intense laser beam, electron oscillatory velocity becomes comparable tothe velocity of light, which modifies the background plasma density profile in a direction transverse to
pump beam axis. The relativistic non-linearity due to increase in mass of the electrons effects the incidentlaser beam, electron plasma wave and back-scattered beam. We have set up the non-linear differentialequations for the beam width parameters of the main beam, electron plasma wave, back-scattered waveand derived SRS back-reflectivity by taking full non-linear part of the dielectric constant of relativisticplasma with the help of moment theory approach. It is observed from the analysis that self-focusing ofthe pump beam greatly affects the SRS reflectivity, which plays a significant role in laser induced fusion.
. Introduction
In recent years, the propagation of high power laser beamshrough plasmas has become a subject of great interest and activity.he interaction of these laser beams with plasmas have led to rapidevelopment in areas like laser induced fusion and charged particlecceleration[1–8]. Due to availability of lasers capable of deliv-ring high power(1018 − 1021W/cm2), its interaction with plasmaecomes a most interesting and important non-linear problem.t such high intensities, the response of plasma free electrons is
ully relativistic (electrons swing in the laser pulse) and highlyonlinear. In the laser plasma coupling process, when a high-ower laser beam interacts with the plasma, various parametric
nstabilities such as self-focusing, filamentation, stimulated Ramancattering, stimulated Brillouin scattering, two plasmon decay,tc.[9–16] take place, and due to these, the energy of the high-ower laser beam is not efficiently coupled with plasma. These
nstabilities can also modify the intensity distribution and thusffect the uniformity of energy deposition. Therefore, the studyf these nonlinear phenomena at high-power laser flux are beingtudied theoretically and experimentally. In particular, stimulatedaman scattering (SRS) governs the amount of laser energy that cane propagated over long distances through plasma. Since, many
aser-matter interaction applications such as advanced radiation
ources, laser plasma accelerators, laser fusion, and relativistic non-inear optics depend critically on the amount of transmitted lasernergy through the plasma. In SRS, the incident laser beam decays
into a scattered wave and an electron plasma wave(EPW). The EPWproduces super thermal electrons that penetrate and preheat thetarget core and the scattered wave represents a substantial amountof wasted energy, i.e., the energy that would otherwise get coupledto the target. Consequently SRS has been very actively studied bothexperimentally and theoretically. Therefore, to ensure the amountof useful and dissipated energy in laser plasma coupling, Ramanreflectivity becomes a very important parameter to decide.
In many theoretical studies, interesting non-linear phenomenasuch as self-focusing and stimulated back scattering have beencarried out separately, ignoring interplay among them. There isno reason to separate the evolution of these instabilities in thenonlinear regime, where they coexist and affect each other. Itbecomes important to investigate and understand the interplayamong various instabilities. In light of considerable current interestin self-focusing and Raman scattering, lot of work has already beendone in the past [17–28]. In most of the above mentioned works,investigations have been carried out in the paraxial approximationdue to small divergence angles of the laser beams involved. In someexperiments, where solid state lasers are used, wide angle beamsare generated for which the paraxial approximation is not applica-ble. 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 [29,30] takes in to account only paraxialregion of the beam, which in turn leads to error in the analysis.In paraxial theory non-linear part of the dielectric constant is Tay-lor expanded up to second order term and higher order terms are
neglected. However, moment theory [31,32] is based on the calcu-lation of moments and does not suffer from this defect. In momenttheory approach, non-linear part of the dielectric constant is takenas a whole in calculations [33–40]. To the best of our knowledge, so
ar no one has used the moment theory approach to investigate thetimulated Raman Scattering. Therefore, the motivation of presentork is to study the effect of self-focusing of gaussian laser beam
n Stimulated Raman Scattering process(SBS) in relativistic plasmaith the help of moment theory approach.
In the present paper, Raman Scattering of a gaussian laseream from a relativistic plasma has been investigated. The pumpave(ω0, k0) interacts with pre-excited electron plasma wave(ω,k)
o generate a scattered wave(ω0 − ω, k0 − k). As a specific case, backcattering for which k � 2k0 has been discussed. The relativisticon-linearity occurs on account of the increase in mass of electrons,hich oscillate at relativistic velocities in an intense laser field. As
result, electrons get redistributed leading to modification in theffective dielectric constant of plasma. Consequently, the pumpeam becomes self-focused. The dispersion relation for electronlasma wave is also significantly modified. The phase velocity of thelectron plasma wave becomes minimum on the axis and increasesway from it. Therefore, if appropriate conditions are satisfied, thelectron plasma wave may also get focused. Since the scatteredntensity is proportional to the intensities of the pump and elec-ron plasma wave, it is therefore expected that the self-focusinghould lead to enhanced back-scattering.
The paper is organized as follows: Section 2 is devoted to theolution of wave equation for the pump beam and derivation ofeam width width parameter of the pump beam with the help ofoment theory approach. Section 3 is devoted to solution of wave
quation for electron plasma wave and derivation of beam widthidth parameter of electron plasma wave. In section 4, the wave
quation for the back-scattered wave is solved by moment the-ry approach and differential equation for beam width parameterf back-scattered wave is derived. Expression for reflectivity ‘R’ ofhe back-scattered is also derived. Finally a detailed discussion ofesults is presented in section 5.
. Solution of Wave Equation for Pump beam
Consider the propagation of a high power laser beam of angularrequency ω0 in a relativistic plasma along the z axis. The initialntensity distribution of the beam along the wavefront at z = 0 isiven by
0.E�0|z=0 = E2
00 exp[−r2/r2
0
](1)
here r2 = x2 + y2 and r0 is the initial width of the main beam. rs the radial co-ordinate of the cylindrical co-ordinate system. Theielectric constant of the plasma is given by
0 = 1 − ω2p
ω20
(2)
here ωp =√
4�nee2
m0is known as the plasma frequency, e, m0 and
e are the charge, rest mass and density of the plasma electronsespectively. On substituting for the relativistic mass m = �0m0,here m0 is the electron rest mass, one obtains
= 1 − ω2p
�0ω20
. (3)
0 is a relativistic factor given by
0 =√
1 + e2E0E�0
m20ω2
0c2(4)
herefore, the intensity dependent dielectric constant is given by
= �o + �(E0.E�0) (5)
4 (2013) 3470– 3475 3471
where �(E0.E�0) represents the non-linear part of the dielectric con-
stant and is represented as
�(E0.E�0) = ω2
p
ω20
[1 − 1
�0
]. (6)
The slowly varying electric field E0 satisfies the following waveequation.
∇2E0 − ∇(∇.E0) + ω20
c2�E0 = 0. (7)
In the WKB approximation, the second term ∇(∇ . E0) can beneglected, which is justified when c2
ω20| 1� ∇2 ln �| � 1,
∇2E0 + ω20
c2�E0 = 0. (8)
One can take
E0 = A(r, z) exp[�{ω0t − k0z}] (9)
where, A(r, z) is a complex function of its argument. The behaviourof the complex amplitude A(r, z) is governed by the parabolic equa-tion obtained from the wave Eq. (8) in the WKB approximation byassuming variations in the z direction being slower than those inthe radial direction.
�dA
dz= 1
2k0∇2
⊥A + (AA�)A (10)
where (AA�) = k02�0
(� − �0) and � = �0 + �(|AA�|2), where �o = 1 −ω2
p
ω20
and �(|AA�|2) are the linear and nonlinear parts of the dielectric
constant, respectively. Also, k0 = ω0c
√�0 and ωp are propagation
constant and plasma frequency, respectively. Now from the defi-nition of the second order moment, the mean square radius of thebeam is given by
< a21 >=
∫ ∫(x2 + y2)AA�dxdy
I0. (11)
From here, one can obtain the following equation.
d2 < a21 >
dz2= 4I2
I0− 4
I0
∫ ∫Q (|A|2)dxdy (12)
where, I0 and I2 are the invariants of Eq. (10)[32]
I0 =∫ ∫
|A|2dxdy (13)
I2 =∫ ∫
1
2k20
(|∇⊥|A|2 − F)dxdy (14)
With[31]
F(|A|2) = 1k0
∫(|A|2)d(|A|2) (15)
and
Q (|A|2) =[
|A|2(|A|2)k0
− 2F(|A|2)
]. (16)
For z > 0, we assume an energy conserving gaussian ansatz for thelaser intensity[29,30]
AA� = E2002
exp
{− r2
2 2
}. (17)
f0 r0 f0
From Eqs. (11), (13) and (17), it can be shown that
I0 = �r20 E2
00, (18)
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472 A. Singh, K. Walia / O
a21 >= r2
0 f 2 (19)
here, f0 is the dimensionless beam width parameter and r0 is theeam width at z = 0. Now, from Eqs. (12)-(19) one obtains
d2f0d2
+ 1f0
(df 0
d
)2
= 2k20
�E200f0
[I2 −
∫ ∫Q (|A|2)dxdy
]. (20)
here = (z/Rd) is the dimensionless distance of propagation andd is the Rayleigh length. Eq. (20) is the basic equation for studyinghe self-focusing of a gaussian laser beam in a nonlinear, non-bsorptive medium. Now, with the help of Eqs. (6),(14)-(17) and20), we get
d2f0d2
+ 1f0
(df 0
d
)2
= 1
f 30
+(
ωpr0
c
)2
.1f0
⎛⎜⎜⎝ 2f 20
˛E200
⎡⎢⎢⎣1 −√
1 + ˛E200
f 20
+ log
⎛⎜⎜⎝√
1 + ˛E200
f 20
+ 1√1 + ˛E2
00f 20
− 1
⎞⎟⎟⎠⎤⎥⎥⎦⎞⎟⎟⎠ .
(21)
he initial conditions for a plane wave front are df 0d
= 0 and f0 = 1 at
= 0. Eq. (21) describes the changes in the beam width parameterf a gaussian laser beam on account of the competition betweeniffraction divergence and nonlinear focusing terms as the beamropagates in the relativistic plasma.
. Solution of Wave Equation for Electron plasma wave
The pump laser beam interacts with electron plasma wave andeads to its excitation. The motion of plasma particles is describedn the hydrodynamic approximation by the fluid equations [41]
∂N
∂t+ ∇ · (NV) = 0 (22)
[∂V
∂t+ (V · ∇)V
]= −e[E + 1
cV × B] − 2�mV − �
N∇P (23)
here N is the instantaneous electron density, V is electron fluidelocity,P = NkBT0 is hydrodynamic pressure, E and B are the electricnd magnetic fields respectively, � is the landau damping fac-
or given by [41] 2� =√
�8
ωp
(k de)3 exp[− 1
2k2 2de
− 32
], where k is
he wave vector of electrostatic wave and de =√
kBT04�N0e2 is debye
ength of plasma.Applying Perturbation approximation,
= Noe + n, V = V0 + v, E = EH + EP (24)
where, n � Noe, v � V0 and EP � EH, where V0 is the particleelocity in presence of high frequency field EH, the plasma isssumed to have no drift velocity. The self consistent field EP ofhe plasma wave, satisfies the following Poisson’s equation.
· EP = −4�ne (25)
Following standard techniques, one obtains the general equa-ion governing the electron density variation,
∂2n
∂t2+ 2�
∂n
∂t− �iv2
th∇2n + ω2p
1�0
n = 0 (26)√kBTo
here, vth = m is the electron thermal velocity. In order to
olve Eq. (26) for n, take n as
= n1(r, z) exp(�(ωt − kz)) (27)
4 (2013) 3470– 3475
Where n1 is the slowly varying real function of r and z, ω andk are the frequency and propagation constant for electron plasmawave. Substituting the value of ′n′ from Eq. (27) in Eq. (26), oneobtains in the Wentzal- Kramers - Brillouin (WKB) approximation
∂n1
∂z= − i
2k∇2
⊥n1 − �Pn1 − �ωn1
k�iv2th
(28)
Where P can be written as P =ω2
pi
2k�iv2th
(1 − 1
�0
).
Now, from the definition of second order moment
< a22 >= 1
I0
∫ ∫(x2 + y2)n1n∗
1dxdy (29)
Where, I0 is zeroth order moment and can be written as
I0 =∫ ∫
n1n∗1dxdy (30)
Now,Following [29,30] solution of Eq. (28) is of the form,
n1 = n00
fexp
[r2
2a2f 2− kiz
](31)
Where, n00 is the axial amplitude of density perturbation of elec-tron plasma wave. f is the dimensionless beam width parameter ofelectron plasma wave. a is the initial width of the electron plasmawave at z=0 and ki is damping factor. Now, from Eqs. (29), (30) and(31), it can be shown that
I0 = �n200a2 exp(−2kiz) (32)
and
< a22 >= a2f 2 exp(−2kiz) (33)
Now, with the help of Eqs. (29) and (33), it can be shown that
d2f
d2+ 1
f
(df
d
)2
= exp(2kiz)
[1
4f 3− 1
4�i
(ωpro
vth
)2
× 1f
[f 20
f 2L1 + f 4
0
f 4L2
]]+ �2ω2fr4
0
2�2iv2
thexp(−2ki(z))
(34)
Where L1 =∫
t˛1−1
[1 −
(1 + ˛E2
00t
f 20
) −12
]dt and L2 =
∫t˛1−1 log(t)
[1 −
(1 + ˛E2
00t
f 20
) −12
]dt
Eq. (34) describes the variation in the dimensionless beam widthparameter f of electron plasma wave on account of the competitionbetween diffraction divergence and nonlinear refractive terms withthe normalized distance of propagation in the relativistic plasmawith f = 1 and df
d= 0 at = 0.
4. Solution of Wave Equation for back-scattered beam
The high frequency electric field EH may be written as a sumof the electric field E0 of the incident beam and Es of the scatteredwave, i.e
EH = E0 exp(iωot) + Es exp(iωst) (35)
where, Es is due to the scattering of the pump beam from the elec-tron plasma wave(i.e Raman Scattering), ′ω′
s represents scattered
frequency. The vector EH satisfies the wave equation
∇2EH − ∇(∇ · EH) = 1c2
∂2EH
∂t2+ 4�
c2
∂JH∂t
(36)
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owtb
A. Singh, K. Walia / Op
here, JH is the total current density vector in the presence ofigh frequency electric field EH. Equating the terms at scattered
requency ′ω′s, we get
2Es + ω2s
c2
[1 − ω2
p
�0ω2s
]Es =
[ω2
pωsn∗
2�0c2ωoNo
]E0 − ∇(∇ · E0) (37)
In order to solve Eq. (37), second term on right hand side haseen neglected by assuming that the scale length of variation ofhe dielectric constant in the radial direction is much larger thanhe wavelength of pump. The solution of Eq. (37) may be obtainedn the form
s = Eso exp(+iksoz) + Es1 exp(−iks1z) (38)
where
2so = ω2
s
c2
[1 − ω2
p
ω2s
]= ω2
s
c2εso (39)
s1 and ωs satisfy phase matching conditions [42], wheres = ωo − ω and ks1 = ko − k. Here Eso and Es1 are the slowly varying
eal functions of r and z. kso and ks1 are the propagation constantsf scattered wave. Using Eq. (38) in (37) and separating terms withifferent phases, we obtain
k2s0E2
s0 + 2iks0∂Es0
∂z+ ∇2
⊥Es0 + ω2s
c2
[εs0 + ω2
p
ω2s
·(
1 − 1�0
)]Es0
= 0 (40)
k2so1E2
s1 + 2iks1∂Es1
∂z+ ∇2
⊥Es1 + ω2s
c2
[εso + ω2
p
ω2s
·(
1 − 1�0
)]Es1
= 12
ω2p
c2
n∗
No
ωs
ωoEo (41)
Now, from Eq. (41), neglecting terms containing space deriva-ives by assuming (ro 2�
ko), one obtains the following equation
′s1 = −1
2
ω2p
c2
n�
N0
ωs
ω0
EE0[k2
s1 − k2s0 − ω2
p
c2
[1 − 1
�0
]] (42)
here E is a unit vector along E.From Eq. (40), again as considered earlier, one obtains in the
KB approximation
∂Es0
∂z= − �
2ks0∇2
⊥Es0 − �PEs0 (43)
Now, from definition of second order moment
a23 >= 1
I0
∫ ∫(x2 + y2)Es00E∗
s00dxdy (44)
here, I0 is zeroth order moment and can be written as
0 =∫ ∫
Es00E∗s00dxdy (45)
he solution of Eq.(43) is of the form,
2s00 = B2
1
f 2s
exp
[−r2
b2f 2s
](46)
Here, Esoo is the real function of r and z. b is the initial dimension
f scattered beam at z=0, B1 is the amplitude of the scattered beam,hose value is to be determined later by applying boundary condi-
ion. fs is the dimensionless beam width parameter of the scatteredeam.
4 (2013) 3470– 3475 3473
Now, from Eqs. (44), (45) and (46), it can be shown that
I0 = �B21b2 (47)
and
< a23 >= b2f 2
s (48)
With the help of Eqs. (44) and (48), one can get
d2fsd2
+ 1fs
(dfsd
)2
= k20r4
0
k2s0b4f 3
s
− k20r4
0 f 20
b4k2s0f 3
s
(ωpro
c
)2[
L3 + r20 f 2
0
b2f 2s
L4
](49)
Where, L3 =∫
t˛2−1
[1 −
(1 + ˛E2
00t
f 20
) −12
]dt and L4 =
∫log(t)
[1 −
(1 + ˛E2
00t
f 20
) −12
]t˛2−1dt.
where fs = 1 and df sd
= 0 at = (z/Rd)= 0. Eq. (49) describes thechange in the dimensionless beam width parameter fs of scatteredbeam on account of the competition between diffraction diver-gence and nonlinear refractive terms as the beam propagates inthe relativistic plasma.
Now, the value of B1 is calculated with the boundary conditionthat Es=0 at z = zc.
Es = Eso exp(+iksoz) + Es1 exp(−iks1z) = 0 (50)
at z = zc. Here, zc is the distance at which amplitude of the scatteredwave is zero. Therefore, at z = zc, one can obtain
B1 = 12
ω2p
c2
ωs
ω0
n00
N0
E00 exp(−kizc)[k2
s1 − k2s0 − ω2
p
c2
[1 − 1
�0
]] fs(zc)f0(zc)f (zc)
× exp(−i(ks1zc)exp(iks0zc)
(51)
with the condition 1b2f 2
s= 1
a2f 2 + 1r20
f 20
. Here f0(zc), f(zc), fs(zc) are the
values of dimensionless beam width parameters of pump beam,electron plasma beam and scattered beam at z = zc.
Now, reflectivity R is defined as ratio of scattered flux to incidentflux and is given by
R = 14
ω2p
c4
ω2s
ω20
n20
N200
× 1[k2
s1 − k2s0 − ω2
p
c2
[1 −
(1 + ˛E2
00exp(−1.0)
f 20
) −12
]]2 [T1 − T2 − T3]
(52)
Where
T1 = f 2s (zc)
f 20 (zc)f 2(zc)
1
f 2s
exp(−2kizc − r2
b2f 2s
) (53)
T2 = −2fs(zc)
f0(zc)f (zc)1
ff0fsexp(− r2
2b2f 2s
− r2
2a2f 2− r2
2r20 f 2
0
)
× exp(−ki(z + zc)) cos(ks1 + ks0)(z − zc) (54)
T3 = 1
f 2f 20
exp(− r2
a2f 2− r2
r20 f 2
0
− 2kizc) (55)
3474 A. Singh, K. Walia / Optik 124 (2013) 3470– 3475
Fig. 1. Variation of beam width parameter f0 against the normalized distance of
p
5
efc
fv
d
iT(ir
eg
fi
tiw
F
p
Fig. 3. Variation of beam width parameter fs against the normalized distance of
propagation = zRd
forω2
p
ω20
= 0.1 and for intensity ˛E200 = 4.0, 5.0, 6.0.
ropagation = z
Rdfor
ω2p
ω20
= 0.1 and for intensity ˛E200 = 4.0, 5.0, 6.0.
. Discussion
The differential Eqs. (21), (34), (49) for the beam width param-ters f0 of the pump beam, f of the electron plasma wave and
s of the scattered beam respectively have been solved numeri-ally for the following set of parameters; ω0 = 1.778 × 1015rads−1,
ω2p
ω20
= nencr
= 0.1, ˛E200 = 4.0, 5.0, 6.0.
Fig. 1 describes the variation of beam width parameter f0 as aunction of dimensionless distance of propagation for differentalues of intensities ˛E2
00 = 4.0, 5.0, 6.0 at a fixed value of plasma
ensityω2
p
ω20
= 0.1. It is observed from the figure that with increase
n the intensity of laser beam, there is an increase in self-focusing.his is due to the fact that the non-linear refractive term in Eq.21) is sensitive to the intensity of the laser beam. Therefore, as wencrease the intensity of the laser beam, refractive term becomeselatively stronger than diffractive term.
Fig. 2 describes the variation of beam width parameter f of thelectron plasma wave against the normalized distance of propa-ation for different values of intensities ˛E2
00 = 4.0, 5.0, 6.0 at a
xed value of plasma densityω2
p
ω20
= 0.1. It is observed from the figure
hat with increase in intensity of the main beam there is decrease
n self-focusing length of electron plasma wave. This is because, as
e increase the intensity of the laser beam, non-linear refractive
ig. 2. Variation of beam width parameter f against the normalized distance of
ropagation = zRd
forω2
p
ω20
= 0.1 and for intensity ˛E200 = 4.0, 5.0, 6.0.
Fig. 4. Variation of Reflectivity R against the normalized distance of propagation
= zRd
forω2
p
ω20
= 0.1 and for intensity ˛E200 = 4.0, 5.0.
term dominate the diffractive term and hence there is decrease inself-focusing length of the beam at higher intensities.
Fig. 3 describes the variation of beam width parameter fs of back-scattered beam against the normalized distance of propagation fordifferent values of intensities ˛E2
00 = 4.0, 5.0, 6.0 at a fixed value
of plasma densityω2
p
ω20
= 0.1. It is observed from the figure that
with increase in intensity of the main beam there is decrease inself-focusing length of scattered beam. This is due to the weaken-ing of diffractive term as compared to non-linear refractive term athigher values of intensity.
Fig. 4 describes the variation of reflectivity R against the nor-malized distance of propagation for different values of pumpbeam intensity ˛E2
00 = 4.0, 5.0 for a fixed value of plasma densityω2
p
ω20
= 0.1. It is observed from the figure that reflectivity of the scat-
tered wave is larger for ˛E200 = 4.0 than for ˛E2
00 = 5.0, which isdue to the fact that self-focusing is appreciably larger in the latercase. Thus, Self-focusing of pump beam leads to decrease in back-scattered flux and hence reflectivity.
6. Conclusion
In the present investigation, moment theory has been used tostudy the Stimulated Raman Scattering(SRS) of laser beam in Rel-ativistic Plasma. Following important observations are made frompresent analysis.
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[41] N.A. Krall, A.W. Trivelpiece, Principles of plasma physics., McGraw-Hill, New
A. Singh, K. Walia / Op
1) The effect of increase of laser beam intensity is to increase theself-focusing of pump beam.
2) The effect of increase of laser beam intensity is to decrease theself-focusing length of plasma wave and scattered wave.
3) There is a decrease in the SRS reflectivity with increase in theintensity of the pump wave.
Results of the present investigation are useful for understandinghysics of laser-induced fusion in which SRS plays a major role as
t scatters a significant fraction of laser energy and thus inhibits itsransfer to the plasma.
cknowledgement
The authors would like to thank Department of Science andechnology(DST), Government of India for the support of thisork.
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