Stimulated Raman scattering of gaussian laser beam in relativistic plasma

Optik 124 (2013) 3470–3475

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Stimulated Raman scattering of gaussian laser beam in relativistic plasma Arvinder Singh ∗ , 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 30 May 2012 Accepted 24 October 2012

Keywords: Self-focusing Relativistic Plasma Electron Plasma Wave Back-reflectivity

a b s t r a c t This paper presents an investigation of Stimulated Raman Scattering of gaussian laser beam in relativistic Plasma. The pump beam interacts with a pre-excited electron plasma wave and thereby generate a back-scattered wave. Due to intense laser beam, electron oscillatory velocity becomes comparable to the 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 incident laser beam, electron plasma wave and back-scattered beam. We have set up the non-linear differential equations for the beam width parameters of the main beam, electron plasma wave, back-scattered wave and derived SRS back-reflectivity by taking full non-linear part of the dielectric constant of relativistic plasma with the help of moment theory approach. It is observed from the analysis that self-focusing of the pump beam greatly affects the SRS reflectivity, which plays a significant role in laser induced fusion. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, the propagation of high power laser beams through plasmas has become a subject of great interest and activity. The interaction of these laser beams with plasmas have led to rapid development in areas like laser induced fusion and charged particle acceleration[1–8]. Due to availability of lasers capable of delivering high power(1018 − 1021 W/cm2 ), its interaction with plasma becomes a most interesting and important non-linear problem. At such high intensities, the response of plasma free electrons is fully relativistic (electrons swing in the laser pulse) and highly nonlinear. In the laser plasma coupling process, when a highpower laser beam interacts with the plasma, various parametric instabilities such as self-focusing, filamentation, stimulated Raman scattering, stimulated Brillouin scattering, two plasmon decay, etc.[9–16] take place, and due to these, the energy of the highpower laser beam is not efficiently coupled with plasma. These instabilities can also modify the intensity distribution and thus affect the uniformity of energy deposition. Therefore, the study of these nonlinear phenomena at high-power laser flux are being studied theoretically and experimentally. In particular, stimulated Raman scattering (SRS) governs the amount of laser energy that can be propagated over long distances through plasma. Since, many laser-matter interaction applications such as advanced radiation sources, laser plasma accelerators, laser fusion, and relativistic nonlinear optics depend critically on the amount of transmitted laser energy through the plasma. In SRS, the incident laser beam decays

∗ Corresponding author. Tel.: +91 9914142123; fax: +91 181 2690320. E-mail address: [email protected] (A. Singh). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.10.031

into a scattered wave and an electron plasma wave(EPW). The EPW produces super thermal electrons that penetrate and preheat the target core and the scattered wave represents a substantial amount of wasted energy, i.e., the energy that would otherwise get coupled to the target. Consequently SRS has been very actively studied both experimentally and theoretically. Therefore, to ensure the amount of useful and dissipated energy in laser plasma coupling, Raman reflectivity becomes a very important parameter to decide. In many theoretical studies, interesting non-linear phenomena such as self-focusing and stimulated back scattering have been carried out separately, ignoring interplay among them. There is no reason to separate the evolution of these instabilities in the nonlinear regime, where they coexist and affect each other. It becomes important to investigate and understand the interplay among various instabilities. In light of considerable current interest in self-focusing and Raman scattering, lot of work has already been done in the past [17–28]. 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 experiments, 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 [29,30] takes in to account only paraxial region of the beam, which in turn leads to error in the analysis. In paraxial theory non-linear part of the dielectric constant is Taylor expanded up to second order term and higher order terms are neglected. However, moment theory [31,32] is based on the calculation of moments and does not suffer from this defect. In moment theory approach, non-linear part of the dielectric constant is taken as a whole in calculations [33–40]. To the best of our knowledge, so

A. Singh, K. Walia / Optik 124 (2013) 3470–3475

far no one has used the moment theory approach to investigate the Stimulated Raman Scattering. Therefore, the motivation of present work is to study the effect of self-focusing of gaussian laser beam on Stimulated Raman Scattering process(SBS) in relativistic plasma with the help of moment theory approach. In the present paper, Raman Scattering of a gaussian laser beam from a relativistic plasma has been investigated. The pump wave(ω0 , k0 ) interacts with pre-excited electron plasma wave(ω,k) to generate a scattered wave(ω0 − ω, k0 − k). As a specific case, back scattering for which k  2k0 has been discussed. The relativistic non-linearity occurs on account of the increase in mass of electrons, which oscillate at relativistic velocities in an intense laser field. As a result, electrons get redistributed leading to modification in the effective dielectric constant of plasma. Consequently, the pump beam becomes self-focused. The dispersion relation for electron plasma wave is also significantly modified. The phase velocity of the electron plasma wave becomes minimum on the axis and increases away from it. Therefore, if appropriate conditions are satisfied, the electron plasma wave may also get focused. Since the scattered intensity is proportional to the intensities of the pump and electron plasma wave, it is therefore expected that the self-focusing should lead to enhanced back-scattering. The paper is organized as follows: Section 2 is devoted to the solution of wave equation for the pump beam and derivation of beam width width parameter of the pump beam with the help of moment theory approach. Section 3 is devoted to solution of wave equation for electron plasma wave and derivation of beam width width parameter of electron plasma wave. In section 4, the wave equation for the back-scattered wave is solved by moment theory approach and differential equation for beam width parameter of back-scattered wave is derived. Expression for reflectivity ‘R’ of the back-scattered is also derived. Finally a detailed discussion of results is presented in section 5.

where (E0 .E0 ) represents the non-linear part of the dielectric constant and is represented as

E0 .E0 |z=0

=

2 E00



exp −r

2

/r02



ωp2



where ωp =

4ne e2 m0

is known as the plasma frequency, e, m0 and

ne are the charge, rest mass and density of the plasma electrons respectively. On substituting for the relativistic mass m =  0 m0 , where m0 is the electron rest mass, one obtains

=1−

ωp2 0 ω02

.

(3)

 0 is a relativistic factor given by



0 =

1+

e2 E0 E0 m20 ω02 c 2

1−



1 . 0

(6)

ω02

∇ 2 E0 − ∇ (∇ .E0 ) +

c2

E0 = 0.

(7)

In the WKB approximation, the second term ∇ (∇ . E0 ) can be 2 neglected, which is justified when c 2 | 1 ∇ 2 ln |  1, ω

0

∇ 2 E0 +

ω02 E0 c2

= 0.

(8)

One can take E0 = A(r, z) exp[{ω0 t − k0 z}]

(9)

where, A(r, z) is a complex function of its argument. The behaviour of the complex amplitude A(r, z) is governed by the parabolic equation obtained from the wave Eq. (8) in the WKB approximation by assuming variations in the z direction being slower than those in the radial direction. 

1 dA = ∇ 2 A + (AA )A dz 2k0 ⊥

(10)

k

where (AA ) = 20 ( − 0 ) and  = 0 + (|AA |2 ), where o = 1 − 0 ωp2

and (|AA |2 ) are the linear and nonlinear parts of the dielectric ω √ constant, respectively. Also, k0 = c0 0 and ωp are propagation constant and plasma frequency, respectively. Now from the definition of the second order moment, the mean square radius of the beam is given by ω2 0

< a21 >=

(x2 + y2 )AA dxdy I0

.

(11)

From here, one can obtain the following equation. d2 < a21 > dz

=

2

4I2 4 − I0 I0



Q (|A|2 )dxdy

(12)

where, I0 and I2 are the invariants of Eq. (10)[32]



|A|2 dxdy

I0 =



1

I2 =

2k02

With[31] F(|A|2 ) = and

1 k0

2

Q (|A| ) =

(13)

(|∇ ⊥ |A|2 − F)dxdy

(14)

(|A|2 )d(|A|2 )

(15)



|A|2 (|A|2 ) − 2F(|A|2 ) . k0

(16)

For z > 0, we assume an energy conserving gaussian ansatz for the laser intensity[29,30] (4)



AA =

Therefore, the intensity dependent dielectric constant is given by

 = o + (E0 .E0 )



The slowly varying electric field E0 satisfies the following wave equation.

(1)

(2)

ω02

ω02



where r2 = x2 + y2 and r0 is the initial width of the main beam. r is the radial co-ordinate of the cylindrical co-ordinate system. The dielectric constant of the plasma is given by

0 = 1 −

ωp2

(E0 .E0 ) =

2. Solution of Wave Equation for Pump beam Consider the propagation of a high power laser beam of angular frequency ω0 in a relativistic plasma along the z axis. The initial intensity distribution of the beam along the wavefront at z = 0 is given by

3471

(5)

2 E00

f02



exp

−

r2 r02 f02

.

(17)

From Eqs. (11), (13) and (17), it can be shown that 2 I0 = r02 E00 ,

(18)

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A. Singh, K. Walia / Optik 124 (2013) 3470–3475

< a21 >= r02 f 2

(19)

where, f0 is the dimensionless beam width parameter and r0 is the beam width at z = 0. Now, from Eqs. (12)-(19) one obtains d2 f0 1 . 2 + f0 d

 df 2 0

=

d



2k02 2 f E00 0





2

I2 −

Q (|A| )dxdy .

(20)

where = (z/Rd ) is the dimensionless distance of propagation and Rd is the Rayleigh length. Eq. (20) is the basic equation for studying the self-focusing of a gaussian laser beam in a nonlinear, nonabsorptive medium. Now, with the help of Eqs. (6),(14)-(17) and (20), we get d2 f0 d 2

+

 2 1 df 0

f0

d

⎛ 1 . f0

⎜ ⎜ ⎝

=

⎡ 2f02 2 ˛E00

⎢ ⎢1 − ⎣

1 f03

+

 1+

 ω r 2 p 0 c

2 ˛E00 f02

i ωn1 ∂n1 = − ∇ 2⊥ n1 − Pn1 − 2k ∂z ki v2th Where P can be written as P =

(28)



ω2

pi

2ki v2

1−

th

1 0



.

Now, from the definition of second order moment < a22 >=

1 I0



(x2 + y2 )n1 n∗1 dxdy

(29)

Where, I0 is zeroth order moment and can be written as

⎛ ⎜ + log ⎜  ⎝

Where n1 is the slowly varying real function of r and z, ω and k are the frequency and propagation constant for electron plasma wave. Substituting the value of  n from Eq. (27) in Eq. (26), one obtains in the Wentzal- Kramers - Brillouin (WKB) approximation

1+

˛E 2

00

f2

+1

0

1+

˛E 2

00

f2



⎞⎤⎞

−1

⎟⎥⎟ ⎟⎥⎟ . ⎠⎦⎠



(21) The initial conditions for a plane wave front are

(30)

Now,Following [29,30] solution of Eq. (28) is of the form,

0

df 0 d

n1 n∗1 dxdy

I0 =

= 0 and f0 = 1 at

r2 n00 n1 = exp − ki z f 2a2 f 2



(31)

= 0. Eq. (21) describes the changes in the beam width parameter of a gaussian laser beam on account of the competition between diffraction divergence and nonlinear focusing terms as the beam propagates in the relativistic plasma.

Where, n00 is the axial amplitude of density perturbation of electron plasma wave. f is the dimensionless beam width parameter of electron plasma wave. a is the initial width of the electron plasma wave at z=0 and ki is damping factor. Now, from Eqs. (29), (30) and (31), it can be shown that

3. Solution of Wave Equation for Electron plasma wave

I0 = n200 a2 exp(−2ki z)

The pump laser beam interacts with electron plasma wave and leads to its excitation. The motion of plasma particles is described in the hydrodynamic approximation by the fluid equations [41] ∂N + ∇ · (NV ) = 0 ∂t



∂V + (V · ∇ )V m ∂t



(22) 1  = −e[E + V × B] − 2 mV − ∇ P c N

and < a22 >= a2 f 2 exp(−2ki z)



ωp

8 (k )3 de

exp −

1 2k2 2

de

the wave vector of electrostatic wave and de =

−



3 2

d2 f 1 + f d 2

(23) ×

, where k is

kB T0 4N0 e2

is debye

length of plasma. Applying Perturbation approximation, N = Noe + n, V = V0 + v, E = EH + EP

(24)

where, n  Noe , v  V0 and EP  EH , where V0 is the particle velocity in presence of high frequency field EH , the plasma is assumed to have no drift velocity. The self consistent field EP of the plasma wave, satisfies the following Poisson’s equation.

∇ · EP = −4ne

(25)

Following standard techniques, one obtains the general equation governing the electron density variation, ∂2 n ∂n 1 + 2 − i v2th ∇ 2 n + ωp2 n = 0 0 ∂t ∂t 2



where, vth =

kB To m

(26)

is the electron thermal velocity. In order to

1 f



 df 2 d

f02

L1 f2

Where



t ˛1 −1

+



= exp(2ki z)



f04

L2 f4

L1 =



(27)

+

t ˛1 −1



log(t) 1 − 1 +

1 1 − 4i 4f 3

 ω r 2 p o vth

2 ω2 fr04 2i2 v2th

exp(−2ki (z))



1− 1+

˛E 2 t 00 f2 0

 −12

!

˛E 2 t 00 f2 0

 −12

(34)

! dt

and

L2 =

dt

Eq. (34) describes the variation in the dimensionless beam width parameter f of electron plasma wave on account of the competition between diffraction divergence and nonlinear refractive terms with the normalized distance of propagation in the relativistic plasma df with f = 1 and d

= 0 at = 0. 4. Solution of Wave Equation for back-scattered beam The high frequency electric field EH may be written as a sum of the electric field E0 of the incident beam and Es of the scattered wave, i.e EH = E0 exp(iωo t) + Es exp(iωs t)

(35)

where, Es is due to the scattering of the pump beam from the electron plasma wave(i.e Raman Scattering),  ωs represents scattered frequency. The vector EH satisfies the wave equation

solve Eq. (26) for n, take n as n = n1 (r, z) exp((ωt − kz))

(33)

Now, with the help of Eqs. (29) and (33), it can be shown that

where N is the instantaneous electron density, V is electron fluid velocity,P = NkB T0 is hydrodynamic pressure, E and B are the electric and magnetic fields respectively, is the landau  damping factor given by [41] 2 =

(32)

∇ 2 EH − ∇ (∇ · EH ) =

1 ∂2 EH 4 ∂JH + 2 c 2 ∂t 2 c ∂t

(36)

A. Singh, K. Walia / Optik 124 (2013) 3470–3475

where, JH is the total current density vector in the presence of high frequency electric field EH . Equating the terms at scattered frequency  ωs , we get



ωs2 c2

∇ 2 Es +



ωp2

1−

0 ωs2



Es =



ωp2 ωs n∗

(37)

In order to solve Eq. (37), second term on right hand side has been neglected by assuming that the scale length of variation of the dielectric constant in the radial direction is much larger than the wavelength of pump. The solution of Eq. (37) may be obtained in the form Es = Eso exp(+ikso z) + Es1 exp(−iks1 z) where 2 kso



ω2 = 2s c

1−

ωp2

ω2 = 2s εso c

ωs2

(39)



εs0 +

ωp2 ωs2

·



1 1− 0



Es0

= 0

(40)

1 = 2

∂Es1 ω2 + ∇ 2⊥ Es1 + 2s + 2iks1 ∂z c

ωp2 c2

εso +

ωp2 ωs2

·



1 1− 0

 Es1

ωs Eo No ωo

n ωs N0 ω0





" E E0 2 − k2 − ks1 s0

ωp2 c2



1−

 1

(42)

0

∂Es0  =− ∇ 2 Es0 − PEs0 2ks0 ⊥ ∂z

<

(43)

2

(x + y

2

∗ )Es00 Es00 dxdy

∗ Es00 Es00 dxdy

2 = Es00

fs2

exp

−r 2 b2 fs2



2 f3 b4 ks0 s



t ˛2 −1 1 − 1 +

˛E 2 t 00 f2 0

df s d

 −12

 ω r 2

p o

L3 +

c

˛E 2 t 00

r02 f02



L 2 4

b2 fs

(49)

!

 −12

dt

f2

and

L4 =

0

! t ˛2 −1 dt.

= 0 at = (z/Rd )= 0. Eq. (49) describes the

change in the dimensionless beam width parameter fs of scattered beam on account of the competition between diffraction divergence and nonlinear refractive terms as the beam propagates in the relativistic plasma. Now, the value of B1 is calculated with the boundary condition that Es =0 at z = zc . Es = Eso exp(+ikso z) + Es1 exp(−iks1 z) = 0

1 ωp ωs n00 2 c 2 ω0 N0

×



(50)

E00 exp(−ki zc ) 2 − k2 − ks1 s0

f (z )

s c  f0 (zc )f (zc ) 1



ωp2

1−

c2

0

exp(−i(ks1 zc ) exp(iks0 zc )

with the condition

(51)

1 b2 fs2

=

1 a2 f 2

+

1 . Here f0 (zc ), f(zc ), fs (zc ) are the r2 f 2 0 0

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 incident flux and is given by R=

2 1 ωp ωs2 n20 4 4 c ω2 N 2 0 00

1

× ωp2 c2



1− 1+

˛E 2 exp(−1.0) 00

 −12

!!2 [T1 − T2 − T3 ]

f2 0

(52)

(44) Where (45)

T1 =

fs2 (zc ) 1 2 f0 (zc )f 2 (zc ) fs2

The solution of Eq.(43) is of the form,



L3 =

2 − k2 − ks1 s0



B12

k02 r04 f02

−

2 b4 f 3 ks0 s

where fs = 1 and



Where, I0 is zeroth order moment and can be written as I0 =

d

k02 r04

=

log(t) 1 − 1 +

Now, from definition of second order moment 1 >= I0

s



(41)

Where " E is a unit vector along E. From Eq. (40), again as considered earlier, one obtains in the WKB approximation

a23

 df 2

2

o

1 2

d2 fs 1 + fs d 2

at z = zc . Here, zc is the distance at which amplitude of the scattered wave is zero. Therefore, at z = zc , one can obtain

Now, from Eq. (41), neglecting terms containing space deriva), one obtains the following equation tives by assuming (ro 2 k  Es1 =−

(48)

With the help of Eqs. (44) and (48), one can get

B1 =

n∗

ωp2 c2

(47)

< a23 >= b2 fs2

Where,



∂Es0 ω2 + ∇ 2⊥ Es0 + 2s + 2iks0 ∂z c

2 2 −kso1 Es1

I0 = B12 b2

(38)

ks1 and ωs satisfy phase matching conditions [42], where ωs = ωo − ω and ks1 = ko − k. Here Eso and Es1 are the slowly varying real functions of r and z. kso and ks1 are the propagation constants of scattered wave. Using Eq. (38) in (37) and separating terms with different phases, we obtain 2 2 −ks0 Es0

Now, from Eqs. (44), (45) and (46), it can be shown that

and

E0 − ∇ (∇ · E0 )

20 c 2 ωo No

3473



T2 = −2 (46)

Here, Esoo is the real function of r and z. b is the initial dimension of scattered beam at z=0, B1 is the amplitude of the scattered beam, whose value is to be determined later by applying boundary condition. fs is the dimensionless beam width parameter of the scattered beam.

exp(−2ki zc −

r2 b2 fs2

)

fs (zc ) r2 r2 r2 1 exp(− − − 2 2) f0 (zc )f (zc ) ff0 fs 2a2 f 2 2b2 fs2 2r0 f0

× exp(−ki (z + zc )) cos(ks1 + ks0 )(z − zc )

T3 =

(53)

1 f 2 f02

exp(−

r2 r2 − 2 2 − 2ki zc ) 2 2 a f r0 f0

(54)

(55)

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A. Singh, K. Walia / Optik 124 (2013) 3470–3475

Fig. 1. Variation of beam width parameter f0 against the normalized distance of propagation =

z Rd

for

2 ωp

ω2 0

2 = 0.1 and for intensity ˛E00 = 4.0, 5.0, 6.0.

Fig. 3. Variation of beam width parameter fs against the normalized distance of propagation =

z Rd

for

2 ωp

ω2 0

2 = 0.1 and for intensity ˛E00 = 4.0, 5.0, 6.0.

5. Discussion The differential Eqs. (21), (34), (49) for the beam width parameters f0 of the pump beam, f of the electron plasma wave and fs of the scattered beam respectively have been solved numerically for the following set of parameters; ω0 = 1.778 × 1015 rads−1 , ωp2 ω2 0

=

ne ncr

2 = 4.0, 5.0, 6.0. = 0.1, ˛E00

Fig. 1 describes the variation of beam width parameter f0 as a function of dimensionless distance of propagation for different 2 = 4.0, 5.0, 6.0 at a fixed value of plasma values of intensities ˛E00 density

ωp2 ω2

= 0.1. It is observed from the figure that with increase

0

in the intensity of laser beam, there is an increase in self-focusing. This 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 we increase the intensity of the laser beam, refractive term becomes relatively stronger than diffractive term. Fig. 2 describes the variation of beam width parameter f of the electron plasma wave against the normalized distance of propa2 = 4.0, 5.0, 6.0 at a gation for different values of intensities ˛E00 fixed value of plasma density

ωp2 ω2

= 0.1. It is observed from the figure

0

that with increase in intensity of the main beam there is decrease in self-focusing length of electron plasma wave. This is because, as we increase the intensity of the laser beam, non-linear refractive

Fig. 4. Variation of Reflectivity R against the normalized distance of propagation

=

z Rd

for

2 ωp

ω2 0

2 = 0.1 and for intensity ˛E00 = 4.0, 5.0.

term dominate the diffractive term and hence there is decrease in self-focusing length of the beam at higher intensities. Fig. 3 describes the variation of beam width parameter fs of backscattered beam against the normalized distance of propagation for 2 = 4.0, 5.0, 6.0 at a fixed value different values of intensities ˛E00 of plasma density

ωp2 ω2

= 0.1. It is observed from the figure that

0

with increase in intensity of the main beam there is decrease in self-focusing length of scattered beam. This is due to the weakening of diffractive term as compared to non-linear refractive term at higher values of intensity. Fig. 4 describes the variation of reflectivity R against the normalized distance of propagation for different values of pump 2 = 4.0, 5.0 for a fixed value of plasma density beam intensity ˛E00 ωp2 ω2

= 0.1. It is observed from the figure that reflectivity of the scat-

0

2 = 4.0 than for ˛E 2 = 5.0, which is tered wave is larger for ˛E00 00 due to the fact that self-focusing is appreciably larger in the later case. Thus, Self-focusing of pump beam leads to decrease in backscattered flux and hence reflectivity.

6. Conclusion

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

z Rd

for

2 ωp

ω2 0

2 = 0.1 and for intensity ˛E00 = 4.0, 5.0, 6.0.

In the present investigation, moment theory has been used to study the Stimulated Raman Scattering(SRS) of laser beam in Relativistic Plasma. Following important observations are made from present analysis.

A. Singh, K. Walia / Optik 124 (2013) 3470–3475

(1) The effect of increase of laser beam intensity is to increase the self-focusing of pump beam. (2) The effect of increase of laser beam intensity is to decrease the self-focusing length of plasma wave and scattered wave. (3) There is a decrease in the SRS reflectivity with increase in the intensity of the pump wave. Results of the present investigation are useful for understanding physics of laser-induced fusion in which SRS plays a major role as it scatters a significant fraction of laser energy and thus inhibits its transfer to the plasma. Acknowledgement The authors would like to thank Department of Science and Technology(DST), Government of India for the support of this work. References [1] S.P. Regan, D.K. Bradely, A.V. Chirokikh, R.S. Craxton, D.D. Meyerhoffer, W. Seka, R.W. Short, A. Simon, R.P.J. Town, B. Yakoobi, J.J. Carroll, R.P. Drake, Laser-plasma interactions in long-scale-length plasmas under direct-drive National Ignition Facility conditions, Phys. Plasmas 6 (1999) 2072–2080. [2] A.Y. Faenov, A.I. Magunov, T.A. Pikuz, I.Y. Skobelev, S.V. Gasilov, S. Stagira, F. Calegari, M. Nisoli, S.D. Silvestri, L. Poletto, P. Villoresi, A.A. Andreev, X-ray spectroscopy observation of fast ions generation in plasma produced by short low-contrast laser pulse irradiation of solid targets, Laser Part. Beams 25 (2007) 267–275. [3] L. Torrisi, D. Margarone, L. Laska, J. Krasa, A. Velyhan, M. Pffifer, J. Ullschmied, L. Ryc, Self-focusing effect in Au-target induced by high power pulsed laser at PALS, Laser Part. Beams 26 (2008) 379–387. [4] T. Tajima, J.M. Dawson, Laser Electron Accelerator, Phys. Rev. Lett 43 (1979) 267–270. [5] P. Jha, P. Kumar, A.K. Upadhay, G. RAJ, Electric and magnetic wakefields in a plasma channel, Phys. Rev. ST Accel. Beams 8 (2005) 071301–071306. [6] C. Deutsch, H. Furukawa, K. Mima, M. Murukami, K. Nishihara, Interaction Physics of the Fast Ignitor Concept, Phys. Rev. Lett 77 (1996) 2483–2486. [7] M.H. Key, et al., Hot electron production and heating by hot electrons in fast ignitor research, Phys. Plasmas 5 (1998) 1966–1972. [8] M. Tabak, J. Hammer, M.E. Glinsky, W.L. Kruer, S.C. Wilks, J. Woodworth, E.M. Campbell, M.D. Perry, R.J. Mason, Ignition and high gain with ultrapowerful lasers, Phys. Plasmas 1 (1994) 1626–1634. [9] B.E. Lemoff, G.Y. Yin, C.L. Gordon III, C.P.J. Barty, S.E. Harris, Demonstration of a 10-Hz Femtosecond-Pulse-Driven XUV Laser at 41.8 nm in Xe IX, Phys. Rev. Lett 74 (1995) 1574–1577. [10] L.M. Goldman, J. Sourse, M.J. Lubin, Saturation of Stimulated Backscattered Radiation in Laser Plasmas, Phys. Rev. Lett 31 (1973) 1184–1187. [11] X. Liu, D. Umstadter, E. Esarey, A. Ting, Harmonic generation by an intense laser pulse in neutral and ionized gases, IEEE Trans Plasma Sci 21 (1993) 90–94. [12] P. Dombi, P. Racz, B. Bodi, Surface plasmon enhanced electron acceleration with few cycle laser pulses, Laser Part. Beams 27 (2009) 291–296. [13] J.L. Kline, D.S. Montgomery, C. Rousseax, S.D. Baton, V. Tassin, R.A. Hardin, K.A. Flippo, R.P. Johnson, T. Shimada, L. Yin, B.J. Albright, H.A. Rose, F. Amiranoff, Investigation of stimulated Raman scattering using a short-pulse diffraction limited laser beam near the instability threshold, Laser Part. Beams 27 (2009) 185–190. [14] W.L.J. Hasi, S. Gong, Z.W. Lu, D.Y. Lin, W.M. He, R.Q. Fan, Generation of plasma wave and third harmonic generation at ultra relativistic laser power, Laser Part. Beams 26 (2008) 511–516. [15] P. Kappe, A. Strasser, M. Ostermeyer, Investigation of the impact of SBS- parameters and loss modulation on the mode locking of an SBS- laser oscillator, Laser Part. Beams 25 (2007) 107–116.

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[16] Y.L. Wang, Z.W. Lu, W.M. He, Z.X. Zheng, Y.H. Zhao, A new measurement of stimulated Brillouin scattering phase conjugation fidelity for high pump energies, Laser Part, Beams 27 (2009) 297–302. [17] M.R. Amin, C.E. Capjack, P. Frycz, W. Rozmus, V.T. Tikhonchuk, Twodimensional studies of stimulated Brillouin scattering, filamentation, and self-focusing instabilities of laser light in plasmas, Phys. Fluids 5 (1993) 3748–3764. [18] C.S. Liu, V.K. Tripathi, Thermal effects on coupled self-focusing and Raman scattering of a laser in a self-consistent plasma channel, Phys, Plasmas 2 (1995) 3111–3114. [19] H.C. Barr, T.J.M. Boyd, G.A. Coutts, Stimulated Raman scattering in the presence of filamentation in underdense plasmas, Phys. Rev. Lett 56 (1986) 2256–2259. [20] D.A. Russell, D.F. Dubois, H.A. Rose, Nonlinear saturation of stimulated Raman scattering in laser hot spots, Phys, Plasmas 6 (1999) 1294–1317. [21] J. Fuchs, C. Labaune, S. Depierreux, V.T. Tikhonchuk, H.A. Baldis, Stimulated Brillouin and Raman scattering from a randomized laser beam in large inhomogeneous collisional plasmas. I. Experiment, Phys. Plasmas 7 (2000) 4659–4668. [22] D.V. Rose, J. Guillory, J.H. Beall, Enhanced Landau damping of finite amplitude electrostatic waves in the presence of suprathermal electron tails, Phys. Plasmas 12 (2005) 014501. [23] W. Rozmus, Nonlinear Langmuir waves in stimulated Raman scattering, Physica Scripta T30 (1990) 64. [24] S.T. Mahmoud, R.P. Sharma, Effect of pump depletion and self-focusing (hot spot) on stimulated Raman scattering in laser-plasma interaction, J. Plasma Physics 64 (2000) 613–621. [25] R.W. Short, A. Simon, Collisionless damping of localized plasma waves in laserproduced plasmas and application to stimulated Raman scattering in filaments, Phys, Plasmas 5 (1998) 4134–4143. [26] T. Matsuoka, et al., Focus optimization of relativistic self-focusing for anomalous laser penetration into overdense plasmas (super-penetration), Plasma Phys. Control. Fusion 50 (2008) 105011. [27] K.C. Tzeng, W.B. Mori, Suppression of Electron Ponderomotive Blowout and Relativistic Self-Focusing by the Occurrence of Raman Scattering and Plasma Heating, Phys. Rev. Lett. 81 (1998) 104–107. [28] S.V. Bulanov, F. Pegoraro, A.M. Pukhov, Two-Dimensional Regimes of SelfFocusing, Wake Field Generation, and Induced Focusing of a Short Intense Laser Pulse in an Underdense Plasma, Phys. Rev. Lett 74 (1995) 710–713. [29] S.A. Akhmanov, A.P. Sukhorukov, R.V. Khokhlov, Self-focusing and diffraction of light in a non-linear medium, Sov.Phys.Uspekhi 10 (1968) 609–636. [30] M.S. Sodha, A.K. Ghatak, V.K. Tripathi, Progress in Optics, Vol. 13, 171, Amsterdam, North Holland, 1976. [31] S.N. Vlasov, V.A. Petrishchev, V.I. Talanov, Averaged Description of Wave Beams in Linear and Nonlinear Media(the Method of Moments), Radiophys. Quantum Electron 14 (1971) 1062–1070. [32] J.F. Lam, B. Lippman, F. Tappert, Self-trapped laser beams in plasma, Phys. Fluids 20 (1977) 1176–1179. [33] M.S. Sodha, S.K. Sinha, R.P. Sharma, The self-focusing of laser beams in magnetoplasmas: the moment theory approach, J. Phys. D: Appl. Phys 12 (1979) 1079–1091. [34] A. Singh, N. Singh, Optical guiding of a laser beam in an axially nonuniform plasma channel, Laser Part. Beams 28 (2010) 263–268. [35] A. Singh, N. Singh, Relativistic guidance of an intense laser beam through an axially non-uniform plasma channel, Laser Part. Beams 29 (2011) 291–298. [36] A. Singh, N. Singh, Guiding of a Laser Beam in Collisionless Magnetoplasma Channel, J. Opt. Soc. Am. B 28 (2011) 1844–1850. [37] A. Singh, K. Walia, Relativistic self-focusing and self-channeling of Gaussian laser beam in plasma, Appl. Phys. B 101 (2010) 617–622. [38] A. Singh, K. Walia, Self-focusing of Gaussian Laser Beam Through Collisionless Plasmas and Its Effect on Second Harmonic Generation, J Fusion Energ 30 (2011) 555–560. [39] A. Singh, K. Walia, Self-focusing of laser beam in collisional plasma and its effect on Second Harmonic generation, Laser Part. Beams 29 (2011) 407–414. [40] K. Walia, A. Singh, Comparison of Two Theories for the Relativistic Self Focusing of Laser Beams in Plasma, Contrib. Plasma Phys 51 (2011) 375–381. [41] N.A. Krall, A.W. Trivelpiece, Principles of plasma physics., McGraw-Hill, New York, 1973. [42] J.F. Drake, P.K. Kaw, Y.C. Lee, G. Schmidt, C.S. Liu, M.N. Rosenbluth, Parametric instabilities of electromagnetic waves in plasmas, Phys. Fluids 17 (1974) 778–785.