Self-focusing of Gaussian laser beam in collisionless plasma and its effect on stimulated Brillouin scattering process

Optics Communications 290 (2013) 175–182

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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

a r t i c l e i n f o

abstract

Article history: Received 16 November 2011 Received in revised form 13 August 2012 Accepted 24 October 2012 Available 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 ionacoustic 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.

Keywords: Self-focusing Ponderomotive force Ion-acoustic wave Back-reflectivity

1. Introduction There has been considerable interest in the non-linear interaction of intense laser beams with plasmas on account of its relevance to laser induced fusion and charged particle acceleration [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) application, 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

n

Corresponding author. Tel.: þ91 9914142123; fax: þ91 181 2690320. E-mail address: [email protected] (A. Singh).

0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.10.063

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. Theoretical 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 electromagnetic 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 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 [29–31] 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

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A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

are neglected. Recently, Sharma et al. [32] used the higher order paraxial theory in which the off-axial part of the laser beam is taken into consideration up to certain extent by including higher order terms and compared their results with that of paraxial theory approach [31]. However, the moment theory [33,34] is based on the calculation of moments and does not suffer from this defect. In moment theory approach, full non-linear part of the dielectric constant is taken as a whole in calculations. This theory has recently been used in the past for studying the effect of selffocusing of Gaussian laser beam on self-channeling, Second harmonic generation [35–37]. So, our motivation of the present work is to study the effect of self-focusing of Gaussian laser beam on stimulated Brillouin scattering (SBS) process in collisionless plasma with the help of moment theory approach. Moment theory is difficult to apply wherever the propagation of more than one wave is involved and therefore one always prefer to apply paraxial theory and higher order paraxial theory, in which the mathematical calculations become simpler as compared to moment theory approach. To the best of our knowledge, so far no one has used moment theory approach to study the SBS process. So, the novelty of the present work is that, we have considered the full non-linear part of the dielectric constant in the present investigation and compared the results with that of paraxial theory approach. The pump wave (o0 , k0) interacts with pre-excited ionacoustic wave (o,k) to generate a scattered wave (o0 o, k0 k). As a specific case, back scattering for which k C2k0 has been discussed. The pump beam exerts a ponderomotive force on the electrons, leading to redistribution of carriers and consequently, the pump beam becomes self-focused. The dispersion relation for ion-acoustic wave is also significantly modified. The phase velocity of the ion-acoustic wave becomes minimum on the axis and increases away from it. Therefore, if appropriate conditions are satisfied, the ion-acoustic wave may also get focused. In Section 2, the wave equation for the laser beam has been set up and differential equations for the beam width parameter of the laser beam have been derived with the help of moment theory and paraxial theory approach. In Section 3, we have set up the wave equation for the ion-acoustic wave and used the moment theory as well as paraxial theory approach to obtain the differential equations for the beam width parameter of the ion-acoustic wave. In Section 4, the wave equation for the back-scattered wave has been set up and differential equation for the beam width parameter of the back-scattered wave have been derived. In Section 5, expression for the reflectivity ‘R’ of the back-scattered beam has been derived. Finally, a detailed discussion of the results has been presented in Section 6.

2. Solution of wave equation for pump beam Consider the propagation of Gaussian laser beam of frequency

o0 and wave vector k0 in hot collisionless and homogeneous plasma along z-axis. When a laser beam propagates through plasma, the transverse intensity gradient generates a ponderomotive force, which modifies the plasma density profile in the transverse direction as   3 m N0e ¼ N 00 exp  a E0  En0 ð1Þ 4 M where a ¼ e2 M=6K B T 0 gm2 o20 . e and m are the electronic charge and mass, M and T 0 are mass of ion and equilibrium temperature of plasma respectively. N0e is electron concentration in the presence of laser beam, N00 is the electron concentration in the absence of laser beam, K B is Boltzman’s constant and g is ratio of two specific heats.

The initial intensity distribution of beam along the wavefront at z¼0 is given by 2 2 2 E0 :E% 0 9z ¼ 0 ¼ E00 exp½r =r 0  2

2

ð2Þ

2

where r ¼ x þ y and r 0 is the initial width of the pump beam and r is radial co-ordinate of the cylindrical co-ordinate system. E0 is the electric field vector of pump beam and E00 is the axial amplitude of the beam. Slowly varying electric field E0 of the pump beam satisfies the following wave equation:

r2 E0 rðr:E0 Þ þ

o20 c2

EE0 ¼ 0

ð3Þ

In the Wentzal–Kramers–Brillouin (WKB) approximation, the second term rðr  E0 Þ of Eq. (3) can be neglected, which is justified when    c2 1 2  51 r ln E   2

o0 E

r2 E0 þ

o20 c2

EE0 ¼ 0

ð4Þ

% where E ¼ E0 þ FðE0 E% 0 Þ , E0 and FðE0 E0 Þ are the linear and nonlinear parts of the dielectric constant respectively. Here   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 o2p N0e E0 ¼ 1 p2 and FðE0 E% 1 o ¼ 4pN 0 e2 =m , p 0 Þ¼ N0 o0 o20

is the electron plasma frequency. Further taking E0 in Eq. (4) as E0 ¼ Aðr,zÞexp½ifo0 tk0 zg

ð5Þ

where Aðr,zÞ is a complex function of its argument. The behavior of the complex amplitude Aðr,zÞ is governed by the parabolic equation obtained from the wave equation (4) in the WKB approximation by assuming variations in the z direction being slower than those in the radial direction

i

dA 1 ¼ r2 A þ wðE0  E% 0 ÞA dz 2k0 ?

ð6Þ

where k0 ðEE0 Þ 2E0

wðE0  E% 0 Þ¼

and

E ¼ E0 þ Fð9E0 92 Þ

2.1. Moment theory approach Now from the definition of the second order moment, the mean square radius of the beam is given by RR 2 ðx þ y2 ÞAA% dx dy ð7Þ /a2 S ¼ I0 From here one can obtain the following equation: ZZ 2 d /a2 S 4I2 4 2 ¼  Q ð9A9 Þ dx dy 2 I0 I0 dz where I0 and I2 are the invariants of Eq. (6) [33] ZZ 2 I0 ¼ 9A9 dx dy

I2 ¼

ZZ

1  2 2k0

 2 9r? 9A9 F dx dy

ð8Þ

ð9Þ

ð10Þ

With [34] 2

Fð9A9 Þ ¼

1 k0

Z

wð9A92 Þdð9A92 Þ

ð11Þ

A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

177

and

and 2

"

Q ð9A9 Þ ¼

2

2

9A9 wð9A9 Þ 2 2Fð9A9 Þ k0

# ð12Þ

k0 ¼

o0 E01=2 c 1

For z 4 0, we assume an energy conserving Gaussian ansatz for the laser intensity [29,30] ( ) E200 r2 % AA ¼ 2 exp  ð13Þ 2 f0 r 20 f 0 From Eqs. (7), (9) and (13) it can be shown that I0 ¼ pr 20 E200

ð14Þ 2

/a2 S ¼ r 20 f 0

ð15Þ

where f 0 is dimensionless beam width parameter and r 0 is beam width at z ¼ 0. Now, from Eqs. (8)–(15) we get   2 ZZ 2 2  d f0 1 df 0 2k0 2 þ ¼ I2  Q ð9E0 9 Þ dx dy ð16Þ 2 2 f 0 dx pE00 f 0 dx where x ¼ ðz=k0 r 20 Þ is the dimensionless propagation distance. Eq. (16) is a basic equation for studying the self-focusing of a Gaussian laser beam in a non-linear, non-absorptive medium. Now, by making use of (1), (10)–(13) and (16) we get  2 2 d f0 1 df 1 op r 0 2 1 þ ¼ 3  2 f 0 dx f0 c f0 dx " " ## ! Z 2 2 f0 a1 E00 expðbtÞ1 exp ð17Þ 1 2 t a1 E200 f 0

Initial conditions of plane wavefront are df 0 =dx ¼ 0 and f 0 ¼1 at x ¼ 0. Eq. (17) describes the change in the beam width parameter of a Gaussian beam on account of the competition between diffraction divergence and non-linear refractive terms as the beam propagates in the collisionless plasma.

Further assuming the variation of Aðr,zÞ as

ð19Þ

and  2 @A20 @S0 @A0 2 þ þ A20 r? S0 ¼ 0 @z @r @r

ð20Þ

Following [29,30], the solutions for Eqs. (19) and (20) can be written as " # E2 r2 A20 ¼ 00 exp  ð21Þ 2 2 f0 r 20 f 0 2

S0 ¼

r b ðzÞ þ F0 ðzÞ 2 0

where 1 df 0 b0 ðzÞ ¼ f 0 dz

3. Solution of wave equation for ion-acoustic wave The laser beam interacts with the ion-acoustic wave (IAW) and leads to its excitation. To analyze the excitation process of ionacoustic wave in the presence of ponderomotive non-linearity. We start with the following set of fluid equations [38]. Continuity equation: @nis þ r  ðN0 V is Þ ¼ 0 @t

ð24Þ

Momentum equation: @V is gi v2th e þ rnis þ2Gi V is  Esi ¼ 0 M @t N0

ð25Þ

ð18Þ

where A0 ðr,zÞ and S0 are real functions of r and z (S0 being the ekional). On substituting A in Eq. (6) and separating the real and imaginary parts of the resulting equation, the following set of equations is obtained:   2   o2p @S0 @S0 1 N 0e 2 þ 2 ¼ 2 r? A0 þ 2 1 @z @r N0 o0 E 0 k0 A0

where f 0 ¼ 1 and df 0 =dz ¼ 0 at z¼0. Eq. (23) describes the change in the dimensionless beam width parameter f 0 of pump beam on account of the competition between diffraction divergence term and non-linear refractive term as the beam propagates in the collisionless plasma.

where nis is the perturbation in the ion density, V is is the velocity of ion-fluid, vth is the ion-thermal velocity, gi is the ratio of specific heat of ion-gas, Gi is the Landau damping factor of the ion wave, Esi is the electric field associated with the generated ion-acoustic wave, satisfying Poisson’s equation:

2.2. Paraxial ray approximation

Aðr,zÞ ¼ A0 ðr,zÞexp½ik0 S0 ðr,zÞ

The parameter b0 may be interpreted as the radius of the curvature of the main beam and F0 ðzÞ is the phase shift, which we do not require for the further analysis as we are interested in the intensity of the laser beam rather than its phase. On substituting Eqs. (21) and (22) in Eq. (19) and on equating the coefficients of r 2 on both sides, we get the following differential equation for the beam width parameter f 0 of the laser beam: !  2 d f0 1 op r 0 2 3 m 2 3 m E200 1 E ¼   a a ð23Þ exp  2 3 4 M 00 4 M f2 f3 c f0 dx 0 0

ð22Þ

r  Esi ¼ 4peðnes nis Þ

ð26Þ

where nes and nis corresponds to perturbations in the electron and ion densities, and are related to each other by following equation: 2 6 nes ¼ nis 6 41þ

31 2 2 k ld 7 7

ð27Þ

N0e 5 N0

where k is the propagation constant for ion-acoustic wave, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ld ¼ kB T 0 =4pN 0 e2 is Debye length. The Landau damping coefficient Gi for IAW is given by [39] 2 0 13 Te rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi sffiffiffiffiffi B C7 k pkB T e 6 m 3 Te T i C7 6 þ expB 2Gi ¼ @ 2 2 2 2 A5 8M 4 M Ti 1 þ k ld 1þ k ld where T e and T i 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 2 2

@2 nis @n N 0e k ld þ 2Gi is gv2th r2 nis þ o2pi n ¼0 @t N0 1þ k2 l2 is @t 2 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where vth ¼ kB Ti=mi is the ion thermal velocity.

ð28Þ

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A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

3.1. Moment theory approach Further taking nis as nis ¼ n1 ðr,zÞexpðiðotkzÞÞ

ð29Þ

where n1 is the slowly varying real function of r and z. o and k are the frequency and propagation constant for ion-acoustic wave. Substituting the value of nis from Eq. (29) in Eq. (28), one obtains in the WKB approximation @n1 i Gon1 ¼  r2? n1 iPn1  2k @z kgv2th

ð30Þ

d

where I0 is zeroth order moment and can be written as ZZ I0 ¼ n1 nn1 dx dy Now, solution to Eq. (30) is of the form " # n00 r2 n1 ¼ exp k z i 2 f 2a2 f

ð31Þ

ð32Þ

ð33Þ

I0 ¼ pn200 a2 expð2ki zÞ

ð34Þ

and 2

/a2 S ¼ a2 f expð2ki zÞ

ð35Þ

Now, with the help of Eqs. (31) and (35), it can be shown that "  2  2 2 2 d f 1 df 1 1 op r o 2 ks ld þ ¼ expð2k zÞ  i 2 3 2 2 f d 4 x g v th dx 1 þks ld 4f " ## 2 4 1 f0 f G2 o2 fr40 ð36Þ I1 þ 04 I2 þ 2 2 2 f f 2g vth expð2ki ðzÞÞ f

a1 E200 t 2

t a1 1 dt

Eq. (36) describes the variation in the dimensionless beam width parameter f of ion acoustic wave on account of the competition between diffraction divergence and non-linear refractive terms with the distance of propagation in the collisionless plasma with f¼1 and df =dz ¼ 0 at z¼ 0. 3.2. Paraxial ray approximation Assuming the variation of nis ðr,zÞ as nis ðr,zÞ ¼ n1 ðr,zÞexp½iðotkðz þ Sðx,y,zÞÞ

0

d

Eq. (42) describes the variation in the dimensionless beam width parameters f of ion-acoustic wave on account of the competition between diffraction divergence term and non-linear refractive term with the distance of propagation in the collisionless plasma with f¼1 and df =dz ¼ 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 of the electric field E0 of the incident beam and Es of the scattered wave, i.e.

r2 EH rðr  EH Þ ¼

!#

f 0

where n00 is the axial amplitude of density perturbation of ionacoustic wave, ‘S’ is the eikonal for the ion-acoustic wave, FðzÞ is a constant whose value will not be required explicitly in further analysis. f is the dimensionless beam width parameters of ionacoustic wave. To obtain an equations for the beam width parameter, we employ the paraxial ray approximation and then equating the coefficients of r 2 on both sides, we obtain from Eq. (38), the following equation for f:  2 2 o2p k0 r 40 d f 3 m 2 E ¼   a 00 2 2 3 dz k a4 f gi k2 v2th 4 M ! 2 2 3 m E200 f 2 2 k ld k r ð42Þ exp  a 0 0 2 2 4 M f2 f4 1þ k l

ð43Þ

where, Es is due to the scattering of the pump beam from the ion acoustic wave(i.e. Brillouin scattering), 0 os 0 represents scattered frequency. The vector EH satisfies the wave equation

and logðtÞ 1exp 

ð40Þ

ð41Þ

EH ¼ E0 expðioo tÞ þ Es expðios tÞ

where " !# Z a1 E200  t a 1 1 t 1exp  I1 ¼ dt 2 f0

I2 ¼

1 2 1 df r þ FðzÞ 2 f dz

0

From Eqs. (31)–(33), it can be shown that

"

ð39Þ

The solution to Eqs. (38) and (39) can be written as " # n00 r2 n1 ¼ exp ki z 2 f 2a2 f

Sðr,zÞ ¼

By the definition of second order moment ZZ 1 ðx2 þ y2 Þn1 nn1 dx dy /a2 S ¼ I0

Z

@n21 @S @n21 2G on21 2  þ þn21 r? S þ 2i  ¼0 @r @r @z gvth k

ð38Þ

where ki is damping factor

where P can be written as 2 2  o2pi ks ld Noe 1 P¼ N0 2kgi v2th 1 þk2 l2 s

one obtains  2   2 2 o2p @S @S 1 N 0e k ld 2 þ ¼ 2 r2? n1 þ 2 1 2 @z @r N0 1þ k2 l2 k n1 k vth d

ð37Þ

where n1 is the slowly varying real function of r and z. o and k are the frequency and propagation constant for ion-acoustic wave, S is the eikonal for the ion-acoustic wave. Substituting for ‘nis ’ from Eq. (37) in Eq. (28) and separating the real and imaginary parts,

1 @2 EH 4p @J H þ 2 c2 @t 2 c @t

ð44Þ

where, J H is the total current density vector in the presence of high frequency electric field EH . Equating the terms at scattered frequency 0 os 0 , we get " # " # o2 Noe o2p os nn o2 r2 Es þ 2s 1 p2 ð45Þ Es ¼ E0 rðr  E0 Þ c oo N o 2c2 oo N o In order to solve Eq. (45), 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 to Eq. (45) may be obtained in the form Es ¼ Eso expð þ ikso zÞ þ Es1 expðiks1 zÞ

ð46Þ

where 2

kso ¼

o2s c2

" 1

#

o2p o2 ¼ 2s eso 2 os c

ð47Þ

A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

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 real functions of r and z. kso and ks1 are the propagation constants of scattered wave. Using Eq. (46) in (45) and separating terms with different phases we obtain "  # o2p @Es0 o2 N 0e 2 2 ks0 E2s0 þ2iks0 þ r? Es0 þ 2s es0 þ 2  1 Es0 ¼ 0 @z N0 c os ð48Þ "

2

kso1 E2s1 þ 2iks1 Es1 ¼

2 p 2 c

 o2p @Es1 o2 Noe 2 þ r? Es1 þ 2s eso þ 2  1 @z No c os

I4 ¼

1 o n os Eo 2 N o oo

ð49Þ

b 0 EE

 # N 0e 2 2 ks1 ks0  2 1 N0 c

o2p

ð50Þ

b is a unit vector along E. where E From Eq. (48), again as considered earlier, one obtains in the WKB approximation @Es0 i ¼ r2 Es0 iPEs0 @z 2ks0 ?

" logðtÞ 1exp 

a1 E200  t 2

f0

!# t a2 1 dt

where f s ¼ 1 and df s =dz¼0 at z ¼0. Eq. (57) describes the change in the dimensionless beam width parameter f s of scattered beam on account of the competition between diffraction divergence and non-linear refractive terms as the beam propagates in the collisionless plasma.

Now, from definition of second order moment Z Z 1 ðx2 þy2 ÞEs00 Ens00 dx dy /a2 S ¼ I0 where I0 is zeroth order moment and can be written as ZZ I0 ¼ Es00 Ens00 dx dy The solution to Eq. (51) is of the form " # B2 r 2 E2s00 ¼ 21 exp 2 2 fs b fs

I0 ¼ p

@E2soo @Ss @E2soo 2 þ þ E2soo r? Ss ¼ 0 @z @r @r

ð52Þ

ð53Þ

With the help of Eqs. (52) and (56), one can get " #  2 2 2 2 2 2 k0 r 40 k0 r 40 f 0 op r o 2 r 20 f 0 d fs 1 df s þ ¼ 2 4 3 4 2 3 I3 þ 2 2 I4 2 f s dx c dx ks0 b f s b ks0 f s b fs

ð61Þ

Here, B1 is the amplitude of the scattered beam, whose value is to be determined later by applying boundary condition. f s is the dimensionless beam width parameters of the scattered beam and satisfies the following differential equation: !  2 2 o2p k20 r 20 3 m 2 k0 r 40 d fs 3 m E200 f s ¼ 2 4 3 2  a E ð62Þ exp  a 2 4 M 00 4 M f2 f4 os es0 dx ks0 b f s 0 0 where f s ¼ 1 and df s =dz¼0 at z ¼0. Eq. (62) describes the change in the dimensionless beam width parameters f s of scattered beam on account of the competition between diffraction divergence term and non-linear refractive term as the beam propagates in the collisionless plasma.

5. Expression for back-reflectivity Now, the value of B1 is calculated with the boundary condition that Es ¼0 at z ¼ zc . Es ¼ Eso expð þ ikso zÞ þEs1 expðiks1 zÞ ¼ 0

ð63Þ

at z ¼ zc . Here, zc is the distance at which amplitude of the scattered wave is zero. Therefore, at z ¼ zc , one can obtain

ð55Þ B1 ¼

2 2

1 2 1 df s r þ Fs ðzÞ 2 f s dz

ð54Þ

and /a22 S ¼ b f s

ð59Þ

Here, Esoo is the real function of r and z, Ss is the eikonal for the scattered wave. Solutions to Eqs. (58) and (59) can be written as " # B2 r 2 ð60Þ E2s00 ¼ 21 exp 2 2 fs b fs Ss ¼

Here, Esoo is the real function of r and z. b is the initial dimension of scattered beam at z ¼0, B1 is the amplitude of the scattered beam, whose value is to be determined later by applying boundary condition. f s is the dimensionless beam width parameter of the scattered beam. Now, from Eqs. (52)–(54), it can be shown that 2 B21 b

By putting Eso ¼ Esoo expð þ iks0 Ss Þ in Eq. (51) and separating the real and imaginary parts one can obtain  2   o2P @Ss @Ss N0e 1 þ ¼ 1 r2? Esoo ð58Þ þ 2 2 2 @z @r N0 eso os kso Esoo

ð51Þ

4.1. Moment theory approach

where " !# Z a1 E200  t t a2 1 1exp  I3 ¼ dt 2 f0

Z

4.2. Paraxial ray approximation

n

1 o2p n% os " 2 c2 N0 o0

and

#

Now, from Eq. (49), neglecting terms containing space derivatives by assuming ðr o b2p=ko Þ, one obtains the following equation: Es1 0 ¼ 

179

1 o2p os n00 " 2 c 2 o0 N 0

ð56Þ

E00 expðki zc Þ f s ðzc Þ expðiðks1 zc Þ # f ðz Þf ðz Þ expðik z Þ 2 o c c s0 c 0 N p 2 2 0e ks1 ks0  2 1 N0 c ð64Þ

with the condition ð57Þ

1 2 2

b fs

¼

1 a2 f

2

þ

1 2

r 20 f 0

Here f 0 ðzc Þ, f ðzc Þ, f s ðzc Þ are the values of dimensionless beam width parameters of pump beam, ion-acoustic beam and scattered beam at z ¼ zc .

180

A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

Now, reflectivity R is defined as ratio of scattered flux to incident flux and is given by R¼

E002 = 2.0

1 o2p o2s n20 1 " " !##2 ½I1 I2 I3  4 c4 o20 N200 2 op 3 m E200 2 2 ks1 ks0  2 1exp  a expð1:0Þ 4 M f2 c 0

0.8

ð65Þ

0.6 f

where 2

I1 ¼

f s ðzc Þ

1

2 2 2 f 0 ðzc Þf ðzc Þ f s

exp 2ki zc 

r2

!

0.4

ð66Þ

2 2 b fs

f s ðzc Þ 1 r2 r2 r2 exp  2 2   I2 ¼ 2 2 2 2 f 0 ðzc Þf ðzc Þ ff 0 f s 2b f s 2a f 2r 20 f 0

0.2

!

expðki ðz þ zc ÞÞcosðks1 þ ks0 Þðzzc Þ I3 ¼

αE002 = 1.0

1.0

1 2 2 f f0

exp 

r2 2 a2 f



r2 2 r 20 f 0

0.0 0.0

ð67Þ

0.5

1.0

! 2ki zc

ð68Þ

1.5 ξ

2.0

2.5

3.0

Fig. 2. Variation of beam width parameter f against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.

1.1 1.0

6. Discussion

αE002 = 1.0

0.9

1.0

αE002 = 1.0 αE002 = 2.0

0.8

f0

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5 ξ

2.0

2.5

3.0

Fig. 1. Variation of beam width parameter f0 against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.

0.8

αE002 = 2.0

0.7 0.6 fs

The differential equations (17), (23), (36), (42), (57) and (62) for the beam width parameters f 0 of the pump beam, f of the ion-acoustic beam, f s of the scattered beam respectively have been solved numerically for the following set of parameters; o0 ¼ 1:778  1014 rad s1 , o2p =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 that arises due to the colisionless non-linearity, represents the nonlinear refraction. The relative magnitude of these terms determines the focusing/defocusing behavior of the beams. Fig. 1 describes the variation of beam width parameter f 0 of the pump beam as a function of dimensionless distance of propagation x for different values of intensities aE200 ¼ 1:0,2:0. It is observed from the figure that with increase 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. (17) is sensitive to the intensity of laser beam. Therefore, as we increase the intensity of the laser beam, refractive term becomes relatively stronger than diffractive term.

0.5 0.4 0.3 0.2 0.1 0.0

0.5

1.0

1.5 ξ

2.0

2.5

3.0

Fig. 3. Variation of beam width parameter f s against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.

Fig. 2 describes the variation of beam width parameter f of the ion acoustic wave against the normalized distance of propagation x for different values of intensities aE200 ¼ 1:0,2:0. It is observed from the figure that the extent of self-focusing of the ion-acoustic wave increases with increase in intensity parameter. This is because, as we increase the intensity of the laser beam, nonlinear refractive term dominate the diffractive term and hence there is an increase in focusing of the beam at higher intensity. Fig. 3 describes the variation of beam width parameter fs of back-scattered beam against the normalized distance of propagation x for different values of intensities aE200 ¼ 1:0,2:0. It is observed from the figure that with increase in intensity parameter the extent of focusing of the scattered beam increases. This is due to the weakening of diffractive term as compared to nonlinear refractive term at higher value of intensity. Fig. 4 describes the variation of reflectivity R against the normalized distance of propagation x for different values of pump beam intensity aE200 ¼ 1:0,2:0. It is observed from the figure that reflectivity of the scattered wave is larger for aE200 ¼ 2:0 than for aE200 ¼ 1:0, which is due to the fact that self-focusing is appreciably larger in the former case. Thus, self-focusing of beams leads to 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 moment theory with dimensionless distance of propagation x for

A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

0.000010 0.000008 αE002 = 2.0

R

0.000006

αE002 = 1.0

0.000004 0.000002 0.000000 0.002

0.004

0.006 ξ

0.008

0.010

Fig. 4. Variation of reflectivity R against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 ¼ 0:6 and for intensity aE200 ¼ 1:0,2:0.

1.1

the moment theory as compared to the paraxial theory on account of participation of off-axis parts. Fig. 6 shows the variation of the beam width parameter f of the ion-acoustic wave with dimensionless distance of propagation x for aE200 ¼ 2:0 for the paraxial theory and the moment theory approach. Solid line corresponds to the moment theory and dotted line corresponds to the paraxial theory approach. It is again observed from figure that there is strong focusing of the beam in the moment theory as compared to the paraxial theory. Fig. 7 shows the variation of the beam width parameter f s of the back-scattered beam against the normalized distance of propagation x for aE200 ¼ 2:0 for the paraxial theory and the moment theory approach. Solid line corresponds to the moment theory and dotted line corresponds to the paraxial theory approach. It is observed from the figure that the self-focusing length is less in the moment theory as compared to the paraxial theory. Fig. 8 describes the variation of reflectivity R against the normalized distance of propagation x for pump beam intensity

αE002 = 2.0

1.0

αE002 = 2.0

0.9

1.0

0.8 0.7

0.8

0.6 f0

181

0.5

0.6 fs

0.4

0.4

0.3 0.2 0.1 0.0

0.2

0.5

1.0

1.5

2.0

2.5

3.0

ξ

0.0

Fig. 5. Variation of the beam width parameter f0 against the normalized distance of propagation x ¼ Z=Rd for aE200 ¼ 2:0 and o2p =o20 ¼ 0:6. (Solid line corresponds to result of the moment theory and dotted line corresponds to that of the paraxial theory approach.)

1.1

0.5

0.0

1.5 ξ

1.0

2.5

2.0

3.0

Fig. 7. Variation of beam width parameter f s against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 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.)

αE002 = 2.0

1.0

0.000010

0.9 0.8

0.000008

0.7 0.6

αE002 = 2.0

f

0.000006

0.5 0.4

R

0.3

0.000004

0.2

0.000002

aE200 ¼ 2:0. Solid line corresponds to the moment theory and dotted line corresponds to the paraxial theory approach. It is observed from figure that there is strong focusing of the beam in

0

8

02

0.

6

01 0.

4

01

0.

2

01 0.

0

01 0.

8

01

0.

6

00 0.

Fig. 6. Variation of beam width parameter f against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 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.)

0.000000

4

3.0

00

2.5

0.

2.0

2

1.5 ξ

00

1.0

0.

0.5

00

0.0

0.

0.1

ξ Fig. 8. Variation of reflectivity R against the normalized distance of propagation x ¼ Z=Rd for o2p =o20 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.)

182

A. Singh, K. Walia / Optics Communications 290 (2013) 175–182

aE200 ¼ 2:0 for the paraxial theory and the moment theory approach. Solid line corresponds to the moment theory and dotted line corresponds to the paraxial theory approach. It is observed from the figure that reflectivity of the scattered wave is larger for paraxial ray approximation as compared to that for moment theory approach. This observation is similar to that of Sharma et al. [32]. Results of present analysis are similar to that Sharma et al. [32] in which the authors compared the results of higher order paraxial approach with that of paraxial theory [31]. In higher order paraxial approach, one additional term in the Taylor series expansion is included in the analysis as compared to the paraxial approach and thus off-axial part of the laser beam is taken into consideration up to certain extent. The beauty of the moment theory approach is that it takes into consideration the complete laser profile instead of only certain part as considered in the higher order paraxial approach and is therefore more realistic.

7. Conclusion In the present investigation, moment theory has been used to study the stimulated Brillouin scattering (SBS) of laser beam in collisionless plasma. Results are compared with the paraxial ray approximation. Following important observations are made from present analysis. (1) Self-focusing of pump wave and ion-acoustic wave is stronger in the moment theory approach as compared to that of paraxial theory. (2) Self-focusing length of back-scattered wave is less in the moment theory as compared to the paraxial theory. (3) Reflectivity of the back-scattered wave is larger for paraxial theory as compared to that for moment theory approach. Results of the present investigation are useful for understanding physics of laser-induced fusion in which SBS plays a major role, as it produces a significant level of back-scattered light and thus leads to poor laser plasma coupling.

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