Transient self-focusing and stimulated Brillouin scattering in laser-plasma interaction

J. Plasma Physics (2001), vol. 66, parts 1&2, pp. 119–133. 2001 Cambridge University Press

DOI: 10.1017/S002237780100126X Printed in the United Kingdom

119

Transient self-focusing and stimulatedBrillouin scattering in laser–plasma

interactionS A L E H T. M A H M O U D,1 R. P. S H A R M A1 and A. K U M A R2

1Centre for Energy Studies, Indian Institute of Technology, New Delhi-16, India2Centre for Advance Technology, Indoor, India

(Received 7 December 2000 and in revised form 21 February 2001)

Abstract. This paper presents an investigation of the time-dependent setting-upof nonlinearity in a collisionless plasma in the presence of a short-duration laserpulse. Using the paraxial-ray approximation, transient self-focusing of the incidentlaser pulse and Brillouin back-scattering have also been studied, including pumpdepletion. The effect of this transient nature on reflectivity has been discussed.

1. IntroductionParametric instabilities such as stimulated Brillouin scattering (SBS), stimulatedRaman scattering (SRS), self-focusing and filamentation have long been of con-cern in inertial-confinement thermonuclear fusion research, because high levels ofscattered light can reduce energy coupling and modify the distribution, affectingthe symmetry of capsule implosion (Kruer 1988). These instabilities have been ex-tensively studied, both experimentally and theoretically (Frycz et al. 1992; Rose1995; Eliseev et al. 1996, 1998; Chirokikh et al. 1998; Fernandez et al. 1998; Hulleret al. 1998; Divol and Philippe 1999a,b; Giulietti et al. 1999; Russell et al. 1999;Cobble et al. 2000). In spite of these extensive efforts, many difficulties remainin the interpretation of these instabilities. Such instabilities can occur whenever ahigh-intensity laser pulse interacts with a long underdense plasma. It is mainlythe ponderomotive force that expels plasma from the axis of the laser beam untilequilibrium against gas-dynamic pressure is established.

A short-pulse laser with Gaussian distribution of intensity propagating through aperformed underdense plasma produces a channel by exerting a radial ponderomo-tive force on the electrons. The electrons and ions, coupled through the space-chargefield, move together to cause ambipolar diffusion. The characteristic diffusion timeτ = r0/cs, where r0 is the beam width and cs is the ion sound speed (Sodha et al.1979). The density profile has a minimum on the axis.

Schmitt and Afeyan (1998) have studied the setting-up of a ponderomotive non-linearity in a plasma in the presence of a high-power laser pulse. Within the paraxial-ray approximation, they have solved the wave equation along with the density re-sponse equation and studied the filamentation of the laser pulse. They conclude thatthe steady-state limit of filamentation is seldom reached in realistic laser–plasmainteraction. The simulations also show that the transient behaviour of filamenta-tion persists even for times large compared with the characteristic hydrodynamic

https://www.researchgate.net/publication/236558819_The_Physics_of_Laser_Plasma_Interactions?el=1_x_8&enrichId=rgreq-837e9232d53da0cf38d26893eeecb372&enrichSource=Y292ZXJQYWdlOzIzMTgwMTAwNjtBUzoyMDE5MDgzMzg4NjAwNDRAMTQyNTE1MDEwODAzNg==

120 S. T. Mahmoud, R. P. Sharma and A. Kumar

time scale. Schmitt and Afeyan (1998) found that the time dependence and smooth-ing of laser non-uniformities is a persistent feature of the filamentation interactionfor times long compared with other characteristic laser–plasma interaction timescales. Schmitt and Ong (1983) investigated transient self-focusing in a plasma dueto thermal and ponderomotive forces using the moment method, but the effect ofself-focusing on SBS has not been considered.

In the present paper, we study the time-dependent setting-up of a nonlinearity,and evaluate the modification in density in the presence of a high-power laser pulse,semi-analytically in a single hot spot in a collisionless plasma. In a typical case,it is demonstrated that the density changes with time. The nonlinear part of therefractive index of the plasma becomes time-dependent, and transient self-focusingoccurs. The phenomena of SBS, which depends on the intensity of incident laser,has not been understood fully so far. Specifically, the effect of self-focusing of theincident laser on SBS has not been studied in detail. The transient nature of self-focusing of the pump laser pulse is expected to affect the SBS process also. Thereforethe SBS back-reflectivity changes with time.

In order to study the above-mentioned phenomena, we first set up the equationsfor the modified background density in the presence of a laser pulse by using thehydrodynamic response of the plasma in Sec. 2. Using the Green function andFourier transform technique, we find an expression for the modified density. Westudy the response of plasma density with time. We set up the equations governingthe dynamics of laser pulse in the presence of the modified density. Using theparaxial-ray approximation, the solution for laser pulse intensity is evaluated inSec. 3. By taking typical parameters for a laser plasma, the intensity of the laserpulse is calculated on the beam axis at different times. In Sec. 4, we solve the coupledequations to study SBS back-reflectivity using the paraxial-ray approximation. InSec. 5 we present our conclusions based on the present investigations.

2. Time-dependent response of plasma densityWe consider the propagation of a laser pulse in a plasma, along the z axis. Theintensity distribution of the laser pulse in a collisionless, hot, unmagnetized andhomogenous plasma at z = 0 is given by

E0E∗0 |z=0 = E2

00(t) exp(−r

2

r20

). (1)

Here r2 = x2 + y2, r0 is the initial beam width and E200(t) is a function of time.

On account of the non-uniform intensity distribution of the laser pulse in a planetransverse to the direction of propagation, the nonlinear (ponderomotive) forcebecomes non-zero and leads to diffusion of electrons and ions. Hence, in the presenceof the laser pulse, the background electron and ion densities may be modified. Herewe derive an analytical expression for the modified electron density in the presenceof a Gaussian laser pulse. For completeness, we first derive the equations governingthe dynamical response of the density in the presence of a laser pulse as follows.

In the quasi-hydrodynamic two-fluid approximation, one obtains the followingequations for diffusion of electrons and ions:

me∂Ve∂t

= −eEs − γekBTeNe

∇Ne − Fe, (2)

Self-focusing and Brillouin scattering in laser–plasma interaction 121

mi∂Vi∂t

= eEs − γikBTiNi∇Ni. (3)

Here the third term on the right-hand side of (2) is given by

Fe = − e2

4me

1ω2

0∇(E0 · E∗0). (4)

The indices e and i refer to electrons and ions respectively, me and mi are theelectron and ion masses, Te and Ti are the electron and ion temperatures, kB isBoltzmann’s constant, ω0 is the angular frequency of the laser pulse, Es is thespace-charge field, and −e is the electronic charge, other symbols have their usualmeanings. The nonlinear force on the ions is not taken into account in writing(3), because its magnitude is less by a factor of me/mi in comparison with thaton electrons. Owing to space-charge effects, electrons and ions move with almostthe same velocity, i.e. Ve ≈ Vi ≈ V in an almost electrically neutral medium withNe ≈ Ni ≈ N , where V is the diffusion velocity of electrons and ions and N is themodified electron and ion density in the presence of the pulse. Adding (2) and (3),and neglecting the electron mass me as compared with the ion mass mi, we obtain

∂V∂t

= −c2s∇NN

− e2

4memi

1ω2

0∇(E0 · E∗0), (5)

where c2s = (γekBTe + γikBTi)/mi. For further analysis, (5) may be supplemented

by the equation of continuity

∂N

∂t+∇ · (NV) = 0. (6)

Equations (5) and (6) may be combined to give

∂2

∂t2

(lnN

N0

)= c2

s∇2(

lnN

N0

)+

e2

4memi

1ω2

0∇2(E0 · E∗0), (7)

whereN0 is the electron density in the absence of the pulse. From (7), we can obtainan analytical expression for N as follows.

We add a damping term phenomenologically. Taking the Fourier transform ofthe resulting equation with respect to space coordinates would give an equation ofthe form

d2φ

dt2+ 2Γi

dφ

dt+ ω′2φ = F (t), (8)

where φ is the Fourier transform of ln(N/N0) and F (t) is the Fourier transform ofthe second term on the right-hand side of (7), given by

F (t) = −e2(k2

1 + k22)

8memi

r2

0

ω20E2

00(t) exp[(− 14 (k2

1 + k22)r2

0f20 ].

In evaluating F (t), we have assumed the solution for E0 as given by (21) in Sec. 3.A Green-function solution of (8) is

φ =1Ω

∫ t

0sin[Ω(t− t1)] exp[−Γi(t− t1)]F (t1) dt1, (9)

where Ω = (ω′2 − Γ2i )

1/2 and Γi is the damping factor. It should be mentionedhere that (9) represents both a driven and a homogeneous solution. Inverse Fourier


transformation of φ gives an equation for N . Equation (9) gives N as

N = N0 exp(− e2

16πmemics

r20I1

ω20

), (10)

where

I1 =∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin

[(k2

1 + k22)1/2cs(t− t1)

]k21 + k2

2

Ω

× exp[−Γi(t− t1)] exp[− 14 (k2

1 + k22)r2

0f20 (t− t1, z)] exp[−i(k1x + k2y)]E2

00(t1).

(11a)

We now consider two special cases of (11a) before proceeding further.First, in the special case of a fast-rising flat-top laser pulse of duration τ r0/cs,

with zero damping (Γi = 0), I1, may be approximated as follows at z = 0 (becausef0 = 1):

I1 =∫ ∞−∞

dk1

∫ ∞−∞

dk2 [1− cos(k21 + k2

2)1/2cst] exp[− 14 (k2

1 + k22)r2

0]

× exp[−i(k1x + k2y)]E200. (11b)

In order to have an appreciation of the evolution of the modified density N , wehave obtained an analytical solution for (11b) at r = 0, I1 can be written as

I1 =∫ ∞

0(1− cos kcst) exp(− 1

4k2r2

0) k dk.

Therefore, using (10), one gets(N

N0

)r=0

= exp(− e2r2

0E200

16πmemic2sω

20

)∫ ∞0

(1− cos kcst) exp(− 14k

2r20) k dk.

Indeed, one finds (N

N0

)r=0

= exp(− e2E2

00

4πmemic2sω

20

)D

(t

t0

)where D(x) = e−x

2 ∫ x0 ek

2dk is Dawson’s integral (Abramowitz and Stegun 1965;

Annou et al. 1996). Therefore, at t =∞, one finds(N

N0

)r=0

= exp(− e2E2

00

8memic2sω

20

),

which is exactly the modified density in the steady-state situation (Kaw et al.1973). The Dawson integral and error function are related to each other; thereforethe integral of (11b) can also be written in closed form in terms of the error function.

If damping is finite, the modified density expression cannot be found in closedform. Therefore, we have evaluated it numerically, and the variation of density withtime at r = 0 is as given in Fig. 1. It is obvious from Fig. 1 that the modified densityis not constant with time. It is also expected that during the propagation of thelaser pulse inside the plasma (at finite z), the self-focusing effects will change thisprofile further on the axis (r = 0) and away from the axis (r 0).

When the modified density is changing with time, the refractive index will alsochange with time. Therefore the nonlinear part of the refractive index should alsochange with time. This may lead to time-dependent self-focusing of the laser pulse.This is studied in detail in the next section.


1.001

1.000

0.999

0.998

0.997

0.996

0.995

0.994

0.9930 50 100 150 200 250 300 350

Time (ps)

NN0

Figure 1. Variation of the plasma density N/N0 with time for the pump laser at r = 0. Thebeam intensity I0 = 1015 W cm−2.

3. Transient self-focusing of the laser pulseThe electric vector of the laser pulse is governed by the wave equation

∂2

∂x2E0 +∂2

∂y2E0 +∂2

∂z2E0 =1c2

∂2

∂t2E0 +

4πc2

∂

∂tJ0. (12)

Here J0 is the total current density in the plasma in the presence of the laser pulse.We have neglected the ∇(∇ · E0) term, which is justified as long as

ω2p

ω20

1ε0

ln ε0 1.

However, for a two-dimensional beam and ∂/∂y = 0, this is identically zero. As-suming the variation of E0 in space and time as

E0 = A(x, y, z, t) exp[i(ω0t− k0z)], (13)

where

k0 =ω0

c

(1− ω2

p

ω20

)1/2

and A is a complex function of space and time, the current density in the plasmacan be written as

J0 ≈ Ne2

meiω0

(A +

i

ω0

∂

∂tA)

exp[i(ω0t− k0z)]. (14)

Substituting the expression for J0 from (14) and E0 from (13) in (12), we obtain

− 2ik0∂

∂zA +

∂2

∂x2A +∂2

∂y2A ≈ω2

0

c2

ω2p

ω20

N −N0

N0A +

2iω0

c2

∂

∂tA. (15)


Further, assuming the variation of A as (Akhmanov et al. 1968).

A = A0(x, y, z, t) exp[−ik0S0(x, y, z, t)], (16)

where A0 and S0 are real functions of space and time, substituting for A from (16)in (15) and separating the real and imaginary parts of the resulting equation, weobtain

1Vg

∂

∂tA2

0 +∂

∂zA2

0 +∂

∂xS0

∂

∂xA2

0 +∂

∂yS0

∂

∂yA2

0 +(∂2

∂x2S0 +∂2

∂y2S0

)A2

0 = 0 (17)

and2Vg

∂

∂tS0 + 2

∂

∂zS0 +

(∂

∂xS0

)2

+(∂

∂yS0

)2

=ω2p

ω20

N0 −NN0ε0

+1

k20A0

(∂2

∂x2A0 +∂2

∂y2A0

), (18)

where

Vg = cε1/20 , ε0 = 1− ω2

p

ω20.

On transforming the variable t to ξ = t − z/vg (17) and (18) assume the followingforms:

∂

∂zA2

0 +∂

∂xA2

0 +∂

∂yS0

∂

∂yA2

0 +(∂2

∂x2S0 +∂2

∂y2S0

)A2

0 = 0 (19)

and

2∂

∂zS0 +

(∂

∂xS0

)2

+(∂

∂yS0

)2

=ω2p

ω20

N0 −NN0ε0

+1

k20A0

(∂2

∂x2A0 +∂2

∂y2A0

). (20)

The solution of these equations can be written as

A20 =

E200(ξ)

f 20 (z, ξ)

exp[− r2

r20f

20 (z, ξ)

], (21)

S0 = 12r

2β0(z, ξ) + φ0(z, ξ), (22)

β0 =1f0

d

dzf0, (23)

where f0 is the dimensionless beam-width parameter. In order to calculate the non-linear phase shift, one requires φ0(z, ξ). However, it is difficult to find an analyticalexpression for φ0. We are more interested in the intensity of the laser than its phase,so we do not need φ0 in further analysis here. Using (21)–(23) in (20) and equatingthe coefficients of r2(= x2 + y2), we obtain an equation governing the beam-widthparameter:

d2f0

dz2 =1

k20r

40f

30− ω2

p

ω20ε0

f0χ, (24a)

where χ is the coefficient of r2 in the expansion of N/N0, given by (because of thesymmetry in I1 along x and y)

χ =∂

∂x2

(N

N0

)= − e2r2

0

16memicsω20

∂I1

∂x2 exp(− e2r2

0

16memicsω20I1

), (24b)

where I1 is given by (11a). In obtaining the expression for χ, N/N0 has beenexpanded in the paraxial region. In fact, N/N0 is assumed to be a function of


7

6

5

4

3

2

1

0.1 0.2 0.3 0.4 0.5 0.6

f0

n

Figure 2. Variation of the dimensionless beam-width parameter of the incident laser pulsef0 with the dimensionless distance of propagation ξ = z/Rd0 (Rd0 = k0r

20) at different times:

———, 10 ps; −−−−, 35 ps; · · · · · ·, 150 ps.

1−r2/r20f

20 and a Taylor expansion is used. In order to study the dynamical setting-

up of ponderomotive nonlinearity and self-focusing of the laser pulse, we simplifyχ (within the paraxial ray-approximation), and we get

d2f0

dz2 =1

k20r

40f

30− ω2

p

ε0ω20

e2

16πmimecsf0r2

0I3

ω20

exp(− e2

16πmemics

r20I′3

ω20

), (25)

where I3 and I ′3 are given by

I3 =∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin[(k2

1 + k22)1/2cs(t− t1)]

k21(k2

1 + k22)

Ωexp[−Γi(t− t1)]

× exp[− 1

4 (k21 + k2

2)r20f

20

]E2

00(t1),

I ′3 =∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin[(k2

1 + k22)1/2cs(t− t1)]

k21 + k2

2

Ωexp[−Γi(t− t1)]

× exp[− 1

4 (k21 + k2

2)r20f

20

]E2

00(t1).

The differential equation governing the beam-width parameter of the pump laserbeam, (25), can be solved numerically, assuming the boundary conditions f0 = 1and df0/dz = 0. The following sets of parameters are chosen for the illustration ofnumerical results, for the case of small finite damping:

λ0 = 1.053µm, Te = 1 keV,ω2p

ω20

= 0.2, r0 = 15µm, I = 1015 W cm−2.

The electron temperature is assumed to be much larger than the ion temperature.The results are presented in Fig. 2 for different times. The time is taken in units ofthe hydrodynamic time (τ = r0/cs).

Figure 2 depicts the variation of the dimensionless beam-width parameter f0 withdistance of propagation at different times. The pump beam becomes focused anddefocused along the beam axis. The position of the focal spot changes in time. It


moves backward in the direction of the laser axis. This movement is a consequenceof the change in the background density with time as depicted in Fig. 1.

4. Stimulated Brillouin scatteringSo far we have considered the nonlinear evolution of the background density of theplasma in the presence of a single laser pulse and the consequent time-dependentself-focusing of the pulse. We shall now consider the interaction of this high-powerlaser pulse with the low-frequency mode of the plasma (ion-acoustic) wave of fre-quency ω and wavenumber k and the resulting SBS wave of frequency ωs andwavenumber ks. In addition to this, on account of the non-uniform intensity dis-tribution of the incident laser pulse in a plane transverse to the direction of propa-gation, the ponderomotive force becomes non-zero and leads to the diffusion of theelectrons and ions. Hence the background electrons and ions densities are modified.The expression for this modified background density has been used earlier whenthe characteristic diffusion time (t0 = r0/cs) is much less than the incident laserpulse duration. When these two times are comparable, the steady-state expressionfor the modified background density is not valid and one should look for the tran-sient response of the background density. Following the treatment in Sec. 2, onecan write down the expression for the modified density as follows (in the presenceof the incident laser pulse and the SBS pulse):

N = N0 exp

[− e2

16πmemics

(r2

0I0e−2g0z

ω20

+r2sIse

−2gsz′

ω2s

)], (26a)

where N0 is the background electron density in the absence of the laser pulse and

I0,s=∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin[(k2

1 + k22)1/2cs(t− t1)]

(k21 + k2

2)Ω

exp[−Γi(t− t1)]

× exp[− 14 (k2

1 + k22)r2

0,sf20,s(t− t1, z)] exp[−i(k1x + k2y)]E2

0,s(t1). (26b)

In writing (26a,b), we have used the solutions for the pump and scattered pulsesas given by (35) and (36). Following standard techniques, the equations governingthe pump laser pulse and the scattered pulse are

∇2E0 +ω2

0

c2

(1− ω2

p

ω20

N

N0

)E0 = −4πe

c2

12∂

∂t(NesVs) +

2iω0

c2

∂A0

∂t(27)

and

∇2Es +ω2s

c2

(1− ω2

p

ω2s

N

N0

)Es = −4πe

c2

12∂

∂t(NesV0) +

2iωsc2

∂As

∂t. (28)

In (27) and (28), E0 and Es are the electric fields of the pump laser pulse and thescattered pulse respectively and Nes is the electron density perturbation in theion-acoustic equation. Using the ion-acoustic equation, one obtains the followingequations after substituting for Nes in (27) and (28):

∇2E0 +ω2

0

c2

(1− ω2

p

ω20

N

N0

)E0 +

e2

4memi

ω2p(N/N0)ω2(Es · E∗s)E0

(−ω2 + 2iΓiω − k2c2s)c2

sc2ω2

0

− 2iω0

c2

∂A0

∂t= 0 (29)


and

∇2Es +ω2s

c2

(1− ω2

p

ω20

N

N0

)Es +

e2

4memi

ω2p(N/N0)ω2(E0 · E∗0)Es

(−ω2 + 2iΓiω − k2c2s)c2

sc2ω2s

− 2iωsc2

∂As

∂t= 0 (30)

Then using the paraxial-ray approximations, one can solve the coupled equations(29) and (30). Expressing

E0 = A0(x, y, z, t) expi[ω0t− k0(z + s0(x, y, z, t))],Es = As(x, y, z, t) expi[ωst− ks(z + ss(x, y, z, t))].

and substituting these into (29) and (30) respectively and separating the real andimaginary parts, one obtains the following set of coupled equations after trans-forming the variable t to ξ0,s = t− z/vg0,s, with vg0,s = cε

1/20,s and ε0,s = 1−ω2

p/ω20,s.

Equation (29) gives

∂A20

∂z+∂A2

0

∂x

∂S0

∂x+∂A2

0

∂y

∂S0

∂y+(∂2S0

∂x2 +∂2S0

∂y2

)A2

0

+e2ωω2

p|As|2(N/N0)A20

4memi2Γic2sω

2sω0cε

1/20

≈ 0 (31)

and

2∂S0

∂z+(∂S0

∂x

)2

+(∂S0

∂y

)2

= −ω2p

ω20

N −N0

N0ε0+

1k2

0A0

(∂2A0

∂x2 +∂2A0

∂y2

)(32)

Similarly, from (30), one obtains

∂A2s

∂z+∂A2

s

∂x

∂Ss∂x

+∂A2

s

∂y

∂Ss∂y

+(∂2Ss∂x2 +

∂2Ss∂y2

)A2s

− e2ωω2p|A0|2(N/N0)

4memi2Γic2sω

20ωscε

1/2s

A2s ≈ 0 (33)

and

2∂Ss∂z

+(∂Ss∂x

)2

+(∂Ss∂y

)2

= −ω2p

ω2s

N −N0

εsN0+

1k2sAs

(∂2As∂x2 +

∂2As∂y2

). (34)

If the variation of As in the perpendicular direction in (33) and (34) is neglected,these equations reduce to the set of equations (6) in Eliseev et al. (1996). Thereforethe model represented by the set of equations (33) and (34) is more realistic for thestudy of self-focusing effects on the SBS process. The solution of (31) and (32) canbe written as

A20 =

E200(ξ0)

f 20 (z, ξ0)

exp[− x2 + y2

r20f

20 (z, ξ0)

− 2g0z

], (35a)

S0 =x2 + y2

21

f0(z, ξ0)df0(z, ξ0)

dz+ φ0(z, ξ0), (35b)


g0 =e2ωω2

p|As|2r=0(N/N0)r=0

16memiΓic2sω

2sω0cε

1/20

. (35c)

Similarly, the system (33) and (34) gives for the back-scattered pulse

A2s =

E2s0(ξs)

f 2s (z′, ξs)

exp[− x2 + y2

r2sf

2s (z′, ξs)

− 2gsz′], (36a)

Ss =x2 + y2

21

fs(z′, ξs)dfs(z′, ξs)

dz+ φ(z′, ξs), (36b)

gs =e2ωω2

p|A0|2r=0(N/N0)r=0

16memiΓic2sω

20ωscε

1/2s

, (36c)

= −GSBS |A0|2r=0(N/N0)r=0

E200Lz

,

where

GSBS =2.5α0(me/mi)E2

00csω2p

c2ΓiLz

and

α0 =e2mi

6kBTeγem2eω

20.

Here z′ = Lz − z, Lz is the interaction length, GSBS is the usual SBS gain for thethree-wave interaction case, Eso is the amplitude of the seed SBS wave at z′ = 0and rs is the beam width of the seed SBS beam at z′ = 0. It should be notedhere that the transient nature of the SBS back-scattered wave arises on accountof the two factors in the gain gs. These are, first, the laser pulse intensity and,secondly, the transient nature of the background density. These two effects com-bined determine the resultant SBS gain, which becomes transient. Consequently,the back-reflectivity is expected to be time-dependent.

In (35a) and (36a), f0 and fs are dimensionless beam-width parameters forthe pump and back-scattered laser pulses respectively. The differential equationsgoverning f0 and fs are

d2f0

dz2 =1

k20r

40f

30− ω2

pf0

ω20ε0

e2

16πmemics

(r2

0I3e−2g0z

ω20

+r2sI4e

−2gsz′

ω2s

)

× exp

[− e2

16πmemics

(r2

0I′3e−2g0z

ω20

+r2sI′4e−2gsz′

ω2s

)](37)

and

d2fsdz2 =

1k2sr

4sf

3s

− ω2pfs

ω2sεs

e2

16πmemics

(r2

0I3e−2g0z

ω20

+r2sI4e

−2gsz′

ω2s

)

× exp

[− e2

16πmemics

(r2

0I′3e−2g0z

ω20

+r2sI′4e−2gsz′

ω2s

)], (38)


1.02

1.00

0.98

0.96

0.92

0.90

0.88

0.86

0.840 50 100 150 200 250 300 350

Time (ps)

NN0

0.94

Figure 3. Variation of the plasma density N/N0 with time for the pump and scattered laserbeams. The parameters of the run are I0 = 1015 W cm−2 and Lz = 600λ0.

where

I3,4 =∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin

[(k2

1 + k22)1/2cs(t− t1)

]k2

1k2

1 + k22

Ω

× exp[−Γi(t− t1)] exp[− 1

4 (k21 + k2

2)r20,sf

20,s

]E2

00(t1) (39)

and

I ′3,4 =∫ ∞−∞

dk1

∫ ∞−∞

dk2

∫ t

0dt1 sin

[(k2

1 + k22)1/2cs(t− t1)

]k21 + k2

2

Ω

× exp[−Γi(t− t1)] exp[− 1

4 (k21 + k2

2)r20,sf

20,s

]E2

00(t1) (40)

The integrated reflectivity R is the ratio of the scattered power∫ |Es(z′ = 0)|2 dr⊥

to the input power∫ |E0(z = 0)|2 dr⊥. Using the solution for the scattered field

from (36a–c), one obtains the following expression for the integrated (over spaceonly) reflectivity:

R =E2s0(z′ = 0)E2

00(z = 0)r2s

r20e−2gs(z=0)Lz . (41)

In order to study the mutual effect of self-focusing of the pump wave and thescattered wave on the SBS back-reflectivity, the same set of parameters as in Sec. 2has been chosen, with seed beam intensity 10−6 of the maximum intensity of thepump and the boundary conditions fs = 1 and dfs/dz = 0 at z′ = 0. The results arepresented in Figs 3–6.

The evolution of the plasma density (26a) in the presence of the pump laser andthe scattered beam is illustrated in Fig. 3. It can be seen that at early times upto 60 ps, there is only one density dip. At times later more than 60 ps, the density


6

5

4

3

2

1

0.1 0.2 0.3 0.4 0.5 0.6

fs

n«

Figure 4. Variation of the dimensionless beam-width parameter of the Brillouinback-scattered laser pulse fs with the dimensionless distance of propagation ξ′(= z′/Rd0) atdifferent times: ———, 10 ps; −−−−, 35 ps; · · · · · ·, 150 ps.

oscillates. This behaviour can be attributed to the competition between self-focusingand the back-scattered SBS and hot-spot motion. It is also obvious from Figs 1 and3 that the density response changes drastically in the presence of the back-scatteredSBS beam.

Figure 4 depicts the variation of the dimensionless beam-width parameter fs ofthe back-scattered beam with the distance of propagation, at different times. It isapparent that the beam becomes focused and then defocused along the beam axis.The location of the hot spot also changes with time. The hot spot moves backwardin the direction of the laser beam. The hot-spot motion affects the plasma densityand hence the SBS gain and SBS back-reflectivity.

Using the results shown in Figs 3 and 4, we calculate the SBS back-reflectivityat different times, as given in Fig. 5. The SBS back-reflectivity depends on theintensity of the back-scattered signal and the SBS gain (36c). The SBS gain, whichdepends on the modified back-scattered density and the response of the pump andscattered beams is expected to change in time. Pump depletion is taken into accountin calculating the SBS back-reflectivity. At early times, the figure shows an initialphase of rapid growth of SBS back-reflectivity, followed by a rapidly oscillatingbehaviour of lower reflectivity at later times. There is competition between self-focusing of the pump laser beam and the back-scattered beam. Moreover, hot-spot motion results in oscillation of the plasma density that affects the SBS gain.Therefore the SBS back-reflectivity changes with time.

In order to estimate the effect of the statistical distribution of the laser-beamintensity on the variation of reflectivity with time, one can make use of the expres-sion for the reflectivity given in (41) and the statistical distribution of laser-beamintensity given by Rose and DuBois (1994) and Tikhonchuk et al. (1996). The curveis obtained by averaging the reflectivity from a single hot spot over the hot-spotintensity distribution at different times. The variation of 〈R〉 with time is depictedin Fig. 6. It is seen that 〈R〉 shows almost the same behaviour with time as in asingle hot spot (Fig. 5), but with a lower level of reflectivity.

The motivation for developing this paraxial theory for the transient self-focusing


0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

01 60 119 178 237 296 355

Time (ps)

R

Figure 5. Variation of the integrated reflectivity R with time. The parameters of the runare I0 = 1015 W cm−2, N/N0 = 0.2, Lz = 600λ0, GSBS = 9.5 and Γi/ωs = 0.06.

of a laser beam and its effect on SBS back-reflectivity is to understand and highlightthe physics of some complicated behaviour of SBS back-reflectivity seen in recentexperimental observations (Cobble et al. 2000) and non-paraxial rigorous numericalsimulations (Frycz et al. 1992; Amin et al. 1993; Eliseev et al. 1995; Huller et al.1996). The results of our semi-analytical paraxial theory presented in Fig. 6 showthat the broad features (the early principal maximum followed by rapid oscillationswith time) are in agreement with the recent experimental observations (cf. Fig. 5of Cobble et al. 2000) and non-paraxial numerical simulations (cf. Fig. 2 of Eliseevet al. 1997). Moreover, we have found that the location of the principal maximumpredicted by the present work is around 0.8τ . Using the experimental parameters,the present work predicts that the principal peak is around 190 ps, while in theexperimental observations (Fig. 5 of Cobble et al. 2000), it is around 185 ps.

The physical mechanisms responsible for this behaviour of SBS back-reflectivityhave not been fully understood in previous work. But by making use of simpleparaxial theory, we have calculated the SBS back-scattered reflectivity in the pres-ence of a single hot spot and have studied the factors governing the magnitude ofthis reflectivity for example, the intensity of the hot spot, the background density,the motion of the hot spots, and the beam size of the pump and scattered beams.Figure 5 shows the cumulative effect of these processes on the SBS back-reflectivityin a single hot spot. Figure 6 shows the cumulative effect of multiple-hot-spot statis-tics on the SBS back-reflectivity.

5. ConclusionsThe present semi-analytical study in the paraxial approximation is perhaps the firsteffort to understand the physics involved in governing the transient nature of SBS


0.003

0.002

0.001

01 60 119 178 237 296 355

Time (ps)

-R.

Figure 6. Variation of 〈R〉 with time. The parameters of the run are I0 = 1015 W cm−2,N/N0 = 0.2, Lz = 600λ0, GSBS = 9.5 and Γi/ωs = 0.06.

back-reflectivity through the intensity of the hot spot, the background density, themotion of the hot spot, and the beam size of the pump and scattered beams (themodel equations used in the present paper also reduce to the set of equations (6) inEliseev et al. (1996) in the steady-state case). Our study reveals that the hot-spotmotion and the competition between the self-focusing and back-scattered beam willaffect the modified background density. The temporal behaviour of the modifieddensity and the beam size of the pump and scattered beams affect the SBS gain,and hence the backward SBS reflectivity shows a behaviour with time as observedin recent laser–plasma interaction experiments (Cobble et al. 2000).

It should also be noted here that the background electron density in the absenceof a laser pulse is assumed to be uniform in the present paper (in the direction oflaser pulse propagation, namely the z direction). This assumption could make acomparison of the present results with some experiments difficult. If this type ofdensity non-uniformity in the z direction is present, the SBS back-reflectivity levelwill be enhanced with increasing inhomogeneity gradient, as has been discussed inthe literature (Tsytovich et al. 1973) for the three-wave interaction case (withoutself-focusing of the laser beam). We also expect that the SBS back-reflectivity levelwill be modified in the non-uniform-density case in the presence of laser-pulse self-focusing. But this is a complex process, because the self-focusing of laser pulse isalso affected. This is outside the scope of the present study, and will be part offuture work.

Acknowledgements

We would like to thank Professor (Ms) Ahuja Raj for helpful discussion of thenumerical part of this paper. This work was partially supported by the Departmentof Science and Technology (India) and the Indian Council of Cultural Relation(ICCR-India).


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Transient self-focusing and stimulated Brillouin scattering in laser-plasma interaction

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