Physics Letters A 371 (2007) 135–139

www.elsevier.com/locate/pla

Nonlinear temporal evolution of plasma waves in backward stimulatedRaman scattering

Gaurav Raj, Ajay K. Upadhyaya, Rohit K. Mishra, Pallavi Jha ∗

Department of Physics, University of Lucknow, Lucknow 226007, India

Received 5 March 2007; received in revised form 3 May 2007; accepted 9 May 2007

Available online 13 May 2007

Communicated by F. Porcelli

Abstract

The phenomenon of laser–plasma coupling is numerically studied by examining the nonlinear coupled temporal evolution of plasma waveamplitude in backward stimulated Raman scattering (BSRS). The theoretical model developed for numerical study takes into account the combinedeffects of relativistic and ponderomotive nonlinearities. The study shows that in BSRS, the phenomenon of nonlinear temporal growth andsaturation of plasma waves strongly depends on laser radiation wavelength. This appears in the form of significant changes in saturation amplitudeand saturation time of plasma waves excited due to long and short wavelength laser waves.© 2007 Elsevier B.V. All rights reserved.

PACS: 52.38.Bv; 52.35.Mw; 52.38.Kd

Keywords: Backward stimulated Raman scattering; Plasma waves; Temporal evolution; Saturation

1. Introduction

Laser–plasma coupling is an important area of research forapplications such as inertial confinement fusion (ICF) [1,2] andlaser–plasma accelerators [3,4]. An important physical processthat affects the efficiency of these applications is stimulated Ra-man scattering (SRS) [5–10], which has been investigated overlast several decades. It is widely known that SRS is an insta-bility associated with laser beams propagating in plasma. Thefundamental process involves propagation of a laser pulse (elec-tromagnetic wave) through plasma, evoking response whichappears in the form of electron plasma (electrostatic) waves.The pump wave mode then parametrically couples with theplasma wave mode, leading to generation of scattered lightwaves (Stokes and anti-Stokes electromagnetic components).When the waves satisfy appropriate frequency and wave num-ber matching conditions, the near resonant interaction of thescattered and plasma waves with the pump wave leads to their

* Corresponding author.E-mail address: prof.pjha@gmail.com (P. Jha).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.05.029

growth at the expense of the pump. The SRS instability requiresthat ωl � 2ωp , i.e. n0 � ncr/4, where ωl (ωp) is laser (plasma)frequency and n0 (ncr) is the initial (critical) plasma density [6].

Considering SRS to be a three-wave process involving thepump, Stokes and plasma waves, most of the earlier workershave studied the growth rate of back and side SRS instabili-ties in linear (nonrelativistic) regime [5,6]. In these studies thegrowth rate is shown to be dependent on parameters such as ini-tial amplitude of the pump wave, the ratio of plasma frequencyto that of the pump and the angle of scattering (θ ) between thepropagation vectors of the pump and scattered waves. Further,recent studies [8,10–12] have shown that the nonlinear growthof SRS instabilities are significantly affected by the relativisticincrease of plasma electron mass under the influence of largeamplitude pump field.

The BSRS [6,7] instability occurs when the scattered lightwave propagates in a direction opposite to that of the incidentpump wave. The BSRS instability can drive large amplitudeplasma waves, which can trap and accelerate properly phasedplasma electrons to ultra high energies in self-modulated laserwakefield acceleration (SM-LWFA) [13,14]. This motivates thestudy of plasma waves excited in BSRS instability with a view

136 G. Raj et al. / Physics Letters A 371 (2007) 135–139

to understanding the process of nonlinear growth and satura-tion of their amplitudes and their dependence on parameters oflaser–plasma system.

In the present Letter, the interaction of laser light with com-pletely ionized homogeneous underdense plasma leading toBSRS instability is studied numerically. The dependence oflaser–plasma coupling on pump wavelength and other parame-ters of laser–plasma system has been obtained. In particular,the nonlinear coupled temporal evolution of plasma wave am-plitude in the BSRS instability has been analyzed. The numer-ical study has been performed using a theoretical model whichsimultaneously includes the effects of relativistic and pondero-motive nonlinearities. The structure of the Letter is as follows:In Section 2, a theoretical model which gives the basic equa-tions governing the three-wave process is developed. A set ofcoupled equations describing nonlinear time dependent evo-lution of amplitudes and phases of the pump, scattered andplasma waves in BSRS instability are derived. Section 3 dealswith numerical solutions and analysis of the results. Conclu-sions are presented in Section 4.

2. Theoretical model

A theoretical model is developed for laser–plasma interac-tions in the mildly relativistic regime. The laser field is assumedto be long enough to evoke response of plasma electrons, whileit is short enough so that the heavy ions of the backgroundplasma may be considered to be immobile. The plasma is con-sidered to be cold and it is assumed that the thermal velocityof plasma electrons is small compared to their quiver velocityunder the influence of the laser field.

2.1. Basic equations

Consider the pump, scattered and plasma wave electric fieldspropagating through preformed plasma, to be represented by

(1)Ej (r, t) = ej

2

[εj e

i(kj .r−ωj t) + ε∗j e

−i(kj .r−ωj t)],

where j = l, s,p, respectively, represents the pump, scatteredand plasma waves. Further, kj and ωj are the propagation vec-tors and frequencies of the three waves, while ej and εj arethe polarization vectors and the electric field amplitudes. Theseamplitudes are slowly varying in space and time.

The general wave equation that governs the propagation ofelectromagnetic and electrostatic waves in plasma is given by

(2)∂2E(r, t)

∂t2+ c2[∇ × (∇ × E(r, t)

)] = −4π∂J(r, t)

∂t,

where E(r, t) (= El + Es + Ep) is the total electric field vectordue to the three waves. J(r, t) (= −env) is the plasma currentdensity due to presence of n plasma electrons per unit volumeeach having charge −e and moving with velocity v (= vl +vs +vp). Further, J has both linear as well as nonlinear contributionsdue to the three waves propagating through plasma. The fieldsare governed by conditions ∇ .El = ∇ .Es = 0 and ∇ ×Ep = 0.

In presence of electromagnetic and electrostatic fields theplasma electron dynamics is given by the Lorentz force equa-tion

(3)∂v∂t

= − eEmγ

− e

mcγ(v × B) − (v . ∇)v − v

γ

(dγ

dt

),

where B (= Bl + Bs ) is the magnetic field vector with Bl,s =c(kl,s × El,s )/ωl,s , m is the rest mass of a plasma electron andγ (= (1 − (v2/c2))−1/2) is the relativistic factor. The electrondensity n can be obtained from the equation of continuity givenby

(4)∂n

∂t+ ∇ . (nv) = 0.

As a consequence of interaction between laser and plasma, theinitial electron density (n0) is perturbed and results in the to-tal density n (= n0 + δn), where δn is the perturbed electronplasma density. Substitution of Eqs. (3) and (4) in the timederivative of the current density gives

∂J∂t

= −en

[−eEm

− 1

2∇(

v2) + eEv2

2mc2+ ev2(v × B)

2mc3

+ ev(E . v)

mc2− ev(E . v)v2

2mc4− v(∇ . v)

]

(5)+ ev(v . ∇δn).

The right side of Eq. (5) describes the linear as well as nonlinearcontributions to the current density. The first term in conjunc-tion with the ambient plasma density n0 gives the linear sourceterm arising due to the quiver velocity of plasma electrons. Thesecond term is due to the ponderomotive force of the radiationfield. The following four terms are additional changes in thecurrent density arising due to relativistic effects, while the lasttwo terms are contributions to the current density due to plasmaelectron density perturbations. These current density contribu-tions significantly affect the growth of SRS instabilities. The setof Eqs. (1)–(5) are the fundamental equations governing laser–plasma coupling in SRS process.

2.2. Coupled temporal evolutions in BSRS process

Stimulated Raman scattering is incited by only those elec-tron plasma waves that satisfy the wave number and frequencymatching conditions, respectively, given by kl = ks + kp andωl = ωs + ωp , where plasma frequency is related to ambientplasma density as ω2

p = 4πe2n0/m. The linearized velocitiesare obtained by using Eq. (1) in Eq. (3) as

(6)vj (r, t) = eej

2imωj

[εj e

i(kj .r−ωj t) − ε∗j e

−i(kj .r−ωj t)].

Considering the pump wave to be linearly polarized along x-di-rection and propagating along z-axis, the excited longitudinalelectron plasma wave will propagate along the z-direction. Fur-ther, for BSRS process the generated scattered wave propaga-tion vector (ks) makes an angle π with respect to the propa-gation vector (kl ) and has the same polarization as the pump.Substitution of fields (Eq. (1)) and linearized velocities (Eq. (6))

G. Raj et al. / Physics Letters A 371 (2007) 135–139 137

in Eq. (5) gives the total source which includes nonlineari-ties due to relativistic and ponderomotive effects. Substitutionof the source terms (Eq. (5)) into Eq. (2), alongwith the con-sideration that the electron plasma density perturbations obeyPoisson equation δn = −∇ . Ep/4πe, leads to nonlinear modecoupled equations for the slowly varying electric field ampli-tudes in completely ionized underdense plasma. The coupledtemporal evolution of the pump wave in presence of relativisticand ponderomotive nonlinearities (upto the fourth order in fieldamplitudes) is given by

∂εl

∂t= i

[L1εsε

∗s + L2εpε∗

p

]εl

(7a)

− [L3 − L4εlε

∗l − L5εsε

∗s − L6εpε∗

p

]εsεp + L7ε

2l ε

∗s ε

∗p,

where

L1 = (3Aξ/ωs), L2 = (Aξωs/ω

2p

),

L3 = A(1 + (ωs/ωp)

),

L4 = (Aξ2/2ωpω2

l

)[2c(ks − 2kl) + 5ωs

],

L5 = (Aξ2/2ωpω2

s

)[3ωs + cks],L6 = (

Aξ2/2ω3p

)[3cks + (3ωs + 2ωp)

],

L7 = (Aξ2/2ωpω2

l

)[(3ωs − 4ωl) + c(2kl + ks)

],

A = (ekpωp/4mωlωs), ξ = (e/2mc),

while kl , ks and kp are the magnitudes of pump, scattered andplasma wave vectors, respectively. The evolution of scatteredwave in presence of the two nonlinearities is obtained as

∂εs

∂t= i

[S1εsε

∗s + S2εpε∗

p + S3εrε∗r

]εs

(7b)

− [S4 − S5εpε∗

p − S6εrε∗r − S8εsε

∗s

]εrε

∗p + S7εpε∗

r ε2s ,

where

S1 = (3Aξωl/2ω2

s

), S2 = (

Aξωl/ω2p

),

S3 = (3Aξ/ωl), S4 = A[1 − (ωl/ωp)

],

S5 = (Aξ2/2ω3

p

)[3ckl + (2ωp − 3ωl)

],

S6 = (Aξ2/2ω2

l ωp

)[2ckl − 3ωl],S7 = (

Aξ2/2ωpω2s

)[c(kl + 4ks) − (3ωl − 4ωs)

]and

S8 = (Aξ2/ω2

s ωp

)[ckl − 3ωl],while the electrostatic electron plasma wave evolves as

∂εp

∂t= i

[P1εsε

∗s + P2εrε

∗r + P3εpε∗

p

]εP

(7c)

+ [P4 + P5εrε

∗r − P6εsε

∗s + P7εpε∗

p

]εrε

∗s + P8εsε

∗r ε

2p,

where

P1 = (Aξωl/ωsωp), P2 = (Aξωs/ωlωp),

P3 = (Aξωlωs/2sω

3P

), P4 = A,

P5 = (−3Aξ2/2ωpω2l

)[c(ks + kl)

],

P6 = (3Aξ2/2ωpω2

s

)[c(ks + kl)

], P7 = (−Aξ2/ω2

p

),

P8 = (Aξ2/2ω3

p

)[c(2kp − kl − ks)

].

In obtaining the set of Eqs. (7), second order derivatives andspatial variations of the field amplitudes have been neglected.The slowly varying amplitudes are considered to be of the form

(8)εj (t) = qj (t)eiαj (t),

where qj and αj are the real amplitudes and phases, respec-tively, and are functions of time. Substituting the respectiveforms of amplitudes given by Eq. (8) in Eq. (7), and equatingthe real and imaginary parts, we get the nonlinear coupled timedependent evolution of amplitudes and phases of the pump,scattered and plasma waves, respectively, as

(9a)∂ql

∂t= −[

L3 − κqlq2l − L5q

2s − L6q

2p

]qsqp cosβ,

(9b)∂qs

∂t= −[

S4 − κqsq2s − S5q

2p − S6q

2l

]qlqp cosβ,

(9c)∂qp

∂t= [

P4 + κqpq2p + P5q

2l − P6q

2s

]qlqs cosβ,

and

(10a)

ql

∂αl

∂t= [

L1q2s + L2q

2p

]ql

+ [L3 + καlq

2r − L5q

2s − L6q

2p

]qsqp sinβ,

(10b)

qs

∂αs

∂t= [

S1q2s + S2q

2p + S3q

2l

]qs

− [S4 + καsq

2s − S5q

2p − S6q

2l

]qlqp sinβ,

(10c)

qp

∂αp

∂t= [

P1q2s + P2q

2l + P3q

2p

]qp

+ [P4 − καpq2

p + P5q2r − P6q

2s

]qlqs sinβ,

where in Eqs. (9) and (10) κql = L7 +L4, κqs = S7 +S8, κqp =P8 +P7, καl = L7 −L4, καs = S7 −S8, καp = P8 −P7 and β(t)

(= αl(t) − αs(t) − αp(t)) is the relative phase difference of thecoupled waves.

3. Numerical results and discussions

The set of Eqs. (9) and (10) are constituted by six first or-der partial differential equations, which physically representthe phenomenon of nonlinear laser–plasma coupling in BSRSinstability. The equations respectively give the nonlinear cou-pled temporal evolution of amplitudes and phases of the threewaves in BSRS process under the combined effects of relativis-tic and ponderomotive nonlinearities. To study and analyze theprocess of nonlinear growth and saturation of these waves inBSRS process, the set of Eqs. (9) and (10) are simultaneouslysolved using the fourth order Runge–Kutta method. The ini-tial conditions used in the numerical solutions are n0 = 1.25 ×1020 cm−3, qs0 ≈ qp0 ≈ 0.01ql0 and αl0 ≈ αs0 ≈ αp0 ≈ 0where ql0, qs0, qp0 and αl0, αs0, αp0 are, respectively, the initialamplitudes and phases of the three waves.

Fig. 1 shows the comparative study of nonlinear coupledtemporal evolution of amplitudes of pump (λl0 = 0.3 µm), scat-tered and plasma waves in present case of BSRS (solid curve)

138 G. Raj et al. / Physics Letters A 371 (2007) 135–139

Fig. 1. Coupled temporal evolution of amplitudes of 0.3 µm laser wave, scat-tered wave and plasma wave with ωp = 6.28×1014 s−1 for BSRS (solid curve)and θ = π/4 side SRS (dashed curve).

Fig. 2. Nonlinear temporal evolution of plasma wave amplitudes for pumpwavelengths λl0 ≈ 1.0 (solid curve), 0.5 (dashed curve) and 0.3 (dotted curve)µm with ωp = 6.28 × 1014 s−1.

and θ = π/4 side SRS (dashed curve) instabilities (Ref. [10]).It is seen that the laser wave amplitudes are attenuated while si-multaneously the scattered and plasma wave amplitudes growwith time to attain a maximum (saturation) [10,11,15] valueand then decrease. The study clearly shows that among the twoinstabilities, temporal growth of scattered and plasma wavesfor BSRS is faster as compared to that obtained in side SRSprocess, leading to early time saturation of scattered and plasmawave amplitudes in BSRS instability. The possible mechanismsof BSRS saturation are wave breaking [16], pump depletion andnonlinear phase mismatch due to relativistic increase of plasmaelectron mass [11]. However, one-dimensional relativistic fluidsimulations indicate that for laser intensities of the order of1017–1018 W/cm2 saturation is caused primarily due to non-linear phase mismatch [11].

Nonlinear temporal evolution of plasma wave amplitudesin BSRS process is shown in Fig. 2. These solutions areobtained for interaction of long wavelength (LW) and short

Fig. 3. Temporal evolution of electron plasma density perturbations for pumpwavelengths λl0 ≈ 1.0 (solid curve), 0.5 (dashed curve) and 0.3 (dotted curve)µm with ωp = 6.28 × 1014 s−1.

wavelength (SW) laser radiations with the plasma, where SWlaser light implies λl0 � 0.5 µm (Ref. [6]). For same intensity(Il0), laser radiation of different wavelengths will be charac-terized by varying laser strength parameters al0 (= (7.32 ×10−19λ2

l0[µm]Il0[W/cm2])1/2) (Ref. [16]). It is observed thatthe saturation amplitude of plasma waves excited due to 1.0 µm(solid curve) laser light is approximately 30% and 66% more ascompared to 0.5 µm (dashed curve) and 0.3 µm (dotted curve)laser radiation, respectively. This is observed since the criti-cal density ncr (= mω2

l /4πe2) for propagation 1.0 µm (LW)laser radiation is close to initial plasma density as comparedto SW (0.5 and 0.3 µm) pump waves, this leads to enhancedtransfer of energy between LW laser radiation and the plasma.Further, the study shows that the saturation time [10,11,15](time taken by plasma wave amplitude to reach its maximumvalue) for 1.0 µm laser light is approximately 65% and 83%less as compared to 0.5 and 0.3 µm (SW) laser waves, respec-tively. This can be explained as for same intensity, the LW laserradiation will lead to larger value of al0 (sufficiently strongrelativistic effects), leading to nonlinear phase mismatch at anearly time as compared to SW laser light cases. Further, usingPoisson equation, the corresponding perturbed plasma densi-ties normalized to ambient density are plotted with time inFig. 3. The figure reveals that for small propagation distance z

(≈ 0.85 µm), laser radiation of 1.0 µm wavelength (solid curve)induces strong density perturbation in the plasma as comparedto 0.5 µm (dashed curve) and 0.3 µm (dotted curve) wavelengthlaser radiations.

The numerical solution in Fig. 4 shows the, nonlinear tem-poral evolution of phases of plasma waves excited by differ-ent wavelength pump waves in BSRS instability. It is observedfrom the figure that for SW 0.5 µm (dashed curve) and 0.3 µm(dotted curve) laser light cases, the plasma wave phases in-crease monotonously in the positive direction. However thephase of plasma wave excited by LW 1.0 µm (solid curve) laserradiation first increases in positive direction during initial stageof evolution and then undergoes direction inversion.

G. Raj et al. / Physics Letters A 371 (2007) 135–139 139

Fig. 4. Nonlinear temporal evolution of phases of plasma wave for pump wave-lengths λl0 ≈ 1.0 (solid curve), 0.5 (dashed curve) and 0.3 (dotted curve) µmwith ωp = 6.28 × 1014 s−1.

4. Conclusions

In the present Letter, the process of nonlinear laser–plasmacoupling is studied by considering coupled temporal evolutionof laser, scattered and plasma wave amplitudes in BSRS insta-bility. The set of six first order partial differential equationsformulated in the study take into account the relativistic andponderomotive nonlinear effects. The numerical analysis showsthat in BSRS instability there is rapid temporal growth of am-plitudes of scattered and plasma waves as compared to side SRS(θ = π/4) process. This leads to early time saturation of ampli-tudes of scattered and plasma waves in BSRS process. Also inBSRS process excited by laser sources having same intensitybut different wavelengths, the nonlinear growth and saturationof plasma wave amplitudes is significantly dependent on thepump wavelength. The saturation amplitude for LW laser waveis large, showing stronger coupling of the laser energy to plasmaas compared to the SW laser cases. The analysis shows that forLW laser waves the relativistic effects will be strong, leading toearly time saturation of plasma wave amplitude as compared toSW laser radiation. The numerical study indicates that for suffi-ciently strong laser radiation interacting with plasma, nonlinear

phase mismatch is the main mechanism for saturation of reso-nantly driven plasma waves in BSRS process. The present studyalso shows that plasma waves associated with BSRS processexcited due to LW pump wave grows to large amplitude as com-pared to SW laser wave case. Consequently for LW laser wavecase, the plasma waves will accelerate the trapped backgroundelectrons to high energies as compared SW laser radiation case.Thus LW laser waves will lead to efficient electron accelerationmechanism in (SM-LWFA) as compared to SW laser light.

Acknowledgements

This work has been done with the financial support of Sci-ence and Engineering Research Council, Department of Sci-ence and Technology, Government of India. The authors thankthe organization for funding the research project.

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