2. Centre de PhysiqueThéorique

1. DAM; Bruyères-le-ChâtelDCSA

1. : M. Casanova

1., 2. : Thomas Fouquet

2. : S. Hüller, D. Pesme

HEDP Summer School

UC Berkeley, 2005

Nonlinear Evolution of Stimulated Raman Scattering

driven by a RPP Laser Beam in a 2D Inhomogeneous Plasma

CPHT

Aim : Study the Stimulated Raman Scattering (SRS) and its saturation via the coupling of the plasma waves generated by this instability, with the sound waves; in the case of an inhomogeneous plasma in density.

2D modelization

case of a linear density profile

laser beam : monospeckle or RPP

Raman instability results of the coupling of an incident laser wave with a plasma electronic wave to give birth to a transverse wave named scattered : it is a three waves coupling process.The conditions of the resoning coupling imply that this instability can be developed in a field called 'under quarter critical'. This field corresponds to ne< nc/4 densities.

0a )exp( 0zikRa )exp( zikR

Rkk ,0 Lk )exp( zikL

RL kkk 0

Model and equations

)()( znnzN hydroerefhydro . )(zNhydro

.erefn

)(znhydro

La Rkk ,0

LSISLRIPLL aazikaaaD )exp(*0

RsISLLIPRR aazikaaaD )exp().( *0

000 )exp().( aazikaaaD sISLLRIP

22

0

22RLISSS aaanD

RamanAutofocalisation

Raman

Raman LDI

Autofocalisation

ponderomotive forces

pump

backscattered

plasma

sound

DLinear propagator

Envelope and paraxial approximation for the pump and backscattered waves

Inhomogeneity

Inhomogeneity

Inhomogeneity

shydros nzna )(

SRS in a 2D homogeneous plasma

Test of the numerical scheme

Monospeckle pump wave

5# f

pst 10 pst 15

221420 /.10.8 cmmWI L =L = m65

Backscattered wave

pst 10 pst 15

Plasma wave

pst 10 pst 15

Gaussian beam

15.0n

n

c

ref

Transmission

2

0

2

0

),0(

),(onTransmissi

ya

yLa 2

0

2

),0(

),0(tyReflectivi

ya

yaR

Reflectivity

Energy conservation OK !

time time

T R

Raman in an inhomogeneous 1D plasma

0 )(0 z « finite length » effect

Resonance conditions cannot be satisfied in the whole plasma slab

),( )()( kD ),,( )()( zkD

),(),(),( 000000 zkzkzk RLRR

: Solutions of parametric resonance conditions in

0zz

)0(k

)()()( 0)0(

)( zkzzkzk

)(zk

.

.

2 effects of distinct nature :

Dispersion relation is local

)()()()( 0 zkzkzkzk LR

« Linear « Linear profile » profile » ::

)())(()( 00

zzzkzkzzz

Monotonous profile of density

Rosenbluth’s result :Finite spatial amplification

LIGRos )( 2

0)( 0 zk

)(

1

zkL

z

PRL 1972

Non robust

Absolute instability

z

Mismatch : /

)(zNhydro.

!!

Difficulty !!!

Numerical difficulties :

Non robustness of the Rosenbluth’s result

Realistic plasma = finite length + inhomogeneous in density

Numerical techniques in order to control the behaviour at the boundaries :

Difficulties = Numerical artefacts

. numerical dampings

. « window » function in the simulation box

absolute instability

If the variation is too fast

SRS without its coupling to the IAWs :inhomogeneous plasma with a linear density profile;RPP laser beam

pump backscattered

plasma

3# f 221520 /.10.5 cmmWI

52 Ros G

2.0DeLk 11.0n

n03.0

c

6.0eT keV

5P

P

c

t = 1ps

t = 2ps

L= L m150=

m65L //plasma

pump backscattered

plasma

Raman

reflectivity

w.o LDI

t = 3ps

t = 5ps

SRS reflectivity saturates at ~ 10% :

-filamentation

-finite amplification gain=

RosGtime

R

pump backscattered

plasma sound

01,0s

s

SRS with its coupling to the IAWs (LDI)* :inhomogeneous plasma with a linear density profile;RPP laser beam

t = 2ps

t = 1psand same parameters as used before

*generalizing in a 2D inhomogeneous plasma, the results investigated by Bezzerides, DuBois, Rose, Rozmus, Russel, Tikhonchuk, Vu

pump plasma soundbackscattered

1,0005,0

Sn

n

t = 3ps

t = 5ps

Ramanreflectivityw. LDI w. LDI

red : Ramanreflectivity

black : Raman

reflectivityw.o LDI

}

Evidence of saturation effects due to LDI

SRS reflectivity saturates at ~ 1.5% when plasma and sound waves are coupled; compared to 10% without coupling to the IAW’s

R

time

Summary/conclusions

*Design of a 2D numerical code :-inhomogeneous plasma-SRS coupled to LDI

*First simulations of SRS with LDI in an inhomogeneous 2D plasma

*Evidence of saturation due to LDI; generalizing in an inhomogeneous plasma the results already seen in a homogeneous one