A multi-D model for Raman amplification

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A multi-D model for Raman amplification

Mathieu Colin, Thierry Colin

To cite this version:

Mathieu Colin, Thierry Colin. A multi-D model for Raman amplification. 2008. �inria-00332462�

A multi-D model for Raman amplification

by M. Colin and T. Colin, Universit´e de Bordeaux

IMB, INRIA Bordeaux Sud Ouest, Team MC2 351 cours de la lib´eration, 33405 Talence cedex, France.

mcolin@math.u-bordeaux1.fr colin@math.u-bordeaux1.fr

Abstract : In this paper, we continue the study of the Raman amplification in plasmas that we have initiated in [5] and [6]. We point out that the Raman instability gives rise to three components. The first one is colinear to the incident laser pulse and counter propagates. In 2-D, the two other ones make a non-zero angle with the initial pulse and propagates forward. Furthermore they are symmetric with respect to the direction of propagation of the incident pulse.

We construct a nonlinear system taking into account all these components and perform some 2-D numerical simulations.

Key words: Zakharov system, Raman amplification, three waves mixing.

1 Introduction

The interaction of powerfull laser pulse with a plasma gives rise to several com- plex multiscale phenomena. It is of great interest since it occurs in the labo- ratory simulations of nuclear fusion (NIF, Laser Mega Joule). One of the key mechanism is the Raman instability that can be coupled with Landau damping (see [1]). In [5] and [6], we have initiated a quite systematical mathematical study of the Raman amplification process in plasma by justifying nonlinear models in 1-D and 2-D. These models relies on the propagation of three kind of waves : the initial laser pulse (K

0

, ω

0

), the Raman component (K

R

, ω

_R

) and the electronic plasma wave (K

1

, ω

_pe

+ ω

1

) where K and ω stands respectively for the wave vector and the frequency and ω

_pe

is the electronic plasma frequency.

In order to be efficient, the interaction has to be a three waves mixing that is the data must satisfy the following relationships :

• The dispersion relation for electromagnetic waves

ω

₀²

= ω

_pe²

+ c

²

| K

0

|

²

, (1.1) ω

²_R

= ω

²_pe

+ c

²

| K

_R

|

²

, (1.2)

AMS classification scheme number: 35Q55, 35Q60, 78A60, 74S20

where c is the velocity of light in the vacuum.

• The dispersion relation for electronic plasma waves

(ω

pe

+ ω

1

)

²

= ω

²_pe

+ v

_th²

| K

0

|

²

, (1.3) where v

th

denotes the thermal velocity of the electrons, see below (section 2.1) for its value.

• The three waves resonance conditions

ω

0

= ω

_pe

+ ω

_R

+ ω

1

, (1.4)

K

0

= K

R

+ K

1

. (1.5)

Even in 2-D, one can find solutions of this system such that K

1

, K

R

and K

0

are colinear. This corresponds to the solution used in [5], [6]. The aim of this paper is to provide a more general study in order to understand the influence of the geometry. We solve (1.1) − (1.5) numerically and show that there exists infinitely many solutions in the plane. We compute the amplification rates asso- ciated to these solutions and show that the backward solution has a maximum amplification rate when it is colinear to the initial pulse, while the most two amplified forward directions make a non-zero angle with the laser pulse and are symmetric with respect to the direction of propagation of the incident pulse.

(see Section 2 and Section 3).

In Section 4, we introduce a nonlinear model taking into account both direc- tion of propagations. First, denote by A

0

the incident laser field, K

0

and ω

0

the associated wave vector and frequency, A

_R1

the backscattered Raman compo- nent K

R1

and ω

R1

the associated wave vector and frequency, A

R2

, the forward Raman component K

R2

and ω

R2

the associated wave vector and frequency, and finally A

R₂^s

the second forward Raman component, K

R^s₂

and ω

R^s₂

the associated wave vector and frequency. We assume that K

0

is colinear to the x − axis and then K

R2

and K

R₂^s

are symmetric with respect to the x − axis. < n

e

> is the low-frequency variation of the density of ions. Furthermore, we put

K

0

= k

0

0

, K

_R1

= k

_R1

ℓ

_R1

, K

_R2

= k

_R2

ℓ

_R2

, K

_R^s₂

= k

R^s₂

ℓ

_R^s₂

, θ

1,1

= K

1,1

· X − ω

1,1

t, θ

1,2

= K

1,2

· X − ω

1,2

t, θ

1,2^s

= K

1,2^s

· X − ω

1,2^s

t.

The system reads in a nondimensional form i

∂

_t

+ c

²

k

²₀

ω

₀²

∂

_x

A

0

+ c

²

k

₀²

2ω

₀²

∆ − c

⁴

k

⁴₀

2ω

₀⁴

∂

_x²

A

0

= ω

²_pe

2ω

₀²

< n

_e

> A

0

(1.6)

− ∇ · E

A

_R1

e

^−iθ1,1

k

R1

| K

_R1

| + α A

_R2

e

^−iθ1,2

+ A

_R^s₂

e

^−iθ1,2s

k

R2

| K

_R2

|

,

i

∂

t

+ c

²

k

₀

ω

R1

ω

0

K

R1

· ∇

A

R1

+ 1 2ω

R1

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²₁

K

R1

· ∇

2

A

R1

= ω

_pe²

2ω

0

ω

R1

< n

_e

> A

_R1

− ∇ · E

^∗

A

₀

e

^iθ1,1

k

_R1

| K

R1

| , (1.7)

i

∂

t

+ c

²

k

0

ω

_R2

ω

0

K

R2

· ∇

A

R2

+ 1 2ω

R2

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²

2

K

R2

· ∇

2

A

R2

= ω

_pe²

2ω

0

ω

R2

< n

e

> A

R2

− α ∇ · E

^∗

A

0

e

^iθ1,2

k

R2

| K

R2

| , (1.8)

i

∂

_t

+ c

²

k

0

ω

R2

ω

0

K

_R^s₂

· ∇

A

_R^s₂

+ 1 2ω

R2

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²₂

K

_R^s₂

· ∇

2

A

_R^s₂

= ω

_pe²

2ω

0

ω

_R2

< n

_e

> A

_R2^s

− α ∇ · E

^∗

A

0

e

^iθ1,2s

k

R2

| K

_R2

| , (1.9)

i∂

t

E + v

_th²

k

₀²

2ω

pe

ω

0

∆E = ω

pe

2ω

0

< n

e

> E + ∇

A

0

A

^∗_R1

e

^iθ1,1

k

R1

| K

R1

| + α A

0

A

^∗_R2

e

^iθ1,2

+ A

0

A

^∗_R^s

2

e

^iθ1,2s

k

R2

| K

R2

|

, (1.10)

∂

_t²

− c

²_s

∆

< n

e

>= 4m

e

m

_i

ω

0

ω

R1

ω

²_pe

∆

| E |

²

+ ω

pe

ω

0

| A

0

|

²

+ ω

pe

ω

_R1

| A

R1

|

²

+ ω

pe

ω

_R2

| A

R2

|

²

+ | A

R^s₂

|

²

. (1.11) The constants c, c

s

, m

e

and m

i

are respectively the velocity of light in vacuum, the acoustic velocity, the electron’s and ion’s mass. The methods of [5] applies and one gets the following theorem .

Theorem 1.1. Let s >

^d₂

+ 3, (a

0

, a

_R1

, a

_R2

, a

_R^s₂

, e) ∈ H

^s+2

(R

^d

)

5d

, n

0

∈ H

^s+1

(R

^d

) and n

1

∈ H

^s

(R

^d

). There exists T > 0 and a unique maximal solution (A

0

, A

R1

, A

R2

, A

R^s₂

, E, < n

e

>) to (1.6) − (1.11) such that

(A

0

, A

R1

, A

R2

, A

^s_R2

, E) ∈

C([0, T [; H

^s+2

)

5d

,

< n

e

> ∈ C([0, T [; H

^s+1

) ∩ C

¹

([0, T [; H

^s

), with initial value

(A

0

, A

_R1

, A

_R2

, A

_R^s

2

, E)(0) = (a

0

, a

_R1

, a

_R2

, a

_R^s

2

, e)

< n

_e

> (0) = n

0

, ∂

_t

< n

_e

> (0) = n

1

.

In Section 5, we perform some numerical simulations in order to illustrate

the phenomena and to emphazise the new directions of propagation.

2 Obtaining a 3-D Raman amplification system

2.1 The Euler-Maxwell system

As noticed in the introduction, the main drawback of the model developped in [5] is the fact that the Raman component and the laser incident field are colinear in the sense that the wave vectors are proportional (in opposite direction). The aim of this section is to get rid of this hypothesis. We will only sketch the computations that are very closed to the ones done in [5]. We start from the bifluid Euler-Maxwell system. The Euler equations are

(n

0

+ n

_e

) (∂

t

v

_e

+ v

_e

· ∇ v

_e

) = − γ

e

T

e

m

_e

∇ n

_e

− e(n

0

+ n

e

) m

_e

(E + 1

c v

_e

× B), (2.1) (n

0

+ n

i

) (∂

t

v

i

+ v

i

· ∇ v

i

) = − γ

i

T

i

m

_i

∇ n

i

+ e(n

0

+ n

i

) m

_i

(E + 1

c v

i

× B), (2.2)

∂

t

n

e

+ ∇ · ((n

0

+ n

e

)v

e

) = 0, (2.3)

∂

_t

n

_i

+ ∇ · ((n

0

+ n

_i

)v

i

) = 0. (2.4)

The Maxwell system is written in terms of the electric-magnetic fields for the study of the electronic-plasma waves (Langmuir waves)

∂

t

B + c ∇ × E = 0, (2.5)

∂

_t

E − c ∇ × B = 4πe (n

0

+ n

_e

)v

e

− Z (n

0

+ n

_i

)v

i

, (2.6)

while the formulation with magnetic potential, electric potential and electric field in the Lorentz jauge is used for the study of the electromagnetic waves (light)

∂

t

ψ = c ∇ · A, (2.7)

∂

_t

A + cE = c ∇ ψ, (2.8)

∂

_t

E − c ∇ × ∇ × A = 4πe (n

0

+ n

_e

)v

e

− Z (n

0

+ n

_i

)v

i

, (2.9)

where Z is the atomic number of the ions. We first perform a linear analysis of system (2.1) − (2.9) and compute the dispersion relations as well as the polarizations conditions. Then using the time enveloppe approximation, we derive a quasilinear system describing the interaction.

2.2 Dispersion relations and polarization conditions.

Since the mass of the ions is much larger than that of the electrons (a ratio of at least 10

³

), the velocity of the ions is smaller than that of the electrons.

Therefore we can neglect the contribution of the ions in the current in (2.6) or

(2.9). We then linearize System (2.1) − (2.9) around the steady state solution 0

and one gets

n

0

∂

_t

v

_e

= − γ

e

T

e

m

_e

∇ n

_e

− en

0

m

_e

E, (2.10)

∂

_t

n

_e

+ n

0

∇ · v

_e

= 0, (2.11)

∂

t

B + c ∇ × E = 0, (2.12)

∂

t

E − c ∇ × B = 4πen

e

v

e

, (2.13) Note that the accoustic part concerning the ions is decoupled from the high fre- quency part concerning the electrons and will be considered below. We look for plane wave solutions to (2.10) − (2.11) under the form e

^{i(K·X−ωt)}

v

e

, n

e

, B, R

. Two kind of waves can propagate :

i) Longitudinal waves for which K is parallel to E (electronic-plasma wave).

They satisfy the dispersion relation

ω

²

= ω

²_pe

+ v

²_th

| K |

²

, (2.14) with

ω

_pe²

= 4πe

²

n

₀

m

e

, v

_th²

= r γ

_e

T

_e

m

e

.

ii) Transverse waves for which K is orthogonal to E (electromagnetic waves).

They obey the dispersion relation

ω

²

= ω

²_pe

+ c

²

| K |

²

. (2.15) Since for our applications, v

th

≪ c, the shape of the graph of (2.14) or (2.15) are very different. Indeed, (2.14) is very flat near the origin compare to (2.15) (see Figure 1).

0 2 4 6 8 10

-10 -5 0 5 10

omega

|K|

c=1 vth=1/10

Figure 1: Plot of the first part of the dispersion relations (2.14) (dashed line)

and (2.15) (solid line) with ω

_pe²

= 1, c

²

= 1 and v

_th²

= 0.01. The plot corresponds

to ω as a function of | K | .

Therefore, even if a precise couple (K

0

, ω

₀

) is imposed for the incident laser field, we have to consider only the frequency ω

_pe

for the electronic plasma wave with a continuous range K of wave vectors. Therefore, the complete solution reads



 

 B E v

_e

n



 

 =



 

 B

||

E

_||

v

_e||

n

_e



 

 e

^−iωpet

+ X

n j=1



 

 B

_⊥^j

E

_⊥^j

v

_e⊥^j

0



 

 e

^{i(Kj·X−ωjt)}

+ c.c. (2.16)

where || corresponds to the longitudinal part and ⊥ coresponds to the tranverse part. Furthermore, for all 1 ≤ j ≤ n

ω

²_j

= ω

_pe²

+ c

²

| K

j

|

²

. (2.17) As usual B

||

, E

||

, v

e||

, n

e

and B

_⊥^j

, E

_⊥^j

, v

^j_e⊥

satisfy some polarization condi- tions that are obtained like in [5] by pluging plane waves in (2.10) − (2.13).

v

e||

= − i iω

pe

4πen

0

E

||

, B

||

= 0, n

e

= − 1

4πe ∇ · E

||

.

For the transverse part, one writes B

⊥

= ∇× A

⊥

and using the Maxwell system in the Lorentz jauge we get

− iω

j

A

^j_⊥

+ cE

^j_⊥

= 0, (2.18)

− iω

_j

E

^j_⊥

+ cK

_j

× K

_j

× A

^j_⊥

= 4πn

0

v

_e⊥

, (2.19)

− iω

_j

v

^j_e⊥

= − e n

e

E

_⊥^j

. (2.20)

Using (2.19) and (2.20), one obtains

− iω

j

E

_⊥^j

+ cK

j

× K

j

× A

^j_⊥

= − i 4πe

²

n

0

m

_e

ω

_j

E

_⊥^j

.

It follows that E

^j_⊥

is orthogonal to K

j

and therefore so do A

^j_⊥

and v

_e⊥^j

. In a 2-D geometry, this leads to look for E

_⊥^j

, A

^j_⊥

and v

^j_e⊥

under the form

E

_⊥^j

, A

^j_⊥

, v

^j_e⊥

K

_j^⊥

| K

j

| , (2.21)

where E

_⊥^j

, A

^j_⊥

, v

^j_e⊥

denote now scalar functions. The polarization relations on these scalar fields read

v

^j_e⊥

= e

m

e

c A

^j_⊥

, E

_⊥^j

= i ω

_j

c A

^j_⊥

. (2.22)

2.3 The weakly nonlinear theory.

We restrict ourself to the 2D case. Since we are interested in the Raman insta- bility, we take n = 2 in (2.16) write



 

 B E v

_e

n



 

 =



 

 0 E

_||

v

_e||

n

e



 

 e

^−iωpet

+



 

 B

0

E

0

v

e0

0



 

 e

^{i(K0·X−ω0t)}

+



 

 B

R

E

R

v

eR

0



 

 e

^{i(KR·X−ωRt)}

+ c.c., (2.23)

where 0 stands for the incident laser field and R for the Raman component.

The equations satisfied by each of the electromagnetic fields B

0

, E

0

, v

e0

and B

R

, E

R

, v

eR

are, using the vector potential A (A=A

0

or A

R

)

∂

_t²

A + c

²

∇ × ∇ × A = − 4πec(n

0

+ n

e

)v

e

. The polarization condition (2.21) leads to

∂

_t²

A − c

²

∆A = − 4πec P

_K⊥

(n

0

+ n

e

)v

e

, where P

_K⊥

is the orthogonal projector onto K

^⊥

. Now we write

A = Ae e

^{i(K·X−ωt)}

+ c.c.

(with K = K

0

or K

R

, ω = ω

0

or ω

R

) and get

∂

_t²

A = ∂

_t²

A e − 2iω∂

t

A e − ω

²

A e e

^iθ

,

∇ A = ∇ A e + iK A e e

^iθ

,

∆A = ∆ A e + 2iK · ∇ A e − K

²

A e e

^iθ

, where θ = K · X − ωt. At the first order

− 2iω∂

t

A e − 2ic

²

K · ∇ A e = 0.

Therefore

∂

_t²

A e = c

⁴

ω

²

K · ∇ )

²

A. e

This is the method used to derive the BBM equation in the water wave theory (see [2] or [4]). We obtain (omitting the tildes)

− 2iω∂

t

A − 2ic

²

K · ∇ A − c

²

∆A + c

⁴

ω

²

K · ∇

2

A + (c

²

K

²

− ω

²

)A

= − 4πec P

K^⊥

(n

0

+ n

_e

)v

e

e

^−iθ

. (2.24)

We now consider a three waves mixing. In a 2-D framework we consider three waves vectors K

0

, K

_R

, K

1

and three frequencies (ω

0

, ω

_R

, ω

1

) satisfying K

0

= K

R

+ K

1

ω

0

= ω

pe

+ ω

R

+ ω

1

, K

0

=

k

0

0

, K

R

= k

R

l

R

, K

1

=

k

1

l

1

. (2.25)

Without loss of generality we assume that the incident laser field propagates along the x − axis. In the right-hand-side of (2.24), the nonlinear term gives using (2.22)

(n

0

+ n

_e

)v

e

= n

0

v

_e

+ n

_e

v

_e

= n

0

e

m

_e

c A + n

_e

v

_e

. Equation (2.24) then reads

− 2iω∂

t

A − 2ic

²

K · ∇ A − c

²

∆A + c

⁴

ω

²

K · ∇

2

A + (c

²

K

²

− ω

²

)A

= − ω

pe

A − 4πec P

_K⊥

n

e

v

e

e

^−iθ

, and we get

i

∂

_t

+ c

²

ω K · ∇

A + 1 2ω

c

²

∆ − c

⁴

ω

²

K · ∇

2

A = 2πec

ω P

K^⊥

n

_e

v

_e

e

^−iθ

.

(2.26) At this step, we have to compute the right-hand-side of (2.26)

n

_e

v

_e

=

< n

_e

> +n

e||

e

^−iωpet

+ c.c.

v

_e||

e

^−iωpet

+ v

e0

e

^iθ0

K

₀^⊥

| K

₀

| + v

_eR

e

^iθR

K

_R^⊥

| K

_R

| + c.c.

, where < n

_e

> denotes the low frequency part of the variation of density of ions.

We have to use the interaction condition (2.25).

• Equation on A

0

. We keep only the resonant terms in the right-hand-side of (2.26), that is

2πec ω

0

P

_K⊥

< n

_e

> v

_e0

K

₀^⊥

| K

0

| + n

_e||

v

_eR

K

_R^⊥

| K

R

| e

^−iθ1

with

n

e||

= − 1

4πe ∇ · E

||

, v

e0

= e

m

_e

c A

0

, v

eR

= e m

_e

c A

R

. Therefore

i

∂

_t

+ c

²

ω

0

K

₀

· ∇

A

₀

+ 1 2ω

0

c

²

∆ − c

⁴

ω

²₀

K

₀

· ∇

2

A

₀

= 2πe

²

ω

0

m

e

< n

_e

> A

₀

− e

2ω

0

m

e

∇ · EA

_R

P

_K⊥ 0

K

_R^⊥

| K

R

|

e

^−iθ1

.

Moreover, for any vector a P

_K⊥

0

(a) = K

₀^⊥

· a

| K

₀^⊥

| K

₀^⊥

| K

0

| . The 2-D equation for A

0

then reads

i

∂

_t

+ c

²

ω

0

K

₀

· ∇

A

₀

+ 1 2ω

0

c

²

∆ − c

⁴

ω

₀²

K

₀

· ∇

2

A

₀

= 2πe

²

ω

0

m

_e

< n

e

> A

0

− e

2ω

0

m

_e

∇ · E

A

R

e

^−iθ1

K

0

· K

R

| K

0

|| K

_R

| . (2.27)

• Equation on A

_R

. A similar computation for A

_R

gives i

∂

_t

+ c

²

ω

R

K

_R

· ∇

A

_R

+ 1 2ω

R

c

²

∆ − c

⁴

ω

²_R

K

_R

· ∇

2

A

_R

= 2πe

²

ω

R

m

e

< n

_e

> A

_R

− e

2ω

R

m

e

∇ · E

^∗

A

0

e

^iθ1

K

0

· K

_R

| K

0

|| K

R

| . (2.28)

• Equation on E. The electronic-plasma part is very similar to that of [5].

We describe briefly the procedure. Using (2.1), (2.2) and (2.6), one has

∂

²_t

E + c

²

∇ × ∇ × E = 4πe∂

t

(n

0

+ n

_e

)v

e

= 4π(n

0

+ n

e

)∂

t

v

e

+ v

e

∂

t

n

e

= 4πe

− (n

0

+ n

_e

)v

e

· ∇ v

_e

− γ

_e

T

_e

m

e

∇ n

_e

− e(n

0

+ n

e

) m

_e

E + 1

c v

_e

× B

− v

_e

∇ ·

(n

0

+ n

_e

)v

e

. Keeping only at most quadratic terms gives

∂

_t²

E + c

²

∇ × ∇ × E = 4πe

− n

0

v

_e

· ∇ v

_e

− γ

_e

T

_e

m

_e

∇ n

_e

− e(n

0

+ n

e

)

m

_e

E − en

0

cm

_e

v

e

× B − n

0

v

e

∇ · v

e

. Writting E = E

||

e

^−iωpet

+ c.c. gives

∂

²_t

E

_||

+ c

²

∇ × ∇ × E

_||

+ ω

_pe²

E

_||

+ 4πeγ

e

T

_e

m

e

∇ n

_e

= 4πe

− n

0

v

_e

· ∇ v

_e

− e(n

0

+ n

_e

)

m

_e

E

_||

− en

0

cm

_e

v

_e

× − n

0

v

_e

∇ · v

_e

e

^iωpet

. Using the polarization condition

n

e

= − 1

4πe ∇ · E

||

,

we obtain

∂

²_t

E

_||

+ c

²

∇ × ∇ × E

_||

− v

_th²

∇∇ · E

_||

+ ω

²_pe

E

_||

= 4πe

− n

0

v

_e

· ∇ v

_e

− e(n

0

+ n

_e

) m

e

E

_||

− en

0

cm

e

v

_e

× − n

0

v

_e

∇ · v

_e

e

^iωpet

. The nonlinear resonant terms are given by

v

_e

· ∇ v

_e

= v

₀

· ∇ v

^∗_R

+ v

_R^∗

· ∇ v

₀

, n

e

E

||

=< n

e

> E

0

,

v

_e

× B = v

0

× B

_R^∗

+ v

^∗_R

× B

0

, v

e

∇ · v

e

= v

0

∇ · v

^∗_R

+ v

_R^∗

∇ v

₀^∗

, since v

0

and v

_R^∗

are orthogonal. But

B

0

= ∇ × A

0

, v

0

= e m

e

c A

0

, then

− n

e

v

e

· ∇ v

e

− en

₀

cm

e

v

e

× B = − e

²

n

₀

m

²_e

c

²

A

0

· ∇ A

^∗_R

+ A

^∗_R

· ∇ A

0

+ A

0

× ∇ × A

^∗_R

+ A

^∗_R

× ∇ × A

0

= − e

²

n

0

m

²_e

c

²

∇

A

0

· A

^∗_R

. Since

A

0

· A

^∗_R

= A

0

A

^∗_R

K

0

K

R

| K

0

|| K

_R

| , it follows that

− 2iω

pe

∂

_t

E

_||

+ c

²

∇ × ∇ × E

_||

− v

_th²

∇∇ · E

_||

= − 4πe

²

n

0

m

e

< n

e

> E

||

− 4πe

³

n

0

m

²_e

c

²

∇ A

0

A

^∗_R

e

^iθ1

K

0

K

R

| K

0

|| K

R

| . The final equation reads denoting E

₀

= E

_||

i∂

_t

E

₀

+ v

_th²

2ω

pe

∇∇ · E

₀

− c

²

2ω

pe

∇ × ∇ × E

₀

= ω

_pe

2 < n

_e

> E

0

+ eω

_pe

2m

e

c

²

∇

A

0

A

^∗_R

e

^iθ1

K

0

· K

_R

| K

0

|| K

R

|

. (2.29)

• Equation on < n

e

>. The accoustic part is the same as in [5] and reads

∂

_t²

− c

²_s

∆

< n

e

>= 1 4πn

i

∆

| E

0

|

²

+ ω

_pe²

c

²

| A

0

|

²

+ | A

R

|

²

, (2.30) where

c

²_s

= γ

i

T

i

m

_i

+ γ

e

T

e

m

_e

.

System (2.27) − (2.30) is the 2-D Raman interaction system.

3 The amplification rates and the most ampli- fied directions.

3.1 Semi-classical asymptotic.

As in [6], we write a semi-classical limit of System (2.27) − (2.30) in order to obtain amplification rates. Denoting E

0

= E e

^{i(K1·X−ω1t)}

and writting

∂

_t

E

0

≪ ω

1

E

0

, ∇ E

0

≪ K

1

E

0

leads to i

∂

_t

+ c

²

ω

0

K

0

· ∇

A

0

= 2πe

²

ω

0

m

_e

< n

_e

> A

0

− i e

2ω

0

m

_e

K

1

· E A

_R

K

0

· K

R

| K

0

|| K

_R

| , (3.1) i

∂

_t

+ c

²

ω

R

K

_R

· ∇

A

_R

= 2πe

²

ω

R

m

e

< n

_e

> A

_R

− i e 2ω

R

m

e

K

1

· E

^∗

A

0

K

0

· K

_R

| K

0

|| K

R

| , (3.2) i

∂

t

− v

²_th

ω

_pe

K

1

· ∇

E + ω

1

E − c

²

2ω

pe

K

1

× K

1

× E

= ω

pe

2 < n

e

> E + i ω

pe

e

2m

e

c

²

A

0

A

^∗_R

K

0

· K

R

| K

0

|| K

_R

| K

1

. (3.3)

Now recall that the third wave (ω

pe

+ ω

1

, K

1

) satisfies the dispersion relation (2.14)

(ω

pe

+ ω

1

)

²

= ω

²_pe

+ v

²_th

| K

1

|

²

, and thus a direct expansion gives

ω

1

≈ v

²_th

| K

1

|

²

2ω

pe

. Then the equation on E reads

i

∂

_t

− i v

²_th

ω

_pe

K

1

· ∇ K

1

· E

| K

₁

| = ω

_pe

2

K

1

· E

| K

₁

| + i ω

_pe

e

2m

e

c

²

A

0

A

^∗_R

K

0

· K

_R

| K

₀

|| K

_R

| | K

1

| . Finally, denoting by

f

0

= ω

_pe

c A

0

, f

R

= ω

_pe

c A

R

, f = K

₁

· E

| K

1

| , Equations (3.1), (3.2) and (3.3) become

i

∂

t

+ c

²

ω

0

K

0

· ∇

f

0

= 2πe

²

ω

0

m

_e

< n

e

> f

0

− i e | K

1

|

2ω

0

m

_e

f f

R

cos(θ), (3.4) i

∂

t

+ c

²

ω

R

K

R

· ∇

f

R

= 2πe

²

ω

R

m

e

< n

e

> f

R

− i e | K

1

| 2ω

R

m

e

f

^∗

f

0

cos(θ), (3.5) i

∂

_t

− i v

_th²

ω

pe

K

₁

· ∇

f = ω

_pe

2 < n

_e

> f + i e | K

1

| 2m

e

ω

pe

f

₀

f

_R^∗

cos(θ), (3.6)

where θ denotes the angle beetwen K

₀

and K

_R

.

3.2 Amplification rates.

Let us now consider that f

0

is fixed (it is a pump wave). System (3.5) − (3.6) becomes

∂

t

f

R

= e | K

1

|

2m

e

ω

_R

f

^∗

f

0

cos(θ), (3.7)

∂

t

f = e | K

1

|

2m

e

ω

_R

f

0

f

_R^∗

cos(θ). (3.8) The amplification rate is therefore proportional to

β = | K

1

|

√ ω

R

ω

pe

| cos(θ) | .

Recall that the incident field propagates along the x − axis and that K

0

=

k

0

0

, K

R

= k

R

ℓ

R

, K

1

=

k

1

ℓ

1

.

Remark that we have ℓ

R

= − ℓ

1

. The dispersion relation (2.14) and (2.15) gives

 



ω

²₀

= ω

_pe²

+ k

²₀

c

²

,

ω

²_R

= ω

²_pe

+ (k

²_R

+ ℓ

²_R

)c

²

,

(ω

pe

+ ω

₁

)

²

= ω

²_pe

+ v

_th²

(k

²₁

+ ℓ

²₁

).

(3.9) We take ω

_pe

as unit for ω and

_ω^c

pe

as unit for k. Introduce α = v

th

c << 1, then System (3.9) can be rewritten into

 



ω

²₀

= 1 + k

₀²

,

ω

²_R

= 1 + (k

²_R

+ ℓ

²_R

),

(1 + ω

1

)

²

= 1 + α

²

(k

²₁

+ ℓ

²₁

).

(3.10) The amplification rate β is given by

β =

p k

²₁

+ ℓ

²₁

p 1 + k

_R²

+ ℓ

²₁

| k

_R

|

p k

_R²

+ ℓ

²₁

. (3.11) For given k

0

, ω

0

satisfying ω

₀²

= 1 + k

₀²

, we therefore need to find the maximum of β(k

R

, k

1

, ℓ

1

) subject to the constraints

ω

_R²

= 1 + (k

²_R

+ ℓ

²_R

), (1 + ω

1

)

²

= 1 + α

²

(k

²₁

+ ℓ

²₁

), k

0

= k

_R

+ k

1

. Replacing k

_R

by k

0

− k

1

in (3.11) gives

β =

p k

²₁

+ ℓ

²₁

p 1 + (k

0

− k

1

)

²

+ ℓ

²₁

| k

0

− k

1

|

p (k

0

− k

1

)

²

+ ℓ

²₁

, (3.12)

with q

1 + k

²₀

= q

1 + (k

0

− k

1

)

²

+ ℓ

²₁

+ q

1 + α

²

(k

₁²

+ ℓ

²₁

). (3.13)

3.3 Conclusions.

The above problem (3.12) − (3.13) is solved numerically. The conclusions are the following ones. For the backscattered component (k < 0), the maximum is reached for ℓ

1

= 0. This means that the most amplified direction corresponds to the case where the Raman field is colinear to the incident laser field (see Figure 2).

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0 1 2 3 4 5 6 ß

l1

Figure 2: β with respect to ℓ

₁

with k < 0

This model is used in [5]. For the forward component (k > 0), one can see in Figure 3 that the maximum of β is reached for ℓ

1

6 = 0. Therefore, the Raman field makes a non-zero angle with the incident laser pulse and gives rise to new direction of propagation. It would be a cone in 3-D.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 ß

l1

Figure 3: β with respect to ℓ

1

with k > 0

The next step is to take into account this new direction.

4 A complete model

4.1 Some basic tools

In order to describe the directions of propagation, one introduces the three wave vectors for the Raman component given by problem (3.12) − (3.13)

K

R1

= k

R1

ℓ

_R1

, K

R2

= k

R2

ℓ

_R2

, K

R^s_2s

= k

_R^s

2

ℓ

_R^s

2

, satisfying

K

0

= K

R1

+ K

1,1

= K

R2

+ K

1,2

= K

R₂^s

+ K

1,2^s

, that is

k

0

0

= k

R1

ℓ

R1

+

k

1,1

ℓ

1,1

,

= k

_R2

ℓ

_R2

+ k

_1,2

ℓ

1,2

,

= k

_R^s₂

ℓ

_R^s

2

+

k

1,2^s

ℓ

1,2^s

.

We then intoduce the Raman frequencies ω

R1

, ω

R2

and ω

R^s₂

solution to ω

0

= ω

_pe

+ ω

_R1

+ ω

1,1

,

= ω

_pe

+ ω

_R2

+ ω

_1,2

,

= ω

pe

+ ω

R₂^s

+ ω

1,2^s

.

Note that (K

0

, ω

0

), (K

R1

, ω

R1

), (K

R2

, ω

R2

) and (K

R^s₂

, ω

R^s₂

) satisfy the disper- sion relation for electromagnetic waves (2.15) while (K

1,1

, ω

1,1

), (K

1,2

, ω

1,2

) and (K

1,2^s

, ω

1,2^s

) satisfy the one for electronic-plasma waves (2.14). That means that we have

ω

_R1

= q

ω

_pe²

+ c

²

(k

²_R1

+ ℓ

²_R

1

), ω

R2

= q

ω

_pe²

+ c

²

(k

²_R

2

+ ℓ

²_R

2

), ω

R^s₂

= q

ω

_pe²

+ c

²

(k

²_Rs 2

+ ℓ

²_Rs

2

).

Since K

2

and K

₂^s

are symmetric with respect to K

0

, choosing K

0

colinear to

the x − axis, we have k

_R2

= k

_R^s₂

and ℓ

_R2

= − ℓ

_R2^s

. Therefore, one has ω

_R2

= ω

_R^s₂

and ω

1,2

= ω

1,2^s

. In the sequel, we replace ω

_R^s₂

and ω

1,2^s

by ω

_R2

and ω

1,2

.

4.2 The equations.

As in Section 2, one gets the following set of equations, assuming that K

0

is colinear to the x − axis,

i

∂

t

+ c

²

ω

₀²

K

0

· ∇

A

0

+ c

²

k

²₀

2ω

²₀

∆ − c

⁴

2ω

₀⁴

K

0

· ∇

2

A

0

= 2πe

²

m

_e

ω

0

< n

e

> A

0

(4.1)

− e

2m

e

ω

0

∇ · E

A

_R1

e

^−iθ1,1

K

0

· K

_R1

| K

0

|| K

R1

| + A

_R2

e

^−iθ1,2

K

0

· K

_R2

| K

0

|| K

R2

| + A

_R^s

2

e

^−iθ1,2s

K

0

· K

_R2^s

| K

0

|| K

R^s₂

| ,

i

∂

_t

+ c

²

ω

R1

K

_R1

· ∇

A

_R1

+ 1 2ω

R1

c

²

∆ − c

⁴

ω

²_R1

K

_R1

· ∇

2

A

_R1

= 2πe

²

m

e

ω

R1

< n

e

> A

R1

− e 2m

e

ω

R1

∇ · E

^∗

A

0

e

^iθ1,1

K

0

· K

R1

| K

0

] | K

R1

| , (4.2) i

∂

t

+ c

²

ω

R2

K

R2

· ∇

A

R2

+ 1 2ω

R2

c

²

∆ − c

⁴

ω

²_R

2

K

R2

· ∇

2

A

R2

= 2πe

²

m

_e

ω

_R2

< n

_e

> A

_R2

− e

2m

e

ω

_R2

∇ · E

^∗

A

0

e

^iθ1,2

K

0

· K

R2

| K

0

|| K

_R2

| , (4.3) i

∂

_t

+ c

²

ω

_R2

K

_R^s₂

· ∇

A

_R^s₂

+ 1 2ω

R2

c

²

∆ − c

⁴

ω

²_R

2

K

_R^s₂

· ∇

2

A

_R^s₂

= 2πe

²

m

e

ω

R^s₂

< n

e

> A

R^s₂

− e 2m

e

ω

R2

∇ · E

^∗

A

0

e

^iθ1,2s

K

0

· K

_R^s₂

| K

0

|| K

R^s₂

| , (4.4)

i∂

t

E + v

²_th

2ω

pe

∆E = ω

_pe²

2 < n

e

> E + ω

pe

e 2m

e

c

²

∇

A

0

A

^∗_R1

e

^iθ1,1

K

0

· K

R1

| K

0

|| K

_R1

| + A

0

A

^∗_R2

e

^iθ1,2

K

0

· K

R2

| K

0

|| K

_R2

| + A

0

A

R₂^s

e

^iθ1,2s

K

0

· K

R^s₂

| K

0

|| K

_R^s₂

|

, (4.5)

∂

_t²

− c

²_s

∆

< n

_e

>= 1 4πm

i

∆

| E |

²

+ ω

_pe²

c

²

| A

0

|

²

+ | A

_R1

|

²

+ | A

_R2

|

²

+ | A

_R^s₂

|

²

. (4.6) Note that since the Raman components are not resonant one another, one gets two distinct equations without coupling terms like A

R1

A

^∗_R2

and the same pro- cedure than in Section 2 can be used. A non dimensional form can be obtained.

We denote p =< n

e

>. Using

_ω¹

0

as time scale,

_|K¹

0|

as space scale and denoting A e

0

= √

ω

0

ω

_pe

c

A

₀

γ , A e

R1

= √ ω

R1

ω

_pe

c

A

_R1

γ , A e

_R2

= √ ω

_R2

ω

_pe

c A

_R2

γ , A e

_R2

= √ ω

_R2

ω

_pe

c

A

R^s₂

γ ,

with

γ = 2m

e

ω

0

ek

0

√ ω

0

ω

pe

ω

R1

, we obtain (dropping the tildes) and introducing

α = r ω

R1

ω

_R2

,

i

∂

_t

+ c

²

k

²₀

ω

₀²

∂

_x

A

0

+ c

²

k

₀²

2ω

₀²

∆ − c

⁴

k

⁴₀

2ω

₀⁴

∂

_x²

A

0

= ω

²_pe

2ω

₀²

pA

0

(4.7)

− ∇ · E

A

R1

e

^−iθ1,1

k

R1

| K

_R1

| + α A

R2

e

^−iθ1,2

+ A

R^s₂

e

^−iθ1,2s

k

R2

| K

_R2

|

,

i

∂

t

+ c

²

k

₀

ω

R1

ω

0

K

R1

· ∇

A

R1

+ 1 2ω

R1

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²₁

K

R1

· ∇

2

A

R1

= ω

_pe²

2ω

0

ω

R1

pA

_R1

− ∇ · E

^∗

A

0

e

^iθ1,1

k

_R1

| K

R1

| , (4.8)

i

∂

_t

+ c

²

k

0

ω

_R2

ω

0

K

_R2

· ∇

A

_R2

+ 1 2ω

R2

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²

2

K

_R2

· ∇

2

A

_R2

= ω

_pe²

2ω

0

ω

_R2

pA

R2

− α ∇ · E

^∗

A

0

e

^iθ1,2

k

R2

| K

_R2

| , (4.9)

i

∂

t

+ c

²

k

₀

ω

R2

ω

0

K

R^s₂

· ∇

A

R^s₂

+ 1 2ω

R2

ω

0

c

²

k

₀²

∆ − c

⁴

k

₀²

ω

_R²₂

K

R^s₂

· ∇

2

A

R^s₂

= ω

_pe²

2ω

0

ω

R2

pA

_R^s

2

− α ∇ · E

^∗

A

0

e

^iθ1,2s

k

_R2

| K

R2

| , (4.10)

i∂

_t

E + v

_th²

k

₀²

2ω

pe

ω

0

∆E = ω

pe

2ω

0

pE + ∇

A

0

A

^∗_R1

e

^iθ1,1

k

R1

| K

_R1

| + α A

0

A

^∗_R2

e

^iθ1,2

+ A

0

A

^∗_Rs

2

e

^iθ1,2s

k

R2

| K

_R2

|

, (4.11)

∂

_t²

− c

²_s

∆

p = 4m

e

m

_i

ω

0

ω

_R1

ω

²_pe

∆

| E |

²

+ ω

_pe

ω

0

| A

0

|

²

+ ω

_pe

ω

_R1

| A

_R1

|

²

+ ω

_pe

ω

R2

| A

_R2

|

²

+ | A

_R^s₂

|

²

. (4.12) Remark 4.1. Note that the only new coefficient is the ratio of the Raman frequencies

^ω_ω^R1

R2

.

5 Numerical simulations.

5.1 The scheme.

We adapt the scheme introduced in [6]. We consider a regular mesh in space.

The fields are approximated by A

i,j

for i = 0, ..., N

x

and j = 0, ..., N

y

. We use periodic boundary conditions that is for all j = 0, ..., N

y

, A

0,j

= A

Nx,j

. In space, we consider centered finite difference discretization for each differential operator. Introducing

X

₀ⁿ⁺¹

= A

ⁿ⁺¹₀

+ A

²₀

2 , X

_Rⁿ⁺¹

1

= A

ⁿ⁺¹_R1

+ A

²_R1

2 , X

_Rⁿ⁺¹

2

= A

ⁿ⁺¹_R2

+ A

²_R2

2 ,

X

_Rⁿ⁺¹s

2

= A

ⁿ⁺¹_Rs 2

+ A

²_Rs

2

2 , X

_Eⁿ⁺¹

= E

ⁿ⁺¹

+ E

ⁿ

2 , X

_pⁿ⁺¹

= p

ⁿ⁺¹

+ p

ⁿ

2 ,

as new unknowns, the scheme reads 2i X

₀ⁿ⁺¹

− A

ⁿ₀

∆t + i c

²

k

²₀

ω

₀²

∂

_x

X

₀ⁿ⁺¹

+ c

²

k

₀²

2ω

₀²

∆ − c

⁴

k

⁴₀

2ω

₀⁴

∂

²_x

X

₀ⁿ⁺¹

= ω

_pe²

2ω

²₀

X

_pⁿ⁺¹

X

₀ⁿ⁺¹

− 1 2

Ψ

ⁿ⁺

1 2

E

X

_Rⁿ⁺¹

1

+ ∇ · X

_E^N+1

Φ

ⁿ⁺

1 2

AR1

e

^−iθ

N+ 12 1,1

k

_R1

| K

R1

|

− α 2

Ψ

ⁿ⁺

1 2

E

X

_Rⁿ⁺¹

2

+ ∇ · X

_E^N+1

Φ

ⁿ⁺

1 2

AR2

e

^−iθ

N+ 12 1,2

k

R2

| K

_R2

|

− α 2

Ψ

ⁿ⁺

1 2

E

X

_Rⁿ⁺¹s

2

+ ∇ · X

_E^N+1

Φ

ⁿ⁺

1 2

ARs2

e

^−iθ

N+ 12 1,2s

k

_R2

| K

_R2

| , (5.1)

where

Ψ

ⁿ⁺

1 2

E

+ Ψ

ⁿ⁻

1 2

E

2 = ∇ · E

ⁿ

, Φ

ⁿ⁺

1 2

AR1

+ Φ

ⁿ⁻

1 2

AR1

2 = A

ⁿ_R1

, Φ

ⁿ⁺

1 2

AR2

+ Φ

ⁿ⁻

1 2

AR1

2 = A

ⁿ_R1

, Φ

ⁿ⁺

1 2

ARs2

+ Φ

ⁿ⁻

1 2

ARs2

2 = A

ⁿ_Rs 2

,

are auxiliary functions. The equations of System (4.8) − (4.12) are discretized in the following way

2i X

_Aⁿ⁺¹

R1

− A

ⁿ_R

1

∆t + c

²

k

0

ω

_R1

ω

₀

K

_R1

· ∇ X

_Rⁿ⁺¹

1

+ 1

2ω

R1

ω

₀

c

²

k

²₀

∆ − c

⁴

k

²₀

ω

²_R

1

K

_R1

· ∇

2

X

_Rⁿ⁺¹

1

= ω

²_pe

2ω

0

ω

_R1

X

_pⁿ⁺¹

X

_Rⁿ⁺¹

1

−

Ψ

ⁿ⁺

1 2

E

∗

X

₀ⁿ⁺¹

e

^iθ

n+ 12 1,1

k

R1

| K

_R1

| , (5.2)

2i X

_Aⁿ⁺¹

R2

− A

ⁿ_R2

∆t + c

²

k

0

ω

_R2

ω

0

K

R2

· ∇ X

_Rⁿ⁺¹

2

+ 1

2ω

R2

ω

0

c

²

k

²₀

∆ − c

⁴

k

²₀

ω

²_R

2

K

R2

· ∇

2

X

_Rⁿ⁺¹

2

= ω

²_pe

2ω

0

ω

_R2

X

_pⁿ⁺¹

X

_Rⁿ⁺¹₂

−

Ψ

ⁿ⁺_E ¹²

∗

X

₀ⁿ⁺¹

e

^iθ

n+ 12 1,2

k

R2

| K

_R2

| , (5.3)