WAVE AND PARTICLE INTERACTIONS WITH A NONLINEAR DIELECTRIC PERTURBATION IN A PLASMA

(1)

HAL Id: jpa-00219249

https://hal.archives-ouvertes.fr/jpa-00219249

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

WAVE AND PARTICLE INTERACTIONS WITH A NONLINEAR DIELECTRIC PERTURBATION IN A

PLASMA

J. Mendonça

To cite this version:

J. Mendonça. WAVE AND PARTICLE INTERACTIONS WITH A NONLINEAR DIELECTRIC PERTURBATION IN A PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-545-C7-546.

�10.1051/jphyscol:19797263�. �jpa-00219249�

(2)

JOURNAL DE PHYSIQUE Colloque C7, suppldment au n07, Tome 40, ~ u i Z Z e t 1979, page C7- 545

WAVE AND PARTICLE WTERACTIONS WITH A N O K M A R DIELECTRIC PERTURBATION IN A PLASMA

J.T. Mendonca.

Cowlex I n t e r d i s e i p l i ~ , In s t i t u t e Superior T e C n i ~ , Lisboa, 1, Portugal.

1. Introduction

In recent years much attention has been devoted to the problem of nonlinear susceptibilities. We discuss here two effects which are related to the susceptibility perturbation induced by a nearly monochromatic wave packet, traveling in an infinite ho- mogeneous and isotropic plasma, due to the non-

linear motion of the particles.

From a macroscopic point of view the wave packet will behave as a dielectric perturbation moving with the group velocity, w = ( a ~ / a k ) ~

,

which can become relativistic if its main frequgncy w is much larger qhan the electron plasma frequency,@

.

P xne waves which can be excited in this plasma $11 interact with this rapidly moving perturbation and will undergo a partial reflection. The result is very similar to the classic relativistic mirror effect, with a significant frequency upper shift of the reflected wave. On the other hand, each particle of the plasma will give rise to a nonstationary electric field when submitted to the dielectric per turbation and the result is the emission of a transition-like radiation. It is the aim of the present work to discuss these wave and particle interactions with the nonlinear dielectric perturb&

tion induced by a strong quasi-monochromatic wave- -packet.

2.Nonlinear dispersion relation

Using Maxwell's equations and the relativistic fluid equations for the electrons we can easily obtain the nonlinear dispersion relation for an electromagnetic mode (w, k) in the presence of a large amplitude -+

electromagnetic wave (wo ,ko): -f

2 2 2

k c = w - Q & (1) where the nonlinear plasma frequency is given by

2 ^mw2

-

^u2^C^(w, wo) Wo (2) 'NL p

and the coupling coefficient relating the two modes is

I

(3)

-

+ + +

k+=

-

k t k and w+

-

= w z w

.

The dispersion relation depends on the value of the density of energy

% -

0

^I

^E(wo,ko)

⁾

and the variation of W _in

0

space and time leads to a nonstationary plasma situation.

3. Relativistic mirror

Suppose now W changing rapidly in a time scale of w-I and oter a distance k-l. be can then assu- me a sharp boundary of infinitesimal thickness connecting the free plasma and the plasma submitted to the quasi-monochromatic pulse (uo, -t k ) . The reflection of an incident field Ei(w, k) on the mo- +O

ving boundary can be calculated using a Lorentz transform to the frame boundary. In this frame we have two different media at rest, with refractive indices (see Fig. 1) :

where w is the Lorentz transformed frequency of the incident wave. how we can use the Fresnel for- mulae to calculate the reflection coefficient, at normal incidence, and we get:

where XNLE C(w ,wo)W0,k1'=w"/c and w" is the £re

-

quency of the reflected wave in the laboratory

~q.(5) shows that the reflection coefficient is always of the order of the nonlinear susceptibility xNL,in the laboratory frame. And we see, from eq.(6), that the frequency of the reflected wave w"

is considerably larger than w, if v

-

^G

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797263

(3)

4. Transition radiation

Let us consider now~&plasma particles interacting with the moving boundary. In the moving frame each

+ 3

particle has velocity =

-

w, if we neglect the influence of the temperature. When the particle goes from the free region, described by Nfl, to the region described by Nr2, it creates a nonstationary electric field whose spectrum is given by

Where 6 = (V'/C)N'~ and @is a form factor describ- ing the angular pattern. This expression is valid at very long distances R from the transition boundary. It can be shown from this equation that the radiation emitted by the particle has a sharp maximum in a direction nearly paralled to the particle trajectory and that significant levels of radiation can only be obtained for to%> to2 _P

.

^Using

eq. (7) we can deduce the following expression for the total energy radiated by a particle, in the laboratory frame:

. -

Where. A = w XNL(uo) ^{and 1}< cx < 3. We see from eq.(7) that E (w') is proportional to the electric charge and is independent of the particle mass.This means that the total energy emitted by the plasma

in nearly zero, unless the plasma is slightly non-neutral. For example, suppose a plasma of density

no

= 1012 crossed by an electron beam of small density, n << no and small velocity,

f

vf<< P, The transition radiation produced by a strong electromagnetic wave-packet which is launched in this plasma is given by:

Even for moderately high energy Wo, such that the nonlinear susceptibility

$ (w

) is of the order of

L 0

(a w /W )2 << 1, we get I m Watt of radiated power,

P 0

with a very weak electron beam, nf/no = 10-~.~ore detailed discussions of these can be found

elsewherely2

.

References

/I/ 3.T. MENDONCA, J. of Plasma Phys. (1979); to be published

/2/ J.T. MENDONCA, Internal. Report CEL-1/79 ,Lisboa (1979).

FIGURE 1

-

Geometry of the problem: S is the laboratory frame; S' is the moving frame;

E ~ ( u , k ) the linear dielectric constant of the plasma; cz(o,k;W ) is the dielectric constant in the presence of the strong wave (uOrko)