SELF-ACTION OF QUASI-OPTICAL BEAMS IN A MAGNETOPLASMA

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SELF-ACTION OF QUASI-OPTICAL BEAMS IN A MAGNETOPLASMA

A. Litvak, A. Sergeev, N. Shakhova

To cite this version:

A. Litvak, A. Sergeev, N. Shakhova. SELF-ACTION OF QUASI-OPTICAL BEAMS IN A MAGNETOPLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-637-C7-638.

�10.1051/jphyscol:19797309�. �jpa-00219299�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n 0 7 , T o m 40, J u i l Z e t 1979, page C7- 637

SELF-ACTION OF QLJASIIOPTICAL BEAMS IN A MAGNETOPLASMA

A.G. Litvak, A.M. Sergeev and N.A. Shakhova.

~ p p ~ i e d ' P h y s i c s I n s t i t u t e , Academy o f S c i e n c e s o f t h e U.S.S.R., Gorky U.S.S.R.

Bonlinesrr effects of raelf-action of beams and self-modulation of electromag- netic wave packets in a magnetoactive plas- ma have a number of peculiarities which make them distinct from the p logo us pro- ceaaea in an isotropic plasma. !Phis mani- fests itself both in change in the nonli- nearity and modification of the disper- sion properties. !Po illustrate the said above we shall give the example of quasi- oqtical beams of lower hybrid waves [I,

Z J

In the frequency range OWL<< ~ < < W H ~

gyrotropy of the striction nonlinearity becomes essential, ioe. the expression for the averaged perturbation of plasma den- sity 6n under the action of a rf field

When writing "[I ) one used a coordinate system depicted in Fig. 1 ; Wpe,i, are the LangtnuAr and cyclotron frequencies of electrons and ions, T and T . a r e their temperatures, B is th8 r l a d density. If the wave dispersion is defined by the thermal motion of particles, then the eq- uation for slow (an scale 2a/~, 1 oom- plex 8mg ^{itude & =} Ex is written as follows f

where ₂

*'~n equation with the analogous nonline- arity for the waves with frequency

W D ~

was obtained in paper [4]

.

'Jre ^and VTi are the thermal electron and ion velocities. The nonlocal nonlinearity part m y determine the aelf-action for sufficiently mall field distribution eca- lee. However, in case the characteristic dimension L 7 ~ ' /OUHolC,, then the laat term in ( 2 ) b y % e neglected and one may pass to investigating the dimensionless equation

i - = - - -

az a x 2

ay:

A similar equation hich differs principal=

ly from the u m l non-one-dimensional Schro- _{- - - - -} dinger equation is true also for eome other wave types in a magnetieed plasma. Here are the examples.

1). A parabolic equation with coeffici- ents changing their signs with the plasma oharacteriatica ia obtained in paper 151

for the wave beam propagating acrose the extepal magneti! field H

.

~ = w , / u 2 , 1 1 = ~ ~ / w the r& of interest is t%e region between curves

U = { - U and u = { - u 2

2). Self-action of electromagnetic waves- whistlers is described by equation ( 3 ) , if the angle between the central wave beam vector and the magnetic field is 8 > 0 *

( t g e * = E ) 151

3). It is easy to understand that (3) is just

.

where &JUI=upi/((+

3

longitudinal and tr%averse (in relation - . . . - . --

to %he magnetic field) wave vector compo- nents, m and M are the electron and ion masses, respectively (in this case coordi- nate Z is the time variable). A similar equation may be obtained analysing the thermal diapersfon 0fPigher hybrid waves in the range upe >3uHe ^•

Let us discuss some peculiarities of the processes described by equation ( 3 ) , resul- ting from the ratios for the moments of'lo- cerlized field distribution6 C21

- -

For collimate; beams 2; (Z="O)= K(z= ^{0) =o, P}

increases monotonically s o n g t e b e y pro- pagation trajectory,but a2 andha

"

( t h t corresponds to the beam crosrs sec- tion area) may first decrease. However, when 2 - 0 0 the beam behaviour is characte- rised by the transve se dimension growth.

1 2 OE IEI?

I xI(l~\-/Gl ^.I-T ) d x d y > o ^{' then for}

the iPitial amplitude distribution exten-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797309

C7- 638 -

ded along the beam cross section for rather greit Z has the form extended along x

.

To understand the qualitative pattern

of the process one ma use the nonaberra- --____

tional approximation. f6.71 and get t e equation for beam dimensions a and % ⁱⁿ

the x and -directions, assuming its form to be onstant (Gaussian for example):

- ^{=- 1}

d2a I 9 d2b I

dZ2 ~ 3 - 6 2 6 ) = g ~ + fl ' ⁽⁴⁾ ~ i g . l . W a v e v e c t o r s u r f a c e f o r l o w e r h y -

9 characterizes the energy flux through brid waves the cross section. Within the framework

of the method used one succeeds in redu- cing (4) to the non-autonomous differen-

tial aecond-order equation 3

d2a 4

- = - - ^IP

d z 2 a 2 J a 2 t , u + ~ z ~ ' ^{(5) /&I}

w e - a j = + - a. and(,,

are the initial beam dimedaiona. In (5) ² the change of a is determined by the two

factors: a diffractional beam divergence leading to the section widening along x

and an inhomogeneous nonlinear defocuaing opposed to the former. It is easy to in- vestigate the action of the contrary ten- dencies in a general case using a numeri- cal calculation, but in some cases it is possible to find an anslytical solution or to point out its important features, Nonlinear self-action of beams with initi- al dimensions a, _< 6, is of primary interest.

Under such condition8 the variables in

equation (5) may be substituted rp=a(y+~~2)"f a ^{-.I -,-+X.J} r = - I a ~ c t & Z, making it autonomous:

w1 d P , - -

,

iw (P3 (pzw

Analysing the phase plane of equation (6) Fig.2. Some examples of beam self-defocu- it is easy to realize that the exact solu- sing.l&(x=o =0,~=0)1=2,5 ; ao=4 ;

tion for a _- has the form _, r , References &.=$f ^;5;40;30.

a = { Y + $ Z ~ U [& ^aqctg i$- ^' * ⁽⁷⁾ I .I?.R.Pereira,A.Sen and A.Bers, Phys.Ruids, where u is the periodic function with pe- 21, 117, 1978.

riod T=~(u(z=~\),determined from ( 6 ) by ~ . A . G . L ~ ~ v ~ ~ , A . M , s ~ ~ ~ ~ c v , I , A . s ~ ~ ~ ~ o v ~ , means of integrals, Two qualitatively dif- ZhETF Letters, 5, Bo.1, 1979.

ferent self-action regimes may be distin- 3. A.G~Litvak, 1zv.WZov-Radiofieika, 9, guished, The process of quasi-one-dimensi- 900,1966.

0-1 self-focusing of the beam along the 4. H.H.Kueh1, Phys.Flwlds, 21, 2120, 1978 direction is realize 5. A.G.Litvak, I?.A.Shakhova, Pizika Plaz-

-6-p (<*=ctp8( _{6. my} V.I.Bespalov, (to be published), A.G,Litvak, V.I.Talanov width osci%atss near in ~b.~doplinear opticsn(~mp.transa~t.f ly growing value, the amplitude of oscilla- 428, n19aukan, ~ovosibirsk, 1968.

tions increasing with 2 and their frequen* 7. V.V.Vorob8ev, Izv,W~ov-Radiofizika,13 cy decreasing; the total number of oscilla- 1905,1970.

tiona N = n / t T P . The maximum focusing possible is deE(crmined from the one-dimen- sional problem a,,, = ao/(29 -4)

When Z ~ Z * the beam self-def o%uses. Fig.2 illustrating the foregoing conclusions shows the change of the electric field am- plitude (E(x=0&=0,1)1 in a beam and of its dimension ong the propagation trajeo- tory z depending on the initial distri- bution parameters.