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JOURNAL DE PHYSIQUE IVColloque C6, supplkment au Journal de Physique 11,Volume 5 , octobre 1995

Beat-Wave Experiments in the Micro Wave Range: Pump Depletion

N.E. Andreev, E.D. Campos*, B. Cros*, J. Godiot*, L.M. Gorbunov*" and G.Matthieussent*

Plasma Theory Division, Institute for High Temperatures of RAS, Izhorskaya S t. 12/19,Moscow, 127412, Russia* Laboratoire de Physique d es Gaz et des P lasmas, Universite'Paris XI-CNR S,91405 Orsay cedex, France** Plasma Theory Division, Lebedev Physical Institute of RAS, Leninsky pr. 53,Moscow,11 7924 , Russia

Abstract. Excitation of electron plasma waves by the mixing of two microwave beams is studied in a laboratory

plasma. The first experimental observation of pump wave depletion is presented. The numerical calculations, based

on one-dimensional model, describing space-time evolution of eight interacting waves, are in good agreement with

experimental data. Estimation of the depletion on the basis of conservation laws is also discussed.

1. INTRODUCTIONThe nonlinear excitation of plasma waves by beating between two electromagnetic waves has a wide

range of applications in plasma physics, including plasma heating ' 9 2 * , plasma diagnostics 4. I,ionospheric plasma studies and modification c6 ], acceleration of electrons C7 9 8 1.Main attention is focused

on the amplitude of the excited plasma waves [ . The variation of pump waves is usualy neglected

although energy is partially transferred from one electromagnetic wave with higher frequency o, o theother w ith lower frequency w, and plasma oscillations with frequency, w = w, - w, , near electron plasma

frequency w , . As a result, the higher frequency wave loses energy and therefore is depleted.

We present here the first experimental observation of pump wave depletion appearing during resonantplasma wave excitation by nonlinear mixing of two electromagnetic waves. A simplified theoretical model

describing the space-time evolution of both electromagnetic and plasma waves is in good agreement withexperimental data. W e also derive approximate expressions for the amplitudes of plasma w aves and for the

depletion of the higher-frequency pump wave.Th e experiment is performed in a large volume ( - 0,8 m3) of unmagnetized plasma, created in a

cylindrical multipolar with following param eters: electron density n = 1,7.lOU~ m - ~ ,eutral

(Ar ) density - 1013 electron tempe rature T,= 3 eV , electron-ion collision frequency v, - 106 s-',electron-neutral collision frequency v, = 106 s-' . The electromagnetic waves are launched inthe chamber by small horns (Fig. 1) . One of them,

with lower frequency 5,6GHz (o, 3,53.101° s-' )

and pulse duration 7, = 1,25 ~s , is introduced

along the chamber axis. The other wave withhigher frequency 9,3 GHz (o, 5,86.1010 s-') has

a pulse duration 7, = 1,Sps and is injected at 30'inclination with respect to the axis of the chamber.

The peak power for both waves is near 200 kW.

The amplitude of the electric fields of both

electromagnetic and plasma waves, as well as the

electron density are measured with a small Figure 1: Schematic diagram of the beat-wave experiment.

cylindrical Langmuir probe ( 4 = 0,4mrn, length Dotted lines indicate the boundaries of the beams.

= 2,Omm) ituated at a distance - 40 cm from side wall and at a distance - 50cm from front wallof the chamber. The measurements of electron density are done approximately 1 ps before the high

frequency pulse. For this the probe is biased at a potential + 10,s V above the plasma potential. T he highfrequency fields in the plasma are detected by a non polarized probe. The signal is splitted and then the

use of narrow frequency filters, followed by quadratic detectors, allows to measure separatly the amplitude

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1995614

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C6-76 JOURNAL DE PHYSIQUEIV

plasma wave am plitude on electron density is seen.

~h~ temporal variation of the square of electricFigure 2: Dependence of the normalized amplitude of plasma

wave on the normalized plasma density.field both for microwave pulses and for plasma

of the electric field of each wave.The abso lute value of the electric fields of electromagn etic waves in the plasma is estimated on the basis

of ion-sound waves excitation by the action of ponderomotive forces produced by a standing

electromagnetic wave [ l1 . As result we evaluate the amplitudes of both h€ waves as El, - b0 4 a

6.104 Vlm and the effective coefficient of reflection from the walls of the chamber is approximatelymeasured to be r - 0,6 .

Electron density variations, of the orde r of 10' in relative value, with temporal scale of the order of

5.1W3 s , arise due to the residual modulation of the power supplies of the multipolar discharge.The amplitude of the detected plasma oscillations shows a sharp resonant dependence with electron

density. Among the measurements of plasma waves amplitude with frequency w = o, - w2 = 2,3.101° s-'we take the one w ith maximum amplitude. W e suppose that the electron den sity in this case is the resonantone. Norm alizing the amplitude of plasma oscillations as w ell as plasma density to these resonant values,

we obtain the points shown in Fig.2. For comparison Fig.2 shows also the theoreticalcurve obtained when plasm a wave saturation is due

wave are shown in Fig.3 in the case of resonant10

0.15

density. The higher-frequency pulse begins , : . a E:

- , ,approximately 0,2 ps later and is completed 0,4 ps

later as compared to the lower-frequency pulse.The plasm a oscillations are exc ited only du ring theoverlapping of pulses. Th e depletion of the higher-frequency wave is observed also only at this time.After turn off of the lower-frequency wave, thehigher-frequency wave restors its amplitude. The

Tim. (r)

depletion of the higher-frequ ency wave takes place

only if the electron density is near to the resonant Figur- 3: ~~~~~~~l variation of the scIuareof microwave @

value and if the am plitude of plasma oscillations is %) and plasma @,3 lectric fields in arbitrary u t s .

sufficiently high.

In order to treat the experimental results we examine a one-dimen sional theoretical model describing thespace-time evolution of eight coupled waves in a uniform plasma of finite lenght L . The subset of

electromagnetic waves contains a pair of right-going waves (El and q ) ith frequencies ol and o, andwith corresponding wave numbers kl and k 2 , and a pair of reflected left-going waves (E, an d E,)withthe same frequencies and opposite wave number. These waves interact with four plasma waves of equalfrequency o = o1- o2 but w ith different wavelengths. One pair o f conterpropagating plasma waves (E,,,and E,, ) has the wave number k = k l - k2 (long waves) as the other one (E,, and E @ ) has the wavenumber K = k, + k2 (short waves).

The se t of equa tions for the slowly varying am plitudes of electromagn etic waves m ay be obtained by the

usual method (see, for example, [2]) and has the form

to resonant density detuning. The maximum plasma

wave amplitude is taken to be determined by the - Idriving pulse duration 1 s . t is seen that the half- $ o., -'

width of the curve corresponds to a density change 4of 0,05 % in relative value. However, the 5 0 . 0 1 - ~

experimental incertitude in determination of both Ithe plasma density and the plasma wave amplitude ' .00'-'

g

doesn't allow to speak of an agreement betweenLo,mo,

, = I z s , z z E I s I i I s z z f i I I I g i g

:i

I < ii: : :

: ' . . - - . - - - -' :

'!

' ' ',: I z e > i. - - i : : : : ._ _ . - - . . . . - - -_ _ _ _ _ _ . _ _ _ _ . _ . . . . - -_ . _ _ , . _ _ - C - . - , - - - r - - - , - - - .

-. _ . _ . _ _ _ _ _. ! .: 2 : I I z j _ _= j i_ _ . - . - - - -I I : I : I 2- - - - . . . . - - - - -, . - : z v - ; ' c f!;;!:!:;;;:- - -' ' # - - - . - - - % - . -

; ' ' I ' ' ' " g ' " ' ' z " : z& ; ; ; , ; & ; ; ; ; ; ! ; ; ; ! ; ; ; ; ; ; ; :_ _ - , ~ . . - , . - - - c - - - I - - - 7 - - -

experimental results and theoretical prediction. o.es o . ~ 0.851 5 3. 1 1.15

No n n - X d --.IN ( n,",_ 1

Nevertheless, a strong resonant dependence of

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where v,,, = cZklJ1wlJ ar e the group velocities of electromagnetic waves with frequencies w l and w,

respectively, qi is the slowly varying am plitude of the electron density perturbation f or the correspon dingplasma wave

ikE,, i kE,, iKE,, iKE,,q 1 = 4 ~ ~ n , e ~3 = - 4 1 ~ n o e ~,= 4 x n , e t ' 4 = - ~

Th e se t of equations for qi , aking under consideration the nonlinearity due to relativistic electron massvariationrlZ , the collisional damping and the frequency mismatch A = (w2- w;)/ 2wZ, has the form

( d a + ~ + i y + - l e i 2t l?6 ) q 3 = a i P ~ l ~ : ,8) qA= a K 2 E 3 E ; , (g)

where y = (v, + v,)/o , or = 2 /4mzolwzw2To so lve this set of coupled equations, (1)-(4) and (6)-(9), we suppose that in initial mom ent of time both

electromagnetic and plasma waves are absent everyw here in region 0 S x S L. The two electromagneticpulses E li (t) and Ezi (t) ar e introduced in a plasma layer through the boundary x = 0 with some delayone with respect to another. The temporal variation of the amplitudes ar e taken to be in accordance with

experimental data.At the boundary x = 0 the conditions El(t) = Eli(t) + rE;(t) and &(t) =Gi ( t ) + rE4(t) are supposed

to be valid. At the other boundary x = L we assume r E l exp(ik,L) = E, e x p ( - a l l ) a n d rE, exp(ik,L)

= E., exp (-ikzL ) wh ere the reflection coefficient r is taken to be real and the same for both

electromangetic waves.To model the experimental conditions we assume in our numerical calculations that the thickness of the

plasma is L = 80 cm, the coefficient of reflection r = 0,5, ol = 6.101° s-' , w, = 3,5.1010 s-' , q, = Go,t,= 1ps. Fig.4 shows the temporal variation of the normalized waveenergy density at frequencies w l

(dark line), w, (dotted line) and w = wl - wz (dashed line) calculated at the point x = 4 1 c m .All values

ar e given in units of the peak incident energy density E1,2/83r where th e ratio of the electro n quiver

velocity to the light velocity ( vlolc), corresponding to the maximum of amplitude Eli t) , s choosen tobe 7,5.104. The mismatch and damping rate are taken to be A = 2.104 and y = 3.104 .

The comparison of experimental results in Fig.3 with the results of a num erical simulation (Fig.4) allows

to speak of a good agreement between them.Having in mind the smallness of the depletion effect we are able to formulate simpIified considerations

on Eqs. (1)-

(9). The amplitudes of plasma waves, Epj,may be in quasistatic approximation (slat < v,o p A ) calculated by means of Eqs. (6) - (9) in which the amplitudes of electromagnetic waves Ej aredetermined by Eqs. (1) - (4) with right hand sides equal to zero. As the result we find

where the values I E.,,, ,, ' I E l , , I [ 1+ r4 21?cos(2k~,~L'' determine the amplitudes of theelectromagnetic waves inside the plasma taking into account the reflection from walls effects. With ourexperimental parameters the last term in square brackets of expression (10) is the major one an d it is due

to the sho rt plasma w ave.The Eqs. (1) - (4) and (6) - (9) allow us to find the conservation laws for th e transverse action (M anley-

Rowe, or photon conservation) and for the energy (with dissipation) by means of a method analogeous tothe one used in paper [' . In stationary case, when the amplitudes of electromagnetic and plasma waves

don't change in time, thes e conservation laws give

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C6-78 JOURNAL DE PHYSIQUE IV

vl a4 (11)- ( I E I I 2 - 1 % 1 2 ) = - 2 Y C I ~ p j2 l a l l l l ~ ~ l ~ ~ L I ' L 1 , l l l / I ~ ~ I ~ ~ ~ I I I I I I I I , I r

0 1 j - 1 1.0 -T h e

E q .( ll ) denotes that the spatial variation in higher o, s - -

frequency photon flux is equal to total dissipation $ -rate of plasm ons. Upon integrating Eq. (11) over x O16 - -

on the length of the plasma we find the full 6 -reduction of the square amplitude of the higher-

0'4 - .-..:...-...--- 2

frequency wave during its propagation from x =0 3 ,2 - . -

to x = L and back- I

AIEI2 = ( I E ~ ( ~ ) \ ~IE1(~) I2) ( IE 3(~ ) I2I ~ ~ ( 0 ) 1 ~ : 0, 0 : -

_ U l ~ . . . l i ~ ~ l l ~ i l l l L l ~ t ~ I . I I I I I I I I I I I I I I ! ~ L C

0 *A I E I ~ ~ Y L - - ' C l ~ ~ ~ 1 ~12) rime ( P I

VI - 1Figure 4: Temporal variation o f the normalized waveen ergy

densities at frequencies w , (dark h e ) w, (dotted h e )and

w = w , -q clashed line). Other data are indicatedTh e value A IE may be used as an estimation in th e text. The plots for frequencies U, d are

of the wav e depletion discussed above. Substituting reduced four times compared to the plot for w,.

in Eq. (12) the expression (10) and using the indicated above set of number we obtain( A / E I ~ / I E ~ ~ ~ ~ ) ~ ~ . I O - ~hich is in quite good agreement with results of the numerical simulations and of

the experiment.Th e observation of the pump-wave dep letion requires two conditions. Fro m one side, the pump d epletion

begins to be important, in the resonant conditions, when the plasma wave has a rather high amplitude.

From the other side, for a to high amplitude, the intense plasma wave undergoes the parametricinstab ilities["] and its regulare structur e is destroied. Hence, the effect of pump-w ave depletion is allowedto be observed in a sufficiently narrow range of parameters.

References[ I ] - M.N.Rosenbluth and C.S.Liu, Phys. Rev. Lett., 29, 701 (1972)[2] - B.Cohen, A.Kaufman and K.Watson, Phys. Rev. Lett., 29, 581 (1972)

[3] - B.Cohen, Phys. of Fluids, 1 7, 496 (1974); B.Cohen et al., Phys. of Fluids, 1 8, 470 (1975); V.Fuchset al., Phys. Rev. Lett., 31, 1110 (1973)[4] - N.Krol1, A.Ron and N.Rostoker, Phys. Rev. Lett., 13, 83 (1964)

[5] - R.Cano, 1.Fidone and B.Zanfagna, Phys. of Fluids, 1 4, 811 (1971); E.S.Weibel, Phys. R ev. Lett.,

37, 1619 (1976); B.Stansfield, R.Nodwell, Ph ys. Rev. Lett., 26, 1219 (1971); L.A.Godfrey, Phys. Rev.

A, 20, 567 (1975)[6] - J.Lavergnat et a]., Geoph. Res. Lett., 4, 417 (1977); K.Rypda1, J. Geophys., 86, 1544 (1981);V.A.Puchkov, J. Plasma Phys., 9, 718 (1983); G.Wei1, Phys. Fluids, 13, 1802 (1970)

[7] - T.Tajima and J.M.Dawson, Phys. Rev. Lett., 43, 267 (1979)[8] - C.E.Clayton et a]., Phys. Plasma, 1, 1753 (1994); F.Amiranoff et al., Phys. Rev. Lett., 68, 48

(1992); Y.Kitagawa eta l., Phys. Rev. Lett., 68, 48 (1992); N.A.Ebrahim, J. Appl. Phys., 76, 7645 (1994)

[9] - B.Am ini, Phys. Rev. Lett., 54, 1163 (1985); P.S.Lee e t a]., Phys. Rev . Lett., 30, 538 (1973);

G.Schrnidt, Phys. F luids, 16 , 1676 (1973); B.Amini et al., Phys. Rev. Le tt., 53, 1441 (1984); B.Amini,

Phys. Flu ids, 28, 387 (1985); C.M.Tang, P.Sprangle and R.N.Sudan, App l. Phys. Lett., 45, 375 (1984)[ lo] - M.Benhassine et al., Rev. Phys. Appl., 19, 545 (1984)

[ l l ] - P.Fischer etal. , J. Phys., 48, 233 (1987)[12] - C.J.McK instrie and D.W .Forslund, Phys. Fluids , 30, 904 (1987)

[13] - Ya.L.Bogomolov et a]., Physica Scripta, 52, 51 (1994)