On Stimulated Mandelstam-Brillouin Scattering
T h e article is a con trib u tion to the discussion on the effects ex p ected from the in teraction o f in ten sive laser radiation and p artially ion ized piasnia, i.e. the spark discharge. The p ossib ility o f stim u la ted M an deistam -B rillou in scatterin g (SM BS) in pa rtially ion ized plasm a, and in absence o f m agn etic field has been discussed. I t has been assum ed th at electrom a gn etic w av e is scattered on the flu ctuation s of den sity in th e neutral gas. T h is w av e freq u en cy has been com p a red w ith th e electrom a gn etic w ave frequ en cy scattered on th e flu ctu a tion s electron den sity, as w ell as w ith the w a v e freq u en cy scattered on the ion sound.
1 . Introduction
Iu any medium the light scattering is produ ced by the interaction between the light wave and the matter.
The Rayleigh scattering may be used to analyse the reciprocal effect between wave and molecule. The energy fraction transferred to molecules by light waves is so small, that no frequency change in the scattered light has been observed for a long time. High resolution spectral methods applied to scattered light allowed to state that beside the lines characteri zed by the same wave length, there are also some weak lines shifted on the left and right, with respect to the main line. This fine struc ture of Rayleigh-lines was described by Bril louin [1] and Mandelstam [2]. According to their explanations the light waves can interact not only with individual molecules of the me dium, but also with the mechanical waves (e.g. ultrasound waves). In this way the energy exchange can be realized as a result of électro striction effect.
In Raman stimulated scattering the shift ¿incident-scattered may approximate several tens
A, thus being different from the SMBS, where
the corresponding values for liquid amounted to 0.05 A and 0.1 A, respectively [3].
The SMBS was observed mainly in liquids, crystals [4-6] and gases [7].
* D epart m ent o f E x p erim en ta l P h ysics, K om en sk y - -U n iversity, S m eralova 2, 88506 B ratislava, C zech o slova k ia.
For SMBS both the energy and momentum conservation hold, i.e.
(Tim) mi = mg + cog, (1 )
(A/A = & -Ti) = &2 + &.S, (2)
where indices 1,2,*$ correspond to the incident, and scattered electromagnetic wave and the sound wave, respectively.
The graphical illustration of the equation (2) is given in Fig. 1. By virtue of the equation (2) and Fig. 1
6 6
!&st = l^ i- ^ 1 = R ism y A ^ s m - , (3) where 6 is the scattering angle.
jound wow
F ig. 1. The v e c to r diagram 1?, = f62 + Tc,s, w h ich im ages the creation o f p h o to n (w a v e v e c to r fig) an d o f scattered p h o to n (w ave v e c to r fig) fro m p rim a ry p h o to n (w a v e v e cto r fcp
In our experiments the difference between mi and mg appeared to be very small and there fore &i = &g has been assumed.
The sound wave (SW) frequency tog = m, — mg is given by the equation (3) and by the
3) definition of the wave vector & = — :
3)g = ---sm —,2%Vc3q . 6
c 2 ' (4)
where
% — index of refraction,
c — velocity of light in vacuum, Pg — SW velocity.
From the classical theory point of view the SMBS can be represented as follows: The alter nate electrical field induces the sinchronously changing deformation which initiates the emis sion of SW. On the other hand, SW modulates the dielectric permittivity of medium, which can lead to the exchange of the energy between electromagnetic waves characterized by different frequencies. This difference in the EM frequen cies is equal to SW frequency.
2 . Calculation o f the laser power flux necessary for the generation o f SMBS
According to YARiv [8] it is possible to obtain an equation for the acoustic field Mg of wave propagating in the r§ direction and for the field of scattered electromagnetic wave
Eg propagating in the direction. By using
simplifying assumption ¡Eg]" ¡E^s we can put
= const. Further, assuming the validity of ojg = A*g - Vg we have for the SW :
dMg
— 7 EiEs (3)
d?g 2p T g " 8 p f s
and for the scattered electromagnetic wave:
dEg rEg
I T 4e Ei Mg, (
6
)where p — density of matter, r — SW absorp tion constant, y — électrostriction coefficient, r — scattering term connected with the finite conductivity of medium c by the relation:
(where — magnetic permeability
of vacuum, e — dielectric constant of medium). The equations (5) and (6) can be modified by introducing a new coordinate ^ according to Fig. 2.
The equations discussed describe an increase (or absorption) of Mg and Eg, respectively, along two arbitrary directions, corresponding to <y. Assuming the increase to be of exponential form we get:
Eg(<?) = EjMc.
(7)
If 1 > 0 , then, according to (7), the SW and electromagnetic wave will be simultaneously amplified in fig and fig directions, respectively.
By assuming 1 > 0 and in view of the rela tions (3), (6), (7) we get:
32 Te
/'" g Eg 7's (8)
F ig . 2. T h e co n tin u ity betw een the distance rg) in the d irection n orm al to the sou n d w av e fron t) an d the dista n ce ?'g (in the direction n orm al to the lig h t w ave
fron t)
where T — modul elasticity, Eg, Eg — distances at which the field is being e-times reduced.
The formula for the threshold intensity is quoted also in papers [9] or [10].
Numerical value of Ei can be calculated from the equation (8). This value is necessary for generation of SMBS. According to Yariv the threshold value of laser power flux for quartz is the following:
Ci ¡E ^ s^ lO " Wm"^.
For the spark discharge in air at atmosphe ric pressure and at the temperature T = 15,000 K we shall obtain:
ci !E,j3 = 3-10's Wm
According to [10], by focusing the intensive laser beam with the laser power output ^10" W we get the laser power flux within the sample ^10^ Wm "". Bearing in mind that power out put of the most powerfull lasers is at most 10" W [11], (ISHCHENKO [12] gives the value 6-10" W), we can suppose that SMBS is genera ted also in gases.
The possibility of SMBS generation is greater if the pressure of the gas used is higher than atmospherical one. Since the plasma may be obtained under extremely high pressure (up to 5-10^ atm) the conditions for SMBS genera tion are more advantagenous.
3. SMBS in partially ionized plasma
In the case of partially ionized, plasma the situation is more complicated than in that of neutral gas, since the light scattering on the fluctuations density of electrons, ions and neutral gas must be considered individually. For the sake of simplicity, we assume additionally that plasma is not situated in magnetic field (in this case spectral lines forma more fine structure).
The light scattering on electrons is known as Thomson's scattering [13]. We are concerned with the frequency qualities of the scattered light. The shape of spectrum depends on para meters of plasma and on the factor
" 6 '
47vDsm —
2
where ^ — wavelength of incident light, / Eo&B^e
is so-called Debye length, — Boltzman constant, Tg — temperature of electrons, JVg — number of electrons in cm\ e — charge of electron.
If a < 1, the classical Thomson's scattering can be defined. From the Doppler's shift of
lines we can find and Tg.
If a > 1, the spectrum of scattering light is determined by collective effects, expressed by increased electrostatic oscillations in plasma, resulting in characteristic changes of the spec trum. The spectrum of the scattering radiation in the full ionized plasma consists of the central ion component and two electron satellites.
In warm plasma the amplitude of electron satellites is smaller, its difference from the basis frequency aq is determined by the disper sion relation for the plasma oscillations. For the given temperature pressure the dispersion rela tion is determined according to [14]
^ple (9)
where
^plc is the plasma frequency,
— matter of electron, is sound velocity.
By the sound velocity we mean the velocity at which the pressure-perturbation is spread. In the case of an isothermic process for the elec tron plasma oscillations we can write
Se
where is sound velocity in electron gas.
The plasma oscillations can occure also under low density condition. Then the sound cannot spread within electron gas, because of the lack of the pressure-transmission mechanism. For the low density of plasma the dispersion equation has the following form:
2 2 , 3^B^e ^ ^ ^
Wg — cjp^-)- ^ - (9a)
Hence, the sound velocity takes the value
" y
Similar analysis can be also made for the ion component. The spectrum shape, produced by the scattering radiation on ions, depends on the qualities of the ion acoustic oscillations. If their damp is small, we can observe the ion component. The fine structure of the latter is determined by the dispersion relation for ion oscillations: ,2 2 ^ C , 2 K Mi --- _ ^---r K Me * i df T T ("i + ^ Me (10) where
= Pg. is the ion sound velocity, Xg, Xj —corresponding exponents of adia- batics,
— temperature in energy units, df — mass of ions.
From the scattered radiation spectrum (Fig. 3) we can get information on the temperature and density of electrons and ions. From the widths of lines the damping of plasma and ion oscilla tions, may be also estimated.
In the partially ionized plasma the scattering of light on the neutral component must be also taken into consideration. If the fine structure of the Rayleigh lines, i.e. SMBS is observed it is useful to compare the frequency of single scattered wavings.
For the spark discharge in the air at normal pressure, JV^ = 2 -10" cm*", 8 = (a = 4.3), the numerical values of frequency have been determined from the relations (9) and (10) and calculated on differences of wavelengths.
F ig . 3. T h e spectrum o f scattered ligh t in th e com pletely- ion ized plasm a in case a = 2.17; A g = 2 -l()n cm ^ S ;
Te = 5 eV
The differences between the incidence and scattered light wavelengths are the following ones
electron sound [(1, — AJgl = 67.23 A, ion sound 1(1, — = 0.351 A.
For the sake of comparison these values were calculated from the relation (4), the velocity of sound being given for electron, ion and neutral
gases. The values and (A —AJi are
identical, and this means that the Thomson's scattering (considering the collective interac tions in plasma and the temperature pressure) is the scattering of light on electron and ion sound.
From the relation ( 4) we have 1(1,-liLctrai gas !
= 0.294 A.
The distance between the lines of light scattered on ion sound and on sound in neutral gas is:
1(1, - Ai),! - ](A, - A,)„.^l = 5.8 -10-' A.
4 . Discussion and conclusion
In view of the performed experiment the lines of light scattered on sound waves of the neutral gas and on the ion sound are the only lines useful in the laser diagnostic of plasma, the difference ( A- ^ i ) i —(A-^i)neutr discussed
earlier being negligible. The present contribu tion points to the fact that it is the matter of two lines which can be distinguished by the high resolution methods.
The laser diagnosis can be applied to plasma only in case jYg ^ 10" cm "\ There are also, moreover, the experimental difficulties associa ted with the synchronization of the laser pulse duration with the life-time of high-pressure plasma (or spark discharge).
Despite the limitations introduced and ex perimental difficulties the laser diagnosis of plasma is one of the best method because one spectrum of scattered radiation allows to make a number of statements on the qualities of plasma.
The article points to the fact that, by consi dering SMBS in plasma, more information on the partially ionized plasma may be obtained.
C on tribu tion à ia dispersion de M a n d e ls ta m - -B r illo u in
Ce trav ail est une con trib u tion à la discussion con cern an t ies effects de l'in tera ction d 'u n ray on n em en t iaser in ten sif et du plasm a ion isé; ii s 'a g it n ota m m en t des décharges par étincelie. On a ex am in é la dispersion stim ulée de M an delstam -B rillou in (8M B S ) possible dans un plasm a ion isé en l'a b sen ce de ch a m p m a gn é tiqu e. On su pposait que l'o n d e électrom a gn étiq u e éta it dispersée par des flu ctu a tion s de densité dans un gaz neutre. L a fréqu en ce de cette on d e a été com p a rée a v ec fréqu en ce de l'o n d e électrom a gn étiq u e dispersée par des flu ctu a tion s d 'électron s et a v ec celle de l'o n d e dispersée par l'o n d e acou stiqu e ion iq u e.
К вопросу о рассеянии Мандельш тама-Бриллюэна Работа связана с дискуссией о б эффектах, которые возможны в результате взаимодействия интенсивного ла зерного излучения и частично ионизированной плазмы, а именно изкровых разрядов. Обсуждена возможность стимулированного рассеяния Мандельштама-Бриллюэна (SMBS) в частично ионизированной плазме при отсутствии магнитного поля. Принято, что электромагнитная волна рассеивается на флуктуациях плотности в нейтральном газе. Частота этой волны сравнена с частотой электромагнит ной волны, рассеянной на флуктуациях электронной плот ности, а также с частотой волны, рассеянной на ионной звуковой волне. References
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