On electromagnetic wave momentum transfer to an inhomogeneous plasma

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On electromagnetic wave momentum transfer to an inhomogeneous plasma

R. Dragila

To cite this version:

R. Dragila. On electromagnetic wave momentum transfer to an inhomogeneous plasma. Journal de

Physique, 1981, 42 (10), pp.1413-1419. �10.1051/jphys:0198100420100141300�. �jpa-00209332�

On electromagnetic ^{wave momentum transfer to an} inhomogeneous plasma

R. Dragila (*)

Laser Physics Laboratory, Department ^of Engineering Physics, Research School of Physical Sciences,

The Australian National University, Canberra, ACT, 2600, ^Australia

(Reçu ^{le 31 mars} 1981, accepté ^le 10 juin 1981)

Résumé.

Nous calculons le transfert d’impulsion d’une onde électromagnétique ^{sur une couche de} plasma inhomogène ^de profil de densité quelconque à l’aide de l’équation ^{d’Euler incluant un terme de force} pondéro-

motrice. Dans le cas limite où le profil ^{de densité du} plasma ^{est une fonction en marche} d’escalier, ^{nous obtenons}

les formules de Fresnel en incidences normale et oblique (polarisation ^{s et} p). ^{Pour un} profil de densité régulier

avec un maximum supérieur à la densité critique appropriée, ^la pression ^induite par la force pondéromotrice

sur le plasma plus ^{dense ne} peut pas dépasser ² w0 (w0 ^est la densité d’énergie électromagnétique ^{de l’onde incidente}

dans le vide) ^{et cette} pression n’est pas modifiée par l’accroissement d’amplitude ^{de l’onde} près de la densité critique.

Dans le cas d’une incidence oblique, ^{on a aussi} calculé les analogues ^des impulsions ^{d’Abraham et de Minkowski.}

Abstract.

Electromagnetic plane ^{wave momentum transfer to an} inhomogeneous plasma slab has been evaluated for an arbitrary reflecting ^and absorbing plasma density profile ^{based on Euler’s} equation of motion but including

a ponderomotive ^{term. In} the limit where the plasma density ^{follows a} step function, the Fresnel formulae for the coefficient of reflectivity have been obtained for both normal and oblique ^incidence (s- ^and p-polarization).

For a smooth profile ^{with a maximum} density higher ^than the relevant critical density, ^the ponderomotive ^force

induced pressure ^on the overdense plasma ^{cannot be} higher than 2 w0 (w0 ^{is the} electromagnetic energy density corresponding ^to the incident wave in a vacuum), ^{and is} independent ^{of the wave} swelling ^{near the critical} density.

Also, ^{for the case of} oblique incidence, ^the analogues of the Abraham and Minkowski momenta have been calculated.

Classification Physics Abstracts 52.35H - 52.40D

1. Introduction.

During ^{the last ten} years ^or so, much consideration has been given ^to the role of the

ponderomotive force in the context of laser fusion

[1-25]. Experiments ^{on laser induced} plasmas ^have

obtained conditions where the electron oscillation energy in the electromagnetic field of the laser radia- tion is comparable with the electron thermal energy,

at least in the vicinity of the critical density ^{where the}

electric component ^{of the} driving ^{field can} undergo significant swelling. ^The conditions are even more

dramatic at oblique ^{incidence when} p-polarized

laser light impinges upon ^an inhomogeneous plasma

and a resonance occurs due to efficient linear coupling

of the transverse electromagnetic ^and longitudinal plasma ^waves [26].

Shearer, Kidder and Zink [1] ⁱⁿ 1970, using ^their

code Wazer, first investigated ^{the effect of the} pondero-

motive force in their _gas dynamic ^{treatment of a} plasma expanding ^{into a vacuum.} They ^{were the first}

to observe both plasma density profile modification with the profile steepening ^{in the} vicinity of the critical

density ^{and the} establishment of the so-called cavitons in an underdense plasma. ^Since then, much work has been done in this field both analytically [3, 14-19] ^and purely computationally ^{by either} particle ^simulation

codes [4-6] ^or gas dynamic ^codes [7-13]. Nevertheless,

a clear answer, to the question of whether the pondero-

motive force can change ^the plasma dynamics signi- ficantly (in ^a primary ^{sense, i.e. without} considering

such secondary effects such as a change ^{in the} plasma absorption properties ^{due to} profile modification etc.),

or drive an overdense plasma compression dynamics,

has not yet ^been formulated. Obviously, ^{these ques-}

tions are related to the basic problem ^{of the momentum}

transfer from radiation to an inhomogeneous plasma.

The proposal [2] ^{that the} ponderomotive ^{force can}

induce a large ^{momentum transfer to} both under- dense P;nh ^{and overdense} P;nt plasmas, which is much

larger ^{than the momentum} Po of the radiation itself

(and ^thus necessarily ^of opposite sign ^{so that the net}

momentum is formally conserved Po ⁺ Pin, ^{= -} Pinh) have, however, ^not been confirmed by ^{our recent} computations for both normal [9] ^and oblique ^inci-

dence and p-polarization [27]. ^{Instead, we have}

observed just ^a redistribution of the momentum in the vicinity of the critical density ^{and a wave-like}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420100141300

1414

pattern ^of density ^and velocity ⁱⁿ the underdense

region, ^{and no} significant net momentum transfer to the overdense plasma ⁱⁿ comparison ^{with the case of} plasma ^free expansion ^{into a vacuum with the} pondero-

motive force switched off.

It is the aim of the present paper ^to sheal more

light ^{on the} problem of electromagnetic wave momen- tum transfer to an inhomogeneous plasma.

2. Plane wave at normal incidence.

Let us consider

an inhomogeneous rigid plasma layer providing ^{a con-}

tinuous transition from a vacuum z _{zo to} ^a homo- geneous plasma, occupying ^a halfspace z > z, > zo.

Let the plasma ^{be assumed} nonabsorbing i.e. with the relevant dielectric constant as follows

where e(z) ^{is an} arbitrary real continuous function (not necessarily monotonic), ^{such that} e(zo) ^{= 1 and} e(zl) ⁼ E 1. An electromagnetic ^{wave, with} amplitude

E ⁼ Eo ^{in the vacuum,} is assumed to be incident

perpendicular ^{to the} plasma-vacuum boundary. ^The plasma and radiation momenta balance equation ^is

well known [2, 17]. ^For normal incidence

with

where _p is a plasma ^momentum density z-component,

, 1

is the time-averaged Maxwellian stress tensor, 1 a

unit tensor, g = ^S >/c2 ^the electromagnetic ^field

momentum density, ( S > ⁼ cEN/8 yr ^{the time-ave-} raged Poynting ^{vector, c the} speed ^of light, ^E = [ E [,

H = 1 H the amplitudes of the electric and magnetic components ^{of the} electromagnetic field and co the

angular frequency of the incident electromagnetic

wave. For the case of a plane ^{wave, the last term in} (2) gives ^no contribution, ^and consequently ^{the net}

momentum transferred from the radiation to the

inhomogeneous plasma (per unit surface area of the

plasma) during ^a time interval t becomes

At z ⁼ _{zo -} 0, as well as at z ⁼ z, ⁺ 0, ^{the medium}

is homogeneous ^and consequently ^{in that} region

H 2 ⁼ eE 2. The _energy density ^{of the} electromagnetic

field in a plasma is, after time-averaging ^{over 2} n/co,

W ⁼ E 2/8 , ^{half of which} represents always ^the

energy density of the electric field while the remainder is complementarily split between the magnetic ^field

energy density ^{and the} oscillation _energy of the charged particles [28]. ^In photon terminology, ^{W =} Nhco,

where N is a local photon density.

Hence, ^{in terms} of the photon density, ^{the momen-}

tum transferred to an inhomogeneous plasma ^is

If R is the coefficient of reflectivity ^{relevant to the} plasma density profile (1), ^then

where No ^{is the} density of the incident photon ^beam

with the assumption ^{of a} nonabsorbing plasma, ^the

net Poynting ^{vector S must be a} constant, and ^conse-

quently

Inserting (6) ^and (7) ^into (5) yields

where ko ⁼ wlc ^and JC(t) represents the number of

photons per unit ^area that entered the plasma-vacuum boundary, ^{z =} zo, during the time interval t.

For a nonreflecting profile (R ⁼ 0), ^{we obtain the}

same momentum p(n)inh ^{as that obtained} by ^Hora [2]

in his WKB treatment. On the other hand, by steepen- ing ^the profile (1) ⁱⁿ the limit Zo --> z, (assuming

max e(z) ⁼ E 1, ^z e ( zo, z, », a plasma density step ^is obtained. As Tz2 ^{is a} continuous quantity, ^{the momen-}

tum P;nh ^is necessarily ^zero, which, according ^{to the} equation (8), ^is possible only ^if

resulting in the well known Fresnel formula.

A property ^{of the WKB solution} [2] ^{is that} P;nh ^is

a function of s(zo) ^and E(z1) only, independent ^{of the} shape ^{of the} plasma density profile. Obviously, ^{it is}

the coefficient of reflectivity ^R which contains the information about the details of the profile shape.

However, ^{it is a common} property ^{of a} whole class

of nonreflecting dielectric profiles (even ^{where the}

WKB is not valid) that E , H depend ^{on z} impli- citly only ^{via e =} E(z), and thus the relevant P1:h ^is

a function only ^{of the} boundary ^{values of e. For} example [28], ^if f ⁼ f(z) ^{is an} arbitrary ^real function,

then the solution of the wave equation

with

reads

and obviously yields ^no reflection.

Let us go back to equation (8), ^{which can} be rewritten to the following ^form

illustrating ^that taking ^{into account} reflectivity ^cannot only modify ^{the WKB} solution, ^{but can even} change

the sign ^{of the momentum} P nh, ^{if R} > RF(n), ^{which is}

the case when max e(z) > el for z E (zo, zl). ^{For a} plasma slab with _E1 ⁼ 1, ^we obtain the obvious result P nh ^{= 2} RX(t) hko.

There is one more very useful form of equation (8)

which leads to a clear physical interpretation. Namely,

the momentum P nh transferred to an inhomogeneous plasma ^{is due to both} the reflection of photons (2 hko ^momentum change per reflected photon) ^and

recoil with an increase of the momentum

of the transmitted photons (- (1 - R)). ^Relation (10)

is consistent with the photon ^momentum

in an homogeneous plasma obtained earlier by ^other

authors (see for instance [29, 2]). Notice that for E 1 --> 0, convergency is provided by ^{the term} (1

R)

in (9).

Thus, ^{in the context} of laser fusion, ^{if we consider}

a plasma density profile generated by ablation of a

target ^{with a maximum} density ^{above the critical} density ^and neglecting absorption (short pulse regimes

at normal incidence), ^the relevant coefficient of

reflectivity ^is obviously ^{R =} 1, ^and consequently

the net momentum transferred to an underdense

plasma ^becomes

where Io is ^the intensity ^{of the incident laser} light.

Hence, ^{there is no net} deconfining force in the under- dense plasma, ^and, therefore, ^no reaction-induced

confining force in the overdense region results from interaction of the ponderomotive ^{force with an} inhomogeneous plasma (where the maximum density

is above the critical density). The ef’ect of absorption

will be discussed in Section 5.

A completely ^different picture ^is obtained, ^when

the plasma represents ^a transparent slab, ^i.e.

e(zo) = e(zl) ^{= 1 and 0} e(z) ^{1 for} zo ^z zl.

Thus, assuming ^the plasma ^{slab to be} nonreflecting (R ⁼ 0), it follows that Ps1ab ^{= 0.} However, plasma

to the left and right ^{of the absolute} plasma density maximum, nmax’ ^can obtain a significant (if

nee - nmax « nee) ^{momentum of} opposite sign, ^lead- ing ^{to some kind of an} explosion, ^driven by ^{a localized} high electromagnetic energy density. ^{Such an effect}

has been demonstrated numerically [30-31] ^{and ana-} lytically [32] ^{on a so-called} bi-Rayleigh-type profile.

3. Plane wave at oblique incidence and s-polariza-

tion.

Our treatment of a plane ^{wave at} oblique

incidence and s-polarization ^is analogous ^{to the case}

of normal incidence (see ^Section 2) ^{but, with} ( Tzz >

in the form

where 0 ⁼ O(z) is the local angle of incidence and is related to 00 ^{via Snell’s law, i.e. E. sin2 0 = sin2} 03B80.

Note that the momentum balance equation ^now

reads

where, however, ^{the last term} yields ^no contribution when a plane ^{wave is} considered. After integration,

the momentum transfer to an inhomogeneous plasma,

with a density profile ^defined by (1), ^becomes

1416

Again, N(zo - 0) ⁼ No . ( 1 ⁺ R(s») ^{and, in a} nonabsorbing plasma, ^the z-component ^{of the} Poynting ^vector

is Sz(z) ⁼ const., i.e.

where 0, ⁼ 0(z ⁼ zl 1 + 0). ^{Here R (s) is the} coefficient of reflectivity ^{of the s-wave relevant to the whole} profile

and is, ⁱⁿ general, unknown and dependent upon the profile ^{details. Hence,}

where JC(0o. t) represents the number of photons per unit surface area that entered the plasma-vacuum boundary,

z ⁼ Zo, during ^a time interval t. Again, in the limit _{zo - zl,} i.e. passing ^{to a} plasma density step ^{at the} plasma-

vacuum boundary, ^{we have lim} P(s)inh ^{= 0, and thus,} according ^to (12),

ZO -+ Z l

we obtain the relevant Fresnel formula. Equation (12) ^can be rewritten in various forms, ^{and we shall} immediately

rewrite it in a form analogous ^to (9), ^i.e.

where the z-component ^momentum increase of the photon after its transition from vacuum to a homogeneous plasma ^becomes

Of course, setting 0. ^{= 0,} equations (13) ^and (14) yield (9) ^and (10), respectively. According ^to (14), photon

momentum is no longer ^{a local} quantity, depending ^on 81 alone. Instead, ^the photon ^{momentum also} depends

on its history and, ⁱⁿ particular, ^on the initial angle of incidence and is

which can be written formally ^as

where _pM ⁼ hkolê, is the Minkowski momentum and PA(OO) ⁼ PM.COS2 00/(e1- ^sin2 00) ^an analog ^{of the}

Abraham momentum at oblique incidence and s-polarization.

4. Plane wave at oblique incidence and p-polarization.

Comparing ^{the cases of} p- and s-polarization

at oblique incidence, ^two principal differences arise :

a) ^with p-polarization, ^a longitudinal component Ez ^{of the} electric field _appears and consequently Tzz Tzz

and is

b) Tzz ) ^{is no} longer continuous at a plasma density step, ^thus giving ^{rise to a nonzero value of lim} P(p) inh.

Zo-Z1 1

Inserting (17) ^{into the momentum balance} equation ^and integrating ^as in sections 2 and 3, ^one obtains

In passing ^{from a smooth} plasma density profile ^{(1) to a} plasma density step, ^{it is} necessary ^to evaluate a quantity

such as

which, ^after straightforward manipulation, yields

Hence, ^{in the} limit _zo ^- _{z1, contrary} to the case of normal incidence or oblique incidence and s-polarization,

we obtain

This « residual » momentum is induced by ^an electrostatic field force affecting the elastic dipoles ^{created at a} polarized sharp plasma-vacuum boundary. ^{For the case of a smooth} plasma density profile (1), ^{this kind of}

momentum transfer is still présent, although it is built _up continuously along ^the profile up ^to the magnitude (20),

and depends only ^{on the} boundary ^{values of e. In the limit} zo ---> Zl, equation (18) yields ^{in this case}

and consequently

which is again the relevant Fresnel formula.

As in the previous sections, ^{one can} introduce the momentum (p) ^{of the photon in an} homogeneous ^dielectric medium, assuming that the relevant momentum increase Ap(P) ^{provides a balance to the} reflected-photon-

induced recoil at a step-like plasma-vacuum boundary. ^{To do} this, ^we first rewrite equation (18) ^{in the} following

form

which is formally ^{the same as} (13) ^{with the} replacement ^{R (s)} ---> R ^(p) (R ^{(s) :0 R (p) in} general). ^{Then, we extract the}

electrostatic term, introducing ^{a new momentum} Pinh ^as

such that lim P;nh ^{= 0. The momentum increase} Ap(p) ^{is now}

zo-z 1

and leads to the relevant photon ^momentum

1418

where

is ^an analog of the Minkowski momentum at oblique ^{incidence and} p-polarization. ^{As seen from (21) and (16),}

the relation PM(03B80) = 03B51 PA(oo) ^{between our} Minkowski and Abraham momentum analogues ^{is the same as}

that between the original ^Minkowski pM and Abraham _PA momenta.

For an unpolarized ^{wave at} oblique incidence, ^{one can} identify ^a photon ^{momentum with the} average momentum as follows

and, ^{one can} certainly ^se

for the momentum transferred to an inhomogeneous plasma (1).

5. Absorbing inhomogeneous plasma.

^{In all the} previous ^{sections, we} have considered a nonabsorbing reflecting inhomogeneous plasma ^{and found the}

relevant momentum transfer from the incident wave

to the inhomogeneous plasma region. ^{In this section,}

we shall demonstrate the effect of absorption ^{on the}

momentum transfer for the case of normal incidence.

We start again ^{from the} equation ^{of momentum}

balance and, ⁱⁿ particular, in the form (5), ^i.e.

Let us assume that it is only ^the inhomogeneous region

that absorbs, ^{with a relevant net} coefficient of absorp-

tion A (not ^affected by reflection), whilst the homoge-

neous regions ^are nonabsorbing. Then, ^{one can set}

and

where R retains its old meaning ^{for a} nonabsorbing plasma. Inserting (23) ^and (24) ^into (22), ^{one obtains}

Hence, ^{for R =} 0, ^we recognize ^{the WKB} solution, modified, however, by ^the positive absorption ^term.

The limit _zo ^- _zi leads again ^to the Fresnel formula,

due to the fact that absorption ^{is a} volume effect and thus lim A ⁼ 0. In a completely analogous way ^to

Zü-Zl

that in section 2, ^{after some} algebra, equation (25)

can be rewritten in the final form

with an obvious physical interpretation. ^{The three}

terms represent respectively ^{recoil to the reflected} photons; ^{the momentum} balance of the photon

momentum increase when entering ^{the dielectric} médium ; and the absorbed photon ^momentum accept- ed by ^the absorbing ^medium. Note, ^that the coefficient

of reflectivity ^{that would be measured, i.e. that} defining

the intensity of the reflected wave is not R itself, ^but Rm=(1-A)R,andalsoAm=A,ifR=0.

The generalization ^for oblique incidence is straight- forward, and thus will not be presented ^here.

6. Conclusions.

In the present paper, ^we have treated the problem ^{of momentum transfer from a} plane ^{wave to an} inhomogeneous plasma layer zo ^z Zl, bounded by ^a homogeneous (noir- reflecting ^or opaque) plasma. ^{Without loss of} gene-

rality, ^{we have assumed a vacuum in the} halfspace

z Zoo Both normal and oblique incidence have been considered, ^{and in the latter case for both s- and} p-polarizations.

The momentum transferred to an inhomogeneous plasma consists of : a) ^{recoil to the reflected} photons ; b) ^momentum balance of the photon ^momentum

increase when entering the dielectric médium ; c) ^the

momentum of the absorbed photons accepted by absorbing ^medium; d) electrostatic-force-induced momentum at oblique incidence and p-polarization.

Assuming ^no absorption ^{and no} reflection, ^our

results are equivalent ^{to the} results of the WKB

approximation, ^{obtained earlier for} normal inci- dence [2].

In the limit _{Zo --+ z} which represents the transition from an inhomogeneous plasma density profile ^{to a} sharp plasma-vacuum boundary, ^and allowing ^{for the} obviously vanishing ^{momentum transfer to the inho-}

mogeneous plasma (from ^the electromagnetic field, except its irrotational part), ^the Fresnel formulae have been obtained in all cases considered.

For the case where the maximum plasma density

in the profile ^is higher than the relevant critical den-

sity (el ⁼ 8(zi) 0), the coefficient of reflectivity (as specified above) ^{R = 1} (R., 1, ⁱⁿ general ; ^see

section 5), ^and consequently ^the momentum trans- ferred to the underdense region ^is

Hence, ^the electromagnetic ^wave induced pressure ^on the overdense plasma ^{cannot be} higher ^{than 2} No hco,

where No hcv ⁼ E’/8 n is defined by ^{the incident}

wave amplitude Eo ^{in a vacuum and is} independent

of the swelling of electric field in the vicinity ^{of the}

critical density. ^That is, ^the electromagnetic ^field

variations in an underdense plasma yield ^a pondero-

motive force-induced redistribution of the momentum

there, ^{but the net} purely dynamic ^{effect on the over-}

dense plasma is determined solely by the radiation pressure E’0/8 n of the incident wave in the vacuum.

Acknowledgments. - The author wishes to acknow-

ledge discussions with Dr. B. Luther-Davies.

References

[1] SHEARER, ^J. W., KIDDER, ^R. E., ZINK, ^J. W., ^{Bull. Am.} Phys.

Soc. 15 (1970) ^1483.

[2] HORA, H., Phys. ^{Fluids 12} (1969) ^182.

[3] GIL’DENBURG, ^V. B., FRAIMAN, ^G. M., ^ZhETP 69 (1975) ^1601.

[4] ESTABROOK, ^K. G., VALEO, ^E. J., KRUER, ^W. K., Phys. ^Fluids

18 (1975) ^1151.

[5] FORSLUND, ^D. W., KINDEL, J. M., LEE, K., LINDMAN, ^E. L., MORSE, R. L., Phys. ^{Rev. 11} (1975) ^379.

[6] ESTABROOK, ^K. G., KRUER, W. L., Phys. Rev. Lett. 40 (1978)

42.

[7] BRUECKNER, ^R. A., JANDA, ^R. S., ^{Nucl. Fusion 17} (1977) ^451.

[8] LARSEN, ^J. T., HARTE, J., Bull. Am. Phys. ^{Soc. 22} (1977) 1106, UCRL-79757.

[9] DRAGILA, R., KREPELKA, J., ^J. Physique ³⁹ (1978) ^617.

[10] ANDREEV, ^N. E., SILIN, ^V. P., STENCHIKOV, ^G. L., ^{Pis’ma v} ZhETP 28 (1978) ^533.

[11] BAUMGARTEL, K., SAUER, K., Phys. ^{Lett. 70A} (1979) ^107.

[12] MONTRY, ^G. R., POWERS, ^L. V., BERGER, ^R. L., ^{Nucl. Fusion}

20 (1980) ^283.

[13] POWERS, L. V., MONTRY, ^G. R., TANNER, ^D. J., BERGER, ^R. L.,

Nucl. Fusion 20 (1980) ^821.

[14] LEE, K., FORSLUND, ^D. W., KINDEL, J. M., LINDMAN, E. L., Phys. ^{Fluids 20} (1977) ^51.

[15] MULSER, P., VANKESSEL, C., Phys. Rev. Lett. 38 (1977) ^902.

[16] VIRMONT, J., PELLAT, R., MORA, P., Phys. ^{Fluids 21} (1978) ^567.

[17] DRAGILA, R., J. Phys. ^{D 11} (1978) 1683.

[18] MAX, ^C. E., MCKEE, ^C. F., Preprint UCRL-79392, ^1977.

[19] DRAGILA, R., LIMPOUCH, J., ^{Czech. J.} Phys. ^B (1980) ^143.

[20] ZAKHARENKOV, ^Yu. A., ZOREV, ^N. N., KROKHIN, ^O. N., MIKHAILOV, ^Yu. A., RUPASOV, ^A. A., SKLIZKOV, ^G. V., SHIKANOV, ^A. S., ^{Pis’ma v ZhETP 21} (1975) ^557.

[21] FEDOSEJEVS, R., TOMOV, I. V., BURNETT, ^N. H., ENRIGHT, G. F., RICHARDSON, ^M. C., Phys. ^{Rev. Lett. 39} (1977) ^932.

[22] MANES, K. R., AHLSTROM, ^H. G., HAAS, ^R. A., HOLZRICHTER,

J. F., ^J. Opt. Soc. Am. 67 (1977) ^717.

[23] ATWOOD, ^D. T., SWEENEY, ^D. W., AUERBACH, J. M., LEE, P. H. Y., Phys. ^{Rev. Lett. 40} (1978) 184.

[24] RAVEN, A., WILLI, O., Phys. ^{Rev. Lett. 43} (1979) ^278.

[25] LUTHER-DAVIES, B., ^submitted.

[26] DENISOV, ^N. G., ^{ZhETP 31} (1956) ^609.

[27] DRAGILA, R., KREPELKA, J., ^submitted.

[28] GINZBURG, ^V. L., Propagation of Electromagnetic ^{Waves in}

Plasmas (Pergamon Press, ^New York) 1964, p. 188 ^{and 478.}

[29] LINDL, J., KAW, P., Phys. ^{Fluids 14} (1971) ^371.

[30] HORA, H., LAWRENCE, ^V. F., ^Laser Interactions and Related Plasma Phenomena, ^{vol. 4B edited} by ^{H. J.} Schwarz and H. Hora (Plenum, ^New York) 1977, p. 877.

[31] LAWRENCE, ^V. F., ^{Ph. D.} Thesis, University of New South Wales, Australia, 1978.

[32] DRAGILA, R., submitted.