IEEE Transactions on Plasma Science, Vol. PS-6, No. 4, December 1978

RADIATION PHENOMENA OF PLASMA WAVES

PART 1. FUNDAMENTAL RADIATION THEORY

Toshiro OhnumaDepartment of Electrical Engineering

Tohoku UniversitySendai 980, Japan

INVITED REVIEW ARTICLEReceived 3/22/78

CONTENTS

1.

I.

PrefaceIntroductionFundamentals for waves

Il-1. Phase, group, and ray velocities11-2 Phase velocity surface, group velocity surface,

ray velocity surface, and refractive index sur-face

111. Fundamental radiation theory in an isotropic plasma

111-1. Radiation theory for electron plasma waves

111-2. Radiation theory for ion plasma waves111-3. Generalized radiation theory in an isotropic

plasmaIV. Electromagnetic radiation from an electric dipole

in a cold anisotropic plasmaV. Quasi-static radiation from a point source in a warm

anisotropic plasmaV-1. General formula for quasi-static radiation fieldV-2. Radiation field in an electron plasma-high

frequency resonance cone

V-2-1. Radiation in a cold electron plasmaV-2-2. Radiation in a warm electron plasma

V-3 Radiation field in a two component (electronand ion) plasma low frequency resonance cone

V-3-1. Radiation in a cold plasmaV-3-2. Radiation in a warm prasmaReferences

PREFACE

Radiation phenomena of plasma waves has been paidmuch attention in the fields of space plasmas and plasmaphysics. Historically, the study is considered to have started

in the 1950's in order to investigate the space phenomenon.In the 1960's, the investigations, mainly on electromagnetic

radiation, had been reported by many authors. In the 1970's,

the study on the radiation properties in plasmas has madea great advance both experimentally and theoretically. Fur-thermore, the investigations have been made in relation toa heating of plasmas for controlled thermonuclear fusion.

Because it is considered an appropriate time now to

review, it is a great pleasure for the author to be able to

present the review paper on the radiation phenomena ofplasma waves. Total contents of this report are as follows:

"Radiation phenomena of plasma waves"Part 1. Fundamental radiation theoryPart 2. Radiation from point sourcesPart 3. Radiation from finite sources

In part 1, the theoretical radiation theory will be reviewedand the fundamental theoretical results will be presented.In part 2, the experiments on radiation from point sources

will be reviewed with the simplified theoretical considerations.In part 3, the experiments on radiation from finite sources

will be presented with the summarized theoretical results.

1. INTRODUCTION

Radiation characteristics of plasma waves have been in-

vestigated theoretically in the early stage, especially, in the

field of the electromagnetic properties of an anisotropicmedium. Many studies on radiation phenomena in the 1950's

and the 1960's had been made on the electromagnetic pro-perties in a cold magnetoplasma. Furthermore, those studies

had been mainly theoretical and the corresponding labora-

tory experiments had not been reported. The electrostatic

properties and the thermal effects on the radiation in plasmashave been investigated mainly in the 1970's. In this intro-

duction, the radiation theories in plasmas are summarizedhistorically.

0093-3813/78/1200-0464$00.75 © 1978 IEEE

464

Bunkin's work (1957) is very famous on radiation pro-blems. He had given a general solution for the problem of find-ing the radiation fields produced by the sources in an aniso-tropic medium [ 1 1 .

In an isotropic plasma, there have been many papers onradiation phenomena. Hessel and Shmoys (1961) indicatedthe excitation of electromagnetic and electron plasma wavesby an oscillating point current element using a hydromagnetictheory [2]. Cohen (1961, 1962) investigated several effectson radiation in an isotropic plasma, namely, the Cherenkoveffect, equivelant sources and metal boundaries [3]. Kuehl(1963) derived the general expression for the electromag-netic field due to an arbitrary monochromatic current sourcein a warm isotropic plasma with a hydrodynamic description[4]. The general results are -specialized to the case of anelectric dipole for transverse, longitudinal electron and longi-tudinal ion waves. Kuehl (1964) extended this problemto obtain the general expressions using kinetic theory [5].A summary of this kinetic theory will be presented in Sec-tion 111-3. On this problem, Chen (1964) also obtained thefields of both the electromagnetic and electron plasma modesexcited by a Hertzian and cylindrical dipole in an isotropicplasma [6]. Seshadri (1965) has also investigated the radiationfrom electromagnetic sources for the electromagnetic mode,the electron plasma mode, and the ion plasma mode [7].The problems of radiation due to a short electric dipole,a uniformly moving point charge and a filament with a tri-angular current distribution were analyzed.

Since the novel paper by Bunkin, the electromagneticradiation in a magnetized plasma has been treated by manyauthors. Kogelnik (1960) investigated electromagnetic radia-tion in a cold magneto-ionic plasma [8]. A formula is givenfor the power radiated by any distribution of alternating cur-rent in terms of the wave matrix. Lighthill (1960) studiedmagnetohydrodynamic waves and other anisotropic motionsand gave the asymptotic behavior at large distances of wavesgenerated by a source of finite or infinetesimal spatial extent[9]. Kuehl (1962) has modified Bunkin's results and investi-gated electromagnetic radiation from an electric dipole in acold anisotropic plasma [10]. He evaluated the far fieldexpressions and found that the radiation characteristicsare highly directive for certain ranges of plasma parameters.Arbel and Felsen (1963) investigated in detail the radiationfrom sources in an anisotropic medium, namely, from generalsources in a stratified medium and point sources in an infinite,homogenuous medium [ 1 1 ] . Furthermore, Wu (1963) studiedthe radiation from elemental magnetic and electric dipolesin a cold magnetoionic plasma [12]. Motz (1965) also calcu-lated electromagnetic radiation from a point magnetic dipole

with arbitrary orientation imbedded in a lossless cold mag-netoionic plasma [131. Mittra and Duff (1965) have numeri-cally calculated the far zone electric field components of a

point electric dipole (parallel to the magnetic field) for variousplasma parameters [141. Freire and Scarabucci (1967) consi-dered small losses by collisions for radiation from an electricdipole by following Bunkin's formulation [151. Snyder andWeitzner (1968) have obtained several far field expansions anda near zero expansion that is useful for discussing the powerrediated by a dipole [16]. GiaRusso and Bergson (1970)present far zone power patterns (radial Poynting vector)by Kuehl's formulation for VLF from arbitrary oriented pointmagnetic and electric dipole sources in a two component,slightly lossy magnetoplasma [171. With the use of a pointpower integral formulation, Wang and Bell (1972) made astudy of the VLF/ELF radiation patterns of arbitrarily orien-ted electric and magnetic dipoles in a cold lossless multi-component magnetoplasma [18] . The radiation patternsillustrated the fact that focusing effects arose from the geo-metrical properties of the refractive index surface.

In those electromagnetic studies in an anisotropic plasma,the most interesting phenomenon was obtained in the analysisby Kuehl (1962) of a point dipole in a cold anisotropic plas-ma. Namely, he showed the existence of the high frequencyresonance cone that the field should become singular on aconical surface. The nature of the singularity in the fieldsis such that it yields an infinite amount of power flow fromthe dipole. This was called the "infinity catastrophe," whichwas clearly solved later [191 by Singh and Gould (1971)with a kinetic treatment of plasmas including electron thermalmotion.

On the effects of electron thermal motions on the radia-tion characteristics, Deshamps and Kesler (1967) first treateda short dipole [20] . They derived a formula for the radiationfield of an arbitrary exciter with a fluid model plasma. Chen(1969) has also investigated radiation characteristics of anelectric dipole in a warm anisotropic electron plasma andderived a formula for the radiated power by means of thedyadic Green's function method [211. Tunaley and Grard(1969) investigated the similar problem in the electrostaticapproximation [221. They noted that the phase velocitywas infinite on a high frequency resonance cone in a uni-axial plasma. Furthermore, Singh and Gould (1971, 1973)reported important results on the effects of electron tempera-ture on the radiation fields of a short electric dipole in auniaxial plasma [19],[23]. They solved the "infinity catas-trophe" of the resonance cone with kinetic description of theplasma medium including the electron thermal motion. Forf > fpe the electron plasma mode (Landau mode) was found

465

to be trapped in some spatial region. This fact was the new

result from the warm effect of the plasma and had not been

reported in the earlier papers on cold plasma theories.In the history of the radiation phenomena, Fisher and

Gould (1971) made an epoch-making observation of the high

frequency resonance cone [24]. Since this important experi-mental verification, radiation theories on the resonance coneshave been presented by many authors. Kuehl (1973) investi-

gated the interference structure near the resonance cone of

an oscillating point charge in a warm magnetized plasmausing the quasistatic approximation [25]. The potential

patterns of a coupled resonance cone and electron Bernsteinwaves were also clearly calculated. Furthermore, Kuehl (1974)investigated the electric field and potential close to the reso-

nance cone of an oscillating point charge and dipole in a warm

magnetized plasma [26]. The potential of the point charge

has a maximum closest to the cold plasma resonance cone

and has the largest interference structure. Chasseriaux (1974,1975) also studied the potential created by an alternating

dipole and the resonance cone potential created by an alter-nating point charge in a warm magnetoplasma [27],[28].Furthermore, Simonutti (1976) showed that the linear tran-

sient response of a cold anisotropic plasma to a point source

under impulse excitation displays a frequency spectrum

having a maxima at the two frequencies determined by the

resonance condition and at the upper hybrid frequency [29].Belan (1977) also investigated plasma ringing associated with

pulsed resonance cones [30]. In a warm magnetized bounded

slab plasma, Grabbe (1977) investigated resonance cones in

detail [31].Resonance cones near the lower hybrid frequency have

also been investigated by many authors with a relation to a

lower hybrid heating of plasmas. Stix (1965) reported radia-

tion and absorption via mode conversion in an inhomoge-nuous plasma [32]. Briggs and Parker (1972) investigatedtransport of rf-energy to the lower hybrid resonance in an

inhomogenuous plasma [33]. Belan and Porkolab (1974)clarified the propagation and mode conversion of lower

hybrid waves generated by a finite source in an inhomoge-nuous plasma [34]. The conversion at the lower hybrid layerof the electrostatic cold plasma waves into propagating hot

plasma waves was calculated. Kuehl and Ko (1975) also

studied the excitation and linear mode conversion of lower

hybrid resonances cones in an inhomogenuous plasma [35].Furthermore, Kuehl (1976) considered nonlinear ponder-motive force effects on mode-converted lower hybrid waves

[36]. In a bounded, inhomogenuous plasma,Colestock and

Getty (1976) studied the excitation and propagation of

lower hybrid waves [37]. Further investigations have been

continued. In addition to those studies, Ohnuma et al. (1977)

investigated electrostatic waves (lower hybrid and ion waves)

near the lower hybrid frequency in a warm magnetized plasma

and found that the lower hybrid wave becomes the ion wave

with weak collisions [38].For frequencies below the ion cyclotron frequency,

Kuehl (1974) showed that the potential of an oscillating point

charge in a magnetized plasma contained a low frequency

resonance cone which was due to the electrostatic ion wave

[39]. Burrel (1975) investigated the low frequency resonancecone structure in a warm anisotropic plasma by an asymp-totic analysis of the electrostatic Green's function [40].The interference structure associated with the cone was

shown to be quite sensitive to the ion temperature. This

fundamental low frequency resonance cone was confirmed

experimentally by Ohnuma et al. (1976, 1977) [41],[42]and Belan (1976) [43]. In addition to the experimentalconfirmation, they descirbed the relation of the resonance

cone to the wave fronts of the radiated wave. Furthermore,Ohnuma et al. investigated the ray velocity surface [42].

11. FUNDAMENTALS FOR WAVES

In this section, fundamental concepts of propagatingwaves are explained. These will become very important,especially in an anisotropic plasma.

11-1. Phase, Group, and Ray VelocitiesDefinitions for the various velocities are given as follow:

phasevelocity u - ' ^ W A .) AV = i k = k+-kl ,p 1k k Ik I k_-'

group velocity\V =g

(1)

k=ck x OA ak A

x y z

DW' ^-6bUJAray velocity 1 kil L

r Cos s k 0

(2)

(3)

tanc~. - dk Ildntai-i = - (4)

where n(6 ) = ck(6 )/c is the refractive index and 0 indicates

the unit vector in the direction of the group velocity. The

magnitude of those velocities are given by

V(p ) = k( s )(5)

466

V (0) = cb w

g Cosa okV'(6)

1 (2 1 V (~). (7)r( P) coscd k cos > pThe relation between tne angles 0, 6 , a, is indicated in Figure1. Those phase, group, and ray velocities are explained as

t

IK

Figure 1. Relation of group (Vg) and ray (Vr) velocities to

wave normal (k).

follow:

A. Phase velocity.An infinite train of waves propagates with the phase

velocity IV p perpendicular to the wave fronts. The directionof IVpdefines the direction of wave normal.

B. Group velocity.The velocity of the envelope of a short train of waves

is given by the group velocity IVgin the direction of the group

velocity, namely, the ray direction. The velocity IVg is thevelocity of energy propagation at least when the mediumis non-dissipative. When one considers a wave packet

1E( ir, t ) = IE( ik ) ei( Ik r ( 1k)t)d lkwith 1k = 1k tUk

0

the group velocity is given for the condition of

a --jjIk-lkr- U( lk)tb

= t =

for Ik= Ik0

namely, Er ag t a Ik

The phase and group velocities on the diagram of the dis-

persion curve are indicated simply in Figure 2.C. Ray velocity.The ray velocity IVr is the phase velocity in the direction

of the group velocity. The magnitude of the ray velocityis the speed at which wave fronts travel along the direction of

the group velocity. If a monochromatic point source radiates

Figure 2. Phase (V ) and group (Vg) velocities in a dispersioncurve.

continuously in all directions, the wave fronts spread out

radially from the source with the ray velocity. IVr is not ingeneral normal to the wave fronts, while IVp is normal to thewave fronts. The relation of the ray velocity to the phasevelocity is shown in Figure 3. From this figure

V = f2- = fX cos0s = V coscl Xp p r r

NT = Vr coscl p

WAVE FRONTS K. \Vp

\Vg \Vr

Figure 3. Relation of wavelength X in the direction of thephase velocity to a wavelength (ray wavelength) in thedirection of the group velocity.

This relation is the same one as Eq. 7. If a point source were

switched on and thereafter radiated continuously, the initialdisturbance would propagate radially with the group velocity

V9. The wave fronts move out radially with the ray velocityinside the signal front. The difference of the phase velocityand the ray velocity comes from the angle a which is betweenthe directions of the group and phase velocities. The directionof the group velocity is in general different from the directionof the phase velocity in an anisotropic plasma. The differenceangle oa is given by a relation of Eq. 4. Several interestingphenomena in an anisotropic plasma appear from the dif-

ference of the directions of those velocities.Now, we will consider Eqs. 1-7 in more detail. The deri-

vation of Eqs. 4 and 6 can be derived from the definition

467

K

of the group velocity as follows:_w bW A A~O'

\V = b Ik = b k k + 0 a 0 0 * (8)g 3lk 6 k k

tanc8 = tan (0 -

1B/MM 1 dkk.~~k - -~ d~' (9)k. 0/ k k d/'

tanI

kd ' (4)

where we used the relation &o= (aco/U )6W + (aco/ak)6k = 0.The magnitude of the group velocity is

Vg \Vg= +(k)2 k

V J(=D/aD)2 + aD/-- 2\ 3kll,/ brJ1)

(15)tan 0 = -

From Eq. 15, one can obtain the magnitude of the groupvelocity and the direction of the group velocity. The phasevelocity Vp ( ) can be obtained from Eq. 5 after a calculationof k(6 ) from the dispersion relation D(,lk) = 0 for a givenpropagation angle 6. From these angles 0 and , namely, a =

0 - eS, one can obtain the ray velocity from Eq. 7 with Eq. 5.

11-2. Phase Velocity Surface, Group Velocity Surface, RayVelocity Surface, and Refractive Index Surface

Four velocity surfaces will be explained with their defini-tions.

/ a 2+ - ktan a

1 axcosa b6k

(10)

v = 1 a k * (6)g cosco\ -6k (6

Next, the derivation of those velocities is explained with ause of the dispersion relation

D ( UJT, Ik ) = 0 . (11)

From this equation, 3 D fi

gV = kD -MD9 al Qa

-DDk = 0,

A oD / D A D Id D

A .,

( A b (IYBecause of \V = k + k = kjV + k V

org 1 bki k g1

the next relations can be obtained;bDD D

V6D a

D6ll k1I/ asThen, tan 0 is obtained as follows:

Vg kD l8Dtan 0= gk=D-Dg911 1 k~

(13)

(14)

A. Phase velocity surface.The phase velocity surface is sometimes called the "wave

normal surface." The normalized "phase velocity surface"is a plot of Vp/C against d in a polar coordinate. The norma-

lization of the phase velocity is sometimes useful when oneuses the characterisitc velocity instead of the light velocityC. This kind of normalization is also useful for other velocitysurfaces. The shape of this phase velocity surface should notbe confused with the shape of a wave-front.

B. Group velocity surface.The normalized "group velocity surface" is a plot of

V /C as a function of 0 in a polar coordinate. The shape isthe same as the shape of a signal front originating from a

point source.C. Ray velocity surface.The normalized "ray velocity surface" is a plot of Vr/C

against 0 in a polar coordinate and is sometimes called "raysurface" for simplicity. Its shape is the shape of a wave frontoriginating from a point source. In relation to a Huygensconstruction, the secondary wavelets will have the shape ofthe ray surface. The normal to the surface is the directionof the phase velocity. This fact is graphically indicated inFigure 4 with the other surfaces.

z| /RAY

G IFigure 4. Wave normal (phase velocity) surface, ray velocitysurface, and refractive index surface.

468

r

I

2

-a

D. Refractive index surface.The "refractive index surface" is a polar plot of n = ck/ci

against Ci If one draws a line from the origin to intersect therefractive index surface, the normal to the surface at the pointof intersection is the direction of the group velocity. Therelation of typical refractive index surface to ray and wavenormal surface is indicated in Figure 4. The relation of thegroup velocity IV9 and the refractive index surface is shown inmore detail in Figure 5. The direction wave normal 1k makes

11wg IK

n (0)Sn

I

Figure 5. Group velocity V9 normal to refractive index surfacen(of), tana = 6n/ni.

an angle at with the normal to the refractive index surface.From this figure, one can easily see the relation tana = -6n/6d,which is indicated in Eq. 4.

These various surfaces will be presented for many plasmamodes in Part 2.

111. FUNDAMENTAL RADIATION THEORYIN AN ISOTROPIC PLASMA

I1-1. Radiation Theory for Electron Plasma WavesFundamental equations for electron plasma waves are

given using a fluid model as follow;

-6t 0V e

\YVe eelE 2Vne- c p

DO t me e no(16)

V-IE = - ene/eo± JPex/o 9

where Pex/Eo is included as an external source term. Theseequations are the continuity and momentum equations for

electrons and the Poisson's equation. From these equations,one can obtain the following equation:

2 1 6 neeU e noe5Pexe c2 at2 c2 e m 2 9V ee e

meC eeo

(17)For a time dependence exp(-icot), this equation becomes

2 2 noe pexV ne +k n m= e

2

(18)

k2 2 2 2. e e

This is an inhomogeneous HelmJloltz equation. The solutionis given by

enoe (iwt ex( dV!)eikl,ne( ir)=m cdV 12-r~e

me 2Oc- Ir,meCe 0 Vt(19)

where the primed coordinate refers to the source. By solvingthis equation for an arbitrary charge distribution, one canobtain the spatial distribution of the electron density pertur-bation ne. For a point source, for example, ne becomes

ikR -iuLtne(R) = 0 e , (20)

meGoCe R

where R is the distance from the point source to an observa-tion point. This equation indicates a spherical electron plasmawave radiated from a point source. For other charge distri-butions, the approximate fields will be presented in Part 3"Radiation from finite sources" in this report.

111-2. Radiation Theory for Ion Plasma WavesFundamental equations for the ion plasma waves are

given by - +niBy +1 OVr. =-r O,

Z \Vi e IEO t mi

(2 1)

0 =- e E - CVn /n,e eo0

V.E = e ( ni ne)/e-o Pex/eo -

For an electrostatic approximation of E = -V6S and a time

dependence of exp(-icot), one obtains the following Helm-

469

holtz equation as a wave equation;

2 + k2I _=__ _ J'exv p ~~2 ,2

p£i 0

D e

2 2 2 2 (22)k =kDe > pi/uJ -1) .De pi

The similar solution to the case of electron plasma wavesis obtained as follows:

>d (2eJiJt rex( r)eikK rn'

4(Oi(rPi)2)LO hr - ir T

(23)where the primed coordinate refers to the source. For a pointsource, for example, the oscillating potential becomes

Wj2 ikR -iwtgf Or) = 4TEe0(W2. .-JuZ) Re(24)

This indicates a spherical ion plasma wave with the geometricaldamping factor of 1/R1.

111-3. General Radiation Theory in an Anisotropic PlasmaKuehl derived general radiation fields of electromagnetic

and electrostatic waves which are radiated from an arbitrarycurrent source. Fundamental equations for the kinetic modelare the Vlasov equation and Maxwell equations

+~ e

- iuf+\v.7 f +- IE-7 f = 0, (25)r me v °

VxVx E _W@2H0%IE = icIV( f++ ex)

IT =-ef \v d\v (26)p

where a time dependence is assumed as exp(-iwt). The Fouriertransform with respect to r of Eqs. 25 and 26 yields

- i wf (1k) +ilk\vf(1k)_ IE(lk)V,vf = 0

lkxlkx E (1k) +w U c-0 E (1k) (27)

= iw e f (lk) vd3v - icp J(k),

where Ik is the Fourier-transform variable and f(lk), IE(Ik),and liex(1k) are the transformed distribution function, electricfield and external current density, respectively. The Fourier-transform f(1k) of a function f(lr) is defined by

f (1k) = f (tr)exp(-ilk-ir) d ir . (28)

From these equations, one obtains

lkxlkxlE (1k) + 2IE (Ik) (29)

2pg e IE (1Ik) -( Vvfo) \vd \v _* Ie4k)=C w

= (o- lk.\v))

where c2 = -1.e0) ' By using the relation Vvfo = (lv/v)afo/av, 2

IkxIkxlE(Ik) + IE (1k)

{ XWPef V\foAvo v(A - lk.\v) dvjE(1k)

= -iUVPf ex(I'k) (30)

with an inverse Fourier-transform of Eq. 30, one can derivea formula of IE(1r) as follows:

IE(ir) =( 3{IE(Ik)exp(ilk.r) d Ik, (31)()2T

It-,(rk) =rJfk2 11 - .1" klkc2Ik!k=1JJ4~-~22IJfx(Ik)i

(32)

1 k2 z2 kVe

D =1- -J ipk2 c2 k3 kV2

This equation yields the general expression of the electricfield IE(lr) wich is radiated from a given external current

iex(lk). Both the electromagnetic component (Dt) and theelectrostatic component (D1) are included. For a point dipoleof |p = pl, the approximate electric field becomes

IE(ir) = IE t(ir) + IE (r) (33)

iktR-iwtIEt(I ) pe0( 1 -u w2) R

0pe

[(R R 2)

-(2 ikt 1 . jk(k+ R sinO 9jR

peiklR-iutE1( ir) = 1R

4Tl e0 ( 1 - W 2pe/( 2) R

k2+ 2ik 2 )cosE r

-( z )JsineJ

470

IV. ELECTROMAGNETIC RADIATION FROM ANELECTRIC DIPOLE IN A COLD

ANISOTROPIC PLASMA

By investigating electromagnetic radiation in a cold aniso-

tropic plasma, Kuehl found [101 an important behavior

that the fields radiated from the source become very largein certain directions for certain ranges of plasma parameters.

The fact is called the high frequency resonance cone. In this

section, his theoretical results are presented. The dyadic

Green's function is used for a derivation of the electromag-netic field of an electric dipole in a plasma. The electric field

IE satisfies the following equation from Maxwell equations;V x V x IE - J2poIKlIE = iw4olJex,

where lJex is the external current density and IK is the di-electric tensor, which will be given later in Eq. 55. Becauseof the linearity of Eq. 34, the general solution may be ex-pressed as

(35)

where IF(Ir, Ir') is the dyadic Green's function and the inte-gration is performed over the volume containing the sourcecurrents. From these two equations, one obtains

fv'[V x V x IU(1r, Ir') - 9ArK.IF(Ir, Ar') - icqLU5(Ar - Jr')]*IJ(Ar')d '=0, (36)

where 6 (Ir - Ir') is the three-dimensional Dirac function and

AU is the unit dyadic. By making the quantity in bracketszero, 2 2

(-V lU+7-U E; ) (m,m)

Poynting vector S = re(lExIH*)/2 are obtained as follow.

For an x-directed dipole,

IE = 4%kRPx Lsiine kRIE sincA i A2kR N

-K1I cosO coso 9 e

K-( COSeK-II sin20 )3/2JK1

2-k Px KLL sin ieAkR

4TIeOR PoK A~~~\iAY2kR -

+ 1i cos9 coso 0 el I

Kl(cos20 + Kll sin20 ) ;'~~~~-

(41)

(42)

- Ak4P(2) /2 [ sin2

K2 2 2

K2 ( co s2q +E1sin2q3) 5/2J (43)

%1~~c- l@)1/

K222(cs KII Si20 1/2Fo (Cosd+ fn sin

For a dipole in the z direction,

(37)

A Fourier transform with respect to Ar is carried out to yield

(k U - Ikik -uW IK ).Ij( 1k, jr')

-ilk-irt=wUXt e

,(38)3

where Pr(lk, Ir') is the transform of I1(jr, Ar'). Taking the

inverse transform, one obtains

_iHAi ik( -ir-) (39

where A1O is the inverse dyadic and IX(jk) = k2IU - Ikik - , o

x K. The field from an electric dipole at the origin can be

found from the dyadic Green's function through the relation

tE (ir) = - icojj(ir, 0).p (40)where Ip is the dipole moment. By using the formula, the

electric field IE, the magnetic field IH, and the time-averaging

2 i Y2'kRIE -k Pz Kiisin9 gee244

47TeoR KL(cos29+K!sin2 0)3/22 i kR

H=-k Pz Ki sinkRI 47Le0OR(K4Y/12(cos2o+Kisinl2e0)4)

4 2 2 2A~ k Pz K11sin 0

32TL2e R2K (K1Mo)'/2

I(o2 Kii .2 5/2 a

(CSG+-sln e)(46)

In both cases, time averaging Poynting vector is radial. Typical

radial Poynting vectors in a polar coordinate are indicated

in Figures 6 and 7.

471

IJE (Jr) = fV,JF(Jr, JO IJex(ir')dV ,

t' I ) ..L .1lkA US( ir - irto

fe/W=O

\ r0--.5crl90~so

0.80.9

00tz

1

0

Figure 6. A verage Poynting vector of an electromagneticwave radiated from a point dipole directed along the mag-

netic field (z). pe/w is the parameter (after [10]).

V. QUASI-STATIC RADIATION FROMA POINT SOURCE IN A WARM

ANISOTROPIC PLASMA

Kuehl derived [25] a fundamental formula for quasi-

static radiation from an oscillating point source in a warm

anisotropic plasma. This formula includes the near fieldof the electromagnetic modes and the field of electrostaticmodes wich are radiated from the source. Although the

180°Figure 7. Average Poynting vector of an electromagnetic wave

radiated from a point dipole directed perpendicular to the

magnetic field (x) (after [10]).

472

. . . . m i

original paper used an electron plasma, the generalized theory

which includes the electron and ion components will be

explained in this report. Furthermore, the application of

theory into typical cases is presented.

V-1. General Formula for Quasi-Static Radiation Field

The following Poisson-Vlasov equations for two compo-

nents (electrons and ions) plasmas are used as the fundamental

equations;

- +\v-Vvfj + m (E +\vxlB).Vf.=0, (47)

j= e, 1

V-E =e(n. -n )/ f (ir)/e- (48)1 e o0 ex o

where Pex(Ir) is the charge distribution, and the quasi-staticapproximation, namely, IE = - v6, will be used. From these

equations, the Fourier-transformed potential is obtained easily

as

ex( Ik)_ 2

e 0 Ik-IK kDIkIKlk

(49)

,f (ir) = 8(ir), so that P ( k)=1, (52)ex ex

In cylindrical coordinate (p, 4, z), the potential can be written

as follows:

-iwt

0( e)-e

4T[2e.0-iu)te

872 c-

(dkee1lz dkkZ

X I o DD(k2k)(53)

where we've used Ik.Ir = (kicos6, k1sin6, k11 ) (pcos4, psin4, z)= kipcos(6 -4) + kiz, dik = k1dk1dk Udd and f02wexp(ikipcos x

(d - )d = 2irJO(kip). From this equation, the followingpotential equation can be obtained.

~ ~ije w I) (-,M-N)xie~~k8 r nl;l 11 DI(k)

where IK is the tensor dielectric constant of the plasma. The where

term D(ki2, k11) is generally given by D(k2 k 1) = 0, Im k > OII Dkk,~~~~~~~~~~-kM1=0ITkI

D( k, kp = k2+4kD2 [1

+ e J EIn z)d(Z.) 3= °,n:~D4 3 03 n13(50)

2 2 2 2 2

--( W -nUW )/k,lVj , V2- 2 Tj/ ,Dj pj j 3C

2 __2 _ o ,(1)U._eB /m.pi mje- CJ o

where ln(X) and Z(a) are the modified Bessel function and the

plasma dispersion function, respectively. The potential 0 (Ir)is given-with the inverse Fourier transform of Eq. 49;

ei.t fex(ke itIk.iri t= 3(lk)2e dlk, (51)

(27T) 3e D (k2,1X

For the potential of an oscillating point charge located at

the origin,

Dt k a) D(k ,k1)1k = k (k)

I nimilV-2. Radiation Field in an Electron Plasma - High Frequency

Resonance ConeV-2- 1. Radiation in a cold electron plasma

For a cold electron plasma, the tensor dielectric constant

K is given by

K± iKh 0

K = E -iKh KL 0

%0 0 K1l

o 22

11 u22P 2ce

(55)

o K = 1 -pe

ui e CLceKh= p

(J(L2 ~-ClC2W (e2 - 2e )

Then, 2 2 2D( kL, kil) = Ik-lK-lk = kLKL+ k ilK

473

(54)

(56)

One can obtain the following solution for the potential field:

(ir) = C(P, Z) (57)-iuJt

e

=4TT@O( K2K 1/2(p°2/K +z2/K1 )1/2'This equation has a singularity along the cone defined by thevanishing of the denominator,

K11sin 0 + K1cos 0 = 0,

surface for both cases of K11K < 0 and K11K > 0 is indicatedin Figures 9. For K11KI < 0, in which the resonance cone

exists, the potential field is shown to be directed in certaindirections. For K11K > 0, in which the resonance cone doesnot exist, the potential field is directed in all directions.The resonance cone exists for the case of the localized field.

(d)

(58)

= tan ?,/z,.This cone angle is called the "high frequency resonance cone";for simplicity, the resonance cone. The cones exist only infrequency regions for K K < 0, namely, in Min. (o co,

(c2 + (Ai )IAP> >> X > 0 (lower branch) and Cuh pe ceMax. (wpe "ce) (upper branch). The lower branch is calledthe whistler resonance cone and the upper branch is theZ-mode resonance cone. The resonance cone angle is expressedin another form;

2 Cov2((Jpe±.+U2eJ2)SI n 2 2 ~~~~~(59)

pe c e

This cone angle is shown in Figure 8 for typical plasma

K11'K~~O 4(&

~ 0

(b) K K >0I'I'

ZN

11

I

Figure 9. Typical wave fronts radiated from a point source

for two cases of KK11 § 0. Relation of the resonance coneto wavelength is shown.

V-2-2. Radiation in a warm electron plasmaThe effects of the electron thermal motion have been

indicated clearly by Singh and Gould [19].When one uses the asymptotic expansion of the Hankel

function with the large argument for the far field and the

saddle point method, equation 54 in V-1 can be written into

(R,)4= eer j (-1) 290 4ITc-0r~LR m (K±(kim))1/'2

Figure 8. Resonance cone angle °c versus

values Co /cp c

clYc for various

parameter values. In order to clarify the relation of the reso-

nance cone angle to the potential field, the typical potential

eiRQ(hlmj 9)DI(k1jm) (sin8G I Q( )j)1/2

(60)

474

20

Cl1 5

C10

5

0 3009c

60°

A

Q(k e)-kL(k)sn+ kcos9

dQ dkLQf _ dQ =_sinO + cosOdk11 dk1l

d2Q d2 k I(11Q, d Q= d- _sinr1 , n=|Q dk2- dk2 0 1>

This equation can be rewritten with the use of ki = ksin6 andk11 = kcos6Sas follows (see Part 2):

e-iWt sin /t 2]TroOrLR m sinO

2 (-1)niT -iRQ(k, 9)ike 2et Q1)1/2(k)1/2 D'

Q(k,G)=kcos(O-#) , k=km

Q " -2 c o s (qg- z)+2 Xs in (9g->d) - kc o s ( @- ).

Singh and Gould investigated the effect of the electronthermal motion in a uniaxial plasma (c+ o°°), in which thetensor dielectric tensor is given by

(61)

9 3k 'I I

.04 .02 0

(1 0 0

IK = EO[0 I1 0 t

%O O Kiii

(62)

K = 51 U pe

2__ ))en~ll~kZ V2 k ve

The polar plots of the phase velocity are indicated in Figures10 and 11 for both cases of < wpe and w > COpe* The

bottom figure in Figure 10 is the extended one of the upper

figure around the original point. For C < wpe the group

velocity is double valued for 0 < OC The double value of the

group velocity arises from the face that two phase velocities(a slow thermal mode and a fast electromagnetic mode)pointing in two directions can have their correspondingvelocities pointing in one and the same direction. For co>

wpe, the new cone which cannot

plasma is indicated. This results

mode, namely, the Landau mode.

appear in a cold electron

from an electron plasma

V-3. Radiation Field in a Two Component (Electron and Ion)

Plasma Low Frequency Resonance ConeV-3- 1. Radiation in a cold plasmaWhen one considers cold electrons and ions, K1 and

K for the electron plasma of V-2-1 can be rewritten as

KL= 1pe pi

2 2 2 _ 2i

~2 2

K11= 1 Pe2 2

(63)

In this case, the resonance cone angle satisfies the same for-mula as that of the electron plasma,

2K1sin + K11Cos 0 = 0 (64)

475

.02 .04Figure 10. Electromagnetic phase (Va) and group (Vi) velo-city surfaces for Cw < Wpe including the effects of electronthermal motion. Bottom figure is the extended figure of theupper figure (after [19]).

/CAN

C/Ve=300(A)/(p,=

.002 .001 0 .001 .002Figure 11. Phase (V ) and group (Vf) velocity surfaces ofpelectron plasma mode for w> cop (after [19]).

The frequency range for the lower branch becomes min.(Copei Cce) > a > CLh (the lower hybrid frequency). Thereexists a lower limit at the lower hybrid frequency for theresonance cone. This cone belongs essentially to the highfrequency resonance cone.

V-3-2. Radiation in a warm plasmaThere exists an electrostatic ion cyclotron wave and an

ion wave near the ion cyclotron frequency. With respect to

these modes, the new resonance cone can be derived. Thiscone is called the "low frequency resonance cone." The term

D(k2, k11) = Ik.IK.Ik for these modes in a fluid model is givenby

2 2 2 wpi -k- LOPik2D("k lkk +k 2- k_k__I-De (02 02. 2(

(65)

where k2De is the Debye wave number and indicates the effect

of the electron thermal motion. With this term, the potentialradiated from an oscillating point source can be obtainedas follows for the frequency range of C< Coci:

2 ~21

-iuit eikD e z-l 1{ (R, 0) = e -e IJe\1K1'41Teo(--Kij)1/2 K1B0

10

z2 32 1 /2( z2

it1 K)(66)

2 2~~~for O0(p /Kj<z /(-K11)

-ith3t ek z2K /je(R,0) e e "kIe _SL 111

411ieo ( -K1) 1/2 K1

1+2 ?2)1/2+ KHl _L

for 0 z2/(K )For the frequency range w> wci,

(67)

< r 7K1.

+(R, 0) = the same equation as (66),

In equations 66-68, K11 and K1 are given by

2.(QU2 (69)CO2 2

K = 1 _pi

w -wciIt must be noted that these definitions are different fromthose of Eq. 55. For the case of C < Cci, the potential be-came singular at p2 /KI + z2 /K11 = 0, namely, at

2 (02 2 ~2 2sirn0 = (C-5pi +U3ci -COcXpi U ci

(70)

This angle corresponds to the "low frequency resonance

cone." This equation has the same form as that of the highfrequency resonance cone (Eq. 59). The theoretical cone

curve in Figure 8 can also be used for the low frequencyresonance cone for coc -+ Coci and Cup - cop.i It must be

noted, however, that this low frequency resonance cone

was derived for C2 < cp .. The typical equi-potential surfaces

of those low frequency waves radiated from a point source

are indicated in Figure 12. Those are very similar to that of

the high frequency resonance cone. As known in the above

476

11

I

Figure 12. Typical wave fronts of electrostatic low frequencywaves radiated from a point source for low frequency reso-

nance cone (after [42] -[44]4).

discussion, the high and low frequency resonance cones

may be thought to be a fundamental and important pheno-mena in an anisotropic plasma.

REFERENCES

1 F.V. Bunkin, Sov. Phys. JETP 5, 277 (1957).2 A. Hessel and J. Shmoys, Symp. on electromagnetics

and fluid dynamics of gaseous plasma, Polytech. Inst.of Brooklyn (1961).

3 M. Cohen, Phys. Rev. 123, 711 (1961); 126, 389, 398(1962).

4 H.H. Kuehl, Phys. Fluids 6, 1465 (1963).5 H.H. Kuehl, J. Math. Phys. 43, 218 (1964).6 K.M. Chen,Proc. IEE 111, 1668 (1964).7 S.R. Seshadri, IEEE Trans. AP-13, 819 (1965).8 H. Kogelnik, J. Res. NBS 64D, 515 (1960).9 M.J. Lighthil, Trans. Roy. Soc. London, Ser. A252,

397 (1960).10 H.H. Kuehl, Phys. Fluids 5, 1095 (1962).11 E. Abel and L.B. Felsen, Proc. Symp. on Electromagnetic

Theory, Pergamon Press, Inc., New York, 1963, p. 391.12 C.P. Wu, IEEE Trans. AP-l1, 681 (1963).13 H. Motz, Radio Sci. 69D, 671 (1965).14 R. Mittra and G.L. Duff, Radio Sci. 69D, 681 (1965).

15 G.F. Freire and R.R. Scarabucci, Int. J. Electr. 22, 65(1967).

16 M.A. Snyder and H. Weitzner, Radio Sci. 3, 943 (1968).17 D.D. GiaRusso and J.E. Bergson, Radio Sci. 5, 745

(1969).18 T.N.C. Wang and T.F. Bell, J. Geophys. Res. 77, 1174

(1972).19 N. Singh and R.W. Gould, Radio Sci. 6, 1151 (1971).20 G.A. Deshamps and O.B. Kesler, Radio Sci. 2, 757 (1967).21 H.C. Chen, J. Appl. Phys. 40,4068 (1969).22 J. Tunaley and R. Grard, Ann. Geophys. 25, 55 (1969).23 N. Singh and R.W. Gould, Phys. Fluids 16, 75 (1973).24 R.K. Fisher and R.W. Gould,Phys. Fluids 14, 857 (1971).25 H. H. Kuehl, Phys. Fluids 16, 1311 (1973).26 H.H. Kuehl, Phys. Fluids 17, 1275 (1974).27 J.M. Chasseriaux, J. Plasma Phys. 11, 225 (1974).28 J.M. Chasseriaux, Phys. Fluids 18, 866 (1975).29 M.D. Simonutti, Phys. Fluids 19, 608 (1976).30 P.M. Belan, Phys. Fluids 20, 649 (1977).31 C. L. G rabbe, Phys. Fluids 20, 599 (1977).32 T.H. Stix, Phys. Rev. Let. 15, 878 (1965).33 R.J. Briggs and R.R. Parker, Phys. Rev. Let. 29, 852

(1972).34 P. Belan and M. Porkolab, Phys. Fluids 17, 1592 (1974).35 H.H. Kuehl and K.K. Ko, Phys. Fluids 18, 1816 (1975).36 H.H. Kuehl,Phys. Fluids 19,1972 (1976).37 P.L. Colestock and W.B. Getty, Phys. Fluids 19, 1229

(1976).38 T. Ohnuma et al., Phys. Rev. A16, 387 (1977).39 H.H. Kuehl, Phys. Fluids 17, 1636 (1974).40 K. H. Burrel, Phys. Fluids 18, 897 (1975).41 T. Ohnuma et al., Phys. Rev. Let. 37, 206 (1976).42 T. Ohnuma et al., Phys. Rev. A 15, 392 (1977).43 P. Belan, Phys. Rev. Let. 37, 903 (1976).44 T. Ohnuma, "Linear wave phenomena in plasmas," in

Textbook of the Plasma Summer School, I nst. of PlasmaPhys., Japan, Aug. 1976.

FURTHER REFERENCES

1 T.H. Stix, The Theory of Plasma Waves, McGraw-Hill,1962.

2 W.P. Allis et al., Waves in Anisotropic Plasmas, MITPress, 1963.

3 K.G. Budden, Radio Waves in the Ionosphere, CambridgeUniversity Press, Carmbridge, England, 1953.

4 L.R.O. Storey, Phil. Trans. Roy. Soc. A246, 113 (1953).

477

(b) W>(.Uci

5 a s i| | al a| I