IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-9, NO. 1, MARCH 1981

D. E. Baldwin and J. L. Hirshfield, Appl. Phys. Lett., vol. 11, p.

175, 1967.S. Kojima, S. Hagiwara, and H. Ogihara, J. Phys. Soc. Jap., vol.20, p. 851, 1965.J. C. Nickel, J. V. Parker, and R. W. Gould, Bull. Am. Phys. Soc.,vol. 10, p. 211, 1965.J. V. Parker, Ph.D. dissertation, California Institute of Technology,Pasadena, 1964.A. E. Aubert, A. M. Messiaen, and P. E. Vandenplas, in 1971 Proc.4th Internat. Conf Int. Atomic Energy Agency, (Madison, WI),vol. 3, p. 513.

[16] A. W. Trivelpiece, "Slow-wave propagating in plasma waveguides,"

San Francisco Press, San Francisco, CA, 1967, p. 69.

[17] M. Camus, Annales Telecommun., vol. 24, pp. 309-362, 1969.

[181 L. P. Mix, L. N. Litzenberger, and G. Bekefi, Phys. Fluids, vol.

15, pp. 2020-2026, 1972.[19] A. M. Messiaen and P. E. Vandenplas, Plasma Phys., vol. 15, pp.

505-533, 1973.[20] J. W. Willis and J. C. Nickel, Phys. Fluids, vol. 17, pp. 214-221,

1974.[211 P. Leprince, Plasma Phys., vol. 14, pp. 523-541, 1972.

Radiation Impedances of Disc-Shaped Antennas foran Electron Plasma Wave

SHIGEO OHNUKI, MEMBER, IEEE, AND SABURO ADACHI, SENIOR MEMBER, IEEE

Abstract-The radiation impedances of disc and circular mesh anten-nas in the high-frequency region were measured. As a result, it is shownthat measured impedances in the frequency region of the electronplasma-wave propagation agree with the theoretically predicted ones,

which are obtained by solving a boundary-value problem.

HE IMPEDANCES of antennas such as wire, sphere, disc,parallel-plate and so on in plasma have been studied theo-

retically or experimentally in many different ways. The imped-ance properties of various antennas have been reviewed [1] .

The impedance characteristics of antennas are important forquantitative measurements of wave potential, and for efficientexcitation of various waves into plasmas. As for disc antennas,we have reported the theoretical formulas for the radiationimpedance of an oblate spheroidal antenna which are obtainedby analyzing the antenna as a boundary-value problem [2].In this paper we report the experimental results of the imped-ances of the disc and circular mesh antennas for an electronplasma wave, and compare them with the theoretical results.The experiments were carried out by using the space chamber.

The experimental setup and block diagram of measurements

are shown in Fig. 1. Two kinds of antenna are used as experi-mental antenna. One is the circular stainless plate (10 cm in

diameter and 0.5 mm in thickness), and the other is the circular

stainless mesh (10 cm in diameter) having 24 lines (0.28 mm in

diameter) per inch. Both plate and mesh antennas were con-

sidered because they may have different properties in the bias

potential region which is near or higher than the plasma poten-tial. While they may be considered to behave as the same be-

Manuscript received June 16, 1980; revised October 29, 1980.

The authors are with the Department of Electrical Engineering,Tohoku University, Sendai 980, Japan.

4m r

E

(N

I

RF Impedance X-YAnalyzer Recorder

Fig. 1. Experimental setup.

cause of the ion sheath surrounding the antennas in the biaspotential region less than the plasma potential. A back diffu-sion type of plasma source is used with argon as a discharge gas

at a neutral pressure (7 - 9) X 10-5 torr. The plasma was quietand satisfactorily homogeneous [3]. The typical plasmaparameters are (2 - 3) X 105 cm-3 in electron density andTe - 3000 K in electron temperature, which were measuredwith a Langmuir probe. In this experiment, a set of square

Helmholtz type of coils (1.6 m X 1.6 m), which is not shownin Fig. 1, was set to compensate the earth's residual magneticfield inside the chamber.To measure the impedance as a function of frequency, we

used an RF impedance analyzer with a sweepable dc antenna

bias voltage of the range of ± 40 V. The disc or circular meshantenna under experiments is connected through a 5042

coaxial cable to the impedance analyzer as shown in Fig. 1.

The impedance is measured after calibrating the analyzer usingthe short, open, and 50-Q standards. An electrical length of

coaxial cable between the analyzer and the antenna under

experiment is automatically compensated by the coaxial cable

which is extended from the reference plane of the analyzer.

0018-9383/81/0100-0016 $00.75 1981 IEEE

[ill

[121

[131

[14]

[151

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OHNUKI AND ADACHI: RADIATION IMPEDANCES OF DISC-SHAPED ANTENNAS

2 3 4 5 6 7

z20 f_eese

v2 3 4 5 6 7f (MHz)

Fig. 2. Measured impedances of the circular mesh antenna in a plasma(upper data) and in a free space (lower data).

|1l 2av10cm Mesh2a 10cm Disk V.-3.6V

10~V, _3.8V

OD f ,e45MHz10poT,-3000K 10h h T,v30T00=KK

x):Measured \esue0 r:gid 3g6) e \3

cold e °ss \ -:o\-:absorptive ° \

10 15 wlw0 2.0 10 1.5 wI 20

Fig. 3. Normalized impedances (o: normalized experimental resistanceand x: normalized experimental reactance) of the disc and circularmesh antennas when the dc bias potential Vb are fixed to the spacepotential of plasmas. The solid anld dot-dashed curves are the nor-malized theoretical impedance of the disc with rigid and completelyabsorptive surfaces, respectively. The broken lines indicate the coldplasma approximation.

Fig. 2 shows the raw data for the impedance of the circularmesh measured with the impedance analyzer mentioned above.The upper figure is the result measured in plasma, and the lowerfigure is the measured in free space. The dc potential of themesh antenna is kept to 3.6 V, the space potential of plasmaby the application of the dc bias potential '7b in order to elimi-nate the ion sheath effect. The very clear antiresonance of theantenna is observed at the plasma frequency of 4.5 MHz. Theradiation impedances of the circular mesh shown in Fig. 2 andthe disc are shown in Fig. 3. In these figures the disc antennais also kept to the space potential of plasma: 3.8 V, which is alittle different from 3.6 V for the case of the mesh, because theplasma used for both experiments was different. The measuredimpedances of the disc and mesh antennas are normalized bythe free-space impedance, that is, the reactance measured infree space. The solid and dot-dashed curves indicate thetheoretical impedances of the disc obtained for the rigid andcompletely absorptive boundary conditions, respectively. InFig. 3, sis the normalized radius of the disc, = a/(V XD)where XD iS the Debye length.

The radiation impedance of the disc (or circular plate) forelectron plasma waves [2] is expressed byZ = [ 1/j47reOa(Xo2/o - 1)]

- {0. E a dOP(-jkpa){R(4) (-jkpa, jO0)

P=o

- j(aukaa/co)R() (-jkpa,j)}] (1)

where ap is determined by solving the following infinitesimultaneous linear equations

00

aado'P (7jkpa)[{1 +j(oauk'a1w') (Qo(iO)IQ'(jO))}p=o

* R(4)(-jkpa, j0) - {X2 Qo((jo)/I2 QO (iO)}* R(4) 1(-jk0a,jO)] = 1, n =0

00

Z apd°P(-1kpa) [{l + j(ctciukpa/(2) (Q n(j0)/Qn(f0))}p=O

*R(4)(-jkpa,j0) - {X2 Qn(j)/c2 Qn(i°)}*R(4) (-jkpa,jO)] = O, n = 1, 2,3 ... (2)

and kp is the propagation constant of the electron plasmawave expressed by

kp= (Wp/U)(w2/( - 1)1/2, U = f3kTe/m. (3)

In (1) and (2), a and wp are the radius of disc and electronangular plasma frequency, respectively. Rop and doP repre-sent the oblate spheroidal radial function of the zeroth order,pth degree, and the expansion coefficients of the angular func-tions, respectively, after the notation of Flammer [4]. Qn isthe second kind of Legendre function. The notation ai is theabsorptive coefficient [5] on the surface of the spheroid. Inparticular, oa = 0 and F2/V3Tr stand for the rigid and the com-

pletely absorptive boundary conditions, respectively. Theexpression of impedance for the disc in (1) is applicable onlyin the frequency region of the electron plasma-wave propaga-tion X > wp. The broken lines indicate the theoretical react-ance of the disc in a cold plasma. We find that the measuredimpedances of the disc and the circular mesh agree fairly wellwith the curves calculated from (1) for the disc antenna. How-ever, the measured radiation resistance of the disc is smallerthan that predicted by (1) and also than that of the mesh.One reason of this is considered as follows. The mesh antennais considered to behave more or less as a transparent grid forthe plasma at the space potential. On the other hand, the discantenna rather easily disturbs the plasma surrounding theantenna when the radius of the disc normalized by Debyelength a/XD is relatively large. Namely, the disc antenna can-not be considered as being immersed in a homogeneous plasma.We have measured the variation of the resistance of the cir-

cular mesh in the frequency region of the electron plasma-wave propagation near the plasma frequency versus the biasvoltage of the antenna. The results are shown in Fig. 4. Themeasured electron current versus the bias voltage is also shownby x in Fig. 4, where Vf and V. indicate the floating and the

17

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-9, NO. 1, MARCH 1981

~10c

0

1-

4)

E

z

2in

100

_t-

100

OUL

Fig. 4. Normalized resistance which is measured in the frequencyregion of the electron plasma-wave propagation and measured elec-tron current of the circular mesh antenna versus the dc bias voltage.Vf and Vs indicate the floating and space potential of plasma.

space potential, respectively. It is very interesting to find theclear peaks of the radiation resistance at the bias voltage whichis equal to the space potential. The phenomenon is in accor-dance with the observed enhancement of the excitation effi-ciency of the electron plasma wave at the space potential [3].

ACKNOWLEDGMENTThe authors wish to thank Prof. T. Takahashi for his generous

help in their experiments using the space chamber of theResearch Institute of Electrical Communication, TohokuUniversity.

REFERENCES

[1][2]

[31

[4]

[5]

K. G. Balmain,Ann. Telecommun., vol. 35, p. 273, 1979.S. Ohnuki, S. Adachi, and T. Ohnuma, J. Appl. Phys., vol. 49,p. 138, 1978.S. Ohnuki, T. Ohnuma, and S. Adachi, Int. J. Electron., vol.39, p. 385, 1975.C. Flammer, Spheroidal Wave Functions. Stanford, CA: StanfordUniv. Press, 1957.K. G. Balmain, Radio Sci., vol. 1, p. 1, 1966.

A New Numerical Method for Asymmetrical Abelinversion

YUICHI YASUTOMO, KATSUYUKI MIYATA, SHUN-ICHI HIMENO, TAKEAKI ENOTO, MEMBER, IEEE,AND YASUTOMO OZAWA

Abstract-A new asymmetrical Abel inversion is presented for applica-tion in asymmetrical data which is obtained from plasma diagnostics(for instance, interferometry or spectroscopy) on the toroidal plasmasystem, and in which asymmetry exists normal to the direction ofobservation.

In this new numerical method a weight function, which was ob-tained by separating the integrated quantity into odd and even parts,was used for determination of the asymmetrical local value.To demonstrate the usefulness of this new method, we set up a hypo-

thetical data set, which resulted in a valid local value.

I. INTRODUCTIONIN PLASMA DIAGNOSTICS it is usually desirable to ana-

lyze the spatial distribution of plasma density and/or radia-tion energy. For these measurements experimental techniquesof interferometry or spectroscopy are often employed; how-ever, the directly obtained data is not a local value but an

integrated one along the optical path. Therefore, to obtain

Manuscript received June 9, 1980; revised October 7, 1980.The authors are with the Department of Nuclear Engineering, Faculty

of Engineering, Hokkaido University, Sapporo, 060 Japan.

the spatial distribution, it is necessary to solve the integralequation

(1)-R -r

I(y) = n(r) dx.--,f 2r

If the plasma is assumed to be axial symmetric, (1) may bereduced to a one-dimensional problem

R n(r) r dr(1')

Equation (l') is referred to as the Abel integral equation.The z axis is in the center of the cylinder normal to the paper,the x axis is in the direction of the light beams, and the y axisis perpendicular to the x axis, as shown in Fig. 1. Variablesr and y represent the distance from the z axis to the measuredpoint, and the perpendicular distance from the z axis to theoptical path, respectively.

It is generally difficult to solve (1); however, if the physicalquantity is symmetric with respect to the z axis, solution of(1') becomes the well-known Abel inversion

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