IEEE Ttanaaction6 on Pa6ma Science, Vot.PeS-4, No,1, Pkatch 1976

SCATTERING FROM A BOUNDED PLASMA

PRAKASH BHARTIAFACULTY OF ENGINEERINGUNIVERSITY OF REGINA

REGINA, CANADA S4S OA2

Received September 24, 1975

ABSTRACT

The electromagnetic scattering from a boundedplasma column in a rectangular waveguide is invest-igated. Using image and boundary value techniques,equations are formulated for reflection, trans-mission and attenuation coefficients. Numericalresults obtained from computations are in good agree-ment with results obtained using alternative form-ulations elsewhere.

1. INTRODUCTION

Electromagnetic wave scattering from a cylind-rical plasma column in a waveguide has receivedconsiderable attention (1-3). More recently Bryantand Franklin (4) have given an approximate solufionfor the problem while Nielsen (5) has formulated anexact solution using model expansion of fields andpoint by point matching of fields and boundaryconditions. However, while the former solution isapproximate and the latter requires considerablecomputer storage they are also restricted to theplasma column being centred in the waveguide.

In this work, Okamoto et al's (6) solution for aferrimagnetic cylinder in a waveguide is extended tothe case of a plasma column in a glass tube. Thissolution uses an extension of Lewin's (7) and Nik-olskii's (8) solution of replacing the plasma columnby multipole current sources and their images withrespect to the guide walls and application of theproper boundary conditions to obtain the scatteredfields. The reflection, transmission and attenuationcoefficients for the plasma column are thus easilycomputed. This formulation requires less computerstorage than Nielsen's solution (5), and results canbe computed easily for the column being located atany position in the waveguide.

2. THEORY

Consider a dielectric tube of inner radius r ,outer radius r2 and dielectric constant C placed ina rectangular waveguide at a distance x0rom theguide wall as shown in Fig. 1. With the tube filledwith a plasma, the complex propagation constant there-in is given by

k = _ TC _ \F (1)

where a is the angular frequency and c is thevelocity of light, and the plasma permittivity £is given by r

pP2

£ = £ 1 -p +

[- (w +iv) Ji

(2)

In eq. (2), is the plasma frequency and v

is the electron coYlision frequency of the plasma.v = 0 therefore denotes a cold lossless plasma, whereas,

v > 0 results in. the general case of a lossy plasma.

The electric field of the propagating dominantmode in the waveguide may be written as

69

'I 7X~e-j ~ZE = Sin - e

a

2 2where S k - (22)

Introducing athe tube centre asexpressed in terms

0n

Ei= E

m=-0

k = 21ro - (3)

cylindrical coordinate system withorigin, the field in (3) may beof cylindrical Bessel functions as

J (k r)e i sin (ma + rx )m o e

a(4)

where a = tan I(X/Sa)

Similarly, the scattered fields in the plasma,dielectric tube and region outside the tube may berespectively written as

EP =E

t ~E - e

m=-O

Es=Em=-0

B J (k r) e-jmim m p (5)

[C (k r) (k r)eJm4(6)m m t m m tJ

A H (k r) e-jmim m o (7)

where kt = w and A, B, Cm, D are fieldt c '~[2 n m m n

coefficients. Hence it remains to determine the co-efficients and the relationship between them to obtaina solution.

An elegant method, of evaluating the scatteredfield Es [6 of equation (7), which will lead to thedetermination of the reflection, transmission andabsorption properties of the plasma column, is toreplace the latter with multipole sources located atthe centre of the tube and their images with respectto the guide walls. These images are located atx = 2an + x Z = 0, (n = 1, 2 . . . ) and

Xn = 2an - x,0 zn = 0 (n =O, +1, 2 . . . ) respect-

ively as shown in Fig. 2. Further, if r and rn n

represents the distance oT a source from the observ-ation point P (x,z) and 4) and 4) the angle with thex axis, then equation (7)nmay benexpressed as

Es= Zm A

(8)

[HL (k r )e jm4)s - (-1) H 2) (k r )e m4s ]

Using the addition theorem for the Hankel func-tions, the scqtteVed field Es may be rewritten interms of (kr,r ) to obtain

I(2) + -jm~+H( (kr0)e 0om 00o

(-l)m + (-l)P}cn co

n=l p=2ac cK

) H(2) ( 1H+P (+) (2) -i'n)n

n=-w p-_

(-l)P Jp (k0ro) H%(p (Q)en)e (9)

where e, = 2k0a In| n = ± 1, ± 2, . . .

Qn = 2k Ian-x0I n = 0, + 1, + 2,

Satisfying continuity of field relationships atthe plasma-tube and air-tube interface, we arrive atan infinite set of equations for the coefficient Aand the relationship between the other field co-efficients. Thus we obtain

C ktrlBm [ktrlJm (kpr) Ym (ktr1) -

results in the coefficients A . To evaluate thereflection, transmission and ibsorp;ion properties ofthe plasma column, E5 in equation (9) must be expressedin terms of waveguide modes, resulting in

A (-l)m 4-

n=l

-jknze jknz sin (nTrxo

k an

- m sin nit ) sin nrx z > 0k a a0

00

and

m=-

co

e-j kn4z I sin (nTrxo +k a

n=l n

m sin 1 n7) sin nrx z < 0o a

where k2 [2 _ 21 1n = L -o("")J (12)

and hence the reflection and transmission coefficientsare given by

kpr1Jm (kpr1)Ym(k tr1)

irk r1BD = -

2 kprJm (kpr1) Jm (ktr)

ktr 2lm(kpr) m ( trt)

(10)m 4A

k asin ( o + m sin1

a

where the primes denote derivatives with respect to theargument, and using these we obtain

T = 1 +- k a sin

m=-C I

(TrX0Tm sin k-) (14)

0

A = C sin (- + ma) +m m a

N

A 6i_n nmn=-N

3. NUMERICAL RESULTS

(11)

m = 0, ± 1, ± 2, . . .

where

B VJ (v) - A uJ (v)m m m m

~~m - (2) -(2)-' (v)uA H (v) -VB Hm m m m

[ tJm(W)Ty (t) wJn (W2)Ym(t)]

+[wf(w)Jm(t) tJm(w)Jm (t) ]

B- [t (w)TY(t) - wJ (w) Y(t)]

+ m(w) Jm(t) tJm(w) Jm(t)]

ao

mm E H(2 (is)s=l

J (u)im(Y (u)

J (u)m

Y (u)m

{( l) m+n

+H(n) (sZ)-(-l) H(2) (Qs)Hm 2n m+n

s=1 s=-X0

and ktr2 ktrl, =kprTruncation of the series of equations in (11)

and subsequent solution of the simultaneous equations

Using the final system of equations for thecoefficients A , a computer program was developed.Input parameters included waveguide and tube dimens-ions, electron collision frequency, plasma frequency,operating frequency and dielectric properties of thetube. The number of terms required for convergence

depend in general on the physical parameters above.For results presented here 10 terms were used.

Fig. 3 presents the results of computation fortwo particular cases of a lossless homogeneous plasma.The computed results are compared with those ofNielsen (Figure 4 of Ref. 5), obtained by an alter-native technique. They compare favourably, thusconfirming the validity and usefulness of the form-ulation. Further, this formulation has the advantageover Nielsen's solution (5) in that it is general forany location of the plasma column in the waveguideand is less complicated.

4. CONCLUSIONS

A general solution for a plasma column locatedat an arbitrary position in the E plane of a wave-

guide has been formulated. The solution is notlimited by the size of the column or its physicalcharacteristics, and has the advantage over previousformulations in that the tube need not be centred inthe guide. Finally, this formulation also requiresless computation time due to its simplicity.

70

ao

m=

+ iPJ (k0r0)e 0p 00o

0

(13)

co

sE Ar..a

00

r =

=--00

5. ACKNOWLEDGEMENTS

This research was supported by the NationalResearch Council of Canada (Grant A8863). Theauthor is grateful to Ms. A. McGillivray for typingthe manuscript.

6. REFERENCES

1. D. Rommel, "Radio Reflections from a Column ofIonized Gas", Nature (London), Vol. 167, 1951,pp. 243.

2. J.H. Battocletti and W.D. Hershberger, "Resonancesin the Positive Column of a Low-Pressure arcDischarge,", J. App. Phys., Vol. 33, 1962,pp. 2618 - 2624.

3. W.D. Hershberger, "Absorption and ReflectionSpectrum of a Plasma,", J. Appl. Phys., Vol. 31,1960, pp. 417 - 422.

4. G.H. Bryant and R.N. Franklin, "Scattering froma Bounded Plasma,", Proc. Phys..Soc., Vol. 81,1963, pp. 531 - 543.

5. E.D. Nielsen, "Scattering by a Cylindrical Postof Complex Permittivity in a Waveguide,", I.E.E.E.Trans. Microwave Theory and Techniques, Vol.MTT-17, 1969, pp. 148 - 153.

6. N. Okamoto, I. Nishioka and Y. Nakanishi,"Scattering by a Ferrimagnetic Circular Cylinderin a Rectangular Waveguide," I.E.E.E. Trans.Microwave Theory and Techniques, Vol. MST-21,1971, pp. 521 - 527.

7. L. Lewin, Advanced Theory of Waveguides. London:Iliafe, 1951.

TEST TUBE \

'Ei4

X-a

8. V.V. Nikolskii, "A Transverse Ferrite Rod in aRectangular Waveguide," Radiotech. Elektron.,Vol. 3, 1958, pp. 826 - 828.

x = 2a +XXI o 'i

xl = 2a-XO

e;r.

7S~~~~~~~;-+x2a+XO

x:=2a-XO -

-I 0~~~~~~~~~~~~~~~

IlD

z

Fig. 2 Location of Image Sources.

r,1 3.95mm E Q a 4.82

r. 5.5 mm WG 10 WAVEGUIDE

NIELSEN

* MULTIPOLE-IMAGE METHOD

3.0

f ( GHz ) - .

Fig. 1 Test Tube Contained Plasma Column in Wave-guide.

Fig. 3 Reflection Coefficient vs. Frequency for aCold Lossless Plasma.

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