ABSTRACT

ELECTROMAGNETIC SCATTERING FROM A
PLASMAPCOATED CYLINDER

BY

Chen Yi Lee

The present study deals with the problem of electromagnetic
scattering from a plasma-coated object. An infinite cylinder of a
finite radius is covered by a layer of inhomogeneous, lossy and hot
Plasma. This plasma-coated cylinder is assumed to be illuminated by
a plane wave with either TE or TH polarization. when the temperature
effect of the plasma is considered, an electroacoustic wave in
addition to the electromagnetic wave is excited in the plasma layer.
The effects of this electroacoustic wave on the electromagnetic
scattering are studied. It is found that if the plasma-coated
cylinder is illuminated by a TM plane wave, no electroacoustic wave
can be excited in the plasma layer. To handle the wave propagation
in the inhomogeneous plasma medium, the stratification method is
applied.

In the analysis, the dipolar, quadrupolar and temperature
resonances have been feund to exist in the plasma layer. The effect
0f various parameters on the electromagnetic scattering are also

ItUdiCde

An experiment was conducted to verify the theory.

ELECTROMAGNETIC SCATTERING FROM A
PLASMA-COATED CYLINDER

BY

Chen Yi Lee

A THESlS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1971

To my parents

“to 8 "1.80 Sun RI. Lee

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation to his
major professor, Dr. K. M. Chen, for his guidance, counsel and
encouragement throughout the course of this investigation.

He also wishes to thank the committee member, Dr. B. Ho, for
his valuable suggestions and help in the experimental part of this
research, and to the other members, Drs. D. P. Nyquist, J. Asmussen
Jr. and W. Denenson, for their time and interest in this research.

Finally, the author especially wishes to thank his wife,
Hsin-huei, for her typing and proofreading the manuscript, as well
as for her encouragement throughout the major part of his graduate
study.

The research reported in this thesis was supported by the

National Science Foundation under Grant GK-2952.

iii

1.

2.

3.

TABLE OF CONTENTS

Page
ACKNOHLEDGMENTS . . . . . . . . . . . . . . . . i i i
LIST OF FIGURES . e . . . . . . . . . . e . . . vi
INTRODUCTION . e . . . . . . . . e . . . . . . 1

SCATTERING FROM A METALLIC CYLINDER SURROUNDED BY A LAYER OF
LOSSY, COLD PLASMA ILLUMINATED BY A TE HAVE e e . e e e 4

Zel Introduction e e e e e e e e e e e e e e e a
2.2 GCOUQETY Of the Problem e e e e e e e e e e e 5
2.3 Fields in Free Space Region . . . . . . . . . . 9
Zea Fields in 61388 "811 Region e e e e e e e e e e 12
2.5 Fields in Plasma Region . . . . . . . . . . . 12
2.6 Matching the Boundary Conditions at Interfaces . . . . 14
2.7 Scattered Field in Free Space Region . . . . . . . 21
2e8 SODQ SPOCIRI C8828 e e e e e e e e e e e e e 23

2.8.1 The Scattered Field from a Plain Plasma Cylinder . 23
2.8.2 The Scattered Field from a Plasma-Coated Dielectric

Cylinder e e e e e e e e e e e e e e 24

2.9 NUNQEICRI RCSUILS e e e e e e e e e e e e e 26

SCATTERING FROM A METALLIC CYLINDER SURROUNDED BY A LAYER OF

LOSSY, HOT PLASMA ILLUMINATED BY A TE HAVE . . . . . . . 44
3.1 Introduction . . . . . . . . . . . . . . . an
3.2 Geo-etry Of the Problem e e e e e e e e e e e 45
3.3 Fields in the Regions of Free Space and Glass Wall . . 49
3e“ FiBldS in HOE P1.8ma Region e e e e e e e e e e 50
3.5 Matching Boundary Conditions at Interfaces . . . . . 59
3.6 Scattered Fields in Free Space Region . . . . . . . 79
3e7 Some SPQCLEI c3883 e e e e e e e e e e e e e 81
3.7.1 Scattered Fields by a Plain Plasma Cylinder . . 81
3.7.2 Scattered Fields by a Plasma-Coated Dielectric
Cylinder e e e e e e e e e e e e e e 82
3e8 Numerical RESUILS e e e e e e e e e e e e e 83

iv

4.

S.

Page

SCATTERING FROM A METALLIC CYLINDER SURROUNDED BY A LAYER 0F
LOSSY PLASMA ILLUMINATED BY A TM HAVE . . . . . . . . 99
4.1 Introduction e e e e e e e e e e e e e e 99
4.2 GCOMEETY Of the Problem e e e e e e e e e e e 100
4.3 Fields in Free Space Region . . . . . . . . . 100
4.4 Fields in Glass Wall Region . . . . . . . . . 103
4.5 FLEIdS in Plasma Region e e e e e e e e e e e ION
4.6 Matching of Boundary Conditions at Interfaces . . . 112
4.7 Scattered Field in Free Space Region . . . . . . 115
4e8 Some SpECial COCOS e e e e e e e e e e e e 116
4.9 NUMEEICOI RCSUIE‘ e e e e e e e e e e e e e 116
EXPERIMENTAL INVESTIGATION OF THE SCATTERING FROM A PLASMA-COATED
METALLIC CYLINDER AND A PLAIN PLASMA CYLINDER . . . . . 121
Sol Introduction e e e e e e e e e e e e e e 121
5.2 Experimental Setup e e e e e e e e e e e e 122
Se3 Experimental Procedure e e e e e e e e e e e 123
5.4 Experimental Results and Comparison with Theory . . . 128
5.4.1 Experimental and Theoretical Results . . . . 129
5.4.2 Comparison Between Experiment and Theory . . . 131
5.5 DISCUSSIOH e e e e e e e e e e e e e e e 133
APPENDIX A THE DECOMPOSITION OF PRESSURE GRADIENT INTO THE D. C.
AND A. C. COMPONENTS . .- . . . . . . . 141
REFERENCES O O O O O O O O O O O O I O O O O 145

Figure

2.1.(a)

2.1.(b)

2.3

2.4

2.5

2.6

2.7

2.8

LIST OF FIGURES

Page

A plasma~coated metallic cylinder illuminated by a
TE wave from the left (cold plasma model) . . . 6

Stratified cold plasma medium . . . . . . . 7

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mp/w)2

for a uniform plasma density distribution

(f 3 2.3 CHI, 6 3 180°, ROI 3 10) e e e e e e 34

Values of A“ of Eq. (2.93) for various values of
n as functions Of (mp/W) e e e e e e e e e 35

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mp/m)2

with glass tubes of various dielectric constants

and with a uniform plasma density distribution

(f 3 2.3 CHI, 9 3 180°, koT.‘ 3 10) e e e e e 36

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mp/m)gve,

for a parabolic plasma density distribution

(f I 2.3 GHz, 9 a 180°, kor = 10, elm = 0.001). . 37

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mp/w)ave.

for a parabolic plasma density distribution

(f 3 2.3 Gal, 6 3 180°, kor 3 10, VIC” 3 0001) e e 38

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mg/w)ave.
for various thickness's of non-uniform plasm

layer and a fixed conductor radius (f 8 2.3 CH2,
9 3 180°, kor 3 10, VIN 3 0e01) e e e e e e 39

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (WY/W)ive.

for various radii of conductor and a fixed g ass wall
radius (f 3 2e3 6H2. e 3 180°. kor 3 10, VI“) 3 OeOI) 40

vi

Figure

2.9

2.10

2.11

3.1.(a)

3.1.(b)
3.2

3.3

3.4

3.5

3.6

3.7

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of ( /W)§ve.
for various density profiles (f I 2.3 GHz, 9 I 1800,
kor 3 10, VIII 3 Deal) e e e e e e e e e e

Theoretical back scattered E field from a plain

plasma cylinder as a function of (mp/w)§ve. for

a parabolic plasma density distribution
(f 3 2.3 GHZ, e 3 180°, VJ” 3 0.001, kor 3 10) e e

Theoretical back scattered E field from a plasma-
coated glass cylinder as a function of (mp/w)2

for a uniform plasma density distribution

(f 3 2e7 6H2. e 3 180°, kor 3 10, V’W 3 OeOOS) e e

A plasma-coated metallic cylinder illuminated by
a TE wave from the left (hot plasma model) . . .

Stratified hot plasma medium . . . . . . .
The interface It r 3 En e e e e e e e e e

Stationary electron density profiles and associated

static Ed fields a e e e e e e e e e e
eCe

Theoretical back scattered E field from a plasma-

coated metallic cylinder as a function of (mp/m)

for a uniform plasma density distribution

(f I 2.3 GHz, vo/c I 0.01, e I 180°, kor I 10) . .

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/w)2
for a uniform plasma density distribution

(f I 2.3 GHz, volc I 0.0133, 9 I 1800, kor I 10) .

Theoretical propagation constant, ke I Be - j: ,
of electromagnetic wave as a function of e
(mp/m)2 for various collision frequencies (fI2.3 GHz)

Theoretical propagation constant, kp e Bp - jap.

of electroacoustic wave as a function of
(mp/m)2 with vo/c - 0.01 and Wm = 0.001 (f a 2.3 GHz)

vii

Page

41

42

43

46
47

6O

63

9O

91

92

93

Figure

3.8

3.9

3.10

3.11

3.12

4.1

4.2

4.3

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/w)2

for a uniform plasma density distribution and for
various values of vo/c (f I 2.3 GHz,0 I 180°,

hot 3 10, VI") 3 0.001) e e e e e e e e e e e

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/m)2

for a uniform plasma density distribution and for
various values of vo/c (f I 2.3 GHz, 9 I 180°,

kor 3 10, vim 3 0.001) e e e e e e e e e e e

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/w)ave.
for a parabolic plasma density distribution with
a3-sublayer model (f I 2. 3 GHz, vo /c I 0. 01, 0 I 180°,
kor 3 10, VI") 3 0. 001) e e e e e e e e e e e

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/w)§ve,
for a parabolic plasma density distribution with

a 3-sub1ayer model (f I 2.3 GHz, volc I 0.0133,
9318000,kr310. V/w30e001) e e e e e e e

Theoretical back scattered E field from a plain
plasma cylinder as a function of (mp/w)ive, for

a parabolic plasma density distribution with

a 3-sub1ayer model (f I 2.3 GHz, volc I 0.0133,

9 3 180°, Rot 3 10, VIC” 3 OeOOI) e e e e e e e

A plasma-coated metallic cylinder illuminated by
a TM wave from the left 0 e e e s e e e e e

Theoretical back scattered E field from a plasma-
coated metallic cylinder for various radii as a
function of (mp/m)§ve. for a parabolic plasma

density distribution with a 13-sublayer model
(fI2.3GHz,eI180°,k°rI10,vlwI0.01) . . .

Theoretical back scattered E field from2 a plain
plasma cylinder as a function of (mp/m)§ve. for

a parabolic plasma density distribution with a
13-sublayer model (f I 2.3 GHz, 6 I 180°,

kor 3 10, VI“) 3 0.01) e e e e e e e e e e e

viii

Page

95

96

97

98

101

119

120

Figure

5.1.1

5.1.2

5.2

5.3.1

5.3.2

5.4.1

5.4.2

5.5

5.6

5.7

5.8

5.9

Cross section view of a rectangular waveguide
with an inserted plasma tube (TE polarization) . .

Cross section view of a rectangular waveguide
with an inserted plasma tube (TM polarization) . .

Experimental setup for the measurement of
scattered inld e s e e e e e e e e e e

The plasma discharge tube inserted in the waveguide

The waveguide and directional coupler with double-
stub tuners at both ends . . . . . . . . .

Experiment set up for the measurement of scattered
field with a TE incident wave . . . . . . .

Experiment set up for the measurement of scattered
field with a TM incident wave . . . . . . .

Experiment results of the back scattered E field
from a plasma-coated metallic cylinder as a function
of the discharge current with a TE incident wave .

Experiment results of the back scattered E field
from a plasma-coated metallic cylinder as a function
of the discharge current with a TE incident wave .

Theoretical back scattered E field from a plasma-
coated metallic cylinder as a function of (mp/m)§ve.
with a TE incident wave and a parabolic density
distribution (hot 3 10' 0 3 1800) e e e e e e

Experimental results of the back scattered E field
from a plain plasma cylinder as a function of the
discharge current with a TE incident wave . . .

Theoretical back scattered E field from a plain
plasma cylinder as a function of (mp/w)£ve. with

a TE incident wave and a parabolic density distri-
bution (1‘0: 3 10, VI” 3 OeOOI, e 3 180°) e e e 0

ix

Page

124

124

125

126

126

127

127

134

135

136

137

138

Figure

5.10

5.11

Page

Experimental and theoretical results of the back
scattered E field from a plasma-coated metallic
cylinder with a TM incident wave . . . . . . 139

Experimental and theoretical results of the back
scattered E field from a plain plasma cylinder
with a TM incident wave. . . . . . . 140

CHAPTER 1

INTRODUCTION

The electromagnetic scattering of the object surrounded by or
immersed in a plasma medium has been a subject received a great deal
of attention from researchers in recent years. The interest was moti-
vated by the problems of electromagnetic wave propagation in the
ionosphere, the electromagnetic reflection from meteor trails and the
radar reflection from a re-entry vehicle.

This dissertation deals with the electromagnetic scattering
from a plasma-coated metallic cylinder and some other related subjects.
Similar problems have been considered by other researchers. In 1952,
Kaiser and Closs‘l) studied the electromagnetic reflections from
meteor trails. In 1955, Keitel (2) investigated the forward electro-
Vmagnetic scatterings by meteor trails. Later, a number of investi-
gators including Ohba, (3) Chen and Cheng (4) and Yeh and Busch, (5)
have studied the electromagnetic scattering from a plasma-coated
metallic cylinder. Many other workers have studied the related
problem and will be cited later. Most of previous workers, however,
assumed the plasma medium as a dielectric with an equivalent dielectric
constant or using so called cold plasma model. It is well known that
if the temperature effect of the plasma is not ignored, an electro-

acoustic wave in addition to an electromagnetic wave can be excited in

the plasma medium. This electroacoustic wave may have significant

effects on the electromagnetic scattering from a plasma-coated object.
Fer this reason, both the cold plasma and hot plasma models are used to
analyze the plasma surrounding the object.

Since the plasma layer which surrounds an object is usually
inhomogeneous, a certain density distribution is assumed for the plasma
medium in the analysis. To solve the problem of propogation of waves
in an inhomogeneous medium, the stratification method is used. The
plasma layer is divided into a number of thin sublayers and a step
function approximation is used to describe the density distribution of
the plasma medium. In the course of applying the stratification method
a difficulty was encountered which led to a series of numerical singula-
rities. These singularities or "mathematical resonances" bear no
physical meanings and were carefully handled in the analysis.

Throughout this study, the macroscopic approach which uses the
hydrodynamic equations instead of the Boltzmannequation is used to
describe the dynamic behaviors of the plasma. The problem was solved
based on the hydrodynamic equations and Maxwell's equations.

In Chapter 2, the electromagnetic scattering from a plasma-
coated cylinder illuminated by a plane wave with the E’field perpen-
dicular to the cylinder is studied based on the cold plasma model. The
plasma layer surrounding the cylinder is assumed to be inhomogeneous.

The same problem is treated in Chapter 3, but based on a much
more complicated hot plasma model. Effects of the electroacoustic wave
are studied.

In Chapter 4, the same plasma-coated cylinder is assumed to be

illuminated by a plane wave with the E field in parallel with the

cylinder. Under this illumination, it is shown that no electro-
acoustic wave can be excited in the plasma layer.

An experimental study on the subject is described in Chapter 5.
Experimental results agree qualitatively with the theoretical results

obtained in Chapters 2, 3, and 4.

CHAPTER 2

SCATTERING FROM A METALLIC CYLINDER SURROUNDED BY A LAYER
OF LOSSY, COLD PLASMA ILLUMINATED BY A TE HAVE

2.1 Introduction

 

The scattering of an electromagnetic wave by a plasma-coated
metallic cylinder when it is illuminated by a normally incident plane
wave with its N field parallel to the cylinder axis is studied in this
chapter. In the analysis, the plasma is assumed to be cold and non-
uniform. Also an equivalent permittivity and a collision frequency

are assigned to describe the characteristics of the plasma.

A number of workers have studied this problem. Tang(6) studied

the backseattering from an infinite cylindrical obstacle coated by a

homogeneous dielectric. In treating the reflection from.meteor trails

(1)

Kaiser and Close considered a meteor trail as a plasma cylinder

(3)

which was then treated as a lossless dielectric column. Ohba and

(4)

Chen studied the scattering from an anisotropic and uniform cylinder

and considered the plasma as a medium with an equivalent tensor
permittivity in the presence of a steady magnetic field. Vandenplas(7)
also studied the same problem but treated the plasma as a medium with
a equivalent complex permittivity taking into account of the collision
loss in the plasma. Yeh and Rusch (5) studied the scattering from an
inhomogeneous plasma cylinder with a differential equation method.

(8)

Pong calculated briefly the radar cross section of a plasma-coated

 

metallic cylinder by the stratification method.

In this chapter the temperature effect of the plasma is
neglected. Due to the existence of a static potential on the metallic
cylinder and other boundaries the density distribution of the plasma is
assumed to be inhomogeneous. The stratification method is used in the
analysis. The inhomogeneous plasma layer is subdivided into a number
of concentric sublayers of sufficiently small thickness compared with
the electromagnetic wave length. The plasma density is assumed to be
constant in each sublayer so that a step function approximation for the
density profile is adopted. The wave equation is, then, solved in each
sublayer resulting in two cylindrical waves with unknown magnitudes and
propagating in opposite directions. The magnitudes of waves are deter-
mined by matching the boundary conditions at the interface of two
ajacent sublayers. This boundary matching process will lead to the
final determination of the scattered fields in space.

To compare with experimental results, a glass wall is assumed to

surround the plasma in theoretical model.

2.2 Geometry of the Problem

An infinitely long metallic cylinder with a radius a and covered
by a layerof non-uniform cold plasma is confined in a glass tube with
inner radius b and outer radius c. This plasma-coated cylinder is
placed along the z axis and is illuminated normally by a plane electro-
magnetic wave with its E’field perpendicular to the z axis and E field
parallel to the z axis (TE wave). The layerof noel-uniform cold plasma
is subdivided into a number of sublayers as shown in Fig 2.1 for the

analysis. These sublayers are counted from outmost sublayer and

 

 

 

 

 

 

Incident wave

Region 1: free space
Region 11: glass wall
Region III: cold plasma

Region IV : metallic cylinder

Fig. 2.1.0) A plasma-coated metallic cylinder
illuminated by a TE wave from the
left. (cold plasma model)

inwardly. For example,the first sublayer is located immediately inside
the glass wall and the last sublayer is located immediately outside the
metallic cylinder. The radius between two adjacent mth and (m+1)th
sublayers is denoted as rm. In the mth sublayer, we assume that the
plasma density is 9o,m’ the collision frequency is uh, the propagation
constant is ke m

The cylinder is assumed to be infinitely long in the analysis so

and the equivalent complex permittivity is gm.

that there is no field variation along the z direction. The angle 9 in
the cylindrical coordinates starts from the x axis and increases in the
counter clockwise direction. The time dependence of exp(jwt) is assumed

and the field of incident plane wave are given by(9)

Ho: . e jkox ' e jkorcosO
a, n
a “50606-9 cos(n6) Jn(kor) (2.1)
i i
"or 3 I109 8 O (2.2)
i i
1ear 3 - 331;? 3% H02
. w n
. $3.07 .5506 om(«5) 11.111010) Jn(k°r) (2.3)
E1 - A i

oe (06° Br Hoz

a, n
- J§0n§o e .n“5) come) Jamar) (2.4)

1

£02 3 o . (ZeS)

In these expressions, the sumscript ”i” represents incident wave.

k0 is the propagation constant of free space and is defined as koew-Ju E .
o o

E is the Neumann factor defined as 6 I 1 when n-0 and 6 = 2
on on on

when niO. Jn(k°r) is Bessel function of the first kind with integer
order n and argument kor. Jn'(kor)is the first derivative of Jn(k°r).

go is the impedance of free space and is defined as £0 Il-EQ- - 120" olms.
o

”o and 6 o are permeability and permttivity of free space respectively.

2.3 Fields in Free Space Region

In the free space region the Maxwell's equations are

vxi‘; - -j.uofi; (2.6)
vxil: - 3.503: (2.7)

where E:, I: are the scattered electric and magnetic fields, and no, 6 o
are the permeability and permittivity of free space respectively. Due
to geometrical syuetry, all fields are synetrical with respect to the
0 I 0 axis.

From Eqs. (2.7) and (2.6) we obtain

VxVxE; I (0211060? . (2.8)

Due to TE polarization of incident electromagnetic plane wave and the
geometry of problem, E field does not have 2 component and is indepen-
dent of z. with these two conditions the lefthand side of Eq. (2.6)

reduces to a 2 component equation as

I
1 as
1: [58; (rt-2:0) - 23’] i - -jmu.o if; . (2.9)

 

Equation (2.9) shows that a is allowed to have 2 component only.

Assuming that 36 I 110; 2, Eq. (2.8) reduces to

 

2 s s 2 s
a 1102 a H“ 3 no: 1:2 n' 0
31-5 + :31: + 1:2305 + ° °' . . (2.10)

10

Equation (2.10) can be solved by the method of separation of variables.

He assume that

3
"oz I H(r) 11(0) (2.11)

where H(r) and R(0) are functions of r and 0 respectively. The substi-

tution Of Me (Zell) in Me (2.10) 128d! LO

im+amnzrz . n2 (2.12)
am n2 am a: °

4—32 "(9) . n2 . (2.13)
8(8) 38

Considering the symmetry of problem and the degeneracy of angle 6, the

solution for 3(0) is

H(6) I cos(n9) (2.14)

where n is an integer.

Equation (2.12) can be rearranged to

 

2 2
34(35)- +--1--3§-(£-)-— + (1- n 2) Mr) . o (2.15)
3(kor) (kr) 3(kor) (Rot)

which is a Bessel equation.

The solution to Eq. (2.15) is a Bessel function of order n with
an argument of kor. Because only an outgoing cylindrical wave is expected
to exist in the free space, the proper solution for Eq. (2.15) is the

second kind of the Hankel function such as
am - 11(2) (1. r) . (2.15)
n o

With Eqs. (2.14) and (2.16), the final solution for Rs; can be written as

11

s °° (2)
"oz I “£20 cos(n0) H n (kor) An (2.17)

where An is a constant to be determined by the boundary conditions.

The corresponding if: field can be found from

ESI 1 VIC?

 

o jmeo o
l 1 3 no: . 1 a Hos: A
a 3:32-— (3:— T) r '- W (V) 9 (2.18)
o 0
or
s °° (2)
Eor :- (0601' “:20 nsin(n0) R n (kor) An (2.19)
s m (2).
1:06 - jgo n50 cos(n0) H n (kor) An (2.20)

n s
where go I’-€2 I 120" ohms and Hg) (kor) is the first derivative of
(2) °
N n (kor) with respect to (kor).
The total fields in the free space region can be obtained by

sunning the incident and scattered fields to be

no: - E0 cos(ne) [é on(-j)n Jn(kor) + 11‘? (kor) An] . (2.21)
not: . Hate . 0 (2.22)
a: - wear Ea nsin(n8) 5 015-1)“ Jn(kor) + 11‘? (Rot) An] (2.23)
E02 I jéo n23: cos(n8) [5 0n(-j)n J;(kor) + H(:)'(kor) An] (2.24)
at - 0 . (2.25)

12

2.4 Fields in Glass Hall Region

In the glass wall region the Maxwell's equations are
V x E8 I -jmp.o MS (2.26)

V x is - jmeo Es E8 (2.27)

where 6 8 is the dielectric constant of glass.
If we allow an incoming and an outgoing (reflected) waves to exist

in the glass well region, the solution for as field can be expressed as

-° A
"a . "322 (2.28)
where
a
(1) (2)
"a: . “so cos(n0) [a n (1182:) an + a n (kgr) an] , (2.29)

In Eq. (2.29). Bn and C“ are the constants to be determined by the

boundary conditions and k8 is propagation constant of glass defined as

1‘s ' kale—s'

The corresponding Eg field has components given by

Egr I file—r 2 nsin(n0) E H(:)(k8r) 3n + R(:)(k8r) C11] (2.30)

o g nIo

5 o °° (1)' (2)'
E89 I 7% “ED cos(n0) [R n (ksr) Bn 4- H n (ksr) C11] (2.31)
3 0. (2032)

E
gz
2.5 Fields in Plasma Region
As mentioned before, the plasma layer is subdivided into a

number of concentric sublayers of sufficiently small thickness. The

plasma density is then considered to be uniform within each sublayer,

13

but it varies from sublayer to sublayer in radial direction. In the mth
sublayer the plasma medium is considered as a frequency and collision

dependent dielectric. The equivalent complex permittivity can be ex-

pressed as
wz. imz. 1.
s. - 6. <1--§‘7>--£2LT m
(1) +‘1ll (0(0) Wm)

where mp m is the plasma frequency associated with density no
0

mth sublayer and is given by

e n
.. ’42! .
mpsm M60 . (2 34)

e and M are the charge and mass of electrons respectively in Eq. (2.34).

in the
,m

Also um is the collision frequency in the mth sublayer.
The field in the mth sublayer of plasma medium can be obtained

from the Maxwell's equations,

V! Em I “50810 am (2.35)
Vx fin ',3‘”§n Em , (2.35)

The components of Em and En fields can be expressed as

, °° (1) (2)
n 2‘. co.(ne) H n (ke' r) 0 n + a n (he. r) rum] (2.37)

m: nIo m m, m
Hm 3 "no 3 0 (2e38)
&
1 (2)
ant I Ii? 2 nsin(n8) [H(n)(ke,mr) Dm,n + H n (hunt) PM; (2.39)

1130

a O I
p‘ (1) (2)
“'9 1 info cos(n0) [H n (hunt) Dm,'n + R m (kept) Fmm] (2'40)

3 3 0 (2 e31)

where Dm,n and Fm,“ are constants to be determined by boundary conditions

14

and ke m is the propagation constant in the mth sublayer of the plasma
0

layer and is given by

ke,m II wfiiogb . (2.42)

If we denote ke,m as

k I B - ja (2043)

and after substituting g5 with Eq. (2.33), Be m and ae m can be
0 9

expressed as

 

2 2 4
Bo w m 2m m w m 5 %
Barf;- 1";'—i‘+[1'_2hT+'-2—25'—2‘] W“)
m +Vm m +vm w (w +vm)
\
2 2 4
Bo { m m [ 2m m m m ] ‘5 55
a -—J-l+-§J—5 + 1- + (2.45)
e" 2 0) +vm (n +v2 m (mzwi)
with

8o ' "’Foéo ° (2.1.6)

Up to this point, the fields in the mth sublayer have been solved.
Similarly, the solutions of the fields in the other sublayers will have
the same ferns as that in the mth sublayer with appropriate change in the

quantities of he, g, 110 and v etc.

2.6 Matching the Boundary Conditions at Interfaces

In the solutions for E; and 3; obtained in Sec. 2.5, there are
two unknown constants to be determined by boundary conditions. In order
to express the constants in one sublayer in terms of the constants of its

adjacent sublayer, it is necessary to have two boundary conditions at the

15

interface of these two sublayers. Consider the boundary at r . rIn
between the mth and the (m+l)th-sub1ayers (refer to Fig. 2.1). The
boundary conditions at this interface are the continuity of tangential

components of if and '13 fields. In symbols,

Hm - “(HUI at r . rIll (2.67)
2'6 3 E(m+l)e at r . rm (2.&8)

01'

a
(1) . (2)
“E0 °°““”[H n (he'll rm) Dunn + H n (ke'm rm) rum]

(2)
1‘11:) Dm-I-l,n + H n (ke,m+l I'.m)Fm+l,n]

(2 .49)

°° (1)
- 2 cosme) H (k
n l: n e,

n+1

and
“'0 0° (1). (2).
31%} 2 cos(n6)[fl n (he,m 1‘”) Drum + H n (ke'm rm) Pam]

two

"'0 m (1). (2).
. 5E “:30 cos(as u n (Rem-H rm) 0M1,“ + a n (5M1 qnn-‘mflfla .
(2.50)

Due to the orthogonality of cos(n6) functions, Eqs. (2.49) and (2.50)
lead to the following matrix equation

(2) "-7

' , (1)
in n (Rem rm) H n (ke,m rm) m,n

1 (1). 1 (2).
LE H n (k8,!!! rm) E H n (kem rug-J Flinn

 

 

 

 

16

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1) (2) ‘ F "
H n (ke,m+lrm) H n (ke,m+lrm) I)m-O-lm
E1 (1)' :1 (2).
+1 H n (hem-firm) gm +1 H n (ke,m+lrm) Fm-i-l,n '
Equation (2.51) gives
-1
(1) (2)
Dllfll H n (ke,mrm) H n (ke,mrm)
- 0 I
1 (1) 1 (2)
Fllfil E H n (ke,mrm) j§Cm H n (ke,mrm)
P (1) (2) _ ' -
H n “hm-urn) H n (ke,m+lrm) DIa-o’l,n
! 13: (1)' E1 (2).
1 H n (Rem-arm) 1 H n (Items-Iris) LEM-1m J '
" _ (2.52)
We can write Eq. (2.52) as
f- "7 "' '-
”m,n DIM-1m
-l
' [Lnotemrmfl En‘kem-I-lrmil (2'53)
.FWL Lam”;
where
P (1) (2) _
H n (kg-rm) H n (ke,mrm)
[L“(k°"'r")] . ' . (2.54)
l (l) 1 (2)
LE 8 n (ke’mrm) E a n mental: .
Similarly at the interface of r I r m-I-l we have
l)m-i-l,n Dm+2,n
-1
F ~ " En‘kem-l-lrm-I-lfl E‘n‘kemn’mufl F (2'55)
lid-1,11 M2,“

 

17

Thus, combining qu. (2.53) and (2.55), the constants D and F in

m,n m,n
the mth sublayer can be expressed in terms of the constants D m+2 n and
3
Ffiz ,n in the (n+2 )th sublayer as
D
m,n
F ' [Ln(ke, m 1‘In] 1[n(k e ,m+1r mil
m n _
' FD
-1 M2,“
[Ln(ke,m+lrm+li| E‘n‘kem-o-Zrm-t-lil ' (2'56)
I"m+2,n

 

 

By carrying out the successive operation of Eq. (2.56) to cover
all the interfaces, the constants of the outmost sublayer can be expressed

in terms of that of the inmost sublayer as follows:

l,n
F ' [Ln(ke,lrl)]11' ke, 21.1)] [L n,(ke 2 1‘2] IE'nO‘ e, 3 r2] H
l,n
Ds,n
”,E'n(ke .-1‘.-1]1En°‘.,.’s-1] F (2.57)
s,n

where the sth sublayer is the inmost sublayer. Equation (2.57) can be

expressed in shorthand as

. (2.58)
F1 ,n Mn(2,1) Mn(2 ,2) is ,n

0
where Mn(i, j) s are the entries of the matrix which is the product of
those [Ln] matrices in Eq. (2.57).
Let's now consider the interface between free space and glass wall

at r I c. The tangential components of E and ii. fields are continuous

18

at this interface. This leads to

H I H at r I c (2.59)
oz g2

12" . .
06 I Ege at r c (2 60)

where q:; and 3;; are given in Eqs. (2.21) and (2.24) respectively and

H82 and £89 are given in Eqs. (2.29) and (2.31) respectively.
With Eqs. (2.21), (2.24), (2.29) and (2.31), Eqs. (2.59) and (2.60) lead

to

- H(:)(k°c) An + 112%?) an + H(:)(k8c) on - €°n(-j)an(koc) (2.51)

1
8

I O
(1) 1 (2)
H n (ksc) an + —— H n (kgc) cn

F?

n I
- 6mm) Jnaoc). (2.62)

(2)' .
- H'n (hoe) An +

we consider next the interface between the glass wall and the
first sublayer of plasma at r I b. The continuity of the tangential

components of f and if fields leads to

_ (1) _ (2) (1) (2)
H n (1:81)) B“ H n (1:81)) Cu + H n (k b) D + H n (he’lb) F

e,l l,n l,n

. o (2.63)

_ 1 (1)' _ 1 (2). E: (1).
-——j_€_8 n n (1.81)) ”:1 “f5: 11 n (ksb) 0,, + j; H n 02,1") ”1,11

’50 (2;
'l' T; H n (ke,1b) F1,“ ' 0. (2.611)

Expressing D1,“,

and (2.64) lead to

F in terms of D , F
l,n a,“ 8

n by Ego (2058), Eqs. (2o63)

l9

(1) (2) ..
- H n (kgb) an - H n (ksb) cn +[Rn(l)] 11$,n +[Rn(2)] 1:8,“ — o
(2.65)
and
- 1 11(1).“ b) n - 1 H(2).(k b) c +[R (3)]1) +[R(A)]F
7?; n g n 7?: n g n n s,n n s,n
3 0 (2.66)
where
Rn(l) = H(!1‘)(ke,1b) Mn(1,1) + H(:)(ke’1b) Mn(2,1) (2.67)
g (1) (2)
Rn(2) H n (ke,1b) Mn(1,2) + H n (he'lb) Mn(2,2) (2.68)
o (1)' 66 (2)'
Rn(3) = l-é—Z H n (ke'lb) Mn(1,l) + E H n (He’lb) Mn(2,1) (2.69)
_ 0 (1). o (2).
an“) E H n (he'lb) Mn(l,2) + E H n (ke’lb) Mn(2,2) , (2.70)

Finally, we consider the interface on the metallic cylinder
surface at r I a. If the cylinder is assumed to be a perfect conductor,
the tangential component of the E field at its surface vanishes. That is

E I O at r I a . (2.71)

36
This leads to

(1)' (2 )'
H n (ke,sa) Ds,n +'H n (ke,sa) Fs,n I 0 . (2.72)

Equations (2.61), (2.62), (2.65), (2.66) and (2.72) can be represented in

a matrix equation as

20

 

 

 

 

 

 

Fqnun) Qn(1,2) Qn(l,3) o o 7 ”A“ ' Rama)"
Qn(2,l) Qn(2,2) Qn(2,3) o 0 En Qn(2,6)
o Qn(3.2) one.» (253.4) (1.53.5) cn - o
0 Qn(4.2) Qn(4.3) Qn(4s") Qn(4".5)L D8,!) 0
_ o o o Qn(5,4) 0155,53“ 88m 0 J
h (2.73)
where
(2)
Qn(1,1) ’- ‘Hn “(00) (2074)
Qn(1.2) = H(:)(kgc) (2.75)
3 (2)
Qn(1.3) H n (kgc) (2.76)
n
Qn(1.6) = Eon(-j)Jn(koc) . (2.77)
(2)'
Q“(Zsl) " " "n (ROG) (2078)
_ 1 (1)'
Qn(2.2) 58 H n (kgc) (2.79)
g 1 (2)'
Qn(2'3) 68 H n (kgc) (2.80)
n ' '
Qn(2,6) = 60n(-j) JnOtoc) (2.81)
(1)
Qn(3,2) 2: - Hn (keb) (2.82)
Qn(3,3) a - “(121)(kgb) (2.83)
011(3).) . 1193(22'18) Mn(1,l) + H(:)(ke,1b) Mn(2,1) (2.84)
Qn(3,5) - $302,111) Mn(l,2) + H(:)(ke’1b) Mn(2,2) (2.85)

05(4.2)
Qn(4,3)
Qfi(4.4)
Qn(4.5)
Qn(5.4)

Qn(5.5)

”(1)

1
8 68 n (Rab)
_ __ 1 (2)
H H " (k‘b)

6. (1)' 66 (2)'
. IT; a n (He'lb) Mn(1,1) +5} H n (keflb) Mn(2.1)

_ e. (1)' 66 (2)'
E H n (He’lb) Mn(1,2) +E H n (ke'lb) Mn(2,2)

, (1)'
H n (ke,8a)

(2)'
3 H n (he'sa) o

21

2.7 Scattered Field in Free Space Region

The constant An is of main interest and can be obtained from

Eq. (2.73) by Cramer's Rule as

'- An1

n

A
n

 

Two determinants,An and Anl’ are given as

An.

and

Qn(1.1) Qn(1.2)
Qn(2,1) Qn(2,2)

 

0 Qn(3,2)
o Qn(4,2)
o o

Qn(1,3) O 0
Qn(2,3) O 0
Qn(3.3) Qn(3.4) Qn(3.5)
0154.3) Qnaom Qn(4.5)

 

(2.86)

(2.87)

(2.88)

(2.89)

(2.90)

(2.91)

(2.92)

(2.93)

22

Qn(1.6) Qn(1.2) Qn(1.3) o o

Qn(2.6) Qn(2.2) Qn(2.3) o o
- o (3.2) (3.3) (3.4) (3.5) (2.94)

Am Qn Q. Qn Q.

o Qn(4.2) Qn(4.3) anam Qn(4.5)

o o o oncsus) Qn(5.5) .

 

 

Finally, the scattered fields in free space region are obtained as

00

s (2)
H02 I Z cos(n6) H n (kor) An (2.95)
nIo
s ._ s 8
H6: - H69 0 (2.96)
N
s 3' . (2)
or = w r 2 nsm(n6) H n (kor) An (2.97)
() nIo
s °° (2).
£09 = Jgo 2 cos(n6) H n (kor) An (2.98)
nIo
s —
Eoz - 0 (2.99)

with An expressed as Eq. (2.92).
For the scattered fields observed at a large distance, Hankel

function can be expressed in its asymptotic form as
-J'(k°r - ‘1’"! - i‘rr)

 

(2) / 2
H n (kor) s: ”‘0‘ e . (2.100)
And the scattered fields at a large distance are then obtained as
8 2 -j(k0r ' h) °° 5551111
H02 I r e 2 cos(n6) e An (2.101)
O n-o
s s
”or 3 806 ' O (2o102)
E8 j 2 1 -j(k0r ' 5:") °° J'lfin‘"
'3 —— e Z nsin(n6) e A
01‘ nicer wéor “.0 n

i o (2.103)

23

’j(k r -‘1(n+1)w - 5m)
= jéo (171;: r 20 cos(na) [- e o

 

 

-. - 1 -
n J(kor 1m 3301
+--—-e A
k r n
o
g 2 ‘j(kol‘ " 2:") 2 53111111 ( )
N / e cos(n6) e A 2.104
O “or n” n
after neglecting the r 2 term
gs g o (2 105)
oz ' '
To derive Eq. (2.104) the relation of
(2). H(2 ) H(2)
n
H n (kor) I Mn+l(k r) +— “O n (kor) (2.106)

is USEdo

2.8 Some Special Cases

In section 2.6 we have developed a theory and a set of five
simultaneous linear equations with five unknowns which can be solved to
determine the scattering from a plasma-coated metallic cylinder. we will
show that with a slight modification this theory can be used to determine
the scatterings by a plain plasma cylinder and by a plasma-coated dielec-

tric cylinder.

2.8.1 The Scattered Field from a Plain Plasma Cylinder

For this case the plasma fills the whole glass tube in the
absence of metallic cylinder. If we let the inmost sublayer be the
plasma cylinder with an extremely small radius and located along the z

axis, the whole plasma cylinder is subdivided into an extremely thin

24

plasma cylinder at the center and a number of concentric sublayers
extended from radius r I 0 to radius r I b up to the glass wall (Refer

to Fig. 2.1). Since the Bessel function of the second kind, Yh(ke,8r),
has a singularity at r I 0, the proper solution for the fields in the
inmost sublayer or the thin cylinder at the center is the Bessel function

of the first kind, Jn(ke 8r),only. This condition can be achieved by
9

setting the constants D and F to be equal, because
s,n s,n

1 (1) (2)
Jn(ke,8r) . 7|}; n (kefir) + H n mafia] . (2.107)

Thus for a plain plasma cylinder, the simultaneous equations are Eqs.

(2.61), (2.62), (2.65), (2.66) and the following equation:

With this set of equations An can be solved and consequently the

scattered field.

2.8.2 The Scattered Field from a Plasma-Coated Dielectric Cylinder
In this case a dielectric cylinder instead of a metallic

cylinder is located in the center of the plasma column. The tangential
component of E field will not vanish on the surface of the dielectric
cylinder. Due to the singularity of the Bessel function of the second
kind, the proper solution for the 8 field in the dielectric cylinder is

.. a

"d I H i I ‘2 cos(n0) Jn(kdr) Gn 2 (2.109)

n-o

where subscript d implies the dielectric regon. CD is a constant to be
determined by boundary conditions. k is the propagation constant of the

d
dielectric cylinder and is given by

25

kd 3 0).] “.06 0 Ed (2o110)

where E'd is the dielectric constant of the dielectric cylinder.

The corresponding § field can be obtained from a Maxwell's

equation,
a jweoed .

to yield the following components

 

N
E = ——-l—— )3 nsin(n9) J (k r) c; (2.112)
dr wEOEdr “-0 n d n
j€5o “’ '
Ede - .jE—E' nEocos(n0) Jn(kdr) Gn (2.113)
E 3 0 o (2.114)

dz

The boundary conditions at the interface between the plasma region
and dielectric cylinder are the continuity of the tangential components

of E and H fields. These boundary conditions lead to

_ (1) _ (2) ..
H n (Ra'sa) 0 H n (ke — o (2.115)

s,n a) Fs,n + ‘Jn(k

,s e,da) G11

and

9
Go (1) o (2) 1 '
Jr; H n 0‘...“ Ds,n “Jr; H n 0‘...“ Fs,n we": Jn(ke,da) ch
= 0 s (2s116)
With this modification, Eqs. (2.61), (2.62), (2.65), (2.66), (2.115) and

(2.116) form a set of six simultaneous linear equations with six un-

knowns which can be solved to determine the scattered fields.

26

2.9 Numerical Results

 

The back scattered E fields from a plasma-coated metallic cylinder,
a plain plasma cylinder and a plasma-coated dielectric cylinder have been
calculated as a function of (mp/m)2. Although in the deveIOpment of
theory the collision frequency v is treated as a variable, in our numeri-
cal calculation 0 is assumed to be a constant for all sublayers for
simplicity. For a laboratory plasma the ionization degree is very low
and the electron-neutral particle collision usually is the predominant
effect. Even though the electron density of the plasma may be non-uniform,
the density of neutral particles can be uniform in the plasma. Thus the
assumption of a constant collision frequency in a non-uniform plasma may
bereasonable. The series solution is produced by summing up the first
four terms only (up to nI3). The accuracy of the numerical results based
on four-term summation is quite satisfactory since these results deviate
less than one percent from the numerical results based on ten-term summa-
tion. The scattered fields are calculated at a distance from the z axiswith
kor I 10 for convenience. Andthese fields are plotted in figures with
its normalized value, Ezlfii where E: and E: are the scattered and incident
fields respectively. From Fig. 2.2 through Fig. 2.6, the dimensions for
the glass tube, plasma layer and central cylindrical conductor are based on
the actual dimensions of the experimental model. Those dimensions, dielec-

tric constant of glass and the operating frequency are given in Table 2.1.

 

 

Operating frequency a(mm) b (m) c (mm) 68 l
5

f I 2.3 GHz 2.158 7 8

 

 

 

 

 

 

Table 2.1 Physical dimensions of plasma tube, dielectric
constant of glass and operating frequency.

27

Figure 2.2 shows the back scattered B field in the direction of
0 I 1800 as a function of (wp/w)2 for various collision frequencies for
the case of a uniform density distribution in the plasma region (barge).
The main resonance occured at (mp/w)2 I 2.58 is the so called the
dipolar resonance which corresponds to the resonance due to the n I 1
term of the series solution. The sharp peak at the right main resonance

(10'7) The hexapolar resonance or nI3

is the quadrupolar resonance.
resonance does not appear in the figure although the resonance does
occur at a higher value of (wp/w)2. It is observed in Fig. 2.2 that
when collision frequency is increased to a value of the order of v/w I 0.5
no resonance appears any longer. Also the quadrupolar resonance seems
to be damped out by the collision more strongly than the dipolar
resonance.
Figure 2.3 is a plot of the determinant An given by Eq. (2.93)
for various values of n as a function of (wp/w)2. Because of rapid
convergence of the series, the terms with n 24 are neglected. The
real and imaginary parts of the determinant are calculated separately.
The determinant is plotted for the region of l <:(wp/w)25; 5 only to
show the locations of resonances. Figure 2.3.1 shows the smooth
behavior of the n-O term of the series. In Fig. 2.3.2, the point of
rapid sign change of the real part of the determinant fer nIl occurs
at (wp/w)2 I 2.58 and it corresponds to the dipolar resonance. A
similar behavior of rapid sign change of the determinant for nI2 case
is shown in Fig. 2.3.3 and this point of rapid sign change corresponds
to the quadrupolar resonance. In Fig. 2.3.4 although the hexapolar

(nI3) resonance is clearly seen, due to an extremely small value of

28

A3 compared with A0, A1

not detectable in the scattered field as shown in Fig. 2.2.

and A2, the effect of hexapolar resonance is

Figure 2.4 shows the effect of the dielectric constant of the
glass tube. Three curves of back scattered B field are plotted for the
glass tube with dielectric constants of Gig I l, 2.5 and 5. The case
of € 8 I 1 is equivalent to the absence of the glass tube. It is
observed that as the dielectric constant of glass tube is increased the

location of the dipolar resonance shifts to a higher value of (mp/w)2

and the separation between the dipolar and quadrupolar resonances
becomes greater. These three curves are plotted with an assumption of
a collision frequency of v/m I 0.001 and an operating frequency of
2.3 GHz.

Figures 2.5 and 2.6 show the back scattered B field from a
metallic cylinder covered by a layer of non-uniform plasma as a function
of (mp/w):ve. which corresponds to the average plasma density. The

density distribution of the plasma layer is assumed to be given by

no,r I no,c[1 - aéggf] (2.117)
where no,c is the plasma density at r ..2§2., and a and p are constants
which are used to adjust the density distribution. The formula (2.117)
gives a similar parabolic density profile adopted by Vandenplas<11>
and Killian (12) in their studies for a plain plasma column. In our

calculation we assign the values of p I 2 and a I 0.6. The layer of

non-uniform plasma is then subdivided into 3, 7 or 13 sublayers with a
constant density assigned in each sublayer. This gives a step function
type of density distribution. The numerical calculations based on this

scheme are shown in Figures 2.5 and 2.6. In these two figures, it is

29

observed that the general shapes of curves and the locations of the
dipolar and quadrupolar resonances remain quite unchanged as the number
of sublayers in numerical calculation is varied. However, a series of
small peaks appears to the left hand side of the main (dipolar)
resonance. The number of these small peaks increases as the number of
sublayer is increased. Obviously, these small peaks can not be physical
since they are created in the process of subdividing the plasma layer.

In the paper by Shohet and Batch (13)

in solving eigenvalues of a
microwave cavity filled with a plasma of variable radial density, a
stratification method similar to our method has been used, and they
observed the number of the eigenvalues increases as the number of
sublayers is increased. They attributed this phenomenon to the
mathematical process involved in the stratification method. Any
resonance associated with the stratification method in a non-uniform
plasma should be termed as a mathematical resonance which has no
physical meaning what so ever. In our analysis, we have found that
each mathematical resonance occurs when the density of a sublayer
approaches toa value when its (up/w is approximately equal to l. The
location of main (dipolar) resonance tends to approach to a value of
(mp/w)2 I 2.8 as the number of sublayers is increased. Nhile the
location of quadrupolar resonance converges to a value of (mp/w)2 I 3.6

as the number of sublayer is increased. The values of (mp/w):v where

e
both physical and mathematical resonances occur is listed in Table 2.2
for the case of collision frequency of vlw I 0.001 and for various

stratifications.

30

 

 

 

 

 

 

 

 

 

Model Mathematical Resonance Dipolar Quadrupolar
2 2 2
(mp/(1))”,e o (mp/w )ave s (mp/w )ave o
Uniform None 2.57 3.25
3 sublayers .8839, 1.089 2.631 3.37
5 CablIyera o8282, 1005M, 1o2726 2o757 3o47
1.387
.8212, .8813, .9814:
13 sublayers l .1417 , 1 .2218, 1 .4421 . 2.804 3.60
1.6024

 

 

 

Table 2.2 Locations of resonances in a plasma layer
coating a metallic cylinder.

Figure 2.7 is a plot of the back scattered B field from a plasma-

coated metallic cylinder as a function of (mp/w):v with various dimen-

e
sions of plasma layer and glass wall but with a fixed conductor radius
of a I 2.158 mm and a fixed operating frequency of 2.3 GHz. The density
distribution is assumed to be expressed by Eq. (2.117) withcz I 0.6 and
p I 2. The calculation was made based on a l3-sublayer model and those
mathematical resonances are ignored inthe figure. Theoretical calcula-
tion shows that the location of the main (dipolar) resonance shifts to
a larger value of (mp/m):ve. as the thickness of the plasma layer is
increased. 0n the other hand, the location of the quadrupolar resonance
shifts to a lower value of (mp/m):ve. as the thickness of the plasma is
increased.

Figure 2.8 shows the plot of the back scattered B field from a

plasma-coated metallic cylinder as a function of (mp/<11):ve with various
0

radii of metallic cylinder while the dimension of glass wall is kept

31

constant with b I 7 mm and c I 8 mm. The density distribution is assumed
to be expressed by Eq. (2.117) with a I 0.6 and p I 2. It is observed
that the amplitude of main (dipolar) resonance remains approximately the
same while the location of the main (dipolar) resonance tends to move

to a smaller value of (mp/m)ive. as the radius of the metallic cylinder
is increased. In the calculation, the l3-sub1ayer model is again used
and the mathematical resonances are ignored in the figure.

Figure 2.9 shows the back scattered B field from a plasma-coated
metallic cylinder with the dimensions given in Table 2.1 as a function of
(mp/m)ive. with various plasma density profiles. Again the 13 - sublayer
model is used and the mathematical resonances are neglected in the figure.
Three different density profiles are assumed for the plasma layer in the
calculation. The first density profile as shown in curve (1) in Fig. 2.9
is a distribution with plasma density increasing linearly from glass wall

(r I b) to the metallic cylinder (r I a) and can be expressed mathemati-

cally by
2r-b-a
no“. I now [1 - 0.4(T)] (2.118)

where nb.c is the plasma density at r ..2;1 .

The second density profile as shown in curve (2) has a density distribu-

tion linearly decreasing from r I b to r I a and can be expressed by

2r-b-a‘
no“ no,c [l + 0.4(—g:;—)] . (2.119)

The third density profile as shown in curve (3) has a combined distribu-

tions of the first and the second profiles. Its density distribution in

the region of baraggg-obeys Eq. (2.118) and for the region of Eggzrza

32

it follows Eq. (2.119). All curves in the figure are plotted with a
constant collision frequency of vlw I 0.01 and the operating frequency
of 2.3 CH2. It is ovserved in Fig. 2.9 that the density profile of the
plasma layer has little effect on the behavior of the back scattered B
field from a plasma-coated metallic cylinder.

Figure 2.10 shows the back scattered B field from a plain plasma
cylinder which has been discussed in section 2.7 as a special case.
The curve is plotted for a plasma cylinder with the dimension of

b I 7 mm and c I 8 mm. The operating frequency again is 2.3 CH2. A

l3-sub1syer model with a density profile given by (11)
n - n [1—0 «if-)2] (2 120)
olr 0pc . b O

is used in the calculation..The behavior of the back scattered B field
from a plain plasma cylinder is similar to that from a plasma-coated
metallic cylinder. However, the locations of dipolar and quadrupolar
resonances of former cylinder tend to move to a larger value of
(wplm)ive. . Not included in Fig. 2.10, the calculations based on 3,
5 or 7-sublayer model were made and the locations of mathematical,
dipolar and quadrupolar resonances obtained based on different sublayer
models are shown in Table 2.3.

Figure 2.11 is a plot of theoretical back scattered B field from
a plasma-coated glass cylinder which was used in the experiment conducted
by Vandenplas .0) The dimensions of the experimental plasma tube are
a I 1.10 mm, b I 3.97 mm and c I 5.25 mm. The dielectric constant of

the glass is assumed to be 68 I 4.3 and the operating frequency is

fixed at 2.7 GHZ. A uniform density distribution and a collision

33

frequency of vim I 0.005 are assumed in the numerical calculation. The

results shown in Fig. 2.11 agree well with the experimental and

theoretical results of Vandenplas.

The main difference between the

numerical results of our theory and that of Vandenplas' theory, which

was based on quasi-static approximation, is that our theory predicts

finite resonance peaks and a quadrupolar resonance while Vandenplas'

theory yields infinite resonance peaks and total absence of quadrupolar

 

 

 

 

 

 

 

 

 

 

 

resonance.
Model Mathematical Resonance Dipolar Quadrupolar
2 2 2
(mp /w)3ve o (mp “Dave s (mp “Dave o
Uniform None 2.9 3.3
3 sublayers .788, 1.061 3.01 3.727
5 sublayers .791, .932, 1.228 3.06 3.954
7 sublayers '8015’ '9 ’ 1'05' 3.07 4.0
1.336 _ A_
0813, 0885, .911.
l3sublayers .995, 1.1074,l.2616. 3.08 4.0
1.5
Table 2.3 Locations of resonances in a plain plasma

cylinder.

 

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CHAPTER 3

SCATTERING FROM A METALLIC CYLINDER SURROUNDED BY A LAYER
OF LOSSY, HOT PLASMA ILLUMINATED BY A TE WAVE

3.1 Introduction

 

In the previous chapter, using a cold plasma approximation we
have developed a theory for the scattering of an electromagnetic wave
from a plasma-coated metallic cylinder when it is illuminated by a
normally incident plane wave with its 3 field parallel to the cylinder
axis. In this chapter, the surrounding plasma medium is assumed to be
hot and the temperature effect or the excitation of an electroacoustic
wave will be considered. Also the plasma medium will be assumed to be
non-uniform.

This temperature resonance, also known as Tonks-Dattner's

resonance, has received attention from a number of investigators. (14' 15’

16) Some problems related to the present one have also been investi-
gated. Crawford and Kino(l7) studied the mechanism of Tonks-Dattner's
resonances excited in a plain plasma discharge tube. Nait(18) studied
the scattering of an electromagnetic wave by a cylindrical object in an
infinite hot plasma. Fejer(19) studied the scattering of an electro-
magnetic wave by a plain plasma cylinder using a differential equation

(11)
method. Vandenplas and Messiah studied the scattering of an

electromagnetic wave from a plasma cylinder using a quasi-static approxi-

mation. There are other investigators who studied similar problems.

44

4S

Hbuever, to our best knowledge, the problem of the electromagnetic
scattering from a metallic cylinder covered by a layer of non-unifbrm,
hot plasma has not been treated elsewhere.

In the development of theory, the collision loss, the excitation
of an electroacoustic wave and the inhomogeneity of the surrounding
plamma layer are all considered. The stratification method is used in
the analysis. The plasma layer is subdivided into a number of concen-
tric sublayers of sufficiently small thickness compared with the
electromagnetic wave length. The plasma density is, then, assumed to
be a constant within each sublayer so that a step function approximation
of density profile is adopted. In each sublayer of plasma, one can
find two electromagnetic and two electroacoustic cylindrical waves with
unknown magnitudes propagating in opposite directions. These electro-
magnetic and electroacoustic waves are coupled at the interface of two
adjacent sublayers. The magnitudes of these waves are determined by
matching the boundary conditions at the interfaces. This boundary
matching process will lead to the final determination of the scatted
fields in free space.

In order to compare with experimental results, a glass wall is

assumed to surround the plasma in the theoretical model.

3.2 Geometry of the Problem

An infinitely long metallic cylinder with a radius a and covered
by a layer of noneuniform hot plasma is confined inia glass tube with
inner radius b and outer radius c. The plasma-coated metallic cylinder
is placed along the z axis and is illuminated normally by a plane

electromagnetic wave with its E field perpendicular to the z axis and

46

 

 

 

§——»~1
H
O I 1

Incident wave

 

Region 1: free space
Region II: glass wall
Region 111: hot plasma

Region IV: metallic cylinder

Fig. 3.1.(a) A plasma-coated metallic cylinder
illuminated by a TB wave from the
left. (hot plasma model)

 

 

o,l

 

47

 

Fig. 3.1.(b) Stratified hot plasma medium.

48

3 field parallel to the z axis (TE wave). The layer of non-uniform hot
plasma is subdivided into a number of concentric sublayers as shown in
Fig. 3.1 for the analysis. These sublayers are counted from outmost
sublayer and inwardly. For example, the first sublayer is located
immediately inside the glass wall and last sublayer is located immediately
outside the metallic cylinder. The radius between two adjacent mth and
(m+l)th sublayers is denoted as rm. In the mth sublayer, we assume that
the plasma density is no,m’ the collision frequency is uh, the propagation
constant of electromagnetic wave is ke , the propagation constant of

electroacoustic wave is kp m' and the equivalent complex permittivity
D

is gns

The cylinder is assumed to be infinitely long in the analysis
so that there is no field variation along the z direction. The angle of
9 in cylindrical coordinates starts from x axis and increases in the

counter clockwise direction. The time dependence of exp(jwt) is assumed

and the fields of incident plane wave are the same as given in Chapter 2.

They are
i 'n n
Hoz - 2 €°n(-j) cos(ne) Jn(kor) (3-1)
n-o
Hi . u1 = o (3 2)
or 00 '
i a i ° (” n .
EOI‘ 3 - 63%; 85 "OZ 3 3%? 2 Eon('j) 118111019) Jn(kor) (3.3)
o o n-o
I3i 8 j a i 8 2° n ' 3 a
00 (060 E '52 jgo nao €on(-j) cos(n0) Jn (kor) ( ° )
Bi . o . (3.5)

49

In this expression the superscript ”i" represents the incident wave.

k0 is the prepagation constant of free space. EEon is the Neumann
factor defined as Eon - 1 when n-0 and eon - 2 when nfio. Jamar) is
Bessel function of first kind with integer order n and argument kor.
J;(k°r) is the first derivative of Jn(kor). 6 o is the permittivity of
free space. go is the impedance of free space and is defined as

go - /-€3 . 120" ohms where “o is the permealflity of free space.
0

3.3 Fields in the Regions of Free Space and Glass Wall

 

In these regions the Maxwell's equations are the same as those
in the cold plasma case (Chapter 2) and thus, fields in these regions
remain the same as that in Chapter 2. ApprOpriate solutions for the
fields in these regions are reproduced from Chapter 2 as follows:

The total fields in free space are

Hots - “go cos(n0) [eon(-j)an(kor) + H(:)(kor) an] (3.6)
not: - "ate - o (3.7)
Ba: - 521:1? ninsifinfi) [e on(-j)an(kor) + H(:)(k°r) an] (3.8)
Beta - jgo “:3: cos(n0) [eon(-j)“J;<kor) + aging) An] (3.9)
so: - o . (3.10)

The superscript ”t” represents the total fields (incident wave plus
0

reflected wave). H(:)(kor) is Hankel function of second kind. 8(izkor)

is the first derivative of H(:)(kor). An is a constant to be deter-

mined by boundary conditions.

50

The fields in the glass wall region are

°° <1) (2)
H52 - n50 cos(n0) [n n (ksr) 3n + H n (ksr) en] (3.11)
Her :- "so - o (3.12)

Egr :- Fla—17 2 nsin(n0) [H(:)(k8r) 3n + H(:)(k8r) On] (3.13)

o g n-o
o °° (If (2)'
£89 j/E—g “ED cos(n0) [H n (kgr) Bn + H n (ksr) Cu] (3.14)
£82 I O (3015)

where k8, the propagation constant of glass, is defined as k8: (‘yuoeoég

with 638 as the dielectric constant of glass. Bn and Cn are the constants

to be determined by boundary conditions.

3.4 Fields in Hot Plasma Regigg_

In the plasma region (bzrza) the plasma is considered as an one-
component electron fluid and the ions are neglected in the equations of
motion. The presence of ions is, however, required to neutralize
electrical charge in the plasma. The stratification method is used in
the analysis. The inhomogeneous plasma layer is subdivided into a number
of concentric sublayers with sufficiently small thickness. The plasma
density is then considered to be a constant within each sublayer but it
varies from sublayer to sublayer in the radial direction. In the mth
sublayer the plasma density is assumed to be no,m° The collision
frequency of a electron with neutral particles of gas is assumed to be
v5. The density deviation of electrons from the mean no m is assumed to

’

be n1 m and the velscity of electrons induced by the fields is assumed
9

tobeV.
m

51

It is assumed that the perturbation of plasma due to the fields
is sufficiently small that the linearized equations are applicable. No
static magnetic field is present in the analysis. The time dependence
of exp(jmt) is assumed and the Maxwell's equations in the mth sublayer

of plasma region are

V x an - - M10 am (3.16)
V x H. ," -eno,m vm + jweo Em (3.17)
v '° ° m
° I - nun—l—
Em 6 (3.18)
o
V-fim - o . (3.19)
The linearized continuity and force equations are
no’mV-vm +jm1’m .. o (3.20)
2
(\«h 4' jw) Vm ' - 3'1- Em - fiVfll'm (3.21)
D

where e and M are the electron charge and mass respectively. V0 is the

r. m. s. velocity of electrons which is considered to be constant through-
out the plasma region and is defined as vO - kT/M where k is Boltzmann's
constant and T is the electron temperature. The last term in Eq. (3.21)
represents the force due to the pressure gradient, and vo-JBiT7i' is valid
on the assumption of adiabatic pressure variation and one dimensional

compression.(20)

In our formulation of the problem there are four unknowns to be

solved. They are m’ “l,m’ Em and Vm . We will determine an and nl,m

52

first and then calculate E; and 3;.

From Eqs. (3.17) and (3.21), we obtain two expressions as

V x V x Hm = - eno,m V x Vm+ waon Em (3.22)
and
-o — - e -o
v X Vm - W VX Em . (3.23)

The substitution of Eq. (3.23) in Eq. (3.22) yields

2
en
~ _ o m . r
v x vx am -[ 74“” Vmfi‘” + Jweo ] v x Em . (3.24)

Expressing ‘J’x E; in terms of a; as in Eq. (3.16), Eq. (3.24) can be

rewritten as

2
w
v x vx Hm .. (0|).oeo[1 + jwwmfiw :lnm (3.25)

where mp m is the plasma frequency defined as
9

w 3 ___‘lz.’.“. (3.26)

Equation (3.25) can be expressed as

8,111 m

vxvxg=

where ke m is the complex propagation constant of the electromagnetic
I

wave in the mth sublayer and is given by

k8,!“ 3 w lilo gm (3.28)

with gm, the equivalent complex permittivity, defined as

53

“,2
I e (1 .- L) - M (3.29)
{m cl: (0 2w“ 2 (”(1112 4A»: ) ]
If we express ke,m as
kenn " Benn " 3 “e... (3.30)
Be,m and (1e m can be determined to be
9
2 a 8
5 . 9.9. 1-m§zn+[1-_21_+—2£Tm T (3.31)
e,m 5 w +v «124-é mzwz-w )
m m
B wz 2m2 ma :5 3
.2... .1h... -__£u_.+___21____
a - 1 + + (3.32)
e’" 5 (0 2mm on 2% 2 w2(w2+ v2) .
Be m and ct.e m are the wave number and the attenuation constant of the

electromagnetic wave in the mth sublayer of the plasma and so is the
wave number of free space defined as 30' - (DIED—é; .

In the mth sublayer both incoming and outgoing (reflected) waves
can exist. Before solving Eq. (3.27) the if fields of incoming and
reflected waves will be assumed to have 2 component only. This
asstmption can be justified from Maxwell's equation and the symmetry
of geometry. Since the incident 3 field has no 2 component and the

symmetry of geometry provides no variation along 2 direction such that

5% . 0, Eq. (3.16) can be reduced to

—o

as
l a «m a _
1-[35 “ Ema) '13-]2 " “5"“. "m ° “'33)

Equation (3.33) shows that am is allowed to have a 2 component only.

Thus, we assume that

-o a
Hm - Hmz . (3.34)

Equation (3.27) can now, be reduced to

2 2

a H a a H

—%£+%_$i+.lf—_%£+kimfimz 0 . (3.35)
3: Br r 66 '

Equation (3.35) can be solved by the method of separation of variables.

We assume that

Hm: I H(r) 11(9) (3.36)

where Hm(r) and 8mm) are functions of r and 9 respectively. The
substitution of Eq. (3.36) in Eq. (3.35) leads to

2

 

2 _
r 3 Hz“) + r BH(r) + k: m r2 .3 n2 (3.37)
H(r) 3r H(r) a: '
1 5214(9) 2
_....._...2_. n . (338}
8(9) 30

Considering the symnetry of the problem and the degeneracy of angle 6,
the solution for MS) is
8(6) - cos(n9) (3.39)

where n is an integer. Equation (3.37) can be rearranged to

2 2
mf+—-L—- M+(l--——Q—-§) H(r) o 0 (3.40)
3(ke'mr) (ke,mr) 3(ke'mr) (ke,mr)

which is a Bessel equation.
Since both incoming and outgoing waves are expected to exist the proper
solutions for H(r) are Hankel function of first kind and second kind.

Therefore, the final solution for "m2 can be written as

SS

. (1) (2)
"ms “.250 cos(n9) [H n (ke.mr)nm,n + H n (ke,mr) Fm,n:l (3.41)
where on and Eh n are constants in the mth sublayer and will be
9 3

determined by boundary conditions. It is noted that in the expressions
of QM,“ and Pi,“ the first subscript m specified the sublayer and the
second subcript n specifies the index of summation.

Up to this point, the magnetic field a; has been determined and

the next quantity to be solved is the density deviation, of

nl ,m'

electrons from the mean. Taking the divergence of Eq. (3.21), we obtain

2
v
0... 8 - as... a... — M 2
(vm + jw)v V“l H V Em no v n1,m . (3.42)
From Eq. (3.20), V71Vm can be represented in terms of nl,m as
jwn
a... I - —-—-L'2
V vIn n . (3.43)
o,m

Substituting the quantities, V‘ {in and V'Em from Eqs. (3.43) and

(3.18). into Eq. (3.42), we obtain a homogeneous wave equation for n1 m
D

2

p,“ 111 'm 0 (3044)

2
V “l,n +k

where kp m is the complex propagation constant of the electroacoustic
0

wave and is expressed by

2 1 2 2 .
kp'm - :i-[Qo - wpm) - 3‘”va . (3.45)
0

If we ex ress k as
p Pom

kpsm ‘ spam - jd$’m ' (3.46)

56

B m and up,” can be determined to be

1 5;
Bp,m - fiv{m2-m n-1»2l:(m - .2)+ (02 v i] F, (3.47)

O

1,
_ l. _ _ 2 2 2 2 5
up,” fi' {2 m + a): NEG» comm) + (o um] } (3.48)
o

 

 

where 8p m and (1p m are the wave number and the attenuation constant of
D 9

the electroacoustic wave in the mth sublayer of plasma medium respective-

IYs
With the symetry condition of «392- - 0, Eq. (3.44) can be reduced
to
2 2
3 n B n b n
--}2-12+-%--—-}-LE+-}2——;-L"—'+kzmm1m = o . (3.49)
at at 1‘ 59 Po 3

Equation (3.49) has the same form as Eq. (3.35). Following the same
procedures as we used in solving Eq. (3.35), the proper solution of

"l,n can be expressed as

n . z: sin(n0)[H(r1‘)(k r) c + "(12'1)(kp,m r) 1mm] (3.50)

I'm an p.11! m,n

where cm 11 and Im 11 are constants to be determined by boundary conditions.
9 9

The density deviation of electron is completely determined. The
next quantities to be determined are the induced velocity of electrons,
V7”, and the electric field, Em . From Eq. (3.17), the E field can be

expressed in terms of Vx in and Vm as

"' om
sm- T150 Vxfi m+-——l--vm . (3.51)

Substituting Eq. (3.51) in Eq. (3.21) and after rearrangement, it yields
VZE

--0

“vii“, 1.,qu Tm +31» Vnm . (3.52)

57

In Eq. (3.52), the quantities Em and 111 m have been determined before.
0

Taking the curl of the magnetic field am as expressed in Eq. (3.41) and

using the symetry of geometry, Vx am can be obtained as

'— . _ 1L. °° (1) (2) .
V x Hm r {ninsin(n9)[ H n (kemrmmm + H n (ke,mr)Fm,n] }r

so ' '
(1) (2) .
.. ke'm{nEocos(n9)[H n (ke,mr) Dm,n + H n (ke,mr)Fm,n]} 9

(3.53)

Similarly, the quantity Val m can be derived from Eq. (3.50) as
9

°° (1i (2)' .
“261nm” [a n (kp'mr) cm,n + a n (kp,mr)1m’n]} r

vnl,m = kp,m{

+ -;-{ 2 ncos(n6)[H(r1‘)(kp, r) G + "(3)05 rnngnj} 8

m m,n ,m
n-o
(3.54)

Substituting Eqs. (3.53) and (3.54) into Eq. (3.52), the induced velocity

CI.

of electrons ,Vm, can be obtained explicitly. The components of Vm vector

 

are
-je a. .
vm - flmem-o-er nins1n(n0)[Rmm(ker, s,n]
”:56“ m co '
- W “)30sin(n0)[Rm,n (kpr, G, 1)] (3.55)
V 9 - gig-7 Eocoflne) [11. (k r, D, 10]
m w I! vm+3w n-o m,n e
2
- W n:’,3':ol'1¢08(fle)[Rm’n(kp‘r:, G, I) :l (3.56)
V 8 0 (3.57)

mz

where

58

Rm’n(ket, D, F) ' H(:)(ke,mr) 0m,“ + H(§)(ke,mT) F5,“ (3.58)
R;,n(kpr, c, I) - H(:).(kp,mr) cm,n + H(r2‘).(kp,mr) 1mm (3.59)
nL’nuer, n, F) - a‘:)'(ke,mr) um,n + n‘ifmemr) Fm,“ (3.60)
Rm'n(kpr, c, I) - H(:)(kp'mr) Gm,n + H(:)(kp’mr) 1m,n . (3.61)

The last quantity to be determined is the electric field E; .

From Eqs. (3.51) and (3.21), the electric field E; can be expressed as

 

2
30"
1am - jmgm :xnm+m§m vn+jw vnlun . (3'62)

Substituting Eq8.(3.53) and (3.54), into Eq. (3.62), the components of

E; field can be obtained as

5‘
Em . din—r ninsin(n9) [summer-,nfifl

 

 

GVZR °°
+ j o p,m 2 sin(n9)[:R. (k r G 1)] (3.63)
w§h(§h+jw) “-0 man P ' '
11‘, m °° .
Emo - EioT‘L' }: cos(n6)[Rm,n(ker, D, U]
n: n-o
2
w" w (3 as)
+ wngvmi'jm)!‘ “.230 ncos(n0)|:Rm’n(kpr, G, 1)] .
E B o (3065)

mz

I
where the expression for those [Rn(k r):] s are given in Eqs. (3.58) to
(3.61). Up to this point, all the relevant quantities, Hi, nl,m’vm and Em,
have been explicitly determined. Those unknown constants associated with
these four quantities will be determined by matching boundary conditions

in the next section.

59

3.5 Hatching Boundary Conditions at Interfaces
Four relevant field quantities in the mth sublayer of the plasma

medium are E;, EL, n1 and V; . Associated with these field quantities,

there are four unknown constants, D F . In applying

m,n’ m,n’ Gm,n' Im,n
the stratification method, the field quantities are matched at the
interface of two adjacent sublayers and the four unknown constants in a
sublayer are expressed in terms of the corresponding four unknown
constants in the adjacent sublayer. Since there are four unknown
constants to be determined, four independent boundary conditions are
needed at the interface. we will derive four independent boundary
conditions from four basic equations used in Section 3.4.

For the needed four boundary conditions, three of them are rather
conventional. They are the continuity of the tangential components of
electric field and magnetic field and the continuity of the particle
flux. The fourth boundary condition is not a trivial one and is not

uniquely known since various forms are used by various workers.(18'21'

22’ 23’ 7) In our study, the fourth boundary condition will be
directly derived from the force equation.
Let us consider the boundary condition at r - rm which is the
interface between the mth and (m+l)th sublayers. .
Equation (3.16) will readily lead to a boundary condition of
tangential component of E field is continuous.
In the present study, this implies that

no"

(”l-)6 .t r - rm 0 (3.66)

Equation (3.17) can be used to derive a boundary condition of

6O

tangential component of if field is continuous
if no surface current is assumed on the interface. In the present study,
this condition gives

H

mz - H(m+l)z at r - rIn . (3.67)

The third boundary condition will be derived from the original form

of Eq. (3.20). The original equation for the continuity of particles is
q one
V-(ne v) + 'SE' - o (3.68)

where ne is the total electron density which is the sum of the average

electron density no and the density deviation n Integrating Eq. (3.68)

1 0
over the pill box as shown in Fig. 3.2, we have

an
"" e
fV‘GleV) dV 4' f-SE- dV 3 0 e (3.69)

If we let the increment dr approach to zero,the

 

limiting case will be

 

lim J‘V-(neV-I.) drdA + 5% lim I nedrdA - 0 .
Fig. 3.2 The interface dr-o dt"0
at r I rn (3.70)

The last term of the above equation approaches zero, provided ne is
finite accross the boundary. Thus, using the divergence theorem, the
above equation can be written as

lim J‘ (navyfi ds . o (3.71)

dr~1>

where n is the unit vector pointing outward of the surface of pill box.

This equation then leads to

ne,m er - ne,m+l v(m+1)r 0 at r - rm (3.72)

61

where subscripts m and m+l identify the sublayers and subscript r denotes
the radial component. Since the total electron density ne m is
D

he,m I “o,m + nl,m a “o,m , (3.73)

under the linearized assumption, Eq. (3.72) can be reduced to

“o,m Vin - no,m+l‘v(m+l)r 0 at r - rm . (3.74)

Equation (3.74) is the third boundary condition to be used in our analysis.
The fourth boundary condition will be derived from the original

form of the force equation. The force equation as expressed in Eq. (3.21)

is a linearized form containing only a.c. component. The original ferce

equation contains both a.c. and d.c. components and can be expressed as

a)? - e-° 1 e-° 1
at-O-vv '3 --i-Ft-B:EVP.3 "'fi'Et'njfivP (3.75)

where v is the collision frequency, P is the pressure and E; is the total
E field including both a.c. and d.c. components. Mathematically we write

Et . EdeCe +5 e (3s76)

The gradient of pressure can be shown in Appendix A to be
VP :- kT Vno + 3 kTan (3.77)

where the term .Vno, gives d.c. component and an gives a.c. component.
Substituting Eq. (3.77) in Eq. (3.75) and taking d.c. component out of,

Eq. (3.75) we have

= - .2.“ - fl.
0 M Ed.c. noM V"‘6 ' (3‘78)

Equation (3.78) implies that there exists a static E field if the

62

stationary plasma is non-uniform. Physically, it means that the
stationary density variation is maintained by a static electric ferce
acting on the electrons. In our analysis using the stratification
method, step density discontinuities are assumed to approximate a
non-uniform density profile. Therefore, delta function type of static
E fields should exist theoretically at the points of step discontinuities
or the interfaces between sublayers. This phenomenon is explained
graphically in Fig. 3.3.

The a.c. component of Eq. (3.75) is

v2

. -* e -* o
(v + 3w) V I - F E - a: an (3.79)
for the exp(jwt) time dependence. Integrating Eq. (3.79) over the pill

box shown in Fig. 3.2, we obtain

J' (vfiMVdv . -—§— J‘Edv‘viffil'; andv. (3.80)

The limting case of Eq. (3.80) as dr approaching zero is

lim J‘(v+3w> '17 drdA . - lim 7} I E drdA - v: J‘ 31-an drdA.(3.81)
drdo drdo o

In Eq. (3.81) the first two terms approach zero since the volume goes to
zero and the quantities in the integrands, V'and E, are finite accross

the boundary. Thus Eq. (3.81) becomes

lim V: f a!" V111 drdA ' 0 e (3.82)
dr-mi 0

Since the step discontinuity of density is balanced out by the static
3 field and the density is constant within each sublayer, Eq. (3.82)

can be expressed as

deCe

 

63

 

 

 

deCe

 

 

 

 

(a) (b)

Fig. 3.3 Stationary electron density profiles
and associated static Ed.c. fields.

64

I“
lie J‘V-fil drdA - o . (3.83)
dr-to o

This step is justified because the singular part of Vno has been taken
into account in Eq. (3.78). Applying the gradient theorem, we have at

the boundary of r- rm

n n
3.1.1.3-..13211 . o atrcrm . (3.84)

o,m no,m-+1
Up to this point, four independent boundary conditions have been
derived in Eqs. (3.66), (3.67), (3.74) and (3.84). Applying the boundary

condition of Eq. (3.67), we have

as Q
E cos(n0) Rm,n(kerm' D, F) s z: cos(n0) Rm+l,n(kerm' D, F) .
3'0 mm
(3.85)
Due to the orthogonality of cos(n0) function, Eq. (3.85) leads to
(1) (2)
H n (ke,mrm) Dm,n + H n (ke,mrm) Fm,n
, (1) (2)
H n (Rem-urn) Dm-l‘l,n + H n (ke,m+lrm) Fm+l,n . (3'86)

The boundary condition of Eq. (3.66) can be used to derive an

expression such as

k 0 k C
e m (l) e m (2)
i H n (ke,mrm) I)m,n + E H n (ke,mrm) Fm,n

 

 

evgn Hall)“p mrm) evzn “(r21)(kp,m rm)
+ 1 G + I
§m(vm+jw) rm m,n {m(vm+jw) rm m,n

k ' k s
_ e m+1 (l ) e m+l (2 )

6S

1)
inn“ (kp,m+lr m

5:+1( “h+1+5m) rm

(2
)G av: nH n)(kp,m+1rm)
m+1,n+

g...,1(vn+1+jw) I‘m

4'" I e

m+1,n

(3.87)

From the boundary condition of Eq. (3.74), we obtain

jen
°'"‘ 2 nsin(ne) R (k r , D, 10]

«gm M(vm+jw) ‘rm rule 111, n e m

v25 k

+-2—-9—2-'— )3 sin(m)[:RIn n(k prm’G' 1):]
§h(vm +jw) n-o

 

jen
o,m-+1
E nsin(ne) (k r . D. F)
(”gm-*1 “(vm 1+”) rm “-0 [R Rm+1,n e m J

V26 k

0 op,m+l g sin(19)
gm+1("m+1+jw) n”

Due to the orthogonality of sin(tfl) function, it yields

+ (kp r , G, 1)] . (3.88)

Rm+1,n m

(1) . (2)
jenolmn H n (ke,mrm) D + Jenoilnn H (k e1mrm)

m,n

F
m,n

gnu) M(vm+jw) rm {mm M(vm+jw) rm

2 k “(I)“ r) 2 k H(2)(k r)

v v
oeomeH n p,mrm G + E

+ o o p,m p1mrm
§m(vm+jw) m,n §m(vm+jw)

I
m,n

(1) (2)
0‘ o,m-I-ln H n (keLm-tlrm)

§m+1w “(vmfl +3111) rm

0 ,m+ln e ,m+lrm) + jen

{1le M(vm+1+jw) rIll “1'“

jen

 

m+1,n

H(l)(k r ) :6

H(2)
p,m-i-lmG + okp,m+l Wn(p,m+lm
m+1,n

g‘n+1(vm+1+jw) §m+1(vm+1+jw)

(2,6 )

okpm-i-l

(3.89)

Im+1,n '

66

From the boundary condition of Eq. (3.84), we obtain

 

 

 

H<:>( “(2%k
ler m G p,m m
no ,m m,n “o,m
”(Ink
. kp.9+1r m +
no,m-fl 9H1“!!!

m,n

p,m+1 m

 

no,m+1

m+1 ,n

. (3.90)

Equation (3.86), (3.87), (3.89) and (3.90) can be written in a matrix

form and after rearrangement we obtain a matrix equation as

 

 

where

-1
l“note,In’kp,1n’rm)]

’—

 

and

Lm,n(1'1) Lm
Lm,n(2’1) Lm
Lh,n(3'1) Lm

I'm,n(l"1) Lm,n

'n(1.2)
,n<2.2>
,n<3.2)

(4,2)

[Ln(ke,m+1 'kp,m+1 ’rm)]

Lm,n(1'3)
L ,n(2'3)
L ,n(3'3)
Lm'n(4,3)

I [Ln (ke,' m' k,p m'r-m)]1[Ln (ke,m+1’kp,m+1’rm):l

Lm,n‘1'“>
Lm,n(2'“>
gm,n<3.a)

Lm'n(4,4)-

 

 

m+1,n
m+1,n

m+1 ,n

__ m+1,nd

 

(3.91)

(3.92)

with

 

Lm+1,n

Lm+1,n

L

m,n

Lm+1,n(1'1)
Lm+1,n(2’1)

(3.1)
(4.1)

(1.1)

Lm,n(1'2)

L ,n(1'3)

Lm,n<1'“)

Lm,n(2'1)

Lm,n(2’2)

Lm,n(2’3)

Lm’n(2,4)

Lm,n(3'1)

L ,n(3’2)

Lm+1,n

ym+1,n

(1.2)
(2.2)

Lm+1,n(3'2)
Lm+1,n(a’2)

H(1

H(2)(k
n e,m

k

elm

§m

’(k

67

Lm+1, n

(1 3)

Lm+1,n(2'3)

Lm+1,n

Lm+1,n

r)

e,mr m

H(:)(k , r

r )

m

emm

(3.3)
(4.3)

)

k 0
e m (2)
1- " n (“ma")

ev 2n H

O

ev2 n H
o

jen

jen

n

<1>(k

r
pjm m

2§m(vm +jw) rm

n

<2>(k

p,m 111

gm (v m+jw) rm

n H(1)(k:
glmnm

o,m

)

r )

r )

gmm M(vm+jw) rm

o,m

(2)
n H n (k

r )

ejm m

gmw M( vm+jw) rIn

3:" it” it" it"

1mam.)
,n<2.4>
,n<3.4>

'n(4.4)

 

(3.93)

(3.94)

(3.95)

(3.96)

(3.97)

(3.98)

(3.99)

(3.100)

(3.101)

(3.102)

(3.103)

b"n(3,3)

Lh,n(3'4)

1n,n(a.1>

Lu'n(6,2)

Lh'n(h,3)

y..n<4.41

I'm-i-l ,n

Ln+1 ,n

(1.2)

Lm+1,n(1' 3)

I'm-O-1 ,n

I"m-O-l. ,n

LII'H. ,n

(1.4)

(2.1)

(2.2)

68

vie 1111(1)“ r )

0 pg: n
(“(gi+1n)
2 n‘z’ck r)

060k p11: n phar-

§m( vnfiM

1’“ fi

0

H(1)(k r )

pm to
n
o,m

"(2)“ r )

p111: III

11
o,m

(1.1) - n‘;’<k r )

e,m+1 m

11‘: )(k r )

9.11144. In
0
0
kelm-1 Ha)“ r )
{n+1 n e,m+1 m
-—!——-—k° “"1 110%): r )
gun-1 n e,m+1 m

(3.104)

(3.105)

(3.106)

(3.107)

(3.108)

(3.109)

(3.110)

(3.111)

(3.112)

(3.113)

(3.116)

(3.115)

Lm‘I-l ,n

Lm+1,

Lm+1,n(

L,m+1 n(

I"111-1-1,n('

I"m-I-l ,n(

Lm+1 ,n

L,m+1 n

(2. 3)

n(2.4)

3,1)

3,2)

3, 3)

3.4)

(4.1)

(4. 2)

Lm+1,n(a’3)

L +1,n(4'4)

V060 p,m+1 n

v06 0 p,m+1

69

(1)(k r )

evonI-ln p,m+1m

§m+1(

vm+1+3w) rm

H(2)(k r)

6V2
or1

§m+1

k1.p,n1-l-1 m
vm+1+jw) I11:

(1)
o,m-11n H n (keLmlrm)

§m+1w ”(Wmfll

jen
+Jw) rm

H(2)(k r )

Jen e, m+1 m

o,m-+1

gmlw M(vm+

2 (1)
k H (k p,m+1rm)

1+311») rm

n+1+51»)

5m+1“

H(2)
k(kp1m+1rm)

gm+1("'xn+1+jw)

(1)
H n (kp,m+1rm)

no ,m+1

(2)
n (kp ,m+1rm)

no ,m+1

(3.116)

(3.117)

(3.118)

(3.119)

(3.120)

(3.121)

(3.122)

(3.123)

(3 .124)

(3.125)

70

Up to this point, it is possible to express four unknown constants of

the mth sublayer in terms of the corresponding four unknown constants

of the (m+l)th sublayer.

the interface of r a r

n+1

By applying the same boundary conditions at

between the (m+l)th sublayer and the (m+2)th

sublayer and following the same procedure, we can obtain a relation

between the unknown constants in the (m+l)th and the (m+2)th sublayers

 

Substituting 0m+

into Eq. (3.91), we can express Dm n’ F
3

D

m+2

 

P

P

I
L m,n

Dm+1,n

l=111.-I-1l.,n

Gm+l,n

Im+1,n

”m,n?

F
m,n

G
m,n

 

-

 

d

-1'
. [Ln(ke ,m+1 ' kp ,m+1 'rm+1)] [Ln (ke ,m+2 ’kp,m+2 ' 1rm-l-l )]

l,n'

,n' FnH-Z,n' Gm+2,n

-1
[Ln(ke ,m+1 ’ kp,m-I-1 ' T:m-I-l )] [ Ln(ke,1‘u+2 'kp,m+2 ’ 1-m+l )]

§m+l,n’

and I

Qm+l,n

m+2,n

and I

my

m+1,n

n’ Gm

" 1
Dm+2,n

Em+2,n

Gm+2,n

_Im+2,n

(3.126)

 

 

as expressed in Eq.(3.126)

,n

and I
m

in terms of

'3 [Ln(ke,m' kp,m' rm)]-1 [Ln(ke,m+l’ kp,m+l' 1‘.m)]

”D _
m+2,n
Irun-Hm

Gm+2,n

 

 

(3.127)

71

Following the same procedure of matching the boundary conditions

at the interfaces of sublayers successively, it is possible to express the

 

 

 

 

unknown constants of Dl,n' Fl,n' Gl,n and Il,n in the first sublayer
in terms of the unknown constants of Ds,n' Pg,n, Gs,n and Is,n in the
last inmost sublayer as
F Dl,n
Fl,n _1
61 n . [Ln(ke,1’kp,l’rl)] [Ln(ke,2'kp,2’r1)]
9
-11,“-
-1
[Ln(ke’2,kp,2,r2)J [Ln(ke'3,kp’3,r2):l . . . .
"D _
s,n
F. n
-1 '
° ° [Ln(ke,s-1’kp,s-l’rs-l)] [Ln(ke,s'kp,s’rs-l)] Gsn
D
-Is,n J .
(3.128)
Equation (3.128) can be expressed in shorthand as
fnl'n Hanan) Mn(1,2) Mn(l,3) Mn(1.4) fps,“
F1,n Mn(2,1) Mn(2,2) Mn(2'3) Mn(2,4) Fs,n
- (3.129)
Gl,n Mn(3,l) Mn(3,2) Mn(3,3) Mn(3,4) Gs,n
_ 11m- _Mn(4,1) Mn(4,2) Mn(4,3) Mn(4,4)_ _15'n_

 

 

 

 

 

 

where Mn(i,j)'s are the entries of the matrix which is the product of
those [LnJIatricea in Eq. (3.128).

Let's now consider the interface between free space and the

72

glass wall at r - c. The continuity of the tangential components of E

and 3 fields leads to

02 8 H82 at r = c (3.130)
and

at = 1: at r = c (3.131)

09 89

t t . .
where H02 and Boo are given in Eqs. (3.6) and (3.9) and ng’ EgO are
given in Eqs. (3.11) and (3.14) respectively. The substitution of Hg;,

t . _
300’ H82 and 389 in Eqs. (3.130) and (3.131) leads to

H(:)(koc) An + H(:)(k8c) 8n + H(:)(kgc) Cn = 6'0n(-j)an(koC)
(3.132)
-H(2)(ko c) An +—1H(1)(kgc) Bn +—1H(:)(kgc) Cn

13." F;

n 0
' 6 0710-1) J11 (kOC) o (3.133)

Next, we consider the interface between the glass wall and the
first sublayer of plasma at r 8 b. The continuity of the tangential

components of H and E fields lead to

”(2%k

(1) H(2) (1)
- an (kgb) an - (R 8b) on + a (k e,1 b) Dl'n+ e.1 b) F1,n

3 0 (30134)

73

(1). _ go H(2) “o H(l)

 

8 8
+ [u o H(2)'(ke H) ev on
'§;H e,1 +w§1201+jw3b
evzn Hm
+w§1(v1+jw)bH (kpp,l b) Il,n = 0

H(1)(kp

b) G

l,n

(3.135)

Two additional boundary conditions for the induced electron

velocity on the rigid surfaces of the glass wall and the metallic

cylinder can be used. Assuming that the surfaces of the glass wall

and the metallic cylinder are rigid, the normal component of the

induced electron velocity at those surfaces can be required to vanish.
(18, 24)

This boundary condition has been used by numerous workers.

Applying this boundary condition to the r component of induced electron

velocity, vr, at r I b and r I a and using Eq. (3.55), we obtain

JEE|H(1)(R

wa + j_r_1_ H(:)(k

b) D

e,l 1,ne,1

+_2___£L1.H(1)(kp b) C

no,1 np,11,n

and

b) Fl,n

0,1

JSE-H(1)(k a) 08 n+ H(z)(ke a) F

wfla n e,s mMa e,s s,n
2 . 2
1! 63 k 1r 6; k
+ o o p,s H(1)(k b) C + o o p,s
no 5 n p,s s,n no 3
9 9

Jr":"€¢>;_5_,1H'p(2)(k
n n

(2)
H n (k

p35

1b) Il,n .
(3.136)
a) Is,n =

(3.137)

0

74

Since the metallic cylinder is assumed to be perfectly conducting,
the tangential component of E field is required to vanish at its surface.

This boundary condition will lead to

k H(l)(k a) D +'K H(2)(k a) F

e,s n e,s s,n e,s n e,s s,n
+ W H n (kms‘m +W "n 0‘9 "01..“ ' 0(3'138)

Seven boundary conditions as expressed in Eqs. (3.132) to (3.138)

contain eleven unknown constants, An, Bn' Cn, D1,“, Fl,n’ 61,“. 11'“:

Ds,n’ Fs,n' Gs,n' and Is,n . However, the constants, D1,“, Fl,n’ Gl,n

I can be expressed in terms of the constants, D , F , G , I

1:“ s,n s,n s,n s,n

in a manner as indicated by Eq. (3.129). With this substitution, we
obtain a set of seven independent equations with seven unknown constants,

, F , G and I . This set of simultaneous equation

An' Bn' cn’ Ds,n s,n s,n s,n

can be expressed in a matrix form as

r'An ‘ "Qn(1,8fl
Bn QnC2.8)
cn o

[an ”s,n ' 0 (3.139)
F o
s,n

as,“ o

1.. 18,1! .1 .. O J

 

 

 

 

where [9n] is the matrix given by

O
[on] - o
0
O

O

 

The matrix entries Qn(i,j)'s are expressed explicitly as follows:

Qn(1.1)
Qn(1.2)
Qn(1.3)
Qn(1.8)
Qn(2.1)

Qn(2,2)
Qn(2’3)

Qn(2.8)
Qn(3.2)

Qn(3.3)

0

0

0

7S

'qn(1,1) Qn(1,2) Qn(l,3) o
Qn(2.1) Qn(2.2) Qn(2.3) 0
Qn(3.2) Qn(3.3) Qn(3.4) Qn(3.5) Qn(3.6) Qn(3.7)
Qn(4,2) Qn(4,3) Qn(4,4) Qn(4,5) Qn(4,6) Qn(4,7)
o Qn(S,4) Qn(S,S) Qn(5,6) Qn(5,7)
0 Qn(6.4) Qn(6.5) Qn(6.6) Qn(6.7)
o Qn(7,4) Qn(7,S) Qn(7,6) Qn(7,7)

- H(:)(koc)
H(:)(k8c)
H(i)(kgc)

e moi)“ Jn(koc)
-H(:;(koc)

1 (11
-—- H n (kgc)

<21
-— H n (kgc)

. n '
€°n(-J) JnOtoc)
(1)
-H n (kgb)

_H(2>

n (kgb)

 

(3.140)

(3.141)

(3.142)

(3.143)

(3.144)

(3.145)

(3.146)

(3.147)

(3.148)

(3.149)

(3.150)

Qn(3.4)
Qn(3,5)
Qn(3,6>
Qn(3,7)

Qn(4,2)

Qn(4,3)

Qn(4.4)

Qn(4,5)

Qn(4.6)

76

(1)
H n (ke,1b)

(1) (2)
. H'n (ke,1b) Mn(1,4) + H n (ke'

- féf H(:)(k8b)

- 7%; H(i)(k8b)

8

“'0 (1). Fan (2).
'Enn(ke111',b)M(11)+ g1 (ke 1b)Mn(2.1)

ev 2n H(1)(k
+ 0 13,1

w§1(vi+jw)b

b)
M (3,1) +

(1)
+ evon H “(131th1 )

Mn(3,2) +
w§1(vi+jw)b

ev2 n H(1)(k
+ o n p,1

w§1(vi+jw)b

b)
M (3,3) +

(1) (2)

H n (ke,1b) Mn(l,l) + H n (ke,1
(2)

Mn(1,2) + H n (kc,

(1) (2)
H n (ke’lb) Mn(1,3) +1H a (Re,

evon H 13,11)

w§1(v1+jw)b

“‘o (1). [3811(2)
. j; a n (kelb)Mn(1’2) + g1 (ke 1b) Mn(2.2)

evZnH
o

evon H n leb

w§1(v1+jw)b

b) Mn(2,1)

1b) Mn(2,2)
1b) Mn(2,3)

1b) Mn(2,4)

(2>(k

(2>(k

(z>(k

Pp
w§1(v1+jw)b

8 “o (l). __<>_LL H(2)
V['§,-1-Hn(ke1.n'b)M(13)+ —§-1- m(k,b) Mn(2.3)

(3.151)

(3.152)

(3.153)

(3.154)

(3.155)

(3.156)

M (491)

(3.157)

M (4.2)

(3.158)

M (4.3)

(3.159)

Qn(4.7)

Qn(5.4)

Qn(5,5)

Qn(5,6)

Qn(5,7)

77

M'0 H(1)(ke 1b) M “(1,4) 4.]: H(2)'e’(k 1b) M (29“)
519:1 §1 n

(1) (2)
av: n H (k b) ev 2n H (k b)
+ P11 M (3,4) .1. o n 2’1 Mn(4.4)
«151‘ ”1””) b n w§1(“1+j'”) b

 

 

(3.160)

<1>(k

jen H b) 59“ H<2>(ke1b)

 

 

e,1 Mn(1:1) + e, Mn(2'1)
wa wa

v26 k H(1)(k

2 H(2)
009.1 11 ”1111) VOER (1)

o 1
Mn(3'1) + plan p, Mnm'l)

 

+

11

0,1 0,1

(3.161)

Jen H(1)(ke 1 b) jen H<2)(k b)

 

 

9’ Mn(1,2) + n e'1 Mn(2'2)
Mb . “Mb

. I
k k b) V e k H (k b)
+ o 0 9.1 n .p.1 Mn(3'2) + 0 0.211 n .911 uh(a,2)

n
0,1

(3.162)
(2)(k

n e,1
n

b) jen H b)

Mn<2.3>

v e k (k b) v e k H (k b)
+ 0 01311“ P11Mn(3,3)+ O ”,1 n P’1 Mn(493)
n0,1 no'l
(3.163)
(1)(k b) jen H(:)(keb1b)
Mn(1’4) + 1 Mn(2,4)
wa “Mb

jen H e,1

 

2 H(1)
+ VO 6 OkPLIH J1

2 (2)
b) v k H b
6.1 1153.4) + 05o ml 11 0‘ 611_ )M <4 4)

n n
0,1

(k

°'1 (3.164)

78

jen H(1)(ke 3)
ansm = '3 (3.155)
wMa

 

jen H(2)(ke sa)

 

 

(6,5) = (3.166)
Qn mfla
2 (1)
v E It I! (k a)
Qn(6,6) = ° ° PIS “ 9'3 (3.167)
11
0,5

2 k H(2)(k a)

 

 

 

, V060 P15 n P15
Qn(6.7) - (3.168)
no,s
Qn(7.4) . ke,s H(:)(ke,sa) (3.169)
Qn(7,5) = ke's H(:)(ke,sa) (3.170)
evon H(1)(kp sa)
(211(796) = (3.171)
(v§+jw) a
ev2 n H(2)(kp a)
o s
Qn(797) 8 p, (30172)

(vg +jw) a .

In Eq. (3.139), the first two rows represent the continuity of Hz and
Be at r a c respectively, the third and the fourth rows represent the
continuity of Hz and Be at r c b respectively, the fifth and the sixth
rows indicate zero normal component of induced electron velocity at

r a b and r I a, respectively, and the seventh row represents zero

79

tangential electric field at the conductor surface at r = a. The seven
linear simultaneous equations as expressed in Eq. (3.139) can be solved

numerically using a computer or by any other method.

3.6 Scattered Fields in Free Space Region
The quantities of main interest in this study are the scattered
fields in free space region. To calculate these quantities, the

constant An is solved from Eq. (3.139) by Cramer's Rule as

Anl
An An 0

The two determinantslfin and [Kn1,are given as

 

(3.173)

Qn(l,l) Qn(l,2) Qn(l,3) o o o o
Qn(2,l) Qn(2,2) Qn(2,3) o o o o

o Qn(3,2) Qn(3,3) Qn(3,4) Qn(3,5) Qn(3,6) Qn(3,7)
[3.“ a o Qn(4,2) Qn(4,3) Qn(4,4) Qn(4,5) Qn(4,6) Qn(4,7)(3.174)

 

 

 

o o o Qn(5,4) Qn(S,S) Qn(5,6) Qn(S,7)
o o o Qn(6,4) Qn(6,5) Qn(6,6) Qn(6,7)
o o o Qn(7,4) Qh(7,5) Qn(7,6) Qn(7,7)
and
Qn(l,8) Qn(l,2) Qn(l,3) o o o o
Qn(2,8) Qn(2,2) Qn(2.,3) o o o o
o Qn(3,2) Qn(3,3) Qn(3,4) Qn(3,5) Qn(3,6) Qn(3,7)
Axxu. .. o Qn(a,2) Qn(4,3) Qn(4,4) Qn(a,5) Qn(4,6) Qn(4,7) 3.175)
0 o o Qn(5.4) Qn(5’5) Qn(5,6) Qn(S,7
o o o Qn(6,4) Qn(6,5) Qn(6,6) Qn(6,7)
o o o Qn(7,4) Qn(7.5) 1,0,6) Qn(7.7)

 

80

where the expressions for Qn(i,j)'s have been given in the previous
section.
After the constant An is determined, the scattered fields in

free space region are obtained as

s ‘n (2)
02 I 2 cos(n9) H n (kor) An (3.176)
n-o

s s ‘
nor = Hoe = o (3.177)
. g (2)
Ear 5,-6.1?miz'ornsirmne)rl H (ko r) An (3.178)
1:056 - jgo n2: ”cos(n6) H(2)(ko r) An (3.179)
53 g o (3 180)
oz '

where r is the distance between the observation point and the cylinder
axis. If the scattered fields are observed at a large distance, Hankel

function can be expressed in its asymptotic form as

2105 r-%mr-%n)
H(§)(kor) as «er e ° , (3.181)
0

and the scattered fields at a large distance are then obtained as

 

S 2 -j(k0 r-h) fl j35m'f
H02 8 m: r e 2 cos(ne) e An (3.182)
0 n-o
s s
not 3 Hoe 3 0 (3.183)
8 F 1 -j(k° r-krr)” 2 ( ) km
E. = j e nsin n0 e A
or wkor wear “-0 n

0 (3.184)

81

co - j (k r-o‘5(n+l )Tr-kfl)
s . / 2 0
£09 3 JQO --—--H E cos(n9) -e
" n-o

 

n -J'(kor-‘5m-’m)
4--——-e :] A
k r n
o
'J'Ckor-hr)” J'lzrm
at. Q / kzr e 2 cos(n9) e An (3.185)
0 "'0 n-o
-.£L
after neglecting the r 2 term.
E8 g 0 o (3.186)
oz
To derive Eq. (3.185) the relation of
<2)' (2) (2)
n
H n (kor) - -Hfi+1 (kor) + E;? H n (kor) (3.187)

has been used.
It is also observed that the only significant scattered fields
in the far zone of the plasma-coated cylinder are 8;;

ratio between these two fields is simply the impedance of free space £0.

and H s and the
02

3.7 Some Special Cases

 

There are two special cases, namely: a plain plasma cylinder and
~a plasma-coated dielectric cylinder which are quite interesting from a
practical viewpoint. These two cases can be solved by modifying the

procedure and solutions obtained in the previous section.

3.7.1 Scattered Fields by a Plain Plasma Cylinder

 

In this case, the whole cylinder is filled with a plasma in the
absence of a metallic cylinder. If we let the inmost sublayer be a

plasma cylinder with an extremely small radius and located along the z

82

axis, the whole plasma cylinder is subdivided into an extremely thin
plasma cylinder at the center and a number of concentric sublayers
extended from radius r I 0 to radius r = b up to glass wall (Refer to
Fig. 3.1). Since the Bessel functions of the second kind, Yn(ke,sr)
and Yfi(kp’sr), have a singularity at r = 0, the proper solutions in
the inmost sublayer are Bessel functions of the first kind. This
condition can be achieved by setting the constants Ds,n and Fs,n’ G

and Is n’ to be equal respectively, because
9

Jn(ke,sr) = é-[n‘i’me'sm + H(:)(ke’sr)] (3.188)
and
_ 1 (1) (2)
Jn(kp’sr) - 1T.[H:n (kp,sr) + H n (kp,sr)] . (3.189)

Thus, the simultaneous equations which give solutions to the scattered
fields from a plain plasma cylinder are Eqs. (3.132), (3.133), (3.134),

(3.135), (3.136) and the following two equations
D - F = 0 (3.190)
G " I g 0 o (30191)

With the set of equations, An can be solved and consequently the

scattered field.

3.7.2 Scattered Fields by a Plasma-Coated Dielectric Cylinder
In this case a dielectric cylinder instead of a metallic
cylinder is located in the center of the plasma cylinder. The

tangential component of E'field will not vanish on the surface of

83

the dielectric cylinder as it did in the metallic cylinder case. The
fields inside the dielectric cylinder are purely electromagnetic and
their amplitude remain finite. Because of the singularity of the
Bessel function of the second kind at r I 0, the prOper solution inside
the dielectric cylinder is the Bessel function of the first kind. Thus,
one additional constant is introduced to describe the fields inside the
dielectric cylinder compared to the metallic cylinder case. However,
the continuity of tangential components of E and H fields provides two
boundary conditions at r I a. Using these two boundary conditions
instead of the boundary condition of zero tangential electric field at
r I a as previously used for the case of a metallic cylinder, we obtain
a set of eight simultaneous equations with eight unknown constants.

The scattered fields from a plasma-coated dielectric cylinder can then

be obtained by solving this set of equations.

3.8 Numerical Results

 

The back scattered 8 fields from a plasma-coated metallic cylinder
and from a plain plasma cylinder have been calculated as a function of
(mp/w)2. In the numerical calculation, the collision frequency v is
assumed to be constant for all sublayers for simplicity (Reason for this
assumption has been given in Sec. 2.9, Chapter 2). The series solutions
are produced by summing up the first four terms only because of the
rapid convergence of series. The scattered fields are calculated at
a distance from the z axis with kor = 10 for convenience. The asymptotic
from of Hankel function is used whenever the argument with real part or

imaginary part becomes greater than 10. The dimensions for the glass

8h

tube, plasma layer and central cylindrical conductor are based on the
actual dimensions of the experimental model. These dimensions and the
dielectric constant of glass and the operating frequency are given in

Table 301 o

 

Operating frequency a (mm) b (mm) [C (mm) 65]

 

 

 

 

 

f I 2.3 GHz 2.158 7 J 8 5 I

 

Table 3.1 Physical dimension of plasma tube, dielectric
constant of glass and operating frequency of
a plasma-coated metallic cylinder.

All the numerical results of the back scattered B field are
plotted with the normalized value, Ei/E: , where E: is the scattered
E field and E: is the incident wave as a function of (mp/w)2 .

Figure 3.4 shows the back scattered B field of a plasma-coated
metallic cylinder in the direction of e I 1800 as a function of (cop/w)2
for various collision frequencies and for the case of a uniform density
distribution in plasma region (bzrza). The ratio of the r.m.s. electron
velocity to the velocity of light in free space, vo/c, is assumed to be
0.01. This is equivalent to a electron temperature of approximately
equal to 200,0000 K. The main resonance, also known as the
dipolar resonance, occurs at the value of (mp/w)2 I 2.34. This
resonance corresponds to the resonance due to n=l term of series
solution. The sharp peak at the right of the main resonance is a
quadrupolar resonance corresponding to the resonance due to n=2 term in
series solution. Three small peaks occured in the region of 0<(mplw)%<1

are the so called temperature resonances due to an electroacoustic wave.

85

These resonances are set up when an electroacoustic wave sets up a
standing wave pattern between the metallic cylinder and the glass wall.
It is observed that when the collision frequency is increased to a
value in the order of vim I 0.5 all resonances disappear. Also the
quadrupolar resonance seems to be damped out by the collision more
strongly than the dipolar resonance.

Figure 3.5 is also a plot of the back scattered B field of a
plasma-coated metallic cylinder as a function of (mp/w)2 for various
collision frequencies. The plasma layer is assumed to have a uniform
density distribution. The ratio vO/c is assumed to be (4/3) x 10.-2
which is slightly greater than the value of vo/c in Fig. 3.4. It is
observed that the effect of collision frequency is to damp out the
resonances and the over all picture is approximately the same as that
in Fig. 3.4. Due to the change of v0 and consequently kp, the number
of temperature resonances reduces to two compared to three in Figure
3.4. The locations of main resonance and quadrupolar resonance in
Figs. 3.4 and 3.5 are slightly different.

Figure 3.6 is a plot of the propagation constant, ke, of an
electromagnetic wave as a function of (cop/w)2 . The expression of
ke is given by Eqs. (3.30), (3.31) and (3.32). The abrupt change of
real part Be and imaginary part ae in the neighborhood of (mp/w)2 = 1
is clearly shown in the figure. The slopes of curves are greater for
the case of lower collision freqnency.

Figure 3.7 is a plot of the propagation constant, kp, of an

electroacoustic wave as a function of (mp/w)2 with vole = 0,01 and

v/w I 0.001. The expression for kp is given by Eqs. (3.46), (3.47)

86

and (3.48). It is observed that in the region near (mp/w)2 = 1 the

real part of kp changes from the order of 103 to the order of 1 while

the imaginary part changes from the order of l to the order of 103 .

The real part, Bp’ represents the wave number and it determines the

wave hngth of an electroacoustic wave. The imaginary part, up, represents
the attenuation constant and consequently it produces a cut off phenomenon
for an electroaccoustic wave when (mp/w)2 becomes greater than 1.

Figures 3.8 and 3.9 are the plots of the back scattered E fields
of a plasma-coated metallic cylinder with a uniform density distritution
as a function of (cop/w)2 . The collision frequency is assumed to be
v/w I 0.001. Various values of r.m.s. electron velocity are assumed.

It is observed that smaller value of volc will lead to more temperature
resonances appearing in the region of 0<<(wp/w)2<fl . This is resonable
since a smaller value of vO/c implies a larger value of kp or a shorter
wave length of an electroacoustic wave. Consequently, an electroacoustic
wave can set up more standing wave modes between a finite distance between
the boundaries. It is also shown that a higher value of vo/c produces
greater amplitudes for the temperature resonances.

Figures 3.10 and 3.11 are the plots of the back scattered B field
of a plasma-coated metallic cylinder with a non-uniform plasma density
distribution as a function of (mp/w):ve. which is the average value of
(wp/w)2 . The collision frequency is assumed to be v/w I 0.001. The
ratio of vole is assumed to be 0.01 and 0.0133 in Figs. 3.10 and 3.11,
respectively. The non-uniform plasma density distribution is assumed

to be the same as that used in the cold plasma case which is mathe-

matically expressed as

87

n I n [1 - 0 6(M)2] (3 192)
o,r o,c ' b+a °
. . b+a .
where 110 c 15 the plasma den81ty at r "75' . It is noted that the
0

value of (mp/w)2 is directly proportional to the density “o,r under a
fixed operating frequency. The stratification method is used and a 3-
sublayer model is assumed. A great care has been taken when performing
the numerical calculation. Due to a great difference between the
magnitude of an electromagnetic wave and that of an electroacoustic
wave, the matrix:[Lfi]as expressed in Eqs. (3.92) and (3.93) becomes
nearly singular. In our calculation, the matrix inversion and multi-
plication have been performed by a double precision technique to avoid
run off error. Without this technique, a single precision method

-would have led to a complete false result. It is noted that there are
mathematical resonances inherently associated with the stratification
method being used (Details of mathematical resonances, refer to page 29,
Chapter 2). It is possible to identify the mathematical resonances
when Figs-3.10 and 3.11 are compared with the corresponding cold plasma
cases where no temperature resonances are possible. Since as we have
discussed in Chapter 2, mathematical resonances bear no physical meaning,
they are to be overlooked. For example, the peak located at the value
of (mp/w):ve. I 0.94 in Fig. 3.10 is a mathematical resonance. For the
cases of uniform plasma density distribution, the temperature resonances
are all shown to be located in the region where (mp/w)2 is less than 1.
HOwever, for a non-uniform.plasma density distribution, temperature

resonances can be set up in the plasma layer in the region where

(mp/waive is greater than 1. The reason is rather evident. Because
0

88

even (mp/m):ve is greater than 1, there are regions of plasma near the
O

boundaries where the local (mp/0))2 is still smaller than 1. Thus, a
standing electroacoustic wave can be set up in these underdense plasma
regions. This mechanism has been discussed by Crawford. 17) The main
resonance of the 3-sublayer model of a non-uniform density distribution
is located at a slightly higher value of (mp/w):ve. compared with the
case of uniform density distribution. The S-sublayer model has also
been calculated. Unfortunately, the asymptotic form of Hankel function
gives a discontinuity in the region where mathematical resonances and
temperature resonances occur. This discontinuity causes the identifi-
cation of temperature resonances from mathematical resonances becoming
a very difficult task. This difficulty will be solved if a computer
subroutine for calculating Hankel functions with a large complex
argument becomes available.

Figure 3.12 is a plot of the back scattered B field of a plain
plasma cylinder with b I 7 mm and c I 8 mm as a function of (mp/w):ve. .
Both cases of uniform and non-uniform density distributions are shown

in Fig. 3.12. The 3-sublayer model is used for the non-uniform density

distribution with a density distribution of

'r 2
no,r a no“: [1 - 0.6(-5-) J (3.193)
being assumed. The resonance located at (mp/w):ve I 0.94 is a mathe-

matical resonance and should be ignored. The general behavior of a
plain plasma cylinder is similar to that of a plasma-coated metallic
cylinder. HOwever, the locations of temperature resonances are quite

different and also the dipolar and quadrupolar resonances are moved to

89

higher values of (wb/w):ve for the case of a plain plasma cylinder.

The numerical calculations are performed by a CDC 6500 computer.

90

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coo

CHAPTER 4

SCATTERING FROM A "STILLIC CYLINDER SURROUNDED BY A
LAYER OF LOSSY PLASMA ILLUMINATED BY A TH HAVE

4.1 Introduction

In the preceding chapters. the scattered field from a plasma-
coated metallic cylinder when illuminated by a TB wave at normal
incidence has been studied. In general, the incident plane wave may
have an arbitrary polarization. In order to study the scattered field
from a plasma-coated cylinder illuminated by a plane wave of arbitrary
polarization, it is necessary to consider the scattering from the same
cylinder when it is illuminated by a TM wave. A TM wave is defined as
a plane wave with its fi field perpendicular to the cylinder axis and
its 3 field parallel to the cylinder axis. The superposition of a TI:
and a TM wave can yield a plane wave of arbitrary polarization.

The analysis in this chapter is similar to that of preceding two
chapters. The stratification method is again used. Also a glass wall
is assumed to surround the plasma in the theoretical model. The
surrounding plasma is assumed to be hot and non-uniform in the analysis.
It is shown in a later section that under the assumed geometry and
boundary conditions it is impossible to excite an electroacoustic wave
in the hot plasma. Thus, the solutions obtained for the case of a
cylinder covered by a cold plasma can also be applied to the hot plasma

case a

99

100

4.2 Geometry of the Problem

The geometry of the problem and the notations of stratified
plasma medium are the same as that described in Sec. 3.2 of Chapter 3.
However, the incident fields are different from that of the previous
chapters. The indicent plane wave in this chapter is assumed to have
a E field paralled to the z axis and a‘fi field perpendicular to the z

axis as shown in Fig. 4.1. The fields of the incident plane wave are

given by
i . -jk x a -jk rcose
302 e o e o
A” n
8 2 6 (-j) cos(ne) J (k r) (4.1)
“-0 on n o
i i
or . £06 - 0 (402)
i j A” n
“or - - “Nor n§o € on(-j) nsin(n6) Jnfltor) (4.3)
Q
i n '
Hoe - - J: “50 eonc-j) 1208016) Jn(kor) (4.4)
i
“02 ' 0 a (‘05)

Mathematical symbols used in Eqs. (4.1) to (4.5) have been explained

in the preceding chapters.

4.3 Fields in Free space Region

 

In the free space region Maxwell's equations are

VXE: I -jw0;: (4.6)
“s
Vx'fi: . jwéoEo (4.7)

where E: and E: are the scattered electric and magnetic fields.

101

 

 

 

 

 

Incident wave

Region 1: free space
Region II 8 glass wall
Region DE: plasma

Region IV: metallic cylinder

Fig. 4.1 A plasma-coated metallic cylinder
illuminated by a m wave from the
lEfts

102

Due to geometrical symmetry, all fields are symmetrical with respect to
the 6 I 0 axis. Since the E field of the incident wave does not have a
2 component, it is reasonable to assume that the scattered B’field

possesses no 2 component and it is independent of 2 variable because of

geometry. Thus, the Eq. (4.7) yields
511’
317352- (r H:o)--1£-E]; - 11160 if: - (4.8)

Equation (4.8) implies that E: is allowed to have a 2 component only.

we can assume that
E 3 E a a (409)
0
From Eqs. (4.6) and (4.7),we obtain a wave equation as

“5 2'8
V X W X £0 . kOEO s (4010)

Substituting Eq. (4.9) in (4.10), it yields

 

 

2 s. s 12 s
a E a E a E
oz 1 oz 1 oz 2 s -

Equation (4.11) can be solved by the method of separation of variables.
Since Eq. (4.11) is the same equation as Eq. (2.10) in Chapter 2, the

solution of Eq. (4.11) can be written as

,°° (2) ‘
oz 2 008(n9) H’n (Rot) An (4.12)

n-o

E

where An is a constant to be determined by the boundary conditions. The

corresponding H: field can be found from Eq. (4.6) to have components

such as

103

w
s (2)
nor . .. 331;? “250 nsin(n6) a n (Rot) An (4.13)
H8 - - :3. cos(ne) “(2).“ r) A (4 14)
09 g “-0 n o n °
s
H62 3 0 o (4.15)

The total fields in the free space region can be obtained by

summing the incident and the scattered fields to be

 

co
t n (2)
£02 - 2 cos(n6)[€°n(-j) Jn(kor) + H n (kor) An] (4.16)
n-o
t t
Ear . E09 ‘ O (4017)
N
t. a _ _ n (2)
or “—51%; “1.30 nsin(ne) [e on‘ j) Jn(k°r) + H n (kor) An]
(4.18)
t _1_ ‘a n ' (2;
"06 = - go “i cos(n9) [E on(-_)) Jn(kor) + H n (kor) An]
(4.19)
t
H02 3 0 o (4.20)
4.4 Fields in Glass Hall Region
In the glass wall region Maxwell's equations are
V x as - ’5‘”. as (4.21)
V x H8 - ”Goes 38 (4.22)

where 6 8 is the dielectric constant of glass.

If we allow an incoming and an outgoing (reflected) wave to exist
in the glass wall region, the solution for E; field can be expressed as

is - saz‘z‘ (4.23)

104
where

E a 2 cos(n8) [H(:)(k8r) Bn + H(:)(k8r) Ca] . (4.24)

32 an
In Eq. (4 .24) , B“ and (.In are the constants to be determined by the
boundary conditions and k8 is the propagation constant of the electro-
magnetic wave in the glass defined as k8 - Ito/E: . The corresponding

is field has components given by

as
81' B - 631;? nfo nsin(n0) [1193080 an «I» H(:)(k8r) Cu] (4.25)
J (6 w ' '
"so - - -———5 z; cos(n8) [u‘;’(x8r) an + H(:)(kgr) on] (4.26)
0 n-o
I O . (4.27)

H
gz

4.5 Fields in Plasma 3323.".

In the analysis , the plasma is assumed to be hot. The possible
excitation of an electroacoustic wave is considered and the density of
plasma is assumed to vary in the radial direction only. As the result
of analysis. with the assumed geometry and the polarization of the
incident wave, it is shown that no electroacoustic wave can be excitated.
Thus, the hot plasma case reduces to the cold plasma case for this
particular polarization of the incident wave. As in the preceding
chapters. the stratification method is used and the same geometry as
that shown in Fig.3'.l'.(b)of Chapter 3 is adopted.

Maxwell's equations in the mth sublayer of plasma region are

II.

V x E‘ (4.28)

3'

Vxfi \7'

m o,m m + (4.29)

I
I
m
:3
La.
6
m
F11

105

." . -.._1.r;’!'. .
V Em 6° (4 30)
V-i? - o . (4.31)

 

“o,m V'Vn*5"’1~,.. . o (4.32)
' 2
( + a»)? - “-9-" v° (4 33)
‘5! j I 1! gm - n.. V7m1,m '
o,m
where
31.1“
v0 . T (4.34)

and all other mathematical symbols have been defined in the preceding
chapters.

From Eq . (4.28), we obtain
VxVx an - ~jmuOVX an . (4.35)

The term, V x am, can be expressed in terms of 9m and Em as given by

Eq. (4.29). Thus, Eq. (4.35) can be rewritten as

VxVxEm . jmuen V +k kit-2 . (4.36)

oo,mm

3‘“

From Eq. (4.33), the induced electron velocity,
explicitly as

m' can be written

 

2
v
'* e - o
vm ‘ vm-I-jm MEm (vm+jw)n°'n vnlm . (“'37)

Substituting Eq. (4.37) in Eq. (4.36), we obtain

(”zozuev

2
m
‘ m ‘4
V xVx Em . w Zopoe 1 +1w1vm+jm JEm ”(2m +1” an l,m o(4.38)

106

Equation (4.38) can be expressed as

-o 2 -o
V x Vx Em ke,m Em + Rmv ml... (4.39)
In Eq. (4.39) the term, R., is defined as
2 2
m uoevo
“a " W ‘“-’*°’

fer convenience. ke,m is the propagation constant of the electromagnetic
wave. in the plasma in the mth sublayer. Equation (4.39) can be
reduced to three coupled scalar equations. After that, effort will be
made to decouple the equations. In order to do this, we consider the
magnetic field, fim' first. The incident 3 field has no 2 component and
the plasma density is assumed to vary in r direction only. Thus, in the
mth sublayer of the plasma the magnetic field, 13“., will not have a 2
component. We can assume that

-s A A
“m I Hfir r + His 8 . (4.41)

Substituting Eq. (4.41) in Eq. (4.28), we have
--0 A .
VxEm - -m°[umr+nuee] . (4.42)

Equation (4 .42) implies the vanishing of 2 component of vector quantity,
V x Em . Also, due to the geometry of the problem, no variation of
the fields along the z axis is assumed, i.s. £2— - 0. The vector quanti-
ty, v xVx Em, can then be expressed as

an 522 .

VxVxEm :- -%[§:(r—£-z-)+%-—a-e¥;l]z . (4.43)

107

with Eq. (4.43), Eq. (4.39) can be divided into three components to

yield three scalar equations as

 

 

R an
2 m 1 m
ke'm m + —r TL: ' 0 (4 04‘.)
R n
2 m l m
k... m0 “E‘s-6" ° (“‘5’
2
32: as w a B
22-43- .+-,-17 ”+14qu - o . (4.46)
Br r or r 88 ’ I

Equation (4 .46) is a homogeneous wave equation for the z couponent of Em

field and its solution is

Q
g .. )3 cos(ne) [32)“. mt) pm
0

(2)
III! I!” n n e, :1

g m "'11

where on n and Fm n are the constants to be determined by the boundary
0 9

conditions. For the components of Emr and 1:

me’ they'are coupled with

electroacoustic term, nlfi, as shown in Eqs. (4.44) and (4 .45). Before
determining Enr and E m9 , we seek the solution for :11"I .
From Eq. (4.33), we have
-0 d v2
(Vm'l’jW) V' Vm :- - 19‘— v- Ell - «5:2;‘72 n1,m s (4.48)
D

The substitution of the quantities V- V. and V - Em expressed in

Eqs. (4.32) and (4.30) in Eq. (4.48) gives a wave equation for n1 m as
9
V2 n +1.2 - o (4.49)

1 ,m p ,m n1 ,m

where kp m is the propagation constant of the electroacoustic wave and
9

its expression has been given in Chapter 3. The solution for Eq. (4 .49)
is

108

_ °° (1) (2) cos(n6)
n “i [a n (kp'mr) c + a u (up r) 1mm] (4.50)

l,m m,n ,m sin(n8)

where Gm,n and Im,n are constants to be determined by boundary conditions.
With Eqs. (4.50), (4.44) and (4.45), the r and 6 components of E; field
can be determined theoretically. Up to this point, E'field and n1 in

any sublayer can be written down explicitly with appropriate labeling

the quantities ke' hp, no, n1, v, etc. Before finding the final

solutions for the E and n1, let us consider the boundary conditons at

the inmost sublayer of the plasma layer, i.e. the sth sublayer. This

will lead to an interesting result and the solutions can be simplified
greatly. Since the metallic cylinder is assumed to be perfectly
conducting, the tangential components of‘E field at the surface of

cylinder vanish. This give,

Ese I 0 at r I a (4.51)

and

E82 3 0 at r . to (4052)

The 9 component of E; field in the sth sublayer can be expressed from

Eq. (ll .45) I8

 

a
a .. - 8 312‘. , (4.53)
89 k2 r 86

e,s

The substitution of Eq. (4.53) in the boundary condition, Eq. (4.51),

leads to

(1) (2)
H n (kp'sa) G + H n (kp a) I 3 0 a ((4054)

s,n ,s s,n

109

This is one of the two equations which determine the constants G8 n and
9

I8 n . we now aim to find the other equation in order to completely
9

specify Gs n and I8 n . From Maxwell's equations in the sth sublayer,
D l

we obtain

 

1 -s c-o
jwp‘onVan en V +jm€ E . (4.55)

The r component of Eq. (4.55) is

- eno's Vs: + 3m 60 Ear . o , (4.56)

since no r component can be obtained from V x V x is as explained in

Eq. (4.43). If E8r is expressed in terms of n according to Eq. (4.44),

1,3
the radial component of the induced velocity of electrons, vsr’ in the
sth sublayer can be expressed as

1“’60 R8 anl s

I -————————-——-L-
V” en 1:2 r or . “’57)
o,s e,s

If the rigid boundary of metallic cylinder is assumed as before, it
requires the normal component of the induced velocity of electrons,

Var, to vanish at r I a. This leads to

"3EFE I 0 at r I a . (4.58)

Substituting Eq. (4.50) in Eq. (4.58), we have

I 0
(1) (2) ,
H‘n (kp’sa) Gs,n + H n (kp,3a) Is,n 0 . (4.59)

Equations (4.59) and (4.54) are the two simultaneous equations needed

to determine the constants, Gs n and Is n . These two equations yield
9 D

trivial solutions of

110

G 3 I 3 0 (4.60)
with the exception that when the determinant ,

(1) (2)
H n (kmsa) H n (kp'sa)

A“ ' (>' ()' (4'61)
1 2
H n (kp'sa) H n (kp'sa)

becomes zero .

The determinant expressed in Eq. (4.61) can be written as

(1) (2)' (2) m'
A - H n (kp'sa) H n (hp a) - H n “p,s” u n (kp'sa) . (4.62)

sn ,s

Using the relation between Hankel function and its derivative such as

H(:)(z) - - 332(2) + 1} n‘i’m (4.63)
and
H(:)(z) I - 1133(2) + -2- H(:)(z) , (4.64)

Eq. (4.62) can be rewritten as

, (2) (1) , <1) (2)
A u n (kp 8a) Hmlmp's.) a n (Rhea) H (kp'sa) (4.65)

an , n+1
which is a Hronskian and is equal ”(25)
A” - - (ii/«(13,50 1‘ o . (4.66)

Since Eq. (4.66) implies that the determinant of Eq. (4 .61) is not equal

to zero, Eq. (4.60) are the only valid solutions for as n and I8 n .
D I

111

Therefore, the electron density deviation,n1's,is zero everywhere within
the sth sublayer.

Next, we will dertermine the 111 in the other sublayers. Consider
the boundary at r I r”1 (refer to Fig. 3.14b))and using the same boundary

conditions as used in Chapter 3, such as

Thngential components of E and H fields are continuous (4.67)
“o,s-1 V(._1)r - "o,s V.r I 0 at r I ran1 (4.68)
n n

ALL]; .— .1112 n 0 at r I rB-l . (4.69)

“0,0-1 no.8
Since n1,s and vsr are zero as indicated in Eqs. (4.60) and (4.57).

Eqs. (be68) and (4.69) yield

H<1>( ”(2%k

) I . 0 (4.70)

p,s-1 rs-l s-l,n p, s-lrs-l

and

8(1)(k+H(2)(k r )1 - o (4.71)

) G p, s-l s-l s-l,n

p, s-lr s-l s-l,n

respectively. Eqs. (4.70) and (4.71) are similar to Eqs. (4.59) and

(4.54), thus, we have

(:91.n - 18-1,“ 0 . ' (4.72)

By successive matching of the boundary conditions at all other boundaries,
we can show the total vanishing of electroacoustic mode in the plasma
medium. Therefore, Maxwell's equations given in Eq. (4.28) to Eq. (4.31)

can be simplified for this particular case to the following equations:

Y7 x rm - -jmu° an (4.73)

112

Vac H. I ngm Em (4.74)
V- am - o (4.75)
V- Hm - o . (4.76)

These equations are the some set of equations used in the cold plasma
case in Chapter 2. Physically, it implies that when a metallic cylinder
coated by a layer of hot, noneuniform plasma is illuminated by a plane
wave of TH polarization, the hot plasma behaves as a cold plasma and no
electroacoustic wave is excited in the plasma layer. Only an electro-
magnetic wave exists in the plasma region. The electric field, E; ,

in the mth sublayer yields only a 2 component and is expressed as

Eq. (4.47). The corresponding fi’field has components given by

N
(1) (2)
Hmr - - E13 nEo nsin(n9)l:fl n (Rent) ”m,n + H n (Refit) Fm'n]("~77)

” 0 0
' (1) (2)
"no - - j./;—E n50 co.(ne)|:n n (ke'mr) pm“ + a n (ke'mr) Fm,n]("'78)

Hm: ' 0a (4079)

4.6 Matching of Boundary Conditions at Interfaces

Since only electromagnetic waves exist in the plasma layer and
other regions, the boundary conditions at interfaces are the same as
that described for the cold plasma case in Chapter 2. Similar matching
process as used in Sec. 2.6 of Chapter 2 can be employed here. However,
in this case, the polarization of the incident wave is different from

the case in Chapter 2, a different set of five simultaneous equations

113

can be obtained from matching the boundary conditions.

can be represented in matrix form as

0

0

0

 

L

where

Qn(3,2)
Qn(4.2)
o

Qn(1.l)
Qh(l,2)
Qn(1.3)
Qn(l.6)
Qn(2,l)
Qh(2,2)
45(2.3)
qn<2.6>
qn(3,2>
Qn(3,3)

Qn(3.4)

"onam onus) 0.5“”
0.52.1) (452.2)

Qn(2.3)

Qn(3.3)

Qn(4,3)
o

—

o o

o o
Qn(3,4) Qn(3,5)
0154.4) Qn(4.5)

 

 

Qn(5.4) ths.5)41 r

(2)
-H n (11°C)

(1)
H n “8‘”

(2)
H n (REC)

ll
6 one-5) anacoc)

(2).
-H n (11°C)
(1).
’68 H n (Rae)
(2).
’68 H n (kgc)

n 0
E on(”31) Jn(kOC)

(l)
-H n (ksb)

(2)
-H n (ksb)

 

 

»_Qn(1.6fl

Qn(2'6)
0

0

L 0

These equations

 

(4.80)

(4.81)

(4.82)

(4.83)

(4.84)

(4.85)

(4.86)

(4.87)

(4.88)

(4.89)

(4.90)

H(:)(ke'1b) un(1,1) + H(:)(ke'1b) Mn(2,l) (4.91)

114

qn(3,s.) - H(:)(ke'1b) snug) + H(:)(ke'lb) un(2.2) (4.92)
0
(1)
05(4'2) - - fiE; n n (ksb) (4.93)
<2>'
Qn(4.3) - - lg; n n (ksb) (4.94)
§ ' 5
1 (1) 1 (2)
Qn(4,4) I L; H n (Re. 11)) M n,(1 1) +£1.11! (Re 11)) Mn(2,1)
(4.95)
g1 (1)' f5“ (2)
Qn(4,5) . j; n n (k b) M n(1, 2) + 60 (k .1, b) mn(2,2)
(4.96)
(1)
045.4) - an (11”.) (4.97)
(2)
°n(5'5) - u n 0‘..." , (4.98)

The elements H(iJ)'s in Eqs. (4.91), (4.92), (4.95) and (4.96) are the

entries of the matrix obtained from the product of the following matrices:
Mn(l,l) M:(1,2)]
Mn(2,1) M (2.2)

[L [11(ke,lr 1)] lD‘n (hear r1)] [Ln (he, 2t.-.2)]1|:I‘n(ke,3r2)J

e a e [Ln(ke,s-er-1)]-l[Ln(ke,8r8'l ] (4.99)
with
(1) (2)
H n (kemrm) H “0‘9 mt“)
[name] = -

(l) (2)
I; H n (ke,mrm) f— H n‘ke,mrm) '

115

4.7 Scattered Field in Free Space Region
The constant An which is the coefficient of the scattered fields
in the free space region is of main interest and can be obtained from

Eq. (4.80) by Cramer's Rule as

 

 

 

Anl
A . c (2.101)
“ A111
The two determinants, A n and Anl’ are given as
Qu(l,l) Qn(l,2) Qn(l,3) o o
Qn(2,l) Qn(2,2) qn(2,3) o o
A“ - o Qn(3.2) one.” Qn<3.4) 0.33.5) (2.102)
0 Qn(4.2) Qn(4.3) Qn(4.4) Qn(4.5)
o o o Qn(5,4) Qn(5,5)
and
Qn(1.6) Qn(1.2) Qn(1.3) 0 0
Qn(2.6) Qn(2.2) Qn(2.3) 0 0
Am - o Qn(3,2) Qn(3,3) Qn(3,4) Qn(3,5) (2.103)
0 Qn(4.2> Qn(4.3) Qn(4.4) Qn(4.5)
o o o Qn(5,4) Qn(5,5) .

 

 

The expressions for Qn(i,j)'s are given in Sec. 4.6.

Finally, the scattered fields in free space region are obtained

IS
12’ - 2:. ( >110)“ A 2 104)
oz c” no 11 or) n ('
nIo
8 8
301' I 300 I 0 (2.105)

116

s °° (2)
not - - Hi; “55° nsin(n0) a n (tor) An (2.106)
c. 0
1100 i: a: cos(n0) H n (kor) An (2.107)
8
H02 . O (2.108)

with An expressed as Eq. (2.101).

4.8 Some Special Cases

The electromagnetic scatterings from a plain plasma cylinder and
a plasma-coated dielectric cylinder will be considered as two special
cases of the problem studied in this chapter. These two cases can be
solved by modifying the procedure and solutions obtained in the previous
section. Since these modifications are similar to that derived in

Sec. 2.8, only numerical results will be presented in the next section.

4.9 Numerical Results

The back scattered E fields from a plasma-coated metallic cylinder
and from a plain plasma cylinder have been calculated as a function of
(ob/m):ve. . In the numerical calculation, the collision frequency v
is assumed to be constant for all sublayers for simplicity (Reason for
this assumption has been given in Sec. 2.9, Chapter 2). The series
solutions are produced by summing up the first four terms only, because
of the rapid convergence of series. The scattered fields are calculated
at a distance from the z axis with her I 10 for convenience. The
dimensions for the glass tube, the plasma layer and the central cylindrical

conductor are based on the actual dimensions of the experimental model.

117

These dimensions and the dielectric constant of glass and the operating

frequency are given in Table 4.1.

 

Operating frequency] a (m) l b (m) c (ma) 6

 

Ul

 

 

 

 

f - 2.3 an: I 2.158 L 7 a

 

Table 4.1 Physical dimensions of plasma tube, dielectric
constant of glass and operating frequency.

Numerical calculation shows that the inhomogeneity along radial
direction has little effect on the back scattered E fields. A 13-
sublayer model has been used to approximate the plasma density distri-
bution which is given by Eq. (3.192) in Chapter 3. The result obtained
for this l3-sub1ayer model is quite similar to that for a homogeneous
plasma layer. Also, the numerical result shows that collision frequency
has only a little effect on the back scattered E field. The results are
plotted with the normalized back_scattered E field,(E:IE: where E: is the

scattered E field and E: is the incident field) as a function of

2

(mp/w)ave e .

Figure 4.2 shows the back scattered E fields from plasma-coated
metallic cylinders of various radii illuminated by a TM plane wave, as

e.
fied plasma medium with a density distribution given by the Eq. (3.192)

a function of (mp/m):v . A l3-sublayer model is used for the strati-

in Chapter 3. The collision frequency is assumed to be v/m I 0.01.
It is observed that the scattered field increases only slightly as the
value of (mp/m):ve. is increased. It is also observed that a metallic
cylinder of smaller radius gives a smaller scattered field, but the

scattered field from a smaller metallic cylinder increases more rapidly

118

as the plasma density is increased. For a uniform plasma layer, the
result is nearly the same as that of the non-uniform case and that is
not shown in the figure.

Figure 4.3 is a plot of the back scattered E field from a plain
plasma cylinder as a function of (mp/01):”. for two different collision
frequencies, who I 0.1 and who I 0.01. The asstmxed density distribution
of the plasma layer is expressed by Eq. (3.193) in Chapter 3. In the
figure, it is observed that the back scattered E field reaches a
minimum when the plasma density reaches at a value of (mp/w):ve. I 1.96.
This phenomenon disappears when the effect of glass wall is neglected.
For a smaller collision frequency the minimum in the back scattered E
field tends to become more outstanding. The effect of the collision
frequency on the other part of the curve is insignificant. In general,
the amplitude of the back scattered E field from a plain plasma cylinder
is smaller than that from a plasma coated metallic cylinder. Also, the
back scattered E field from a plain plasma cylinder with uniform density
distribution is nearly the same as that of a non-uniform density distri-

bution case.

119

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CHAPTER 5

EXPERIMENTAL INVESTIGATION OF THE SCATTERING FROM A PLASMA-COATED
METALLIC CYLINDER AND A PLAIN PLASMA CYLINDER

5.1 Introduction

 

The electromagnetic scatterings from a plasma-coated metallic
cylinder when illuminated by incident TE and TM waves have been
studied theoretically in the preceding chapters. Although there have
been a number of theoretical studies on the subject conducted by
researchers in this area, very few experimental studies have been
reported in the literature. Vandenplas(7) has conducted experiments
on the scatterings from a plain plasma cylinder and a plasma-coated

(26) are

glass cylinder. However, to our best knowledge, Ho and Chen
the only researchers who have investigated experimentally the scatter-
ing from a plasma-coated metallic cylinder.

Our experiment on the scattering from a plasma-coated metallic
cylinder has been conducted inside of a waveguide While Ho and Chen
conducted their experiment in free space. Keriar and Heissglass(27)
performed the experiment using a similar technique we used but they
studied a plain plasma cylinder only.

The main task of this experimental investigation is to measure
the back scattered fields from a plasma-coated metallic cylinder as a

function of plasma density under a fixed operating frequency. This

experiment has been performed for the purpose of checking theoretical

121

122

results developed in Chapters 2, 3 and 4. The resonance phenomenon and
general behavior of the scattered field predicted by the theory have

been qualitatively confirmed by the experiment.

5.2 Experimental Setup

 

The plasma-coated metallic cylinder was constructed by install-
ing a metallic cylinder in the center of a cylindrical mercury~vapor
discharge tube.

Two long mercury~vapor plasma tubes with outside and inside
diameters of 16 and 14 mm were constructed. Installed in the center
of one of the plasma tubes was a metallic cylinder of 4.316 mm diameter
and 120 mm length. The plasma density was varied by sweeping the
discharge current from zero to 600 mA which corresponded to a plasma
density of 3.4 x 1011/cm3. The pressure of plasma was about 1 u Hg.

In the experiment, the positive column parts of the plasma tubes
were inserted into a S band rectangular waveguide (l ll/32”x 2 27/32”)
through holes on the waveguide wall. Two different arrangements for
the plasma tube and the waveguide as shown in Figs. 5.1.1 and 5.1.2
have been considered in the experiment. In Fig. 5.1.1, the plasma tube
is inserted through the narrow walls of the waveguide. This arrange-
ment provides a situation of a plasma-coated metallic cylinder illumi-

nated by a TE wave when the waveguide is excited by a TE mode. In

10
Fig. 5.1.2, the plasma tube goes through the wide walls of the wave-
guide. For this case, the E field of the waveguide is in parallel with

the plasma tube, thus, provide the situation of a plasma-coated metallic

cylinder illuminated by a TM wave.

123

The schematic diagram of the experimental setup to measure the
back scattered fields from a plasma-coated metallic cylinder and a
plain plasma cylinder is shown in Fig. 5.2.

The back scattered field is fed into Channel 1 of the dual-
vertical-input of an oscilloscope and the transmitted wave (incident
and forward scattered waves) is fed into Channel 2 of the vertical
input. The horizontal input of the oscilloscope is swept with a 60-
cycle voltage which is linearly proportional to the discharge current
of the plasma tube. Since the discharge current of the plasma tube
is approximately proportional to the plasma density, the intensities
of back scattered and transmitted waves can be plotted as functions
of the plasma density directly on the oscilloscope.

Figures 5.3.1, 5.3.2, 5.4.1, and 5.4.2 are photographs of the

experimental arrangements and setups.

5.3 Experimental Procedure

 

To minimize the error caused by unmatched loads, the following
preparations were made before the measurement.

(1) Remove the plasma tube from the waveguide and plug the
holes on the waveguide wall to maintain an unperturbed situation for
the waveguide.

(2) Turn the microwave oscillator to a desired operating
frequency (2.3 Gas in our experiment) with a l KHz square wave modu-
lation applied. Turn the horizontal sweep of the oscilloscope to
"internal sweep.” Two sets of square waves with different amplitudes
will appear on the oscilloscope. The reflected wave appears on

Channel 1 and the transmitted wave on Channel 2.

124

/-v-—Waveguide Hole
M t lli
K C§lInderc: I Glass Wall

L

\

[Plasma Tube

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5.1.1 Cross sectional view of a rectangular waveguide with
an inserted plasma tube. (TE polarization)

 

 

 

Metallic Hole
Y Cylinder /- /_ Glass Wall

\

[Llama Tube

 

 

 

 

 

 

 

/— Waveguide

 

 

 

Fig. 5.1.2 Cross sectional view of a rectangular waveguide with
an inserted plasma tube. (TM polarization)

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125

126

 

Fig. 5.3.1 The plasma discharge tube inserted
in the waveguide.

 

Fig. 5.3.2 The waveguide and directional coupler
with double-stub tuners at both ends.

 

Fig. 5.4.1 Experilental set up for the measurement of
scattered field with a TE incident wave.

 

Fig. 5.4.2 Experiment set up for the measurement of
scattered field with a TM incident wave.

128

(3) Adjust the double-stub tuner l to obtain a maximum
amplitude for the display on Channel 1. By this way a highest sensi-
tivity is obtained for the receiving system for the reflected wave.

(4) Adjust the double-stub tuner 2 to obtain a minimum
amplitude for the display on Channel 1. This implies that the
reflection from the waveguide system in the absence of the plasma
tube is minimized or the waveguide is terminated by a matched load.

(5) Thrn off 1 KHz square wave modulation while keep the
microwave oscillator on at the same frequency as in procedure (2).

(6) Insert the plasma tube back to the waveguide. Start
the plasma tube and connect the sweeping anode voltage to the horizon-
tal input of the oscilloscope.

Two curves appearing on the oscilloscope are the reflected
waves and the transmitted wave displayed as functions of discharge

current which is proportional to the plasma density.

5.4 Experimental Results and Comparison with Theory

Experiment have been conducted with two plasma tubes. One is
a plasma-coated metallic cylinder and the other a plain plasma cylinder.

Their dimensions are given in Table 5.1.

 

 

 

Plasma-coated metallic cylinder Plain plasma cylinder
a (m) b (m) 0 (mm) b (min) c (M)
2.158 7 8 7 8

 

 

 

 

 

 

 

Table 5.1 Physical dimensions of plasma tubes.

129

The dielectric constant of glass wall is assumed to be 5 in the theo-
retical calculation. The operating frequency is fixed at 2.3 CH: and
the discharge current is varied in the experiment. Since the plasma
density is proportional to the discharge current and (mp/w)2 is
proportional to plasma density, the displays on the oscilloscope are
the plots of the intensities of the reflected and transmitted waves as
functions of (mp/w)2. Thus, these displays can be compared directly

with the theoretical results calculated in the preceding chapters.

5.4.1 Experimental and Theoretical Results

Figures 5.5 and 5.6 are experimental results of the reflected
waves (back scattered fields) from a plasma-coated metallic cylinder
when illuminated by a TE wave as functions of the discharge current or
the plasma density. The transmitted wave (incident and forward
scattered waves) is also shown in the lower part of Fig. 5.5.(a).

It is noted that the sensitivities of the oscilloscope for the trans-
mitted and reflected waves were adjusted separately to obtain symmetri-
cal displays. Double tracing of the displayed curves is due to
hysteresis phenomenon of the plasma. Figure 5.5.(b) shows the
reflected wave only.

To show the detailed behavior of the curve of the reflected
wave, the reflected wave was measured carefully at lower and higher
discharge currents separately. Figure 5.6.(a) shows the detailed
behavior of the reflected wave in the lower discharge current range.
Figure 5.6.(b) shows the same quantity for the higher discharge current

range. In these two figures, the dipolar resonance occurs at 210 mA

130

of the discharge current and the quadrupolar resonance occurs at 275 mA.
The temperature resonances are shown in the left hand side of the
dipolar resonance in Fig. 5.6.(a) appearing as small ripples.

Figure 5.7 shows the theoretical result of back scattered E
field from a corresponding plasma-coated metallic cylinder when
illuminated by a TE wave. Two values of collision frequency are
assumed to be v/m I 0.001 and u/w I 0.01. Figure 5.7.(a) is obtained
based on a cold plasma theory and a l3-sublayer model. Figure 5.7.(b)
is obtained based on a hot plasma theory with a 3-sublayer model and the
vo/c ratio of 0.01 Both in Figs. 5.7.(a) and 5.7.(b), the non-uniform
plasma density distribution is assumed to be expressed by Eq. (3.192)
of Chapter 3.

Figure 5.8 shows the experimental results of the back scattered
E field from a plain plasma cylinder when illuminated by a TE wave. The
dipolar resonance is identified as the highest peak at the right of
Fig. 5.8.(a) while a number of temperature resonances appearing at the
left. The quadrupolar resonance is shown at the right of the dipolar
resonance in Fig. 5.8.(b).

Figure 5.9.(a) and 5.9.(b) are the theoretical results of the
back scattered E field from a plain plasma cylinder plotted as a
function of (mp/m):ve. based on a cold plasma theory and a hot plasma
theory, respectively. The ratio vO/c is assumed to be 0.0133 in the
hot plasma theory. In both theories a non-uniform plasma density
expressed by Eq. (3.193) of Chapter 3 is assumed.

Figure 5.10 shows the results of the back scattered E field

from a plasma-coated metallic cylinder when illuminated by a TM wave.

131

Experimental result on the reflected wave is shown in Fig. S.lO.(a) as

a function of the discharge current. As we have discussed in Chapter 4,
for an incident TH wave, the cold and hot plasma theories both predict
the same solution for the scattered field. Therefore, the theoretical
back scattered B field shown in Fig. 5.10.(b) applies both for the cold
and hot plasma cases. The collision frequency is assumed to be vim

I 0.01 in the theoretical calculation. The non-uniform plasma density
distribution is assumed to be expressed by Eq. (3.192) of chapter and

a l3-sublayer model is used.

Figure 5.11 shows the results of the back scattered B field
from a plain plasma cylinder when illuminated by a TM wave. Experi-
mental results on the reflected and transmitted waves as functions of
the discharge current are shown in Fig. 5.ll.(a). The theoretical
result on the reflected wave is shown in Fig. S.ll.(b). The non-
uniform plasma density is assumed to be expressed by Eq. (3.193) of

Chapter 3 and a.l3-sublayer model is used in theoretical calculation.

5.4.2 Comparison Between Experiment and Theory
For the case of the back scattered E field from a plasma-
coated metallic cylinder illuminated by a TE wave, the dipolar and
quadrupolar resonances are predicted both by the cold and hot plasma
theories. The temperature resonances are predicted only by the hot
plasma theory. All those dipolar, quadrupolar and temperature reso-
nances are observed in the experiment as shown in Figs. 5.5 and 5.6.
In the hot plasma theory, the theoretical value of (m

Paq

mp q and mp d are the plasmm.frequencies corresponding to the quadru-
9 9

polar and dipolar resonances, is 1.27. With the cold plasma theory

lmp d)2, where
3

132

the value of (mb q/mb d)2 is found to be 1.24. EXperimentally as in
I

Fig. 5.5.(b), the value of Tall where I and I are the discharge

(1’ q d

currents corresponding to the quadrupolar and dipolar resonances, is
observed to be 1.28. Thus, for the location of the dipolar and
quadrupolar resonances, experiment and theory come to a very good
agreement. As for the shape of the curve, the quadrupolar resonance
observed in the experiment is a rather smooth peak while the theory
predicts a sharp peak. The temperature resonances are seen to occur
to the left of the dipolar resonance. Those resonances occured at
plasma densities where the value of (map/m)2 are less than 0.7 are not
detected in the experiment.

For the back scattered E field from a plain plasma cylinder
illuminated by a TE wave, the temperature resonances are observed
experimentally as shown in Fig. 5.8. The dipolar resonance occurs at
the extreme right of Fig. 5.8.(a). The quadrupolar resonance occurs
at a discharge current of I - 440 ms in Fig. 5.8.(b). The magnitude
of quadrupolar resonance in this case appears to be larger than the
case of a plasma-coated metallic cylinder. The theoretical prediction
of the temperature resonances gives weaker amplitude than that observed
in the experiment. This discrepancy is probably due to the inaccuracy
in the theoretical analysis sincea computer subroutine for calculating
Hankel functions with a large complex argument is not available.

For the case of the back scatterings from a plasma-coated
metallic cylinder and a plain plasma cylinder when they are illuminated
by a TM wave, theory and experiment agrees very satisfactorily as

evidenced in Figs. 5.10 and 5.11. For both cases of a plasma-coated

133

metallic cylinder and a plain plasma cylinder, no resonance is observed

experimentally or theoretically.

5.5 Discussion

The experimental investigation of the scattering from a plasma-
coated metallic cylinder when illuminated by a TE or a TM weve gives
satisfactory results which compare fairly closely with the theoretical
results based on the hot plasma theory. The cold plasma theory also
gives a fair agreement with experiment but it fails to predict the
temperature resonances.

The mmin difficulty encounted in the theoretical study is the
.mathematical resonances associated with the stratification.method.
This has been discussed in Sec. 2.9 of Chapter 2. To avoid the mathe-
matical resonances, it may be worthwhile to apply the differential

equation method‘2'5'7)

to our problem if this study is to be
extended in the future.

Finally, the experiment in this research was conducted in a
weveguide while our theory assumed the free space situation. In spite
of these differences, theory and experiment agree quite satisfactory.
For a future extention, it may be suggested to conduct the experiment

in free space.

 

 

Reflected
T Wave
>~.
u
-r'4
2 Transmitted
3 Have
5
'D
,—4
0
M
Eh
1 4J l l l J
120 180 240 300 360 mA
-* Discharge Current (Plasma Density)
(a) reflected and transmitted waves
Reflected
Have

Field Intensity -—>

 

 

l l l l l l l
120 180 2&0 300 360 mA
—+ Discharge Current (Plasma Density)

(b) reflected wave
Fig. 5.5 Experimental results of the hack scattered E field from a

plasma-coated metallic cylinder as a function of the
discharge current with a TE incident wave.

135

Reflected
Have

t

 

Field Intensity -—»

1 l I 1 l l l l
100 160 220 280 340 mA
‘—> Discharge Current (Plasma Density)

(a) lower discharge current range

Reflected
*—
Have

  

Field Intensity —+-

   

L, l 1 l l 1 1 l 1 J
180 2A0 300 360 GZO mA
-—e Discharge Current (Plasma Density)

(b) higher discharge current range
Fig. 5.6 Experimental results of the back scattered E field from

a plasma-coated metallic cylinder as a function of the
discharge current with a TE incident wave.

136

(Fa;

v/m 3 OsOOl

1

ta
.3.“

° v/m g 0e01
O.4_

 

 

 

 

J
0 l 2 3

(a) cold plasma theory with 13-sub1ayer model

Vim ' 0e001

E VJm 3 0001

0.3.—

o.2 _ ' ’

Del ’

(,I--.‘ALk IV ,‘ 7 l I l 2
0 1 2 3 4 —.-(mp/m) ave .
(b) hot plasma theory with 3-sublayer model and vo/c = 0.01

 

Fig. 5.7 Theoretical back scattered E field from a plasma-—coated ‘
metallic cylinder as a function of (mp/w)ave. with a TE lnrldent
wave and a parabolic density dist. (kor = 10, e 2 180°)

Fifi. 5.8

137

 

 

>

A.)

'4

2 Reflected
3 Have
I:

H

'U

v-l

O

-.-a

la.

1 I 1 i 1 I 1 l
140 200 260 320 380 mA

—> Discharge Current (Plasma Density)

(a) dipolar and temperature resonances

Reflected
4—
Have

Field Intensity-—>

 

l l i l

1
460 nm

l l
140 220 300 380
-—r Discharge Current (Plasma Density)

(b) dipolar, quadrupolar and temperature resonances

FXperimental results of the back scattered E field from a
plain plasma cylinder as a function of the discharge
current with a TE incident wave.

138

 

 

 

 

 

8e

 

 

 

 

1
m -
o
IEII
o
0.4
0.3
0.2
0.1 (\
0
2
-—~- (mp/w)“
(a) cold plasma theory with 13-sublayer model
A
0 1 2 3 a —-e (m /m)2
P ave.

(b) hot plasma theory with 3-sublayer model and vole 8 0.01333

Fig. 5.9 Theoretical back scattered E field from a plain plasma
cylinder as a function of (m long,“ with a TE incident wave
and a parabolic density dist? (kor = 10, Wt» =90! , 0 = 180°)

139

 

 

 

 

Reflected
t Have
m
M
-H
m
c
o
H
:
ya
u
H
O
q-l
m
1 J 1 L4, 1 l l l
112 156 200 244 288 mA
-—> Discharge Current (Plasma Density)
(a) experiment result
~s
1
IE.
i
0
lg |
0.3 '//
0.2 —
0.1 —
0 0 11 2 1 31 (w [(9)2
° ' ° p ave.

(b) theoretical result (6 I 180, kor = 10, v/w = 0.01)

Fig. 5.10 Experimental & theoretical results of the back scattered E
field from a plasma-coated metallic cylinder with a TM
incident wave.

140

 

 

 

 

Reflected
Have
I
>\
U
'4
C
r.
O
H
c
H
3 Transmitted
3'. “Have
a.
l l l l I 1 L I
100 200 300 600 500 mA
—.Discharge Current (Plasma Density)
(a) experiment result
IE‘l *
_£L
IE‘I
o l-
0.15 -
0.1 _
0.05
I I l I
o o 1 2 3 a —_. (m /m)2

p ave.
(b) theoretical result ( e - 180°, k01- - 10. v/m - 0.01)

Fig. 5.11 Experimental and theoretical results of the back scattered
E field from a plain plasma cylinder with a TM incident
wave.

APPENDIX A

THE DECOMPOSITION OP PRESSURE GRADIENT
INTO THE D.C. AND A.C. COMPONENTS

 

APPENDIX A

THE DECOMPOSITION OF PRESSURE GRADIENT
INTO THE D.C. AND A.C. COMPONENTS
The pressure gradient,V7P, of Eq. (3.75) is considered and
decomposed into the d.c. and a.c. components as follows:
If a static pressure (d.c. case) is concerned, the pressure
is established by an isothermal process. That is, the temperature
of the gas is fixed to a constant value, and we have

P - nek‘l‘ am)

where T is the fixed temperature of gas, ne is electron density of the
plasma and k is Boltzmann's constant.

If an external force disturbes ne in such a way that
ne(;,t) - nod-3 + n1(¥,t) ‘ (A.2)

and ne is a fast function of time, such as a high frequency dis-
turbance, then the temperature of gas is not fixed simply due to the
fact that not enough time is allowed to exchange energy in the gas to
keep the temperature constant. In such case, the adiabatic law is

used e that 13

Pt: I constant (A.3)

where )/ is the ratio of specific heat such that

C I
CV m

142

143

with m I degree of freedom of gas.

In a high frequency plasma oscillation, the motion of electron
is usually in one direction only. So that we can assume m I 1. This
leads to Y :- 3.

Eor the case of a small r.f. perturbation, as in Eq. (A.2),

the relationship between the pressure and the electron density is
Pnmy I P n"y I constant (A.S)
since initially P I P and n I n . Then
0 e o
n
.9. V
P . P°( n ) 0 (A06)
0
Pois the static pressure established by an isothemal process, so that
PO 3 nokT 0 (A07)

From Eq. (A.6), we have

vr- V[P0<;39’J
O

n n
eY eY
'(a:)VP°+POV(a-;) . (A.8)

The substitution of Eq. (Ad) in Eq. (A .8)lleads to

n n
VPI(1+;§)YVP°+POV(1+E-l-)'Y

 

0
n1')’ “1 7-1
.(1+a:)kTVno+klho[Y(l+-fi:)
nVn -nVn
. o l 2 l o] (A.9)
o

01'

 

144

n

n n n
VP Ikr[( 1 +3-1- )T- Y( 1 +3-1- )Y'1-;1-]Vno +'Yk'l‘ ( 1 +5-1- )y"“§7n1 .
O O O O
(A.10)

If n1<<no , Eq. (A.10) yields
VP - kTVno + YkTan . (A.ll)

In the above expression, the terms, kT Vno and YkT Val , are the

d.c. and a.c. components of the pressure gradient respectively.

 

REFERENCES

 

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

REFERENCES

T. R. Kaiser and R. L. Closs, ”Theory of Radio Reflections from
Meteor Trails : I," The Phil. Magazine, Ser. 7, Vol. 43, No. 336,
pp. 1 - 32, Jan. 1952.

G. H. Keitel, ”Certain Mode Solutions of Forward Scattering by
Meteor Trails," Proc. of IRE, pp. 1481 - 1487, Oct. 1955.

Y. Ohba, "Diffraction by a Conducting Circular Cylinder Clad by
an Anisotropic Plasma Sheath," Canadian J. of Phys., Vol. 41,
pp. 881 - 889, 1963.

H. C. Chen and D. K. Cheng, "Scattering of Electromagnetic Waves
by an Anisotropic Plasma-Coated Conducting Cylinder," IEEE Trans.
Antennas and Propagation, Vol. AP - 12, pp. 348 - 353, May 1964.

C. Yeh and W. V. T. Rusch, ”Interaction of Microwaves with an
Inhomogeneous and Anisotropic Plasma Column," J. of Appl. Phys.,
Vol. 36, No. 7, pp. 2302 - 2306, July 1965.

C. C. H. Tang, "Backscattering from Dielectric-Coated Infinite
Cylindrical Obstacles,“ J. of Appl. Phys., Vol. 28, No. 5,
pp. 628 - 633, May 1957.

P. E. Vandenplas, "Electron Haves and Resonances in Bounded
Plasma,” Interscience Publishers, N. Y., 1968.

K. Pong, "Computation of Backscattering Cross Section of Plasma-
Clad Metal Cylinders by the ‘Layer Method'," IEEE Trans. Antennas
and Propagation, pp. 138 - 140, Jan. 1968.

J. A. Stratton, "Electromagnetic Theory,” McCraw-Hill Book Co.,N. Y.,
p. 374, 1941.

N. Herlofson, ”Plasma Resonance in Ionospheric Irregularities,"
Arkiv For Pysik, Vol. 3, pp. 247 - 297, Feb. 1951.

P. E. Vandenplas and A. M. Messian, "Equations of a Hot Inhomogeneous
Plasma Model-II, Perturbed Scalar Pressure Approximation," Plasma
Physics (J. of Nuclear Engery, Part c), Vol. 6, pp. 459 - 468, 1964.

T. J. Killian, ”The Uniform Positive Column of an Electric Discharge
in Mercury Vapor," Phys. Review, Vol. 35, pp. 1238 - 1252, May 1930.

146

 

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

147

J. L. Shohet and A. J. Hatch, ”Eigenvalues of a Microwave Cavity
Filled with a Plasma of Variable Radial Density,“ J. of Appl.
Ph’... V01. 41, N00 6, pp. 2610 - 2618, May 1970.

A. Dattner, "Plasma Resonance,” EricssOn Tech., Vol. 3, No. 2,
pp. 309 - 350, 1957.

J. V. Parker, J. C. Nickel and R. H. Gould, "Resonance Oscillations
in a Hot Nonuniform Plasma,” The physics of Fluids, Vol. 7, No. 9,
pp. 1489 - 1500, Sep. 1964.

F. H. Crawford, ”Internal Resonances of a Discharge Column,” J. of
Appl. Phys., Vol. 35, No. 5, pp. 1365 - 1369, May 1964.

F. H. Crawford and G. S. Kino, ”The Mechanism of Tonks-Dattner
Resonances of a Discharge Column,” Proc. of the Sixth International
Conference on Ionization Phenomena in Cases, Paris, July 1963.

J. R. wait, ”Scattering of Electromagnetic and Electroacoustic Waves
by a Cylindrical Object in a Compressible Plasma,” Radio Science

J. of Research NBS/USNC-URSI, Vol. 69D, No. 2, pp. 247 - 256, Feb.
1965.

J. A. Fejer. ”Scattering of Electromagnetic Waves by a Plasma
Cylinder,” The Physics of Fluids, Vol. 7, No. 3, pp. 439 - 445,
Mar. 1964.

L. Spitzer, ”Physics of Fully-Ionized Gases," Interscience, 1956.

M. H. Cohen, ”Radiation in a Plasma. 11. Equivalent Sources,”
Phys. Review, Vol. 126, No. 2, pp. 389 - 397, April 1962.

A. H. Kritz and D. Mintzer, "Prepagation of Plasma Haves across a
Density Discontinuity,” Phys. Review, Vol. 117, No. 2, pp. 382 -
386' Jan. 1960.

K. A. Haynes and D. Kahn, ”Have Propagation across a Two-Fluid
Plasma Density Discontinuity,” The Physics of Fluids, Vol. 8,
No. 9, pp. 1681 - 1688, Sept. 1965.

8.3..Seshadri, I. L. Morris and R. J. Mailloux,”Scattering by a
Perfectly Conducting Cylinder in a Compressible Plasma,” Canadian
Jo 0f PhYUe' V01. 42' pp. 465 - “76. March 1964.

"Hand Book of Mathematical Functions," Dover Publications, Inc.,
N. Y.

8. Ho and K. M. Chen, ”Electroacoustic Resonance in Plasma Layer
Surrounding a Metallic Cylinder,” Proc. of IEEE, Vol. 56, No. 9
pp. 1600 - 1602, Sept. 1968.

 

148

(27) B. Keriar and P. Heissglas, ”Plasma Microwave Interaction,”
J. of Appl. Phys., Vol. 36, No. 8, pp. 2479 - 2484, Aug. 1965.