AN INVESTIGATION OF THE ELECTROMAGNETIC . V
BEHAVIOR OF A MICROWAVE PLASMA SOURCE OVER A I
WIDE RANGE OF PRESSURES ANO FLOW RATES ‘   ‘. '

. Dissertation for the Degree of Ph. D.'
MICHIGAN STATE UNIVERSITY
RAGHUVEER MALLAVARPU
1976 ‘

 

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This is to certify that the

thesis entitled
AN INVESTIGATION OF THE ELECTROMAGNETIC

BEHAVIOR OF A MICROWAVE PLASMA SOURCE
OVER A WIDE RANGE OF PRESSURES AND FLOW RATES

presented by

Raghuveer Mal lavarpu

has been accepted towards fulfillment
of the requirements for

Ph.D degree in Electrical Engineering

RAXVMAL

Major professor

 

 

Date M I

0-7639

 

ABSTRACT
AN INVESTIGATION OF THE ELECTROMAGNETIC BEHAVIOR

OF A MICROWAVE PLASMA SOURCE OVER A WIDE RANGE
OF PRESSURES AND FLOW RATES

By

Raghuveer Mallavarpu

The electromagnetic behavior of a microwave plasma source is ex-
perimentally examined over a wide range of pressures and flow rates.
The plasma source under study is a variable length microwave cavity
partially filled with a cold, lossy, isotropic, homogeneous plasma column
enclosed in a quartz tube. A small signal linear theory which neglects
the flow rate of the gases, but takes into account the pressure dependence
of the discharge, is used to qualitatively explain some of the observed
experimental results.

Three different types of experiments are conducted on the plasma-
cavity to demonstrate its use as a versatile microwave plasma source.
Two plasma cavities, i.e., Cavity 1 and Cavity 2 operating at incident
powers of 30W and 1.2 KW respectively, are used to perform these experi-
ments. Cavity 1 maintains a stable, low pressure, non-flowing plasma.
It is suitable for examining the linear and non-linear behavior of the
plasma and is amenable to conventional plasma diagnostics. A high
pressure (over 1 atm.), flowing plasma can be maintained in Cavity 2 and
is suitable for studying the absorbed power characteristics of the

plasma source.

Raghuveer Mallavarpu

In the first of these experiments, unusual EM phenomena of the
plasma source are examined using Cavity l. The presence of two plasma
density operating points for a single eigenlength in the TE:11 mode is
experimentally verified. This unusual behavior is explained as result-
ing from the presence of a backward wave region in the w - k diagram of
the TE:1 mode for non-zero losses. The coupling of EM energy to space
charge waves, resulting in the formation of long plasma columns, is ex-
perimentally demonstrated and this phenomena is qualitatively explained
from the eigenlength vs. (mpe/w)2 and (palm) curves. Short wavelength
standing waves of the TMOOp mode are also excited, as predicted by the
theory. Two different types of sidebands are detected in the reflected
power spectrum of the TE:11 mode, and the occurrence of one of these is
qualitatively explained as being caused by a fluctuation in the average
plasma density as a function of time.

The second phase of the experimentation involves diagnostics on
the plasma source. The average electron density, N0, and the electron
temperature, Te, are measured over the pressure range 0.04 - 20 torr
using the TM010 frequency shift, double floating probe and disc diagnostic
techniques. A knowledge of these parameters is used in calculating the
effective electric field Ee and the reduced field Ee/p in the plasma.

The shape of the No’ Te, E8 and Ee/p vs. pressure curves follow the same
general variation with pressure as those exhibited by other investigators
in He gas.

In the third type of experiments, the absorbed power characteristics
of the plasma source are examined over a wide range of pressures, flow

rates, and incident power levels. The absorbed power characteristics of

a flowing plasma are shown to be significantly different from those of a

Raghuveer Mallavarpu

non-flowing plasma. The absorbed power is shown to increase directly

as a function of the flow rate at low flow rates and then reach a satura-
tion at high flow rates. This phenomena is qualitatively explained as
being caused by the shielding effect of the high density plasma layer
adjacent to the quartz tube.

The variation of the absorbed power as a function of pressure is
shown to be more uniform in this plasma source than in other types of
plasma sources. No tuning stubs are required to provide an impedance
match between the external microwave system and cavity. Plasmas can be
sustained in argon gas beyond 1 atm. and the gas flow rate can be varied
to as high as 14,000 cc/min. It is possible to couple more than 90% of
the incident power to the plasma by optimizing the length, discharge

pressure and the gas flow rate-of the plasma source.

AN INVESTIGATION OF THE ELECTROMAGNETIC BEHAVIOR
OF A MICROWAVE PLASMA SOURCE OVER A WIDE RANGE
OF PRESSURES AND FLOW RATES

By

Raghuveer Mallavarpu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering and Systems Science

1976

To my parents

Mr. & Mrs. M. Nageswara Rao

ii

ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude to his major
professor, Dr. J. Asmussen, for his able guidance, encouragement and
invaluable advice during the course of this investigation. Special
thanks are also due to the other members of the guidance committee,
Drs. K. M. Chen, D. P. Nyquist, M. Siegel and G. L. Pollack, for their
time and interest in this work.

The author also acknowledgestfluasupport of Detroit Edison Company,
Kimberly-Clark Corporation, and the Division of Engineering research

at Michigan State University for financing part of this research.

iii

Chapter

LIST OF FIGIJRES O O O I O O O C C O O O O O O O 0 O

I

II

III

IV

TABLE OF CONTENTS

INTRODUCfI ON C O O O O O O O O O C O O O O O O O O 0

DERIVATION OF THE MODES ON A CONCENTRIC LOSSY COLD
PLASMA ROD INSIDE A CYLINDRICAL METAL WAVEGUIDE. . .

2 1 Introduction. . . . . . . . . . . . . . . . . .
2 2 Basic Equations . . . . . . . . . . . . . . . .
2.3 Derivation of the Characteristic Equation . . .
2 4 w - k Diagrams. . . . . . . . . . . . . . . . .
2 5 Summary . . . . . . . . . . . . . . . . . . . .

POWER ABSORBED IN A PLASMA WAVEGUIDE . . . . . . . .

3.1 Introduction. . . . . . . . . . . . . . . . . .
3. 2 Power Flow and Power Absorbed for TM

Waveguide Modes . . . . . . . . . . .p. . . . .
3.2.1 TM Mode . . . . . . . . . . . . . . . . . .

00
3.2.2 TMO1 Mode . . . . . . . . . . . . . . . . .
3.3 Power Flow and Power Absorbed for TB
Waveguide Modes . . . . . . . . . . . . . .
3.4 Power Flow and Power Absorbed for TEll.
Waveguide Modes . . . . . . . . . . . . . .

ELECTROMAGNETIC AND SPACE CHARGE MODES OF A COLD
LOSSY PLASMA ROD INSIDE A CYLINDRICAL CAVITY . . .

4 1 Introduction. . . . . . . . . . . . . . . . . .
4.2 Theoretical Cavity Modes. . . . . . . . . . . .
4.2.1 Classification and Description of the Plasma-

Cavity Modes. . . . . . . . . . . . . . . . .
4.2.2 Rotationally Symmetric Modes, TM

011
and TEOll O O O 0*. Q 0 O 0*. O O O O O O O 0
4.2.3 Dipole Modes: TE and TM . . . . . . . .

111 101
4.2.4 Space Charge Modes TMOOp' . . . . . . . . . .

4.3 Summary . . . . . . . . . . . . . . . . . . .

EHERIMENTAL SYSTEMS O O O O O O O O O O O O O O O 0

iv

12
22
61
63
63

66
7O

73
75

80

88

88
93

93

94
95
97
99

100

Chapter

VI

VII

1 Introduction. . . . . . . . . . . . . . . .
2 The Two Plasma-Cavity Systems . . . . . . .
.2.1 The #1 Plasma-Cavity System . . . . . . .
2 2 The External Microwave System for

Cavity 1. . . . . . . . . . . . . . . . .
5 2 3 Diagnostic Circuits for Cavity 1. . . . .
5.2.4 The #2 Plasma-Cavity System . . . . .

5 2 5 The External Microwave System for

Cavity 2.. . . . . . . . . . . . .
5.2.6 Diagnostic Circuits for Cavity 2.

EXPERIMENTAL PERFORMANCE OF THE MICROWAVE
PIOASMA S OURC E O O O O O O O C O O O O O I O O 0

Introduction. . . . . . . . . . . . . . .

.1 Introduction. . . . . . . . . . . . . .

2 .Excitation of Two Resonant Plasma-

Density Operating Points for a

Single Resonant Length. . . . . . . . . .

6.2 3 Formation of Long Plasma Columns. . . . .

6.2.4 Excitation of Short Wavelength (A g=1cm)
Standing Waves. . . . . . . . . . g. . . .

6.2.5 Detection .of Sidebands in the Reflected

Spectrum. . . . . . . . . . . . . . . . .

6.3 Measurement of No, Te and Calculation of E8

in the #1 Plasma-Cavity . . . . . . . . . .
3.1 Introduction. . . . . . . . . . . . . . .
3 1.1 Use of a Microwave Cavity to Measure
Electron Densities. . . . . . . . . . .
2 Floating Double Probe Method. . . . . .
.3 Measurement of the D.C. Electrical
' Conductivity. . . .'. . . . . . . . . .
6.3.1.4 Evaluation of ve and Ee from No

and Te . . . . . . . . . . . . . . . .

6.3.2 Experimental Results . . . . . . . . . .
6.4 Absorbed Power Characteristics of the
Microwave Plasma Source . . . . . . . . .
6.4.1 Introduction. . . . . . . . . . . . . . .
6.4.2 Experimental Results. . . . . . . . . . .
SUMMARY AND CONCLUSIONS. . . . . . . . . .
APPENDIX A O O O‘ O I O O O O O 0 O O O O O O O 0

LIST OF REFERENCES 0 O . O O O O O O O O O C O Q

6 1 . .
6.2 Linear and Non-Linear Operation of Cavity 1 .
6 2
6 2

Page

. 100

100
100

104
105
107

110

. 112

114

. 114

116
116

. 117

119

126

129

132
132

133
133

136

138

141
163
163
164

185

. 189

191

Figure
2.1

2.23

2.2b

2.3a

2.3b

2.4a

2.4b

2.4c

LIST OF FIGURES

An infinitely long metal waveguide with three con-
centric dielectric regions. Region 1: Cold, lossy,
homogeneous, isotropic plasma of radius a. Region
2: Quartz tube of outer radius b. Region 3: Air
space from p = b to p = c. Region 4: The outer
boundary of the metal waveguide. . . . . . . . . .
w/wpe vs. B/kc for theTM00 mode. fpe I 6.0 GHz,

(ripe/wc I 3.47. Waveguide dimensions: a I 0.3 cms.,
b I 0.3875 cms., c = 5.08 cms. . . . . . . . . . .

w/OOpe vs. OL/kc for the TM00 mode. fpe = 6.0 GHz,

(ope/wc I 3.47. Waveguide dimensions: a I 0.3 cms.,

b a 0.3875 ems. , C a 5.08 ems. O O C I I O O C C O O

w/wpe vs. B/kc for the TM00

fpe I 4.5 GHz, (ope/Luc I 2.6. Waveguide dimensions:

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . . . .

mode for w/w S 0.9.
pe

w/wpe vs. a/kc for the TM00 mode for w/wpe S 0.9.

fpe I 4.5 GHz, wpe/wc = 2.6. Waveguide dimensions:

a I 0.3 cms., b I 0.3875 cms., c = 5.08 cms. . . . .

w/wpe vs. B/kc for the TM00

f I 6.0 GHz, w /w I 3.47. waveguide dimensions:
pe pe c

mode for w/w S 0.9.
pe

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . . . .

(Io/wpe vs. a/kc for the TM mode for w/wpe S 0.9.
f I 6.0 GHz, w /w I 3.47. Waveguide dimensions:
pe pe c

a I 0.3 cms., b = 0.3875 cms., c I 5.08 cms. . . . .

<
(IO/wpe vs. B/kc for the TM00 mode for LII/wpe 0.1.
f I 6.0 GHz, w /w I 3.47. Waveguide dimensions:
pe pe c

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . . . .

vi

Page

13

26

27

28

29

30

31

32

Figure
2.4d

2.5a

2.5b

2.6

2.7a

2.7b

2.8a

2.8b

2.8c

2.8d

2.8e

2.8f

(IO/wpe vs. OI/kC for the TM00

f I 6.0 GHz, w /m I 3.47. Waveguide dimensions:
. pe pe c
a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . .

w/w vs. B/kc for the TM00 mode for (IO/wpe

pe

f I 10.5 GHz, f /f I 6.1. Waveguide dimensions:
pe pe c

mode for w/w
pe

< 0.1.

S 0090

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . .

w/w vs. OI/kC for the TM

P9 00

fpe I 10.5 GHz, Lupe/O0c I 6.1. Waveguide dimensions:

mode for w/w
pe

S 0.9.

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . .

Field distribution and charge perturbation for
circularly symmetric surface wave on isotropic

12

plasma column. Reproduced from Trivelpiece . .

Real part of plasma dielectric constant er vs. w/w

and v /w . f I 6 GHz. . . . . . .
e pe pe

Imaginary part of plasma dielectric constant

61 vs. w/wpe and ve/wpe' pe

01 mode. fpe

m [w I 6.1. Waveguide dimensions: a
pe c

b I 0.3875 cms., c I 5.08 cms. . . .

w/wpe vs. B/kc for the TM

w/wpe vs. OI/kc for the TM01 mode. fpe

wpe/wc I 6.08. Waveguide dimensions:
b I 0.3875 cms., c I 5.08 cms. . . . .

w/wpe vs. B/kc for the TM01

f I 6.0 GHz.

a

mode for w/w
pe

10.5 GHz,
0.3 cms.,

10.5 GHz,

I 0.3 cms.

< 0.24.

fpe I 10.5 GHz, (ope/wc I 6.L.Waveguide dimensions:

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . . .

w/wpe vs. a/kc for the TM01

f I 10.5 GHz, w /w I 6.08. Waveguide dimensions:
pe pe c

mode for w/w
pe

S 0.20.

a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. . .

Real part of plasma dielectric constant at vs. w/w

and v /w . f I 10.5 GHz. . . . . .
e pe pe

Imaginary part of plasma dielectric constant 61 vs.

w/w and v /w . f I 10.5 GHz. . .
pe e pe pe

vii

33

34

35

38

4O

41

44

45

46

47

48

49

Figure

2.9a

2.9b

2.103

2.10b

2.10c

2.10d

2.10e

2.10f

3.1

3.2

3.3

w/wpe vs. B/kc for the TM01 mode. fpe I 20 GHz,

-w e[wC I 11.6. Waveguide dimensions: a I 0.3 cms.,

P
b I 0.3875 cms., c I 5.08 cms. . . . . . . . . . .

(II/mpe vs. OI/kc for the TM01 mode. fpe I 20 GHz,

Lupe/wc I 11.6. Waveguide dimensions: a I 0.3 cms.,

b I 0.3875cms., c I 5.08 cms. . . . . . . . . . . . . .

it

*
III/wpe vs. B/kC for TM10 and TE11 modes. fpe 5.3 GHz,

cope/wc I 3.07. waveguide dimensions: a I 0.3 cms.,

b I 0.3875 cms., c I 5.08 cms. . . . . . . . . . . . . .

*
w/w vs. o/k for TM and TE* modes. f I 5.3 GHz,
pe c pe

10 ll

(ape/wc I 3.07. Waveguide dimensions: a I 0.3 cms.,
b I 0.3875 cms., c I 5.08 cms. . . . . . . . . . .

*
. < .
w/wpe vs B/kc for the TE11 mode for (IO/OI)pe 0 35 and

Ve/wpe I 0.2.fpe I 5.3 GHz. (ope/u)C I 3.07. Waveguide

dimensions: a I 0.3 cms., b I 0.387 cms., c I 5.08 cms.

*
. < .
w/wpe vs OL/kc for the TE11 mode for w/wpe 0 35 and
I = = B
Ngu$e 0.2. fpe 5.3 GHz, Lope/wc 3.07. Waveguide

dimensions: a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms.

Real part of plasma dielectric constant Er vs. (IO/OIpe and
V lb) 0 f = 503 GI'IZO o o o o o o o o o o o o o o o
e pe - pe

Imaginary part of plasma dielectric constant e vs.

1
w/w and v /w . f = 5.3 GHz. . . . . . . .
pe e pe pe

Arbitrarily-shaped metallic waveguide of unit length
81 and 82 are waveguide cross sections at Z I 0 and

z a 1. C . O . C . . C O C 0 O C O C O O O C O O O O 0
Normalized absorbed power Pa/IAI2 vs. w/wpe and

00
a I 0.3 cms., b I 0.3875 cms., c I 5.08 cms. fpe=6 GHz .

ve/wpe for the TM mode. Waveguide dimensions:

Normalized absorbed power in the resistor Pa/III2 vs.

w/2fl for a parallel RLC circuit driven by an ideal

current source. L I IODH, C I 10 pf. . . . . . . . . . .

viii

SO

51

54

55

56

57

59

6O

64

71

72

Figure

3.4

3.5

3.6

4.1a

4.1b

4.2

5.1

5.2

5.3

5.4

Normalized absorbed power P [IAIZ vs. w/w and v /w
a pe e pe

for the TM01 mode. Waveguide dimensions: a I 0.3 cms.,
b I 0.3875 cms., c I 5.08 cms., fpe I 10.5 GHz.

Normalized absorbed power Pa/IAI2 vs. (IO/wpe and
ve/wpe for the TE01 mode. fpe I 10.5 GHz. Wave-

guide dimensions: a I 0.3 cms., b I 0.3875 cms.,
c I 5.08 cms. . . . . . . . . . .

Normalized absorbed power Pa/IAI2 vs. (IO/mpe and
*

ve/wpe for the TE11 mode. fpe I 5.3 GHz. Wave-

guide dimensions: a I 0.3 cms., b I 0.3875 cms.,

c = 5.08 ems. C C O O O O C O O O O C O O O C O C

Resonant length LS vs. the plasma density (mpe/w)2

* *
00p’ ”011' “101’ 3“" TE‘111'
Operating frequency fixed at w/2n I 3.03 GHz.
Waveguide dimensions: a I 0.3 cms.,b I 0.3875 cms.,
c g 5.08 ems. O O I O O O I O O O O O I O O O O O Q C

for the modes TM

Resonant length LS vs. (Lupe/w)2 for the cold plasma
*
modes, TMIO0p and TMlOp' Operating frequency
w/ZN I 3.03 GHz. Waveguide dimensions: a I 0.3 cms.,
b I 0.3875 cms., c I 5.08 cms. . . . . . . . . . . . .
Resonant length vs. plasma density (tripe/w)2 and
ve/w for a cavity of radius 10.146 cm., plasma
radius of 1.248 cm., and an operating frequency

Of 2045 GHZ. O O O O I O O O O O O O O O O O O O 0

Cross section of the cylindrical cavity, 30 watt

microwave source, and plasma diagnostic circuits . . . .

Resonant frequency versus cavity size. Waveguide

dimensions: c I 5.08 cms. . . . . . . . . . . . . . . .

w/ZN vs. fpe and ve/w for the TM010 mode. Wave—

guide dimensions: a I 0.3 cms., b I 0.3875 cms.,

c - 5.08 ems. O O O O O O O O O O O O O O O O O O O O O

The #2 plasma-cavity system. (a) Cross section
of plasma cavity (b) Isometric drawing. . . . . . . .

ix

74

79

86

90

91

92

101

102

106

108

Figure

5.5

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8a

The external microwave system for the #2 plasma-
caV1ty. C C O O O O O C O C O O O C C O O O O O O O O 0

Comparison of the theoretical and experimental
curves of the cavity resonant length LS vs.

2 *
(wpe/w) for the TE111 mode. Waveguide dimen
sions: a I 0.3 cms., b I 0.3875 cms., c I 5.08

ems I O O C C O O O O O O O O O C O O O O I I O O O O O 0

Plasma being ejected out of cavity as a result
of efficient coupling to the space charge wave
through an electromagnetic excitation of the

*
cavity (TE111 or TM011 mode). . . . . . . . . . . . . .

Absorbed power vs. the discharge pressure for the
TMOll mode, with the incident power P1 held constant.

The cavity resonant length Ls I 8.6 cms. . . . . . . . .

Absorbed power vs. the discharge pressure for the
TM011 mode, with the incident power P1 held constant.

The cavity resonant length LS I 8.1 cms. . . . . . . . .

Absorbed power vs. the discharge pressure for the

TE“;11 mode, with the incident power P1 held constant.

The cavity resonant length Ls I 5.92 cms. . . . . . . . .

Excitation of 8 half-wave resonances of the TMOOp

mode for an off resonance electromagnetic excitation
ofthe cavity (TE111 mode). . . . . . . . . . . . . . ..
Excitation of 3 half-wave resonances of the TM00p

mode: for a large off resonance electromagnetic

excitationof the cavity (TE:11 mode) . . . . . . . . . .

Sidebands detected on the spectrum analyzer in the
TE;11 mode with the operating frequency m/2fl I 3.03 GHz.
Cavity resonant length LS I 6 cms. Discharge pressureI

12 mtorr. Power absorbed I 14.2 watts. Plasma electron
density (tripe/w)2 fluctuated between 2.04 - 3.16. Fre-

quency scale: 0.2 MHz/cm. . . . . . . . . . . . . . . .

Less

111

118

120

122

123

124

127

128

130

Figure

6.8b

6.9

6.10s

6.10b

6.10c

6.1la

6.11b

6.11c

6.11d

6.12s

Sidebands detected on the spectrum analyzer
in the TE;11 mode with the operating frequency

w/2n I 3.03 GHz. Cavity resonant length LS I

6.02 cms. Discharge pressure I 12 mtorr.
Power absorbed I 14.1 watts. Plasma electron
density (wpe/w)2 I 2.95. Frequency scale:

2.0 WIZICmO O O O O O O O O O O O O O O O O C
The effective collision frequency in A as a
function of velocity and frequency. Reproduced
from Whitmer and Hermanso. . . . . . . . . . . . . . . .
Normalized electron density (Lupe/w)2 vs. dis-

charge pressure Po and the absorbed power Pa

using the TM010 frequency shift method. . . . . . . . .

Effective electric field Ee vs. discharge pressure

Po and absorbed power Pa’ from the TM010 frequency
Shift method 0 O O O O O O O O O O O O O O O O O O O O 0

Reduced field Ee/Po vs. discharge pressure Po and
absorbed power Pa’ from the TM010 frequency shift
me thOd O O O O O O O O O O O C O O O O O O O O O O O 0 0
Typical floating double probe I-V characteristics

at a discharge pressure of 86 mtorr. Absorbed
power Pa 8 20 w. 0 O O O O O O O I O I O O O O O O O O 0

Typical floating double probe I-V characteristics
for pressures above 1 torr. Absorbed power Pa I

20 w. 0 O O O O O O O O O O O O O O O O O O O I O O O 0

Typical floating double probe I-V characteristics
at a discharge pressure of 140 mtorr and different
absorbed power levels. . . . . . . . . . . . . . . . .

Typical floating double probe I-V characteristics

at a discharge pressure of 730 mtorr and different
absorbed power levels. . . . . . . . . . . . . . . . . .
Normalized electron density (wpe/w)2 vs. discharge
pressure Po and absorbed power Pa’ from the floating

double probe measurements. . . . . . . . . . . . . . . .

xi

Page

131

140

142

143

144

146

147

148

149

151

Figure Page

6.12b Electron temperature Te vs. discharge pressure PO
and absorbed power Pa’ from the floating double

probe measurements. . . . . . . . . . . . . . . . . . 152

6.12c Effective electric field Ee vs. discharge Po and

absorbed power Pa’ from the floating double probe

measurements . . . . . . . . . . . . . . . . . . . . 153

6.12d Reduced field Ee/Po vs. discharge pressure Po and

absorbed power Pa’ from floating double probe mea-
surements. o o o o o o o o o o o o o o o o o o o o o o o 154

6.133 Normalized electron density (Lupe/m)2 vs. discharge
pressure Po and absorbed power Pa, from measurements

using discs. . . . . . . . . . . . . . . . . . . . . . . 156

6.13b Electron temperature Te vs. discharge pressure Po

and absorbed power Pa, from measurements using discs . . 157
6.13c Effective electric field E vs. discharge pressure

Po and absorbed power Pa’ from measurements using

discs. 0 O O O I O O O O I O O O O O O O O O O 0 O O O O 158

6.13d Reduced field Ee/Po vs. discharge pressure Po and

absorbed power Pa’ from measurements using discs . . . . 159

6.143 Normalized electron density (Lupe/w)2 vs. discharge
pressure Po and absorbed power Pa over the pressure

range 40 mtorr - 20 torr, from three different di-
agnostic methods. . . . . . . . . . . . . . . . . . . . 160

6.14b Electron temperature Te vs. discharge pressure Po
and absorbed Pa over the pressure range 0.1 - 20

torr, from two different diagnostic methods. . . . . . . 161

6.14c Reduced field Ee/Po vs. discharge pressure Po and
absorbed power Pa over the pressure range 0.04 -

20 torr, from three different diagnostic methods. . . . 162

6.15 Absorbed power Pa vs. eigenlength L8 and different
discharge pressures Po’ for the TEO11 mode plasma
of Cavity 2. Flow rate I 1000 cc/min., Incident

power P1 8 1.2 KW. O O O O I O O O O O O O O O O O O O O 166

xii

Figuge Page

6.16 Power absorbed Pa vs. eigenlength LS, and flow rate
(cc/min) for the TE011 mode plasma of Cavity 2. Dis-
charge pressure, PC I 10 torr. Incident power Pi I
12KW 169

2
6.17. TE011 resonant length vs. plasma density (wpe/w)

and collision frequency (De/w) for Cavity 2. w/2n
2045 GHZO o o o o o o o o o o o o o o o o o o o o o o o 170

6.18 Power absorbed Pa vs. discharge pressure Po and inci-
dent power Pi for the TE112 mode plasma of0 Cavity 2.
Flow rate I 147 cc/min. Gas: argon. . . . . . . . . .. 173

6.19 Power absorbed Pa vs. discharge pressure Po and inci-

dent power Pi for the TE112 mode plasma of oCavity 2.

Flow rate I 520 cc/min. Gas: argon . . . . . . . . . . 174

6.20 Power absorbed Pa vs. discharge pressure Po and inci-
dent power Pi for the TE112 mode plasma of oCavity 2.

Flow rate I 955 cc/min. Gas: argon. . . . . . . . . . 175

6.21 Power absorbed Pa vs. discharge pressure Po and inci-

dent power P1 for the TE112 mode plasma of0 Cavity 2.

Flow rate I 1480 cc/min. Gas: argon. . . . . . . . . . 176
6.22 Power absorbed Pa vs. discharge pressure Po and inci-

dent power P1 for the TE112 mode plasma of0 Cavity 2.

Flow rate I 2500 cc/min. Gas: argon. . . . . . . . . . 177
6.23 Power absorbed Pa vs. discharge pressure Po and flow

rate for the TE112 mode plasma of Cavity 2. Incident

power?1 I 966 W. Gas: argon. .,. . . . . . . . . . . 178

6.24 Power absorbed Pa vs. discharge pressure PO and flow
rate for the TE112 mode plasma of Cavity 2. Incident
power P1 I 759 W. Gas: argon. . . . . . . . . . . . . 179

6.25 Percentage peak-absorbed power vs. flow rate for the
*
.TEll2 plasma of Cavity 2. Incident power Pi I 966 W . . 180

xiii

Figure

6.26

Plot of absorbed microwave power vs. pressure for
argon plasmas at various incident powers. Repro-
duced from Bosisio et.al.54. . . . . . . . . . .

xiv

CHAPTER I

INTRODUCTION

Microwave discharges have been investigated for over 25 years1 and
applications have been suggested in the areas of plasma chemistryz,
laser dischargesB, and RF heating in thermonuclear fusion research4.
Despite this interest, experimentally versatile steady-state microwave
discharges at pressures greater than 1 torr have not been available until
recentlys. In most investigations the microwave plasma formation is
restricted by the metal coupling structure. For example, certain cou-
pling structures produce plasmas that are small in comparison to the
electromagnetic wavelength6’7, resulting in maximum plasma volumes of

only about 50 cmZL. Also, many experiments can only operate over re-

stricted pressure ranges7’8, and are able to achieve densities which are
only slightly higher than the critical density NC, corresponding to
plasma resonance at the exciting frequency8’9. Furthermore, most plasma
coupling structures are complex and difficult to analyze electromagnet-
ically6’7’8. This electromagnetic complexity makes direct measurement
of the plasma characteristics, such as electron density,temperature,
etc., difficult without disturbing the plasma and electromagnetic fields.
Thus, from an experimental and practical point of view, there is a need
for an RF plasma source which is able to provide a high variable plasma
density, which is capable of maintaining a microwave discharge over

large changes in background pressure and gas flow rate, which is able to

be analyzed electromagnetically, and which is amenable to diagnostic

measurement techniques for the quantification of microwave plasma para-
meters. In addition, many industrial applications will require a plasma
source having a volume much larger than 50 cm3.

Recently, a large-volume microwave plasma source was developeds.
By means of a slowg traveling-wave structure, microwave energy of
up to 2.5 kW was coupled to a 19 mm outside diameter quartz tube obs
taining plasmas exceeding 2000 cm3. This microwave plasma system was
shown to possess a good match at many different incident power levels
without external tuning systems for pressures of 1 torr up to 1 atm.
in argon gas. Thus, this system appears to overcome the tuning prob-
lems of previous microwave plasma sources while producing a large-volume
plasma over a wide pressure range.

In this thesis the experimental properties of still another micro-
wave plasma source is describedlo. Here the plasma is part of a cylin-
drical waveguiding system and either fast or slow waves are able to
sustain a plasma. The waveguiding system is terminated at one end by
a fixed short and at the other by a variable short, consequently, forming
a plasma cavity. Microwave power is then coupled to the resonances of
the plasma cavity, imposing the requirement for a physical length tuning
mechanism. However, this tuning is a fast, simple adjustment that seems
justified in view of the higher electric field (an important parameter

in many plasma chemistry experiments) obtained at the cavity resonances.

A linear, lossy, cold plasma theory is used in modeling the micro-
wave plasma source and is described in Chapters II-IV. Attempts are
made in Chapter Vfilto qualitatively explain the experimental results by

comparing with this theory.

The basic equations which will be used to mathematically describe
a cold, lossy, homogeneous, isotropic plasma are introduced in Chapter
II. From this set of equations a dispersion relation for the propaga-
tion of electromagnetic and cold plasma modes is derived for a quartz-
enclosed plasma concentrically located in an infinite metal waveguide.

w - B and w - a diagrams are explained for the different modes, with
the electron density and the electron-neutral collision frequency as
the variable plasma parameters. Regions of forward and backward waves
are identified on theta-lcdiagrams.

The Poynting vector is evaluated in Chapter III for the lossy plasma
modes that were discussed in Chapter II. From this Poynting vector a
normalized power absorbed per unit waveguide length is derived for the
modes. The power absorbed is studied as a function of the frequency with
the electron density and the electron collision frequency as the variable
plasma parameters. Volume and surface resonances of the plasma are
identified and the power absorbed by the plasma at the resonantfrequen-
cies is discussed.

The characteristic equation derived in Chapter II is used in Chapter
IV to obtain an eigenlength vs. plasma density and electron collision
frequency plot fortfiuzvarious modes of a cylindrical plasma—cavity. The
behavior of these modes at low and high values of the plasma densities
and for a range of collision frequencies is discussed. Slow-wave and
backwarddwave resonances are identified on the eigenlength plot. The
region of intersection of EM resonances with slow wave resonances is also
discussed.

The experimental apparatus is presented in Chapter V. The design

of two different cavities used in performing experimental work from

10 mtorr to 500 torr is discussed. The driving microwave circuits and
the probe and microwave diagnostic circuits are explained.

The experimental results are discussed in Chapter VI. The actual
experiments are done in two different cavities. Experiments are carried
out in the pressure range 10 mtorr to 20 torr and incident power levels
up to 30 W with Cavity 1. In experiments on Cavity 2, the pressure range
is l torr - 500 torr, flow rate is varied from 0-2500 cc/min and incident
power levels up to 1.3 KW are used. Eigenlength curves of the TE*111
mode are verified and the coupling of EM and plasma modes is demonstrated.
Non-linear effects in the form of sidebands intfluareflected spectrum of
the TE:11 mode are observed. The reason for the occurrence of one of
these sidebands is explained.

For many experiments involving applications of a gas discharge, it
is important to evaluate an effective electric field in the plasma. The
determination of this quantity requires a knowledge of the electron
density and the collisional frequency in the plasma. These plasma para-
meters are measured as a function of pressure in the range 100 mtorr to
20 torr using microwave, probe and disc diagnostic techniques. The
limitations of these methods and the accuracy of the results is also
presented.

Experiments on a flowing gas are conducted on Cavity 2, which is
capable of handling incident powers of up to 1.3 KW. Power absorbed
vs. cavity eigenlength are obtained for the TE011 mode with flow rate
and the gas pressure as the variable parameters. Another set of experi—
ments are conducted with a TE:12 mode where the absorbed power is

measured as a function of the gas pressure with the incident power and

the gas flow rate as the externally variable parameters. All these

experiments show that the absorbed power characteristics of a flowing
plasma differ significantly from those of a non-flowing plasma.
Chapter VII summarizes the work presented in the thesis and suggests

further investigations where required.

CHAPTER II

DERIVATION OF THE MODES ON A CONCENTRIC
LOSSY COLD PLASMA ROD INSIDE A CYLINDRICAL METALMWAVEGUIDE

2.1 Introduction

The general problem of propagating waves in a plasma waveguide has
been examined by several workers. The problem can be divided into two
specific cases which depend on the exact plasma volume occupying the
waveguide:

1) A plasma totally filling the waveguide, and

2) A plasma partially filling the waveguide.

Allis, Buschsbaum and Bersll obtained the w - k diagram for the
various propagating modes in a lossy, cold plasma completely fillingra
circular waveguide. Only EM waves can propagate in such a plasma-
waveguide system. Surface wavesare not present because they require
charge accumulation on the plasma surface.

Trivelpiece12 and Gould made a quasistatic analysis to show that
surface wave propagation is possible in a circular waveguide partially
filled by a concentric,cold lossless plasma rod. Agdur and Eneander13
solved the complete electromagnetic problem by deriving the characteristic
equation for the waveguide modes and solved for the cavity resonances for
several important resonant modes; i.e., TElll’ TMOlO’ etc. Shohet14 ex-
tended their results to include electron-neutral collisional losses and

solved for cavity resonant frequency as a function of electron density

and collisional losses.

wave propagation along warm plasma columns in free space and par-
tially filling circular waveguides has been examined by Diament, Granat-
stein and Schlesingerls, Vandenplasl6 and Fredericksl7. They have shown
that the cold plasma theory is not valid in and beyond the region where
the phase velocity equals the electron thermal velocity and that a warm
plasma description is more accurate. They also describe warm plasma
modesls, Tonks-Dattner or temperature resonances, and demonstrate experi-
mentally the coupling of EM modes to these resonances.

In this chapter, solutions to the characteristic equation of a lossy
plasma waveguide are studied in greater detail than in previous investi-
gations. For example, the dispersion curves for a number of propagating
modes in such a plasmadwaveguide are obtained for both the lossless and
the lossy cases. These curves will show that in addition to forward EM
and cold-plasma waves, it is also possible to have frequency regions
where backward waves propagate. Using this complete electromagnetic
description, the power absorbed per unit length has also been calculated

and is discussed in detail in Chapter III.

2.2 Basic Equations

 

Consider a cold, homogeneous, isotropic, lossy plasma of infinite
extent. An electromagnetic analysis of such a plasma is well known and
can be carried out by treating it as a material with a complex dielectric
constant. The latter is easily derived from a consideration of the
cold plasma fluid equations and the associated Maxwell's equations that
describe the electromagnetic fields in the plasma.

A set of basic fluid equations obtained from the first two velocity
moments of the Boltzmann equation for both electrons and ions are shown
below. In these equations the pressure termois neglected since Te - T1==CL

in the cold plasma approximation.

3Ne +
3—t—+ V. (NeVe) = 0 2.1

a?

__£.+ m (6 V)? = - e(E +‘V IKE) - (V V ) 2 2
Ineat: e e' e e ee me .
BN

1 +
a: +v. (NiVi) — o 2-3

a?

____i_+ (i? V)_\7=e(-E+-V 35-0937) 24
“‘13:; “‘1 1° 1 1" iimi '
where

N = electron number density

ion number density

2
I

macroscopic electron velocity

m<+
I

<+
I

macroscopic ion velocity

electron mass

5
(D
I

m = ion mass

9.

= electric field

WIN
I

magnetic field
e 8 electronic charge
' v = effective electron-neutral particle collision frequency,

v 3 effective ion-neutral particle collision frequency.

Equations 2.1 to 2.4 express the conservation of mass for electrons,
conservation of momentum for electrons, conservation of mass for ions and
conservation of momentum for ions,respectively. The electron-neutral
particle collisional losses within the plasma are accounted for by a non-
zero value of ve. The associated Maxwell's equations in such a plasma

are:

+ 3%
Vx E =-IJO'5? 2.5
(N 'N)
V. E = e -$-—£- 2.6
E
.0
+ + 3?;
VxH ' 8(N1Vi - NeVe) + 80 _t 2.7
v . fi = 0. 2.8

B . e‘E 2.9
O

+ +

B t “0H . 2.10

These relations are valid under the assumption that the electrons and
ions are free and are present in a free space medium, i.e., there is no

bound charge present as in a dielectric.

Assuming a time-variation of the form ejmt and expressing the vari-
able quantities as the sum of a dc term and a perturbed ac term, the

variables become:

N = N + n ejwt 2.11
e oe e

N = N + n ejwt 2.12
i oi i

+ + jwt

Ve = Ve e assuming no dc electron drift 2.13

V1 =‘Vi ejmt assuming no dc ion drift 2.14

E = E ejwt assuming no static electric field 2.15
'15 , 'fi e3”t

assuming no static magnetic field. 2.16

IO

Noe and N01 are the unperturbed electron and ion particle densities re-

spectively, and they are equal for both electrons and ions in a homo-

geneous plasma, i.e., N0e 8 N01 = No'

Using Equations 2.11 through 2.16, the first velocity moment

equation for electrons as given by Equation 2.2 can now be reduced to,

-> + ->
jwV m = q E - v m V . 2.17
e e e e e e

+ + -> +
In Equation 2.17, the (Ve x B) term and the non—linear term (Ve.V)Ve
have been neglected. These approximations are justified provided we

study those wave solutions for which Ve<<V¢, where V is the phase velo-

¢
city of the wave. The Ve x B term can be simplified by using Maxwell's

curl equation for E, to become:

+

V
.13

V¢

where 6 is a unit vector in the direction of propagation.

+
x fi x E

The non-linear term (Ve.V)Ve upon simplification, becomes:
A +
(Ve.n)Ve

e V¢

Under the assumption that Ve<<V¢, (perturbation theory). these terms are

 

jwm

very small compared to other terms in Equation 2.2: i.e.,

 

(v .fiW
(V2 x h x E)<<E and jwm e e
«b e W:

and the omission of these terms from Equation 2.17 is justified. Cold

<<jwmeVe

plasma theory, however, fails if V ”Vt is the electron thermal

¢ h h

velocity. In that case, a warm plasma description is more accurate.

where Vt

From Equation 2.17, the macroscopic electron velocity is easily

obtained as,

11

+ eE \
ve a -(mer + Va); , 2.18

 

The high frequency plasma phenomena are of interest in this study.
Thus, under the high frequency approximation, the plasma ion frequency
mp1 is very small compared to the operating frequency w. The motion of
the ions can be considered to be negligible and,therefore, the contri-

bution to the current density,J,comes entirely from the electron motion.

Thus,

3 = -N 6‘; o 2019
0 8

Using Equation 2.18 in Equation 2.19,

2+
+ Noe E

J:
me(jw + ve)

 

2.20

Since also 3 = OE, an equivalent plasma conductivity 0 can be defined as,

N e2(v - jw)
m (v + w )
e e

 

Substituting forCI from Equation 2.21, into Maxwell's curl equation for

the magnetic field in the plasma, we have

 

- ++E~ 1+ 0 E
- jweoE O — jweo jweo 2.22
w2 v w2
or Vxfi=jw€o l-i—Z-j-f—i-fig-z- E 2.23
v + w v + m
e e
N e2

 

where w =
P

12

Equation 2.23 is of the form V x R = jw E E. Thus, we can define an

equivalent dielectric constant for the plasma as,

 

 

! w; v .3 1 ‘
e e e
s = s 1 - -————-—- - j ——- 2.24
° v2+w2 w v2+w2
L e e
=€o(€r+J€i)
where 2 2
we . ve we
€r= 1"J-2—T and 61“?»- 2 2 -
(Ve + w ) (ve +1w

For a lossless plasma, ve = 0 and 8 is purely real.

2.3 Derivation of the Characteristic Equation
We next consider an infinite metal waveguide of inner radius c, con-
centric with a homogeneous, isotropic lossy plasma rod of radius a, en-
closed in a thin quartz tube of radius b. A generalized characteristic
equation, dependent on the plasma parameters wpe and vs and describing the
eigenmodes of such a waveguide,will be derived here. The effect of the
finite thickness of the quartz tube will be included in this derivation.
Since the plasma is assumed homogeneous, isotropic and lossy, Max-
well's Equation 2.5, 2.6, 2.8, and 2.23 describe the electromagnetic
behavior within the plasma rod. The problem then reduces to a boundary
value problem of three concentric dielectric regions in a metal wave-
guide as shown in Figure 2.1. Region 1 is the homogeneous, isotropic,
lossy plasma; region 2 is the quartz dielectric that encloses the plasma;
and region 3 is free space. The theory for such a problem is well knownls.
The fields in the three regions are expressed in terms of electric and
1’ me

magnetic scalar potential functions, wel’ wez, $83, wm and me. A

 

Figure 2.1.

An infinitely long metal waveguide with three concentric di-
electric regions. Region 1: Cold, lossy, homogeneous, iso-
tropic plasma of radius a. Region 2: Quartz tube of outer
radius b. Region 3: Air space from p I b to p - c. Region
4: The outer boundary of the metal waveguide.

14

scalar Helmholtz equation in cylindrical coordinates is formulated for
each of the potential functions in the three regions. These equations
are solved subject to boundary conditions on the perfectly conducting
boundary r = c, at the free space-quartz interface r = b and at the
quartz-plasma interface, r = a. Expressed in terms of a general w, the

scalar Helmholtz equation is,

2 2
_]_'._§_ 9.3.4). +lfl +M+k2w=0 2.25
Q 3p 3p p2 3¢2 322

where k relates the longitudinal and radial propagation constants as

k2 = wzus =‘k2 + k2.
p z

The general solutions to the magnetic and electric potentials in each of

the thrée regions are,

ml -jkzz
w = AJ (k r) cosn¢ e 2.26
n pl
m2 -jkzz
w = [BJ (k r) + CN (k. r)] cosn¢ e 2.27
n 02 n 02
m3 ‘ -jkzz
w = DJ (k r) + EN (k r) cos n¢ e ' 2.28
“ p3 “ p3
-jk z
wel = FJ (k r) sin n¢ e z 2'29
n p2
w = [CJ (k r) + HN (k r) sin n¢ e 2-30
“ p2 “ 02
93 -ijZ 2 31
W = [iJn(kp r) + KNn(kp r)] sin n¢ e .

3 3

In each of the three regions, the associated electric and magnetic

fields for the TM and TE modes can then be derived from the following

general relations:

 

TM Mode
E =71Klé—2'fl
p y apaz
E = 1 fl
¢ 9 p awaz
2
E =7ch +k2 19m
2 y 822
H = lain:
0 03¢
awm
H =-———
w 30
H = 0
2
TE Mode
E =_l§llfi
o 0 M
E =21»:
¢ Bo
E = 0
2
H =l_a_2_‘p_e_
p 2 8032
H nififi
¢ 39 3¢32
2
H -%<§-—+k2>we
z z 2
32

where 9 = jwe and 2 = jwu.

.32

.33

.34

.35

.36

.37

.38

.39

.40

.41

.42

.43

In each region the potential functions must also satisfy the separation

relations:

l6

2 2 2 w 2
Region 1. kpl + k2 - k1 — (c) 81 2.44
2 2 2 w 2
Region 2. kp2 + kz = k2 = (E) 62 2.45
Re ion 3 k2 + k2 = k2 - (252 2 46
g ° 03 z 3 c ° ’

81 is the plasma permittivity and is given by Equation 2.24, and 82 is
the permittivity of the quartz tube.

The solutions to the Helmholtz equation must satisfy the following

boundary conditions:

1) The tangential components of E and H must be continuous across
the plasma-quartz tube interface; i.e., at p = a, Ez, Hz, E¢
and H¢ must be continuous.

2) The tangential components of E and H must be continuous across
the quartz tube - free space interface; i.e., at p = b, Ez, Hz,
E¢ and H45 must be continuous.

3) The tangential components of E must vanish on the outer metal
boundary; i.e., at p = b, E2 = 0, E¢ = 0.

In this class of problems the fields cannot be divided into pure

TE or TM modes; thus, both Ez and H2 will be necessary to satisfy the
boundary conditions for any given mode. This will ensure a general so-
lution. Except for the case of rotionally symmetric modes, which can be
thought of as pure TE or TM, the other modes are hybrid in character.

Using equations 2.26 through 2.46, subject to the boundary condi-

tions as given above, we arrive at the following 8 simultaneous equations

2 2 2
Pk J (k a) G k Jn (kp a) + H kp Nn (kp a) 2.47

01 “ D1 D2 2 2 2

17

2 2

 

 

 

 

 

 

 

 

 

A8 k J(k a)=B€ R J (k a)+C€ k N(k a) 2.48
2 p1 n pl 1 p2 n oz 1 92 n 02
kzn
Ak J'(k a)+F J(k a)=Bk J' (k a)+Ck N'(k a)
pl “ D1 ”“03 “ 01 p2 “ 02 p2 “ 92
kzn kzn
+ G J (k a) + H N (k a) 2.49
whoa n p2 wuoa n p2
kzn kzn kzn
A J(k a)+FkJ'(k a)=B J(k a)+C N(k a)
wela n pl pl n 01 weza n p2 weza n p2
+ G R J' (k a) + H k N' (k a) 2.50
02 “ D2 p2 “ 02
c k2 J (k b) + H k2 Nn (k b) = K'kz F2 2.51
02 n 02 02 02 03
B 83 k2 Jn (k b) + C€3 k2 N (k b) = E'ezk2 F1 2.52
02 02 02 n 02 D3
kzn kzn
B k J' k b + C k N' k b + G J k b + H N
02 n ( 02 ) 02 n ( 02 ) wuob n ( 02 ) wuob n (kpzb)
kzn
= ' '
E kp3F3 + K ‘ “mob F2 2.53
kzn kzn
B---—J (k b)+C————N (k b)+Gk J' (k b)+
w€2b n 02 w€2b n 02 02 n 02
kzn
' = ' ————— + K' k 2.54
H kpan (kpzb) E ws3b F1 03 F4
where
' = - 2.55
E E/Jn (kp c)

3

18

I _ _ I
K - K/Jn (RQ c) 2
3
F1 = Jn (kp b) Nn (kp C) - Jn (kp C) Nn (kp b) 2
3 3 3 3
= ' _ 0
F2 Jn (kp b) Nn (kp C) Jn (kp C) Nn (kp b) 2.
3 3 3 3
= I _ I
F3 Jn(kp b) Nn(kp C) Jn(kp C) Nn (kp b) 2.
3 3 3 3
= I I _ I I -
F4 Jn (kp3b) Nn (kp3c) Jn(kp3c) Nn(kp3b). 2

For a non-trivial solution to the simultaneous equations 2.47 through
2.54, the determinant of the coefficients must be identically zero.

Hence, it follows that,

.56

.57

58

59

.60

19

Ho.

 

« s as
1:1

 

«Q c «Q
An xv.z x

N O

Q a :3
3 ézn~
cx

 

 

«a cmo
2 5% v.

«Q c 0:3
3 scan

«
c x

 

Q a m :3
A... in

 

c x:

An

 

 

 

«Q c NQS
S 5an
c x
«Q a «Q
An xv.z x
« «

Q c Q
is is 2 «saw

Q c m a:
CW 52'!

c x:

«a «u
C 3
S éspu
ax

 

«Q c «Q
An xv.fi x

N N

Q c
is so 4 Min

u

 

 

 

20

Upon simplifying the above determinant we obtain the following

characteristic equation for the three dielectric boundary value problem

 

 

 

I
k

81 kp2 Jn( p1a) (
-—— e k F F - s k F F ) - (e k F F -s k F F )
e2 kpl Jn(kpla) 3 02 3 7 2 Q3 1 8. 3 oz 3 5 2 Q3 1 6

I
kpz Jn(kpla)

k F F -k F F > - (k F F - k F F )
kpl Jn(kpla) ( 03 2 8 02 4 7 p3 2 6 oz 4 5
2 2 (k2 4 k2 )2
=nkz pl 92 [ekkFFFF-ekzFFF2
wzu 6 82 k4 R2 3 p2 D3 2 3 7 8 3 92 3 4 7
° 2 ‘1 D2

2 2
+ Ezkpzkp3F1F4F7F8 - 62k p3F1F2F8 ]

 

 

 

 

2 2 2
k - k J' k
n2k2z ( 02 93) 2 2 “( 018) k k
+ 2 2 2 2 2 'Ezkp FleFs + J (k a) (52 + 51) p p F1F2F5F7
w poszb kp kp kp l n 01 1 2
1- 2 3
2 2 2 2 2
J' k - -
n( p a) n2R2 (kp kp )(kp kp )
_ 1 2 2 z 1 2 2 3
2 E1kp F1F2F7 + 2 4 4 2 ‘
Jn(kp a) 2 w Hoezab kp k k
1 1 92 p3
n2k2
2 2 2 z 2 2 2 2 2
2e k k k F F F F + (k - k ) (k - k ) F1F2F7
2 D1 D2 D3 1 2 9 1° wzuoab D1 02 D2 03

2.62
where,

21

F5 = J;(kp2a) Nn(kpzb) - N;(kp2a) Jn(kp2b) 2.63
F6 = J;(kp2a) N;(kpzb) - Ng(kpza) Jh(k02b) 2.64
F7 = Jn(kpza) Nn(kpzb) - Nn(kpza) Jn(kp2b) 2.65
F8 = Jn(kpza) Nn(k02b) - Nn(kpza) JA(kpzb) 2.66
F9 = J;(kp§O Nn(kpza) - N;(kpza) Jn(kpza) 2.67
F10 = Jn(kpzb) N;(k02b) — Nn(kp2b) J$(kpzb). 2.68

For rotationally symmetric modes, n = O and the right side of
Equation 2.62 becomes zero. The left hand side is a product of two ex-
pressions, each of which when equated to zero yields the TM and TE mode
solutions, respectively. When n + 0 the entire Equation 2.62 must be used
in obtaining a solution, since the modes are hybrid in nature; i.e., they
havelxnfll Ez and H2 field components. If the enclosing quartz tube
thickness were assumed zero, i.e. if we set b = a inISquation 2.62 and
if Ve = 0, then we would obtain the two dielectric problem characteristic
equation, as has been derived by otherslB.

It is apparent that for finite losses within the plasma almost all
the terms in the transcedental equation above are complex, including the
Bessel's functions which are functions with complex arguments. Details

of time numerical techniques involved in evaluating the roots of such an

equation are explained in Appendix A.

22

Since the plasma under study is lossy, k2, in general, is complex,
i.e., k2 = E3-’ja, where B is the propagation constant and a the attenua-
tion constant in the z - direction. By solving the characteristic equa-
tion for w vs. kz, with wpe and Ve specified,(u—E3andlu-11diagrams can
be generated for each mode. Interesting electromagnetic properties can
be obtained from such diagrams and these will be the subject of discus-

sion in the following section.

2.4 ka Diagrams

 

This section presents the w - k diagrams for the various propagating
modes in the inhomogenous waveguide. This diagram gives a useful insight
into the fast wave and slow wave solutions to this boundary value problem.
It explains some modes that exist without a plasma and the transformation
of electromagnetic modes in the presence of the plasma and also predicts
the appearance of modes that do not exist in the empty waveguide.

Both fast wave (V¢3>c) and slow wave (V¢‘<c) solutions are found to
exist in this type of waveguide, because of the coaxially located di-
electric enclosed plasma rod. In general, both TE and TM modes are
needed to satisfy boundary conditions. Thus, the modes are hybrid in
nature, except in the rotationally symmetric case where TE and TM modes
uncouple and exist independently. The propagating modes for this bound-
ary value problem can be broadly classified into two categorieslo’13’16’l7.

1) Fast wave electromagnetic modes: These are empty waveguide

 

electromagnetic modes that have been perturbed by the presence of the
plasma and have a phase velocity greater than the speed of light. They
can be further subdivided into two classes.

(a) TEom or TMom: m > 0. These are rotationally symmetric modes

 

and hence, they are pure TE or TM in nature. They are labeled the same

23

13,16,17. At

as the corresponding modes in the empty circular waveguide
high plasma densities, i.e., wpe>>w, and all losses these modes approach
the coaxial waveguide mode that has the same index as the empty circular
waveguide modelo. At high frequencies, i.e., w>>wpe, and all losses these
modes approach the corresponding empty circular waveguide modes.

(b) EESm-QE—Inzm: n>02 m>0. These are hybrid modes, the asterisk
denoting their hybrid nature. They are labeled with respect to the TE
or TM modes in a circular waveguide, into which they reduce when the
plasma density goes to zero. At high plasma densities and zero losses,
mode in a coaxial wave—

nm+1)
guide with a metallic center conductor. At high plasma densities and fi-

* *
the TE (TM )mode approaches the TM (TE
nm nm nm

* * *
nite losses, however, the TE (TM >mode approaches the TE (TM )

nm nm nm nm
coaxial waveguide mode. At high frequencies, and finite losses the

* *
TEmu (TMnm) mode approaches the empty circular waveguide TEmu (mum) mode.

2) Slow wave plasma modes: These modes do not exist in an empty
waveguide and have a phase velocity smaller than the velocity of light
over a wide range of plasma densities. They can be further subdivided
into two classes of modes.

(a) TMOO: This mode is circularly symmetric and does not exist in

 

the empty waveguide. However, it is classified with respect to the
circularly symmetric TM modes of an empty waveguide. This is also called
a surface wave or a space charge wave. At large plasma densities and low
losses the TM00 mode approaches the TEM mode of a coaxial waveguide with
its phase velocity approaching the speed of light, whereas at large
plasma densities and high losses it deviates from the TEM mode and becomes
more of a slow wave.

*
(b) TMno' These are hybrid modes that are ¢ dependent. They are

 

slow waves in a lossless plasma and for small losses. At high

24

plasma densities they asymptotically approach the coaxial TEn1 mode.

*
For finite losses, however, the TMno mode becomes degenerate with the
*
TEn1 mode and its asymptotic behavior at high plasma densities and high

*
frequencies, is same as that of the TEn1 mode.

The following sub-sections deal with a detailed study of the nor-

malized w - k diagrams, i.e., w/wpe vs. B/kc and 111/wpe vs. a/k: with

*

m /we and ve/wpe as variable parameters, for the TM 10,

pe 00’ TMOl’

* .
TE 11 modes. we and kC are the cutoff frequency and cutoff wave

TM

number,respectively,of the TE11 mode in an empty circular waveguide.
These diagrams have been plotted for experimental waveguide dimensions:
a - 0.3 cms., bl= 0.3875 cms., c = 5.08 cms. The experimental operating
frequency §% = 3.03 GHz is also shown in these curves. For these wave-

(0

~59- = 1.73 GHz and kc = 0.362 cm‘l.

guide dimensions, 2n

TM00 mode.

Space charge or electromechanical waves propagate in a stationary
plasma of finite cross-section, due to a perturbation either of the
average volume charge density or of the surface charge densitylz. Pro-
pagation of slow waves in a cold plasma due to a volume charge pertur-
bation is possible only in the presence of acbwzmagnetic field. Surface
waves can propagate on a quartz-enclosed plasma rod, partially filling
a metal waveguide, with or without a d-c magnetic field.

The TM00 mode is a surface wave that propagates in the absence of
a d-c magnetic field. The dispersion relation for the TMOm mode can

be obtained as a special case of equation 2.62, by setting n = 0. The

resulting equation is shown 0n the following page.

25

 

 

k J'(k a) k k
.51 O2 0 p1 = e3 02 F3F5 ‘ E2 p3F1F6 2 69
s R J (k a) e k F F -‘e k F F '
2 pl 0 61 3 62 3 7 2 03 1 8
where,

F1, F3, F5, F6, F7 and F8 are as given in equations 2.57, 2.59, 2.63

to 2.66 with n = 0. The first root (m = O) of equation 2.69 is the

solution to the TM00 mode, the second root (m = 1) that of the TM01

mode and so on. The TM00 mode has basically three field components:

longitudinal and radial electric fields Ez and ED respectively, and

a circular magnetic field H¢ . Ez would be predominant for those fre-

quencies when the TM00 mode is a space charge wave and E2 would become

zero whenever the TM00 mode approaches the TEM mode.

The w - k diagram for the TM mode is obtained from equation

00

2.69, with ve/wpe and w elwC as running parameters. Several interesting

P
features of the plots of w/wpe vs. B/kc (figures 2.2a, 2.3a, 2.4a, 2.4c,

2.5a) and m/wpe vs. OL/kC (figures 2.2b, 2.3b, 2.4b, 2.4d, 2.5b) will
now be explained. Figures 2.2a and 2.2b show the lossy TM00 waves for

fpe - 6.0 GHz and ve/wpe = 0.1 over the entire frequency range. In
Figure 2.23, the w - k diagram is separated into four regions. Region
1 is the slow wave portion of the curve. In region 2, the TM00 mode is

a backward wave. Region 3 shows that the TM00 mode is almost cut off in
this frequency range. In region 4, this mode is an EM wave which
eventually becomes asymptotic to the light line. Figures 2.3a, 2.4a,
2.5a show the w/wpe vs. B/kc diagrams and figures 2.3b, 2.4b, 2.5b show
the w/wpe vs. OL/kc diagrams for the different plasma densities of f e I

4.5 GHz, 6.0 CH2 and 10.5 GHz and for a range of collision frequencies.

The ve/wpe s 0.03 curves in figures 2.5a, and 2.5b have the same

w/wPe

10

26

 
    

///{:ght Line

 

 

/ 4“

 

 

 

2
=- +
l l l l- l l l I l
4 8 12 16 20 24 28 32
B/kc

Figure 2.2a. w/wp vs. B/kC for the TM00 mode. fpe=6.0 GHz. wpe/wcs3.47.

e

Waveguide dimensions: a - 0.3 cms., b = 0.3875 cms., c =
5.08 cms.

27

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32

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6.0 GHz, wpe/wc I 3.47. Waveguide dimensions: a I 0.3 cms.,

b I 0.3875 cms., c I 5.08 cms.

33

 

 

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6.0 GHz, Lupe/wc I 3.47. Waveguide dimensions: a I 0.3 cms.,
4b I 0.3875 cms., c I 5.08 cms.

Figure 2.4d. w/wpe vs. OI/kC for the TM

34

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36

collisional loss as the ve/wpe = 0.052 curves in figures 2.43 and 2.4b
and the ve/wpe = 0.07 curves in figures 2.3a and 2.3b. Similarly, the
ve/wpe = 0.1 and Ve/wpe = 0.3 curves in figures 2.5a and 2.5b have the
same collisional loss as the v /w - 0.175 and v /w = 0.525 curves

e pe e pe
respectively, in figures 2.4a and 2.4b. Figures 2.4c and 2.4d are
enlargements of the circled areas of figures 2.4a and 2.4b, respectively.
They shOw the low-frequency (In/wpe <0.l) details for fpe = 6.0 GHz.

12

(a) Lossless plasma or ve/wpe = . Trivelpiece has investigated

 

this case extensively. The w - k diagram (figures 2.3a, 2.43, 2.4c,

2.5a) show that when Ve/wpe = 0, propagation is possible from zero fre-
. r____

quency up to the slow wave resonance frequencym=wpe / v1+ke, where

Re is the relative dielectric constant of the quartz tube. For ke = 1,

i.e., free space or no quartz tube,co=wpeI’/—. The effect of the quartz

tube is essentially to lower the frequency of the slow wave resonance.

=0) is a slow wave and lies entirely to the right

00
of the light line. As w‘+ mpg/“1+ e’ both the phase velocity and the

The TM mode (v /m
e pe

group velocity V¢ and V8 respectively, approach zero. Thus, at
w =wpel/I:E;’ this mode become a non-propagating space charge oscilla-
tion. However, as w+wpe//T:E;, there are limitations to the cold plasma
theory. For Te ¥ 0, i.e., a warm plasma, and as B/kC approaches the
thermal velocity line, warm plasma effects will alter the results and
for still large B/kc's collisionless Landau damping becomes predominant.
In this region, the cold plasma theory no longer holds.

This mode can propagate only if the plasma partially fills the
metallic waveguide since its propagation depends on surface charge accumu-
lation. This would require that a dielectric region separate the plasma

column and the circular waveguide. In most practical cases, the

37

separating dielectric region is a combination of free space and a di-

electric tubing containing the plasma. The electric field and charge

12
distribution on such an isotropic plasma column are shown in Figure

2.6. At frequencies close to the slow wave resonance frequency wzw I

pe

Vl+ke, B is large. If Btzl, where t = thickness of the quartz di-

electric, then the fields would be confined entirely to the dielectric
and the plasmalz. The phase velocity at these frequencies is, thus,
determined by the thickness of the dielectric, the value of the dielec-
tric constant ke, and the properties of the plasma. At low frequencies
or high plasma densities, w/wpe <<l, Ba is small (Figures 2.3a, 2.4a,

and 2.53) and the effect of the quartz dielectric is negligible. The
plasma assumes conductor-like properties and fields exist primarily in
the region between the plasma and thezmetallic boundary. The surface
currents on the plasma penetrates a thin outer shell into the plasma
just as surface currents on the inner wall of the outer metallic wave-
guide penetrates a skin-depth layer. The TMOO mode,in this case,approaches
the TEM mode. The TM00 lossless curve gets closer to the light line with

increasing fpe and, as a result, the frequency range of the TEM mode

operation also increases.

Trivelpiece12 has shown that it is possible to have a backward
wave region for the lossless case, by using a thin dielectric of high
permittivity. However, the dielectric thickness, permittivity and
waveguide dimensions chosen here are such that this region does not
exist in theta-ESdiagrams of figures 2.3a, 2.4a, and 2.5a.

(b) Lossy plasma or ve/wpe'i-g' In practical gas discharges
where energy losses due to electron-neutral collisions cannot be ig-
nored, it is important to study the propagation of space charge modes

as ve, the effective collision freqeuncy is varied. The study of the

38

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39

lossy TM00 mode is divided into three frequency regions: (w/wpe)< 0.1,
0.1<(w/wpe)<1/ Vl—lrké, and (w/wpe)>l/ /T_:7ké.

The low frequency behavior, m/wpe<0.l, for a typical plasma density
fpe = 6.0 GHz is shown in figures 2.4c and 2.4d. AS density increases,
the lossless curve shifts towards the light line. For small losses,
ve/wpe f 0.1, the difference between the lossy curve and the lossless
curve is negligible. As Ve/wpe becomes greater than 0.1, the deviation
of the lossy curves from the lossless curve is more significant. At
low frequencies and high collisional losses (ve/wpe=0.525, 1, 10) the
departure of the curves from the lossless TEM mode of operation is ap-
parent (figures 2.4c and 2.4d). This is consistent with the values
of the complex dielectric constant for frequencies in the region
(w/wpe) f 0.1 (figures 2.7a and 2.7b)because at high collision frequen-
cies the plasma properties approach that of a lossy dielectric. The
attenuation curves at low frequencies are shown in figures 2.3b, 2.4d,
and 2.5b.

For frequencies in the range 0.10<w/wpe<l/ Vlik; , the deviation
of the small loss curves, ve/wpe<0.1, from the lossless curve is negli-
gible (figures 2.3a, 2.4a and 2.5a). The wave damping due to collisions
in this frequency range and for ve/wpe<0.l is also small as shown by
the w/wpe vs. a/kc curves (figures 2.3b, 2.4b and 2.5b). For instance,
for the following given plasma parameters: fpe=6.0 GHz, ve/wpe=0.05
and w/wpe = 0.35, a wave of unit amplitude is attenuated in a distance
of one wavelength to 0.285. The approximate frequency range
0.25<w/wpe < ll/Elkg— would correspond to a region where an experimental

generation of space charge waves would be pOSsible.

40

1.0 r—

0.5 " .5

0.3

2 1.6

 

 

-20 _. o

 

 

-100

Figure 2.7a. Real part of plasma dielectric constant 8r vs. w/w and

pe
v /w . f = 6 032.
e pe pe

41

-10 -— \

 

 

-1 - v/w = 0.525
pe
0.30
.175
.0100
_10-1 l I l l l
0 0.2 0.4 0.6 0.8 1.0 1.2
w/w
pe

Figure 2.7b. Imaginary part of plasma dielectric constant 51 vs. III/Lupe
and ve/wpe' fpe 8 6.0 GHz.

42

The lossy curve, unlike the lossless curve, does not become asymp-
totic to w =wpe/Jl:k;, at high values of B/kc' A non-zero value of
(Va/mp8) removes the singularity19 of the lossless curve at the fre-
quency (w/wpe) 8 l/ /T:k;. Therefore, in the presence of losses, (B/kc)
decreases with frequency for w>wpel Vii—g. It crosses the light line,
becomes a fast wave and at very high frequencies becomes asymptotic to
the light line, (see figure 2.23). The lossy curve thus, becomes a
backward wave in the frequency range 1/ /T;k; f w/wpe130.9 (figures
2.3a, 2.4a and 2.5a). This backward wave region, however, is highly
damped as shown by the 00/00pe vs. OI/kc curves (figures 2.3b, 2.4b, and
2.5b) and is no longer a true wave phenomenon. For example, for the
following given plasma parameters: fpe=6.0 GHz, ve/wpe=0.l and 00/0)pe
= 0.44, a wave of unit amplitude is attenuated in a distance of one
wavelength to 0.0031. These curves also indicate that for very small
losses, say Ve/wpe f 0.01, it might be possible to have a frequency
range where the backward space charge waves are not altogether damped.
Computation problems prevented the generation of such curves.

As seen in figure 2.2b, beyond w=wpe / /l:k;, OI/kc increases with

w/wpe for a given ve/wp , reaches a maximum then decreases and eventually

e
tends to zero at very high frequencies. At very high frequencies, w/wpéx>l,
all the lossy TMO0 curves asymptotically approach the light line.

Figures 2.2a and 2.2b show this behavior for fpe=6.0 GHz, and ve/wpe

= 0.1. This high frequency part of the curve is consistent with the

value of the complex dielectric constant of the plasma at such frequen-
cies and losses, (figures 2.7a and 2.7b). For an fpe=6.0 GHz and a

given (ve/wpe), er-*l and ei-*0 as w becomes very large. At these fre-

quencies, the plasma waveguide becomes an empty circular waveguide and

the TM00 mode behaves as a circular waveguide mode.

43

TM01 mode:

This is a rotationally symmetric mode and, hence, purely trans-
verse magnetic. Its characteristic equation is given by equation 2.69.
Its field pattern is easily analyzed even in the presence of the plasma.
It has radial and longitudinal electric fields ED and E2, respectively,
and a circular magnetic field H¢. The dispersion equation is solved
for w/w vs. B/k with w /w and v /w as variable parameters.

pe c pe c e pe
Figures 2.8a, 2.9a, 2.8b and 2.9b show w/wpe vs. B/kC and w/wpe vs.
a/kc for fpe = 10.5 GHz and fpe = 20.0 GHz, respectively, and a range

of losses. Figures 2.8c and 2.8d show the low frequency details of

the 00/0)pe vs. B/kc and 00/0.)pe vs. a/k diagrams for fpe = 10.5 GHz.

 

c
(a) Lossless plasma or ve/w 829. As seen in figure 2.83, for

P
fpe=10.5 GHz, this mode is cut off below w/wpe = 0.23 or f = 2.42 GHz.

An increase in the plasma density to fpe = 20.0 GHz as shown in figure
2.9a, raises the cut off frequency to w/wpe = 0.134 or f = 2.68 GHz.

The TM01 coaxial mode cut off frequency for the waveguide dimensions

chosen is f = 2.92 GHz. Thus, the TM cut off frequency is seen to

01
approach f = 2.92 GHz at very high plasma densities. The dispersion

curves (figures 2.8a and 2.9a) are typical fast-wave electromagnetic
solutions when ve/wpe=0. At high frequencies, the curves asymptotically

approach the light line and become the TM empty waveguide mode.

01

(b) Lossy plasma or ve/w f . As shown in figures 2.8a and 2.9a,

 

pe
at high frequencies all the loss curves are asymptotic to the light line

and deviate very little from the lossless curve. Also, the collisional
damping at these frequencies is negligible, see figures 2.8b and 2.9b.
As 00/0)pe approaches the region w/wpez0.23 in figure 2.8a, the deviation

of the lossy curve from the ve/wpe - 0 curve is more significant. The

w/wpe

.40

.36

.32

.28

.24

.20

44

/Light Line

 

 

 

”q I 96 /
05’
/ See enlarged /
I?“ circle above /
I l l 1 1 l
0 .4 .8 1.2 1.6 2.0 2.4 2.8

e/kc

Figure 2.83. w/mpe vs. E3/kC for the TMO1 mode. fpe = 10.5 GHz, (ope/00C =

6.1. Waveguide dimensions: a a 0.3 cms., b 8 0.3875 cms.,
c = 5.08 cms.

45

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0 0.1 0.2 0.3 0.4
B/kc
Figure 2.8c. w/wpe vs. B/kc for the TM01 mode for 00/0)pe < 0.24.

 

 

46

 

 

 

 

10.5 GHz, (ope/00C - 6.1. Waveguide dimensions: a = 0.3 cms.,

b = 0.3875 cms., c = 5.08 cms.

f

pe

w/wpe

0.20

0.18

0.16

0.14

0.12

 

 

47

 

 

0.10 0.1
0.08
0. 06
0. 04
0.02 ..
-0. 4 -0. 6 .of8 -1: 0 -1.L2 -1.'4 -1.l6 -1.'8

Figure 2.8d.

o/k
C

(II/00pe vs. OI/kc for the TM01 mode for 00/00pe S 0.20. fpe =
10.5 GHz, (ope/wc - 6.08. Waveguide dimensions: a - 0.3 cms.,

b 8 0.3875 cms., c - 5.08 cms.

48

 

 

 

 

 

-100

Figure 2.8e. Real part of plasma dielectric constant er vs. w/w and

v /w . f = 10.5 GHz.
e pe pe

49

 

 

-10 -
-101
e.
1
-1...—
\)_/wpe =
1.0
3.0
..0
-101-— ' 10.0
0.1
-102 J l I 1L I TJ
O .2 .4 .6 .8 1.0 1.
w/wPe
Figure 2.8f. Imaginary part of plasma dielectric constant 61 vs. w/wpe
and V /w f = 10.5 GHz.

e pe‘ pe

2

w/wpe

.20 -
.18 -
.16 .

.14
_ ve/wpe
03
012 I-
.10 ..
008 [I

 

 

 

 

Figure 2.93.

 

 

50

/
/

a See enlarged /

'circle above

/

B/k
c

///light Line

{(U/Zfl = 3.03 GHz

 

 

w/w vs. B/k for the TM mode. f = 20 GHz, w /m
pe c pe pe c

01
= 11.6. Waveguide dimensions:
c = 5.08 cms.

a = 0.3 cms., b = 0.3875 cms.,

 

w/wpe

 

51

 

Figure 2.9b.

 

w/w vs. a/k for the TM
pe c

01
11.6. Waveguide dimensions:
c = 5.08 cms.

mode. f = 20 GHz, w /w =
pe pe c
a = 0.3 cms., b = 0.3875 cms.,

52

lossy curve has no cut off frequency. B/kc decreases with w/wpe and
becomes zero when w/wpe = 0, as shown in figures 2.8c and 2.9a. The
attenuation per unit wavelength for all the lossy curves for frequencies
below 00/00pe = 0.21 in figure 2.8c and below 00/00pe = 0.115 in figure
2.9a is very high. For instance, for the following given plasma para-
meters: fpe=10.5.GHz, Ve/wpe = 0.1 and w/wpe = 0.20, a wave of unit
amplitude is attenuated in a distance of one wavelength to 7 x 10-23.
The curves for small losses ve/wPe = 0.1 (figures 2.8a and 2.9a) almost
coincide with the lossless curves above the cut off frequency. As
ve/wpe is increased through the range 1 to 100 (figure 2.8a), the dif-
ference between the lossy and lossless curves above the cut off fre-
quency is more apparent. For very high losses ve/wpe = 100 (figure
2.8a), the mode appears to be practically cut off for frequencies below
w 5 0.208 wpe or f = 2.2 GHz. So with the plasma densityfixed and as
ve/wpé*” the cut off frequency of this mode approaches that of the TM01
circular waveguide mode. This behavior is consistent with the value of
the complex dielectric at (w/wpe)~0.208 and fpe= lO.5(Hh:(figures 2.8e
and 2.8f). As (ve/wpe) increases at this frequency,6r+l and €i+0'

At high frequencies, i.e.,alarge w/wpe, €f+19 €1+0 (figures 2.8e
and 2.8f) and this mode approaches the TM01

plasma. As w+0, Er remains finite and €{*dw. The plasma becomes a

waveguide mode without a

lossy conductor and the TM01 mode is, thus, highly damped at very low

frequencies.

* *
ggll and TMlG_modes.

The dispersion relation for these modes is obtained by putting

 

n = l in equation 2.62. Because of their hybrid nature, these modes

53

have all the possible electric and magnetic field components, i.e.,
Ep, E¢, Ez and Hp, H¢, Hz, respectively. The dispersion equation is
solved for w/wpe vs. B/kc with (ope/u)c and Ve/wpe as variable para-
meters. Figures 2.10a, 2.10b, 2.10c and 2.10d show the w/wpe vs. B/kC
and w/wpe vs. 0L/kC for a typical plasma density of fpe = 5.3 GHz.
Several features of these curves will now be explained.

*
(a) Lossless plasma or V /w =0. For f =5.3 GHz, the TE mode
e pe—- pe

11

has a cut off frequency at w = 0.58 wpe for the guide dimensions of a

= 3 cms.,b = 0.3875 cms. c = 5.08 cms. This mode has the lowest cut
off frequency in both the circular and coaxial waveguides. It is a
typical fast electromagnetic wave, eventually becoming asymptotic to
the light line. At very high plasma densitities, the cut off frequency

of this mode approaches the TM coaxial mode cut off frequency.

11
(w/wpe= 0.68).

The TM:O mode, which does not exist in the empty circular waveguide,
has a cut off frequency at w = 0.33 wpe for waveguide dimensions: a=0.3
cms., b = 0.3875 cms., c = 5.08 cms. B/kC increases with w/wpe and this
mode behaves like a fast wave up to B = 1.86 kc. For B>l.86 kc, this
mode becomes a slow wave and eventually becomes asymptotic to the fre-
quency of dipole resonance. This frequency obtained from the quasi

static formula16, a 2
1 - (E9

2 a 2
1 + fig)

1 + e

2
a C -
(wpe/w) l + 2

v—II—

1- (-3)2
—-—-————+e:2

1 + (-3)2

 

L

is wzwpe‘IVB . In the formula, a = plasma radius, b = outer radius of

54

.28 mo.m u o .680 mmwmd u n :95 m.o u m

 

"mcowmcmafip wwwswm>m3 .mo.m u o3\ma3 .Nmu m.m n mam .mmcoa flame cam CHEF How ox\m .m> oa3}... .moH.N muawwm
a» an
0
E
q.~ ctm 04 NA w.o «.0 o
_ a
_ _ _ \\ _.uS.~ 3sz 0% \I‘Ig
.30
\ 2, ..

so. 9/

 

 

\. e4
/ m
\ ads I To W
\ /.0.. w
)0
\ N30 mo.m u =~\3
an Ho.
" Q I 0
mx: 3\3\ o u 0 30? .HHmH o o

\

0:3 EmS\

\

 

55

umcowmaoEwu mufiswm>m3

0
ll

0
II

.mao wo.m u o ..mao mnwm.o n n ..mEo m.o n m

 

 

 

 

 

 

 

u on on o
.ao.m 3\ 3 .Nmo m.m u u .mmsoa away new owns you x\s .m>
o
is
9.01 q.ol m.o1 «.01
_ a _ _
ma3\m> .one
um
":0 mo.m u :~\3
ma 0 oH Ho.o mnaxm> _
3\ > . «E. .
.III .IIIAHmmmmW .III III. III. III. III. III. III
a
w 3\0> .meh

 

 

 

on

3\3

om.o

oc.o

om.o

00.0

mo.o

.noH.N mpswum

adm/m

56

.mao wo.n u o..mao mmwm.o u n ..mao m.o u m "macamcmawu mvwswo>m3

 

 

9
.Ba u 0&0 3 .55 ms "use as a 3&0» Ba 26 v 323 you 88 A”: 2: 08 as; .9, 83? .03.~ 8%:
o
{a
mm. v. mm. m. mm. N. md. fl. mo.
11 _ . . . . _ . . H.
1 S.
I. N o m
I
m
.0
a
1 mm.
m 0
on o
N .O H 3\ 3

 

57

 

u on
.mEu wo.m u o ..mEo mmwm.o u n ..mao m.o a m "msoamcmaww mvwswm>m3 .no.m I 3\ 3

 

u on
.Nmo m.m I mam .~.o u masxflzwcm mm.o v 0a3\3 now 0608 mee onu you x\d .m> 3\3 .poa.~ ouswwm
ox\d
oo .o: ow .0: or .o- oo .o: cm .0. ow .o: om .o: om .o: 3 .o:
J11 . . . q . . . . 0.0

on

 

58

dielectric tubing, and 82 = permittivity of the dielectric. Without the
quartz dielectric this mode would have been asymptotic to the line

w - wpel/2—. As w approaches wpel/3—, the phase and group velocities

V¢ and V8 approach zero and the space charge oscillations are highly at-
tenuated (Figure 2.10s). However, as w + wpe / /3-, B/kC becomes large
and the cold plasma theory is not valid in this region. Warm plasma

theory considerably modifies the dispersion curve near V¢ = Vth’ where

Vth - electron thermal velocity. For still larger B/kc's when the wave-

length becomes the same order of magnitude as xde’ the debye length,
collisionless Landau damping becomes predominant. Thus, the present

*
theory is not valid in that region for the lossless TM mode.

10
(b) Lossy plasma or ve/wpe ¥ 0. When Ve/wpe is non-zero, the

*
singularity19 of the TM 10 mode at the frequency of dipole resonance is

 

* *
removed. The TE 11 and the TM 10

the presence of losses.

modes have one distinct solution in

As shown in Figure 2.10a, for very small collisional losses
= > . o
(Ve/wpe 0.01) and for w 0 334 wpe the w/wpe vs B/kC curve closely
*
follows the lossless TM 10 curve up to w = 0.55 wpe. ForuP’0.55 mpe,

B/kc first decreases then increases and the dispersion curve becomes

*
asymptotic to the lossless TE mode solution. A backward wave region

11
is, thus, seen to exist in the frequency range 0.565wpefi w 3 0.575 wpe'
The behavior of this curve below the frequency w = 0.33 wpe is shown in
Figure 2.10c. The lossy curve for small losses has no cut off frequency.
B/kc goes to zero as 00/00pe tends to zero.

At higher losses (Va/wpe = 0.2), the backward wave region is not too

prominent, but the asymptotic behavior of this curve outside the fre-

quency range 0.55 wpe<w<0.060 wpe remains the same. An examination of

59

1.0

 

 

1.0

 

 

Figure 2.10e. Real part of plasma dielectric constant Er vs. w/wpe and

v/w . f 85.36112.
e pe pe

60

 

 

~102P-
-10 "'
-1 III-—
G1 1.0
0.2
-10"1-
0.06
.01
_10- 1 J 1 1 1 a
0 .2 .4 .6 .8 1.0 1.2
- w/w
pe
Figure 2.10f. Imaginary part of plasma dielectric constant £1 vs. w/w and
pa
0 /w . f = 5.3 GHz.

e pe pe

61

the w/wpe vs. OI/kc curve (Figure 2.10b) reveals that there is a peak in
attenuation in the backward wave region defined by 0.55 wpe<w< 0.60 wpe'
Attenuation in this region is a maximum for low effective collison fre-
quencies. In the limit of ve/wpé+0, the attenuation-r”, and the TE*11
and TM*10 modes become distinct. The backward wave region suggests
that it might be possible to excite unstable waves in this frequency
range by operatingén: low discharge pressures and driving the plasma
with strong RF fields.

Attenuation is almost negligible for w>0.60 wpe and in the range
0.55 mpe<w<0.33 wpe for the ve/wpe's shown in Figure 2.10b. Below w =
0.3300pe collisional damping rises steeply (Figure 2.10c).

The asymptotic behavior of these curves is consistent with the
values of the plasma complex dielectric constant as a function of the
frequency (Figures 2.10e and 2.10f).

For fpe = 5.3 GHz and with (ve/wpe)+w the cut off frequency of the
TE*

mode approaches that of the TE circular waveguide mode at

11 11
(w/wpe - 0.326). An examination of Figures 2.10e and 2.10f reveals that
at w/wpe = 0.326, €r+l, and €1+0 as (Va/(ope)+00 and the plasma waveguide
becomes an empty waveguide. As (w/wpe) becomes large, ef+l and €£+0
*
for a fixed collision frequency (Figures 2.10e and 2.10f) and the TE 11

mode becomes the will circular waveguide mode.

2.5 Summary

The dispersion curves of this chapter describe the propagation char-
acteristics of EM and space charge waves in a waveguide partially filled
with a concentric lossy plasma rod. The w - k diagrams for a lossless

plasma waveguide have been studied by otherslz’l7. In this chapter, the

62

emphasis has been on the study of dispersion curves in the presence of
collisional losses and how these curves differ from the curves for the

lossless case.
The introduction of losses removes the singularity19 of the lossless

' *
TMO0 mode at the slow wave resonance frequency and of the lossless TM 10

*
11 and TM 10

modes have one distinct solution to the characteristic equation in the

*
mode at the dipole resonance frequency. Furthermore, TE

presence of losses. Also, a backward wave region appears in the lossy
*

dispersion curves of TM00 and TE 11 modes. The asymptotic behavior of
the dispersion curves for very large and very small values of the nor-
malized frequency (w/mpe) and the normalized collisional frequency

(v /w ) has been described.

e pe
The properties of the dispersion curves of this chapter will be

used in the following chapters in studying the power absorbed in a lossy

plasma waveguide and in examining the eigenmodes of a plasma cavity.

CHAPTER III

POWER ABSORBED IN A PLASMA WAVEGUIDE

3.1 Introduction

The propagation characteristics of EM and space charge waves in a
plasma-waveguide (see Figure 2.1) was described in the last chapter.
The energy carried by these waves is absorbed by the plasma due to col-
lisions between electrons and neutral particles. A useful insight into
the energy relationships in a plasma waveguide can be obtained by
applying the complex Poynting theorem to this problem. Assuming an

e3“)t variation and a small signal approximation, the general form of

this theorem is:

* *
v.05'fixfi )-jw(15eE2-15uH2)=1§E-3. 3.1
Integrating the above equation over a volume of unit length of a
metallic waveguide (see Figure 3.1), we have
_1*
J

aIExh‘*-ads-30[ (geEZ-apuz)dv=s[‘€. dv 3.2
+

S S V V

1 2

where 81 and 82 are the waveguide cross-sectional areas at z = 0 and
z - l and V represents integration over the waveguide volume. The con—
tribution to the surface integrals from the perfectly conducting metal-
lic surface of the waveguide is zero.

Equation 3.2 can be further specialized for the case of a unit

length of the plasma waveguide, as shown in Figure 2.1, Chapter II.

63

64

   
 

Perfectly conducting
metallic surface.

Figure 3.1. Arbitrarily-shaped metallic waveguide of unit length. S1

and S2 are waveguide cross sections at Z = 0 and Z = l.

65

Substituting for J from the plasma conduction current for electrons

(Equation 2.20, Chapter II) and rearranging the terms we have:

2

N e
* A

ngEx'fi - zds-jmJ (keE2-%uH2)dv-ij15——2—°———T E2 dv
8+8 v (Ve+w)

1 2v N e2 1 2

=15] $79—2— Ezdv 3.3
(v +1»)
v e

2
where vl represents integration over the entire plasma-waveguide volume
and v2 represents integration over the plasma volume. The real part of

the first term on the left hand side of the equation i.e.,

*
L5ReJ-153xu11 °’zds,

s + s
1 2
represents the time-average power transferred to the unit-length plasma
waveguide volume and is equal to the collisional heating term on the
right hand side of the equation, i.e., the time average power transfer-
red to the plasma particles in v2 .
.a .3 * A
8 Im J E x H - 2 ds,
81+ 32
is equal to the sum of the time-averaged stored electric, magnetic and
kinetic energies per unit time as given by the second and third terms
on the left hand side of the equation.

In this chapter the expressions for the time average power

absorbed/unit length are derived from

P=15ReJExH-’z‘ds 3.4
+

66

t
P 11

The numerical results of Chapter II are used in studying the behavior of

for the TMO modes.

, p = 0, l, 2, etc., TEop’ p = l, 2, 3, etc., and TE
Pa as a function of the plasma parameters (w/wpe) and (ve/wpe). It
will be shown that, in general, the absorbed power per unit length is a
direct function of the losses in the plasma. For modes that have an
electric field component normal to the cylindrical plasma column volume,
surface resonances are observed. It is also shown that the power ab-

sorbed at all resonances varies inversely as the collision frequency.

 

3.2 Power flow and_power absorbed for TMop waveggide modes

A cross-section of the quartz-enclosed plasma in a cylindrical
waveguide is shown in Figure 2.1. The TMOp wave functions and the
electric and magnetic field components for regions 1, 2 and 3 of the
plasma—waveguide are obtained by putting n=0 in equations 2.26, 2.27,
and 2.28 and using equations 2.32 through 2.37. Assuming a time vari-

jwt

ation of the form e , these equations are:

Region 1: o < r 5,a
-jk 2

ml 2
= AJ k r e 3.5
1P 0‘ 01 >
-jkzz
I - ‘ .
Er (Akzkpl/wel)Jo (kplr)e 3 6
-jkzz
= — ' '
H¢ Akp Jo (kp r)e 3 7
1 1
2
jAk p1 'jkzz
E = - ---- J (k r)e 3.8
2 wel 0 pl

67

Region 2: a S r S b

 

 

m2 [ ] —ijz
w - BJo(kp r) + CNo(kp r) e 3.9
2 2
kz —jkzz
= — —— ' '
Er we [ka Jo(kp r) + Ckp N0 (kp r)] e 3.10
2 2 2 2 2
-jkg:
= — ' I
H¢ kpleJo (kp r) + CNo (kp r)] e 3.11
2 2
- 2
Jkpz -jkzz
E2 = - we [BJo(kp r) + CNo(kp r)] e 3.12
2 2 2
Region 3: b S r S c
m3 -jkzz
w = [DJo(kp r) + ENo(kp r)] e 3.13
3 3
k2 -jkzz
= _ __ I I
Er we [ka Jo (kQ r) + Ekp No (kp r)] e 3.14
3 3 3 3 3
.ijZ
= — ' I
H¢ kp [DJo (kp r) + ENo(kp r)] e 3.15
3 3 3
. 2
Jkpa -jkzz
E2 = - was [DJo(kp3r) + ENo(kpar)] e - . 3.16

The above equations involve five arbitrary constants: A, B, C, D,
and E. The constants can be expressed in terms of one arbitrary con-
stant, namely A, by applying the boundary conditions: continuity of E2

and H at r=a, r=b, and Ez-O at r=c. In terms of A, these constants

¢

are: B-F A, CIF

A, E=F A

12 14 13

where,

F11 = (kplF6/kp2F1) 3.17

and

12

13

14

#11
II

'11
ll

'11
ll

#11
ll

I11
ll

'11
ll

Using the Poynting vector the time-average power flow in the z

= ' -
(€2kp3/E3kp2) F2N0(kp2b) F3No(kp b).

68

(kplF7/kp2Fl)

(k01F5F6J0(k03C)IkpaFlFB)

-F13No(kp3c)/Jo(kp3c)

I _ I
Jo (kpza) No(kpza) Jo(kpza) No(kp2a)

Jo(kp3c) No(kpab) - Jo(kp3b) No (kp3C)

Jo(kpac) N;(kpab) - No (kp c) J; (kp b)

3 3

I _ I
Jo (kpzb)No(kpzb) No(kp2b) Jo (kpzb)

I _ I
(Ezkpllelkpz) Jo(kpla) N0(k02a) .Jo(kpla) No(kp a)

Jo(kp a) J;(kpla) - (ezkpllelkpz) Jo(kp a) J;(kpza)

2

2

direction Pz, can now be written as:

.g*

Pz=1§ReJExH-2da

S

1

2

3.18

69

= %Re [ P21 + P + P ] 3-28

k k k a
I I*
e of Jo(kplr)Jo (kplr) r dr 3.29

 

k k
zwlAl{-E;-E;-:* -2az [F F* b ' '*
wez e 11 113 Jo(kp2r) Jo (kpzr)r dr

+ F F* bJ'(k ) N'*(k r) r d
11 12 o p2r o p r

a 2

b
* . .*
+ F12F11 ‘{ No(kp r)Jo (kp r) r dr

3 2 2
b
+ F F* N'(k r) N'*(k r) r dr] 3 30
12 12 o p 0 O .
a 2 2
c 20
P - * d
23 - ErH¢ a
b o
k* k c
2 D O z
= __3__3__. '202 * . ,*
ZNIAI was e [F14F14 b4 Jo(kpar) Jo (kpar)r dr

c
* ' '*
+ F14F13 I Jo(kp r)No (kp r) r dr
b 3 3
c

* t 0*
+ F13F14 I No(kpar)Jo (kpar) r dr
b

70

c
* , ,* _

+ F13F13 I No(kp3r) No (kpar) r dr], 3.31
b

The power absorbed per unit length g1 for the TM,Op mode is:

P = (P

a z ) 3.32

 

z=0 -Pz z=l

 

As is evident from the expressions for P21, P22, and P23, the power ab-
sorbed/unit length R; is a function of the square of the amplitude of

2
the wave function in the plasma IAI , the incident frequency w, the
P21’

P22, P23 are numerically integrated and plots of the normalized power

electron plasma frequency wpe and the collisional frequency ve.

2
absorbed/unit length, Pa/IAI , vs. the plasma parameters (w/wpe) and
(ve/wpe) for a fixed wpe are obtained for the TM00 and TM01 modes.

These are explained in detail in the sections that follow.

3.2.1 'IEOO mode

2
F1 0 O =
gure 3 2 shows Pa/IAI vs (w/wpe) for (Ve/wpe) 0.05, 0.1, 0.175,
0.3, 0.525, and fpe = 6 GHz for the TM00 mode. The (w/wpe) vs. (B/kc)
and (w/wpe) vs. (a/kc) diagrams for these values of the electron density
and the normalized collisional frequency were shown in Figures 2.4a and

2.46.

It is clear from Figure 3.2 that the Pa/IAI2 curves for the TM00
mode have two resonances. One of these occurs near the frequency of
slow wave resonance (w/wpe) - 1/ /I_;fk;'- 0.46 while the other re-
sonance occurs at (m/wpe) = l.

The resonance at (w/wpe)= 0.46 peaks only when the normalized col-

lisional frequency is very small, i.e., when (ve/wpe) - 0.1, 0.05, etc.

10

10

 

10

10

71

 

 

v /w 0.1
e pe
0.175
L
0.05
.525
l 1 l l J
0.2 0.4 0.6 0.8 1.0
00/00pe

TMCO mode.

Figure 3.2. Normalized absorbed power Pa/IAI2 vs. w/w

pe

h
and ve/wpefor t e

Waveguide dimensions: a=0.3 cms., b-0.3875 cms.,
c-5.08 cms. f

pe

- 6 GHz.

72

104 __ R = 10K0

10 r-

 

 

10 r“

 

 

 

 

 

P R - 1000
a
2
III
101 -
100 l 1 l
106 107 108 109

w/Zw Hz.

Figure 3.3. Normalized absorbed pOwer in the resistor Pa/III2 vs.
w/ZH for a parallel RLC circuit driven by an ideal
current source. L - lOuH, C = 10 pf.

73

This indicates that surface waves can be strongly excited when the
frequency and plasma density are such that (w/wpe)=0.46 and when the
collisional frequency is very small.

The resonance at the plasma frequency wzwpe is a volume resonance.

The expressions for P P

21, 22, P23 (Equations 3.29, 3.30, 3.31) reveal

that these quantities reach a maximum in the neighborhood of w = wpe,
giving rise to a peak in the absorbed power. It should be noted that
these curves were computed with constant A. In practice, however, this
does not occur because the coupling of the incident power varies with
plasma density. '

A study of the two resonances described above shows that the losses
in the plasma increase with decreasing ve/wpe. This behavior is analo-
gous to the power absorbed by a resistor at resonance in a series or -
parallel RLC circuit driven by an ideal voltage or current source, re-
spectively. Typical curves for absorbed power in a resistor of a para-
llel RLC circuit driven by an ideal current source are shown in Figure

3.3. As damping in the circuit goes to zero, i.e., as Raw, the ab-

sorbed power in the resistor at resonance becomes very large.

3.2.2 “TM0l_mgd§

The power absorbed curves for this mode are shown in Figure 3.4
for the collisional losses (ve/wpe) = 0.1, 1.0, 3.0, 6.0 and 10.0. The
electron plasma frequency fpe is fixed at 10.5 GHz. The (w/wpe) vs.
(B/kc) and (m/wpe) vs. (a/kc) curves for the same electron plasma fre-
quency were shown in Chapter II in Figures 2.8a, 2.8b, 2.8c and 2.8d.

The behavior of the power absorbed curves of this mode can be
understood by studying the variation of the plasma permittivity 61 with

the frequency. The real and imaginary parts of e vs. (w/wpe) are

l

74

10’-

 

 

 

 

 

l—
v /w -10.0
e pe
1 I 1, l l l
0 2 4 .6 .8 1 0 1 2
w/m
pe
Figure 3.4. Normalized absorbed powerPa/IAI2 vs. w/wfie and ve/mpe for
the TM01 mode. Waveguide dimensions: a-0.3 cms., b80.3875

cms., C=5.08 cms. fpe = 10.5 GHz.

75

shown in Figures 2.8e and 2.8f. By comparing Figures 2.8e and 2.8f
with Figure 3.4, it is evident that the power absorbed is a direct
function of the lossiness of the plasma. This mode also has a volume
resonance at w = wpe, and is prominent for small collisional losses

(ve/w e = 0.1),

P

 

3.3 Power flow and power absorbed for TEop waveguide modes

Consider the cylindrical plasmadwaveguide with regions 1, 2, and

jwt

3 as shown in Figure 2.1. Assuming a time variation of the form e ,

the TEop wave functions and the electric and magnetic field components

for the three regions of Figure 2.1 are:

Region 1: o < r 5 a

el -jkzz
W = AJo(kp r) e 3.33
1
-jkzz
= I
E¢ A kp Jo(kp r) e 3.34
1 1
-jk 2
H = -A k k J' (k r) e z 3.35
r p z o p
_____l__ 1
NH
0
-jk 2
H = -jA k2 J (k r) e z 3.36
z p o p
1 1
mu
0
Region 2: a S r S b
-jk z
082 = [BJ (k r) + CN (k r)] e z 3.37
o p 0 O
2 2
-jkzz
= ' ' 3.38
E¢ kp [BJo(kp r) + CNo(kp r)] e
2 2 2
-kpzkz -jkzz 9
=-————-— ' ' 3.3
Hr mu [BJo(kp r) + CNo(kp r)] e _

O 2 2

76

 

 

 

-j kgz -jkzz
Hz = mu [BJO(kp r) + CNo(kp r)] e 3.40
o 2 2
Region 3: bSrSc
e3 -jkzz
w = [DJo(kpar) + ENo(kp3r)] e 3.41
-jk z
E¢ = kp [DJ6(kp r) + EN;(kp r)] e z 3.42
3 3 3
-kp3kz I I -ijz
Hr = mu [DJo(kp r) + ENo(kp r)] e 3.43
o 3 3
.jkp3 -jkzz
Hz = who [DJo(kp3r) + ENo(kp3r)] e . 3.44

The above equations involve five arbitrary constants: A, B, C, D,
and E. All the constants can be expressed in terms of A by applying the

following boundary conditions: continuity of E¢ and H2 at r - a, r = b,

and E¢ = 0 at r a c. The arbitrary constants in terms of A are: B

FllA, C e Fle, D = F13A, and E a F14A’ where
F11 = (kp F6/kp F1) 3.45
1 2
F12 = (kp.F7/kp F1) 3.46
1 . 2
= _ I I .
F13 No(kpac) F14/Jo(kp3C) 3 47
= ' .
F14 (kszSJo(kp3c)Fll/kp3F8) 3 48

and

77

= I _ I
F1 Jo(kp a)No(kp a) Jo(kp a)No(kp a) 3.49
2 2 2 2
3 ' _ I
F2 Jo(kp c)No(kp b) Jo(kp b)No(kp c) 3.50
3 3 3 3
3 I I _ I I
F3 Jo(kp c)No(kp b) Jo(kp b)No(kp c) 3.51
3 3 3 3
= ' _ 0
F5 Jo(kp b)No(kp b) No(kp b)Jo(kp b) 3.52
2 2 2 2
= ' I - 0
F6 (kp /kp ) Jo(kp a) No(kp a) Jo(kp a)No(kp a) 3.53
1 2 1 2 1 2
= I _ I
F7 Jo(kp a)Jo(kp a) (kp /kp )Jo(kp a)JO(kp a) 3.54
2 1 1 2 1 2
= I _ I 7
F8 (kpalkpz)F2N°(kpzb) F3No<kpzb) , 3.55

Using the expressions for E and H in the Poynting Vector, the time

average power flow in the z direction Pz can be written as:

 

a 20 b 20 c 20
z
= 5 Re I I -E H rdrd¢ + I I -E H*r dr d¢ + I J -E H*rdrd¢
¢ r ¢ r 0 r
o o a o b o
= 1 o
1 Re [P21 + P 2 + P23] 3 56
where
* i:
k k a
21501 01 z -202 ' .* 57
P 1 = ZNIAJ -—&uo OJ Jo(kplr)Jo (kplr) rdr 3.

78

ka k; k; b
P = __L__2__ -20£z * I I*
22 mm W e F111” Jo(kp r)J0 (10 r) r dr
0 a 2 2
b

k ' , '*
+ F11F12 34 Jo(kpzr) No (kpzr) r dr

b
* , ,*
+ F12F11 J No(kpzr) Jo (kp r) r dr
a 2

b
* *
I 1
+ F12F12 34 No(kp2r) No (kpzr) r dr I 3.58

*
k k c

2 D 0
P23 ZNIAI -Zfi;-—- e [F13F13 b1 Jo(kpar)Jo (kpar) r dr

*

c
i 1*
F13F14 bJ J o(kp r) No (kp r)r dr

3 3

c
* *
i I
+ F14F13 b4 No(kpar) Jo (kpar)r dr

* C *
i I
+ F141?14 J, No(kpsr) No (kpar)r dr] . 3.59
b

The power absorbed per unit length Pa, for the TEop mode is given
by Equation 3.32. A comparison between the absorbed power expressions
for the TM01) mode (Equations 3.29 - 3.31) and the TEop mode (Equations
3.57 - 3.55» reveals that these expressions are very similar. This is
due to the dual nature of these two modes. Er and H contribute to the

¢

power flow in the TMop mode while E and Hr do the same in the TEop mode.

¢

 

79

 

 

 

 

 

10-31—
10"“-
P
a
2
IAI
10'5—
6.0
10'6r-
w
10-7 L I L I I J
0.0 0.2 0.4 0.6 0.8 1.0 1.2
011/mpe

Figure 3.5. Normalized absorbed power Pa/IAI2 vs._w/0Ipe and Ve/wpe for the

TEOl mode. fpeg 10.5 GHz. Waveguide dimensions: a=0.3 cms.,

b=0.3875 cms., c=5.08 cms.

80

The normalized power absorbed per unit length Pa/IAI2 for the TE01

mode is plotted against (uluhe) for (veflwpe) = 0.1, 1.0, 3.0, 6.0 and
10.0 and fpe = 10.5 GHz as shown in Figure 3.5.

The variation of the plasma permittivity 81 (see Figures 2.8e and
2.8f), with frequency explains the behavior of the normalized absorbed
power Pa/IAIZ. It is evident that in generaltimanormalized absorbed
power is a direct function of the losses in the plasma. No surface
or volume resonances are observed for this mode, since there is no

electric field component normal to the plasma column.

 

3.4 Power flow and power absorbed for;EE:1 waveguide modes

In this derivation, the thickness of the quartz tube enclosing the
plasma is neglected due to the complexity of the three region problem.
Thus, there are only two dielectric regions in the plasma-waveguide.
Region 1 is the cold, lossy, homogenous, isotropic plasma of radius a.
Region 2 is the free space extending from r = a to r =lb, the perfectly
conducting boundary of the cylindrical waveguide. Assuming a time varia-
tion of the form ejwt, the TE:1 wave functions and the electric and mag-

netic field components for the regions 1 and 2 can be written as:

Region 1: o < r S a

1 'jk 7'
Wm = AJl(kp r) cos¢ e z 3.60
1
e1 -jk z
0 = DJ1(kp r) Sin¢ e z , 3.61
1
D Akplkz ' -jkzz 62
Er = -[; J1(kp r) + T J1(kp r)] cos¢ e 3.
1 1 1
A kz _jkzz
= ' -——— °
E¢ Eflcle1(kplr) + weir J1(kplr)] sin¢ e 3 63

81

 

 

 

 

-°A 2 -jkzz
E2 = Eg—Ikp J1(kp r) cos¢ e 3.64
1 1 1
Dk 1kZ A -jkzz
= — ' —
Hr who J1(kplr) + r Jl(k01r) sin¢ e 3.65
Dkz -jkzz
H = - + 1 ,
¢ mu r J1(kplr) Akle1(kplr) cos¢ e 3 66
k2 .
. pl -3kzz
Hz = -JD'5fi—' J1(kp r) sin¢ e 3.67
0 1
Region 2: a S r S b
m2 -jkzz
w I [BJ (k r) + CN (k r)] cos¢ e 3.68
1 D l D
2 2
e2 -ijZ
w = [EJ1(k r) + FN (k r)] sin¢ e 3.69
D 1 D
2 2
.. _ l
Er - [r . (EJ1(kp r) + FNl(kp r))
2 2
kpzkz -jkzz
' I I
+ we (BJl(kp r) + CNl(kp r)) cos¢ e 3.70
2 2 2
- ' k + m' k r )
E¢ [kp (EJ1( p r) 1( p )
2 2 2
k -jk z
+ 2 BJ (k r) + CN (k r)) sin¢ e z 3 71
we r 1 p 1 p '
2 2 . 2
k2

Dz -jkzz
E2 = -j-BE: [BJl(kpzr) + CN1(kp r)] cos¢ e 3.72
2

82

kp k2
H =- —1—— (EJ'(k r) +FN'(k r))
r mu 1 D 1 D
0 2
*‘k z
1 J z
+ r .(IBJl(kp r) + CNl(kp I») 5160 e 3.73
2 2
k
H¢=- wu—ZPN(EJ(k r)+FNlp(k r))
2 p2
-jkzz
' I
+ kp(B J1(kp r) + 011102p r)) coscb e 3.74
2 2 2
2
koz -szz
Hz = -J‘afi— [EJl(kp r) + FN1(kp r)] sin¢ e - 3.75
0 2 2

The above equations can be expressed in terms of the arbitrary con-
stant A by applying the following boundary conditions: Continuity of
E¢, Ez, H¢ and H2 at r = a, and E¢ = 0, E2 = 0 at r = b. The arbitrary

constants in terms of A are: B = F A, C= A, D = F E= F A, and

13A:

12 F11 15
F = F14A, where
F ==- e k? / e k2 F J (k a) J (k b) 3.76
11 2 p 1 p 1 1 p l p
1 2 1 2
F12 = - F11 (N1(kplb) / J1(kp2b)) 3.77

Fl3 . [(Flzk p Ji(kp a)- kp Ji(kp 1a) + ka 11 Ni(kp a)) /

 

83

g _ 2 . 2
F14 [F13kp J1(kp a)J1(kp b) / kp F2] 3.79
1 1 2 2
= _ I I
F15 [F34N1(kpzb) / Jl(kp2b)] 3.80
and
F1 = [Jl(kp a)N1(kp b) - J1(kp b)Nl(kp a)] 3.81
2 2 2 2
___ I _ I
Fé’ [Jl(kp2a)N1(kpzb) N1(kpzb)Jl(kp;b)], 3.82

\

Using the Poynting vector the time-average power flow in the z

direction,Pz,can be written as:

\ _I A * A
P2 = kRe E x H . 2 da
8

 

a 20 V b 20
22 E H* E H* d d + I (E 8* E H*) d 86
e ( r ¢ ¢ r)r r ¢ r ¢ ¢ r r r
o o a o
= 1 .
1Re[le +'Pz2] 3 83
where
a 20
* E H* d d
le = (ErH¢ ¢ r)r r ¢
0 o
1 * J k )J*(k r)
2 F F k k a 1‘ p r 1 p
-2az l3 13 z z 1 l
= 111 11 e —-——-—— + —— 8:
mp we r
o 1 o

2 *
k k k a
* p z z *
+ F13k + —7l-—-' J J1(kp r)Ji (kp r) dr
p1 (0116: O 1

l
0 1

84

 

 

F* k k k*
13 01 z z a *
+ 7- + F 1‘ JJWR r) J (k r)dr
m U 5 13 D1 0 1‘ pl 1 pl
0 1
k k* k F * k k* k*
01 pl 2 13F13 p p1 z a
+ —-—-———-——- I I
(1)8 + mu J J1(kp r)Jl *D(k r)r dr 3.84
1 ° 0 101
and b 2n

P _ * *
22 -aJ of (ErH¢ - E¢Hr)r dr d¢

it
b J1 (kp r) 31 (kp r)

2 2 dr
r

e-Zaz 15 15212 122

mu we
0 2

 

2 F F* k* +F F* k
=IAIF J
a

F* k*
*
15 142 z 2 dr

 

*
b J k . N k
F12F11 wez r
a

W<FT
Flelsk:z k k: b
4- F

2 '*
15F12kp2 + z J1(kp r)J1 (kp r)dr

 

w H 5 a 2 2
o 2

 

(1)1182 a 2 2

14 1521] 122

(DE

 

F12F14kpzk“ k b *

'

-+ F15F11k22+ 2 I J1(kp r)N1 (kp r) dr
+<T dr

*
*
F F* k* +F F k z) b N1(kp r”10“) r)
A 2
a

b
F F* k* +F F* k *
+ 14 14 z 11 11 z) I N (k r)N (k r)

 

1 O
0011 (D8 r d r
0 2 2
* * F11F15kz k 2*kpz b '
+- F14F12kp + 2 J N1(kpr)J1(kpr)dr

85

F F* k k xk

* * 11142

+ F14F11kp2 +

 

Mr)N *(k r) dr
wT ME 92

 

 

F12F15kpz k k:
+ F F k J'1(kp 2r)J: (kp r)dr
p2

 

 

 

 

 

2
Z
(: 03116: 15 12 92
O 2
F12F14k pzz2k k*
+ 2 + FlsFllkp2 J'1(kp 2r)N1(kp r) dr
w u 6 D2
0 2
*
Flelzkpzkpzzk F15F15kp 2kpzzk b *
+ I I
we + mu I J1(kp r)J1 (kp r) r dr
2 0 a 2 2
F12F11kp 2kpzkz F15 F14kp k zpzkz b *
' ' 0
+ me + wuz I J1(kp r)N1 (kp r) r dr
2 0 a 2 2
*
k k k b
* 92 z z ' *
4- FllFIS -—7-——- +'F14F12kpz I N 1(kp r) J1(kp r) dr
w u e 2 2
0 2 a
*
k k k b
/ * 92 z z ' *
-+ \F11F14--—7-——- + F14F11k92J N1(kp r) N1(kp r) dr
w u e 2 2
0 2 a
kk*k k*kk* b
+ F F* 92022+1“ F* pzpzz N'k )J'*(k ) d
1112 we 1415 can 1(pr1 prrr
2 0 2 2
a
1:131: FF**:kkk b
* 92 p2 z 14 14 92 p22 ' '* .
+- FllFll -——ZE::—_ -+ wuoz a] N1(kpzr)N1 (kpzr) r dr .

3.85

86

 

 

 

 

 

 

 

10 -—
.01
103 1-
\
102 *-
ve/wpe-
pa 1.0
2
101-—
.06
0.06
0.01
0.01
1—
10-1 1 l J l l J
0.1 0.3 0.5 0.7 0.9 1.1 1.1

111/mpe

Figure 3.6. Normalized absorbed power Pa/IAI2 vs. w/wp 'and ve/wp for the

*
T811 mode. fpe - 5.3 GHz. Waveguide dimensions: a-0.3 cms.,
b=0.3875 cms., c=5.08 cms.

87

*
The power absorbed/unit length Pa’ for the TE11 mode is given by

Equation 3.32. The integrals involved in the expressions for P21 and

P22 are numberically integrated and a plot of the normalized power

2
absorbed/unit length, Pa/IAI , vs. the plasma parameters (w/wpe) and
(ve/wpe) for a fixed fpe are obtained for this mode. Figure 3.6 shows

this plot for f = 5.3 GHz and (v /w ) = 0.01, 0.06, 0.2, and 1.0.
pe e pe

*
11

together with the variation of the plasma permittivity as a function of

A study of the power absorbed curves of the TE mode (Figure 3.6)
frequency (Figures 2.10e and 2.10f) shows that for most frequencies the
power absorbed varies according to the lossiness of the plasma. A
plasma volume resonance is present at w = w e’ as in the TM00 and TM01
modes. Also, seen in Figure 3.6 is a small peak for the (ve/wpe) = 0.01
curve in the neighborhood of the dipole resonance. This resonance does

not appear to be prominent for the plasma-waveguide dimensions chosen

here.

CHAPTER IV

ELECTROMAGNETIC AND SPACE CHARGE MODES OF A COLD LOSSY
PLASMA ROD INSIDE A CYLINDRICAL CAVITY

4.1 Introduction

 

The dispersion curves of Chapter II are useful to develop an under-
standing of the electromagnetic and cold plasma modes of a microwave
plasma cavity. The literature reports a considerable amount of work
that has been done relating to this problem. Brown and Rose20 discussed
the effects of plasmas in microwave cavities in the early 1950's . Later
Agdur and Eneander13 theoretically solved the problem of a lossless cold
plasma cylinder concentrically located in a cylindrical cavity. Shohetlé
extended their results to include lossesauuiobtained the eigenfrequencies
of a plasma cavity. Fredericksl7 obtained the eigenfrequenciesznuieigen-
lengths for coldanuiwarm plasma modes in.a lossless plasma partially fill-
ing a cylindrical cavity. Leprince21 has studiedtflmacold and warm plasma
resonances of a cylindrical plasma located inside acylindrical cavity.
Additionally, he extensively studied some observed parametric instabilities

in the plasma. More recently, Moisan, Beaudry, and Leprince22 have re-

ported the production of long plasma columns in a re-entrant cavity.

The plasma cavity under study here is formed from a length LS of the
plasma~waveguide considered in Chapter II. As before, the motion of the
ions is neglected (wpi=0) and the plasma is assumed to be homogeneous,
isotropic and lossy. The waveguide is fitted with metal ends. One end

is movable so that the length LS can be varied. The metal boundary ends

88

89

set up standing waves along the plasma column-cavity system, with the
longitudinal propagation constant for any particular mode given by

B3 p Zfl/ZLS, where p = 0, l, 2. If the driving frequency w is fixed for
a given value of p, the characteristic equation (Equation 2.62) can be
solved for kz with (Lupe/w)2 and ve/w as variable parameters. k2, in
general, is complex and is written as: k2 = B-ja where B = longitudi-
nal propagation constant and a = longitudinal attenuation constant. The
cavity length LS can be obtained from 8 provided the damping of the

standing waves in a unit wavelength is negligibly small i.e., if e-2a==l.

*
00p’ TM011’ TE011’ TM 10p,

Eigenlength curves are generated for the TM
TE:11 and TM:11 modes, and are plotted for two different experimental
waveguide dimensions in Figures 4.1 and 4.2. Figure 4.1a shows the
eigenlength curves for a = 0.3 cms., b 8 0.3875 cms., c - 5.08 cm., and
an operating frequency of f B 3.03 GHz. Figure 4.2 shows the eigenlength
curves for a - b = 1.248 cms. (quartz thickness is neglected), c = 5.08
cms. and an operating frequency of f . 2.45 GHz. Different portions of
these curves correspond to fast waves, slow waves, or backward waves as
described in the w - k diagrams of Chapter II. Also of special interest
are the regions where two different modes intersect, indicating where
cavity resonances are degenerate.

Figure 4.2 is a very general eigenlength curve since it includes all
the possible fundamental modes. Figure 4.13 is a more specialized curve

TM ) are cut off for the chosen oper-

*
011’ 111
ating frequency, f = 3.03 GHz. A detailed description of each of the

since some of the modes (TE

eigenlength curves of Figure 4.13 is given in the next section. Most of

the descriptions are also applicable to the curves of Figure 4.2.

in cms.

S

R esonant length L

Figure 4.1a.

*- TM

90

1
—

011

 

 

 

 

 

 

 

TM004. Ve/m = 0'1

 

 

 

 

 

 

-.__1
a,
M. I I
T 101, I I
ve/w=0. 1
L l 71 i L LJ
0.1 l 2 345 10 20 50 100 200
Z
(wpe/w)

Resonant length L8 vs. the plasma density (nope/w)2 for

*
the modes TMOOp’ TMOll’ TMlOl’ and TElll'
frequency fixed at w/Zfl - 3.03 GHz.

Operating

Waveguide dimensions:

a 8 0.3 cms., b = 0.3875 cms., c = 5.08 cms.

in cms.

Resonant Length L

14-
12_
M
10..
II
N
8
\
0
0..
3
8.. V'
6-

N
"l I
--1 . o
0' 0 II
5 é’ \3
\ >0 )0
A” ~ g
\O In C)
O O Q
O O 5
Eu

 

 

 

91

       

           
    

 

 

 
 

e)

/m)2 = (1+k

    

(00pe

   
 

 

TEM

1
T v e/u) :_O-.-l--

hd
001’

 

 

Figure 4.1b.

 

111
100 1000

1
5 10(wpe/w)2

Resonant length LS vs. (mpe/w)2 for the cold plasma modes,
*

TMOOp and TMIOp'

Waveguide dimensions:

5.08 cms.

Operating frequency w/ZN = 3.03 GHz.

8 - 0.3 cms., b - 0.3875 cms., c =

92

1040 -
9.8

9.6

9.2

9.0

8.8

8.6

lENGYH IN CM
N
‘

N
O

 

I
0
6.0- H/ * ,/
- T TM
58_ wzh w1 ////
/o
5.6- /

 

 

|
l
I
54. | YM001
l
5.2 v | /
|
5.0 - | I
I
48 |
1
L I l l A I ‘4 '
""01 1.0 10 (‘11 $100 1000 10000

Figure 4.2. Resonant length vs. plasma density (Lupe/w)2 and ve/w for a

cavity of radius 10.146 cm., plasma radius of 1.248 cm., and
an operating frequency of 2.45 GHz.

93

4.2 Theoretical Cavitnyode

 

4.21. Classification and Description of the Plasma-Cavity Modes

It is convenient to classify the resonant lengths of the plasma-
cavity into three groupslo.

1) "Perturbed" cylindrical cavity resonances.

These resonances are the empty cavity electromagnetic resonances
which are "perturbed" by the presence of the plasma. Thus, the forward
and backward traveling waveguide modes associated with these resonances
always have a phase velocity greater than the speed of light, i.e., they
are fast wave electromagnetic modes. This group of resonances can be
further divided into two subclasses of modes:

. > .
3) TE or TMOmp’ m 0

Omp
These resonances are pure TE or TM resonances since they are<b inde-

pendent and thus, they are labeled with respect to the TE or TM modes on

a circular waveguide. At high plasma densitites and for high losses,

these modes approach the coaxial mode that has the same index as the

empty cavity.

*

*
b) TE orTM ;n>0,m>0
nmp nmp

These modes are hybrid resonances. They are labeled with respect
to the TE or TM modes in an empty circular waveguide into which they de-
generate when the plasma density equals zero. It is useful to note that

*
for high plasma densities and zero losses, (i.e., Ve/w-O) the TEmn
*
resonance (or TMnmp resonance) approaches the TMInmp resonance (or

TE resonance) in a coaxial waveguide with a metallic center con-

11,111+]. ,P

ductor. The TE:11 mode shown in Figure 4.1a belongs to this family of
modes. However, for large values of collision frequency, the resonant
length increases, decreases, and then increases again returning to the

TE111 coaxial cavity resonant length.

94

*
2) Cold—plasma resonances (TM001 and TMh01’ n

These resonances do not exist in the empty cavity and are affected

>0)

by the finite electron temperature only when the wave phase velocity V

 

 

43
is approximately equal to the thermal velocity Vth = VEET;7E;_ and
w > wpe. Thus, they are called cold-plasma modes.
Each of the modes classified above are now described in detail.
4.2.2 Rotationally Symmetric Modes, TM011 and_TE011
(a) Lossless Plasma. ve/ w = 0. As shown in Figure 4.1a, the TMO11
mode is comprised of a single half-wave resonance of the TM waveguide

01

mode discussed in Chapter II and has an empty cavity resonant length of
L8 = 7.32 cms. With increase in plasma density (mpg/w)2, the eigenlength
increases slowly at first and then increases steeply in an almost linear
fashion. At very high plasma densities, when the plasma rod behaves more
like a conductor, the eigenlength curve becomes horizontally asymptotic
to the TMO11 coaxial cavity resonant length line. This region occurs at
LS . 8 cms. for the cavity with dimensions a = b = 1.248 cms. and c =
10.146 cms., as shown in Figure 4.2. For the eigenlength curves of
Figure 4.1a, the TMO11 coaxial cavity resonance occurs at LS = 19.1 cms.
and is not shown in the latter figure.

For the operating frequency and waveguide dimensions of Figure 4.1a,
the TE011 mode is cut off. It is shown in the more general curve of
Figure 4.2. It has an empty cavity resonant length of L8 - 9.02 cms.

In a manner similar to the TM011 mode, the TE011 mode at high plasma
densities, approaches the TE011 coaxial cavity resonance of LS - 9.55 cms.

(b) Lossy plasma ve/g_i_g. As shown in Figure 4.1a, the eigen-

length of the TMO11 mode at low plasma densities, (nope/00)2 - 0.1, is

95

LS = 7.32 cms. The asymptotic behavior and the general form of the
curve is almost identical to the lossless curve. For a given eigen-
length, however, the lossy curve operates at a higher plasma density
than the lossless curve. At very high plasma densities it becomes
asymptotic to the TMO11 coaxial cavity resonant length LS = 19.1 cms.

The behavior of the eigenlength curves is consistent with the w - k
diagrams for the TM01 mode shown in Figures 2.8a and 2.9a. The effect
of the quartz tubing has been to reduce the empty cavity resonant length
from LS z 7.42 cms. to L8 = 7.32 cms.

For dimensions of the experimental cavity of Figure 4.13, this mode
appears to have a large variation in both the resonant length and the
electron density. The eigenlength curves suggest that it might be possi-
ble to vary the length of the plasma column from 7.4 cms. to 18 cms. and
the average plasma density from (Lupe/00)2 = 3 to at least (Lope/01)2 = 100,
if the required RF input power were available.

The behavior of the lossy TE mode resembles that of the TM

011 011

mode. As shown in Figure 4.2, at high plasma densities the lossy curves

become asymptotic to the TE011 coaxial resonance given by L8 = 9.55 cms.

* *
111 M101

,. *
(a) Lossless plasma. ‘Vefln = 0. As shown in Figure 4.1a, the TE

4.2.3 Dipole modes; TE

 

 

11

mode has an empty cavity resonant length of L8 = 5.99 cms. The lossless
eigenlength curve at first rises slowly with increasing plasma density,
than very steeply in the neighborhood oftnpe < /310 and eventually be-
comes vertically asymptotic to the dipole resonance line given by
«upe/w)2 = 3. If the effect of the quartz had been neglected, this curve
would have been asymptotic to «ape/002 - 2. For densities beyond

(upefin)2 = 3, this eigenlength curve would normally be horizontally

96

asymptotic to the TM coaxial mode eigenlength line17. The TM

111 111

mode, however, is cut off at the chosen experimental operating frequency,
f = 3.03 GHz. The general case is shown in Figure 4.2 where the loss-

*
less TElll curve asymptotically approaches the TM coaxial resonance,

111
LS = 9.55 cms. at very high plasma densities. The TM*01 mode, (see

1

Figure 4.13), i.e., the first half wavelength resonance of the TM?0 mode,
does not exist in the empty cavity. It is present only in the presence
of a plasma and for densities wpe > V3 m. It is vertically asymptotic
to the line, (cape/00)2 = 3. With increase in plasma density the eigen-
length increases and the curve eventually becomes-asymptotic to L8 =
5.925 cms., the TE111 coaxial cavity resonant length. In the neighbor-
hood of (cape/0.1)2 = 3, this mode is essentially a slow space charge wave
as described in Chapter II. It behaves as a fast electromagnetic wave
for L8 2 5.8 cms. The TMIOp modes, for p = 2, 3 ... are also vertically
asymptotic to (wpe/w)2 = 3, and horizontally asymptotic to L8 = 5.96 p,
p = 2, 3, etc., respectively, as shown in detail in Figure 4.1b.

(b) Lossy plasma,¥ve[g_i_g. For this case the TM* mode and the

101
TE:11 mode have identical eigenlength solutions. This mode can best be
studied by further subdividing the losses.

1. Small losses ve/w = 0.1. As shown in Figure 4.13, the low plasma
density [(wpe/w)2 a 0.1 ] eigenlength is L8 - 5.99 cms. The eigenlength
increases slowly with plasma density up to (nope/w)2 - 1.5. In the neigh-
borhood of (tape/w)2 = 3, the eigenlength increases, then decreases sharply
and rises again. It eventually becomes asymptotic to L8 - 5.925 cms.,
the TE111 coaxial cavity resonant length. The region in which the eigen-
length decreases with increasing plasma density (2.8 S («ape/00)2 S 3.3)

corresponds to a backward wave (see Figure 4.13). The backward wave

*

region for the TE11

waveguide mode was described in Chapter II.

97

It is useful to observe that for (nope/(0)2 < 3 and for (wpe/w)2 > 3,
there are operating regions where it is possible to have two plasma den-
sities for a fixed eigenlength. As shown in the w/wpe vs. OL/kc diagrams
for this mode (Figure 2.10b) wave attenuation is high in the backward
wave region.

2. High Losses ve/w = 3.0. At such high collision frequencies,

 

the resonant length of the TE:11 mode is relatively insensitive to varia-
tions in the plasma density. For small values of (cope/w)2 the plasma is
essentially a lossy dielectric and the eigenlength is close to that of
the TE111 empty cavity mode, L8 = 5.99 cms. As (Lupe/00)2 becomes large
the plasma becomes a lossy conductor. The resonant length of this mode
approaches that of the TE111 coaxial cavity mode, LS I 5.925 cms.

It is clear from Figure 4.13 that the eigenlength is almost constant
for density variations above (cope/w)2 > 10. This is true for all colli-
sion frequencies. This region would be suitable for operating RF dis-
charges with practically no tuning of the lengthlo. It would seem that
a plasma with pressures varying from low to high values can be formed in
this region without jumping out of resonance. This would make this mode

practical for flowing and fluctuating plasmas in plasma chemistry appli-

cations or as a general RF plasma sourcelo.

4.2.4 Space Charge Modes TM00p

These modes do not exist in the empty cavity. In the presence of
the plasma, however, they exist over a wide range of resonant lengths.
Figure 4.13 shows the eigenelength curves of several modes, TMOOZ’ TMOO3,
TM in the region of experimental operation i.e., LG I 5.2 cms. to

004

L8 = 8.0 cms. It is in this range of resonant length, in which these

*
modes intersect with two electromagnetic modes TM011 and TElll' These

98

regions of mode intersection are of special experimental importance. It
will be discussed in experimental sections to follow.
Small losses Ve/w = 0.1. The behavior of these modes is best

studied by looking at the eigenlength curves of TM for very small

00p

losses, say ve/w = 0.1. At high plasma densities the TM curves for

00p
p = 1, 2, 3 etc., are horizontally asymptotic to the TEMp resonant length
given by L8 = p 4.95 cm. as shown in Figure 4.1b. With a decrease in
plasma density the eigenlength of these modes falls steadily at first

and then drops very steeply for small decreases in plasma density. The
almost vertical drop of the TMO0p eigenlength curve with decreasing

plasma density continues until it approaches the region wpe > V11k; m,
where ke = 3.78 is the quartz dielectric constant. In this region the
curve reaches a point of zero slope, after which the eigenlength increases
very sharply with 3 further decrease in the plasma density. This curve
cannot approach the horizontal asymptote of an EM mode, because that

mode is cut off at the operating frequency, f = 3.03 GHz, (Figures 4.13
and 4.1b). The TM002 mode is not shown in its entirety in Figure 4.13.

This same type of behavior, however, is true for all losses and a typical

eigenlength curve for TM , ve/w = 0.1 is shown in Figure 4.10.

002
The TM00p lossless curves for p = 1, 2 ... almost coincides with
the TM00p ve/w = 0.1 curves; the difference being that the lossless curves

eventually become asymptotic to the line (Lupe/w)2 I (1+ke) I 4.78, and
do not rise again with a decrease in plasma density. The behavior of
TM00p curves is consistent with the w - k diagrams for the TM00
modes described in Chapter II (Figures 2.2, 2.3, 2.4 and 2.5).

waveguide

As can be inferred from the 111/00pe vs. a/kc (Figures 2.3b, 2.4b and

2.5b) curves of Chapter II for the TM modes, only the forward space

00

99

charge region (00pe > 4.78m) with small collisional losses (Va/w = 0.1)
has low damping. The backward wave region, i.e., when the eigenlength
increases with decreasing plasma density, is not of significance as a
wave phenomena because of heavy damping. However, it still might be

possible to experimentally couple energy to the plasma in this region.

4.3 Summary

The lossy cold plasma theory of Chapter II has been used here to
obtain the eigenlengths of EM and space charge modes of a microwave-
plasma cavity. The eigenlength curves suggest that it might be possible
to operate the plasma-cavity over a wide range of electron densities,
collision frequencies and in a number of different modes. The primary
EM modes of operation are TMOll’ TE011 and TElll' A plasma source oper-

ated in these modes has applications in plasma chemistry experiments and

in pumping gas lasers. Also, the TM mode, which as electric fields

011
in the form of an axial quadropole, is capable of exerting radio fre-
quency forces on 3 plasma for confinement purpose323.

The backward wave region in the lossy TE:11 eigenlength curves sug-
gests that it might be possible to have two plasma density operating
points for a single eigenlength. Also, of interest are the regions where
the eigenlength curves of the space charge modes TM00p intersect with
those of the EM modes TM011 and TElll’ suggesting that it might be possi-
ble to couple the EM energy to space charge waves. These observations
and the usefulness of the plasma-cavity as a RF plasma source will be

experimentally studied in Chapter VI.

CHAPTER V

EXPERIMENTAL SYSTEMS

5.1 Introduction

 

Two microwave cavities have been used to experimentally study the
linear resonances, the non-linear behavior and the absorbed power char-
acteristics of the microwave plasma source. Both cavities are cylindri-
cal, have a sliding short for tuning and a screened window for viewing
purposes. The two cavities, however, differ in size. Cavity 1 has a
radius of 5.08 cms. and operates at the available incident power of 30W
while cavity 2 has a radius of 10.15 cms. and operates at incident powers
of up to 1.3KW.

5.2 The Two Plasma-Cavity Systems

 

5.2.1 The #1 PlaSma—CavitygSystem

A cross-section of Cavity 1 is shown in Figure 5.1. This variable
length cavity has been designed to excite several resonances in the 2 to
4 GHz frequency range. The waveguide dimensions have been chosen so
that when operating in the TE:11 cavity mode the ratio of the resonant
length to radius is approximately one for a 3.03 GHz signal. A cavity
radius of 5.08 cms. satisfies this design requirement.

Several of the resonant modes of this cylindrical cavity are shown
in Figure 5.2. In this figure, the resonant frequency is shown as a
function of the cavity length. Also, shown in Figure 5.2 are a few of
the coaxial cavity modes. These modes are of interest because a very

dense cylindrical plasma has electrical properties similar to a metal

rod inside the cavity.
100

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Cavity 1 is fitted with a mechanical device that smoothly varies
the length and a micrometer that measures changes in cavity length of
the order of .01 mm. This precise tuning arrangement enables the varia—
tion in cavity length for any changes in plasma conditions and thus en-

ables the study of the cavity eigenlength as a function of the plasma

*

density. This is especially useful in verification of the TE111

eigen-
length curves (see Chapter VI).

The cylindrical cavity is driven by the microwave system shown in
Figure 5.1 by a miniature radial or loop type of probe coupling. The
radial probe which excites radial electric fields in the cavity is suit-

1:

able for exciting the TE111 mode. The loop type of coupling can be used

for generating an efficient TMO11 plasma. The power coupled to the
plasma can be adjusted by varying the depth of the probe into the cavity
and in case of the loop probe, by also adjusting the orientation of the
loop.

At the chosen frequency of w/2n = 3.03 GHz, this cavity can be
operated in the TE* TM0 and TM00p modes. The TM*

*
111’ “1101’ 11 101 3"“ TMoo

modes exist onlyixxthe presence of the plasma. It is important to study

P

the behavior of the EM modes TE:11 and TM011 over a wide range of pres-
sures, because of their applications in areas such as plasma chemistry
and possible applications to plasma confinement and gas discharge lasers.
All the low pressure work in this thesis has been done on plasmas
generated in Cavity 1. This plasma-cavity system can be pumped down to
a pressure of 10 mtorr and has a tight vacuum control. Thus, this system.
is ideally suited for studying warm plasma modes such as the Tonks Datt-

17
ner resonances , which occur at very low pressures and very low plasma

*
densities. When driven in the TE111 mode, the fields in the cavity are

104

overwhelmingly dipolar; and thus, the monopole and quadropole fields,
which are excited in the usual scattering experiments when studying TD
resonances, should not be present24’25. Also, non-linear phenomena
arising out of strong RF-bounded plasma interactions and any hysteresis
effects of the plasma can be conveniently examined. The #1 plasma-

cavity is also ideally suited for generating space charge waves, as will

be demonstrated in Chapter VI.

5.2.2 The External Microwaveg§ystem for Cavity l

A line diagram of the experimental system is shown in Figure 5.1.
The incident microwave signal at a frequency of 3.03 GHz is generated by
an external cavity klystron which is connected to an oscillator syn-
chronizer to form a phase-lock loop. The synchronizer locks the klystron
oscillator to a harmonic of the oscillating frequency of an internal
reference crystal, to produce a frequency stable microwave source. Such
a frequency source is necessary, since the plasma media is dispersive and
the microwave circuits are frequency sensitive.

The stabilized RF power from the klystron is delivered to a variable
attenuator, which controls the input power to a traveling wave tube amp-
lifier (TWT). By adjusting the variable attenuator, the amplified output
power from the TWT can be varied from 0 to 30 watts. An isolator protects
the TWT's output helix from being damaged by reflected microwave power.
Most of the experimental work on the Cavity l was done at an incident
frequency of 3.03 GHz. However, 3 Variable frequency source was employed
whenever a number of different frequencies were required.

A pair of directional couplers were used to measure incident and

reflected power from the cavity. The spectrum of the reflected power

105

could be analyzed by connecting a spectrum analyzer to the reflected

signal directional coupler.

5.2.3 Diagnostic Circuits for Cavity l

The #1 plasma-cavity system is provided with circuits for measuring
the average plasma density and the electron temperature from a discharge
pressure of 10 mtorr to 20 torr.

The microwave probe circuit shown in Figure 5.1 uses the TM 0 fre-

01
26,27,28,29

quency shift technique to measure the average plasma density.

This method can be used in the range 10 mtorr to 200 mtorr; the upper
limit on the pressure being determined by the requirement that Vez<<w2,
where w, is the TMOIO diagnostic frequency. The power from a 2 - 4 GHz
backward wave oscillator is fed to a 3-port circulator. The output from
the second port of the circulator is coupled via a miniature coaxial
probe to the cylindrical cavity so as to excite the TMOlO mode. The re-
flected power from the cavity comes out of port 3 of the circulator, and
is sent through a 2600 MHz low-pass filter, which serves to cut off the
cavity driving frequency (3.03 GHz). The output of the filter is recti-
fied and fed to an oscilloscope, which is frequency swept by the END.

The eigenfrequency variation of the TM mode with the average

010
plasma density and the normalized collision frequency is shown in Figure
5.3. The properties of cavity modes, including the TMOlO mode, that are
used for plasma diagnostic purposes are further discussed in Chapter VI.
For density and electron temperature measurements above 100 mtorr
two double Langmuir tungsten probes are used. These probes are inserted

through either ends of the quartz tube. The probes can be longitudinally

moved so as to adjust the depth of the probe into the plasma. The probe

106

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107

data can be analyzed by the usual double-probe theory to obtain numbers
for electron density and electron temperature30’40.

For pressures above 1 torr the probe data is usually not reliable.
The plasma number density at higher pressures is obtained from the D.C.
conductivity of the plasma. This is measured by introducing discs longi-
tudinally into the quartz tube so as to be barely immersed in the plasma7.
Plasma diagnostics using probes and discs is further discussed in Chapter
VI.

The average luminosity of the plasma can be monitored by attaching

a photometer to a one mm hole drilled in the side of the cavity. By cali-
brating the photOmeter reading against a known density measurement of the

plasma using another technique, it is possible to measure changes in the

electron density by observing changes in the light intensity of the plasma.

5.2.4 The #2 Plasma-Cavity System
This plasma-cavity system has been used to investigate chemical re-
actions in microwave discharges, but is general enough to be useful for
other applications such as a high-power light souce, an active plasma
for lasers, and a source of free radicals for chemical lasers. The
plasma is formed in the quartz tube (inside radius equals 1.25 cm.) which
is located on the axis of the cylindrical plasma-cavity system (inside
radius equals 10.15 cm.) as shown in Figure 5.4a. The outer walls of
the cavity are water cooled by tubing attached to the exterior wall sur-
face shown in Figure 5.4b, and the quartz tube can be forcedeair cooled
when necessary. A screened viewing window provides a visual viewing
port. Many small holes are cut into the exterior walls providing coaxial

ports required for microwave diagnostic measurements.

108

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109

Depending on which cavity mode is excited, two adjustable loop or
probe coupling systems located in the cavity side allow the RF energy
to be introduced into the cavity. These adjustable coupling systems
behave like a variable impedance transformer. Thus, by adjusting the
depth of a probe (or rotating the loop or by varying the loop size), the
plasma-cavity impedance can be continuously adjusted.

The cavity length, which is continuously adjustable up to 35 cm.,
can be varied as the plasma conditions change. The inside cavity length
could be determined from a centimeter scale, which was adjusted to read
absolute cavity length by using the short length adjustor as a reference
(see Figure 5.4). The variable cavity length allows the production of a
variable high-density plasma inside the cavity operating from pressures
of 1 torr to 1 atm (argon gas). This retuning can be achieved even under
widely varying plasma loading conditions, and in practice, the length
changes required to retune the cavity from no load to full load vary,
depending on the cavity mode, from as much as l - 2 cm. to less than
several millimeters. This small retuning length results from the fact
that the ratio of plasma diameter to cavity diameter is much less than
one. Thus, the presence of the plasma inside the cavity only slightly
perturbs the cavity resonance frequency.

Since the plasma fills only part of the cavity, the plasma and the
external metallic cavity walls combine to form the resonant circuit.

This plasma-cavity system is a high-frequency equivalent of Babat's RF
plasma systemsBz. However, one cannot think of system performance in
terms of lumped circuit elements, i.e., inductors and capacitors, because
of the short electromagnetic wavelength of microwaves. The plasma-cavity

system is a distributed parameter resonant circuit since the length of

110

the plasma can be of the same order of magnitude or larger than the free-
space electromagnetic wavelength. Thus, an active RF discharge has cer-
tain short sections of the plasma behaving similar to an E or an H
discharge (as defined by Babat32).

The fact that the plasma does not fill the entire cavity allows an
RF plasma to be sustained inside the cavity with densities much greater
than the critical density. If the plasma were allowed to fill the entire
cavity, the plasma density would be limited to approximately the critical
density or less (except in the region near the coupling ports where

evanescent fields exist), since all homogenous isotropic waveguide modes

are cut off at frequencies below the plasma frequencyll.

5.2.5 The external microwave system for Cavity 2

A line diagram of the external RF system is shown in Figure 5.5.
The RF signal source is a fixed frequency, 2.45 GHz magnetron oscillator
capable of delivering 1.3 kw of continuous wave power. As shown in
Figure 5.5, the RF power from the oscillator is incident on a power
divider circuit. This power divider is constructed from three 3-dB side-
wall hybrid couplers, two matched dummy loads, and a tandem sliding
short3l. By moving the sliding short, the power delivered to the power
divider output arm can be continuously varied from zero to full magnetron
output. This divider is designed so that the input port will always be
matched for any sliding short position if the other three ports are
matched.

The power output from the divider passes through a three-port
circulator, through the incident and reflected directional couplers, and
is incident on the plasma-cavity system. The power that is reflected

from the cavity travels back into the dummy load attached to the third

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112

arm of the circulator. Thus, the circulator protects the magnetron from
the large amounts of reflected power that may be present during certain
phases of experimentation. Furthermore, it allows the magnetron to work
into a matched system independent of the plasma-cavity load. The two-
directional couplers monitor the power in the main waveguide arms and
thereby provide a means to measure the incident, reflected, and absorbed
power in the cavity.

The power incident on the cavity is fed into a coaxial system, and
an adjustable length, radial probe, or loop couples the power into the
resonant modes of the plasma-cavity system. The cavity diameter and ex-
citation frequency were chosen to allow only one cavity mode to be excited
for a given cavity length. Coupling power into a given mode was accom-
plished by orientation of the loop or depth of the probe. Tuning stubs
which are used in some plasma-cavity experiments will not be required
since adjustments in the coupling probe and cavity length will produce

the impedance match necessary for all modes of operation.

5.2.6 Diagnostic circuits for Cavity 2

A diagnostic cavity mode can also be excited in addition to the
driving mode to measure the average plasma density26. This circuit is
also shown in Figure 5.5. The diagnostic input is coupled into the
cavity through a small coaxial coupling loop inserted into an appropriate
coupling hole shown in Figure 5.4b. ‘This loop is positioned to minimize
the coupling of the RF sustained mode energy out of the cavity and maxi-
mize the coupling of the diagnostic mode energy into the cavity. The
length independent TM

010 and TM020 modes are convenient for diagnostic

purposes and the average plasma density can be obtained from standard

113

26 '
perturbation formulas or from exact computer solutions of the lossy

plasma-cavity model discussed in Chapter IV.

CHAPTER VI

EXPERIMENTAL PERFORMANCE OF THE MICROWAVE PLASMA SOURCE

6.1 Introduction

In this chapter the experimental properties of the microwave plasma
source are examined over a wide range of pressures, namely 10 mtorr -
500 torr. The type of experiments conducted on this source can be
broadly divided into three categories:

1) Observation of any unusual electromagnetic phenomena such as
the appearance of sidebands and the qualitative explanation of these
phenomena using the linear theory derived in Chapters II, III, and IV.

2) Measurement of the plasma parameters No and Te’ and the calcu-
lation of an effective electric field, Ee’ in the plasma.

3) Study of the absorbed power characteristics of the plasma source
and its dependence on background pressure, gas flow rate, incident power
and eigenlength.of the cavity.

Cavity 1 has been used in the pressure range of 10 mtorr - 20 torr
to examine the linear and non-linear electromagnetic properties of the
plasma cavity. This is described in Section 6.2. Eigenlength curves of
the TE:11 mode are verified and space charge waves are excited in the
cavity. Two different types of sidebands are detected in the reflected
spectrum and speculation as to the cause of these is presented.

In applications of this plasma source in plasma chemistry, or gas
discharge lasers, it is important to measure the effective electric

field Be vs. the discharge pressure. This parameter is obtained by

114

115

measuring No and Te’ i.e. the average electron density and electron tempera-
ture of the ionized gas. The low power and the efficient low pressure
operation of Cavity 1 make it ideally suitable for plasma diagnostics by
the conventional probe30, disc7, and frequency shift technique326’33-36.
The measurement of No’ Te over the pressure range 40 mtorr - 20 torr,

the calculation of Ee’ and the accuracy and limitations of these mea-
surements are described in section 6.3.

The absorbed power characteristics of the plasma source as a func-
tion of the background pressure, gas flow rate, incident power and eigen-
length of the cavity are examined in section 6.4.. Cavity 2 is used to
perform these experiments, because of its operation over a broad pressure
range (1-500 torr), large variation in flow rates (0-14,000 cc/min) and
high incident power levels (up to 1.3 KW). These experiments are per-
formed on two different modes:

a) TEOll' In this mode, the eigenlength is dependent on (cope/w)2
and (ve/w).

b) TE:12. The eigenlength is almost independent of (tape/w)2 and
(ve/w) for critical densities beyond 10 (see Figure 4.2).

The theory of Chapters EIthrough IV is used to qualitatively explain
some of the experiments described in this chapter. This theory cannot
fully explain all experimental phenomena because of its restriction to:
1) small signal linear phenomena, 2) ionized gases with a zero flow

rate. However, the theory does take into account the collision frequency

losses, i.e., the pressure dependence of the experimental phenomena.

116

6.2 Linear and Non-Linear Operation of Cavity 1

6.2.1 Introduction

Cavity l is driven by a 30W, 3.03 GHz microwave source as shown in
Figure 5.1. The RF plasma source is excited by a Tesla coil, and the
linear resonances of the plasma-cavity are observed by monitoring the
reflected power. At the fixed operating frequency of 3.03 GHz, it is
possible to excite the following plasma cavity modes: dipole EM modes,
TE:11 and TMIOl’ the rotationally symmetric EM mode, TMOll’ and the
space charge modes TMOOp’ p = l, 2, 3 ... etc. The empty cavity resonance

*
of the TE111 mode occurs at L8 = 6.0 cms. and theTE coaxial cavity reson-

lll

ance at LS = 5.92 cms. The TM011 mode has an empty cavity resonance at

LS = 7.42 cms. (see Figures 4.1 and 5.2).

Based on experiments conducted in the low pressure regime, (lO mtorr-
l torr) four different types of experimental results are reported here.

They are:

l) The presence of two resonant plasma densities for a single

7:

resonant length while operating in the TE111 mode.
2) Ejection of the plasma from the ends of the cavity while oper-
*
ating at specific resonant lengths in the TE111 and TM011 modes and in

a specific range of plasma densities and collision frequencies.
3) Formation of short wavelength standing waves (£52 1 cm.) in
the plasma when the plasma cavity operates in the off resonance region
*
of the TE111 or TM011
4) Detection of two different types of sidebands in the reflected

modes.

power spectrum.

Each of these experimental phenomena is explained in detail on the

following pages.

117

6.2.2 Excitation of Two Resonant Plasma-Density Operating Points for
a Single Resonant Length

 

It was shown in Chapter IV that the resonant length vs. (wPe/w)2
for a low loss TE:11 mode first rises, falls and then rises again to
become asymptotic to the TE111 coaxial mode. The theoretical curve for
ve/w = 0.1, is shown on an expanded scale in Figure 6.1. Experimental
curves at discharge pressures of 12 mtorr and 85 mtorr are also plotted
on Figure 6.1. All electron density measurements in this experiment
have been made using the TM010 frequency shift method.

The experiment is conducted by starting the discharge at a pressure
of 100 mtorr, an incident power of 30W, resonant length of 6 cms. and
then tuning the resonant length to 5.92 cms. by maximizing the photometer
reading. The discharge pressure is now gradually reduced to a low value
say 12 mtorr. Such a low pressure is necessary to provide a suitable
comparison with the low collisional loss (ve/w - 0.1) theoretical curve.

At a pressure of 12 mtorr and with the maximum avaliable incident
power of 26W, the resonantly-tuned operating point is at L8 = 5.90 cms.
and (mpg/w)2 - 6.42. As the incident power is gradually reduced and the
resonant length tuned at the same time, the dashed curve as shown in
Figure 6.1 is obtained. At (lope/w)2 = 4.8 and LS - 5.85 cms., the plasma
cavity abruptly jumps to the operating point of L8 = 5.85 cms. and
(cope/w)2 8 1.73 (not shown in figure). This point corresponds to a low

it

density non-maximized resonant length operation in the TE111 mode.

As the resonant length is increased so as to retune the plasma, the
operating point shifts to L8 = 6.05 cms. and (wPe/w)2 = 3.26. The
incident power is gradually reduced, but the operating point does not
shift. However, once the incident power falls below a certain threshold,
the operating point jumps abruptly to L8 - 6.05 cms. and (nope/w)2 = 1.99.

lfldsexperimentally verifies the fact that for a single resonant length

118

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119

*
operation in the TE111 mode it is possible to have two different plasma

densities. If the resonant length and the incident power are increased
the operating point moves to L8 = 6.1 cms. and (wpe/w)2 = 3.12. When

the incident power is once again gradually reduced below a certain thres-
hold power level, the operating point jumps to L8 = 6.1 cms. and (wpe/w)2
= 2.25. At very low incident powers the plasma can be resonantly sus-
tained at L8 = 6 cms. and (cope/(u)2 = 1.37.

The dotted curve is drawn to fit all the experimental points.
Obviously, an exact match cannot be obtained between the 12 mtorr experi-
mental curve and the ve/w = 0.1 theoretical curve. However, the experi-
mental curve closely follows the theoretical curve and has the same
general shape. It was not possible to sustain a plasma in the region
below L8 = 6 cms. and in the density range 3 f (Lupe/m)2 §_3.5, where the
theoretical resonant length decreases with increasing plasma density.

A similar experiment is conducted at a pressure of 85 mtorr and a
curveznsshown in Figure 6.1 is obtained. The double density region in
the neighborhood of (cope/w)2 = 3 does not appear to be prominent at this
pressure. This experimentally verifies the theoretical eigenlength

curves for collision frequencies greater than 0.1 (see Figure 4.1).

6.2.3 Formation of longgplasma columns
The RF plasma in the 30W cavity is electromagnetically excited in
the TE:11 or TM011 modes. The cavity length is tuned for resonance.
It is found that while operating in the pressure range of 20 mtorr -
2 2
200 mtorr and an electron density range of (wpe/w) = 5 to (mpg/w) = 15,
the plasma is ejected out of the cavity at either ends (see Figure 6.2).

It is possible to more than double the plasma volume internal to the

cavity in this mode of operation. The length of the plasma column

120

Ejected plasma

  
  
   

Cylindrical cavity with a radius of 5.08 cms..

Quartz tube

/— Sliding short

 

 

 

 

 

 

 

SOQ miniature coaxial line

Figure 6.2. Plasma being ejected out of cavity as a result of
efficient coupling to the space charge wave throu h
an electromagnetic excitation of the cavity (TE111

or TM011 mode).

121

ejected out of the cavity is a direct function of the power input to
the cavity.

As might be expected, there is an enhanced power absorption by the
plasma in the region where the plasma column is ejected. This is
clearly demonstrated in Figure 6.3. This figure shows an experimental
plot of the absorbed power in watts vs. the discharge pressure in tort
for the TMO11 mode with the resonant length'LS held constant at 8.6 cms.

The curves have been obtained for different incident powers P = 25W,

1
20W and 15W. The curves clearly show two peaks in the absorbed power,

one in the low pressure region (around 80 mtorr) and one in the high
pressure region (around 90 torr).

The low pressure peaks occur at the same time that the plasma is
ejected out of the cavity, the absorbed power in this region being
directly proportional to the length of the plasma column ejected out of the
cavity. More than 90% of the incident power is absorbed by the plasma
in this region of operation. The high pressure peak in the absorbed
power curves is a result of increasing collisional losses in the TM011
plasma. At very high pressures when the plasma assumes conductor-like
properties, the power absorbed gradually decreases.

A similar set of curves for the TMO11 mode but, at a different
length, L8 = 8.1 cms., are shown in Figure 6.4. Again, distinct peaks
in the absorbed power can be noted at low and high pressures. These
two set of curves for the TM011 mode, i.e., at Ls = 8.6 cms. and L8 =
8.1 cms., demonstrate that the plasma cavity when excited in this mode
proves to be a useful RF plasma source that can be operated over a wide

range of resonant lengths, pressures and electron densities. No tuning

of the resonant length is required.

122

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125

A plot of the absorbed power vs. the discharge pressure has also

been experimentally obtained for theTE:11 mode and this is shown in
Figure 6.5. The dark curves show the effect of holding the resonant
length constant at L8 8 5.92 cms. At high pressures, the usual peaks in
the absorbed power are obtained as a result of collisional losses in the
plasma.7n30btain an ejection of the plasma out of the cavity at low
pressures, a tuning of the cavity resonant length is required. The dot-
’ted curves show enhanced power absorption as a result of resonant length
tuning.

The ejection of the RF plasma column out of the cavity at low
pressures and the associated enhanced power absorption for both the TM011
and TE:11 modes can be explained as resulting from the coupling of EM

energy to the space charge modes. This can be understood by studying

*
the general plot of the eigenlength vs. (cope/w)2 for the EM modes TElll’

*
TMlOl’ TM011 and the space charge modes TMOOp (see Figure 4.1). It is
observed from this plot that the TM space charge resonances intersect

00p

the TE:11 mode resonant curves at L8 = 5.92 cms., and in the plasma den~
sity range of S : (Lupe/(0)2 : 15. They intersect the TMOll mode re-
sonant curves in approximately the same density range, but over a wider
range of resonant lengths.

Experimental measurements of resonant length, density and pressures
indicate that ejection of the plasma out of the cavity takes place in
the regions where the TE:11 and TM011 mode resonances intersect with the
TM00p space charge wave resonances. The plasma-cavity is electromagneti-
cally excited in the TE:11 or TMO11 modes. Under the right operating

conditions of density, length and pressure, the EM energy is non-linearly

coupled to a space charge traveling wave. This surface wave, which is

126

radially effenescent, propagates along the plasma column and ionizes the
gas as it travels.

The 0 - k diagrams of Chapter II (Figures 2.3a, 2.43 and 2.5a) indi-
cate that the space charge mode propagates for frequencie310<cupe/VE:E; ,
where ke is the dielectric constant of the quartz tube. Here ke = 3.78.
With the operating frequency at 3.03 GHz, it should be possible to propa-
gate this mode for (wPe/w)2 Z.(1+ke) = 4.78. Experimental measurements

of the electron density by the TM frequency shift method (see section

010
6.3.1.1) verify this fact. Moisan et.al.22 have produced similar type
of long RF plasma columns at lower frequencies by exciting a surface
wave in a re-entrant cavity and using a capacitive type of coupling.
Thus, in a cylindrical cavity EM mode energy can be transferred to space
charge wave energy. This coupling is most efficient when the EM mode is

resonantly excited at low pressures, and thus, provides a large volume

RF plasma source at low pressures (20 mtorr - 200 mtorr).

 

 

6.2.4 Excitation of Short Wavelength (Ag-1cm) Standing Waves
The theoretical resonant length curves (see Figure 4.1a) indicate
that it may be possible to excite space charge standing waves, i.e.,

*
111’ TM011

resonant curves with the TM00p resonant curves. Experimental observations

reported below appear to support the excitation of these standing waves.

TMOOB’ TM009’ etc., as a result of the intersection of the TE

Experimental observations have always shown the following to be

*
true. Let the RF plasma be electromagnetically excited in the TE111
mode so as to efficiently couple to the traveling space charge wave.

This case is illustrated in Figure 6.2. The resonant length L8 is 5.92

cms. Then if the resonant length is now gradually increased to say LS

*

resonance
111 ’

= 6.1 cms., so that the plasma-cavity is slightly off TE

127

Cylindrical cavity of inner radius 5.08 cms.

Quartz tube

///—-"‘ Sliding short

 

  

 

 

®I®I®lfil®IfiIfifiifi$

f

 

 

 

509 miniature coaxial line

Figure 6.6. Excitation of 8 half-wave resonances of the TM00p mode

for an off resonance electromagnetic excitation of the
cavity (T8111 mode).

128

Cylindrical cavity of inner radius 5.08 cms.,

Quartz tube

Sliding short

 

 

 

 

 

 

E/r
j ,

509 miniature coaxial line

 

 

 

Figure 6.7. Excitation of 3 half-wave resonances of the TM00p mode

for a large off resonance electromagnetic excitation

of the cavity (TE:11 mode).

129

then the plasma contracts in volume and takes on alternate bright and
dark spots. This is illustrated in Figure 6.6. The length of each of
the bright spots is approximately % cms. The bright spots correspond

to the short wavelength standing waves of a TM mode; Ag being roughly

00p
1 cm. Eight standing wave resonances of a TM00p mode are shown in
Figure 6.6. As the resonant length is further increased to say L8 =

*
6.3 cms., the TE111 mode becomes more effenescent and only three standing

wave resonances of a TM00p mode are observed. This is illustrated in
Figure 6.7. A further increase in the length causes the plasma to become
extinguished.

These experiments on the coupling of energy to the space charge
modes indicate that it is possible to excite short wavelength standing
waves of a TM00p mode. These modes are excited when varying the cavity

length along a TM00p resonant length curve (see Figures 4.1a and 4.1b).

6.2.5 Detection of sidebands in the reflected spectrum
The RF plasma in a cylindrical cavity provides an excellent means
for exciting non-linear signals. At least two different types of side-

bands have been detected in the reflected power spectrum at a pressure

*

of 12 mtorr while operating in the TE111 mode. These spectra are shown

in Figure 6.8a and 6.8b.
The spectrum of Figure 6.83 has a sideband separation Af = 240 khz,

a resonant length L8 8 6.0 cms., and an absorbed power of 14.2 Watts.

010
(section 6.3.1),indicatadthat the value of (nope/w)2 fluctuated between

Simultaneous density measurements with the TM frequency shift method

2.04 and 3.16 as a function of time. A plot of these experimental results on

Relative Amplitude

Figure 6.8a.

130

 

 

 

 

I 1 1 l l I l l
3.03 GHz

Frequency

*
Sidebands detected on the spectrum analyzer in the TB 111

mode with the operating frequency w/Zfl = 3.03 GHz. Cavity
resonant length L8 = 6 cms. Discharge pressure = 12 mtorr.

Power absorbed = 14.2 watts. Plasma electron density
(cope/w)2 fluctuated between 2.04 - 3.16. Frequency scale:

0.2 MHz/cm.

Relative Amplitude

Figure 6.8b.

131

 

 

 

 

l l I I l I I I |
3.03 CH2

Frequency
*
Sidebands detected on the spectrum analyzer in the TE111

mode with the operating frequency w/2fi = 3.03 GHz. Cavity
resonant length L8 = 6.02 cms. Discharge pressure =

12 mtorr. Power absorbed = 14.1 watts. Plasma electron
density (Lupe/w)2 = 2.95. Frequency scale: 2.0 MHz/cm.

132

*

the TE111 theoretical resonant curve for (ve/w) = 0.1 (Figure 6.2) indi-
cates that the occurrence of these sidebands is caused by a rapid flucua-
tion of the operating point along the same resonant length line. This

phenomenon might be a result of the fact that it may not be possible to

*

locate a stable operating on the backward wave region of the lossy TE111

mode i.e., the region where the eigenlength decreases as (wpe/w)2 in-
creases (see Figures 4.1 and 6.1).
Figure 6.8b displays another spectrum that was detected in the re-

*
flected power of the TE111 mode. This occurs at the following operating

parameters: LSA= 6.02 cms., discharge pressure = 12 mtorr, power ab-
sorbed = 14.1 Watts. The measured average electron density, (wpe/w)2,
was 2.95 and the sideband separation Af was 4 MHz. An external signal
at f = 4 MHz was launched directly into the plasma by a miniature co-
axial probe so as to excite signals at 3.03 GHz i;n 4 MHz. It could
not be clearly determined whether any amplification of the externally
excited signals occurred in the neighborhood of the sidebands of the
spectrum. The exact nature of this instability is not clearly under-

stood and needs further investigation.

6.3 Measurement of No’ Te and Calculation of Ee in the #1 Plasma-Cavity

 

6.3.1 Introduction

Several methods have been used to experimentally measure the average
electron density and electron temperature of a microwave discharge. Each
of the methods, however, are applicable to different ranges of discharge

pressures. Three commonly used techniques are described on the following

pages.

133

6.3.1.1 Use of a Microwave Cavity to Measure Electron Densities.

The cavity is usually designed to operate in the TM or

010
27,28,29 .
TE modes. The TM mode den31ty measurement is effective
011 010
for low electron-neutral collision frequencies, i.e., the condition
2 2 33

ve <<w should be satisfied . w is the TMO10 driving frequency.

The TE011 frequency shift technique is another commonly used method
for measuring electron densities. Eigenfrequency vs. (wpe/m)2 plots10
can be used to determine the electron density provided an estimate of
the electron-neutral collision frequency is available. For situations
where the collision frequency can be neglected, a TEOll frequency shift

26,35,36

technique using a perturbation analysis may be employed This

mode may also be used for.measurements in the high-electron density
range. For such measurements the TE011 coaxial resonance of the cavity
may be used as a reference frequency34.

Both these techniques can be employed on D.C. or various kinds of
RF discharges. In the microwave plasma cavity, however, it is possible
to have a multi-mode cavity operation33, i.e., plasma excitation in
one mode, TE:11 or TMOll’ and an electron density measurement with the
TM010 or TE011 modes. The electron temperature cannot be measured with
the cavity method and only the average value of the electron density can

be determined. The diagnostic mode power input should be small compared

to the primary discharge power input so as not to disturb the plasma

characteristics.

6.3.1.2 Floating Double Probe Method
In this well known method30 two probes connected across a variable
floating potential, are immersed into the plasma. The distance between

the probes should be large compared with the Debye length to avoid an

134

interaction of the plasma region perturbed by the two probes. In using
this method to measure the plasma parameters in a microwave discharge,
the probes should protrude only slightly into the plasma region so as
not to distort the RF fields. Furthermore, effective shielding is re-
quired, especially in high-powered microwave discharges, to prevent the
radiation of RF energy away from the active plasma region and into the

external surroundings.

The average electron density and the electron energy are obtained
from the I-V probe characteristics as proposed by Johnson and Malter30.
Their theory is valid only when the discharge pressure is low enough so
that the mean free path is much larger than the probe or sheath dimen-

sions.

Several probe theories have been suggested for collision-dominated
gases. However, these theories are valid only for certain probe geome-
tries, such as spheres37 or ellipsoids38 and are, therefore, applicable
to special situations.

The characteristics of plasma-cavity I, which operates under the
available incident power of 30W, can be conveniently monitored by the
floating probe method at low discharge pressures (1 torr and below).
This experimental system was described in Chapter V. The probes are
1.523 mm in diameter and protrude 1 cm. into the active plasma region.
The electron temperature Te is evaluated from the slope of the lnIe-V

30,39,40

curve while the electron density is obtained from a formula that

is dependent on Te and the saturation ion current iri' This is given by:

16 iri 6.1

(T 95

e

N = 4 x 10
o

135

for the probe dimensions used here. No is obtained in electrons/cmB.

The D.C. floating I-V probe characteristics made here are likely to
be modified to some extent by the RF fields of the microwave discharge.
This problem has been examined by several workers for RF discharges
operating in the frequency range l—lO MHz.

CrawfordAl, Garscadden and Emeleusaz, Chen43 have all studied the
effect of fluctuations in No’ kTe and space potential VS on the time-
average Langmuir probe characteristics in RF or unstable plasmas. Their
results indicate that if kTe is a constant for a given set of plasma
conditions, then the slope of the I-V curve is unchanged by the fluctuat-

ing quantities and the electron temperature measurement is unaffected.

Sugawara and Hatta44 have shown that if kTe cannot be assumed constant
in an RF discharge, then its average value cannot be obtained from the
slope of the I—V characteristics. It has been assumed for probe mea-
surements described in this chapter that kTe is constant. Garscadden
and Emeleusbz, Boschi and Magistrelli45 have demonstrated that for large
amplitude RF oscillations the shape of the I-V characteristic near the
saturation region tends to be rounded as compared to that of an ideal
discharge.

The classical method of plasma diagnostics by the floating double
probes suffers from two drawback343: (1) only a small number of elec-
trons in the tail of the velocity distribution is sampled (2) the I-V
probe characteristics are likely to be modified by the stray capacitance
between the floating probe circuitry and ground.

Gagne' and Cantin46’47

have experimentally demonstrated that the
D.C. floating probe characteristics in their RF discharge tend to be

modified by the RF voltage. These modifications include the enhancement

136

of the ion current by as much as 40% and an apparent increase in the
electron temperature by a factor of 2—3. The perturbation of the probe
characteristics is directly related to the amplitude of the RF voltage
existing between the probe and the plasma. This voltage was minimized
by making the ratio Zp/ZS very small, where Zp is the impedance of the
plasma-probe junction or RF sheath and Z8 is the overall impedance be-
tween the probe collecting surface and the reference ground plane. These
experiments, however, do not explain the physical mechanism responsible
for the observed modifications of the probe I-V characteristics. Also,
41-45

these results are not totally consistent with the work done by others

regarding modifications of the Langmuir probe characteristics by RF fields.
All the work reported in the literature has been done in the low

RF region. The effect of these observations on the probe characteristics

of the 3.03 GHz microwave discharge is not clear. The probe diagnostics

*
111

For this mode, the electric fields are most strongly concentrated at the

conducted on Cavity 1 have been done using the TB mode of operation.
center of the cavity and away from the two probes. It appears that
further investigations are required, especially for microwave discharges,
to demonstrate any RF perturbation of the probe characteristics and if

so to obtain the physical mechanism responsible for it.

6.3.1.3 Measurement of the D.C. Electrical Conductivity

Two flat brass electrodes of surface area 0.203 cm2 have been used
to determine the D.C. electrical conductivity and the electron tempera-
ture of the microwave discharge in the #1 plasma-cavity. This method
has been used to obtain the plasma characteristics up to a discharge

pressure of 20 torr.

137

This measurement technique is similar to the one used by Maksimov7
to obtain electron densities and energies in a microwave helium dis-
charge at an operating frequency of 3.0 GHz.

The voltage applied to the plasma between the brass electrodes
divides itself across two regions: a uniform plasma zone and small
saturation regions adjacent to the electrodes. Under the assumption
that the charged particles are lost in volume recombination, Thomson
obtained the total voltage drop across the electrodes (V) as a function

of the current density (i) in the uniform plasma zone as:

V=Ai+Bi2 . 6.2

The linear term describes the voltage drop in the uniform plasma
zone and the quadratic term gives the voltage drop in the sheath regions.
Maksimov7 finds that the non-linear term in the above equation does not
make a significant contribution to the voltage drop. The presence of
cathode emission under the impact of metastable atoms and under the
action ofultravioletradiation of the discharge considerably reduces the
length of the saturation region. Thus, the non-linear term in the above
equation is neglected and the entire plasma zone is assumed to be uniform
for purposes of calculating the electrical conductivity.

The D.C. electrical conductivity of a weakly ionized gas is given

by the following expression:

 

6.3

where No - average electron density

5
ll

mass of electron

(D
II

charge of electron

C
II

electron-neutral collisional frequency-

138

The electron temperature Te is obtained from the slope of the I-V
characteristics of the discs and is used to calculate Ve at the given
pressure from the curves50 shown in Figure 6.9.

Possible sources of error in this method are that the probe theory
has been applied to the disc I-V characteristics to obtain the electron-
temperature and a uniform voltage drop has been assumed across the elec-
trodes. Thus, the experimental numbers obtained from this method cannot

be relied upon for their accuracy, but can be taken as representing the

orders of magnitude of the plasma parameters.

 

6.3.1.4 Evaluation of v and E from N ”and T

- . e -- e --- o -—- e

The dependence of electron density and electron energy on the
average power density, Wa, can be used to calculate the effective elec-
tric field Ee and the reduced field Ee/p as a function of the discharge
pressure p. The average power absorbed per unit volume of a cold micro-
wave plasmal, expressed in terms of the electric field strength E(r),

is
2 2 2
ne(r)e ve E(r)

a - 2m 0
e e

 

W 6.4

2 2
(w + ve )

where,

ne(r) spatially dependent electron density.

0 8 operating frequency

C
II

effective electron-neutral collision frequency.

Integrating the above equation over the plasma volume, we obtain

an expression for the power absorbed Pa as:

 

2 v: 2
B . a 605
Pa J Wadv 2m v 2 2 I ne(r)E (r)dv.
v e e (w + ve )

139

Ve is assumed to be independent of position. An effective electric field

Ee is defined for the entire plasma volume so that

2 __ 2 _ 2
I ne(r) E (r) dv — Ee J ne(r) dv — NoEe V 6.6
v v

where N0 = average plasma density. Therefore, in terms of Ee’ Wa becomes:

 

 

 

N ezv 2E 2 02 e v E2

o e e _pe o e e
Wa - 2 2 + 2 = 2 2 fif‘kijr . 6'7

meve(m ve ) w (1 + (If) )
Ee can, therefore, be evaluated from:
2
2Wa (l + (ve/w)

E = . 6.8

 

2
(mpg/w) Veeo

A knowledge of the average electron density, the electron temperature,
and the type of gas are, thus, necessary to determine the effective

field strength Ee' ve, the electron-neutral collision frequency can be
estimated from the electron temperature. The effective field strength,
as defined here, can be treated as a figure of merit field strength.

This definition is likely to yield meaningful results in the low pressure,
low density regions where the electric field could be assumed to be
uniform over the cross-section of the plasma. In a collision-dominated
plasma with high electron densities, the electric fields would penetrate
only a skin-depth thickness into the plasma. The electric field strength
in such a plasma would be large at the boundary, but negligibly small in
the interior. In such a situation, a measurement of the spatially de-

pendent field E(r) would prove to be useful.

The effective electron-neutral collision frequency is obtained from

the velocity dependent collision frequency vm(v), as

140

 

1010-.— 109 8
' 11 10
- w/P =10 ...
o
I
h
I _ 6
ARGON 0 10
I... I
I
77‘ 1090- ’
i” I
:3 n
a I
no b I
\ I
0 -
> x
‘V \
108:- \ I
I
p I
\ i
P
I
p 9,,“ (Ramsauer and Kollath,
Vm/Po Frost and Phelps)
107 J L 1 1 L I l 1 l I

 

0.1 1.0 10
VELOCITY(VVOLTS).

Figure 6.9. The effective collision frequency in A as a function of
velocity and frequency. Reproduced from Whitmer and

Hermanso.

 

 

 

Vm(v) 3 Bfo
f” 2 2 v 3v dv
o [vm(V)+w]
Ve = . 6.9
I” 1 v3 Bfo
2 2 -—- dv
o [Um (v) + m ] BV

The above equation can be integrated for different gases assuming
a Maxwellian distribution for fo.‘%n(v)iscalculated from measured cross-
sections, available from the data of Ramsauer and Kollath48, Frost and
Phelpsag, provided electron temperature numbers are available from

separate measurements. As an example, Figure 6.9 shows a plot of

ve/po vs. the RMS value of the velocity, <v2>= 3kTe/me, for different

values of w/po in argon gas. This curve has been reproduced from Whit-

mer and Herrmann50 and is used in the next section in estimating ve.

6.3.2 Experimental Results

*
111

operation of Cavity 1 over the pressure range 40 mtorr to 20 torr. Using

The plasma parameters No and Te have been measured in the TE mode
argon gas as outlined earlier three different measuring techniques have

been used to cover this pressure range.

1) Discharge pressure 40 mtorr - 240 mtorr. The TM010 frequency
shift method was found to be effective in this range. Beyond a pressure
of 240 mtorr the TM010 mode was highly damped because of increasing
collisional losses in the plasma.

The normalized electron density vs. discharge pressure for three
different absorbed powers Pa = 16W, 12W and 8W are shown in Figure 6.103,
for this pressure range. As expected, the electron density steadily
increases with pressure and absorbed power. At a pressure of 235 mtorr

and for P8 = 16W, the value of No is 1.65 x 1012 cm3.

142

  

 

 

14 F—
12-—
10r-
N
s
\
0
D.
a
8 r
P
a
8 w
A ------ 12 w
o------ 16 w
6 I—.
4 I J I 1 1 I
0 40 80 120 160 200 240

P , mtorr
0

Figure 6.10s. Normalized electron density (mpg/w)2 vs. discharge pressure

Po and the absorbed power Pa using the TM frequency shift

010
method.

143

 

 

20 1—
16 ‘-
z
\
>
‘1‘ 12 -
O
....
X
M” P
a
I---- 16 w
v---- 12 w
o---- 8 w
8 I.—
4 I I J
0 80 160 240

Figure 6.10b.

P , mtorr
0

Effective electric field Ee vs. discharge pressure Po

and absorbed power Pa, from the TM010 frequency shift
method.

 

 

44 r-
144
40 -
32 "
I
5.
2'.
\
q.
l
c>
04
x
o
a.
\
m0
' P
a
V---- 12 W
I---- 8 W
8 F'
I
0 l l j
80 160 240
P0’ mtorr

Figure 6.10c. Reduced field Ee/Po vs. discharge pressure PO and ab-

sorbed power Pa, from the TM

010 frequency shift method.

145

Since the electron temperature cannot be measured using this tech-
nique, a value of 90,0000K has been used for Te on the pressure range
40 mtorr - 240 mtorr. This estimate for T8 has been obtained from probe
data measurements to be described later.

Plots (fl? Ee and Ee/p calculated from equation 6.8 are shown in
Figures 6.10b and 6.10c. respectively. Ee 3nd Ee/p decrease with in-
creasing pressure 3nd electron density and appear to be almost inde~
pendent of the absorbed power. Ee varies from 2 x 104 V/M at 45 mtorr

to 6.3 x 103 V/M at 235 mtorr for 3 Pa = 16W.

2) Discharge pressure 140 mtorr to 1.05 torr. The double floating
probe method has been used to measure the electron density and electron
temperature in this range. At very low pressures (below 100 mtorr) and
in the high pressure regime (above 1 torr) the probe data did not provide
reliable results. A typical double probe I-V curve at a pressure of
86 mtorr is shown in Figure 6.113. It is clear that the saturation
regions are not clearly defined, thereby causing an error in the density
calculation. At high pressures, the saturation current was too small
as a result of the collision-dominated plasma. This is demonstrated by
two I-V characteristics at pressures of 5 torr and 11 torr where the
ion saturation current decreases with an increase in pressure from 5 to
11 torr, see Figure 6.11b. In the pressure range 100 mtorr to l torr,
the I-V curves are more conventional, with well defined linear and sat-
uration regions. A typical set of such curves is shown for different

power levels and different pressures in Figures 6.llc and 6.lld.

146

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147

 

0
q_ 4 o
g . .
’4. (
aub— 2 '
g
/
1 n ) 1 n
" l l I T ”
-10 -5 ,/ 5 10
U V, volts
0
O
; __ -2 P
o 0
' o -—-- 5 torr
' a -—-- 11 torr
0 -_ -4

 

Figure 6.11b. Typical floating double probe I-V characteristics for
pressures above 1 torr. Absorbed power Pa = 20 W.

ma

148

 

' 20 40
/

i V, volts
,d--2

 

P
a

A ---- 20 W
o---- 16 w
o---- 12 w

Figure 6.11c. Typical floating double probe I-V characteristics at a
'discharge pressure of 140 mtorr and different absorbed

power levels.

149

 

 

 

O
6 4&—
6'
g 4 .L. .
H
{I
a
g
2 4 l
J.
«F : § ' ; .L >
-40 -20 20 40
, V, volts
1
O
uni-"4
‘P-6
Pa
1 -8
o--—- 16 W
A---- 12 W
‘P-lO

 

V

Figure 6.11d. Typical floating double probe I-V characteristics at
.a discharge pressure of 730 mtorr and different ab-
sorbed power levels.

150

The experimentally measured electron density and electron tempera-
ture vs. discharge pressure for P8 = 20W, 16W, 12W are shown in Figures
6.12a and 6.12b. It is observed from Figure 6.12a that in the pressure
range 100-400 mtorr, the electron density decreases and then rises again
with pressure. As explained in section 6.2.3, the unusual behavior in
this low pressure range is believed to be a result of coupling of the

*
TE electromagnetic energy to the TM

111 00p space charge wave. Normally,

this space charge wave would propagate along the plasma column and ionize
the gas as it travels (see Figure 6.2). However, in this diagnostic
technique, the location of the two probes (see Figure 5.1) prevents the
plasma volume from increasing longitudinally. As a result, the absorbed
power per unit volume increases causing a peak in the electron density

vs. pressure curve. Beyond 400 mtorr, the electron density rises steadily
with pressure and appears to saturate around 1 torr. However, probe
measurements beyond a pressure of 800 mtorr tend to yield lower than
actual values for the electron density because at these pressures, the
collisionless probe theory may not be applicable. At 800 mtorr and Pa =

20W, the value of No is 1.85 x 1012 cm3.

The electron temperature (see Figure 6.12b) appears to be almost
independent of the power absorbed by the plasma, but decreases steadily
with pressure. At 1 torr, Te is approximately 60,0000K. The effective
electric field (Figure 6.12c) and the reduced field Ee/p (Figure 6.12d)
are also independent of the power absorbed, but decrease with an in-
crease in electron density and pressure.

The floating double-probe measurements obtained follow the same

general variation with pressure as those obtained by Maksimov7 in a

151

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155

3.0 GHz Helium microwave discharge and by Bell55 in a 20 MHz Helium
discharge. Since argon gas has been used in the plasma cavity, an
absolute comparison between the results of this section with those of

Maksimov and Bell is not possible.

3) Discharge pressures above 1 torr. Most of the commonly used
techniques for measuring the plasma parameters (probes, frequency shift
method) fail in this pressure region because of the collision-dominated
plasma.

However, a simple technique using discs to measure the conductivity
has been found to be useful in estimating the electron density up to a
pressure of 20 torr. Discs of different cross-sections were tried and
a disc of almost the same cross-sectional area as that of the quartz
tube of Cavity 1, gave numbers for electron density that seemed to match
the results obtained by the double probe measurement at a pressure of

l torr.

The results of the electron density measurement from conductivity
calculations are shown in Figure 6.13a. At a pressure of 20 torr, and
Pa = 20W, the value of No is 7.84 x 1012/cm3. The electron temperature
was estimated from the slope and the saturation current of the I-V
characteristics of the discs. The results obtained by this method match
those obtained by the probe measurements at l torr. The electron tempera-
ture is slightly dependent on the power absorbed and reaches a uniform
value as pressure increases.

Plots of Ee and Ee/p vs. pressure are shown in Figures 6.13c and
6.13d, respectively. Both the effective field and the reduced field
reach a steady value at a pressure of 20 torr. At this pressure Be is

2.5 x 103 V/M and is independent of the power absorbed.

156

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159

 

  
  
  
      

 

 

 

60 F-
50 ‘-
40 "
30 _‘
O
Q)
20 —
P
a
o--—- 20 w
A ---- 16 W
10 _. I‘---- 12 W
J J 4 I 4
0 4 8 12 16 20

Figure 6.13d. Reduced field Ee/Po vs. discharge pressure Po and absorbed

power Pa’ from measurements using discs.

(wpe/w)

Figure 6.14a.

 

160

  

 

60-— _____. '
TMOlO frequency Shlft
—---— Floating Double probes
——--— Discs
0 --P = 16W
a
0 ---P = 12w
50... a . 0/
/' '
40— /
' 0
/ -’
30 I 0/
/ ,0/ °
20_ ,/ /
,0" /
10_ " I'D/ /
---O~‘ ,’ ’0’ ./
“\1V x
o“‘O----O-"o o/
0 l J l l l i I l
0.04 0 l 0 2 0 5 1 2 5 10 20
P , torr
0

Normalized electron density (cope/w)2 vs. discharge pressure
Po and absorbed power Pa over the pressure range 40 mtorr -

20 torr, from three different diagnostic methods.

 

161

 

 

12 '-
Floating Double Probes
---- Discs
O-—-P =16W
a
0--—P =12w
a
10 ‘-
8—
3‘ Q
~3' \
l \
S \
x 6 -— \
a) \
\
E" \
\
R \
\ \
\ \
4— \ \
\ ‘\ ,—C\
\ \ ’0’ \
\\\‘O—-” -Q ‘6
\o-——-—-'°' \
b
‘2__
o l I l I l l l
0.1 0.2 0.5 1.0 2 5 10 20
P , torr
0
Figure 6.14b.

Electron temperature Te vs. discharge pressure PO and ab-

sorbed Pa over the pressure range 0.1 - 20 torr, from two
different diagnostic methods.

 

 

162

 

TMOIO frequency shift

----- Floating double probes
——-— Discs
.---Pa = 16W

0---P = 12W
3

 

2 \
é \
E 'x
\
i 104 L- b
O \
E: \
(D \s
[-11 5 r- .\ \
o
2 - \
103 0 \
5 r— \.
2 ~ \
102 L 1 l 1 J 1 I v)
.04 .1 .2 .5 1 2 5 10 20
P , torr
0

Figure 6.14c.

Reduced field Ee/Po vs. discharge pressure Po and absorbed
power Pa over the pressure range 0.04 - 20 torr. from three

different diagnostic methods.

163

For purposes of comparison, the plasma parameters and the reduced
fields obtained frrmxthethree different measurement techniques are dis-

played over the entire pressure range in Figures 6.143, 6.14b, and 6.14c.

6.4 Absorbed Power Characteristics of the Microwave Plasma Source

6.4.1 Introduction

 

The #2 plasma-cavity system has been used to study the absorbed
power characteristics of the microwave plasma source. The plasma-cavity
was tested in argon gas. The quartz center tube was connected to a
vacuum system which could be pumped down to several microns of Hg. Once
evacuated to low pressures, argon was inserted into the system and plasma
initiation was achieved by 1) adjusting the pressure of the argon gas
somewhere between 0.5 to 10 mm Hg., 2) applying the high microwave power.
and 3) length tuning the empty cavity for resonance at the oscillator
frequency of 2.450 GHz. The microwave source and a cross-section of

Cavity 2 are shown in Figures 5.4 and 5.5.

Gas breakdown was achieved near precalculated cavity resonant
lengths. Once breakdown was accomplished, the pressure, RF power,
cavity length, and cavity coupling adjustments could be varied to pro-
duce the desired plasma conditions. With a little experimental experi-
ence, this tuning procedure could be achieved in less than one-half
minute without fear of extinguishing the plasma. By varying the input
gas flow and exhaust pumping rate, a plasma could be established in the
flowing gaseous environment. The flow rate could be continuously varied
from zero to over 14,000 cm3/minute, the upper limit of the experimental
flow measurement capability.

The plasma cavity could be operated in many different cavity re-

sonances simply by adjusting the cavity length and input coupling. For

164

example, the following empty cavity modes have eigenfrequencies equal

to 2.450 GHz when the cavity length is varied from 6 to 20 cm.: TElll’

TM T

TM011’ TE211’ TE011’ E212’ TE012’ TM112’ TE113’

etc. With the proper cavity length and RF coaxial coupling (i.e., loop

111’ TE112’ TM012’

or probe), the plasma cavity system will sustain a plasma on any of these

modes. However, only the TM , and TE modes, where p = 1, 2,

Olp’ TE01p

or 3 were studied extensively.

llp

All three of these modes were able to maintain microwave plasmas
in argon gas over a wide range of pressures and flowing gas conditions.
The appearance of the plasma varied considerably as the pressure and
power were varied. At pressures less than 5 mm Hg, the discharge could

be resonantly sustainedSI’52

in a TD resonance or in a pure electro-
magnetic resonance. In either case, the discharge took on a dull to

bright reddish pink astimzpower was increased. 'In the pressure range of

5 to 80 mm Hg, the discharge color varied from dull pink to blue to a

bluish white at high power levels (1 KW). At pressures of 80 mm Hg or
more, thin intense discharge streamers appear following paths which

have high displacement currents in the absence of a plasma. When ex-

citing the TEOln resonance at pressures greater than 200 mm Hg hot

32,53

plasmas, similar to low-frequency induction plasmas , can be formed

inside the quartz tube.

6.4.2 Experimental Results
Two different types of_experiments were performed to study the ab-
sorbed power characteristics of the argon plasma maintained in Cavity 2.
1) In the first of these experiments, the absorbed power of the
plasma was studied as a function of the cavity length with the gas dis-

charge pressure (po) and the gas flow rate as the variable parameters

165

(see Figures 6.15 and 6.16). These experiments were conducted using the
TE011 mode, because of its usefulness in plasma chemistry experiments and
also because of its gradual variation in eigenlength vs. electron density
(see Figure 6.17). Because of equipment limitations, only the following
parameters have been measured:hntheseexperiments: the gas flow rate,
discharge pressure and the length of the cavity. It was not possible

to make density and temperature measurements using the conventional

probe and TM frequency shift diagnostic techniques. Probe measure-

010
ments on the high pressure plasma of Cavity 2 yielded erroneous results
because of high electron-neutral collision frequencies. Also, the in-
sertion ofa.probe into the plasma of Cavity 2, caused a significant
amount of power to be coupled to the external surroundings. This was
considered to be a safety hazard. The use of the TM010 frequency shift
method was also limited by the high pressures employed in Cavity 2. The

failure of these methods and the absence of more sophisticated techniques
required that all arguments used in explaining the experimental phenom-
ena be based on the above-mentioned experimental quantities and on esti-
mates of the electron density and temperature.

In the absence of direct measurements of the electron density, col-
lision frequency, rate of ionization and other parameters involving a
flowing gas, the arguments presented below are speculative. However,
these explanations may provide the background on which further theoreti-
cal and experimental research may be based to fully understand this
problem.

Figure 6.15 displays the experimental absorbed power Pa vs. the
cavity eigenlength LS for the TED mode. All experimental points in

11

Figure 6.15 were measured at a constant flow rate of 1,000 cc/min. and

166

 

 

 

 

 

 

 

 

900 "'

800 _

700 ’—

600‘-

P
o

500 r-
0')
III 100 torr
cc
3
a}:

400 '—

70
40

300 20

200 I-

100 '- . L4

...__.. Region l___..' rfi Region 2
I I I I I I I ,J
9.0 9.05 9.10 9.15 9.20 9.25 9.30 9.35 9.40
L , cms.

Figure 6.15. Absorbed power Pa vs. eigenlength LS and different discharge

pressures Po’ for the TE mode plasma of Cavity 2. Flow

011

rate = 1000 cc/min,.Incidentpower Pi = 1.2 KW.

167

each curve represents the power absorbed vs. cavity length for a differ-
ent pressure.

Two regions of operation can be clearly distinguished in these
curves. In region 1, which covers the approximate eigenlength 9.0 cms -
9.1 cms., the absorbed power increases as a function of the discharge
pressure. In region 2, which includes eigenlengths from 9.2 - 9.4 cms.
the absorbed power increases inverselyansthe discharge pressure. These
experimental observations can be explained by studying the theoretical
TEO1 absorbed power vs. (w/wpe) curve for different (ve/wpe), as
described in Chapter III and shown in Figure 3.5. Note that the curve
of Figure 3.5 has been plotted for a plasma density of fpe = 10.5 CH2
and for the waveguide dimensions of Cavity 1. However, the behavior of
the curve is general enough, so that it can be used to qualitatively

explain the variation of absorbed power and plasma density vs. pressure

in Cavity 2.

In region 1 of Figure 6.15 the cavity operates close to the empty
. cavity resonance. The electron density in this region is small and,
hence, (m/wpe) becomes large. This experimental region of operation
corresponds to the right end of the theoretical TE01 absorbed power
curves shown in Figure 3.5. In this region of the theoretical curves,
the normalized absorbed power increases with the collision freqeuncy.
(Compare curves for (ve/wpe) = 0.1 and 1.0). This type of behavior would
be more apparent at still higher values of (Va/mp8). This seems to justify
the experimental observations made in region 1 of Figure 6.15.

In region 2 of Figure 6.15, the electron density is large and, hence,
(w/mpe) is small. This region of experimental operation would lie to

the left of the theoretical TE01 absorbed power curves (Figure 3.5). In

168

this part of Figure 3.5, the normalized absorbed power increases inversely
with the collision freqeuncy. This theoretical behavior explains the
experimental observations made in region 2 of Figure 6.15.

Figure 6.16 shows the absorbed power, Pa’ vs. the cavity eigenlength,
LS, with the discharge pressure held constant at 10 mm. Eachcfifthe curves
of Figure 6.16 have been plotted for different gas flow rates. These
curves show that for a given flow rate, the absorbed power reaches a peak
at a certain eigenlength. For example, when the flow rate is zero, the
peak occurs at LS = 9.15 cms. and the absorbed power is only 490W. As
the flow rate increases, the absorbed power peak shifts towards higher
eigenlengths and for flow rates of 250 cc/min and beyond, the absorbed
power peak remains at LS = 9.35 cms. At the same time, the absorbed
power also increases with the flow rate and is highest for a flow rate
of 2000 cc/min.

In providing a qualitative explanation for the shift in cavity
eigenlength, with changes in flow rate, the eigenlength plot of the TE011
mode for Cavity 2 (Figure 6.17) will be used. Before using the curves

of Figure 6.17, it must be noted that the experimental TE empty

Oll
cavity and coaxial resonances are lower by .05 cms. than the correspond—
ing theoretical values due to the finite thickness of the plasma enclosing
quartz tube, holes in the cavityiknidiagnostic purposes, wall imperfec-

tions, etc. Hence, .05 cms. should be added to all TE experimental

011
eigenlengths for a direct comparison with the theoretical curves of
Figure 6.17. The TEOl1 eigenlength curves have been evaluated for a

zero flow rate. The applicability of these curves to a non-zero flow
rate situation is not clearly understood. It is expected that a non-zero

flow rate, would modify the values of electron density and the collision

frequency from their zero flow rate values. However, a flowing plasma

169

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1000
900 '-
800 "‘
700 "
o
S
600 '-
m I
S
3 500 “‘
'63
0..
o
o
...4
400 '- Flow rate=0
o o c
300K a o 8
\ N
\
200 "
100 *-
l l l 1 l l l Alj
9 00 9 05 9 10 9 15 9.20 9 25 9 30 9 35 9 40
L ,cms.

Figure 6.16. Power absorbed Pa vs. eigenlength Ls’ and flow rate (cc/min)
for the TE011 mode plasma of Cavity 2. Discharge pressure,

P0 = 10 torr. Incident power Pi = 1.2 KW.

170

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171

still has a discrete electron density, temperature and collision frequency,

i.e. , its steady state operation can be represented as a single point on

Figure 6.17. Assuming that the electron temperature may vary between
20,0000K and 90,0000K, the normalized electron neutral collision fre-
quency, (VB/w), can be estimated from Figure 6.9 to vary between 1 and
3.5. This band of collision frequencies would correspond to the shaded
region of operation shown in Figure 6.17.

The experimental curves of Figure 6.16 can best be explained by
classifying the curves of different flow rates into three different
groups: 1) Flow rate = 0 cc/min. 2) Flow rate = 60 cc/min. 3) Flow
rates = 100, 250, 500, 1,000, 2,000 cc/min.

1) The peak-absorbed power for a zero flow rate occurs at LSC =

9.20 cms., and the operating point lies on line A, as shown in Figure 6.17.

LSC is the corrected eigenlength.

2) With an increase in the flow rate to 60 cc/min., the eigenlength
of the peak of the curve shifts to LSC = 9.35 cms. The peak-absorbed
power also increases as a result of the increase in the ionized gas
volume. This contributes to a rise in the electron density and the
operating point lies along line B (Figure 6.17).

3) A further increase in the flow rate to 100 cc/min. causes the
peak absorbed power to increase from 560W to 790W. The new eigenlength
of the peak of the curve occurs at LSC = 9.395 cms. This dramatic rise
in absorbed power is attributed mainly to the increase in the volume of
ionization because of the large flow rate. The operating point now

- lies along line C (Figure 6.17).

The flow rate is now increased gradually in the range 250 cc/min.

to 2000 cc/min. It is observed that for this large variation in the

172

flow rate, that the eigenlength of the peak-absorbed power remains con-
stant at LSC = 9.40 cms. and the peak—absorbed power varies only from
900W to 950W. A possible explanation for this strange behavior is that
the substantial increase in the flow rate may not cause a proportional
increase in the volume of the ionized gas.

As the flow rate is varied in the range 0'to 250 cc/min, the volume
of the ionized gas increases directly with the flow rate. This fact has
been experimentally verified by observing that the light intensity of
the gas increases with the flow rate. This implies that there must be
a large increase in the absorbed power (see Figure 6.16 for flow rates
0, 60, 100, 250 Cc/min.) and, hence, an increase in the electron density.
The electron density at 250 cc/min. may be sufficiently large to form
a high density plasma next to the inside of the quartz tube. This
plasma would prevent the penetration of the fields into the internal
volume of the gas and thus, have the effect of preventing the rest of
the gas from ionization. This shielding effect of the outer layer of
the plasma allows a large volume of the incoming gas to bypass the ex-
citation region in the cavity. Thus, a dramatic increase in the flow
rate from 250 cC/min. to 2000 cc/min. has little or no effect on the
peak power absorbed by the plasma.

2) In the second type of experiments the behavior of the absorbed

*

power of a TE112 mode plasma was studied as a function of the discharge

pressure with the incident power P and the gas flow rate as the vari—

i

able parameters. The values used for the incident power P were 966W,

1
759W, 552W and 276W. A family of such curves was obtained by varying
'the pressure with flow rate constant. The flow rate values used were

147, 520, 955, 1480, and 2500 cc/min. These curves are shown in Figures

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Flow Rate, cc/min

*
Figure 6.25. Percentage peak-absorbed power vs. flow rate for the TE112

plasma of Cavity 2. Incident power Pi,= 966 W.

181

6.18 through 6.22. The length of the plasma-cavity during operation
was held constant at LS = 13 cms.

A study Of the curves in Figures 6.18 - 6.22, shows that for a given
flow rate and incident power, the absorbed power has a relatively flat
response vs. the pressure. A minor peak in the absorbed power is seen
to be present in all curves for constant flow rates. The pressure at

which this peak occurs usually increases with the incident power Pi

see Figure 6.22.

It is generally true that for a given incident power, the absorbed
power increases with the flow rate. There is an exception to this be-
havior at a flow rate of 520 cc/min. as demonstrated in Figure .

6.24. The curves of Figures 6.23and 6.24 have been plotted for incident
powers of Pi = 966W and 759W. A plot of the percentage of peak-absorbed
powervs. flow rate for a given incident power is shown in Figure 6.25.
13m: curve of Figure 6.25 shows that the peak-absorbed power rises steeply
vfijilthe flow rate at first and then reaches a saturation level. This
behavioris consistent with the increase in peak-absorbed power with

flow rate as described for the TE mode (see Figure 6.16). The satura-

011
tflxnlof the peak-absorbed power at high flow rates is probably caused by

timeshielding effect of the boundary plasma layer, as described earlier.

Comparison with other RF Discharges. Similar experiments were performed
by Dorman and McTaggert8 USing a resonant cavity and by Bosisio et.al.54
using a slow wave structure. Their experiments were performed on argon
gas for a zero flow rate and the results are shown in Figure 6.26. Dor-
mand and McTaggert worked at 915 MHz and used 250W of applied power. The.

slow wave structure of Bosisio et.a1. employed incident powers of up to

2 KW at a frequency of 2.45 GHz.

 

182

 

GAS: Argon
P =2000W
1200 .. 1 .
/'/ ‘.‘-\
I: '- o’ '\ extinction
Q . 1 1.40214 \
3 J J’. . \
L 800 .av I -
3 /. \
O 1 000W -‘
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0‘ .‘o e

B ././ / \.
n I “} \
H 0’. \
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’
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0.1 1 10 100 1,000

Figure 6.26.

PRESSURE-TORR

Plot of absorbed microwave power vs. pressure for argon
plasmas at various incident powers. Reproduced from

Bosisio et.al.54.

183

A comparison of their results with the work performed on Cavity 2
reveals the following:

1) The absorbed power variation as a function of the pressure is
more uniform for the TE:12 microwave discharge of Cavity 2 than for a
similar type of discharge in the slow wave structure of Bosisio et. al.54
The flat response of the absorbed power curves especially at high pres-
sures or high values of (Va/w) is a result of two phenomena. The first
reason for the flat response of the experimental absorbed power curves
can be obtained from typical TE:1 absorbed power vs. w/w e curves, shown
in Figure 3.6. It is apparent from these curves that the theoretical

absorbed power varies very slowly with (w/wpe) for (ve/wpe) = 1.0. This

behavior is true for all collision frequencies greater than (V /w ) =

e e
1.0. Thus, the plasma waveguide itself is not resonant with respect to
(ve/wpe) and (w/wpe).

*
112 mode is such that beyond a

Secondly, the property of the TE
critical density of 10, there are no resonances in the eigenlength plot,
(Figure 4.2), and the shift in eigenlength vs. plasma density and col-
lision frequency is very small. It must be noted that the curves of
Figure 3.6 and 4.2 have been derived fromaazero flow rate condition and
for the waveguide dimensions of Cavity 1. Hence, these curves cannot
fully explain the experiments that have been conducted here with a non-
zero flow rate. However, they can be used as a qualitative guide to
understand how efficiently the incident power can be coupled to the
plasma in Cavity 2. The absorbed-power curves of Bosiso et.al. (Figure
6.26) for a non-flowing argon discharge have a fairly distinct peak and
this usually occurs at 40 torr. The reason for the occurrence of this

absorbed-power peak at such a low pressure is attributed to the type of

coupling used in their system.

184

Argon plasmas in microwave cavities, can be maintained at pressures
over an atmosphere. However, in results presented on Cavity 2, experi-
mental limitations prevented the measurement of pressures beyond 500
torr. In experiments conducted on the slow wave structure by Bosisio
et.al., the plasma could be sustained only up to a pressure of 500 torr
with the incident powers being as high as 2 KW.

2) The absorbed power curves of Cavity 2 clearly show that the
coupling of the incident power to the discharge is a function of the

gas flow rate. For an incident power of P = 966 W, the efficiency of

i
this coupling rises from approximately 50% to as high as 79% as the

flow rate increases from 147 cc/min. to 2500 cc/min. (see Figure 6.25).
In these experiments, the eigenlength of the cavity mode was held con-
stant at LS = 13.0 cms. It is possible to couple more than 90% of the
incident power to the plasma, by optimizing the length, discharge
pressure, and the gas flow rate. Bosisio et.al., in their experiments
on non-flowing argon plasmas report a maximum efficiency of 60% using
incident powers of up to 2 KW.

Thus, these experiments conducted on Cavity 2 demonstrate that in.
the TE:12 cavity mode, the absorbed power is relatively insensitive to
changes in electron density and collision frequency. The incident power
can be coupled Very efficiently to the plasma. No tuning stubs are
required to provide an impedance match between the external microwave

system and the cavity.

CHAPTER VII

SUMMARY AND CONCLUSIONS

The electromagnetic behavior of a microwave plasma source has been
experimentally examined over a wide range of pressures and flow rates.

A qualitative explanation has been provided to account for the experi-
mental results using a linear, lossy, cold plasma theory. The latter
theory, however, does not provide a rigorous explanation for the experi-
mental phenomena because of its restriction to: a) small signal linear
phenomena, b) ionized gases with a zero flow rate. 'In spite of its
limitations, the theory does take into account losses in the plasma due
to electron-neutral particle collisions and thus, explains the pressure
dependence of the experimental results.

Three different types of experiments were conducted on the plasma-
cavity to demonstrate its use as a versatile microwave plasma source.
The linear and non-linear behavior of the plasma cavity in the low-
pressure region was examined using Cavity l. The presence of two plasma
density operating points for a single eigenlength in the TE:11 mode was
experimentally verified. The coupling of EM energy to the space charge
waves was demonstrated and this phenomena was qualitatively explained
using the eigenlength vs. (wpe/w)2 and (ve/w) curves obtained from the
lossy, linear, cold plasma theory. A more rigorous theory, however, is
required to explain the non-linear nature of this coupling mechanism.

It was also observed that long plasma columns can be produced in the

space charge mode of operation, with the length of the plasma column

185

186

increasing as the power input to the cavity was increased. Short wave-

length standing waves of the TM00p mode were also generated in the plasma

cavity.

Further non-linear phenomena in the form of sidebands were detected

*
111

One set of these sidebands was qualitatively explained as being caused

in the reflected power spectrum of the TE mode plasma of Cavity l.

by a fluctuation in the average plasma density as a function of time,
i.e., a shift in the operating point along a constant resonant length line
of the TE:11 mode. The reason for the occurrence of the other set of
sidebands was not clearly determined and an examination of the theory of
the interaction of strong RF fields with bounded plasmas may explain the
latter instability.

An effective electric field vs. discharge pressure (40 mtorr-20 torr)
was evaluated in the TE;11 mode plasma of Cavity l, by measuring the
plasma parameters No and Te' The TMOlO frequency shift method, double
floating probe and disc techniques were employed to determine these
parameters. The probe and disc methods yielded results that were repre-
sentative of the orders of magnitude of the plasma parameters and not
their exact values. The shape of the No’ Te, Ee and Ee/p vs. pressure
curves followed the same general variation with pressure as those ex-
hibited by Maksimov7 and Bell55 in He gas. The effect of microwave
radiation on the probe and disc characteristics needs to be examined
further. Plasma diagnostics on Cavity 2 using the conventional techniques

proved to be unsuccessful. The TM frequency shift method and the

010
probe techniques failed as a result of the high pressures and power
levels employed in Cavity 2. Furthermore, the probes coupled significant

amounts of power from the cavity into the surroundings and presented a

187

safety hazard. Thus, more sophisticated techniques, such as plasma

spectroscopy, are needed to evaluate the plasma parameters of Cavity 2.
The absorbed power characteristics of the microwave plasma source

were examined over a wide range of pressures, flow rates and incident

power levels. These experiments were conducted on a length dependent

*
111

The absorbed power characteristics of a flowing plasma were shown to be

mode (TEOll) and in the length independent region of the TE mode.
significantly different from those of a non-flowing plasma. The absorbed
power was shown to increase directly as a function of the flow rate at
low flow rates and then reach a saturation at high flow rates. This
phenomena was qualitatively explained as being caused by the shielding
effect of the plasma layer adjacent to the quartz tube. Thus, the by-
pass region of the cavity quartz tube needs to be reduced to prevent
saturation of the absorbed power at high flow rates. Also, a more exact
theory incorporating the velocity of the ionized gas needs to be derived
to fully explain the absorbed power characteristics of a flowing plasma.
The absorbed power variation as a function of the pressure of the
microwave cavity discharge was found to be more uniform than a Similar
type of discharge in the slow wave structure of Bosisio et.al.54 The
incident power could be coupled very efficiently to the plasma and no
tuning stubs were required to provide an impedance match between the
external microwave system and the cavity. The slow wave structure of
Bosisio et.al.54 absorbed a maximum of 60% of the incident power and a
plasma could not be sustained beyond a pressure of 500 torr. In the
plasma cavity, it was possible to couple more than 90% of the incident
power to the plasma by optimizing the length, discharge pressure, and

gas flow rate. Operation of this plasma cavity beyond 1 atm. has been

observed.

188

Thus, the microwave plasma source described in this thesis has been
shown to operate efficiently over a wide range of pressures, flow rates,
incident power levels and in different electromagnetic modes. This
source has been shown to be amenable to plasma diagnostics at low power
levels and for pressures up to 20 torr. At high pressures and high power
levels it should be possible to conduct diagnostics using spectroscopic
techniques. The simplicity of the plasma cavity structure has made it
possible to conduct a linear electromagnetic analysis of the microwave

plasma source, incorporatingtfiuacollisional losses within the plasma.

APPENDIX

APPENDIX A

In this appendix, the numerical techniques used in evaluating the
roots of the characteristic equation of a cylindrical waveguide partially
filled with a cold, lossy, isotropic plasma (See equation 2.62), are
briefly discussed. A study of equation 2.62, reveals that this equation
is transcendental in nature and involves Bessel functions with complex
arguments. A main program and two subroutines are used to evaluate the
roots of this equation.

FROOT is a subroutine that searches for roots of the transcendental
equation F(Z) = 0 by an iterative process. The argument 2 is, in general,
complex. COMBES is a subroutine that evaluates the real and imaginary
parts of a Bessel function for a given order and complex argument. DIS-
PERSION is a main program which together with FROOT and COMBES evaluates
a typical m - B or w - a diagram (See Chapter II), i.e., it obtains the
complex propagation constantlasfor a given frequency w, as a function of
the plasma parameters wpe and Ve' HYBRIDMODE is another main program
which together with FROOT and COMBES obtains the eigenlength for a given
cavity mode as a function of wpe and ve’ with the driving frequency w
fixed (See Chapter IV).

An initial guess value for a given mode is required to obtain the
dispersion curves of Chapter II or the eigenlength plots of Chapter IV.
For example, to evaluate the eigenlength curves of the TE011 cavity mode
(Figure 4.2), the ‘value of kZ corresponding to the empty cavity resonant

length of this mode is used as the initial guess value. The TE011 curve

189

190

for a given ve/w is then obtained for different values of (wpe/w)2 by
increasing (cope/w)2 in small increments. The size of the increment de-
pends on the value of the gradient of the curve at a given point and,
therefore, on the ability of FROOT to converge. The size of the incre-
ment is decreased until convergence is eventually obtained. The resonant
length of the TE011 curve at high values of (Lupe/w)2 and for a given
ve/w is then compared with the resonant length of the corresponding TE011
coaxial cavity mode. These two values are found to match. Thus, the

empty cavity and coaxial cavity resonant lengths of a given mode serve

as checkpoints in determining the accuracy of the numerical solutions.

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