University of Tennessee, Knoxville University of Tennessee, Knoxville

TRACE: Tennessee Research and Creative TRACE: Tennessee Research and Creative

Exchange Exchange

Masters Theses Graduate School

5-1990

Development of the High Vacuum Steady-State Orbitron MASER Development of the High Vacuum Steady-State Orbitron MASER

Mark S. Rader University of Tennessee - Knoxville

Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes

Part of the Electrical and Computer Engineering Commons

Recommended Citation Recommended Citation Rader, Mark S., "Development of the High Vacuum Steady-State Orbitron MASER. " Master's Thesis, University of Tennessee, 1990. https://trace.tennessee.edu/utk_gradthes/769

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu.

To the Graduate Council:

I am submitting herewith a thesis written by Mark S. Rader entitled "Development of the High

Vacuum Steady-State Orbitron MASER." I have examined the final electronic copy of this thesis

for form and content and recommend that it be accepted in partial fulfillment of the

requirements for the degree of Master of Science, with a major in Electrical Engineering.

Igor Alexeff, Major Professor

We have read this thesis and recommend its acceptance:

J. D. Tillman

Accepted for the Council:

Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official student records.)

To the Graduate Council:

I am submitting herewith a thesis written by Mark S. Rader entitled "Development of the High Vacuum Steady-State Orbitron MASER". I have examined the final copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Electrical Engineering.

We have read this thesis and recommend its acceptance:

I

J D.{.dk.,.

Accepted for the Council:

_. Vice Provost and Dean of The Graduate School

STATEMENT OF PERMISSION TO USE

In presenting this thesis in partial fulfillment of the requirements for a

Master's degree at The University of Tennessee, Knoxville, I agree that the

Library shall make it available to borrowers under the rules of the Library.

Brief quotes from this thesis are allowable without special permission,

provided that accurate acknowledgement ofthe source is made.

Permission for extensive quotation from or reproduction of this thesis

may be granted by my major professor, or in his absence, by the Head of

Interlibrary Services when, in the opinion of either, the proposed use of the

material is for scholarly purposes. Any copy or use of the material in this

thesis for financial gain shall not be allowed without my written permission.

Date if - ;;t - ~ ()

Development of The High Vacuum

Steady-State Orbitron MASER

A Thesis

Presented for the

Master of Science

Degree

The University of Tennessee, Knoxville

Mark S. Rader

May 1990

i

To my wife, Faye, with love

ii

Acknowledgements

I wish to thank Professor Igor AlexefT and Francis F. Dyer for the help

and support they 'have given me. I also wish to thank Doctor Robert Barker at

the Air Force Office of Scientific Research for providing funding for the Air

Force summer student program at University of Tennessee, Knoxville.

iii

Abstract

The Orbitron MASER is a device which can be used to generate R.F.

radiation. This device has been operated for many years as a gas filled pulsed

device. This thesis describes a brief history, design and operation of several

steady-state, hot cathode Orbitron MASER. It also touches on a pulsed

Orbitron MASER and the effects the hot cathode has on the radial electric field

profile. A detailed derivation of emission frequency, gain and saturation is

given in Appendix.

iv

TABLE OF CONTENTS

CHAPTER PAGE

1.

2.

3.

HISTORICAL NOTES .................................. .

A Brief History Of The Hot Filament Orbitron MASER .... .

STEADY-STATE ORBITRON EMISSIONS ............... .

1

1

3

Abstract of Steady-State Orbi tron Emission ................ 3

Introduction

Theory

Steady-State Emissions ................................. .

Pulsed Emissions ...................................... .

Conclusion

STATIC ELECTRIC FIELD CALCULATIONS ............ .

3

4

6

17

20

21

Static Electric Field Simulation Of The Orbitron MASER ... 21

Simulation Outputs ..................................... 22

Conclusion From Electrostatic Simulation ....... ;......... 29

Conclusions About The Orbitron MASER .................. 29

REFERENCES ......................................... 30

APPENDICES: Basic Theory Of The Orbitron MASER ..... 32

Introduction To The Theory Of The Orbitron MASER ....... 33

Dynamics Of Orbital Electrons ........................... 33

Negative Mass And The SelfInstability ................... 38

Theory For An Electron Ring ............................. 39

Theory For An Electron Spread In Radius (Vlasov Theory) ... 50

v

CHAPTER PAGE

. Saturation Of Growth And Physical Limits To Harmonic Emission ................................... 56

Electron Feed Mechanisms ............................... 58

Appendix References .................................... 60

VITA 61

vi

LIST OF FIGURES

FIGURE

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

Orbitron Schematics .................................... .

First Orbitron Spectra

Comparison of Spectra

4 Wire Spectra ......................................... .

Seven Wire Spectra ..................................... .

Schematic of Double Cavity System ...................... .

Model Control of Orbit ron ............................... .

Output of Hewlett Packard Spectra Analyzer .............. .

Output ofIntegra Spectrum Analyzer .................... .

PAGE

5

7

9

10

11

13

14

15

16

2.10 Schematic of Lined Orbitron .............................. 19

3.1 CoaxialVectorElectricField ............................. 23

3.2 Coaxial Contours of Constant Electric Field ................ 24

3.3 Coaxial Contours of Constant Potential .................... 25

3.4 Electric Field Vectors With Hot Filiment .................. 26

3.5 Contours of Constant Electric Field With A Hot Filiment .... 27

3.6 Contours of Constant Potential With A Hot Filiment ........ 28

A.1 Basic Orbitron Schematics ............................... 34

A.2 Schematic Motion of Electron Clumps ..................... 37

A.3 Schematic of Traveling Wave Line 40

A.4 Unspread Dispersion Relationship ........................ 47

A.5 Dispersion Curves With A Spread In Radius ............... 51

A.6 Landau's Dumping Contour .............................. 53

vii

Chapter 1

Historical Notes

A Brief History of the Hot Filament Orbitron MASER

I became involved with the University of Tennessee Plasma Science

Laboratory in the Summer of 1985, when I was hired under a special grant by

the Air Force Office of Scientific Research. This was the first year the summer

fellowship was offered, and it was only available to undergraduates. I was

hired to work on Orbitron MASER's with Dr. Igor Alexeff.

At the time I was hired, the project was coming under cri ticism by

Hughes Research Laboratories. It was their opinion that the device would

never operate in the steady-state and in a hard vacuum. 1.1,1.2 Fred Dyer was, at

the time, working on a high vacuum Orbitron using a filament on the outside

of a rectangular box. This produced spot wire heating, and some very

intermittent R.F. radiation. While we were operating this, I noted that the

emission from the device did not correspond to the experiment being off or on

but rather if the ion gauge tube was being used. This shocked us all, and after

running the tube by itself, with a DC potential across the anode and grid, we

confirmed it was operating as a Orbitron. This tube had a thick spiral grid, a

single wire central anode, and a hot electron source outside the grid.

In the tube described above, the electron source ran in a direction parallel

to the central anode, with enough space between the wires of the grid to allow

free electron flow but fine enough to trap low frequency R.F. From this device,

I decided the easiest method, to inject electrons into the system and trap the

maximum amount of high frequency R.F., was to make a hot filament Orbitron

as described in the paper in the chapter that follows. For this work, I was

1

appointed one of the five 1987 I.E.E.E. Nuclear and Plasma Sciences Society

Graduate Student Fellows.

·The second chapter of this thesis is a reprint of an earlier paper published

by myself in the Transactions on Plasma Science in January of 1987. It is one

of two known journal articles on steady-state high vacuum Orbitron's. The

other is a Physical Review Letter by Burke, Manheimer and Ott at the Naval

Research Laboratories and the subject is covered by them in depth in a book

entitled High Power Microwave Sources. This book is edited by Victor

Granatstein and Igor Alexeff and with the exception of the listed article, most

of the journal literature has been written by myself or Dr. Alexeff. There are

also a few abstracts published in the Bulletin of the American Physical Society

and the IEEE Nuclear and Plasma Science Society proceedings by Dr. Robert

Schumacher at Hughes Research Labs, Malibu CA but nothing of journal

quality.

2

Chapter 2

Steady-State Orbitron Emissions

A bstract of Steady-State Orbitron Emission

The orbitron maser has been operated at a pressure of 2 X 10-6 in the

steady-state. Electrons are supplied to the device by an oxide-coated tungsten

cathode placed inside the cylindrical cavity. The plasma-free emission

corresponded to harmonically related steady-state narrow lines. The

fundamental (lowest frequency) line corresponds to a resonance in the cavity

system, which could be observed with a grid-dip meter.

Introduction

On September 3, 1985, the first steady-state high-vacuum Orbitron

maser~!.l emissions were observed. These emissions were from a device at a

pressure of 5 X 10-6 torr produced by an oil diffusion pump trapped by a liquid­

nitrogen-cooled baffie. The electron feed for this device was provided by an

axial hot cathode in a cylindrical cavity. To demonstrate that no plasma was

present to produce plasma oscillations. as claimed by others, 2.2,2.3 we

monitored the presence of plasma in the cavity of this tube and many of the

subsequent devices with a Langmuir probe. This device has since undergone

many design revisions as the design theory has been better understood. These

changes include multiple anode wires, changing the electron feed system, and

the suppression of low-frequency resonances by excluding large orbit electrons

from the device. This paper will describe these experiments and others using a

gas-filled pulsed Orbitron.

3

Theory

The Orbitron operates by means of a very simple physical mechanism.

Electrons are injected into a coaxial system in which there is a potential well

present. These electrons, because of their angular momentum, orbit the

central wire, and generate radiation as they lose energy and fall closer to the

wire. The physical mechanism by which the Orbitron generates radiation is

known as a negative mass instability. This is the same mechanism by which

gyro tons operate, but important differences lie in the types of electron

confinement used in this device and in the electron kinetic energies involved.

In order to explain these differences and how this device works, one must look

at the basic Orbitron configuration as illustrated in Fig. 2.1(a). The internal

structure of the Orbitron is characterized by a wire radius ro, a wall radius r},

and a difference in potential V between the inner conductor (V wire) and the

outer cavity (V wall). The difference in potentials is such that V wire > V wall.

Electrons injected into this system are electrostatically confined in a potential

well and orbit the central wire in a natural population inversion. This

population inversion is caused by the loss of electrons from the system through

impact on the wire. This system is also naturally negative mass unstable and

so can use electrons with a low kinetic energy.2.1 This is in contrast to the

relativistic energies required by gyrotrons to achieve a negative mass

instability. The gyrotron also requires an electron dumping mechanism

needed to achieve a population inversion while the Orbitron has this as a

natural result of its confinement system. The term negative mass comes from

the fact that, in this system, as the electrons lose energy they fall in orbital

radius and gain orbital angularvelocity.2.4

4

»

ANODE WIRE

GLASS SHIELD

a. Basic Schematic

0-50V

.-.......-_____ ....J - + 1--____ _

'LANGMUIR PROBE

" VOLTMETER

b. Hot Filament Schematic

Orbitron Schematics

Figure 2.1

5

ox IDE COATED TUNGSTEN FILAMENT

0- 5KV

The Orbitron also has an advantage in frequency range over the

gyrotron. For circular electron orbits, the highest angular frequency (W)2.1 an

Orbitron will produce is given by

This only differs from highly elliptical orbits by a factor of ltv'2. So the

frequency is only limited by the highest voltage available and the size of the

center wire.2.5 In a gyrotron the frequency is limited by the magnitude of the

magnetic field that can be produced. The highest possible angular frequency of

a gyrotron is eB/me. A pulsed magnetic field of approximately 24 T is the

strongest magnetic field that has been obtained in this type of device. This

corresponds to a wavelength of about 113 mm.2.6

Steady-State Emissions

On September 3, 1985, the first high-vacuum orbitron emissions were

observed. These emissions have led to a series of proof of principle

experiments. There have been three distinct groups of these experiments, and

these a~e what will be explored.

In the first group of experiments, open cavities between 5 and 10 cm in

length were used with a diameter of between 1 and 2 cm. These cavities

generated a series of harmonics for which the fundamental was determined by

an external cavity cable resonance. Fig. 2.2 shows the first of these

experiments. It was operated at a pressure of 5 X 10.6 torr at a wire potential

6

REFERENCE LEVEL

o

, I

I

1

HP3555A SPECTRUM ANALYZER WITH YIG FILTER

I = 40 MA

0

-10

-2.0

-30

-40

-50

-60

-70

2.Ghz

V - 500 VQ!ts p - 5 X 10 torr r = .075 MM

Q

First Orbitron Spectra

Figure 2.2

7

dB

of 500 V and a current in the center wire of 40 mAo The radius of the center

wire was 3 mils or 0.075 nun.

This cavity and subsequent cavities of this group of experiments were

monitored for the presence of plasma with a Langmuir probe. There were no

indications of a plasma expect for a few microamperes of electron current,

which originated from the hot filament. The next step was to increase the /

number of anode wires from one to two. This was done in the belief that this

would increase the power output and the frequency range.

Fig. 2.3 is a comparison between one and two center wires. The pressure

was at 5 X 10-6 torr and the voltage in the top spectral pictures was L2kV,

while at the bottom it was 800 V. Both had a current of about 34 mA on the

center wire. Some nonharmonic frequencies can be seen in the two-wire

spectral picture. These go away as the current or the voltage is increased and

what appears to be phase locking occurs between the wires.

The number of center wires was increased to four and the power and the

apparent phase locking increased significantly, but more importantly, the

frequency of emission went up sharply. Fig. 2.4 is an example of this. The

input operating voltage to this device was 500 V, and the current through the

center wire was 40 mAo The radius of the central wire was 3 mils or 0.075 mm.

Under these operating conditions the highest circular frequency is 8.3 GHz,

and so this is within a factor of 1.5 of the theoretical value.

In the next experiments, the number of center wires was increased to

seven. It significantly increased the output amplitude, as seen in Fig. 2.5, with

less current required for what appears to be phase locking. In Fig. 2.5 the

center wire voltage was 500 Vat 20 mAo The pressure was 2 X 10-6 torr.

8

REFERENCE LEVEL

REFERENCE LEVEL

o

o

II III

I I

i I I I

I I I i i

i I i I I

I i !

i i

II III I, I I . I ,

I J I I I I I

HP3555A SPECTRUM ANALYZER WITH YIG FILTER

a. One Wire Spectra

I I .• I I~ I I I III. II IlIlhl III Id I II. 1.11

HP3555A SPECTRUM ANALYZER WITH YIG FILTER

h. Two Wire Spectra

Comparison of Spectra

Figure 2.3

9

I 0

-10

- 20

-30

i

I -40 dB

-50 i

I -60

I -70

2Ghz

0

-10

-20

-30

-40 dB

-50

-60

-70

2Ghz

REFERENCE LEVEL

I II I II

I I

I I I II I I I

0

-10

-20

-30

III 11 -40

-50

-60

-70

2.05 6.15Ghz HP3555A SPECTRUM ANALYZER

WITH YIG FILTER

REFERENCE LEVEL = -30dBm

I = 40 MA V = 500 Volts r = .075 MM o

4 Wire Spectra

Figure 2.4

10

dB

n"nr

REFERENCE LEVEL

o

I

HP3555A SPECTRUM ANALYZER WITH YIG FILTER

ro= !. 5 mil or .0375 mm REFERENCE LEVEL = - 30 dBm

v = 500 Volts I = 20 MA

-6 P - 2 X 10 torr

Seven Wire Spectra

Figure 2.5

11

0

-10

-20

-30

-40 dB I -50

-60

-70

2Ghz

In the second group of experiments an attempt was made to control the

fundamental frequency by changing the cavity structure. In order to do this, it

was necessary to reduce the cavity radius so that large low-frequency orbits

would not fit inside the cavity at low to moderate voltages. This was

accomplished by constructing the double cavity system shown in Fig. 2.6.

This device met with limited success. While it was possible to control the

fundamental frequency with a cavity mode, as demonstrated in Fig. 2.7, it

required an accelerating voltage on the secondary cavity and had very small

wire currents.

In the third group of experiments, the configuration was changed back to

the original design with a cavity length of 15 em, but with the cavity radius

reduced. This worked extremely well. The frequencies observed corresponded

to the fundamental TEM modes of the cavity. This was observed on two

receivers, as demonstrated in Figs. 2.8 and 2.9. The power output from this

device was greater than previously observed and in some cases was seen to be -

10 dBm at one frequency, as seen in Fig. 9, with a correction for horn and cable

losses included. This is the power observed impinging on the receiving hom,

and not that radiated over all space. Assuming that the power is radiated

uniformly over free space and knowing the horn area (0.391 m), the radius

from which the data were taken (0.15 m). and the power used by the device (4.2

W), it is possible to estimate the efficiency of the device which is about 0.01

percent, but this is only the efficiency at one frequency. To get the overall

efficiency one must take into account the power radiated at other frequencies.

In these devices, it has been found that a difference in potential of only 40

V is required for noticeable emission to occur with a starting current of about 8

rnA, but this starting current was reduced to the order of 0.5 rnA or less as the

12

« U') ,

'" 0

c (I)

E ..2 1 II-u:

I If '" j' .- ;11 ,

:Ii "

"~I

"

I "

" I , I', ,'I "I " i " 1:1 "

,II II, I,' 'I ,I, "~I 1;1 ~ II c;'"

\ 03: ,II

"

., \

" " 'I' ~O

\ I', 'I, "

II; --0

,I, III "

U c \ :11 1'1

, 'II

(1)--

;Il 1 ~I '"

tiJ!S: , " " I,

" " I

E E u U

:!2 r0

J I

'. I , , 0

f \ " I \ ,'I ',I (I)

f \ I" ',I i5 I

\ 1;1 " :E I

\ 1', , :;1 \ II, 01

I \

I ,I, ,', .~

I \ ::1 II, '0

I \ ,'I :i 1- "

, I ~ I IJ>

~ ,'I .E I .

I ,---1. " I

I I

~ I II I ,

E r E i L> I'-

U U')

Schematic of Double Cavity System

Figure 2.6

13

REFERENCE LEVEL 0

-10

-20

-30 I , -40

! !

-50

-60

-70

o 1.25Ghz HP3554A SPECTRUM ANALYZER

ro= 3mil or .075mm P = I x 10-4TORR

Vcovity = 500V V=2KV I wire = 2ma

Model Control of Orbitron

Figure 2.7

14

dB

REFERENCE LEVEL

o

I; 0

1

I -10

-20 i

-30

I -40 I

i 1 I

i -50

-60

-70

2Ghz HP3555A SPECTRUM ANALYZER

WITH YIG FILTER

V=IKV P= 3)(IO·~TORR REFERENCE LEVEL = - 20dBm ro= 3mil or .075mm

dB

Output of Hewlett Packard Spectrum Analyzer

Figure 2.8

15

.3

I i

I

i I ~ I •

J'\

INTEGRA SPECTRUM ANALYZER

V = 1.3KV P = I xIO-STORR ro = 3mil or .075mm

20

10

0

- 10

- 20 dBm

-30

- 40

- 50 i

2.3Ghz

Output of Integra Spectrum Analyzer

Figure 2.9

16

voltage was raised to 300 V or higher. This start current appears to be highly

dependent upon the Q of the resonant system.2.5 To eliminate the possibility of

signal imaging and spurious responses in our spectral analysis, a YIG-tuned

filter or other filter designed to eliminate aliasing was attached to the

spectrum analyzers when necessary.

Pulsed Emissions

The pulsed Orbitron appears to work by the same mechanism as the

steady-state Orbitron except that electrons are supplied by a cold cathode

discharge which ionizes any gas in the tube. Ions are collected by the cathode

wall and electrons from the gas are trapped in orbit around the positive central

wire and cause the Orbitron effect. The gas pressure in these tubes ranges

from 50 to 0.1 mtorr, depending on the tube.

While the operating mechanism of the steady-state version of this tube

appears to be well understood, there are still some questions about the

operating mechanism of the pulsed device.2.2. 2.3 To dispel these doubts, there

has been a series of experimental attempts to ensure that the operating

mechanism is properly understood.

The experimental work included probing the working tube with a

microwave signal to determine the plasma cut-off frequency. The reason for

this was that one theory for the emission mechanism predicts a emission

frequency of 2wp, where wp is the electron plasma frequency. A microwave

beam was used to probe the plasma in the Orbitron MASER and find the value

for wp. At the same time, the emission was monitored. We started probing the

tube with a beam frequency of,28 GHz and found that the frequency was cut

off. The frequency was then raised until the beam showed no signs of cut off.

17

The uninterrupted signal occurred at 36 GHz. This corresponds to an electron

number density of 1.44x 1020 electrons/m3• We also observed the frequency

emission during the probing of this device using an indium antimonide crystal

as a detector with a set of special high.pass filters and supplementary mesh

filters to determine the band of output. This method demonstrated that the

frequency of output was at least 133 GHz. This gives a minimum emitted to

electron plasma frequency ratio of3.69 with a maximum error of + 35 percent.

We have also observed ratios of 5 to 1 or greater and ratios of less than 3.5.

This suggests that plasma oscillations are not the fundamental emission

process. Also tried was lining 3/4 of the cathode tube with a high·voltage

insulating material, as shown schematically in Fig. 2.10, to eliminate two­

beam plasma interactions which were suggested as the fundamental emission

mechanism.2.2,2.3 In this experiment, it was found that with all other

conditions the same, there is little to no change in the emission characteristics.

In the pulsed gas-filled tube, a much higher voltage can be used, as well

as a thinner wire. The reason for this is that, in the steady·state, the central

anode of the Orbitron can only lose heat by radiation. If the operating

conditions for any size wire reach a certain value, the thermal loading causes

the wire to melt. For the pulsed tube, this is not a factor, since the thermal

inertia of the wire prevents heat buildup. In this mode of operation, detection

of submillimeter wavelengths is routine, and we have obtained radiation at 1

THz (0.3 mm). The peak microwave power output is aout 1.5 W at 1 THz and

about 50 W at frequencies around 100 GHz. Efficiencies range from 1 percent

at 3.5 GHz to 0.001 percent at 1 THz.2.7 While the power may seem low, it must

18

f

MYLAR

".---- - ---'I ' ~ \

\ ---1- -------,-- ---r----I~-

I

MYLAR

Schematic of Lined Orbitron

Figure 2.10

19

r f

be remembered that many of these frequencies are emitted simultaneously

from one tube with the same power levels.

Conclusion

It has been demonstrated that the steady-state Orbitron is an operational

device, and that it appears to operate in the way our assumptions predict.

There is evidence that the pulsed Orbitron also works according to the same

theory. Both the pulsed and steady-state Orbitrons offer an alternative to the

main-line concepts now in use for high frequency microwave production. In

these experiments, many of the concepts of the steady-state orbitron have been

introduced and shown to hold true. While the steady-state frequency limit

demonstrated is not yet that of the pulsed Orbitron, it is believed that this can

be improved to higher frequency ranges.

20

r

Chapter 3

Static Electric Field Calculations

Static Electric Field Simulation Of The Orbitron MASER

One assumption made in previous chapters, is that the radial electric

field is minimally affected by the potential of the hot filament. It is posible to

check this by computer simulation. For the case of a pure coaxial system,

assuming it to be infinitely long, the radial electric field is known to be that

given in equation #3, but there is no known simple analytic solution to the

radial electric field of a coaxial line with a secondary wire of known potential

inserted into the system.

It would be of great value to be able to model these situations using a

finite element analysis code to see how much of a deviation the added wires

causes the radial electric field to change from the single wire analytic case. To

do this analysis we will use a general purpose finite element analysis

electromagnetics code from the MacN eal-Schwendler Corporation called

M.S.C. Maggie3.1, which is operated on an IBM Personal Systems 2, Model50z.

This code is two dimensional and gives a numeric solution to several

electromagnetic properties of a system. In this case we will be solving for the

static vector electric field and static electric potential at points in the system.

The input to the code is made by a graphical sketch made on the

computer screen. This sketch is to scale and in terms of absolute units.

Boundary conditions, such as potentials, are then added to surfaces and

material properties, of each area, are listed. From this information, the code

then generates a finite element mesh matrix, which it uses to solve for various

electrical and magnetic properties of the situation. In this case we will be

21

solving for the static electric field properties. It can then outputs this

infonnation in tabular or graphical fonn.

In all simulations presented in this chapter, the outer wall is grounded,

the anode wires are at 1,000 volts, and the hot filament is at 5 volts. The cavity

radius is 4 millimeters and the wire radii are all .05 millimeters "4 mills".

Simulation Outputs:

The first simulation presented, figures 3.1 - 3.3, is the simulation of a

pure coaxial cavity. The simulation output consists of three parts. The first

part, figure 3.1, shows the vector electric field of a simple coaxial device. It can

be seen that all of the field lines point radially outward from the central anode.

The second part, figures 3.2, shows a contour plot of constant electric field. The

third part, figure 3.3, is a contour of constant electrical potential. These

contours are circular with the center being the anode wire.

The second simulation presented, figures 3.4 - 3.5, is the simulation of a

coaxial cavity with a hot cathode added. This center of this wire was

positioned at radius of 3 millimeters. Figure 3.4 shows the vector electric field

with the hot cathode added. It can be seen that the field lines are concentrated

near the hot cathode but in general are pointed radially outward. This is

especially evident near the anode wire. Figure 3.5 is a contour plot of constant

electric field magnitude. It is circular for the most part with the major

deviation being near the hot cathode, but our major assumption still holds that

the electric field is minimally affected by the hot cathode in the region near the

central anode. Figure 3.6 shows the plot of constant potential. These contours

are also circular, as in the first case, except there is a small deviation near the

hot filament.

22

r- I:'- r- I:'- c..o c:.D c..o c.o c..o c.o c:.D c.o c.o 0:. ($) t.::IC ($) t.::IC as:) t.::IC as:) ($) as:) ($) t.::IC ($) t.::IC (5)

+ + + + + + + + + + + -'- + I W ~ W ~ ~ ~ ~ w J;;W w ~ ~ ~ w IJ") c:.D r- eo c..o r- CT'I ($) N .. L/":) r- eo .... .... t.::IC era eo CT'I t.::IC ..,...; CI") ~ IJ") c:.D r- eo r-.. cw:! .... t.::IC r- I:'- (,Q IJ") ~ CI") N .... t.::IC .. ..... ..,...; ..... ..,...; CT'I eo r- c.o It:I .. t"':) N ..,...; M . . . . . . IS) t.::IC ($) t.::IC as:) t.::IC as:) ($) as:) ($) t.::IC ($) t.::IC (5)

1i:~~!1[~; _1:;lijZl\" It~~

Coaxial Vector Electric }t'ield

Figure 3.1

23

N .r-

n o ~ >< .... ~ - n o =

s- c: Io:

rj ~

....

0 tro

. "'+

:0

c: n

ci 0

t.:l

=

•

ttl

a,...,

..... ~ =

.....

t:r.l

;"

~ .....

'1 .... ~ Io:rj .... (!

) Ci:

8. 1

415E

+87

8. 1

315E

+B?

8. 1

214E

+87

B.1

114E

+B?

8. 1

813E

+87

B.91

Z6E+

B6

8.B

121E

+8G

I

8.71

17E

+86

~t.;~ i:~;

8. G

l12E

+8G

8.

518

7E+8

6 B.

4HIZ

E+B6

10. 3

098E

+86

, B

I 2B

93E+

B6

8. 1

888E

+86

\

~~~--"------

///

. ~

\~ ~ ~--

-

)

CSl CI:)

+ ~

• '11""'1 N N ('I'l "'1:1"1 Ir.) U") ~ t"'- CX) co N CI:) CI:) . . . . . . . . . 0"1 CI:) CI:) ~ (D 0"1 N In co .... "I:t1 t"'- aw C,'I') CI:) CI:) N 'o:J4 <.0 0"1 '11""'1 t:'I':) (D co CI:) ~ U") <.0 CI:) .... 0"1 co t"- ~ <.0 Ir.) 'o:J4 t:'I':) C,'I') N .... t"-

CI:) ......

Coaxial Contours of Constant Potential

Figure 3.3

25

"'" .... N

~

Ci'

t: ~ ~

~

tr:j -(T> (') s: (') "'" .....

(T

> -Q. <:

(T>

(') s- ~ ~ _. .....

::r ::r: 0 ..... "'" ..... -.... 9 (T

> 1::1 .....

8.1Z

94E

+87

I'.' 8.119

4E+8

7 "..

8.18

95E

+87

8. 9

95ZE

+86

8. 9

951E

+86

8. 7

96ZE

+86

18.6

967E

+86

~

8. 5

9?lE

+86

8. 4

916E

+86

8. 3

981E

+86

I'~;:I!

~~~ 11

I·29

86E

+86

8.

199

8E+8

6 8.

995

ZE+8

5 8.

?84

5E-l

l

\ ,~-,.~tlrf,-t~T!tf-t. j.

. I

~~.~

.... f~·~

" ,

/~-.

I.

.... y.' ~(

~\ ,I .

r ~' ~

\; r, f

'I.

f. "''

't. oi

.

. ,~~\

~,,\

~\lf

,r~(

~~W1

M71:

Ac

",t~ \

.\ '\

,~,

~f t ,

..I~

'~ ~

t

,~I/

!,.1

i ./

\ .. f",

'J..

.~. \"

\,

\ • ~.

t t Im

~ " •

f f '"

~'Ill

I'

" ....

"" ,\

"

~,l~

f

• ,I

f'''

41

·'.

I'"

,(

" ''''" \

'\ \'

\~ ,

1ft';

If'

)' ~~

l II

l

•.• f

/'

.' f

.... "

,.\ \

\ '\

~ , ~'

t ."f

, J ,'.~

,

I /

/ /'

.. J

' ~

,,' ......

, '"

, 'I

' ~

I ..

,4

'.

..

l ,,~

~ \

:i.. ""...

' ..

....

" ~

-. "

. 'l"

I " .' ;'

... 1'1 ,.;

.«..

.' ,.

#

.« ,'.

"'"

.'

'\

"',,

,.la

...... "

i"

'>I;I

.,~

,,""

~i

""""

.... ,.,

'\

'Ii ,\

I l!

flf

/'

~.".

t-....

.......

~,,'\. \

\ \

~ '.fl t

~.,

....'.;'

. .'

.. ' ,.:

# • ..

,. ,-

! ""

'.

' ...

....

• \\

" I

r 1

1'"

If,

'"

,.f!

' t.~

''''_'''~''''

\\

,r~"'!"'/r''''7''

"If'

•

......

'"1

>,

...

...., '\

"

f J'..

!If

.rtf

<i

t' ..

T'

•• .r

~

1 "'''

' .......

'h

..,.

. ..

....

....

'-

t .J<

' •• '"

."

• ~

•• r

.rr-

... _.

...,~

""'."

'?-...

. """

lL\..

....,.

~.

'·M.#·~ .

. .,..r .

. """

.,_ ....

. ..F'

..."

Jtp~

...

I .....

.. 1I

f!woI

,. "'1

iV<"_

..

....

....

... ~ "\

"'

, ~ i

,~.~! \~

.K

..... 'f"

" _

fIK

-~

____

-fill"

....

..

-' ,~..

.#

..

...

........

... -..

-~"-

....

...

-11'

-...

.....

....

...

<"~: ~

l p '

:? ..

. ~..

......

...

....

---

..".JJ

I&

.. -lI

P'

-..,

.

... _

...

....

. "9

;""

....

.. _

__

..........

. .....

.. ,

'.' .

... ~.~:".

~'l" ~ _

__

_ -"'

....

... --'"

-~-

~

....-'

+

_ +

-..

.-.. IF

"" ............... """"

'"

"'!:.~

~r::.,

,~.,

.~~~

~" -e

a ..

....

....

-.

.. .

...

-""'

..

....

....

....

....

...

.-JlW

" 1

>. '

or-

,..-

.r-

'lI'"

' ..

....

.,

1).

' ......

. I"

;" ".

, '..

"'.....~,..

-11

' .,

..

-..,

. _

_".

~

... ~

.. ~".

"......

. .....

. ...

... .

...-

.,.-

,...,.

.,.

. . .r

.r "

."~

'j "\

""

" "" £

. ..... "" _

__

_

.,..

. ..

...

I' '.-

-.,.

.-...

. .r.....,

, ..,

.. .

" ".'

,.

r-'

,,4

r'..

,.,

,'" ,I

i~.}

I

\ \

'.....

......

......

......

......

""

" ...

.. _

_

l'

.• ,W

,-.....

It'

"r .....

" ,.'J

r

'" '.

-.,

..

....

....

....

..

-..

. ..

. .,

..-

..,,,,

, ",

,'

....

ti

l.,

It"

\ \\

,....

-.,.

'"

....

. •

r 11/

>" .. ' ,.

.-"

,,> .. , f

l·.r'

,r ~

+ t.

""...

......

....."

,,....

.. --.;.'

,w

' ".

, ..

,.,

,-

If'

I ~I

i J

l ~\

~ \.

'\.

, ...

... ""'"

"'I

. ".

l .C

· (/

fr'

.. I

f ,J

.• ' ,i

" tic

;;

'"

I' ~

~ • '\

1\

'.

.....

...

...,,.

\.-

,,.

•• p

",

, " ~l)!

I'·

\;

\.\

\..

......

"

~ ... '

Jt"

if'-

,.

Ii .. "

-'

'fi"

.,

t. t.

\.".

. ~ "

...... ,l'

\.-

'\.'"

,. . .'"

/,,'.1~';'

~ ~

~ \

\\'" \\

\\..

'\ "\

" ~~

•. <,

.. ,.".r~J

\ ,-'

""

"-....

. ,. "

,.'1

-'

'r

" \ \~,

'-...

7'

~ .. "·"''':r/l.f~+'1\

\'

\ ..

-. ) ..

'

~"

".'"

.'

' ,

,~ I

I I

I ~ \

\ \

\ \

'"

'l-..'

,.,

,,'

"'''r

\'

~

'-'

" ,'/

I

,.' i

i +

t ~

+ ~

\ \

\, ) .

...... ~

,P

;J't

"\.~

,'"

..• '~ ..

..,.

:.

r;

I~L~l

.1.,

---.

..._

;,

T

I r

\~." .

. -~;

. -..t ~

J, ,j.

J-.~"" .,

~_.,11

~"

"'-"

!I

r

N

-.J

('1 o S- O

C ;J o ~

('1 o =

til .... $l) =

.... "%

j ~

....

. ('I

) ctQ

. ("

;I

C

~

'"S

....

('I

) ("

;I

c,..,

"%

j .

-. c:.n

('I

) s: ~ ~ $l

) ::c o .... "%j ..... -..... e ('I

) :; ....

'!'~ B

.129

4E+8

7 >

,l r.

8.11

94E

+87

~

rJJ

B.1

895E

+87

8. 9

952E

+86

8.B

957E

+8G

8.

796

2E+8

6 I 8

.696

7E+

86

'~~

8. 5

971E

+06

8. 4

97G

E+8G

'1l

)'1l,

, ,~1

e.39

81E

+86

8. 2

986E

+86

Ie. 1

998E

+86

i 8.

9952

E+8

5 e.

784

5E-1

1

-

r"---~~-, _

_

-------

----

. ~

.~~

...... ,,-

." iI'-'~

""

/ . .--

'~'"

"-,

1/

,_.-~_ ....

... _.~.

_f·':;

",~-·-

-........ -

-.--_

",-,

""'\

.

• ",_,~

'... ,,\

V

"'",

~"""""_..

,'_

I.

.....

.. ' \

I c.,

. /'

. \

," .....

. /

I ~

.J

\ •

'\

-' ,

\ .. i.····

~"-...

. \

.r" .....

~ ... ~~'

.........

\\

.' .,

.' '"

\ /

.... -'"

\ /

/ ~\

~ I,.

\ /

,t'''''

'\

{ f

( 1

\ i

I )

I i

I ./

;

./

)' .'

~

I ~?

I .

, ~

I ~\

,~ .. , _

__

__

",,--~-~.,.

J'/

\\

/ .. / ..

\~\

.. /'

\~

.•.•. /

..•. /

~~

~

"<,;"'::c~ ...

:..-=

"'/

,

CD CD + ~

..-4 N N ('t:J "'112"1 '" Lt') Ii.O ro- \X) eo N CD CD . . . . . 0 r::1"I CD CD M U) r::1"I N U'j ex:) .-4 -.:t' ro- a;:) C'I".I CD CD N ... c.o 0"1 ..-4 ('t:J U) ex:) CD M Lt') c.o CD ..-I Cf't eo t"- (,f.) c.o '" ... ('t:J C'I".I N .... t"- o

a:;J

Contours of Constant Potential with a Hot Filament.

Figure 3.6

23

, I

Conclusion From Electrostatic Simulation

The Orbitron MASER is a coaxial device that can be used to generate

high frequency radiation, but it requires a source of electrons to roduce this

radiation. One possible source of electrons is a hot, oxide-coated filament

running parallel to the central anode. Using the MSC Maggie Code3 .1, I have

built an electrostatic model of the vector electric fields and the potential inside

the cavity. These simulations were done for the pure coaxial case and for the

case involving a hot filament near the outer walL These simulations indicate

that the hot filament does not affect the electric field near the central anode

and minimally affects the field near the outer wall, except in the region near

the filament. In this region there can be field reversal, and the contours of

constant potential are distorted.

Conclusions About The Orbitron MASER

In the previous chapters I have explored, in theory, experiment, and

simulation, the Steady-State Orbitron MASER. From these explorations, I

have come to the following conclusions.

1. The Steady-State Orbitron MASER works and follows the theory

presented in the Appendix and Chapter 2.

2. Smaller diameter cavities give better frequency control.

3. l.\1.ultiple anodes boost the frequency and power output.

4. The hot cathode has a negligible effect in the regions near the central

wire.

From these conclusions, I can safely say that the Steady-State Orbitron

MASER, if further design is done, can be used as a practical source of low

power high frequency radiation.

29

1 I

REFERENCES

30

.,

J

1

1

References

1.1 R. W. Schumacher and R. J. Harvey, in Proc. 1984 IEEE Int. Conf.

Plasma Sci. (IEEE Pub. 84ch 1959-8), page 109.

1.2 R. W. Schumacher and R. J. Harvey, Bull Amer. Phys. Soc., 29, page

1179, October 1984.

2.1 1. AlexefTand F. Dyer, Phys. Rev. Lett., 45, page 351,1980.

1. AlexefT, IEEE Trans. Plasma ScL, PS-12, page 2880,1984.

1. AlexefT, Phys. Fluids, 28, page 1990, June 1985.

2.2 D. Chernin and Y. Y. Lau, "Stability of Laminar Electron Layers," Phys.

Fluids, 28, September 1984.

2.3 J. M. Burke, W. M. Manheimer, and E. Ott, Phys. Rev. Lett., 56, page

2625, June 16, 1986.

2.4 G. Nusinovich (Inst. Appl. Phys., Gorky, USSR), private correspondence,

December 1985.

2.5 1. AlexefT, F. Dyer, and W. Nakonieczny, Int. J. Infrared Millimeter

Waves, vol. 6, no. 7,p.481,1985.

3.1 The MacNeal-Schwendler Corporation, 815 colorado Boulevard Los

Angeles, CA 90041-1777.

31

Appendices:

Basic Theory of the Orbitron MASER

32

J

I

I

1

Introduction to the Theory of the Orbitron MASER

The Orbitron MASER is a device, which can be used as a source of high

frequency radiation. In its most basic physical form, it consists of a cylindrical

metal outer shell, which is cylindrical in form, and a thin metal wire, which is

positioned along the axis of the metal shell. This device is shown in figure

A.la. It can be described by an inner radius (ro), which is the radius of the

inner wire, and outer radius (q), which is the radius of the outer shell.

It is possible to produce high frequency rf radiation from such a simple

device by applying a high potential between the inner wire, and outer shell.

This must be done in such a way that the wire is biased positive with respect to

the outer shell. The potential applied between the two surfaces creates a

strong radial electric field, which will trap negatively charged particles

(electrons) in orbit around the positively biased wire. In this device the outer

shell acts as a microwave cavity and the entire device resembles a coaxial line.

These orbiting particles are negative mass unstable and couple to cavity modes

within the shell-wire system. This mode coupling causes phase bunching and

power ·radiation from the system. The frequency of this radiation is inversely

proportional to the radius of the particle orbit, and the highest frequency at

any particular voltage is proportional to the radi us of the central wireA .1•

Dynamics of Orbital Electrons

The Orbitron MASER utilizes electrons trapped In orbit around a

positive central electrode to produce high frequency rf radiation. It

accomplishes this trapping by means of a logarithmic electrostatic potential

well, which is created when a high positive potential is applied to the central

anode wire. These electrons are strongly attracted to this central anodeby the

33

) ) )

1 (

\

1

a. Basic Orhitron Schematic

0-50V

)-- ------- - - - --­-::;jr----+-- - - - --- - - - - - - - - -1---.--

ANODE/ WIRE

OXIDE COATED TUNGSTEN FILAMENT

o -5KV

h. Typical Hot Wire Orhitron

Basic Orhitron Schematics

Figure A.I

34

J

1

\ i

1

strong electric field but miss impacting on the wire, because of any initial

angular velocity these particles acquire before entering the region of strong

electric' field. This gives rise to a cloud of electrons orbi ting the central wire.

It is possible to calculate the frequency of these particle orbits (cu) for the

case of circular orbits. This is accomplished by using a simple force balance,

where the outward centrifugal force of the electron is balanced by the inward

attractive force of the positive wire. The outward force on the particle can be

expressed as

(1)

where me is the mass of an electron, v is the azimuthal velocity of the particle

and r is the particles orbital radius. The inward force on the electron caused by

the potential applied between the cavity and wire can be expressed as

F. = ZeE I (2)

where Z is the sign and charge state of the particle, e is the charge of a electron

and E is the value of the electric field. These two terms must balance, if one

wishes the particles to orbi tA.2•

In the case of a coaxial line, it is possible to arrive at a analytic formula

for the electric field at any point between the wire and the cavity wall. This

electric field can be expressed as

(3)

35

where V is the potential between the inner and outer conductor, r is the radius

from the central axis, and In represents the natural logarithm of the term

enclosed, which is the ratio of the outer to the inner radius.A.3•

We can insert equation 3 into equation 2, thus the inward force on the

electron becomes

(4)

This inward force is balanced by the outward force in equation 1, we can allow

these two forces to balance each other, such that

(5)

This is the force balance equation for a stable circular orbit around a positive

central wire placed inside a cylinder. It is possible to solve this equation for

the frequency of particle rotation.

Let us take equation 5 and multiply both sides by 1/r and at the same

time divide both sides by the mass of an electron, me. This yields

(6)

where v 2/r is the square of the radian rotational frequency w. If we take the

square root of both sides we find that

(7)

36

) , ,

1

0

0

0

0 0

0

0

~~ Angular Motion I)

'0

0 0

0

I. Repulsive Force Between Charge Clumps

2. Change In Orbital Path

Schematic Motion of Electron Clumps

Figure A.2

37

)

Thus the frequency of rotation, in a stable orbi t is inversely proportional to the

radius of the orbit and proportional to the square root of the voltage. This is

the stable orbital frequency of a single particle, and it is the frequency of

radiation for a multiple particle system.A.1•

Negative Mass and the SelfInstability

If we investigate equation 7, we find that it gives rise to stable orbits, ifit

is used for a system with no collective interactions between particles or for a

single charged particle system. The Orbitron MASER utilizes many charged

particles to produce R.F. radiation. This leads to a situation where the

equations derived are valid, but the interaction between the particles

themselves and the electromagnetic field of the cavity cause the orbits to

shrink, the particles to emit radiation, and the particles to phase bunch.

Let us look at a ring of charge clumps in orbit around a positive wire, as

shown in figure A.2. On the average, these clumps have the same net charge,

but due to statistical fluctuations, one of these clumps can contain more

charge than the others, so it exerts a net force on the clumps in front of and

behind it in the orbiting ring. This causes the clump in front to gain angular

energy, and in order to absorb this energy, it moves outward in orbital radius

and slows down in angular velocity. Thus, that it moves back in angular phase

such that it overrides the pushing clump and forms a larger clump.

Counterwise the rear clump loses angular energy, drops in radius, speeds up in

angular velocity and moves forward in phase space such that it also overrides

the pushing bunch and forms a still larger clump. It seems strange at first,

that particles losing energy should speed up and particles gaining energy

should slow down but in rotating systems, where the angular velocity is not

38

constant, this is possible. Systems in which this occur are called negative mass

unstableA.2• A.4, A.S, A.6. This is the fundamental process which causes particle

bunching in the orbitron MASER, and the energy contained in these bunches

goes as the number of particles sq uaredA.1, A.4, A.5, A.6.

This, under normal circumstances, would create a continuum of rising

and falling particles, which would emit and absorb radiation at a broad band of

frequencies but these particles also interact with the resonances of the outer

cavity. This interaction causes the particles to bunch at specific orbital radii

and emit at these resonant frequencies or at harmonics of these frequenciesA.3 •

The following three sections were derived by Professor Igor Alexeffin 1985A.6 •

Theory For an Electron Ring

Up to this point we have been examining systems that contain very few

charged particles. Now let us look at a circular ring of electrons. This ring has

a spread in radius r, so that the ring has a thickness and is two dimensional.

This spread in orbital radius will lead to a spread in frequency, the temporal

gTowth rate, and saturation of the device.

One important part of this device is the coaxial cavity resonator. It is

possible to make a simplified model of the resonator, by using a forward wave

line, as shown in figure A.3. We assume that the line supports the dc electric

field that contains the electron ring, but interacts with the ac fields produced.

The justification for use of a line instead of a cavity resonator is that an

oscillating rJ. electric field in a cavity resonator can be replaced by the sum of

clockwise and counterclockwise rotating electric fields. Only the electric field

rotating in the same direction as an electron interacts strongly with it. As

39

Schematic of Traveling Wave Line

Figure A.3

40

required for cylindrical geometry, distances around the line are measured in

radians.

Consider an electron beam element contained inside an element of the

line that is ~'a" m2 in area by il8 radius long. A value n of the electron beam

density (per m2 per radian) attracts and confines an element of charge

comprising -Zeanil8 C. liberating an equal but opposite charge to flow to the

circuit elements. Obviously, the current from this beam segment is

The other currents in the circuit are easily represented:

dV(e) i = -c--4 dt

(8)

(9)

(10)

(11)

If an currents are represented as derivatives, they can be added according to

Kirchhofrs current law:

di1 di2 di3 di4 - + - + - + - =0, dt dt dt dt

(12)

41

( or

(13)

Next, we assume that the capacitances and inductances are derived in

terms of units per radian, as follows:

c = C 6.8, o

(14)

L = L 6.8. (15) o

By letting ile - 0, a differential equation results:

(16)

This is the first of three equations required.

The second equation refers to the change of orbital frequency with

energy. In circular motion in a logarithmic potential well, we have

-ZeV o -----

r (17)

We note that the linear velocity v is independent of the depth of the particle in

the well,

42

(18)

Also, the kinetic energy of the particle is independent of the particle depth in

the well,

2 mv ZeV () -=----

2 21n(r:!r/ (19)

Consequently, any work extracted from the orbital motion of the particle must

come from the particle's descent into the well. IfEs is an electric field tangent

to the electron orbit (VIm), we obtain

ZeV o dr

(20)

But w:::: voir, so

dw v 0 dr - =-dt / dt (21)

Thus we derive an equation relating the rate of change of angular frequency to

an aximuthal electric field,

(22)

43

1 (

The term dw/dt is the actual change of angular frequency of a given particle,

aw/at is the observed change of angular frequency in a fixed region of

observation, and w(aw/a8) is a correction term to compensate for faster particles

replacing slower particles in the region of observation which gives a false

acceleration. This is the second of three equations needed to solve the problem.

The third equation required is simply the conservation of particles

moving in a ring,

an aw an - + n - + W - = O. at ce de

(23)

Here, iln/at is the change of electron density in the region of observation,

n(aw/a8) is the loss of electrons caused by electrons leaving the region at higher

velocity than those entering, and wean/de) is the loss caused by a larger density

leaving the region than entering. The reason that three equations are

required is that the above three equations have three free variables, n, w, and

V. The electric field Eo is derived from V as follows:

1 av E = - --e r de' (24)

The above three equations in n, w, and V are linearized and reduced to

algebraic expressions by the following expressions:

i!lS - w2tl W = Wo + wle , (25)

44

joe - w2tl

n == no + nle . (26)

(27)

Here, wo is the orginal frequency given in equation 7, (.)1 is a parameter, W2 is

the new orbital frequency caused by the electron ring-line interaction, and 1 is

a harmonic number. The electron density n is composed of a large constant

density no and a small fluctuating density nt. Concerning the azimuthal

potential V, we assume no dc potential but only a small fluctuation VI, as any

dc potential is shorted out by the inductors.

When the above expressions are placed into our basic equations, which

are then linearized, we obtain

(28)

(29)

(30)

Using these three equations to eliminate nl, (.)1. and VI. we obtain our final

equation,

45

(31)

The above equation resembles that for a two-stream interaction, with one

major difference-the second term is always negative. If Z is negative for an

electron, Vo must be positive to confine it, and conversely. This results in the

negative-mass characteristic of the system, in which an angular electric field

that decelerates an electron increases its angular velocity.

A plot of the left-hand and right-hand sides (figure AA) reveals a quite

different graph than that for the conventional beam-plasma interaction. Note

that the possibly unstable roots lie above the harmonics of the electron orbital

frequency, lwo, where I is the integer harmonic.

A calculation of the system gain may be made by noting that the unstable

roots lie near lwo, or

t.w = w - Iw 2 0' (32)

where 8.w is small compared to W2 and lwo.

The fourth-order equation, Eq. (31), may be written in terms of powers of

8.w and studied as successive approximations, as shown below. Let

12 --=A LC '

o 0

Zea12 (,} n ___ 0_0_ =B.

Co V ofln(rirl)

46

(33)

(34)

' ..

----------------~------------~~--------W7 o two

Unspread Dispersion Relationship

f<'igure AA

47

Then our equation is

If we let ~w = W2 - lwo. then we get

This equation is rationalized as follows:

2 2 + 6.w(2Blw) + B1 w . () 0

(35)

(36)

(37)

As ~w is small relative to W2 and lwo. the higher power terms are

smaller, and the equation can be solved by successive approximations. as

shown below:

(38)

This violates our assumption [Eq. (32)] that ~w < lwo. and so is not a valid

solution.

Next we try the second-order case,

2 2 2 2 2 - 81 w 10 w - A + B) = 6.w . o ()

(39)

48

/

This approximation yields a small value for 6(1) because one never gets enough

beam density, so B is small. As B < A, we can drop it from the denominator to

obtain.

22222 - BI w /0 w - A) = 6.w . o 0

(40)

The above is complex and large if A<F(1)o2. Thus, maximum growth occurs

when

A (41)

or when the electron electrostatic gyrofrequency is slightly larger than

(l~Co)ll2. We note that the units of l~Co correspond to rad2/sec2 , which is

what is required in cylindrical geometry.

(42)

or

(43)

or

49

-1

+i ~31 v'3

2 (44)

The absolute value sign in Eq. (44) removes confusion caused by the quantity

inside always being negative. The results correspond to one root being real

and slightly below lwo. and two being complex, and slightly above Iwo. in good

agreement with figure A.4.

The fourth-order approximation need not be considered. as figure A.4

shows that the fourth root is so far from 1wo that the approximation that aw is

small is not valid. One can verify that these approximations are properly

ordered in magnitude by substituting the above solution back into the basic

equation [Eq. (34)]. One then finds that the zero-order and third-order terms

are larger than the first. second, and fourth-order terms. There are no higher­

order terms.

Theory For An Electron Spread In Radius (Vlasov Theory)

We next extend the formalism to a distribution in radius of the electrons.

As an insight into the process, let us consider two electron rings closely spaced

in radius and of equal density. The orbit frequencies are lwo and lWl. A graph

of the dispersion function is shown in figure A.5. We find two sets of unstable

roots. One set lies above lwo Iw} (above the distribution function) and

corresponds to a pair of roots depending on the total density of the two beams.

The second pair lies between the two beams and corresponds to waves

occurring inside the distribution function.

50

' ...

--------------~o----------~--~-------- W L (WO [W,

Dispersion Curves With a Spread in Radius

Figure A.5

51

We can proceed to a continuous distribution function by the superposition

of many beams. We rewrite the dispersion function as follows. Let

D

so

12 A= -­

Le' o 0

Let the density of a given beam be written as

n = N f(61 )il61 . o 0 0 0

A superposition of many beams leads to the integral below:

1.

Let W2 - lwo = LlW and one obtains

~ __ DN f [(612 -- il61)2llr f[ (612 -- il61)1l) (-- dilwll) = l. 61

22

0 (il61)2

52

(45)

(46)

(47)

(48)

(49)

Wz -=========~o~======~r-r=======~~wo

Landau's Damping Contour

Figure A.S

53

The above integral may be evaluated by expansion in a Taylor series;

( W2 - .:lw \ ( W2 ) af(w/l) ( .:lw )

f I )=1 I" + ~ - -I + .... (50)

The constant term in Eq. (50) is nonzero. The integral containing it does

not diverge to infinity because aUl is complex and does not go to zero for

integration along the axis. This term yields the roots corresponding to the pair

lying above IUlo and lUll in figure A.5.

The first derivative term in Eq. (50) corresponds to the solution ofV1asov

as modified by Landau. Because of causality, one assumes no infinite solution

at t = - 00, This means that the root is assumed to be above the x axis.

Following Landau, one goes halfway around the pole, as shown in figure A.6A.S,

The result is as follows:

f [«,)2 - .:l('»)l1f f' [ (,)2 - A('»)/ll( - .:lUll!) (- dAwI1)

(Aw)2

(51)

We assume that the main part of the real dispersion function comes from

the resonator. This occurs because A>DNoUlo2, since the beam density No is

always smalL We obtain

(52)

This can be rewritten as follows:

54

(53)

As the complex term can be considered small, we replace tU2 by ltUo. to get

A

(54)

This can be rewritten as shown:

A

(55)

We leave f' in terms of tU2I1 to emphasize that the distribution function can

interact with harmonics of the electron orbit frequency.

We expand the right-and left-hand sides, keeping only the first two terms

of each:

= w2 + 2iw w. + .... (56) r r I

Next, equate real and imaginary parts:

1 A = w2 or + -- = w

r - VL C r' (57) o 0

55

or

2

(Zeal \ ( (2) n 2 2

wNwI' --r C V nn(r ir )) 0 0 . 1 )2

o 0 t

or, since Wr ;:::: W2 ::::: lwo,

2w w., r l

(58)

(59)

(60)

The quantity in parentheses is always negative, as a negative Z requires Vo be

positive for electron confinement. Consequently, if aflaw is positive at (wvl), Wi

is negative, and the wave damps. This is understandable in an energy model,

as the deeper an electron lies in the potential well, the higher its frequency,

Le., the lowest-energy states have the highest frequency.

Saturation Of Growth And Physical Limits To Harmonic Emission

The saturation of the growth for the various I modes is of interest,

because it gives some insight as to the relative power output at the various

harmonics. For the ring model, this is done by noting that as work is extracted

from the electrons, they drift inward, change their frequency of rotation, and

move away from the resonance condition which yields high gain. A change of

gain of a factor of 2 is considered as having moved "off resonance". Such a

56

treatment has been used in the study of the saturtion of an electron beam­

plasma interactionA.9 and the study of the calutron isotope separator.A.lO

To compute the frequency shift, we insert a aw of half the value of that

given by Eq. (44) into Eq. (40), and solve for the frequency shift as shown

below:

(61)

Since we are no longer at resonance, the cubic term may be neglected, as may

be shown by inserting aw/2 back into Eq. (38).

A little algebra reveals the following result:

I Zean I U3 1 fV3 0 2 2

- 2 C V In(r I rl) Wo = Wo - L C == 2w}jw.

o 0 '1 0 0 (62)

Thus, the frequency shift for detuning is the same for all harmonics. Since all

harmonics saturate at the same level, we would expect the faster growing,

higher-frequency harmonics to preferentially radiate the power.

The gain of the system can be controlled by changing the impedance z of

the line. As z = (LoICo)ll2, and Wo = (l/LoCo)ll2 we find that zWo = l/Co•

Factoring Co-ll3 from Eq. (44), we can rewrite it as shown below to demonstrate

the change of gain with z:

57

-1

{ ! +i ~31 I Zean I va

o lw 413 zll3 = aw. 2V iJn(rrjr

1) 0

Y3 1 -1 (63) ~

2

Thus, by proper line (or cavity) design, the gain may be increased while the

frequency (.)0 is kept constant.

In the ring model, growth is reduced when the wavelength on the beam

becomes comparable to the diameter of the line, Va, and electric fields from a

charge clump extend directly to the next clump without intersecting the line.

Thus, there is a geometrical high-frequency limit comparable to that found in

traveling wave tubes. However, an instability in the bare electron ring now

appears although it grows more slowly.A.l1

For the Vlasov case, the problem appears to be similar to that involving

Landau damping, and one would expect the velocity distribution function to

develop a plateau in beam angular velocity (.) in the neighborhood of (.)0 ==

l/(LoCo) 112. As in the case of Landau damping, one expects the width of the

plateau to depend on the amplitude of the wave.A.12 Thus, we suspect that a

rapidly growing wave might create a large plateau and abosrb energy from a

large fraction of the orbiting electrons.

Electron Feed Mechanisms

The Orbitron MASER requires electrons to generate R.F. radiation. To

date there have been three electron feed mechanisms proposed and tested.

They are electrons removed from an ionized gas inside the device, a hot oxide­

coated axial filament, and an electron beam injected from an external source.

58

Of these three electron feed mechanisms, the plasma source is the simplest and

the oldest.

Electron injection by a plasma source is the first method used to supply

electrons to the Orbitron MASERA.l. This method is accomplished by keeping

the gas pressure in the tube above the minimum pressure where ionization

will occur, and exciting the device with a high voltage pulse. This causes the

gas in the device to ionize and the electrons from the gas to go into orbit around

the wire. Utilizing this type of feed mechanism, radiation at a frequency of 1

THz has been achievedA.1• The other two methods utilize a hot electron source

in a high vacuum.

The two hot electron source methods are similar in nature. They utilize a

hot oxide coated source to inject electrons into the cavity~wire system, where

there is no residual gas to ionize. The external injection method, preposed by

Burke, Manheimer and OttA.a, utilizes a electron gun positioned outside the

orbitron to fire a stream of electrons along the axis of the Orbitron MASER.

This method has been successfully tried in the People's Republic of ChinaA.1•

The second hot electron source method, designed by myself in 1985, utilizes a

thin, secondary, oxide-coated hot cathode positioned near the outer wall to

provide electrons to the system. This is shown schematically in Figure A.lb.

These steady-state hot cathode devices have been the basis of my thesis

research and the results, reprinted from the IEEE Transactions on Plasma

Science, are give in chapter 2. This work was published jointly with Professor

Igor Alexeff, and Mr. Fred DyerA·7 •

59

Appendix References

A.l High-Power Microwave Sources Victor 1. Granatstein and Igor Alexeff

editors, (1987) Artech House Norwood, MA, ISBN: 0-89006-241-2 pages

293-305.

A.2 Basic Electromagnetic Fields, HerbertP. Neff, Jr., (1987) New York, NY,

ISBN: 0-06-044783-4 pages 66.

A.3 High-Power Microwave Sources Victor L. Granatstein and Igor Alexeff

editors, (1987) Artech House Norwood, MA, ISBN: 0-89006-241-2 pages

243-292.

AA Igor Alexeff and Fred Dyer, Physical Review Letters, #45 page 351,

(1980).

A.5 Igor Alexeff and Fred Dyer IEEE Transactions on Plasma Science, Vol.

PS-8, Page 163 (1980).

A.6 Igor Alexeff, Physics of Fluids, Vol. 28, page 1990, (1984).

A.7 Mark Rader, Fred Dyer, and Igor Alexeff, IEEE Transactions on Plasma

Science, Vol. # PS-15, page 56, (1987).

A.8 1. Alexeff and M. Rader, International Journal of Electronics, 1990, Vol.

68, #3, page 385.

A.9 V. D. Shapiro, Sov. Phys. JETP 17,416 (1963).

A.I0 I. Alexeff, IEEE Trans. Plasma Sci. PS-6, 539 (1978).

A.11 1. Alexeff, IEEE Trans. Plasma Sci. PS-12, 280 (1984).

A.12 N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics

(McGraw-Hill, New York, 1973), p. 531.

60

VITA

The author was born on January 8, 1963 in Folsum CA. He attended

Knoxville Central High school from August of 1978 to June 1981, where he

graduated with Honors.

In the fall of 1981 he was admitted to the University of Tennessee,

Knoxville in the Electrical Engineering Department. He graduated in March

of 1986 with a Bachelor of Science in Electrical Engineering. During this time

the author began work for the UTK Plasma Science Laboratory under the

AFOSR Summer Fellows Program.

In April of 1986, he was admitted to the Master of Science in

Engineering, and has held a research assistantship with Dr. Igor Alexeff ever

since. His work includes computer modeling, theoretical explorations, and

experimental investigations.

The author received his M.S. in Electrical Engineering in 1990.

61