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An experimental investigation of a tandem relativistic backward wave oscillator - travelling wave tube amplifier system for generating high-power microwaves

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A n experim ental in vestigation o f a tandem relativistic backw ard
w ave oscillator - travellin g w ave tu b e am plifier system for
generating high-pow er m icrowaves
Barreto, Gilberto, Ph.D.
Cornell University, 1992
Copyright ©1992 by Barreto, Gilberto. All rightB reserved.
U MI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
© Gilberto Barreto 1992
ALL RIGHTS RESERVED
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
BIOGRAPHICAL SKETCH
It was a dark and stormy night
in the Queens borough of New York
(well, actually it was early morning light
when they popped the champagne cork)
th a t 14th of Ju ly in 1962
when, kicking and screaming, he came into view.
Now out in the open his parents soon realized
th a t indeed he was massive; for owing to his size
the blubbery m ass of their future Comellian,
Gilberto Barreto, was bom by Caesarian.
Twenty-two years afterward
he was dressed up in cap and gown
listening to speeches he’d already heard
from guest speakers of world renown
who showered the Class o f’84
with grim Orwellian metaphors.
But in his happiness he ignored those mystics,
for clutched in his hand was an A.B. in Physics
granted to him by Columbia University:
home of th e Lions, owners of the City.
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He continued his studies there,
albeit in a different department,
eating the local diner fare
and living in a studio apartm ent.
He studied plasm as and fluids and electromagnetic theory;
he even took statistical mechanics (though it made him weary).
Then, soon after his final grades were recorded,
in October 1985 he was finally awarded
(in a simple little ceremony with friendly secretaries cheering)
an M.S. in Applied Physics & Nuclear Engineering.
In 1986 he weighed a num ber of possible fates
then chose the option th a t seemed most rational.
He flew all the way to California, the Golden State,
to work for a company called Physics International.
At PI he learned all about beams and pulsed power;
but best of all was: they paid him by the hour!
When his interest turned to switches, the plasm a opening kind,
he joined those charged with making experimental finds
to support the company’s latest money-making ideology:
the space-saving wonder of inductive store technology.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
B ut eventually he could no longer endure
the oft-hindering lack of a doctoral degree.
So as soon as his source of funding was secure
he packed his bags for Cornell University.
It was Ja n u a ry 1990 - quite early in the year
when he boldly p u t aside his fears
and set off to m eet his new Ithaca buddies
a t the famous Laboratory of Plasma Studies,
where he began his research and experimental meditation
on the wonders of high-power microwave generation.
And so th a t brings us to the present,
w ith his heartfelt genuine wish to explain
th a t his tim e here has been mostly pleasant
despite the cold and freezing rain.
The question of graduation now lies with his committee
and the official who insures his thesis is pretty.
And if they deem him sufficient in his talents,
and the B ursar can find no lingering balance,
then like Pinnochio his dream will come true,
and he’ll be a real “Dr.” come Summer o f’92.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Dedicado a mis padres
Ernesto y Ana Lucia Barreto
con todo m i am ory cariho
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ACKNOWLEDGMENTS
I would like to th an k my advisor, Professor Charles B. W harton, for
providing me w ith the opportunity and funds to conduct the research pre­
sented here in his laboratory. I would also like to thank the other m em bers
of my advisory committee, Professors David A. H am m er and John A.
Nation, for their help and advice.
I am especially grateful to Dr. Jennifer M. Butler for taking me
under her wing while she was still a very busy graduate student a t Cor­
nell, and for continuing to guide me from across the country while p u rsu ­
ing h e r own career as a very busy professional a t the H ughes Research
Laboratories. H er im portant work in the field of high-power microwaves
paved the way for my own research. This dissertation would indeed have
been impossible without her advice, guidance, and friendship.
Thanks also to all those who withstood my endless pleas for techni­
cal help and advice, especially Alan Dunning, Leo B rissette, Steve Glidden,
G arry Bordonaro, and H arry Orton. And for patiently teaching me the
m ysteries of numerical simulation, I am indebted to Professor Niels F.
Otani.
Over the years I have been very lucky to learn from a variety of
excellent teachers. I thank both my former advisor, Professor C.K. Chu of
Columbia University and Dr. John R. Goyer of the Physics International
Company not only for their wisdom, but also for their kind continued in te r­
est in my career.
L ast b u t certainly not least, I thank my family for th eir never end­
ing love and support. I would never have been able to achieve th is goal
w ithout their emotional (and financial) help. I love you all very much.
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TABLE OF CONTENTS
Biographical S k e tc h .............................................................................................iii
D edication...............................................................................................................vi
A cknow ledgm ents............................................................................................... vii
Table of C o n te n ts ............................................................................................... viii
List of T a b le s.......................................... ............................................................. xii
List of F ig u re s .................................................................................................... xiii
C h a p te r 1: I n t r o d u c t i o n ...................................................................................1
1.1 Microwave Sources................................................................................... 1
1.2 Survey of High-Power BWO and TWT R esearch ............................... 5
1.3 Purpose of Investigation & Outline....................................................... 8
C h a p te r 2: B a sic T h e o ry o f th e B a c k w a rd W ave O s c illa to r a n d
T ra v e lin g W ave T u b e A m p lif ie r .......................................... 11
2.1 G eneral Properties of the Electron B eam .......................................... 11
2.1.1 Equilibrium Condition for Annular Electron B eam s............. 11
2.1.2 Space Charge Lim itations........................................................... 15
2.1.3 Space Charge Waves on the Electron B e a m ............................ 18
2.2 Linear Theory of Relativistic BWOs and TW Ts............................... 20
2.2.1 Electromagnetic Waves in a Cylindrical W aveguide............. 20
2.2.2 Electromagnetic Waves in a Slow Wave S tru ctu re................. 22
2.2.3 BWO and TWT Instabilities........................................................ 27
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2.2.4 The Compton and Ram an Operating Regimes........................ 30
2.3 N onlinear Effects: Pulse Shortening................................................... 31
C h ap ter 3: G eneration and C haracterization o f th e
R ela tiv istic E lectron Beam .................................................. 34
3.1 The Omni Pulsed Power System ......................................................... 34
3.1.1 The M arx B an k ............................................................................. 36
3.1.2 The Pulse Forming L ine.............................................................. 36
3.1.3 The Radial Resistor and Vacuum Diode................................... 39
3.1.4 The Guiding Magnetic F ie ld ....................................................... 41
3.2 Characterization of the Vacuum Diode and Electron Beam
43
C h ap ter 4: E xperim ental C haracterization o f th e R ela tiv istic
B ackw ard Wave O sc illa to r ................................................... 49
4.1 Cold Tests: The BWO Passband.......................................................... 49
4.2 The Experimental S etup....................................................................... 50
4.3 Experim ental R esu lts........................................................................... 55
4.3.1 Microwave Frequency.................................................................. 55
4.3.2 Microwave P ulsew idth ................................................................ 57
4.3.3 Microwave Power & Efficiency................................................... 60
4.4 The RF Sever.......................................................................................... 66
C h ap ter 5: E xperim ental C haracterization o f the
Tandem BWO-TWT S y s te m ................................................... 69
5.1 The TWT Slow Wave Structure........................................................... 69
5.2 Cold Tests: The BWO-TWT P assb an d ............................................... 71
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5.3 The Experimental S etu p ....................................................................... 71
5.4 Experim ental R e su lts........................................................................... 75
5.4.1 Microwave Frequency.................................................................. 75
5.4.2 Microwave P u lsew id th ................................................................ 76
5.4.3 Microwave Power & Efficiency................................................... 77
5.4.4 TWT G ain....................................................................................... 80
5.5 Experim ental Results w ithout the RF Sever...........................
81
C h a p te r 6: C o n c lu s io n s ................................................................................. 84
6.1 Sum m ary of Experim ental Investigation.......................................... 84
6.1.1 BWO C haracterization................................................................ 84
6.1.2 The Tandem BWO-TWT System................................................ 85
6.2 Suggestions for F uture W o rk .............................................................. 86
A p p e n d ix A: E x p e rim e n t D iag n o stics ..................................................... 89
A .l Electron Beam Diagnostics................................................................. 89
A.1.1 The Diode Voltage M onitor......................................................... 89
A. 1.2 Beam C urrent Monitors-Rogowski Coils.............................. 91
A. 1.3 The Axial Magnetic Field M onitor......................................... 95
A.2 Microwave Diagnostics......................................................................... 97
A.2.1 Microwave Detectors................................................................... 97
A.2.2 The Er Coupler Probe.................................................................. 99
A.2.3 The Chebychev Directional C oupler...................................... 101
A p p e n d ix B: D e s c rip tio n o f th e BWOSIM C ode ................................. 104
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B .l The Particle and Field Equations..................................................... 104
B.2 Time Integration of the Particle and Field E q u a tio n s................. 105
B.3 Condition for Numerical Stability.................................................... 108
B.4 Initializing the BWOSIM Code......................................................... I l l
B.5 The M ain Timestep Loop................................................................... 113
B.6 Num erical Checks and Diagnostics.................................................. 115
B.7 Results using BW OSIM ..................................................................... 116
B.7.1 Test of Electromagnetic Wave Propagation........................... 116
B.7.2 Cold Beam in a Smooth-Walled W aveguide......................... 118
B.7.3 Slightly Warm Beam in a Slow Wave S tru c tu re .................. 123
B.8 C onclusions.......................................................................................... 123
References ...........................................................................................................125
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LIST OF TABLES
T a b le 1.1 Microwave Band Designation .......................................................... 2
T a b le 4.1 Dimensions of the BWO slow wave stru ctu re..............................49
T a b le 4.2 Typical operating param eters for BWO experim ents................56
T a b le 5.1 Dimensions of the TWT slow wave structure.............................. 69
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LIST OF FIGURES
Figure 1.1 Basic Sketch of (a) a Two-Cavity Klystron, (b) a M agnetron, and
(c) a Helical Traveling Wave Tube................................................ 3
Figure 2.1 Geometry for an annular beam of radius rf, in a cylindrical
waveguide of radius rQwith applied axial m agnetic field J30. 12
F igu re 2.2 Dependence of If, on % for tq = 1.4 cm,
= 0.75 cm, and various
cathode voltages............................................................................. 17
F igu re 2.3 Geometry for a relativistic BWO or TWT..................................23
F igu re 2.4 Cold dispersion diagram for the TM0i and TM 0 2 modes in a
slow wave structure with r 0 = 1.4 cm,
= 0.2 cm, and zq = 1.1
cm......................................................................................................23
F igu re 2.5 BWO and TWT instabilities: dispersion diagram shows the in ­
tersection of the beam’s slow space charge wave with the TM0i
waves of two different slow wave stru ctu res............................ 27
F ig u re 2.6 MAGIC simulation of a beam -structure system with tq = 1.4
cm, r i = 0.2 cm, Z q
= 1 .
1 cm, no. of ripples = 22,
= 0.75 cm, I^
= 1.5 kA, <))c = -350 kV, and Bz = 16 kG. (a) E z field versus time
a t center of structure and r = r^. (b) Corresponding Fourier
transform showing oscillation frequency...................................29
F igu re 2.7 Phase space plot from MAGIC sim ulation showing momentum
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versus axial length for BWO (a) before saturation and (b) a t
saturation........................................................................................33
F ig u re 3.1 Main components of the Omni pulsed power system ...............35
F ig u re 3.2 Omni Marx bank: capacitors are 0.33 pF/75 kV; charging and
grounding resistors are 23 k£2 and 12 k£2, respectively; and the
output inductor is 6 pH ................................................................ 37
F ig u re 3.3 Diagram of the radial resistor and vacuum diode region
40
F ig u re 3.4 Typical voltage response from the magnetic field
secondary coil................................................................................. 42
F ig u re 3.5 Axial profile of the magnetic field with and w ithout the extra
windings added to the solenoid...................................................42
F ig u re 3.6 Typical waveforms of the (a) diode voltage and (b) beam cur­
ren t.................................................................................................. 44
F ig u re 3.7 Plot of 1^ versus V 3/2 for a 1.6 cm diam eter cathode and AK
gaps of 1.89, 2.21, and 2.53 cm; the magnetic field was 24 kG
for all cases..................................................................................... 45
F ig u re 3.8 Plot of diode perveance versus AK gap showing H d depen­
dence................................................................................................46
F ig u re 3.9 Dependence of (a) Ib and (b) Ib/Ilim on yb f°r a 1-6 cm diam eter
cathode and AK gaps of 1.89, 2.21, and 2.53 cm..................... 47
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F ig u re 3.10 Dependence of electron beam power on y^ for a 1.6 cm diam eter
cathode and AK gaps of 1.89, 2.21, and 2.53 cm...................48
F ig u re 4.1 BWO cold test: (a) diagram of the test setup and (b) TM0i pass­
band for the structure...................................................................51
F ig u re 4.2 Diagram of the BWO characterization experim ent................. 52
F ig u re 4.3 Diagram showing the oversized waveguide and microwave fre­
quency/power diagnostics.............................................................54
F ig u re 4.4 Typical diagnostic waveforms: (a) diode voltage, (b) current en­
tering BWO, and (c) current exiting BWO................................56
F ig u re 4.5 Microwave frequency measurement: (a) voltage response of the
reference and delay microwave detectors; (b) the delay line cal­
ibration curve................................................................................. 58
F ig u re 4.6 Dependence of BWO (a) frequency and (b) bandw idth on
59
F ig u re 4.7 Dependence of BWO microwave pulsew idth on y^................... 60
F ig u re 4.8 Voltage response of the reference line microwave detector
whenfy is (a) 1.48, (b) 1.55, and (d) 1.64....................................61
F ig u re 4.9 Dependence of BWO peak microwave power on y^.................. 62
F ig u re 4.10 Dependence of BWO peak efficiency on y^............................... 63
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F ig u re 4.11 Dependence of BWO peak microwave power on axial magnetic
field for constant beam energy of % = 1.50............................64
F ig u re 4.12 Voltage response of microwave detector when the axial m ag­
netic field is (a) 10.5 kG, (b) 12.6 kG, and (c) 16.8 kG
65
F ig u re 4.13 Diagram showing the location of the RF sever....................... 66
F ig u re 4.14 Diagram showing the electron beam pattern on the nylon w it­
ness screen in the beam -structure alignm ent tests
67
F ig u re 4.15 Dependence of BWO peak microwave power on % with RF sev­
e r................................................................................................... 68
F ig u re 5.1 Cold dispersion diagram of both the BWO and TWT struc­
tures.................................................................................................70
F ig u re 5.2 BWO-TWT cold test: the TM01 passband (a) without and (b)
with the RF sever in place...........................................................72
F ig u re 5.3 Diagram of the tandem BWO-TWT experim ent.......................73
F ig u re 5.4 Diagram of TWT structure showing the tapered ripples
74
F ig u re 5.5 Dependence of frequency on y^ for the BWO-TWT.................. 75
F ig u re 5.6 Dependence of microwave pulsewidth on y^ for the BWOTWT.................................................................................................76
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F igu re 5.7 Voltage response of the microwave detector when y^ is (a) 1.48,
(b) 1.51, (c) 1.55, and (d) 1.57 for the BWO-TWT.................... 78
F igu re 5.8 Dependence of (a) peak microwave power and (b) peak efficien­
cy on yb f°r the BWO-TWT........................................................... 79
F igu re 5.9 Dependence of TWT gain on frequency...................................... 80
F igure 5.10 Dependence of microwave pulsewidth on y^ for the BWO-TWT
w ithout the RF sever.....................................................................82
Figure 5.11 Dependence of peak microwave power on y^ for the BWO-TWT
without the RF sever..................................................................83
Figure 6.1 Sketch of the 3.5 inch diameter, multiwave tandem device. 87
F igu re 6.2 Dispersion diagram for the three slow wave structures in the
multiwave tandem device.............................................................88
F igure A.1 Diagram of the diode voltage monitor m ounted in the PFL. 90
F igu re A.2 Setup for calibrating the diode voltage m onitor...................... 90
F igu re A 3 Diagram of a Rogowski coil with an RC integrating
network............................................................................................92
F igu re A 4 Sketch of the semi-rigid coax Rogowski coil and cross-section
of the stainless steel Rogowski coil holder.............................94
F igu re A 5 Setup for calibrating the Rogowski coil..................................... 94
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F igu re A.6 Diagram of the B-dot loop and styrofoam cylinder in the axial
magnetic field solenoid...............................................................96
F igu re A.7 Dependence of the axial magnetic field on charging voltage. 97
F igu re A.8 Setup for calibrating the crystal detectors................................ 98
F igu re A.9 Calibration curve for the HP X424A crystal detector used in the
reference line for microwave power m easurem ents
98
Figure A.10 Sketches of the Er coupling probe............................................100
Figure A.11 Setup for calibrating the E r coupler probe.............................100
Figure A.12 Swept frequency response of the E r coupling probe............ 101
Figure A. 13 Sketch of a basic directional coupler.......................................102
Figure A-14 Swept frequency response of the Chebychev directional cou­
pler...............................................................................................103
Figure B .l Advancement of the particle and field quantities in time. . 106
F igu re B.2 Spatial arrangem ent of the field quantities on each grid. . 106
F igu re B.3 Shape of particles in the BWOSIM code..................................107
F igu re B.4 Diagram showing the main tim estep loop...............................114
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F ig u re B.5 Radial dependence of the field quantities at t = 0
and
2
= 0........................................................................................ 117
F ig u re B.6 Fourier transform of the Ez field a t r = Tq/2.....................
F ig u re B.7
118
Phase space plots showing a cold beam entering a smoothwalled waveguide a t (a) t = 0.5 nsec, and (b) t = 1.5 nsec. 120
F ig u re B.8 Phase space plots showing the cold beam in a smooth-walled
waveguide a t (a) t = 0 nsec, and (b) t = 10 nsec......................121
F ig u re B.9 Phase space plots when grid size is halved a t (a) t = 0 nsec, and
(b) t = 7 nsec..................................................................................122
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CHAPTER 1
INTRODUCTION
In recent years, the wide-ranging field of microwave generation has
enjoyed renewed and increased interest. This is prim arily due to the substan­
tial im provem ents being pursued in such traditional microwave application
areas as communications, plasma heating, m aterial processing, and particle
acceleration. These improvements generally require power levels much
greater th a n those available from conventional microwave sources. W ith the
development of pulsed power technology for generating intense relativistic
electron beams, high-power microwave sources have become possible, giving
rise to fascinating new technologies.
1.1 M icrow ave Sources
The term microwave generally refers to th at section of the electromag­
netic frequency spectrum which lies between 1 and 300 GHz. Although histor­
ically microwave radiation was first generated artificially by Heinrich Hertz
in the 1880s,1 extensive research of microwaves did not occur until the 1940s.
The development of radio altitude determination and ranging (radar) systems
during World War II was the primary catalyst for most microwave research.
At th a t time, the microwave frequency range was divided, for security rea­
sons, into the now common letter-designated bands shown in Table 1.1.
In general, radiative emission from an electron beam interacting reso­
nantly w ith an electromagnetic field forms the basis for all devices th a t gener1. H. Hertz, E lectric Waves, Macmillan (London), 1893.
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1
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Table 1.1 Microwave Band Designation
Band
Frequency
(GHz)
Wavelength
(cm)
L
1-2
15-30
S
2-4
7.5-15
c
4-8
3.75-7.5
X
8-12
2.5-3.75
Ku
12-18
1.75-2.5
K
18-27
1.1-1.75
Ka
27-40
0.75-1.1
Millimeter
40-300
0.1-0.75
ate or amplify microwave radiation. Typically in these devices, the electrons
are forced to oscillate in a direction transverse to the beam velocity by
imposed periodic or crossed fields. Consequently, conventional microwave
sources (also referred to as microwave tubes) can be categorized by the partic­
ular method used to cause this resonant interaction. Brief descriptions of the
three most common sources are given below.2,3
° Klystrons: Invented in 1939 by two brothers, Russell and Sigurd Varian, the
klystron was the first device to use resonant cavities in conjunction w ith an
electrical circuit.4 In a two-cavity klystron such as th a t shown in Figure
1.1(a), electrons passing through a small opening in the first resonant cavity
are alternately accelerated or decelerated by the field in the cavity. Thus the
2. A.S. Gilmour, M icrowave Tubes, Artech House (Norwood, MA), 1987.
3. A.J. Baden Fuller, M icrowaves, Pergamon Press (Oxford), 1979.
4. R.H. Varian and S.F. Varian, J. Appl. Phys., 10, 321 (1939).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 3 -
(a)
Cavities
K
-I
+t
P
Ph
A
gap
(b)
©
Cavity
B-Field
Electron
Orbit
(c)
B-Field
K
A
£
+t
Figure 1.1 Basic Sketch of (a) a Two-Cavity Klystron, (b) a M agnetron, and (c)
a Helical Traveling Wave Tube.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 4 -
electron beam becomes bunched in the small drift space between the two cav­
ities. This bunched beam then excites an amplified electromagnetic wave
when it passes through the second cavity. In the case of a reflex klystron, only
one cavity is used. A reflector cathode is positioned a t the end of the drift space
causing the beam electrons to reverse direction and pass through the resonant
cavity again. Hence the reflex klystron acts as an oscillator.
• Magnetrons'. The m agnetron was used as the microwave source for most of
the rad ar systems used in World War II and is commonly found in microwave
ovens today. Unlike the klystron, the magnetron is a device in which the elec­
trons interact with the electromagnetic fields over an extended region. As
shown in Figure 1.1(b), it consists of a cylindrical cathode surrounded by sev­
eral coupled resonant cavities which serve as the anode. Electrons em itted
radially from the cathode are deflected by an applied magnetic field, which is
perpendicular to the electron flow. The electrons thus follow trochoidal orbits
about the cathode and excite microwave oscillations in the anode cavities.
• Traveling Wave Tubes: The traveling wave tube amplifier was invented by
Kompfner in 1942 and later improved and analyzed by Pierce.5 In a traveling
wave tube, the electron beam is guided by an applied m agnetic field through
the center of a slow wave structure, such as the helix shown in Figure 1.1(c).
In contrast to the magnetron, the magnetic field is applied parallel to the elec­
tron flow. Because the slow wave structure is capable of supporting electro­
magnetic waves with phase velocities less than the speed of light in vacuo,
electrons can lose their kinetic energy to the microwave fields of the slow wave
structure when the wave’s phase velocity coincides with the beam velocity.
The electron beam can thus interact with the electromagnetic fields continu­
5. J.R. Pierce, Traveling Wave Tubes, Van Nostrand (Princeton, NJ), 1950.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 5 -
ously along the length of the slow wave structure. The traveling wave tube
am plifier and its oscillator version, the backw \rd wave oscillator (also occa­
sionally referred to as the carcinotron), are described in greater detail in
C hapter 2.
Throughout the 1960s and early 1970s, the invention of various solid
state devices (such as the Gunn oscillator) caused the focus of most microwave
research in the
USA
to tu rn from source development to m anufacturing con­
cerns. In terest in microwave sources was renewed in the late 1970s as the
need for higher power levels increased in the communication, radar, and
plasm a fusion communities. Advances in pulsed power technology made it
possible to use intense relativistic electron beams with energies approaching
the electron rest mass (511 keV) in conventional microwave tube designs, thus
giving birth to the field of high-power microwaves.
Although it has no official definition, the qualifier “high-power” typi­
cally describes power levels > 100 MW. Relativistic versions of the klystron,6
m agnetron,7 and backward wave oscillator8 have all achieved microwave
pulses with peak powers > 1 GW and beam-to-microwave generation efficien­
cies of 30-50%. Improvements are expected from the various international
research efforts in high-power microwave generation.
1.2 S u rvey o f H igh-Pow er BWO and TWT R esearch
Over the past twenty years, the Backward Wave Oscillator (BWO) has
attracted much attention due to its ability not only to convert electron beam
energy into microwave radiation simply and efficiently, but also to operate
6. Y.Y. Lau et al., IEEE TYans. Plasm a Sci., 18, 553 (1990).
7. G. Bekefi and T. Orzechowski, Phys. Rev. L ett., 37, 379 (1976).
8. V.S. Ivanov et al., Zh. Tekh. Fiz., 51, 970 (1981).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 -
over a broad band of frequencies while m aintaining a narrow bandw idth. In
1970, experiments by N ation9 a t Cornell University were the first to use a rel­
ativistic electron beam in a BWO - several m egawatts of X-band microwave
power were produced. Since then there has been extensive research in highpower BWOs. Experim ental studies (the majority of which have been per­
formed in the former
U SSR )
have been successful in increasing BWO output
power from several hundred m egaw atts10,11 to levels exceeding 1 GW 8-12-13
with frequencies ranging from 9-33 GHz and a peak efficiency of 35%. Theo­
retical studies, especially those which implement a linear fluid analysis,14,15
have found excellent agreem ent with BWO experiments in the low-current
regime (see Section 2.2.4).
As experiments continue, higher powers and efficiencies have been
sought by increasing the current of the BWO’s driving electron beam; how­
ever, it has been shown th a t space charge effects can lim it such high-current
vacuum BWO system s.16 One method to overcome this lim itation, which was
first tried by Tkach et al. in the mid-1970s,17 is to fill the BWO's slow wave
structure with a plasm a in order to neutralize steady space charge
effects.18,19
9. J.A. Nation, Appl. Phys. L ett., 17, 491 (1970).
10. N.F. Kovalev et al., P is’m a Zh. Eksp. Tear. Fiz., 18, 232 (1973).
11. Y. Carmel et al., Phys Rev. Lett., 33, 1278 (1974).
12. Yu. F. Bondar’ et al., Fiz. Plazm y, 9, 383 (1983).
13. E.B. Abubakirov et al., P is ’m a Zh. Tekh. Fiz., 9, 533 (1983).
14. J.A. Swegle et al., Phys. F luids, 28, 2882 (1985).
15. J.A. Swegle, Phys. F luids, 28, 3696 (1985).
16. N.E. Belov et al., Fiz. Plazm y, 9, 785 (1983).
17. Yu. V. Tkach et al., Fiz. Plazm y, 1, 81 (1975).
18. L.S. Bogdankevich et al., Usp. Fiz. N auk., 133, 3 (1981).
19. Y. Carmel et al., IEEE Trans. P lasm a Sci., 18, 497 (1990).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 -
High-power BWOs have also been made more compact as evidenced by
the Plasm a Assisted Slow-wave Oscillator (PASOTRON™ ) developed by Hughes
Research Laboratories.20 In the
PASOTRON,
the self-magnetic field of an elec­
tron beam passed through a plasma-filled slow wave structure is used to con­
fine the beam radially, thereby eliminating the need for a typically bulky
external m agnetic field.
F u rth er improvements in both vacuum and plasma-filled systems
require a b etter understanding of the BWO’s nonlinear properties. Although a
nonlinear theoretical model has not yet been developed, detailed experimental
studies of such nonlinear phenomena as the BWO’s saturation mechanism
and its dependence on the driving electron beam have recently begun.21
In contrast to high-power BWOs, there has been, until recently, rela­
tively little work involving Traveling Wave Tube (TWT) Amplifiers th a t use
intense relativistic electron beams to achieve high-power levels. In the
USA,
experim ents a t Cornell represent the only work published to date th a t focuses
on high-power TWTs. In experiments by Shiffler 22,23 an
RF
signal produced
by a m agnetron is fed into a TWT driven by a relativistic electron beam; the
TWT’s gain and efficiency are then characterized as a function of the input *.i:'
frequency, beam current, and magnetic field. In experim ents by B utler 24,25 a
high-power BWO is used to introduce an input
RF
signal into twin parallel
TWTs; the structures are powered simultaneously by three independent rela­
tivistic electron beams. The gain, efficiency, and phase coherence of the twin
20. R.W. Schumacher et al., Proc. o f 18th IEEE Int. Conf. P lasm a Sci., (1991).
21. J.M. Butler et al., IEEE Trans. Plasm a Sci., 18, 490 (1990).
22. D. Shiffler et al., Appl. Phys. Lett., 54, 674 (1989).
23. D. Shiffler et al., IEEE Trans. Plasm a Sci., 18, 546 (1990).
24. C.B. Wharton and J.M. Butler, SPIE Proc., 1226, 23 (1990).
25. J.M. Butler, Ph.D. Dissertation, Cornell Univ., (1991).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 8 -
TWTs are characterized as a function of BWO frequency, beam energies, and
m agnetic field.
In the former
USSR,
TWTs have been used prim arily in coupled oscilla-
tor-amplifier systems designed to generate very high-power levels (>1 GW). Of
particular interest are the tandem systems pioneered by Bugaev et al. at
Tomsk26 and B ratm an et al. a t Gorky27 th a t use a single relativistic electron
beam to drive both a BWO and a TWT (or two BWOs operating in different
modes) connected in series. Also, in recent informal discussions, A.V. Smorgonskii has mentioned some non-tandem, high-power TWT experiments by the
signals into
group a t Tomsk to find novel methods for feeding input
RF
TWTs.28 In one such experiment, a parabolic reflector and
TE-TM
vertor are placed a t the input end of the TWT to feed in an
RF
mode con­
signal from a
magnetron. The TWT’s driving electron beam is injected through a hole in the
reflector.
1.3 P u rp ose o f In vestigation & O utline
The experimental investigation presented here is of a high-power, ta n ­
dem BWO-TWT system th a t is very sim ilar to th a t developed by Bugaev et
a l,26 Interest in this type of oscillator-amplifier system stems from the fact
th at, ideally, an array of TWT amplifiers could be driven by a single BWO to
feed an array of high-power antennas. Such an antenna array could be oper­
ated over a broad band of frequencies while m aintaining a narrow bandwidth;
furtherm ore, a controlled phase relationship between each source could be
26. S.P. Bugaev et al., P is’ma Zh. Tehh. Fiz., 19, 1385 (1983).
27. V.L. Bratman et al., P is’ma Zh. Tekh. Fiz., 14, 9 (1988).
28. C.B. Wharton, (private communication).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 -
easily m aintained, allowing this phased array antenna to be steered in any
direction with minimal loss of directivity.
Although these BWO-TWT devices have been shown capable of produc­
ing up to
15
GW of microwave power (the record to date) with
50%
efficiency,2 6
the pulsewidth of the microwave radiation from these tandem devices is very
short when compared to th a t of the voltage and/or current pulses th a t produce
the driving electron beam. For example, in the record setting Tomsk experi­
m ent mentioned above, microwave pulses as short as
observed when using voltage pulses of up to
1
10
nsec
(FW H M )
were
psec duration. This phenom e­
non of pulse-shortening has been reported by various authors working with
relativistic BWOs and other high-power oscillators ,2 1 ,2 9 ,3 0 but it has not been
observed by those working with relativistic TWTs.2 3 ,2 5
The tandem BWO-TWT system described here was therefore designed
to dem onstrate the feasibility of producing high-power (-100 MW) pulses of Xband microwaves having pulsewidths th a t are comparable to the electron
beam duration (-100 nsec). This combination of long pulses and high powers
is accomplished by means of an RF sever placed between the BWO and TWT
th a t reflects and attenuates electromagnetic waves propagating in either
direction while allowing the electron beam to pass through unimpeded from
the BWO to the TWT. In tandem systems lacking the RF sever, the BWO can
receive feedback from the TWT, which in turn allows the BWO to reach saturatio n -a condition which has been shown to lead to pulse-shortening . 2 1 With
the sever, BWO power levels can be kept low to avoid saturation. The high
power levels sought are consequently achieved by driving the TWT with a
modulated beam.
29. A.F. Aleksandrov et al., Zh. Tekh. Fiz., 50, 2381 (1980).
30. S.P. Bugaev et al., Dokl. A kad. N auk S S S R , 276, 1102 (1984).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 0 -
In this dissertation, experimental results obtained with the system
described above are presented and analyzed. An overview of the basic BWO
and TWT theory is given in Chapter
2
followed by a description of the pulsed
power system used to generate the relativistic electron beam in C hapter 3.
Experim ents to characterize the basic properties of the BWO alone are dis­
cussed in C hapter 4; the tandem BWO-TWT experim ents are discussed in
C hapter 5. Finally, conclusions are presented in C hapter
6
along with sugges­
tions for future work.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER 2
BASIC THEORY OF THE BACKWARD WAVE OSCILLATOR AND
TRAVELING WAVE TUBE AMPLIFIER
In both the Backward Wave Oscillator (BWO) and Traveling Wave Tube
(TWT) Amplifier, an electron beam is guided through a slow wave structure by
a longitudinal magnetic field. Instabilities resulting in the generation of
microwave radiation can develop in this beam -structure system when the
velocity of the beam ’s space charge wave coincides with the phase velocity of
the structure’s electromagnetic modes.
2.1 G eneral P rop erties o f th e E lectron Beam
2.1.1 Equilibrium Condition for A nnular Electron Beams
In the BWO and TWT discussed in this dissertation, an an n u la r (i.e.,
hollow) electron beam is used as the energy source for generating and amplify­
ing microwave radiation. B ut before discussing the details of this p articular
type of electron beam-slow wave structure system, it will be instructive to con­
sider a simpler system consisting of an annular electron beam passed through
the center of an evacuated, smooth-walled, cylindrical waveguide.
As this
unneutralized
annular
beam
propagates
through
the
waveguide, it experiences a force from its own electromagnetic fields causing
it to expand radially .3 1 However, an external magnetic field can be applied
parallel to the beam to counteract this radial expansion. The m inim um m ag­
31. B.B. Godfrey, IEEE Trans. P lasm a Sci., P S-7, 53 (1979).
-
11
-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 2 -
netic field necessary to m aintain the beam’s equilibrium radius can be found
by treatin g the beam as a laminar, unneutralized, relativistic, electron fluid.
The cylindrical geometry for this system is shown in Figure 2.1: the
beam is a th in annulus with inner and outer radii
and rb, respectively, and is
guided through a waveguide of radius r 0 by an axial magnetic field B 0. Within
this annulus, the beam has a density nb given by
rit
coj
(2.1)
where e0 is the perm ittivity of free space, e and m are the electron charge and
mass, respectively, and co6 is the plasm a frequency of the beam. (Note: SI units
are used throughout unless otherwise specified.)
A macroscopic fluid analysis of this system necessarily implies th a t the
Larm or radius of each electron’s helical path m ust be much sm aller th an the
beam radius, such th at
Figure 2.1
Geometry for an annular beam of radius rb in a cylindrical
waveguide of radius r 0 with applied axial m agnetic field Bo-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 3 -
ym ve
- B - .r „
2
2
where y = [ 1 - (ve/ c ) - (vz/ c ) ]
—
-^2
(2.2)
is the standard relativistic factor, ve
and vz are the azim uthal and axial electron velocities, respectively,and c is the
speed of light in vacuo. If this condition holds, then it is possible to describe
the system in term s of the macroscopic fluid equations
^ + V •n x = 0
at
at
( v V )v = - — (E + v x B )
m
(2.3)
(2.4)
and Maxwell’s equations
V.E =
(2.5)
eo
V ■B = 0
(2.6)
9B
V xE = - 3 at
(2.7)
VxB = p0J + p 0 e0^
(2.8)
where v is the electron fluid velocity, E and B are the self-consistent equilib­
rium electric and magnetic fields, J = - n e x is the electron current density,
and |i 0 is the permeability of free space. When w ritten in cylindrical coordi­
nates, (2.4) becomes
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 4 -
ym v l
■e{Er + ue [ ( B0 + B z) -
vzB q]}
= 0.
(2.9)
In the system being considered here, the beam ’s axial velocity is much
greater th a n its azim uthal velocity (vz » v e). As shown by Miller , 3 2 this condi­
tion can be used to reduce (2.9) to the quadratic equation
u
yb(o%-n(oe + ^
— fo)*u d u =
( 7br) J
( 2 . 10 )
0
where co0 = uQ/ r is the beam’s angular velocity, Q = e B 0/ m is the electron
-
1 /2
cyclotron frequency, yb = [ 1 - (vz/c) ]
is the effective beam energy, and
(op is defined as
-
0
< r < ri
0
ri<r<rb
a * ( r) =
( 2 . 11 )
rb < r <r o
0
Solving (2.10) for co0 yields the following equation
(r) =
Q.
2
1
±
1
1/ 2 "
co2 /
-
1
-
1 I
vr j
ri < r < r b.
( 2 . 12 )
Clearly the beam will be in radial equilibrium only if the roots of this equation
are real. Thus, from the radical in (2.12), the minimum m agnetic field neces­
sary to m aintain the beam’s equilibrium radius is given by
mn
o> T^T
Eo >b
d 2
2
'
1
-
-
32. R.B. Miller, Intense Charged Particle Beams, Plenum Press (New York), 1982.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.13)
- 1 5 -
The typical electron beam used in the experiments presented here has
n = 10 18 m "3, r j r b = 0.9, and yb = 1.50. Thus from (2.13), a magnetic field of at
least 0.4 Tesla (= 4 kG) is needed to m aintain the beam ’s equilibrium radius.
The field actually used in the experiments is approximately 4-5 tim es greater
th a n this minim um field.
2.1.2 Space Charge Limitations
Independent of the applied magnetic field, the am ount of beam current
th a t can propagate through the cylindrical waveguide is lim ited by the elec­
trostatic potential depression resulting from the unneutralized beam ’s space
charge. As shown by Brejzman et al.,33 this lim it can be found from energy
conservation argum ents as follows: the scalar potential §b(r) associated with
the an n u lar beam within the waveguide is found by solving the Poisson equa­
tion
p(r)
(2.14)
where p(r) is the radial charge density of the beam. At a point sufficiently far
from the ends of the waveguide, the beam potential is given by
l n ( r 0/ r b)
^ (r)
2
ti t 0ub l n ( r 0/ r )
0
<r<rb
rb < r < r 0
(2.15)
where vb and Ib are the beam velocity and current, respectively.
Assume th at the electron beam is generated by a cathode a t some
potential <J>C< 0. If the cathode is located a t one end of the waveguide, then the
total energy of the beam a t the point of injection is
33. B.N. Brejzman and D.D. Ryutov, Nuclear Fusion, 14, 873 (1974).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
where yc = 1 + |et()cj/m e 2 is the relativistic factor a t the cathode. Using (2.15)
and (2.16), the total energy of the beam a t a point w ithin the waveguide can be
found from conservation of energy to be
(2.17)
Solving (2.17) for I b gives
j
[kA]
(2.18)
where vb has been expressed in terms of yb. Equation (2.18) relates the beam
current to the effective beam energy. In experiments w here it is not possible
m easure yb directly (such as those described in this dissertation), (2.18) may
be used to determ ine the value of yb implicitly from m easurem ents of the beam
current, diode voltage, and cathode diameter.
1 /* \
The maxim um of I b occurs when yb = yc such th a t
max
The current
(2.19)
is known as the space charge lim iting current; it represents
the maximum steady state current th at can flow through the waveguide.
Thus, if an electron beam with a current Ib > IUm were injected into the
waveguide, the potential depression due to the electron space charge would be
so great th a t only those electrons with sufficient kinetic energy would pass
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 7 -
through. The rem aining electrons would be reflected back towards the injec­
tion point due to the formation of a virtual cathode.
It is interesting to note th a t the same value of yb may be obtained using
very different voltages and currents. This is shown graphically in Figure 2.2.
<h [kV] = 400
C
----4
-
3
~
/
350
300
250
200
0.8
Figure 2.2 Dependence of Ib on % for r 0 = 1.4 cm, rb = 0.75 cm, and various
cathode voltages.
The plot also shows th a t yb is a double-valued function of Ib. For example,
given a beam current of 3 IcA and a cathode voltage of 350 kV, yb can be either
1/I
1.07 or 1.36. Values of yb to left of the maximum ( 1 < yb <y^' ) correspond to a
low energy beam with a high charge density, whereas those to the right
(y^/ 3 <Y6 <Yc) correspond to a high energy beam with a low charge den.ity.
Fortunately, ambiguities can be avoided since the low energy/high charge den­
sity values have been proven to represent a physically unstable condition . 33
T h at is, the beam prefers the high energy/low charge density state. In the
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 8 -
work presented here, values for yb are understood to correspond to this pre­
ferred state.
2.1.3 Space Charge Waves on the Electron Beam
The electron beam is capable of supporting various electrostatic waves
th a t arise when its equilibrium is disturbed by small am plitude perturbations.
For example, longitudinal bunching of the electrons results in space charge
waves (also known as Langm uir waves), whereas the helical motion of the
electrons gives rise to cyclotron waves. The generation (and amplification) of
electromagnetic radiation in a BWO (and TWT) is critically dependent on the
excitation and growth of space charge waves on the driving electron beam. It
is therefore instructive to determ ine the dipersional nature of these waves.
The general equilibrium dispersion of a cold (i.e., monoenergetic) elec­
tron beam whose fields vary as exp iUiz-mt) is given by
co = k v b
(2 .2 0 )
where co, k, and vb are the angular frequency, wavenumber, and beam velocity,
respectively. Assume at first th a t the beam is neutralized by an infinitely large
background plasma. For this case, it can be shown via a simple, first-order
perturbation analysis 3 4 th a t
co = k v b±(ab
(2 . 2 1 )
where co6 is the plasma frequency of the beam. Equation (2.21) describes two
space charge waves on the beam; notice th at their dispersion relation is essen­
tially th a t of the equilibrium case with a correction resulting from the electron
34. N.A. Krall and A.W. Trivelpiece, Principles o f Plasm a Physics, McGraw-Hill (New
York), 1973.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 9 -
beam density. Although the two waves travel a t the same group velocity
(vg = vb), th eir phase velocities differ as shown below
Ufy
9
In this equation, the
since
—1
1
.
vb
r.
(2.22)
± (cob/(a)
solution corresponds to a fast space charge wave,
> vh for this case. In order for this fast wave to grow, the beam elec­
trons m ust be accelerated-it is thus referred to as a positive energy mode. The
“+” solution corresponds to a slow space charge wave, since v ^ < v b. Because
the slow wave is excited when the beam electrons sire decelerated, it is
referred to as a negative energy mode. It will be shown th a t the slow space
charge wave plays a significant role in both BWOs and TWTs.
Thus far only the case of a neutralized beam has been considered. A
complete, linear perturbation analysis of the more relevant case of an unneu­
tralized, relativistic, electron beam propagating in a cylindrical waveguide
was done by Ramo , 35 who found th at the dispersion relation for the space
charge waves is given by
,2 2
2
k zc -CO
V
y 2
^
(2.23)
+ k 2z) c 2 - co2 )
where k ± is the transverse wavenumber, kz is the axial wavenum ber given by
k z = p 2 + (co/c)2,
and p is determ ined from the transcendental equation
35. S. Ramo, Phys. Rev., 56, 276 (1939).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.24)
- 2 0 -
p J 0 (kLrb) _ I Q(pr 0) K 0 (p r b) - 10 (prb) K 0 (p r 0)
k 1J 1(k1rb) ~ I 0 {pr0) K 1 (prb) + I X(prb) K 0 (pr0)
(2.25)
where J 0 and J x are normal Bessel functions, I 0, / lf K 0, and K 1 are modified
Bessel functions, and co6 is the relativistically corrected plasm a frequency for
a beam of density nb
(2.26)
As was found with the neutralized beam case, the unneutralized, relativistic
beam supports the two space charge waves described by (2.23): the
case is
the fast wave, and the “+” case is the slow wave.
2.2 L in e a r T h e o ry o f R e la tiv is tic BWOs a n d TWTs
2 .2 . 1
Electromagnetic Waves in a Cylindrical Waveguide
In order for an electromagnetic wave to propagate down the cylindrical
waveguide of Figure 2.1, it m ust satisfy Maxwell’s equations, (2.5M2.8), as
well as the boundary conditions imposed by the waveguide’s shape. The dis­
persion relation for the bounded wave propagating in vacuum can be found as
follows: assum e the wave is azim uthally symmetric and traveling in the zdirection so th a t its electric and magnetic fields have the general form
E (r, z , t) = E (r) exp [i (k zz - tot) ]
(2.27)
H (r,z,t) = H ( r ) e x p [i (kzz - t ot ) ].
(2.28)
Implicit in Maxwell’s equations are the two wave equations
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 1 -
V2E z = -(co/ c ) 2E z
(2.29)
V2H z = -(co/ c) 2H z
(2.30)
where E z and H z = B z/\xQ are the axial electric and magnetic fields, respec­
tively. The norm al (i.e., eigen-) modes for the electromagnetic waves described
by (2.29) and (2.30) can be decomposed into two types: (i) the Transverse Elec­
tric (TE) modes, which have E e, H n and Hz field components, and (ii) the
Transverse Magnetic (TM) modes, which have E n Ez, and H Q components .36
Notice th a t if there is a beam in the waveguide, only the TM-modes have the
possibility of perturbing the beam’s space charge waves, since the TE-modes
lack an axial electric field component.
Consider now an azim uthally symmetric TM-wave propagating in the
empty waveguide. Substituting (2.27) into (2.29)
(2.31)
where the Laplacian has been w ritten in cylindrical coordinates, and
(co / c ) 2 - k 2z is the wavenumber at cutoff. Equation (2.31) is the wellknown Bessel equation whose solutions have the general form
E z {r) = A J n (kcr) + B Y n (kcr)
(2.32)
where A and B are arbitrary constants of integration, and J n and Yn are nthorder Bessel functions of the first and second kind, respectively. Because Ez
m ust be continuous and finite at the center of the waveguide, and the tangen­
tial field m ust vanish a t the waveguide wall, (2.32) reduces to
36. J.D. Jackson, Classical Electrodynam ics, 2nd ed., Wiley (New York), 1975.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-2 2 -
J n&cr o) = °-
(2.33)
Furtherm ore, if |i„m are the zeros of J n, then
C
(2.34)
where coc is the cutoff frequency for the waveguide. The dispersion relation for
the azim uthally symmetric TM0m waves is therefore given by
(2.35)
2.2.2 Electromagnetic Waves in a Slow Wave Structure
The phase velocity (coIk) of the TM-wave described by (2.35) is always
g reater th an the speed of light (v^ > c); consequently, there can be no interac­
tion between it and the space charge waves on the beam. Because the opera­
tion of a BWO or TWT is critically dependent on this type of interaction, the
geometry of the waveguide is altered to “slow down” the electromagnetic
wave. This special type of waveguide is known as a slow wave structure.
In most relativistic BWOs and TWTs the slow wave structure used is
not a helix, as shown in Figure 1.1(c), but ra th e r a cylindrical waveguide
whose wall radius rw is perturbed sinusoidally about an average radius r 0 (as
shown in Figure 2.3) such th a t
rw (z ) = ro + r i.sinA0z
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.36)
Figure 2.3 Geometry for a relativistic BWO or TWT.
2
co = k c
20
co = k v.
“
TM
o
>
u»
10
-
TM
5 -
0
2
4
6
8
10
12
14
Wavenumber (rad/cm)
Figure 2.4 Cold dispersion diagram for the TM 01 and TM 02 modes in a slow
wave structure with r 0 = 1.4 cm, r 1 = 0.2 cm, and z 0 = 1.1 cm.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 4 -
where th e perturbation has a period z 0 =
2
n / h 0 and an am plitude r x, which
is typically small so th a t r^/r^ = e « 1 .
Because the slow wave structure is periodic, the dispersion of electro­
m agnetic waves in the structure is also periodic such th a t
co(k0) = 0 ) (k0 + n h 0)
( 2 .3 7 )
where k 0 is the ripple wavenumber, and n is an integer. This can be seen
clearly in Figure
2 .4 ,
which is a cold dispersion diagram showing the TM 01 and
TM 02 modes for a slow wave structure with r 0 =
1 .1
1 .4
cm,
=
0 .2
cm, and z 0 =
cm. (These dispersion curves were generated num erically using the
CYLWG
code w ritten by A. Bromborsky .)3 7 Here the term “cold” refers to the absence of
an electron beam in the structure. The dotted line (co = kc) plotted in Figure
2 .4
is known as the light line- it represents the dispersion of an unbounded
electromagnetic wave. Notice th at the TM-modes shown have regions lying
below this line where v<!><c.
Superimposed on the plot in Figure
2 .4
is the beam line co = k v b, which
represents the equilibrium dispersion of an electron beam with velocity vb =
2 .2 7 x 1 0 8
m/sec. The beam’s fast and slow space charge waves straddle this
line diverging from it by an amount proportional to the beam ’s space charge.
Additionally, the slope of the beam line may be characterized by yb. The beam
line in Figure
cathode a t
2 .4 ,
-3 5 0
which represents a beam current of
1 .6
kA generated by a
kV, has a corresponding yb of 1 .5 3 .
The dispersion of TM0m waves in a slow wave structure th a t includes
the presence of the beam (i.e., warm dispersion) can be determ ined using the
linearized small-amplitude perturbation analysis used by Swegle . 14 (This
approach is described fully in Reference 14-only the m ain points are dis­
37. A. Bromborsky and B. Ruth, IEEE Trans. Micro. Theory Tick., MTT-32, 600 (1984).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 5 -
cussed here.) Because of th e structure’s periodicity, each quantity related to
the TM-waves can be expanded in a Floquet series ; 3 8 for example
oo
Ez n (r,z,t) =
E z (r)exp [ i ( k nz - at )]
(2.38)
n = -<»
where k n = k 0 + n h 0 and - h Q/ 2 < k Q< h 0/ 2 . As in the previous case of a
smooth-walled waveguide, substituting (2.38) into (2.29) gives the Bessel
equation (2.31) with solutions given by
E z (r) =
A nJ o ( k cnr )
° ~ r ^ rb
. B nJ 0 (k cnr ) + C n Y 0 (k cnr )
(2.39)
rb < r ^ rw
where A n, Bn, and Cn are constants. The radial electric field component of the
TM0m wave is found simply from Maxwell’s equations to be
£ r„<r,z,t) = ^
(2.40)
” cn
where k%n = (co/c ) 2 - k 2n .
After applying the boundary conditions across the beam and a t the slow
wave structure wall using (2.39) and (2.40), the following linear equation is
found
oo
I
DmnA„ = 0.
m, n -
The m atrix elements Dmn are given by
38. R. Courant and D. Hilbert, M ethods o f M athem atical Physics, Vol.l, Wiley-Interscience (New York), 1939.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.41)
- 2 6 -
Dm„
TTLTl =
(a ? -km
ry,k„c2\
n
7
(
k„„C
'S2
( 2 '4 2 >
*
- 'L 'V ^ c O , ) ! ) =
0
where the param eter a, which contains the influence of the beam ’s space
charge, is defined as
a = — Ulb ■
(2.43)
with (3b = (ub/c) and Ib in kA, and
71
T^n =
J 0 [kcnr0 ( l + e s i n u ) ] e x p [ i ( . n - m ) u ] d u .
(2.44)
-7 1
y
A sim ilar expression for I mn applies with J 0 replaced by F0.
Equation (2.41) has a nontrivial solution when the determ inant of the
m atrix D vanishes
det D = 0.
(2.45)
Equation (2.45) is the desired linear dispersion relation for TM0m waves th at
includes the presence of the electron beam in the slow wave structure. In prin­
ciple, (2.45) involves an infinite matrix; in practice, the dispersion relation is
determ ined by truncating D to manageable dimensions and then calculating
(2.45) numerically. In calculations done by Swegle, it was found th a t 5 x 5
matrices with - 2 < m , n < 2 were typically sufficient for systems of in tere st . 14
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 7 -
2.2.3 BWO and TWT Instabilities
An electron beam in a slow wave structure provides a source of energy
for driving the instabilities th a t generate and amplify microwave radiation in
BWOs and TWTs. As mentioned before, these instabilities occur when the
phase velocities of the beam’s slow space charge wave and the structure’s TMwaves coincide. In a dispersion diagram such as th a t shown in Figure 2.5, this
occurs in the general region where the slow space charge wave line intersects
the TM-wave dispersion curves. Solutions to the dispersion relation in this
coupled beam -structure resonance region can be described by a complex fre­
quency co = (0 r + i(0 ., where co, >
0
provides a m easure of the instability’s
growth rate.
TW T
Instability
“ BWO
Instability
Slow Space
C h a rg e W ave
4
0
2
3
4
5
6
7
8
W avenum ber (rad/cm)
Figure 2.5 BWO and TWT instabilities: dispersion diagram shows the inter­
section of the beam’s slow space charge wave with the TM 01 waves
of two different slow wave structures.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 8 -
In a BWO, the point of intersection occurs where the structure wave
has negative group velocity (dco/ dkQ< 0 ), i.e., the wave travels “backw ards”
with respect to the beam velocity. The BWO instability is absoiute-i.e., its
temporal growth ra te is the same everywhere along the stru ctu re length-and
does not require an input
RF
signal, since it can develop from noise carried on
the beam. In a TWT the intersection occurs where the structure wave has pos­
itive group velocity (<&)/dkQ> 0), so the wave travels with the beam. The TWT
instability is convective-i.e., its spatial growth rate increases along the struc­
ture length-and does require an input
RF
signal to develop.
The excitation of the negative energy slow space charge wave a t these
intersection points causes the beam to bunch as electrons are decelerated. The
energy lost by the decelerated electrons is systematically transferred to the
electromagnetic fields of the structure’s TM-waves, which fu rth er enhances
the beam bunching, which then enhances the wave-fields, etc. As this positive
feedback loop continues, the amplitudes of the TM-wave fields increase expo­
nentially a t an oscillation frequency corresponding to th a t of the beam -structure resonance. This is shown graphically in Figure 2.6, which is tak en from a
numerical sim ulation of a BWO beam -structure system done by B utler 2 5
using the 2-1/2 dimensional particle-in-cell
(PIC)
code
MAGIC.39
Notice th a t
although the BWO’s axial electric field E z increases exponentially from t = t Q,
as shown in Figure 2.6(a), it does not increase indefinitely. For tim es t > tsat,
the BWO is saturated, since the amplitude of the E z-field rem ains a t a con­
stan t level. Once a BWO (or TWT) has reached saturation, nonlinear effects
become im portant so th a t the system can no longer be described by the linear
model developed in Section 2.2.2.
39. B. Golpen et al., M ission Research Corp., Report No. MRC/WDC-R-068, 1983.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 2 9 -
a)
0 ,0
4>,0
e.O
TIME
12.0
10.0
(a)
b)
a
L_
o'
I—
0.0
1 0 .0
20.0
FREQUENCY
Figure
2 .6
(H z)
simulation of a beam -structure system with r 0 = 1 .4 cm,
r j = 0 .2 cm, z 0 = 1 .1 cm, no. of ripples = 2 2 , rb = 0 . 7 5 cm, Ib = 1 .5 kA,
<pc = - 3 5 0 kV, and Bz = 1 6 kG. (a) Ez field versus time a t center of
structure and r = rb. (b) Corresponding Fourier transform showing
oscillation frequency. [From Reference 2 5 by permission.]
MAGIC
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 3 0 -
2.2.4 The Compton and Raman Operating Regimes
The n atu re of the BWO/TWT instability can be very different depend­
ing on the am plitude of the beam current passing through the slow wave
structure. The previously defined space charge lim iting current, (2.19), can be
used to define two distinct operating regimes for the BWO and TWT. 15
In the low current or Compton regime, where Ib «
single particle
effects dom inate the interaction between the electrons and the electromag­
netic fields . 19 Because the am ount of beam space charge separating the fast
and slow waves is small, the two space charge waves interact with the struc­
tu re wave together, making the instability essentially a three-wave process.
Linear theory has been very successful in modeling BWOs and TWTs operat­
ing in the Compton regime; comparisons of experim ental and numerical
results show excellent agreem ent . 1 4 ,4 0 ,4 1
In the high current or Raman regime, where Ib ~ Ilim, collective effects
dominate the interaction . 19 In this case, the beam space charge is large, so the
separation between the fast and slow space charge waves is greater. As a
result, the two waves interact separately with the structure wave m aking the
instability a two-wave process. Linear theory has proven insufficient in the
Ram an regime. Because the system has reached satu ratio n in this regime,
growth rates predicted by the linear model for BWO instabilities do not agree
with experim ental d ata .2 1 ,2 5 BWOs and TWTs are operated in the Raman
regime for m ost high-power applications; there is therefore a need for a non­
linear theory.
40. G.T. Leifeste et al„ J. Appl. Phys., 59, 1366 (1986).
41. J.A. Sw egle, Phys. Fluids, 30, 1201 (1987).
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- 3 1 -
2.3 N on lin ear Effects: P u lse Sh orten in g
One of the nonlinear effects characteristic of relativistic BWOs operat­
ing in the Ram an regime is the previously mentioned phenomenon of pulseshortening (see Section 1.3). It was originally proposed th a t pulse-shortening
results from having the BWO instability develop to such a point th a t the elec­
tron beam becomes unstable and then strikes the wall of the BWO structure,
thereby term inating oscillation. However, in experiments by Aleksandrov et
a l . , 2 9 in which the beam’s proximity to the structure wall was m easured, the
same degree of pulse-shortening was observed for significantly different beam
radii.
A more recent explanation for pulse-shortening proposed by B utler 2 1 ,2 5
is based on the saturation mechanism in relativistic BWOs. As a BWO
approaches saturation, the wave field amplitudes increase exponentially as
electrons are decelerated. Because of the absolute nature of the BWO instabil­
ity, the cold beam a t the entrance of the structure encounters a wave field th a t
is growing in time. This has two main consequences: first, the energy source
for the beam -structure resonance is eventually depleted as increasing num ­
bers of incoming electrons reflected by the wave fields become trapped in the
deep potential wells associated with the large am plitude fields. Together the
trapped and incoming electrons increase the space charge depression of the
beam until the beam current in the structure is eventually reduced to a level
below th a t necessary for m aintaining the BWO instability. Second, the
trapped electrons may eventually become untrapped and directed a t the beam
source. The incoming electrons may then collide with these electrons as well
as the trapped electrons causing the initially cold beam to become therm alized
even to the point of turbulence. Actual pulse-shortening is most likely due to a
combination of these two effects.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 3 2 -
Results from numerical simulations done by Butler support the p arti­
cle-trapping model for pulse-shortening . 2 5 Using
MAGIC,
it is possible to sim u­
late “non-ideal”, high-power BWOs (i.e., finite structure length, beam width,
and guiding magnetic field). Particle effects are monitored with phase space
plots such as those shown in Figure 2.7, which correspond to the same BWO
described in Figure 2.6. The development of the slow space charge wave and
beam bunching can be seen in Figure 2.7(a) as beam electrons are decelerated.
When the BWO reaches saturation, shown in Figure 2.7(b), the electrons are
fu rth er spread out in phase space with some having velocities less th an or
equal to zero as predicted.
In addition to
MAGIC,
a one-dimensional
PIC
code called
BWOSIM
(writ­
ten by Gilberto Barreto and Niels Otani) is currently being developed a t Cor­
nell University. It is hoped th at this much sim pler code will reproduce the
MAGIC
results, thereby instilling even more confidence in the particle-trapping
explanation for pulse-shortening. A description of the
BWOSIM
code and some
prelim inary results are given in Appendix B.
If particle-trapping is indeed responsible for the pulse-shortening in
high-power BWOs, then it should not contribute to pulse-shortening in highpower TWTs. This is due to the convective nature of the TWT instability,
where the wave fields grow largest a t the end of the structure; any trapped
electrons are prevented from influencing the incoming beam since they are
swept away from the interaction region. The trapped electrons may still con­
tribute to the TWT’s eventual saturation, but to date there have been no
reports of experiments with high-power TWTs exhibiting pulse-shortening. It
is this characteristic of the TWT th a t makes the long-duration/high-power
pulses observed with the tandem BWO-TWT possible (see C hapter 5).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
-3 3
a)
C/3
'e '
I
0.43 5
0 .0 0 0
z
b)
Cm)
•
:T
c/3
*. •
E
Z
2 .7
._>.p 4
<* .*•*» V : " . ^ f ?»•.*'
*. ...4,rt-f ' A**?'
0.435
0.000
Figure
.3^
i r - — r -■&-■• r?.
. . . •- r • • **« . .. .•
.••••—
(m )
Phase space plot from MAGIC simulation showing m om entum ver­
sus axial length for BWO (a) before saturation and (b) a t s a tu ra ­
tion. [From Reference 2 5 by permission.]
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited without p erm ission .
CHAPTER 3
GENERATION AND CHARACTERIZATION
OF THE RELATIVISTIC ELECTRON BEAM
A fundam ental p art of all BWOs and TWTs is the electron beam th a t is
guided through the slow wave structure by a longitudinal m agnetic field. The
use of intense relativistic electron beams in these devices became possible
w ith th e advent and rapid development of pulsed power technology. The
beam s are produced by applying a high voltage pulse to a specially designed
vacuum diode, which can then be characterized as a function of the im pressed
voltage, the resulting beam current, and the surrounding magnetic field.
3.1 T he O m ni P u lsed P ow er System
The relativistic electron beam used in the experim ents described in this
dissertation was generated using a slightly modified version of the Omni
pulsed voltage system . 4 2 A diagram of the m ain components of this system is
shown in Figure 3.1. In general, the system works as follows: the capacitors of
the oil-filled M arx bank are charged in parallel by an external power supply; a
trigger pulse then discharges the capacitors in series im pressing a large nega­
tive voltage pulse on the water-filled, coaxial Pulse Forming Line
(PFL).
The
center conductor of this line is divided into two sections by a sulfur hexafluoride (SF6) gas switch, which is pressurized so as to self-break ju st before the
42. S. Hum phries Jr. et al., Laboratory of Plasma Studies Report, L P S 220, (Cornell U n i­
versity), 1977.
-34-
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Reproduced
with permission
of the copyright owner.
Lucite
Interface
Lucite
Interface
Radial
Resistor
Gas Switch
Further reproduction
Cathode
prohibited without permission.
Water-filled
Pulse Forming Line
Oil-filled
Omni M arx Bank
Vacuum
Pump
Figure 3.1 Main components of the Omni pulsed power system.
- 3 6 -
peak of the impressed voltage pulse. When the electrical breakdown of the gas
closes the switch, a voltage pulse is then delivered to the second section of the
pulse line, which is term inated by the vacuum diode. A more detailed descrip­
tion of the m ain parts of this pulsed power system follows.
3.1.1 The Marx Bank
The Omni Marx is an oil-filled, 4 k J bank capable of delivering voltage
pulses of up to 750 kV. A diagram of the bank is shown in Figure 3.2. Ten
0.33 pF/75 kV capacitors (Maxwell Series S-31164) are charged in parallel by
a 25-50 kV power supply through 23 k£l resistors and grounded through
12 k£2 resistors. Both the charging and grounding resistors consist of fuelgrade Tygon tubing filled with a 16:1 w ater to copper sulfate (C uS04) solution
and sealed on both ends with polished copper electrodes. The Marx is erected
when a command trigger from a PT-55 trigger generator (Pacific-Atlantic
Company) closes the nitrogen (N2) gas switches th a t connect the bank’s five
stages in series. These switches are typically pressurized to 14 lbs/in2. A 6 pH,
copper tubing coil is placed a t the output of the Marx to protect the bank from
possible reflections. The polarity of the high voltage output is negative.
3.1.2 The Pulse Forming Line
A water-filled, coaxial transmission line is used to shape (determine the
width of) the voltage pulse th at is generated by the Marx bank and delivered
to the vacuum diode. As shown in Figure 3.1, a lucite interface separates the
oil-insulated Marx from the water-insulated
PFL
PFL.
The center conductor of the
is connected to the high voltage output of the Marx bank through the cen­
te r of this graded interface.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 3 7 -
- HV
Nitrogen Gas
Sw itches
O
O
SWITCH
TRIG IN
PWR
SUPPLY
C = C harg in g R esisto r
G = G rounding R esistor
Figure 3.2
Omni Marx bank: capacitors are 0.33 |iF/75 kV; charging and
grounding resistors are 23 k£l and 12 kQ, respectively; and the out­
put inductor is 6 pH.
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- 3 8 -
In gen era l, th is type o f PFL h a s a ch a ra cteristic im p ed a n ce g iv en by 32
( ro u te A
Tz
( 3 1 )
7-—
V
in n er J
where e = 81 is the dielectric constant of w ater (deionized), and router and rinner
are the outer and inner radii of the coaxial line, respectively. The Omni PFL
has a characteristic impedance of 9.1 O; its outer and inner radii (14.6 cm and
3.73 cm, respectively) were chosen to insure th a t the w ater provides sufficient
insulation to prevent breakdown for the electric fields used. M artin 4 3 has
shown th a t the breakdown electric field E br varies approximately as
** = ^ r o
tM V /cm i
(3-2)
where t is the time in (isec for the electric field intensity to change from
0.63E br to E bn A is the surface area of the inner conductor in cm2, and k = 0.6
for w ater when the inner conductor is charged negatively.
An SF 6 insulated output switch divides the PFL’s center conductor into
two sections. The switch is pressurized according to Paschen’s Law
44
so th at
its self-breakdown voltage is slightly less than the peak of the M arx’s voltage
pulse (to minimize switch jitter). When the switch closes, a voltage pulse is
delivered from the first to the second section of the line whose pulsewidth x is
determ ined by the length l x of the first section
x = —
c
lv
1
where c is the speed of light in vacuo. For the Omni system 2x ~ 100 nsec.
43. J.C. M artin, AWRE Internal Report No. SSWA/JCM/703/27, 1970.
44. J.D. Cobine, Gaseous Conductors, McGraw-Hill (New York), 1941.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
(3.3)
- 3 9 -
3.1.3 The Radial Resistor and Vacuum Diode
As shown in Figure 3.1, the second section of the
vacuum diode load. A maximum of h alf the
the diode, but only if the
PFL
P FL ’s
PFL
is term inated by a
voltage can be transferred to
and diode impedances (ZPFL and Zdiode, respec­
tively) are matched. Because Z diode is typically 80-100 Q. and Z PFL = 9.1 fi, a
radial resistor is placed between the
and diode, as shown in Figure 3.3, to
PFL
b etter m atch the two impedances. The radial resistor consists of a 2 inch
length of the coaxial line th a t is isolated from the
PFL
and vacuum diode
regions by two lucite interfaces and filled with a conductive solution of w ater
and sodium thiosulfate (Na 2S 20 3 *511:20). Its impedance Z radia[, which can be
changed by varying the concentration of the solution, is in parallel with Z diode
so th a t the total load impedance Z toad seen by the
Z
7
PFL
is
7
_
r a d ia l dio d e
load ~ 7
T~7
•
^ r a d ia l + ^ diode
a
\
To m easure the voltage im pressed on this load, a capacitive-resistive
voltage divider (also known as a Shipm an probe) is positioned 20 cm from the
radial resistor and mounted in the
P F L ’s
outer wall, as shown in Figure 3.3.
(The voltage monitor is described in Appendix A. 1.1.)
On the vacuum (2
x
10- 5 Torr) side of the radial resistor, the
P F L ’s
cen­
te r conductor is terminated, via a tapered extension, by a high-grade graphite
cathode. The purpose of the taper is to aid in matching the
PFL
and diode
impedances; it is also made long enough to insure th a t the cathode is
im m ersed in the strong axial m agnetic field generated by the solenoid shown
in Figure 3.3.
When the
P FL’s
voltage pulse is delivered to the cathode, electrons are
em itted from the cathode tip via field emission 4 4 and then accelerated through
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Reproduced
with permission
W ATER-FILLED P U L S E
FORMING LINE (PFL)
TO VACUUM PUM P
of the copyright owner.
S O LU TIO N -FIL LE D
RADIAL R E S IS T O R
SO LEN O ID
M AGNET CO ILS
Further reproduction
S T A IN L E S S STEEL
ANODE
TO
MARX
BANK
I
£>O
___U
G R A P H IT E
CATHODE
D R IF T
TUBE
prohibited without permission.
VACUUM
R O G O W SK I COIL
VOLTAGE MONITOR
LUCITE IN SU LA TO RS
Figure 3.3 Diagram of the radial resistor and vacuum diode region.
R O G O W SK I
C O IL
-4 1 -
a foil-less anode. The cylindrical cathode is machined to have a knife-edged
surface as shown in Figure 3.3, so th a t the emitted electrons form an annular
beam. The stainless steel anode, which also has a knife-edged tip, is hollow so
th a t the electron beam may pass on through to the slow wave structures. To
vary the anode to cathode
(AK)
gap separation, spacer rings are added beneath
the anode and/or cathode. The diode/beam currents are m easured by
Rogowski coils (see Appendix A.I.2) whose locations are shown in Figure 3.3.
3.1.4 The Guiding Magnetic Field
The axial magnetic field used to confine the electron beam radially is
generated by a solenoid th a t is 45.7 cm long and 8.9 cm in diameter. The
pulsed field is produced when a capacitor bank consisting of sixteen 29.2 (0.F/
14.5 kV capacitors (Tobe NRG Type XN273-3A) is charged to a voltage of 3-10
kV and then discharged by command through the solenoid coil. The strength
of the magnetic field is hence determined by the bank’s total capacitance, the
charging voltage, and the coil’s inductance. M easurem ents show th a t on aver­
age the solenoid has a 2.1 kG/kV calibration (see Appendix A .I.3). The mag­
netic field itself can be varied from 0-26 kG. Shot-to shot m easurem ents of the
field can be made from a 5 turn secondary coil th at is wound around the sole­
noid. Figure 3.4 shows the typical voltage response from this coil. It shows the
risetim e of the magnetic field to be ~ 2.5 msec.
The experim entally m easured axial profile of the m agnetic field is
shown in Figure 3.5. (These measurements are described in Appendix A .I.3.)
Notice th a t the field is greater in the region where the diode is located. This is
due to a set of extra windings added to one side of the solenoid coil by past
experim enters who had hoped to increase the field penetration into the diode
region. In the geometry of the present experiments, the extra windings cause
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 4 2 -
1 m se c /d iv
Figure 3.4 Typical voltage response from the magnetic field secondary coil.
with extra windings
1.2
~
extra windings disconnected
1 -
2
1
£
0.6
"
;
Diode Region
o -U
0
Slow Wave Structures
1 0
20
30
40
50
60
Axial Position (cm)
Figure 3.5 Axial profile of the magnetic field with and w ithout the extra wind­
ings added to the solenoid.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 4 3 -
the diode to be situated in a region with a magnetic field gradient. This unfor­
tunately causes the electron beam generated by the diode to have a transverse
velocity component. As can be seen in Figure 3.5, the field gradient exists even
when the extra windings are disconnected; however in th a t case the field
strength a t the diode is less th an th a t found in the slow wave structure region.
Experim ents confirm th a t the beam is better behaved with the extra windings
connected, so for all the experiments described here, the extra windings were
left connected.
To insure th a t the electron beam is produced in a magnetic field of suffi­
cient strength to confine the beam radially, the Omni system tim ing is
adjusted so th a t the Marx bank is triggered when the magnetic field pulse is
a t a maximum. The voltage pulse from the Marx is delivered to the diode
approximately 500-700 nsec after the system is command triggered; because
the period of the pulsed magnetic field
(~ 1 0
msec) is very long by comparison,
the beam can be generated and propagated through the slow wave structures
(~ 1 0 0
nsec) while the magnetic field is maximum and constant in time.
3.2 C h aracterization o f the Vacuum D iode an d E lectron Beam
As shown by Butler ,2 1 the basic BWO/TWT properties of frequency,
power, efficiency, etc. can be conveniently characterized as functions of y^-the
electron beam ’s effective energy. It is therefore useful to know how the basic
diode properties of geometry, voltage, and current affect the beam and, more
importantly, how they vary with yb.
F irst consider the diode geometry. In any space charge lim ited diode,
the generated current I is related to the diode voltage V by the expression
I = P V 3/2.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
(3.5)
- 4 4 -
Equation (3.5) is the well-known Child-Langmuir relation 4 5 ,4 6 where P, the
diode perveance, is a function of the diode geometry only. For example, in the
sim plest case of a diode composed of infinite parallel plane electrodes,
f2 e je \
1/2
(3.6)
P =
dr
where (in SI units) A is the cathode area, d is the AK gap separation, e0 is the
perm ittivity of free space, and e and m are the charge and m ass of the elec­
tron, respectively.
The exact relation for the perveance of the cylindrically symmetric
“cookie-cutter” diode of Figure 3.3 is not known. Nonetheless, values for the
perveance can be determ ined experimentally using (3.5) and m easurem ents of
the diode voltage and beam current (sample traces are given in Figure 3.6).
(a)
350 kV /div
0.78 kA /div
(b)
50 n sec/d iv
Figure 3.6 Typical waveforms of the (a) diode voltage and (b) beam current.
45. C.D. Child, Phys. Rev., 32, 492 (1911).
4 6 .1. Langmuir, Phys. Rev., 2, 450 (1913).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 4 5 -
Cathode Diameter = 1.6 cm
0. 9
AK Gap
«
■ 1.89 cm
o
2.21 cm
A 2.53 cm
0.7
1. 4
1.6
1.8
2
2.2
2.4
Diode Voltage 3/2 (xlO8 V3/2)
Figure 3.7 Plot of Ib versus V 3 /2 for a 1.6 cm diam eter cathode and A K gaps of
1.89, 2 .2 1 , and 2.53 cm; the magnetic field was 24 kG for all cases.
Specifically, when I b is plotted versus V3/2 as shown in Figure 3.7, the data lie
on a line whose slope is the diode’s perveance. (Note: the external magnetic
field is equal and constant for all cases.) Furtherm ore, the general dependence
of the diode’s perveance on geometry can be ascertained by considering the
related case of space charge limited electron flow between symmetric concen­
tric cylinders, for which Langmuir and Compton 4 7 found
2 tcr
4 7 . 1. Langmuir and KT. Compton, Rev. Mod. Phys., 3, 191 (1931).
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
(3.7)
-46-
1 0
Cathode Diameter =
1 .6
cm
8
>
I
6
£
8
4
1 /d
tV
Oh
2
1 /d'
0
1
1. 5
2
2.5
3
3.5
4
AK Gap (cm)
Figure 3.8 Plot of diode perveance versus AK gap showing 1/d dependence.
where r is the cathode radius. Notice th a t the perveance varies inversely with
d in this case and not with d? as in the parallel plane case of (3.6). As shown in
Figure 3.8, a plot of the perveance data versus AK gap separation confirms the
1Id
dependence.
The diode’s geometry, voltage, and current determ ine the effective
energy yb of the electron beam produced; using (2.18), yb can be calculated from
m easurem ents of these quantities. Experimentally, the dependence of beam
current on yb can be found for various cathode geometries by varying the diode
voltage. This is shown in Figure 3.9(a) where changes in diode geometry
am ount to varying the AK gap, since the same diam eter cathode is always used
(and hence the ratio r j r b is kept constant). The data in Figure 3.9(a) also show
th a t values for yb from 1.44 to 1.65 (which corresponds to a ~ 110 keV range of
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 4 7 -
1. 5
Cathode Diameter =1.6 cm
1. 4
AKGap
1. 3
■
1.89 cm
o
2.21 cm
a
2.53 cm
1. 2
1.1
1
*
a
A
0. 9
o
O
A
0.8
l
0. 7
l
l
I
l
I
I
I
1. 45
1.
0. 36
I
A
*
1 ----1---- 1---- 1----]
I
I
I
I
I
1. 5
1— }----1---- 1
I
I
'
I
I
1
1 . 55
I
I
I
I
|
I
I
I
I
I
1. 6
I
I
I
|
I
I
I
I
I
I
I
I
I
I
I
L-
1.7
1 . 65
<
Cathode Diameter = 1.6 cm
0.34
AKGap
■ 1.89 cm
0.32
J
o
0. 3
A 2.53 cm
AO
O
o
A
0.28
o o o o
A
I
I
I
I 1 I
1. 45
I
I
!
I
1. 5
I
I
o
o
A
A
0.26
1. 4
2.21 cm
A
I i l l !
1. 55
A
A
I L_J
1. 6
A
A
I I 1 I j
1. 65
I 1 I l_
1. 7
Yb
Figure 3.9 Dependence of (a) Ib and (b)Ib/I lim on yb for a 1.6 cm diam eter cath­
ode and AK gaps of 1.89, 2.21, and 2.53 cm.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 4 8 -
energies) can be obtained with this diode.
As discussed in Section 2.1.2, the space charge depression suffered by
the electron beam results in the lim it on beam current given by (2.19) for the
cylindrical geometry considered here. The degree of space charge depression,
which is m easured by the ratio /*//«m, varies nonlinearly with yb as shown in
Figure 3.9(b). Furtherm ore, since the electron beam power Pb is a function of
h,
Pb = I bV,
(3.8)
Pb is also limited by the beam ’s space charge depression; its variation w ith yb
is shown in Figure 3.10. An im portant consequence of this result is th a t the
efficiency of a BWO/TWT- defined as the ratio of microwave output power to
electron beam pow er-has a maximum value th a t varies with yb.
n — |— i— i— i— i— |— i— i— i i— |— i— i
600
i
j— |— i— i— i
i |— i— r
C athode D iam eter = 1.6 cm
A K G ap
§j
500P
_
u
°
%
(2
O
400
O o
0
&
a
o
A
A
A
A
■
1.89 cm
o
2.21 cm
& 2.53 cm
4
300
200
l
1. 4
I
l
I
I
l
1. 45
I
l
'
I
1. 5
I
I
I
I
I
I
1. 55
I___L—I__ I
1. 6
I
I
I
I
L_l
1 . 65
I
I
L
1. 7
Figure 3.10 Dependence of electron beam power on yb for a 1.6 cm diam eter
cathode and AK gaps of 1.89, 2.21, and 2.53 cm.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER4
EXPERIMENTAL CHARACTERIZATION OF THE
RELATIVISTIC BACKWARD WAVE OSCILLATOR
In order to optimize the microwave output from the proposed coupled
oscillator-amplifier system, it is necessary to perform first a series of experi­
m ents involving the BWO alone before running the tandem BWO-TWT device.
The objective of these experiments is to characterize the BWO used in the ta n ­
dem device by determining how its basic properties of frequency, pulsewidth,
power, and efficiency vary as a function of yb-th e effective energy of the elec­
tron beam. In one p art of these experiments, the strength of the axial m ag­
netic field th a t guides the beam is also varied.
4.1 Cold Tests: The BWO Passband
The dimensions of the BWO slow wave structure are given in Table 4.1
(see Figure 2.3 for explanation of geometry); curves showing the dispersion of
TM0i and TM 02 waves in this structure are given in Figure 2.4.
Table 4.1 Dimensions of the BWO slow wave structure.
Dimension
(cm)
Average radius, r0
1.4
Ripple depth, rj
0 .2
Ripple period, z 0
1 .1
Length
16.5
-4 9 -
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 5 0 -
Before running the actual BWO experiments, the frequency passband
of the slow wave structure is determined in a simple bench test. The setup for
this cold (i.e., no electron beam present) test is shown in Figure 4.1(a). A probe
connected to a programmable sweep generator (Wiltron Model 6637A) is used
to launch electromagnetic waves into one end of the BWO structure. A
straight antenna probe a t the opposite end picks up the signal and sends it to
a network analyzer (Wiltron Model 562), where the swept-frequency response
of the BWO is displayed. The launching/receiving probes are carefully placed
along the axis of the structure so as to excite/detect the azim uthally sym m et­
ric TM0m modes.
The results are shown in Figure 4.1(b). The TM0i passband of the BWO
structure is found to be approximately 8-13 GHz. The TM 02 mode is also
detected a t ~17 GHz, though its strength is ~15 dB down from the TM 01 mode.
4.2 The E xp erim en tal Setup
A sketch of the BWO characterization experiment is given in Figure
4.2, which also shows the location of various diagnostic monitors. (Note: the
diagnostic monitors and their calibrations are discussed in the Appendix A.)
As discussed in C hapter 3, a relativistic annular electron beam produced by
the vacuum diode passes through the hollow anode into the BWO slow wave
structure. A strong axial magnetic field prevents the electrons from striking
the wall as it guides the beam through the BWO and drift tube structures.
The beam is “dumped” in a section of stainless steel waveguide lying ju st
beyond the magnetic field region. Calibrated Rogowski coils situated a t the
entrance and exit of the BWO are used to measure the beam current. A capac­
itive-resistive voltage divider located behind the vacuum diode region (see
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-51(a)
Sweep
Generator
0-18 GHz
Microwave
Absorber
Network
Analyzer
Pick-Up
Probe
EMwave
Launch Probe
Microwave
Absorber
(b)
1
2
I
TM.
TM
-4
6
-7
2
4
6
8
1 0
12
14
16
18
Frequency (GHz)
Figure 4.1 BWO cold test: (a) diagram of the test setup and (b) TM 01 passband
for the structure.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Reproduced
with permission
of the copyright owner.
SOLENOID C O IL S FOR
AXIAL MAGNETIC FIELD
STA IN L ESS S T E E L
ANODE
: ROG OW SKI
COIL
dB SIGNAL SAM PLER
$
BW O
yi
ROG OW SKI C O IL
|
H cA M
DUMP
REGION
prohibited without permission.
1
Figure 4.2 Diagram of the BWO characterization experiment.
J
t:
[
1
T O MICROWAVE
-5 2 -
“ “ ’’^ G R A P H IT E
CATHODE
DRIFT TUBE
J]
Further reproduction
smmmwmmmsmm.
- 5 3 -
Figure 3.3) is used to m easure the diode voltage.
As described in Chapter 2, the microwave radiation generated in the
BWO propagates “backwards” with respect to the beam velocity (i.e., towards
the diode region); however, by making the diode’s hollow anode small enough,
it acts as a waveguide beyond cutoff (for X-band frequencies) and the micro­
waves are reflected back downstream to the output waveguide. An E r probe
located ju s t after the BWO can be used to couple to the radial electric field of
the microwave output with an average coupling coefficient of —45 dB.
In order to avoid electrical breakdown resulting from the large fields
associated w ith the high-power microwaves, a large diam eter circular
waveguide is used a t the output end of the experiment. As shown in Figure
4.3, the in n er diam eter of this oversized waveguide increases adiabatically
over a 27 cm length up to a thin mylar window, which serves as the vacuumto-air interface. A Chebychev array multihole directional coupler 4 8 with an
average forward coupling coefficient of -50 dB and a directivity of 25 dB is
used to sample the microwave output ju st beyond the m ylar window. Unfortu­
nately the microwaves cannot be radiated into an open area for far-field m ea­
surem ents due to space limitations. Consequently a large cone of broad band
microwave absorbing m aterial is placed a t the end of the oversized waveguide
to term inate the system.
As shown in Figure 4.3, the sampled signal from the Chebychev m ulti­
hole coupler is divided into reference and delay branches by a 3 dB splitter;
the signal in the delay branch travels through an extra 41 m eters of standard
X-band waveguide. The signals are further attenuated and then fed to rectify­
ing crystal diode detectors. The response of these calibrated detectors is dis­
played on an oscilloscope enabling frequency, power, and pulsew idth measure48. L.M. Earley et a i , Rev. Sci. Instr., 5 7 ,2 2 8 3 (1986).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Reproduced
with permission
of the copyright owner.
DELAY LINE
CHEBYCHEV
M U L T I-H O L E
D IR E C T IO N A L
C O U PLER
Further reproduction
MICROWAVE
OUTPUT FROM
BWO
OVERSIZED
WAVEGUIDE
VACUUM |
R E F E R E N C E L IN E
- 5 0 dB
XI
EL
AIR
TT
MYLAR
WINDOW
TT
MICROWAVE
ABSORBER
prohibited without permission.
D IO D E
DETECTOR
V A R IA B L E
A T TE N U A TO R
D IO D E
DETECTOR
V A R IA B L E
A T TE N U A TO R
Figure 4.3 Diagram showing the oversized waveguide and microwave frequency/power diagnostics.
I
£I
cn
- 5 5 -
m ents. The detectors and oscilloscopes are put in a F araday cage to shield
them from possible electric field transients.
4.3 E xp erim en tal R esu lts
In these BWO experiments, the frequency, pulsewidth, power, and effi­
ciency are found as functions of yfa, which is varied by changing the cathode
voltage and/or the diode AK gap to alter the beam current. The typical operat­
ing param eters used in these experiments are listed in Table 4.2. Sample
oscilloscope traces are shown in Figure 4.4 for the diode voltage and the cur­
ren t entering and exiting the BWO. The waveforms are typically 100 nsec in
duration.
4.3.1 Microwave Frequency
One of the general properties of BWOs th a t make them an attractive
source for driving systems of amplifiers, such as might be found in a phased
array antenna (see Section 1.3), is th a t they can be tuned electronically over a
wide frequency range while m aintaining a narrow frequency bandw idth.
The operating frequency of the BWO is determ ined from the tim e delay
between the reference and delay signals mentioned above. Examples of these
signals are shown in Figure 4.5(a). In general, the group velocity vB of an elec­
trom agnetic wave propagating in a dispersive medium is dependent on the
frequency of the wave. Because the delay line consists of a known length of
standard WR-90 rectangular waveguide, the frequency f of the microwaves
can be found from the time delay At using
At = —
v„
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(4.1)
- 5 6 -
Table 4.2 Typical operating param eters for BWO experiments.
Diode voltage
250-400 kV
Diode AK gap
1.5-2.5 cm
Beam current
0.8-2.5 kA
Beam diameter
1.5 cm
Beam energy (yb)
Magnetic field
1.20-1.65
24 kG
343 kV /div
0.65 kA /div
(b)
0.78 kA /div
50 nsec/div
Figure 4.4 Typical diagnostic waveforms: (a) diode voltage, (b) current enter­
ing BWO, and (c) current exiting BWO.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 5 7 -
where I = 41 m eters, and
2t 1/2
^
= c
1
- ff±
f
(4.2)
where c is the speed of light in vacuo, and fc is the cutoff frequency for the
waveguide (fc = 6.56 GHz in WR-90 waveguide). The delay line calibration
curve is shown in Figure 4.5(b).
The data plotted in Figure 4.6(a) show th a t the frequency of the BWO
can be tuned between 10-12 GHz by varying yb. This frequency range corre­
sponds to the TM 01 passband of the structure. The fact th a t the frequency
increases for increasing yb can be understood by re-examining the BWO’s dis­
persion curve (Figure 2.4): as yb increases, the slope of the beam line also
increases so th a t it intersects the structure’s TM 01 curve a t higher and higher
frequencies.
M easurem ents of the frequency bandwidth, plotted in Figure 4.6(b),
show th a t it is a t worst ~400 MHz. To verify th a t the BWO is operating in a
single mode, a few of the experiment shots are ru n with a tuned high-pass fil­
ter placed a t the output of the Chebychev directional coupler so as to elimi­
nate any signal below 14.5 GHz. With the filter in place, there are no high
frequency signals detected indicating th a t the BWO is indeed operating in the
TM 01 mode only.
4.3.2 Microwave Pulsewidth
The pulsewidth of the microwave power pulse is determ ined by m easur­
ing the width of the reference signal pulse a t its 3 dB point. The phenomenon
of pulse shortening (see Sections 1.3 and 2.3) is quite apparent when the
microwave pulsewidth is plotted versus yb, as shown in Figure 4.7. The pulse-
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 5 8 -
REF
30 mV/div
10 mV/div
D ELAY
50 nsec/div
(b)
1
I
I
f = f [ 1 - (1/cAt)2] -1/2
1
1 0
~
160
165
170
175
1 80
1 85
190
At(nsec)
Figure 4.5 Microwave frequency measurement: (a) voltage response of the ref­
erence and delay microwave detectors; (b) the delay line calibra­
tion curve.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 5 9 -
(a)
11.5 ~
09
#0 900
10.5 -
1 o1—
1. 3
1.2
1. 4
1.6
1.5
1. 7
T'b
(b)
500
400 ~
©
300 -
I
I
S
:
2 0 ° T
1 0 0
Oq
“
1 .2
1. 3
1.4
1.5
1.6
1. 7
Figure 4.6 Dependence of BWO (a) frequency and (b) bandw idth on yb.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 0 -
1
3
T)
60 -
&
*
“f 20
^
“
1. 4
1.45
1.5
1.55
1. 6
1.65
1.7
Figure 4.7 Dependence of BWO microwave pulsewidth on yb.
width is observed to decrease from ~85 nsec to ~30 nsec as the energy of the
beam (whose duration is -100 nsec) is increased. In Figure 4.8, the signal
from the reference line microwave detector is shown for various beam ener­
gies. From this data it is clear th a t a beam with a small value of yb is neces­
sary to achieve long microwave pulsewidths when running the tandem BWOTWT device.
4.3.3 Microwave Power & Efficiency
The peak power of the microwaves generated by the BWO is deter­
mined for each shot by m easuring the peak voltage from the reference line’s
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 1 -
50 n sec/d iv
Figure 4.8 Voltage response of the reference line microwave detector when y*
is (a) 1.48, (b) 1.55, and (c) 1.64.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 2 -
calibrated detector. All the attenuation present in the line (e.g., from the vari­
able attenuator and the Chebychev directional coupler) is taken into account.
The data, plotted in Figure 4.9, show th a t for small yb, the BWO’s peak
power increases as the beam energy increases. However, for yb > 1.53, the
power drops significantly from a maximum of -58 MW. The same effect was
previously observed by Butler et a l . 2 1 who reported no increase in microwave
power beyond a certain value of yb when operating a very sim ilar BWO struc­
tu re under slightly different conditions. This reduction in microwave power
may be partially explained by the decreased space charge depression of the
electron beam a t larger yb values (see Section 3.2 and Reference 21). T hat is,
for given diode and BWO geometries, the ratio IbHum decreases as yb increases
(see Figure 3.9). This consequently affects the coupling between the beam’s
slow space charge wave and the structure’s TM 01 mode.
70
1— I— I— I— I— I— I— I— I— I— f
60
o 1—
1. 4
1. 45
1. 5
1.55
1.6
1. 65
1. 7
Figure 4.9 Dependence of BWO peak microwave power on yb.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 3 -
The peak efficiency r| of the BWO in converting beam power into micro­
wave power is defined as
mw
(4.3)
where P mw is the peak microwave power and Pb is the beam power given by
(3.8). The peak efficiencies corresponding to the d ata in Figure 4.9 are plotted
as a function of yb in Figure 4.10 below. As with the power data, the peak
microwave generation efficiency of the BWO is found to decrease from a maxi­
mum of ~12% for yb > 1.53.
1—i—i—i—r
i—i—|—i—r
4
2
0
*—
1.4
l
I
l
1.45
!
I
I
I
1. 5
I
I
I
I
I
I
1.55
I
I
L
1 .6
1.65
\
Figure 4.10 Dependence of BWO peak efficiency on yb.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1. 7
- 6 4 -
As reported by various authors, a relativistic BWO’s microwave power
output can be affected by varying the strength of the axial magnetic field .4 9 ,5 0
This is observed in Figure 4.11 where the peak microwave power is plotted as
a function of magnetic field for a constant beam energy of yb = 1.50. Notice th a t
the power level is significantly reduced a t field strengths of -10 kG. The qual­
ity of the power pulse is also affected as can be seen in Figure 4.12, which
shows the voltage response of the reference line microwave detector for three
values of magnetic field strength.
The dip in power results from the excitation of the beam’s slow cyclo­
tron wave (see Section 2.1.3) by the BWO structure’s lowest order TE modes
6
5
I
4
£
>
I
s
o
2
1
5
1 0
15
20
25
30
Axial Magnetic Field (kG)
Figure 4.11 Dependence of BWO peak microwave power on axial magnetic
field for constant beam energy of yb = 1.50.
49. G.G. Denisov et al., Int. J. Infrared M illim eter Waves, 5, 1389 (1984).
50. R.A. Kehs et al., IEEE TYans. P lasm a Sci., PS-13, 559 (1985).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 6 5 -
1.3 MW/div
3.6 MW/div
11.3 MW/div
50 nsec/div
Figure 4.12 Voltage response of microwave detector when the axial magnetic
field is (a) 10.5 kG, (b) 12.6 kG, and (c) 16.8 kG.
when the magnetic field strength is approximately 10-15 kG. As shown by
B utler ,2 5 this slow cyclotron wave/TE mode instability (which is of the abso­
lute type) competes with the existing slow space charge wave/TM mode in sta­
bility resulting in a reduction of the total microwave power output. When the
magnetic field strength is greater than ~15 kG, the effect of this instability is
nullified since the frequency a t which it can occur lies below the cutoff fre­
quency for the lowest order TE mode th at can be supported in the structure.
In order to avoid this cyclotron damping in the BWO and BWO-TWT experi­
m ents, the axial magnetic field strength is henceforth m aintained a t 24 kG.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-6 6 -
4.4 The RF S ever
As discussed in Section 1.3, the component of the proposed tandem
BWO-TWT device th at allows long pulses and high power levels to be achieved
is the
RF
sever, whose purpose is to prevent microwaves produced in the BWO
from entering the TWT while allowing the beam to pass through its center.
The ability of the
RF
sever to cut ofFattenuate the BWO output is thus critical
to the success of the BWO-TWT system.
For these experiments, the
RF
sever is placed a t the output end of the
BWO as shown in Figure 4.13 (compare with Figure 4.2). The sever itself con­
sists of a 3.2 cm long graphite tube whose 1.9 cm inner diam eter is chosen so
th a t the sever acts as a lossy waveguide beyond cutoff reflecting and a tte n u a t­
ing electromagnetic waves with frequencies below -12 GHz. The in n er diam e­
ter is however large enough to allow the 1.5 cm diam eter beam to pass
through its center. The ends of the
RF
sever are tapered so th a t the beam expe­
riences a gradual change in impedance when passing from the BWO to the
sever to the drift tube (or TWT). The taper also reduces reflections.
-40 dB SIGNAL
i SAMPLER
RF SEVER
BWO
i
V
kkst
rogowski
.....
Figure 4.13 Diagram showing the location of the
COIL
RF
I
sever.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
RF
- 6 7 -
NYLON SCREEN
ANNULAR
ELECTRON
BEAM
DRIFT TUBE
BEAM & STRUCTURES
ALIGNED
BEAM HITTING
WALL
Figure 4.14 Diagram showing the electron beam p atte rn on the nylon witness
screen in the beam-structure alignm ent tests.
To determ ine if the beam is in fact passing through the sever cleanly
(i.e., w ithout striking the sever wall), a few shots are taken w ith a nylon “wit­
ness” screen positioned across the end of the drift tube in the plane perpendic­
u lar to the beam. When the electrons pass through the witness screen, the
energy deposited in it discolors the nylon enough to leave a pattern, as shown
in Figure 4.14. From this pattern it can be determ ined if the beam and the
structures are properly aligned. (Note: the nylon for these alignm ent tests was
graciously donated by former LPS secretary Sherry Biesecker from her exten­
sive “pantyhose-with-runs-in-them” collection.)
W ith the
RF
sever in place and the beam properly aligned, the BWO’s
peak microwave power output is significantly reduced as shown in Figure
4 .1 5 .
By comparison with Figure
4 .9 ,
the
RF
sever is found to atten u ate the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-6 8 -
70
s<-u<
I
0H
>
40|r
I
307
o
%
1
20
CL,
1or
1.4
1.45
1. 5
1.55
1.6
1.65
1.7
Figure 4.15 Dependence of BWO peak microwave power on yb w ith RF sever.
power flowing downstream by approximately 15-20 dB. Although further
attenuation is possible by increasing the length of the RF sever, the size of the
coupling section (which links the BWO to the drift tube or TWT) currently lim­
its the length of the sever to 3.2 cm. The attenuation may also be increased by
malting the sever from a microwave absorbing m aterial. One such sever made
from a sample of light weight carbon-fiber m aterial given to C.B. W harton by
Russian experimenters during a recent trip is found to have an attenuation of
~40 dB in cold tests. Unfortunately, outgassing of the carbon-fiber m aterial
m akes it nearly impossible to use in these experiments with the present vac­
uum system. The m aterial is also very fragile and difficult to machine.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER 5
EXPERIMENTAL CHARACTERIZATION OF
THE TANDEM BWO-TWT SYSTEM
The aim of these coupled oscillator-amplifier experim ents is to produce
pulses of microwave radiation of both high power and long duration. This is
achieved w ith the help of the RF sever, which is placed between the BWO and
TWT. A single electron beam is used to drive both structures. As with the
BWO characterization experiments, the basic properties of the tandem BWOTWT device are found as functions of yb so th a t the microwave output can be
optimized by simply adjusting the beam energy.
5.1 T he TWT S low Wave Structure
The dimensions of the TWT slow wave structure are given in Table 5.1
(see Figure 2.3 for explanation of geometry). In comparison with the BWO
structure (see Table 4.1), the TWT has a deeper ripple depth and shorter rip-
Table 5.1 Dimensions of the TWT slow wave structure.
Dimension
(cm)
Average radius, r 0
1.3
Ripple depth, rj
0.34
Ripple period, z 0
0.80
Length
15.9
-69-
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
30
- TM ,
£
S
1 5
“
TM ,
5
BW O
TW T
_
0
1
2
3
4
5
6
7
8
W a v en u m b er (rad/cm )
Figure 5.1 Cold dispersion diagram of both the BWO and TWT structures.
pie period. These differences in geometry manifest themselves in the disper­
sion of the structures’ electromagnetic waves as dem onstrated in Figure 5.1,
which is a cold dispersion diagram showing the TM 01 and TM 02 modes for both
the BWO and TWT structures. Notice th at the TWT curves are flatter th an
those of the BWO so that, as discussed in Section 2.2.3, the beam line co = kvb
intersects the TWT’s TM 01 mode where the group velocity (dw/dk) is positive.
The beam line in Figure 5.1 has corresponding beam energy of yb = 1.53.
The TWT described above is chosen for the tandem experim ents
because-of the slow wave structures available-its dispersion characteristics
are the most compatible w ith those of the BWO. T hat is, given the beam ener­
gies available, its range of TWT operating frequencies coincide best w ith the
BWO operating frequencies. Also, this particular TWT structure was built by
Butler
25
for use in experimental conditions fairly sim ilar to those used here;
thus, the present results m ay be conveniently compared with past data.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 7 1 -
5.2 C old Tests: The BWO-TWT P assband
Before running the actual experiments, the frequency passband of the
coupled BWO-TWT structure is found using the same cold test method
described in Section 4.1. The swept frequency response of the BWO-TWT is
shown in Figure 5.2(a); the TM 01 passband is found to be approximately 9.512 GHz, which is slightly narrower than th a t of the BWO structure alone (see
Figure 4.1). Furtherm ore, when the
RF
sever (see Section 4.1) is placed
between the BWO and TWT structures, the power transm itted in the TM 01
passband is attenuated by ~35 dB, as shown in Figure 5.2(b).
5.3 T he E xp erim en tal Setup
The experim ental setup for the tandem BWO-TWT experim ents is the
same as th a t used for the BWO characterization experim ents except th a t now
the drift tube in Figure 4.2 is replaced with the TWT structure, as shown in
Figure
5 .3 ,
and the
RF
sever is placed in the section th a t couples the BWO to
the TWT. In addition, the Rogowski coil th at previously m easured the BWO
entrance current is removed to allow the BWO-TWT system to fit w ithin the
axial m agnetic field region.
As the electron beam passes through the BWO structure, it becomes
bunched (i.e., density modulated) along the axial direction due to its interac­
tion with the electromagnetic wave fields in the structure (see Section 2.2.3).
After the modulated beam passes through the hole in the
RF
sever, it enters
the TWT structure and interacts with the TWT’s wave fields. The beam
excites an electromagnetic wave in the structure whose am plitude is conse­
quently amplified along the TWT’s length. (Notice th a t driving the TWT with
a pre-bunched beam eliminates the need for an input
RF
signal.)
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 7 2 -
(a)
-20
-50
-6 0
“
-7 0
-80
4
2
8
6
1 0
12
14
16
18
Frequency (GHz)
(b)
-20
-40
-7 0
-80
2
4
6
8
1 0
12
14
16
18
Frequency (GHz)
Figure 5.2 BWO-TWT cold test: the TM 01 passband (a) without and (b) with
the RF sever in place.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Reproduced
with permission
of the copyright owner.
SOLENOID C O IL S FOR
AXIAL MAGNETIC FIELD
Further reproduction
S T A IN L E S S S T E E L
ANODE
\
-45 d B SAMPLER
<WAAAf\AAAAAAAAATL
TW T
BW O
G R A P H IT E
CA T H O D E
JWAAAAAAAAAAAAAr
RO G O W SK I COIL
prohibited without permission.
BEAM DUMP
REGION
Figure 5.3 Diagram of the tandem BWO-TWT experiment.
- 7 4 -
-^MIAAAAAAAAAAAA^- w ^AAAAAAAAAAAAaapFigure 5.4 Diagram of TWT structure showing the tapered ripples.
When the beam enters and exits the TWT structure, it may encounter
changes in characteristic impedance th a t consequently result in signal reflec­
tions. A problem common to amplifiers is the ability of these stray reflections
or feedback to cause oscillations within the structure. Therefore, to improve
the impedance match of the TWT structure a t its two ends, the ripple depth is
gradually tapered along the structure length as shown in Figure 5.4. For the
TWT of Table 5.1, the ripple depth is tapered over 5 ripple periods on each
end. This tapering technique has been used successfully in the experim ents of
B utler
25
and Shiffler .22
In general, the TWT’s gain is defined as the ratio of output to input
microwave power. Thus, using the linear theory discussed in Section 2.2, the
gain G can be calculated from
G = 20 lo g (e t,z) = 8.68A-Z
[dB]
(5.1)
where /e, is the wavenumber of the TWT instability, and z is the length of the
structure. Experimentally, the TWT’s gain can be determ ined on a shot-byshot basis by comparing the microwave signals sampled by the E r probe a t the
TWT’s input end (see Figure 5.3) and the Chebychev directional coupler a t the
output end (see Figure 4.3).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 5 -
5.4 E xp erim en tal R esults
As in the BWO experiments of Chapter 4, the BWO-TWT’s characteris­
tic properties of frequency, pulsewidth, power, and efficiency are found as
functions of yb, which is varied by changing the cathode voltage and/or the
diode
AK
gap to alter the beam current. The typical operating param eters
rem ain the same as those described in Section 4.3.
At the start, a series of shots are taken using the nylon witness screen
described in Section 4.4 to verify th at the beam and structures are properly
aligned. Thereafter the screen is removed, and the following results are found.
5.4.1 Microwave Frequency
The operating frequency of the BWO-TWT is determ ined using the dis­
persive delay line technique described in Section 4.3.1. The data plotted in
2.5
12
©
1.5
1 1
1.45
1.5
1.55
1.6
1 .65
1.7
\
Figure 5.5 Dependence of frequency on yb for the BWO-TWT.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 6 -
Figure 5.5 show th a t the tandem device can be tim ed over a ~ 1 GHz range of
frequencies th a t lie in the TM 01 passband of the coupled structures. Tuned fil­
ters help verify th a t the device is operating in the TM 01 mode only-signals
w ith frequencies greater th an 14.5 GHz are not detected.
5.4.2 Microwave Pulsew idth
As discussed in Section 2.3, the pulse shortening phenomenon th a t is
observed with the BWO should not occur in the TWT due to the convective
nature of the TWT instability. In fact, when the 3 dB pulsewidth of the BWOTWT’s microwave power pulse is plotted versus yb, as shown in Figure 5.6, it is
found to decrease by ~ 20 nsec as the beam energy is increased. This repre­
sents much less severe pulse shortening when compared to the ~ 55 nsec
decrease observed over the same range of beam energies in the BWO-alone
1 0 0 — i— i— i— i— |— i— i— i— 1— |— i— 1— 1— i— |— 1— t—
1— I- 1 1— 1— 1— 1— |— i— i— r— 1—
<u
«
fc
40
20
-
0
1. 4
j
i i I i i i.
1. 45
j
1. 5
i i I i i i i L
1. 55
1. 6
1. 65
1. 7
Figure 5.6 Dependence of microwave pulsewidth on yb for the BWO-TWT.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 7 -
experim ents (see Figure 4.7). The fact th a t some pulse shortening still occurs
in the BWO-TWT probably results from the RF sever’s lim itations in reflecting
and attenuating the BWO’s microwave output completely. Also feedback from
the TWT can help the BWO reach saturation (see Section 2.2.3), which conse­
quently leads to pulse-shortening .2 1
5.4.3 Microwave Power & Efficiency
As described in Section 4.3.3, microwave power is m easured in the over­
sized waveguide using the Chebychev directional coupler. Sample signals from
the microwave detector are shown in Figure 5.7 for various values of yb.
The most obvious consequence of running the tandem BWO-TWT
experim ent is the dram atic increase in microwave output power compared to
th a t produced by ju st the BWO. As shown in Figure 5.8(a), peak levels in
excess of 100 MW are observed. This represents a gain of 10-15 dB over levels
obtained using the BWO alone (compare data above with th a t plotted in Fig­
ure 4.9). The peak efficiencies, defined by (4.3), corresponding to the d ata in
Figure 5.8(a) are plotted as a function of yb in Figure 5.8(b). A maximum
beam-to-microwave efficiency of ~ 35% is observed.
Notice th a t both the peak power and efficiency drop significantly from
their respective maximum values for yb > 1.53, as observed in the BWO char­
acterization experiments. As with the BWO, this may again be attributed to
the poorer coupling between the beam’s slow space charge wave and the TWT
structure’s TM 01 mode th at results from the beam’s deceased space charge
depression a t larger yb values.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 8 -
35 M W /d iv
(a)
(b)
o
50 n s e c /d iv
Figure 5.7 Voltage response of the microwave detector when yb is (a) 1.48, (b)
1.51, (c) 1.55, and (d) 1.57 for the BWO-TWT.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 7 9 -
(a)
1 40
1 20
g
100“
80
40 ~
1.4
1.45
1 .5
1 .55
1 .6
1 .65
1.5
1.55
1.6
1.65
1 .7
(b)
40
35 r
30 r
i
o>,
C
<
u
'5
S
w
:
25 r
2 0 r
:
1
10
1.4
1.45
1.7
Figure 5.8 Dependence of (a) peak microwave power and (b) peak efficiency on
yb for the BWO-TWT.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-805.4.4 TWT Gain
The high power levels obtained using the tandem BWO-TWT are, of
course, achieved by the TWT amplifier, which is driven with a pre-bunched
beam. As mentioned in Section 5.3, the gain in microwave power due to the
TWT is determ ined on a shot-by-shot basis by comparing the power m easured
a t points before and after the amplifier. Gains of up to 15 dB are observed,
which is in agreem ent with gains m easured by Butler
25
for the same TWT
structure. However, when gain is plotted as a function of the operating fre­
quency, as shown in Figure 5.9, the TWT is found to be effective only over a
narrow band of frequencies. In particular, gains of 10-15 dB are found only for
frequencies of 11.4-11.7 GHz.
20
5 “
Frequency (GHz)
Figure 5.9 Dependence of TWT gain on frequency.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 8 1 -
The dependence of the TWT gain on frequency can be understood by re­
examining the dispersion diagram (Figure 5.1) for the TM 01 mode in the TWT
structure. Since the point where the beam line intersects the structure’s TM0i
mode determ ines the operating frequency of the TWT, varying the beam
energy changes the microwave frequency. In particular, as yb decreases, the
slope of the beam line becomes flatter, and the intersection point moves “up”
the TWT’s TM 01 curve to higher frequencies. Eventually the intersection point
will pass over the peak of the TM 01 curve and come down on the opposite side
of the curve where the structure wave’s group velocity is negative. This of
course changes the nature of the beam -structure interaction from a TWT- to a
BWO-type instability-w hich explains why the TWT gain drops for frequencies
below ~ 11.55 GHz. For frequencies above ~ 11.55 GHz, the gain drops because
of the lim its of the structure’s TM 01 passband as well as the corresponding
decrease in BWO efficiency (i.e., its ability to m odulate the beam) for large yb
values.
5.5 E xp erim en ta l R esults w ith out the RF S ever
To examine the effectiveness of the
RF
sever, a few experim ental shots
are taken w ith the sever removed. In this way the oscillator and amplifier are
completely coupled with electromagnetic radiation able to pass from one struc­
ture to the other in either direction. This experim ental configuration is sim ilar
to the m ultiwave tandem experiments of Bugaev et a l ,2 6 with the m ain differ­
ences being th a t the Russian experiment uses larger diam eter structures (for
multimode operation) and electron beams of higher current (5-25 kA) and
longer duration
(1
(isec).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 8 2 -
From the microwave pulsewidth data plotted in Figure
5 .1 0 ,
it is clear
th at, w ithout the RF sever, the tandem BWO-TWT device succumbs to pulse
shortening as the beam energy is increased. The very short (<
pulsew idths observed for yb >
1 .5 0
40
nsec)
are most likely a result of feedback from the
TWT into the BWO helping the oscillator achieve saturation.
It is also interesting to note that, without the
power level (Figure
(Figure
5 .1 1 )
RF
sever, the microwave
is less than th a t obtained with the
RF
sever in place
In fact, it appears th a t the TWT is not amplifying the BWO’s
5 .8 ).
microwave output. These reduced power levels are probably due to a combina­
tion of (i) a reduction in the BWO’s efficiency resulting from its reaching sa tu ­
ration quicker than normal and (ii) anomalous behavior of the amplifier
resulting from oscillations excited in the TWT structure.
1
80
I
60
1
&
20
1. 4
Figure
5 .1 0
1 . 45
1. 5
1. 55
1. 6
1 . 65
1 .7
Dependence of microwave pulsewidth on yb for the BWO-TWT
without the RF sever.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Peak Microwave Power (MW)
- 8 3 -
1 20
60“
1. 4
1. 45
1. 5
1.55
1 .6
1 . 65
1 .7
Figure 5.11 Dependence of peak microwave power on yb for the BWO-TWT
w ithout the RF sever.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER 6
CONCLUSIONS
Results from the experiments described in this dissertation demon­
strate the feasibility of generating long pulses of high-power microwave radia­
tion using a special tandem BWO-TWT system. In order to achieve even
higher power levels, a multiwave version of the tandem system is currently
being developed.
6.1 Sum m ary o f E xperim ental In vestigation
The m ain purpose of this experimental investigation is to dem onstrate
th a t long pulses of high-power microwave radiation can be generated by a single-beam, modulator-TWT amplifier system. In this case, a relativistic BWO is
chosen as the electron beam m odulator because of its high efficiency and wide
frequency range. Pulse-shortening in the BWO is avoided by using a low
energy beam. Also the BWO is separated from the TWT by an RF sever
designed to minimize the electromagnetic coupling between the oscillator and
amplifier. High output power levels are thus achieved by driving the TWT
with a modulated beam.
6.1.1 BWO Characterization
The experiments described in Chapter 4 characterize the frequency,
pulsewidth, power, and efficiency of the relativistic BWO th a t is used to modu­
late the electron beam in the tandem device. O perating only in the fundamen-
-84-
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- 8 5 -
tal TM 01 mode, the BWO can be electronically tuned between 10-12 GHz by
varying yb, the beam ’s effective energy. When operated a t an optimum beam
energy of yb ~ 1.53, the BWO is found capable of producing single microwave
pulses of 60 MW peak power a t 11.5 GHz with 12% peak efficiency. At this
beam energy, the microwave pulsewidth is approximately 40 nsec. (The elec­
tron beam duration is typically 100 nsec.) By using lower beam energies,
microwave pulsewidths of up to 80 nsec are possible w ith peak efficiencies
£
10%.
As expected, the BWO’s output power and efficiency are also affected by
the strength of the guiding axial magnetic field. Cyclotron dam ping effects are
found to be m ost notable a t fields of - 10 kG. By operating the BWO with a
24 kG magnetic field, these damping effects are minimized.
Finally, the effectiveness of the graphite
RF
sever is determ ined by plac­
ing the sever a t the output end of the BWO and m easuring the transm itted
power. The sever is found to attenuate the BWO’s output power by 15-20 dB.
Although a longer sever would increase the attenuation, the length of the
present
RF
sever is limited, since the physical length of the tandem device
m ust fit within the axial magnetic field solenoid.
6 .1 . 2
The Tandem BWO-TWT System
As described in Chapter 5, the full tandem BWO-TWT device with the
RF
sever in place is shown capable of producing microwave power levels in
excess of 100 MW. In particular, when the tandem device is operated a t an
optimum beam energy of yb ~ 1.53, single microwave pulses of 120 MW peak
power a t approximately 11.5 GHz with 35% peak efficiency are observed. At
this beam energy, the microwave pulsewidth is approxim ately 45 nsec. Longer
microwave pulsewidths are obtained by using a lower energy beam. For exam­
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-86ple, when a 100 nsec electron beam of energy yb ~ 1.49 is used, microwave
pulses of ~ 80 MW at 11 GHz w ith a peak efficiency of ~ 20% and a pulsew idth
of ~ 55 nsec are observed.
The high power levels obtained with the tandem device result from
driving the TWT with a pre-bunched beam. The TWT gain is found to be 1015 dB over the narrow band of frequencies between 11.4 and 11.7 GHz. The
effectiveness of the amplifier is limited by the dispersive characteristics of the
TM 01 wave in the TWT’s slow wave structure.
When the tandem BWO-TWT is ru n w ithout the
RF
sever, the device
succumbs to pulse shortening, as expected. For beam energies of yb = 1.53,
microwave pulsewidths of ~ 30 nsec are observed. Also, without the
RF
sever,
the power output from the tandem device is significantly less th an th a t
obtained with the sever in place. At the optimum beam energy of yb = 1.53,
microwave pulses of only ~ 50 MW are observed. It would seem the TWT does
not provide any gain of the BWO output when the
RF
sever is absent. This
m ay be a result of oscillations excited in the TWT structure caused by the elec­
trom agnetic coupling of the BWO and TWT.
6.2 S u ggestion s for F u tu re Work
From the data presented in C hapter 5, it may be concluded th a t the fea­
sibility of producing high power (> 100 MW), X-band microwave pulses with
pulsew idths comparable to the electron beam duration
(1 0 0
nsec) has been
dem onstrated. Although the power levels obtained with this fundam ental
mode, tandem BWO-TWT device are modest compared with the gigaw att lev­
els reported from other high-current devices,8,12,13,26 higher power levels can
be expected from the larger scale version of the tandem device currently being
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 8 7 -
ELECT RO N
BEAM
BWO
RF
SEVER
TWT
Figure 6.1 Sketch of the 3.5 inch diameter, multi wave tandem device.
developed a t Cornell University .5 1 A sketch of the this m ultiwave tandem
device is shown in Figure 6.1.
The slow wave structures in the multiwave tandem device consist of
rounded-edge, an n u lar disks of stainless steel th a t fit inside a 3.5 inch diam e­
ter pipe. The separation between the disks can be varied by using different
size spacer rings. In addition, different diam eter disks are used to create a
tapered TWT structure, as shown in Figure 6.1. A 2 inch diam eter annular
electron beam will be used to drive the device.
By increasing the structure diameter, it is possible to operate the device
in higher order modes. A dispersion diagram for a tandem device consisting of
an amplifier (A) and two oscillators (B and C) is shown in Figure 6.2. The two
BWOs operate in different modes (one is in the TM 02 mode and the other in
the TM 03 mode) so th at the combined oscillator is mode locked-i.e., it selects
only one frequency.
51. C.B. Wharton et at., Laboratory of Plasma Studies Report, L P S 92-3, (Cornell U niver­
sity), 1992.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-88-
N
r
o
TM:
r~
0
0
LL
(400keV)
0
1
2
W avenum ber
4
3
[rad/cm]
5
Figure 6.2 Dispersion diagram for the three slow wave structures in the
m ulti wave tandem device.
To achieve the larger beam diam eter and current, the 4.4 k J Maxwell
M arx bank previously used for the Microwave Amplifier Experim ent (MAX) 2 5
will be used. This system, (which is described in detail in Reference 25) can
produce a 60-100 nsec electron beam.
By increasing the total current of the driving electron beam to ~ 12 kA
while m aintaining the current density low (to avoid the beam space charge
depression of the beam’s effective energy’ from becoming too great), beam
power levels of ~ 4 GW should be possible. Thus, if the tandem device is even
only 25% efficient, microwave power levels of ~
1
GW may be obtained.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
APPENDIX A
EXPERIMENT DIAGNOSTICS
The diagnostic equipment used in the experim ents described in Chap­
ters 3, 4 and 5 can be divided into two main categories: (i) electron beam diag­
nostics for m easuring the beam voltage and current as well as the axial
m agnetic field strength, and (ii) microwave diagnostics for m easuring the
power, pulsew idth, and frequency of the single-pulse microwave output.
A.1 E lectro n Beam D iagnostics
A. 1.1 The Diode Voltage Monitor
As mentioned in Section 3.1.3, the voltage im pressed on the vacuum
diode is m easured with a capacitive-resistive voltage divider th a t is positioned
20 cm behind the radial resistor and built into the
PFL’s
outer wall (see Figure
3.3). A diagram of the voltage monitor is shown in Figure A.I. It consists of a
small cylindrical barrel th a t is almost completely enclosed in polyurethane. As
shown in Figure A.l, there are two capacitances related to the monitor: Ci is
the capacitance of the water-insulated gap between the monitor and the
PFL’s
center conductor, and C2 is the capacitance of the polyurethane-insulated gap
between the monitor and the
PFL’s
outer conductor. The voltage Vmon mea­
sured by the monitor is thus given by
V„o„ =
<A-«
-89-
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- 9 0 -
P F L C EN TE R
CONDUCTOR
PFL OUTER
CONDUCTOR
POLYURETHANE
Figure A .l Diagram of the diode voltage monitor mounted in the PFL.
S C R E E N RO OM
HV PULSE
THYRATRON
PULSER
4.7 kO
O SCILLO SCO PE
HV PRO BE
R E SI S TO R
Figure A.2 Setup for calibrating the diode voltage monitor.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 1 -
The size of the barrel is made so th a t the ratio C2/Ci is large enough to insure
th a t monitor voltage can be easily m easured with an oscilloscope.
The diode voltage monitor is calibrated using the setup shown in Figure
A.2: a thyratron pulser (Maxwell Model 40051) is used to send a high-voltage
pulse down the
PFL’s
center conductor while a calibrated high-voltage probe
(Tektronix Model P6105) m easures the thyratron’s output (typically 6-7 kV).
The voltages measured with the probe and the diode voltage monitor (through
a 4.7 k £2 resistor) are displayed on a digitizing oscilloscope (Hewlett Packard
54200P), which is placed in a screen room to protect it from electric field tra n ­
sients. The voltage monitor signal is sent to the oscilloscope via a double­
shielded 50 Q. cable. W ith this configuration, the diode voltage m onitor is
found to have a
6 8 .6
kV/V calibration.
A .I.2 Beam Current Monitors-Rogowski Coils
The electron beam current is determined a t one or two places along the
beam ’s axial path by using Rogowski coils, 5 2 which actually m easure the
change in flux of the azim uthal magnetic field corresponding to the beam cur­
rent. A standard Rogowski coil consists of a solenoidal m agnet coil whose ends
are brought together to form a torus, as shown in Figure A.3; the torus has a
uniform cross-sectional area A, and the solenoid has a constant n num ber of
turns per unit length. Assuming the azim uthal magnetic field J30 varies very
little over a single turn spacing, such th at
|AB0|
«n,
"e
5 2 . 1.H. Hutchinson, Principles o f P lasm a Diagnostics, Cambridge Univ. Press (Cam­
bridge), 1987.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(A.2)
- 9 2 -
Figure A.3 Diagram of a Rogowski coil with an RC integrating network,
then the total flux <t>in the Rogowski coil is given by
d> = raj* \ j B ■d \ dA
A
(A.3)
11
where d \ is the line element along the solenoidal axis. Using Ampere’s Law,
(j>B • d \ = p07
I
(A.4)
where I is the total current encircled by the Rogowski coil, and |i 0 is the Per­
m eability of free space, (A.3) can be simplified to
<t> = n\inA I.
d<t>
The voltage V = - r - measured by the Rogowski coil is thus
KJbl
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(A.5)
Equation (A.6 ) is integrated electronically using a passive R C network,
as shown in Figure A.3, in order to get a signal proportional to I. Thus, for
times t « RC
n\iQA
V(t) = W
/(°-
(A ,?)
The Rogowski coils used in these experiments are made from 0.085 inch
diameter, semi-rigid coaxial cable as follows. A thin strip of PVC tape is wound
in a spiral along the length of the semi-rigid cable-the spacing between tape
windings determ ines the num ber of turns in the coil. The tape-wound cable is
then placed in a ferric chloride (FeCl3) solution, which etches away th a t p a rt
of the cable’s copper outer conductor th a t isn’t covered by the tape. Following
this etching process, the
PVC
tape is removed to reveal a long copper spiral
along the outside of the cable, as shown in Figure A.4. One end of the spiral is
then soldered to the cable’s center conductor providing a retu rn p ath for the
coil, which cancels out any contribution from the Bz field. Before bending the
cable into a circle, a protective sleeve of heat-shrink plastic is p u t around the
cable to prevent it from touching other metal surfaces. The coil is then
inserted into a metal holder that, when mounted, allows penetration of the
electron beam’s azimuthal magnetic field while shielding out electric fields. A
cross-section of the holder is shown in Figure A.4. The holder also protects the
coil in the event the electron beam becomes unstable and strikes the structure
wall.
The Rogowski coil is calibrated using the setup shown in Figure A.5.
With the Rogowski coil mounted in place, a stripped length of RG - 8 cable is
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 4 -
ROGOWSKI COIL
HOLDER
SEMI-RIGID COAX
ROGOWSKI COIL
I
■
ELECTRON
BEAM
Figure A.4 Sketch of the semi-rigid coax Rogowski coil and cross-section of the
stainless steel Rogowski coil holder.
THYRATRON
PUL SER
500 O
SC R E E N
ROOM
REF
TEST
RO G O W SKI
O SC IL LO SC OPE
RC INTEGRATOR
1
Figure A.5 Setup for calibrating the Rogowski coil.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 5 -
passed through the coil’s center. The cable is also passed through the center of
a self-integrating, commercially calibrated (10 A/V) Rogowski coil before being
term inated. A current pulse is sent down the cable length by discharging the
Maxwell thyratron pulser through a 250 Q load of high-voltage resistors con­
nected to one end of the cable. Comparison of the test and reference Rogowski
coil signals is done on a digitizing oscilloscope (HP-54200A).
Using this configuration, the Rogowski coil m easuring the BWO
entrance current (see Figure 4.2) is found to have a 1.30 kA/V calibration
when using a 4.35 psec integrator into a 100 O termination; the coil m easur­
ing the BWO exit current has a 3.11 kA/V calibration when using a 4.24 psec
integrator into a
100
Q term ination.
A. 1.3 The Axial Magnetic Field Monitor
The strength of the axial magnetic field used to confine the electron
beam radially is m easured using a “B-dot” monitor .5 2 As shown in Figure A.6 ,
the monitor itself consists of a 5.5 cm diam eter wire loop m ounted in the cen­
ter of a styrofoam cylinder th a t fits inside the spool of the axial m agnetic field
solenoid. The B-dot loop m easures the change in flux
of the axial m agnetic
field Bz such th at
do
dB
d i = N A it
(A-8)
where N is the num ber of tu rns in the loop, and A is the area enclosed by it.
(The name “B-dot” refers to the time derivative of Bz in (A.8 ).) The voltage
m easured by the loop, when integrated through an RC network, is propor­
tional to the field strength. Thus for times t <<RC,
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- 9 6 -
AXIAL FIELD
SOLENOID
B-DOT
LOOP
STYROFOAM
CYLINDER
Figure A .6 Diagram of the B-dot loop and styrofoam cylinder in the axial mag­
netic field solenoid.
V(t ) = ^
Bz {t).
(A.9)
An axial profile of the magnetic field, such as th a t shown in Figure 3.5,
is obtained by sliding the styrofoam cylinder to different axial positions along
the length of the solenoid.
As discussed in Section 3.1.4, current passes through the solenoid wind­
ings when the magnetic field capacitor bank is discharged. The calibration of
field strength to charging voltage with the B-dot loop positioned 25 cm from
the diode-side end of the solenoid is shown in Figure A.7. Notice th a t the cali­
bration factor is slightly different when the solenoid’s extra windings are dis­
connected.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 7 -
1—1 1 |
30
-
2 5
§
-a
'3
£
u
•■d
0>
—
“
■
1 1
©
extra windings
disconnected
o
with extra
windings
20
1 1 |
1 i
i
1
1
|
l
1 I
9
9
9
O
-
O
-
9
-
0
-
O
9
15
0
-
-
8
-
e
8
o
-
1
-
•
o
_
1 0
i
1 1 i
1 i
... i_i ..I.,
4
i
— . .. 1
i
1 1
6
1
8
i
J l ...
1
1 0
12
Charging Voltage (kV)
Figure A.7 Dependence of the axial magnetic field on charging voltage.
A.2 M icrow ave D iagn ostics
A.2.1 Microwave Detectors
In all the microwave power, pulsewidth, and frequency m easurem ents
described in Chapters 4 and 5, a sampled RF signal is converted into a voltage
pulse by using crystal diode microwave detectors.
The detectors used here are Hewlett Packard Model X424A crystal
diode detectors th a t are mounted into rectangular WR-90 waveguide hous­
ings; their optimum frequency range is 8.2-12.4 GHz. These detectors have an
approximate square-law voltage response-i.e., their output current is propor­
tional to the square of the waveguide’s electric field and hence directly propor-
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- 9 8 -
Sw eep Generator
11.5 GHz
15 dBm
V ariab le
A ttenuator
Crystal
D etecto r
DVM
Figure A.8 Setup for calibrating the crystal detectors.
_1 ,
1 0 0 0 F-1— 1— 1— '— I— 1— 1— r
1 1 , , ,
r
l
I
1 00
1
9bfl
3
I
1
o
I
I
I
I
I
!
I
L.
X
5
_l
I
10
.]
I
I
I
15
l
1
I
L.
20
Power (dBm)
Figure A.9 Calibration curve for the HP X424A crystal detector used in the
reference line for microwave power m easurem ents.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 9 9 -
tional to the power in the waveguide. To calibrate the detectors, the setup
shown in Figure A .8 is used.
A programmable sweep generator (Wiltron Model 6637A) is used as an
RF source; an 11.5 GHz cw-signal is used in these calibrations. Although the
power from the generator is constant a t 15 dBm, the power fed to the detector
is varied by using a variable attenuator (Hewlett-Packard Model X382A). The
voltage response of the detector (term inated in 50 £1) is then m easured on a
digital voltage meter. As an example, the voltage response of the microwave
detector used in the reference line for power, pulsewidth, and frequency mea­
surem ents is shown in Figure A.9.
A.2.2 The E r Coupler Probe
As shown in Figures 4.2 and 5.3, a simple E r coupler is used to sample
the microwave signal a t either the BWO’s output or the TWT’s input. As
shown in Figure A. 10, the coupler itself consists of a straig h t pickup probe
th a t couples to the radial electric field in the waveguide. It is made from a
short piece of 0.25 inch diam eter semi-rigid coaxial cable th a t is fit inside a
solid brass cylindrical sleeve. Although this type of coupler is non-directional,
it does have the advantage th a t its coupling coefficient can be easily varied by
adjusting the penetration depth of the probe into the waveguide.
The setup shown in Figure A. 11 is used to find the coupling coefficient
of the Er probe. With the E r probe mounted in place, a TM 01 wave is launched
down the circular waveguide by a straight probe antenna th a t is connected to
the Wiltron programmable sweep generator (much like the passband m ea­
surem ents described in Section 4.1). As the RF source is swept from 2-18 GHz,
the signal picked up by the E r probe is displayed on the W iltron network ana­
lyzer. In addition, the straight probe antenna is also connected to a -1 0 dB
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
Figure A. 10 Sketches of the E r coupling probe.
Network
A n alyzer
Probe
Sweep
Generator
- 1 0 dB
Directional
C oupler
M ic ro w a v e
A b sorber
Figure A. 11 Setup for calibrating the E r coupler probe.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
-101 —
•
10
Reflected Signal
-
-20
E Probe Signal
-70 r
9
1 0
1 1
1 2
13
14
Frequency (GHz)
Figure A.12 Swept frequency response of the E r coupling probe.
directional coupler (Omni Spectra Model PN 2020-6625-10) as shown in Fig­
ure A.11 so th a t any reflections can be displayed on the network analyzer also.
Results are shown in Figure A.12: the Er probe is found to have an average
coupling coefficient of approximately -45 dB over the frequencies of BWO and
BWO-TWT operation. The reflected power is less th an 10% requiring only a
small correction to the probe signal to account for standing waves.
A.2.3 The Chebychev Directional Coupler
As discussed in Section 4.2, measurements of the microwave power are
made in the oversized circular waveguide by using a Chebychev directional
coupler.4 8 In general, a directional coupler is a device th a t can distinguish
between forward and reflected waves propagating in a waveguide. This is
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
-102-
TERMINATION
COUPLING
A
HOLES
>
-------- ►
B
Figure A. 13 Sketch of a basic directional coupler.
accomplished by placing a series of holes in the waveguide wall th a t are a dis­
tance Xg/4 apart, where Xg is the wavelength of the wave in the waveguide. The
directivity of the coupler is a result of the hole positioning: the wave fields cou­
pled through the holes are reinforced in the positive direction and cancelled in
the reverse direction. Using the basic directional coupler shown Figure A.12,
the coupling coefficient K of the coupler is given by
K = 10 log
fPc\
td B ].
(A.10)
\ PA J
Furtherm ore the directivity D of the coupler is defined as
D = lOlog
(P L
c\
-1 0
log
(P c \
[dB ].
(A .ll)
Aj
In the Chebychev directional coupler, the diameters of the coupling
holes are varied in order to increase the frequency bandwidth over which the
coupler can sample signals efficiently. The holes are designed so th a t the rela-
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
- 1 0 3 -
0
Reflected Signal
1 0
20
30
40
50
Forward Arm Signal
60
70
80
9
1 0
1 1
12
13
14
Frequency (GHz)
Figure A. 14 Swept frequency response of the Chebychev directional coupler.
tive am plitudes of the voltages coupled a t each hole are proportional to the
coefficients of a Chebychev polynomial. 38
Calibration of the Chebychev directional coupler is achieved with the
sam e basic setup used to calibrate the Er probe (see Figure A. 11). Results are
shown in Figure A. 14: the coupler is found to have an average forward cou­
pling coefficient of approximately -50 dB and a directivity of 25 dB.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
APPENDIX B
DESCRIPTION OF THE BWOSIM CODE
In essence,
BWOSIM
is a particle-in-cell
(PIC)
code which sim ulates an
infinitely thin, annular, relativistic electron beam guided through a sinusoi­
dally rippled waveguide by an infinite axial magnetic field. The beam elec­
trons are one-dimensional such th a t only the z-components of the electrons’
velocity uz and momentum p z (i.e., the particle quantities) are used. On the
other hand, the electromagnetic fields are three-dimensional. The
BWOSIM
code deals only with the E r ,EZ, and B Qfield components however, since only
TM-type waves will interact with the electron beam in this idealized system.
B .l The P article and F ield E quations
The
BWOSIM
code advances the beam electrons through the system by
using the Lorentz force equation
dPz = eEz
dt
(B.l)
which links the electrons' momentum to the surrounding axial electric field.
The fields created by the beam are in tu rn calculated from Maxwell’s curl
equations, expressed here in cylindrical coordinate form. (Note: rationalized
Heaviside-Lorentz units are used throughout unless otherwise indicated.)
3i r = - c T z '
-1 0 4 -
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited without perm ission.
(a2 )
where J z = enbvz is the current density of the electron beam of density nb, and
c is the speed of light in vacuo.
B.2 l i m e In tegration o f th e P a rticle and F ield E q uations
Considering the form of (B.l)—(B.4), a simple leap-frog difference
method
53
is used in
BWOSIM
to integrate the equations in time. The particle
and field quantities are advanced in tim esteps of At as shown in Figure B .l
(where n is the timestep index). Spatially,
BWOSIM
uses a square m esh where
each grid has dimension Ar by Az. Figure B.2 shows how the field quantities
are arranged on each grid.
Using this convention, the field equations (B.2)-(B.4) can each be w rit­
ten in difference form as follows
. T - i . / l + l
/ ITT
\
Tl
- (Er)tm
,
/
D
\
^
=
At
1 / 2
. rj
\
fl
+
1 / 2
1 / 2 ~ (B |>) I „ - 1 /2
°I
(E
)n+ 1
- (E
)n
V *'/+l/2,m +l/2
v 2; / + l/2,m
At
Az
+ l/2
_
C
(B.5)
^
gj
(r) i + 1 /2
/ \
/ T"> \ Mi*
/ \ / n \ fl +• 1 /2^ .
+ 1/
1 /J
2d
( ( r ) u i (B q)Ul
l+hm
+1/2 _~ (r ) i rn_'»,l
(B q) L m + 1 / 2 ^
Ar
/
n+ 1/2
+ 1 /2 , m + 1 / 2
53. C.K. Birdsall and A.B. Langdon, Plasm a Physics Via Computer Sim ulation, McGrawHill (New York), 1985.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 0 6 -
Er
Jz
Er
Jz
E
E
-§> tim e
n+1
n+1/2
n-1/2
At
Figure B .l Advancement of the particle and field quantities in time.
)------------------- o------------------- }
Er
Er
O-.......... ... Bo
<
Er
E,
@
E,
o
........
E
B„
, t—----------------- o
Ar
E
ri>
Az
Figure B.2 Spatial arrangem ent of the field quantities on each grid.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 0 7 -
l,m+
1/2
l , m + 1/2
At
_
= C
I + 1/2, m + 1/2
f
1/2. m + 1/2 '
Ar
V
(B.7)
/ 1
-c
Az
\
where I and m are the radial and axial indices, respectively.
The particle and field quantities are coupled together in (B .l) and (B.3)
through the momentum p z and current density J z, respectively. Calculating J z
involves “weighing” the contribution of each particle’s velocity to the current
density a t each grid point. In the
BWOSIM
code, a first-order
PIC
weighting
53
is used so th a t a t the n+1 /2 timestep, the contribution from th e j'th particle, as
depicted in Figure B.3, is given by
S(z)
z m-1,
zm
z.j
z m+ 1,
Figure B.3 Shape of particles in the
z m+2,
BWOSIM
code.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-108-
zm
z m +1
(B.8)
where A is a normalizing constant, m is the grid index, Pz = vz/c is the p arti­
cle’s normalized velocity, and S(z) is essentially the shape of the particle
(which in this case is triangular).
Advancing the particles with (B.l) also requires a weighing of quantities. To determ ine p z, the contribution of each grid’s Ez field is weighed to the
momentum of each particle so th a t (B.l) becomes
z m —Z m -
1
1
From (B.9) the velocity of the particles can be found since p z = 7 m vz.
Using the dimensionless momentum t,z = p z /me, then
(B.10)
which then allows the particle’s axial position z to be determined from
(B .ll)
B.3 C ondition for N um erical S tability
The leap-frog difference method described above is accurate to second
order in both time and space. Its stability however is dependent on the size of
the grid and timestep used. In general, a hyperbolic partial differential equa­
tion of the form
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 0 9 -
% + iM - = 0
at
ax
(B.12)
where f = f ( x , t ) and a is a constant, can be w ritten in leap-frog difference
form as
ra + l
/k
h i
~lk
At
-
1
/■ f t i
rn
U k + l- lk - 1
+U
2Ax
=0
(B.13)
where k is the grid index. It can be shown th a t such a difference method is sta­
ble only if
uAt
^ -< 1 .
Ax
(B.14)
The inequality in (B.14) is known as the Courant condition for stabil­
ity . 53 The particular Courant condition corresponding to the hyperbolic equa­
tions used in
BWOSIM
can be found by considering the case of a TM-type
electrom agnetic wave propagating through an infinitely long cylindrical
waveguide of constant radius. Assume the electric and magnetic fields have
the form
E z (r, z, t)
= E z0j 0 (kcr ) e x p [ i( ( M - k z ) ]
(B.15)
E r (r ,z ,t)
= E rQj l (kcr)exp [i ( w t - k z ) }
(B.16)
B Q(r ,z ,t) =
(kcr)exp [t ((£>t-kz) ]
(B.17)
where jo(kci') and j\{kcr) are Bessel-like functions th a t have the following prop­
erties
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-110W , , 1/ 2 - W
, . 1/2 =
w
'
( B 18)
miQ>
( r ) [+l/2Ar
The constant
(R19)
kc is determ ined by the boundary conditions
a t the
waveguide wail. If the waveguide has a radius r = a, then from (B.15)
j Q(kca) = 0.
(B.20)
This implies th a t kc= K /(lmaxAr) where a = lmaxAr, and i f is a constant th at
specifies the zero of the j 0 function.
Substitution of (B.15MB.17) into (B.5MB.7) yields the following set of
equations
coA£
kAz
co£r 0 d if( — ) = c£Beod if( — )
(B.21)
c o ^ 0 d i f ( ^ ) = - ic k cB Q0
(B.22)
kAz
(aAt
ckcE zo = i ckE r0d if ( ) - O)jBe0 d i f ( —— )
where dif(x)
(B.23)
Equations (B.21MB.23) can be reduced to the single
equation
co2 dif2 ( - £ - ) ~ (^ c) 2 dif2
= (ckc) 2.
(B.24)
Notice th a t in the limit where Az and At go to zero, (B.24) reduces to the sim2
2
r2 2
pie dispersion relation for a bounded electromagnetic wave k c = co - k c .
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-111Equation (B.24) can be rew ritten as follows
cAt
. 2 kAz
n <^ “ ) J
', 2 , ,2 .
Ai
(B .2 5 )
In order for co to be a real quantity (i.e., to avoid instabilities), the following
inequality m ust hold
.2 , , 2
c
A2
. 2 .kAz
m <T )
(B.26)
Clearly the largest value th a t £ can have in this inequality is k max = idAz, such
th a t
(B.27)
which simplifies to
,
1
cA*
2
> - s2- aJ (iL,„„Ar
r~)
2
(tAz )
2
•
(B.28)
The inequality in (B.28) is thus the Courant condition for the difference
method used in the
BWOSIM
code.
B.4 In itia lizin g th e BWOSIM Code
When running
BWOSIM,
the operator begins by providing the following
information in the input data file
• the average radius (r0), perturbation size (rx), period (z0), and length (L ) of
the slow-wave structure
R e p r o d u c e d with p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-112• the radius (rb), initial energy (Win), initial current (/,„), and effective energy
(y6) of the electron beam
• the tim estep size (At) and num ber in') of tim esteps to run
After calculating the grid sizes Ar and Az from the input data, BWOSIM
then verifies th a t the Courant condition (B.28) is satisfied. If the condition is
violated, the code notifies the user and then comes to a halt. The operator m ay
vary the absolute maximum num ber of grids in the r and z directions as well
as the maximum num ber of particles to be used in the simulation.
If the Courant condition is satisfied, BWOSIM then defines the inner
wall of the structure by making the maximum num ber of grids in the radial
direction lmax a function of the axial position m using (2.36). T hat is,
2 k Az (m + 1/2 )
l r n a x ( m ) = ^0 + / l s i n
(B.29)
W hen considering the special case of a smooth-walled cylinder, the period of
the structure is set equal to structure length so th a t lmax(m) = l0 for all m.
Having set up the structure, BWOSIM then determ ines the initial values
of the particle and field quantities. Beginning with the initial velocity of the
beam electrons, BWOSIM assumes the electrons have no kinetic energy when
they are produced at the cathode such th at
(B.30)
where (3m is the initial normalized velocity. The initial normalized m om entum
%in is then calculated as follows
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 1 3 -
t
= ~f==== •
(B.31)
To determ ine the initial current density (J2),„, the initial beam current
(Ib)in is divided by the area of a thin ring of radius rb and thickness Ar such
th a t
C (h )
v
uAr /O
(2rb +‘VAr)
■
(■J
where C = 3 x
1 0 12 /47 c is
Z)
z
• =
»»
(B -3 2 >
' -A
the constant which converts (J z)in into units of stata-
mps/cm2.
Finally
BWOSIM
sets the initial values of the E r ,E Z, and
B 0 fields
along
the boundaries and within the structure. The initial values may be changed
w ithin the program.
B.5 The M ain T im estep Loop
In the m ain timestep loop, the particle and field quantities are calcu­
lated and advanced as described above. (The loop is summ arized pictorially in
Figure B.4.) Those quantities which are evaluated a t the half-timestep inter­
vals (Be, J z, (32, and E;2) are only advanced by At /2 on the zeroth tim estep (t = 0).
Thereafter they are advanced by the full tim estep At.
Also on the zeroth timestep, there are no particles in the system, so at
t = 0, the m ain loop begins by injecting a set of particles. Thereafter (t > 0), it
advances the positions of the particles already in the system before injecting a
new set of particles. The particle positions are advanced using only a halftim estep in (B.10). They are advanced by another half-tim estep later in the
loop.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-1 14 -
Advance
Start here
when t = 0
Advance
Inject
Particles
Calculate
Calculate
Calculate
Inject
Particles
Advance
Figure B.4 Diagram showing the main tim estep loop.
To inject particles, the code assumes there is a semi-infinite line of par­
ticles stretching from z =
to z = 0 (i.e., the left edge of the structure). The
particles are equally spaced in intervals of length Az inj
AzinJ = J
l.
(B.33)
aug
where L is the length of the structure, and N aug is the pre-set, average num ber
of particles allowed in the system (whose value can be changed w ithin the pro­
gram). These particles are all moving in the +z direction w ith velocity cP,„, and
as they enter the structure, they are “born” into the sim ulation.
With particles in the system,
BWOSIM
then advances th eir normalized
momenta ^ using (B.9) and calculates their new normalized velocities Pz from
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 1 5 -
(B.10). It then determines the current density J z from (B.8 ) and the B 0 field
from (B.7) for the resulting distribution of particles.
Following these half-timestep calculations,
BWOSIM
advances the p a rti­
cles’ positions using a half-timestep in (B.10) and injects a new set of particles
into the system. The Er and E z fields are then calculated from (B.5) and (B.6 ),
respectively. At this point, the tim estep is advanced to t n + 1 = tn + At, and the
loop repeats itself.
B.6 N um erical Checks and D iagn ostics
Every time the particle positions are advanced,
BWOSIM
runs a self­
check to insure th a t extraneous particles are not included in the sim ulation.
Specifically, when a particle is found which has z < 0 or z > L,
BWOSIM
will
transfer th a t particle to a new position far to the left of the system’s boundary
(i.e., to a position with z « 0). In this way the particle is essentially removed
from the simulation. This sorting is useful for avoiding effects th a t result from
particles piling up ju st outside the system’s boundaries. Another self-check is
ru n every Atsrl timesteps when
BWOSIM
sorts through all the particles th a t
have been born into the simulation and updates the num ber N p which are
actually within the system’s boundaries. If N p is greater than the maximum
num ber of particles allowed, the code sends an error message to the operator
and then stops. The value of Atsrl can be changed within the program as
needed.
Every Atpu timesteps,
BWOSIM
records particle and field values by m ak­
ing two types of plots. The first type is a phase-space scatter plot of % versus z,
and the second is a history plot of E z versus
2
and J z versus z. Examples are
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
— 116 —
given in the next section. The value of AtpU can also be changed w ithin the pro­
gram as needed.
BWOSIM
also has the capability of storing the plots mentioned above as
H ierarchial D ata Form at (HDF) files, which can be displayed sequentially as a
movie of the simulation using the graphics application NCSA Image. (The
HDF form at and NCSA Image were developed by the National Center for
Supercom puting Applications.) The program lines which create these frames
may be commented out to save memory space.
Finally, the Ez field a t a given position z = zpU is recorded in a d ata file a t
the end of every timestep. The operating frequency of the sim ulated BWO is
determ ined by taking a Fourier transform of this E z time history. The value of
zpit can be changed within the program.
B.7 R esu lts u sin g BWOSIM
The simulations th a t have been performed using the
BWOSIM
code can
be separated into (i) diagnostic runs involving a smooth-walled waveguide and
(ii) full BWO simulation rims involving the slow wave structure.
B.7.1 Test of Electromagnetic Wave Propagation
To insure th at
BWOSIM
functions properly, a few simple runs using the
smooth-walled waveguide case are performed as diagnostic checks. The first
involves running the code without particles to determ ine if the field quantities
are being advanced correctly. This is done by launching a TM0i wave a t one
end of the cylindrical waveguide and then m easuring its frequency after it has
propagated to the other end of the waveguide.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
—117 —
By assum ing a (co)sinusoidal t and z dependence, the field components
of a TM 01 wave in a cylindrical waveguide of radius r0 can be easily shown to
be
E z = E 0J 0 (2.405—) cos oat cos k z
ro
Er =
Ba =
0
krn
(B.34)
r
(2.405 —) coscoisin/jz
rq
a.t Uu
(B.35)
a>r«
r
, (2.405 —) sin(titcoskz.
2.405c u 1
r0
(B.36)
Equations (B.34MB.36) can be used to set up the initial conditions in
the waveguide. In particular, a TM0i wave was launched w ith r 0 = 1.6 cm,
co = 2;t (11 GHz) = 6.91 x 10 10 rad/sec, and k = 0.573 rad/cm. The radial depen­
dence of the initial field quantities is shown in Figure B.5.
1
0.8
0. 6
0 .4
0.2
0
0.2
0 .4
0.6
0. 8
Figure B.5 Radial dependence of the field quantities a t t = 0 and z = 0.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-
118 —
i i i r~ 1 T"TT[T 1 T T i r t
1.2
—i—l—i—i—|—1—1—I—1—
t
1
T M 01
E z/Eo
0.8 0.6
_
0. 4
0.2
0
*i i
i
10
Li_i i—i—1—L. 1 1 L.Lj_i_i_i_
15
20
25
Frequency
(G H z)
Figure B .6 Fourier transform of the Ez field a t r = r j 2.
After running the code with the above conditions, a fast Fourier tra n s­
form of the Ez field a t half the waveguide radius (Figure B.6 ) confirms th a t the
wave rem ains in its initial TM0i mode since there is only one frequency com­
ponent, and it is at 11 GHz. Also, a quick check of the fields’ radial dependence
shows th a t they haven't varied from their initial values.
B.7.2 Cold Beam in a Smooth-Walled Waveguide
Having ascertained th a t the fields are being advanced correctly, the
next diagnostic run introduces a cold (monoenergetic) beam of particles into
the smooth-walled cylindrical waveguide. As explained in C hapter 2, there is
no BWO-type interaction between the beam and the structure wave in this
geometry-i.e., the particles should pass through the waveguide unaffected.
With the initial values of the fields set to zero,
BWOSIM
is run using the follow­
ing d ata in the input file:
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 1 9 -
• r 0 = 1.6 cm, r x = 0.0 cm, z 0 = 16.5 cm, L = 16.5 cm
• rb = 0.7 cm, Wirl = 350 keV,
= - 1.5 kA, yb = 1.30
° At = 10- 1 2 sec, n = 1000
The grid in this case is such th at Ar = 0.032 cm and Az = 0.165 cm. In
Figure B.7 the phase space plot (3* versus z is shown a t various tim es for this
run. U nfortunately these plots show th a t the particles are not only slowing
down b ut also reversing direction fairly early in the simulation.
The problems encountered in the case above are seemingly associated
with the particles entering an empty waveguide. To avoid this problem, BWO­
SIM can be ru n with the particles already in the waveguide a t the sta rt of the
sim ulation. This is done by injecting and advancing particles before the begin­
ning of the m ain time-step loop (that is, for tim esteps t < 0). Of course, in this
case the fields are no longer zero at t =. 0 ; their initial values are determ ined
by the initial beam current and velocity.
Figure B . 8 shows phase space plots a t three different tim es when BWO­
SIM is ru n with the same input param eters shown above b u t with particles
already in the waveguide a t t = 0. Notice th a t a t t = 0, the beam is monoenergetic with a velocity of 0.64c. However as the sim ulation progresses, the beam
becomes slightly thermalized as evidenced by the 0 . 1 c spread in particle veloc­
ities. In fact this spread is simply (twice) the therm al velocity of the beam,
which can be used to determine the beam’s tem perature (and Debye length).
This therm alization of the beam (which does not get any worse - i.e., stabilizes
a t vth = 0 . 1 c) is due to a beam-grid instability in the code and is purely num er­
ical in origin .5 3 To demonstrate the instability’s num erical character, BWOSIM
is ru n using the same physical param eters as used above - only the num ber of
grids in the z-direction was increased five times such th a t Az = 0.033 cm. (The
tim e-step size is consequently decreased to 7 x 10- 1 3 sec in order satisfy the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-120Phaaa apaao a t
1
tlraoatap
500
.■
0
6
4
a
a
-.a
-i.a
f
tfl
za
31
41
5f
71
00
tlnaatap
1900
a
Phaao
spooa
at
a
.6
4
a
s
2
- 1.0
0
10
20
30
40
50
68
78
00
111
x
Figure B.7 Phase space plots showing a cold beam entering a smooth-walled
waveguide a t (a) t = 0.5 nsec, and (b) t = 1.5 nsec.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-121-
Phdae
apdco
at
t fnestep
(3
10
.0
.2
(a)
a -
- .4
-
.6
-.0
■ ■ 1 > i .i
10
29
30
Phdoe
i
■ i
49
50
apdce
dt
J
6fl
70
t inea tep
■ 1__ i__ L60
90
1fl0
xixx
0
6
y W \ V] ! [ /' j
)
V
)
I I t Tf \ * *'!
4
2
0
2
6
6
0
Figure B .8
3
50
60
70
60
130
Phase space plots showing the cold beam in a smooth-walled
waveguide a t (a) t = 0 nsec, and (b) t = 1 0 nsec.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
-122-
Phaaa opaca
it
trnaatap
8
a
.a
,k
.4
.a
(a)
I
2
4
.8
8
»
388
198
Phaao
ip u a
at
288
398
338
tfnaatap
498
****
1.8
.0
.8
.4
.a
09
«
s
(b) “
-.2
- .4
-
,4
-.8
-
1.8
■
58
W
198
338
338
4U
498
318
I
Figure B.9 Phase space plots when grid size is halved a t (a) t = 0 nsec, and (b)
t = 7 nsec.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
- 1 2 3 -
C ourant condition). The results are shown in Figure B.9. Notice th a t although
there is therm alization of the beam, it is now only half as severe as before. The
tem perature of the beam is thus controlled by the grid size.
B.7.3 Slightly Warm Beam in a Slow Wave Structure
With the bugs described above in mind, the BWOSIM code is now run
with a slow wave structure. Particles are allowed to fill the structure before
the m ain time-step loop. The following input param eters are used:
• r0 = 1.4 cm, ri = 0.2 cm, z 0 = 1.1 cm, L - 16.5 cm
• rb = 0.7 cm, Win = 350 keV, Iin = - 1.5 kA, yb = 1.30
• At =7
x
10~ 13 sec, n = 1000
The grid in this case is such th a t Ar = 0.032 cm and Az = 0.033 cm.
Unfortunately, there does not seem to be any difference between this ru n and
th a t shown in Figure B.9. Movies made of E z contour plots seem to indicate
th a t there is some kind of wave propagating radially as well as axially, b u t its
origin is as yet not understood.
B.8 C onclusions
The one-dimensional
PIC
code
BWOSIM
has been w ritten in hopes of sim ­
ulating the saturation mechanism of relativistic BWOs operating in the
Ram an regime. Despite (or perhaps due to) its simplicity, the code is not yet
successful in simulating BWO systems. Diagnostic runs, however, seem to
indicate th a t the individual parts of the code relating to the fields and p a rti­
cles are working correctly, with some acceptable error due to num erical in sta ­
bilities. F uture work will concentrate on improving the m anner in which the
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
in itia l an d boun d ary conditions o f th e fields in a slow w a v e stru ctu re are rep­
r e se n te d in th e BWOSIM code, as w ell a s e lim in a tin g th e beam -grid in sta b ility
th a t ten d s to th erm a lize w h a t should be a cold beam .
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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