Generation of Coherent High-Power Microwave
Radiation with Relativistic Electron Beams
by
Wen Hu
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Master of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1994
Massachusetts Institute of Technology 1994. AR rights reserved.
I
Author.....................................
ri_ ...................
D partment of Physics
May 18, 1990
Certified by .............................
/
I . . . . . . . . . . . . .. 1.1
.. . . . . . . .I.
George Bek fi
Professor of Physics
Thesis Supervisor
A ccepted by .........................
: ..............................
George F. Koster
Chairman, Graduate Committee
MASSACHUS
Mt INST17UTE
OF TECHNOLOGY
MAY2
1994
LIBRARES
Generation of Coherent High-Power Microwave Radiation
with Relativistic Electron Beams
by
Wen Hu
Submitted to the Department of Physics
on May 18, 1990, in partial fulfillment of the
requirements for the degree of
Master of Science in Physics
Abstract
In this thesis, a two-dimensional, nonlinear theory of the double-stream cyclotron
maser is developed in a cylindrical configuration. A dispersion relation which describes small-amplitude, cyclotron space-charge waves on two copropagating gyrating
electron beams in a finite axial magnetic field, is derived and used to analyze the cyclotron two-stream instability. The sensitivity of the instability growth rate to axial
momentum spread of the electrons within each beam is investigated. The saturation levels of the instability are obtained from simulations for parameter regimes of
experimental interest.
In addition 7 the characteristics of beam modulation in a 33 GHz relativistic
klystron amplifier(RKA) are studied experimentally. The first stage of the RKA
has a novel compact input cavity in which an explosive emission cathode generates
a kA, 350 kV annular electron beam with 19 cm radius and 2 mm thickness. The
beam is confined by a 1 - 17 kG solenoidal magnetic field produced by a superconducting magnet. Amplitudes of beam modulation up to 5% are observed, depending
on the input power. The measured beam bunching growth is found in good agreement
with linear theory. Also, a two-dimensional particle-in-cell code, MAGIC, is used to
simulate the experiment. The experimental and simulation results are also in good
agreement.
Thesis Supervisor: George Bekefi
Title: Professor of Physics
Acknowledgments
I wish to express my deepest gratitude
to Professor George Bekefi for his enormous
help in my research and support for my graduate studies at MIT.
I would like to thank Dr. Chiping Chen for his guidance in my theoretical research
and many physical insights he gave me.
Also, I would like to thank Ms. Palma Catravas, my collaborator in the experiment
for her ground breaking work in the laboratory and many insights.
Mr. Ivan Mastovsky provided me with invaluable help whenever I had problems
in the research.
Contents
1 Introduction
9
2 Theory of Double-Stream Cyclotron Masers
2.1
Linear
Theory
2.2
Linear and Nonlinear Description of the Double-Stream Cyclotron Maser
in Cylindrical
Geometry
System
. . . . . . . . . . . . . . . . . .
19
Formulation
. . . . . . . . . . . . . . . . .
19
. . . . . . . . . . . . . . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . . . . . . . .
26
The Mathematical
2.2.2
Particle
2.2.3
Wave Equation
2.2.4
Electrostatic Dispersion Relation for the Double-Stream Cy-
Equations
Maser
Amplifier
. . . . . . . . . . . . . . . . . . . . .
29
Numerical Analysis of the Nonlinear Evolution in the Double-Stream
Cyclotron
Maser
Amplifier
. . . . . . . . . . . . . . . . . . . . . . . .
3 The 33 GHz Relativistic Klystron Amplifier Experiment
4
16
. . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
clotron
2.3
for an Infinite
16
30
38
3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
Experim ent
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.2.1 Relativistic Klystron AmplifierExperiment Setup . . . . . . .
42
3.2.2
51
Relativistic Klystron Amplifier Experimental Results . . . . .
Conclusions
63
Bibliography
65
4
List of Figures
1-1 Schematic of a double-stream cyclotron maser. (a) Cross section of the
maser showing the two annular beams and (b) overall view of the system.
1-2 Schematic dispersion diagram for a two-stream system in a finite axial
magnetic field showing unstable regions in the vicinities of the intersections of dashed lines. Here, the lowest order modes with the cyclotron
harmonic number
=
0
±1 are plotted, whereas higher order modes
are om itted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1-3 Schematic diagram of a relativistic klystron amplifier with an input
cavity
and a passive
or output
cavity . . . . . . . . . . . . . . . . . . .
13
2-1 Intensity growth rate as a function of frequency for the choice of system
parameters listed in Table 21. The solid curve is from linear theory
and the circles are from simulations using equations 2.42 - 2.44) and
(2.58)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2-2 Normalized wave amplitude as a function of the interaction length z
at f = 40 GHz for the choice of system parameters listed in Table 21.
='p,,2/P,,
.
2-3 Phase space of the beam electrons at (a) z =
corresponding
33
to figure 2-6 .
and (b) z = 200 cm,
. . . . . . . . . . . . . . . . . . . . . . .
34
2-4 Intensity growth rate as a function of the fractional axial momentum
spread
o-plp,
= o-p,,j1p,,
. . . . . . . . . . . . . . . . . .
37
3-1 Envelope of the beat wave resulting from the superposition of the slow
and fast space-charge
waves at the same frequency
5
.
. . . . . . . . . .
40
3-2 Experimental setup for the input section of the MIT 33 GHz relativistic klystron amplifier showing the input cavity, electron gun, drift tube
and diagnostic
3-3
Typical
probes .
voltage
. . . . . . . . . . . . . . . . . . . . . . . . . .
pulse trace .
. . . . . . . . . . . . . . . . . . . . . . .
41
44
3-4 Axial profile of the magnetic field provided by the superconducting
m agnet .
3-5
Typical
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
magnetron
pulse trace .
. . . . . . . . . . . . . . . . . . . . .
45
46
3-6 Azimuthal profile of the axial rf electric field E., at the cavity gap at
a radial position corresponding to the annular electron beam radius.
The rf electric field E, is measured with a small electric dipole probe.
47
3-7 Measured power reflection as a function of frequency in the absence of
the electron
3-8
(cold tests) . . . . . . . . . . . . . . . . . . . . . . .
49
Schematic diagram for the directional coupler setup for measuring the
input
3-9
beam
and reflected
Schematic
diagram
rf power
. . . . . . . . . . . . . . . . . . . . . . .
for the probe
calibration
system . . . . . . . . . . .
50
52
3-10 Measured coupling constants for the electric probe as a function of
probe depth.
Dashed lines show the operating point chosen for the
experim ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3-11 Measured coupling constants for the magnetic probe as a function of
probe depth.
Dashed lines show the operating point chosen for the
experim ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3-12 Current modulation signals measured by successive magnetic probes
B17
B27 B3, and B4, for a 150 nsec pulsed annular electron beam at
kA and 350 kV, and 500 kW input rf power. The probes are located
6 cm, 8.5 cm, 11 cm, and 13.5 cm from the center of the cavity gap,
respectively
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3-13 Frequency spectrum of the magnitude of the leading spikes in the current modulation signal pulses, and the corresponding unloaded cavity
Q: (a) measured with the third magnetic probe; (b) measured with the
third electric probe. Both probes are 11 cm from the center of the gap.
6
57
3-14 Frequency spectrum of the percentage current modulation in the flat
portion of the beam pulses, and the corresponding loaded Q: (a) measured with the third magnetic probe; (b) measured with the third
electric probe. Both probes are 11 cm from the center of the cavity gap.
3-15 Percentage current modulation as a function of the square root of the
input rf power, measured with the third electric probe. . . . . . . . .
59
3-16 Percentage current modulation growth as a function of beam propagation length for a
kA beam and with 10 kG and 17 kG guide field
respectively: (a) measured with
5 electric
probes .
magnetic probes; (b) measured with
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3-17 Percentage current modulation growth as a function of beam propagation length for a beam current higher than
17 k
guide field respectively, measured with
7
kA and with 10 kG and
electric probes.
. . .
62
List of Tables
2.1
Parameters used for the numerical analysis of the double-stream cyclotron
3.1
maser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Experimental parameters for the MIT 33 GHz Relativistic Klystron
Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
8
Chapter
Introduction
Relativistic klystron amplifiers(RKAs) and double-stream cyclotron masers are
devices that generate high-intensity, pulsed coherent electromagnetic radiation. The
novel mechanisms of generating high-power microwave and millimeter-waves have
been vigorously sought after in recent years. While the RKA is attractive because it
is capable of producing intense microwave at and beyond hundreds of megawatt power
level, interests in the double-stream cyclotron maser are due to the need of millimeterwave sources that would utilize only mildly relativistic
and moderate magnetic field
<
0 kV) electron beam
< I kG).
The double-stream cyclotron maser is [1 a coherent radiation source in the millimeter to centimeter wavelength range. A schematic of such a device is shown in
figure 1-1. It utilizes two mildly relativistic electron beams copropagating along an
applied uniform magnetic field with an inverted population in the perpendicular momentum.
In comparison with other coherent radiation sources in this wavelength
range, such as the cyclotron autoresonance maser and the gyrotron, one of the advantages of the double-stream cyclotron maser is that it can operate in the regime of
both low beam voltage
100 kV) and low magnetic field (- I kG), thereby providing
an attractive alternative among its class.
The operating principle [1] of the maser is based on the stimulated interaction
between the fast space-charge cyclotron wave on one beam and the slow space-charge
9
(a)
- - WALL
WAVtUU1Ut
.... . _...
btArV1 ;e
I
ELECTRON
MAGNETIC
(b)
BEAM
TO .
ACCELERATOR
bl-'Ll
I
ANUDL
SOLENOID
WAVEGUIDE
CATHODE
Figure 1-1: Schematic of a double-stream cyclotron maser. (a) Cross section of the
maser showing te two annular beams and (b) overall view of the system.
10
cyclotron wave on the other beam. Such an unstable interaction, which we refer to
as the cyclotron two-stream instability 2 leads to the stimulated bunching of the
gyrophases of the electrons relative to the phase of an exponentially growing wave.
In fact, the cyclotron two stream instability belongs to the class of multiple species,
quasi-electrostatic streaming instabilities well known in plasma physics 3.
Figure 12 shows a schematic dispersion diagram for such a two-stream system,
where there are an infinite number of unstable regions appearing in the vicinities
of the intersections of the cyclotron modes of one beam with those of the other
beam. Because the resonance frequency of the cyclotron two-stream interaction is
proportional to the cyclotron frequency and is inversely proportional to the difference
in the axial velocities of the two beams, the double-stream cyclotron maser can operate
at high frequencies without the need of either high beam voltage or high magnetic
field [1].
In this thesis, the linear and nonlinear coupling of axisymmetric cyclotron spacecharge waves on two weakly relativistic electron beams copropagating in a uniform
magnetic field is investigated in a cylindrical geometry. When the Doppler shift in
frequency is mch greater than the cyclotron frequency, the coupling is shown to be
primarily electrostatic, which differs qualitatively from an electromagnetic type of
coupling.
The linearized equations of motion in a wave-particle model are used to derive a
dispersion relation for the electrostatic coupling of two gyrating beams. The linear
stability properties are analyzed. It is found that, like the conventional cyclotron
autoresonance maser (CARM
4 [5], the instability growth rate decreases rapidly
with increasing axial velocity spread within each electron beam. Good agreement is
found between the stability analysis and computer simulations in the linear regime.
The saturation level of the instability is obtained from simulations for parameter
regimes of experimental interest.
In contrast I-lothe double-stream cyclotron maser, the relativistic klystron amplifier only has one electron beam. Figure 13 shows a schematic RKA. Its operating
principle is based on the interaction between the slow and fast space-charge waves on a
11
I. I = Li
I
ti
I
fl
ac
W2
0
nc
Ii
Figure 12: Schematic dispersion diagram for a two-stream system in a finite axial
magnetic field showing unstable regions in the vicinities of the intersections of dashed
Enes. Here, the lowest order modes with the cyclotron harmonic number I = 0, ±1
are plotted, whereas higher order modes are omitted.
12
INPUT
CAVITY
I
IT
CA]
z
I
\-
DRIFT TUH7'
SOLENOID
Figure 13: Schernatic diagram of a -relativistic klystron amplifier with an input cavity
and a passive or output cavity.
13
relativistic electron beam with negligible transverse electron motion 6 The beating
of two space-charge waves with different axial wavenumbers but the same frequency
produces a beat wave with a sinusoidally varying envelope, whose amplitude reaches
a maximum at a distance of approximately 14 of the beat wavelength from the input cavity at which the waves are excited. A passive cavity may be placed at the
maximum to further enhance the beating and the associated current oscillations for
intense microwave generation.
Relativistic klystron amplifier research has been pursued mainly at the Naval Research Laboratory
7 8 9
The experimental parameters are significantly different
from those of conventional klystron amplifiers and are characterized by an intense
annular electron beam in the 500 M
10 kA range, generated from a field-immersed
explosive electron gun. The most remarkable feature of such an ultra high-current
RKA is high-gain operation without use of multiple passive cavities.
The 33
Hz relativistic klystron amplifier research conducted at the Massachusetts
Institute of Technology is driven by the need for high power radiation sources at the
frequency range of
-
GHz for direct energy applications. From the physical point
of view, the motivation of the MIT RKA program is to address the scalability of
the RKA to high frequencies, thereby determining the extent to which the results
obtained at 13 GHz by the NRL group are applicable. The designed system of the
RKA consists of three cavities: the input cavity, the passive cavity and the output
cavity. Unlike the NRL designs, the first cavity of the MIT RKA is a novel compact
coaxial cavity with a large lip at the gap to increase the field and beam coupling. The
RKA is driven by an annular beam of
kA and 35OkVwith a duration of 200 ns As
expected the first cavity has produced modulation amplitudes up to 5% of the total
beam current, depending on the input power.
In Chapter 2 we will focus on the theoretical aspects of the double-stream cyclotron maser. In Section
we will review the small-signal theory and the linear
dispersion relation for an infinite system. In Section 2 we will present a mathematical formulation of the double-stream cyclotron maser in a cylindrical configuration
based on a two-dimensional, self-consistent, fully nonlinear wave-particle model. The
14
model is used to study the feasibility of the double-stream cyclotron maser in particular and the cyclotron two-stream instability in general. In Section 3 we will present
computer simulation results and compare them with the linear theory.
In Chapter 3 the experimental procedure and setup for the RKA are described
in detail. In Section 31, descriptions are given of the Marx generator, the NEURUS
transmission line, the explosive electron gun, the input cavity, the drift tube, the
focusing superconducting magnet, and the input magnetron system. Methods used
in the beam diagnostics are also discussed.
In Section 32, results from experimental measurements and the MAGIC simulations are discussed, and comparisons between theory and experimental results are
made.
A summary of the present thesis is given in Chapter 4.
15
Chapter 2
Theory of Double-Stream
Cyclotron A4asers
Use of a relativistic electron beam gyrating and axially drifting in a uniform axial
magnetic field to generate intense, coherent electromagnetic radiation at millimeter
and centimeter wavelength has been well documented in terms of both theory and
experiment. Te novel scheme of the double-stream cyclotron maser arises in recent
years [1]. Its schematic is shown in figure 1-1. Unlike conventional one-beam sources
such as the gyrotron and the cyclotron autoresonance maser, the instability growth
in this device does not rely on the relativistic effects of two copropagating beams, but
on the mutual interaction of charge density perturbations of the two beams traveling
at two different speeds. The advantage of this novel device over conventional onebeam sources is that it can achieve much higher frequencies with less stringent beam
parameters.
2.1 Linear Theory for an Infinite System
For a single relativistic beam gyrating and drifting in an uniform axial magnetic
field B11,the emission occurs at the Doppler-shifted gyrofrequencies given by
16
W
where
[ -
=
=
+31
I)-y' (MIT),
11
(M
2 3...),
= 1
(2.1)
eB11/moy) is the electron gyrofrequency, m is the harmonic number,
vjj1,c) - (v I Ic)']-11',
11= v11c, and
11=
1,32]-1/2.
11
In a double-stream cyclotron maser, as shown in figure 1-1, two wen mingled
electron beams with different axial velocities v11,1andV11,2
are spun up by means of
a tapered wiggler magnetic field, so that the gyromotion of the beam electrons is
induced in a axial uniform magnetic field B1. The beams have transverse velocities
v ,, and
V 12.
The space-charge wave in this system has wave vector k which has
both transverse and axial components k, and k1j, where k
is determined by the
beam radius and the transverse dimension of the wave guide.
By solving the linearized relativistic Vlasov equation and the Poisson equations
in free space [10], one can come up with a dispersion relation which can be used to
determine the interaction frequency and the instability growth rate. A neutralizing
background of infinitely massive ions is assumed so that a simple equilibrium is constructed for a two-beam system. Let fo -_ fo(v 1, vil) be the unperturbed velocity
distribution function for each beam. It has been shown that the space-charge waves
satisfy a linear dispersion relation of the form
KL(kw
= (k K k)/k 2
(2.2)
= 0,
where K is the dielectric tensor and its longitudinal component
00dviv,
1: ffo
beams
KL
00
X
m(k-L/v-)
EI
M=_00 (w -
[k2 _(kjjvjj +
+
kjjvjj
f.
0"
K
is given by [11]:
2
dv ILop
2irfo
V
d [J,2. (p)
dp
MQ)
MQ)2/
C2]j2(p)
11
2
(w - kjjvjj
MQ)
(2.3)
07
17
where J(p)
is the mth Bessel function with the argument p _- k-v i /Q.
For simplicity, we assume two completely superimposed beams which fill the entire
cylindrical wave guide with radius a. In this case 12], k_ = ul,,/a where ul,, is the
sth root of Jj(u = 0. Then it followsthat
= 1's I /fla = uj,rda,
where
rL
i
(2.4)
the electron Larmor radius. For the lowest transverse magnetic TMoj
mode and a beam distribution function of the form
fO(VL'Vjj
=
1
8( I - V I 8(vll -
011),
(2.5)
_7rv_OL
we obtain, from equation 2.3), the linear dispersion relation for an infinite wave:
KL-
= -
(DO
E
W2
beamsm=-00 V
x
[k2
f
I kllvll+mfl 12]j2
II
C -
2
P - kjjvjj- MQ)
2 f jm2
_
_ J'2
"+Jj
(2.6)
29(w - kivil - MQ)
For a one-beam system, this relation shows that around each harmonic mQ
w - kjvjj, there are two narrow propagation bands, one above and one below each
integral multiple of Q. The propagation band extends approximately ±WpJ,,(p on
either side of mQ. The lower (slower wave) branch is a negative energy wave 13],
while the upper (fast wave) branch is a positive energy wave.
In the double-stream cyclotron maser , however, two beams with slightly different
axial velocities interact with each other as illustrated in figure 12, and phase velocity
synchronism can be achieved at low plasma density. For the case of almost paraxial
propagation (i.e., k1l> ki ), the second term in equation 2.6) can be set to be zero
when p is so chosen as to maximize Jm(p). Then the dispersion characteristics for
each uncoupled beam cyclotron mode are given approximately by:
w _- kliv, + mQ
18
(2.7)
w -_ k JV2
n
where n and m are non-zero integers. For n = -m and
(2.8)
vi'
V2,
equations 2.7) and
(2.8) yield the resonant frequency for the mode coupling:
V2 + V1
W
Solving for complex
Q V
-
V1
2
2(,311-yll)
(2.9)
in equation 2.6) with real k and assuming that the two
beams have equal plasma frequencies, one then obtains [1]:
Im(w) - wJn(p)12-yjjI.
(2.10)
Likewise, for the convective cyclotron two-stream instability in an amplifier system,
the spatial growth rate Im(k) can be derived from the dispersion equation 2.6).
2.2 Linear and Nonlinear Description of the DoubleStream Cyclotron Maser in Cylindrical Ge-
ometry
In order to study nonlinear effects and growth saturation levels and to understand
the nature of the elestrostatic coupling of the cyclotron two-stream instability, we
present here a two-dimensional, self-consistent, fully nonlinear wave-particle model in
cylindrical geometry.
2.2.1
The Mathematical Formulation
We consider two concentric annular beams of electrons gyrating and copropagating
axially in an applied uniform magnetic field Boe,, (figure 1-1). The axes of the beams'
annuli coincide with that of a perfectly conducting, cylindrical waveguide through
19
which the electron beams propagate. Assuming that the equilibrium self-electric and
self-magnetic fields of the beam to be negligibly small, we describe the unperturbed
beams by the following equilibrium distribution function:
fO(x1P =
o.(XP
=
ci=1,2
where 2r f Gc,(rg)rgdrg =
_'1%
I.
=1,2
e,
G .(rg)F.(p
p.),
(2.11)
and 27rf F, (p , p.)p dp dp = 1. In equation 2. 11), _e
is the electron charge, pz and p
(p2X + p2)1/2
Y
are the electron axial and transverse
momentum components respectively, rg is the electron guiding center radius, and
Ic, and V = f vFcd'p
(a =
2
are the current and average axial velocity of the beam a
One should note that variables rg, pz and p_Lare the constants of the
unperturbed single electron motion in the uniform magnetic field BOezand therefore,
the equilibrium distribution function fo solves for the zeroth-order Vlasov equation.
To derive a complete set of ordinary differential equations describing the selfconsistent, slowly varying, axial evolution of cyclotron space-charge waves in a single
frequency, stationary state, double-stream cyclotron maser amplifier, we express the
axisymmetric wave fields in terms of a vector potential of the form
A(r, z, t = Az(r, Z' t)ez
(2.12)
along with the associated scalar potential
= Orl Z' t),
(2.13)
which can be uniquely determined from the Lorentz gauge. Here the axial component
of the vector potential is defined by
00
Az(r, Z'
j
E A.(z)C,.Jo(k,,,r)
n=1
x
exp i 1z kzn(z')dz - wt)] + c.c.,
20
(2.14)
and the transverse component A,(Ir,Z,t) and Ao(r, z, t) are ignored.
In equation
(2.14), the notation "c.c." denotes complex conjugate. The electric and magnetic field
perturbations are then uniquely determined from equation 2.14) under the Lorentz
gauge condition.
In the regime of interest, the neglect of the transverse components of the vector
potential is justified because the coupling is primarily through electrostatic forces due
to a large Doppler upshift of the cyclotron frequency.
In equation
2.14), the index n designates a TMon type of transverse-magnetic
mode, and Jo(x) is the zeroth-order Bessel function of the first kind. The transverse
field profile Jo(k inr) corresponds to that of the vacuum TMOn mode, provided that
the boundary condition Jo(k nb =
k
=
is satisfied. Here, b is the waveguide radius,
1b, An is the nth zero of Jo(x), and
kLn
CLn
71/2 An IJO(9n)
(2.15)
is a normalization constant. In this model, the coupling between the wave fields and
the beam density and current perturbations is described by the slowly varying wave
amplitudes An(z) and the slowly varying axial wave numbers kn(z). The axisymmetric vector potential in equation 2.14) is expressed as a superposition of a complete
set of the vacuum TMOn modes and thereby allow us to treat both electrostatic waves
and electromagnetic waves on the same footing 14] [15].
To derive a complete set of equations governing the cyclotron two-stream instability, we start from the Maxwell's equations. Under the Lorentz gauge condition
'go + V A =
(2.16)
the wave electric and magnetic fields are given by
1
aA
&
B (r, 0, t)
V x A.
21
(2.17)
(2-18)
And the wave equations for
and A are
__T
(V2
1 02
C2
t,)o
_ -)A
(,72
1 i92
C2 bt2
47rp,
(2.19)
--47r J.
(2.20)
C
From equation 2.14) and the Lorentz gauge condition [equation 2.16)], we express
the scalar potential
as
C
2
)Ey-A.(z)CL,.,Jo(kinr)
'W
z
n
X expli( fo Zkz(z')dz
From equations
- wt)] + c.c..
(2.21)
2.14), 2.17), 2.18), and 2.21), we express the small signal electric
and magnetic fields as the following:
8E
=
2
C I nk InJo(k Inr)-An(Z)
C
'W
n
OZ
x expli( fo Zkz(z')dz - wt)] + c.c.,
8E
=
(2.22)
(-)(-)E C nJO(kInr)(- a2+ -)An(Z)
C2
i
c
2
'W
exp[i(
8B9
LO2
n
(9Z2
k,,(z')dz
k
(')EC-Ln
2
n
- wt)] + c.c.,
(2.23)
r)A
_LnJO'(k1n
x expli(fo Zkz(z')dz
n(Z)
- wt)] + c.c..
(2.24)
It is proven useful to express the wave fields in the gyro-coordinate defined by e I
cos( -
)er + sin( -
)eo and e = - sin(
22
-
)er + cos( -
)eq. Carrying out
the derivative with respect to z, we come up with the expressions for 8 I 8E,5, 8 I
8BO, and 8E,
--
8E, cos(
-
) + 8Eq sin(
dA
= -(z)(c)EC,.k,.(
2 W nj
x exp(iA,) + c.c.,
8EO
=
-
)
' + ikzAn)Yln(rg, rL)
dz
(2.25)
8E, sin - ) + 8Eecos( - )
)I:Cln
-(')(
2
W
dAn
-In(
n
kz An) Zln (rg rL)
dz
(2.26)
x exp(iA,) + c.c.,
C2kz2
ic2dk
-)An+- dzzAn+ 2i kz
EP
2c
C2
W
nI
c2d2A
---
W2 dZ2
8B..L
W2
W2
W2
dAn+
dz
]CJ-n;rin(rg7rL)
x exp(iA,) + c.c.,
(2.27)
= 8B, cos( - ) + 8Bosin( - )
i
2
8BO =
E
C 1nk InAnZin(rg)
n,1
8Brsin(
-
rL)
) + 8Bo cos(
exp(iAl) + c.c.,
-
(2.28)
)
1
= -I:ClnkinAnYin(rg7rL)xexp(iAl)+c.c.,
(2.29)
2 nI
where the following notations has been introduced
XIn (rg , rL)
Yln(rg, rL)
-- J (klnrg)Jl(klnrL)5
1
JI(kJ-nrg)JI(kl
kJ-nrL
23
(2.30)
nrL),
(2-31)
Jj(kInrg)Jj'(kInrL),
-71n(rg7rL)
Aj(zOOgt)
(2.32)
kz(z')dz - wt + 1 - 1g + 17r
fo Z
(2-33)
2
With the above framework , we proceed subsequently with the derivation of particle
and wave equations for the double-stream cyclotron maser.
2.2.2 Particle Equations
Under the assumption that the guiding-center drift is negligibly small (i.e., rg
const and 0
2 const), the dynamics of an electron can be described by three
variables: the energy, Irnoc2I the axial momentum, pZ, and the azimuthal angle,
0 = tan-'(p,/p.,).
From the Lorentz force equation, the dynamics of individual
electrons is governed by 16]
d7
e
_2j(ptE,
dt
dp.
e(Ez +
dt
Substituting equations
d-j
d'
-
do
QC
dt
-y
PZ nl=_Oo
+a" cos A, -
[a'n sin Ai
(2.34)
(2.35)
-/MOC
(EO +
Pi
z B-L
pi Bz
'YMOC
'IMOC
2.25)-(2.29) into equations
PI
__--:--
n
pzEz)7
0
kzan cos AI]Yln
234)-(2.36) yields:
00
nl=-oo
an sin A, - 2 kz a' sin A,I XIn
Z
n
24
(2-36)
k I n7
-
2)an
Z
cos Al
(2-37)
dp.
fil
- E ancosAlYj- 'Y E00 p - ka,,cosAl
-
Pz nl=-oo
di
fiz nj=-00
z
+a"n cos Al - Pan
sin Al - 2kza'n sin Ail X1,n
z
(2.38)
n
dO
di
-
+ PzP
fiz
00
E
1 n=-o,
la cos Al
(Pz - k )an sin Al] 21,n
n
ly
(2-39)
z
With the electrostatic approximation:
Yin Zin
(2.40)
07
or:
E = EO Bk
B, -_
(2.41)
the particle dynamics equations for the double-stream cyclotron amplifier are further
simplified as
dOnl
dT
(2.42)
znm
d^1
di =
PZ
100 =-co
EM k In
Xnl
w;,-
Equations
d2an
2
-kzn)an
+
d2
COS'Onl
+
n=1
dkzn an
-
Pz
2kzn
(2.43)
dan ) sin'Onl],
di
(2.44)
constant.
2.42)-(2.44) describe the dynamics of an individual beam electron. In
the simulation, we solve numerically 6N of such equations of motion for N macroparticles representing the electrons of each beam , where N = 1024 is typical. The phase
Oni is defined by,
'Onl(Z
0
g I t)
fo z
kzn(z')dz - wt
25
I arctan( Py - 10g+
PX
17r
2
)7
(2.45)
and represents the Ith harmonic gyrophase of the electron relative to the phase of the
nth mode. Here
const is the polar angle of the guiding center of the electron
orbit. nj(rgrj= J(kl nrg)JI(ki
nrL)
is a geometric factor and rL
radius of the electron. The normalization is such that
pi
fiz
=p-L/mc,
M= e = c=
=
zlc,
is the Larmor
kzn
=
kzn 1W,
pz/mc, 1c = 2clw = eBolmcw, etc., which is equivalent to setting
1, where m is the electron rest mass, and c is the speed of light in
vacuum.
2.2.3
Wave Equation
From the Maxwell's wave equation 2.20) for the vector potential A, we have
(V2
1 a2
-)8Az= --47r
8Jz)
C2at2
c
(2.46)
where
Uz
enabGc,(rg)vczo
f
6(t - r(to, z))dto,
00
(2.47)
Ci=1,2
z
,r(to,z) = to
dz'
fo vz (to, Z),
(2.48)
f Gc,(rg)rgdrgdOg -- 1,
n,,,,b
(2.49)
= number of electrons per unit axial length n the ath beam, (2.50)
vazo = the average velocity of the ath beam.
Inserting the expression for Az equation 214) into
26
V2
A.,,,we find
(2.51)
V2 A,,
02 + V2
( OZ2
1
)SA,,
Eld2A,,
2
z
nr) x exp[i[
)C-LnJO(kl
k2 ,Al
k A + ik'A,]
dz
dZ2
n
X(
2ik.,dAn
fo z
k,,dz'- wt)] + .C..
(2.52)
Substituting equation 2.52) into equation 2.46) yields
Ejd2A
n
x
2
4re
c
Multiply equation
dZ2
dAn
+ 2ikz dz
C 1nJo(k
E
2
kAn+ik'An-k
z
z
Inr) x exp[i(
00
In
An
2
-AnI
C2
k,,z - wt)] + c.c.
8(t - r(to, z))dto.
(2-53)
2iC nJo(k Inr) exp[-i( fo zkzz - wt)],
(2.54)
a=1,2
n'G'(rg)v'
b
fo z
2
zo f
co
2.53 by
we integrate equation 2.53) over
27r/wA f 2w f
r" rdrdO,
(2.55)
noting that
rdrO = r9dr9d0g,
(2-56)
and
C-LnJo
(kInr
= CInEJ1(kInrL)JI(kInrg)
I
x exp[-iA + i fo zk.,dz - wt].
27
(2-57)
This leads to the wave equation 14]:
2
ddj2
a,
+-1- k2zn- k2 jan + i(2k,,n
dan
di + andk.n
di
00
E E gn.(Xnl
exp(-iOnl)).
(2-58)
I=-oo i=1,2
for the double-stream cyclotron maser amplifier.
Equation 2.58) describes the self-consistent evolution of both the normalized wave
amplitude an(i) and the axial wave number kzn(i)
. The dimensionless coupling
constant in equation 2.58) is defined by
gnc = 87r( ck-Ln 3( C I n 2 a
W
k In
IA
where IA = mc'/e
(X),, = N-1
(2.59)
= 17kA is proportional to the Alfven current.
The notation
Xj denotes the ensemble average over the particle distribution of
the beam a.
The average electromagnetic power flow (i.e. Poynting flux) through the wave
guide cross section at the axial distance z can be calculated by the following formula:
W
P (Z) -
2
47r 27r
dt f do-(8E x 8B) ez.
(2.60)
(V-L80)
(2-61)
Using the relation
e,, (8E x 8B)
V 8Az),
we find that
P(Z)
47
f27rlwdt f do,[V,(80'V, 8A,,
W
2 r0
ck2
n
An
dz
47
C
dt
In
2
8OV2 8Az]
dO(-)f
[C I nJo(k I nr)]'
W
kz A, sin o I An sin p,
28
(2.62)
where
= jo' k(z')dz - wt. Carrying out the integrals, we finally obtain
PW
E Pn(Z),
(2.63)
n=1
where
PO
Pn
87r
w
w
)2(
ck-Ln
k1n 2a2
CLn
Z)
(2.64)
n
is the power contributed by the nth mode and Po = m2cl/e2 = 87 GW. Moreover,
the average radio frequency e-beam. power flow is given by
P
PO(
=
a=1,2
From equations
IC,)((-Y).
1).
IA
(2.65)
2.43), 2.58), 2.59), and 2.63)-(2.65), it follows that the total rf
(radio frequency) power is conserved, i.e., P(z) + P(z) -_ constant.
The mathematical formulation presented above is readily used to examine the
stability properties and saturation levels of the cyclotron space-charge waves on two
relativistic electron beams copropagating in a finite magnetic field and cylindrical
wave guide. In principle, the present formulation is also applicable in regimes where
the gyromotions of the beam electron is negligible (i.e., Bo -+ oo), as in the two-stream
relativistic klystron amplifier configuration 17] [18]. The latter case is obtained by
setting the cyclotron harmonic number I equal to zero for each beam. In section 23,
we will present results of a single-mode analysis in which we let n be a single integer
but allow the cyclotron harmonic number
to assume integers 0, ± 1, ±2, ... .
2.2.4 Electrostatic Dispersion Relation for the Double-Stream
Cyclotron Maser Amplifier
In the small signal regime, we linearize equations
derive a dispersion relation.
w 2/C2
2.42)-(2.44), and 2-58), and
Making an electrostatic approximation (kZ2 + k I2
), and choosing the equilibrium distribution function in equation 2.11 as
29
>
1
G. (r.)
2-xr.c.
.8(r - r
9
(2-66)
9
and
F.(p p. = 2-xp I8(,
I
I 8(p. - P..),
(2.67)
it is readily shown that the dispersion relation for the nth mode can be expressed as
0"
nL(Lo, k )
2k2
C
_kElm,
ZVZa
1=-oo a=1,2
Q,/-yc,)2 0.
(2.68)
In
In equation 2.68) we have neglected terms of order (w - kzvz,,,-
Neglecting
of this term is well justified whenever k I Ik < 1, as is the case of the experimental
interest. The normalized coupling constant is defined by
61n =
4
'( Ia )( J1(kLnrg.)J1(k1nrLc,))2
_ya'3.c'_f2
IA
Za
which is related t gna defined in equation (2.59). Here,
)3z =
za/-fa P., a -
I
(2.69)
k I nbJo(k 1nb)
yOL)za
p2Zoe
rLa
(I + Pa
+ p2I J1/2'
Z
p i ./mQ,,
is the electron
Larmor radius for beam a, and J(x) is the first-kind Bessel function of order 1. The
electrostatic dispersion relation 268) has been derived previously 19] using kinetic
theory in the context of a coherently gyrophased electron beam 19] 20].
2.3 Numerical Analysis of the Nonlinear Evolution in the Double-Stream Cyclotron Maser
Amplifier
We have developed a computer simulation code to integrate equations 2.42)-(2.44)
and 2.58) numerically. The code has been benchmarked against the linear theory
presented in section 22.4. The results of the benchmark simulations are summarized
30
in figures 21 - 23 for the choice of system parameters listed in Table 21, where the
TM02
SC=
mode is chosen to maximize the coupling strength. The self-field parameter
2-ybul 2b/-y2
J12 is
Zb
estimated to be 02 for each beam, where wpbis the nonrelativistic
plasma frequency. Such a small value of self-field parameter may well justify neglect
of the equilibrium self-electric and self-magnetic fields of the beams, as assumed
throughout the derivations.
The linear intensity growth rate is plotted in figure 21 as a function of frequency.
The solid curve is obtained from the electrostatic dispersion relation
268), and
the open circles are results from computer simulations using equations 2.42)-(2.44)
and 2.58) which are based on the full Maxwell's equations rather than the Poisson
equations alone. The plots shows that there are two unstable regions, separated at
the chosen resonant frequency f = 34.4 GHz at which the system is stable. The
maximum growth rate is 40 dB/m, which occurs at f = 26 GHz and f = 40 GHz.
The fact that there is good agreement between the computer simulations and the
electrostatic dispersion analysis, as shown in figure 21, illustrates that the coupling
of the cyclotron two-stream instability is indeed primarily electrostatic.
The axial evolution of the normalized wave amplitude is plotted in figure 22 for
f = 40
Hz I agreement with the linear theory, an intensity gain of 41 dB/m is
obtained in the exponentially growing region from z =
0 cm to z = 150 cm. The
oscillations at the beginning of the interaction ( < z < 0 cm) are due to launching
(insertion) losses. At saturation, the electromagnetic power flow P2(z) in the axial
direction is rather low <
KW) compared with the dc beam power because the wave
is primarily electrostatic.
Therefore, in order to assure an efficient operation of the
maser, an effective input and output scheme must be sought.
The computer simulated phase space of the electrons
2-3, corresponding electrons at the initial position z =
(021,'Y
i
shown in figure
and at the final position
z = 200 cm. The open circles in figure 23 represent the macroparticles in beam
and the open squares represent the macroparticles in beam 2 At z =
[figure 23
(a)], the gyrophases of the macroparticles in both beams are slightly bunched relative
to the wave phase so that a small wave amplitude can be assigned initially in the
31
50
E
co
a
40
W
t
Ir 30
-r
0Cr
0
3-1
20
co
z
z
W
10
0
20
30
50
40
FREQUENCY
(G HZ)
I
Figure 21: Intensity growth rate as a function of frequency for the choice of system
parameters listed in Table 21. The solid curve is from linear theory and the circles
are from simulations using equations 242 - 244) and 258)
32
=3
d
0
LLJ
10-5
I
Q-
-6
icE
10
LL)
3:
0
LIJ
-7
10
N
I
ir1E
Cr
0Z
-8
10
0
50
100
150
INTERACTION LENGTH (cm)
Figure 22: Normalized wave amplitude as a function of the interaction length z at f
= 40 GHz for the choice of system parameters listed in Table 21.
33
200
L.13D
1.320
.)00000000000000000000oooooooooooooooooooc-
1.305
x
1.290
1.275
booooooooooooooooooooooooooooooooooooooo
>n
77'
0
2V
q2 e
1.6.5D
(b)
1.320
.0
0
0
0
0
0
0
0 0 0 00
I
0-
i.305
>1
1.290
1.275
.0
0
1J..,-13 IC10 v %
I
I
0
I
0
I
0
0
0
I
I
0
0
0
I
V
I
27T'
'P21
Figure 23: Phase space of the beam electrons at (a) z =
corresponding to figure 26.
34
and (b) z = 200 cm,
Resonant frequency
33.4 GHz
TM02 (n = 2)
Operating mode
Waveguide radius
2.54 cm
503 G
Axial magnetic field
Beam
I
Current
10 A
Voltage
137 kV
V.
I
0.2
0.8 cm
C
radius (rg I
Harmonic number I
-Guidifig-center
Coupling constant
I
I.OX 10-3
121 .
Beam 2
Current
10 A
.163 kV
Voltage
0.2
0.8
VI 21C
Guiding-ce'nter radius (rg)2
Harmonic number I
Coupling constant
122
cm
I
9.2 X 10-4
Table 21: Parameters used for the numerical analysis of the double-stream cyclotron
maser.
35
simulation. At z -_ 200 cm [figure 23 (b)], the gyrophases of the macroparticles in
beam I (the open circles) are well bunched at the phase
=
021
-_ 37r/2, whereas
the gyrophases of the macroparticles of beam 2 (the open squares) are well bunched
at the phase of
=
02-1
_-
7r/2. One beam bunching out of phase relative to the
other beam is a, characteristic of the unstable two-stream interactions.
Having found good agreement between theory and numerical simulations with
a special choice of the distribution function defined in equation 211),
266) and
(2.67), we have used the simulation code to examine the sensitivity of the growth
rate to axial velocity spread within each electron beam. In particular, we loaded the
particles in each beam with a Gaussian axial momentum distribution. The result is
shown in figure 24, corresponding to the choice of system parameters used in figure
2-1 at the upper maximum gain frequency f
fractional momentum spread defined by opzlpz
40 GHz. The horizontal axis is a
upzllpzl
=
O'pz2 /Pz2
where
Z.
is
the standard momentum spread width for beam a. It can be seen in figure 24 that
the growth rate decreases rapidly with increasing spread due to the fact that there is
a large Doppler upshift in the frequency. This sensitivity to axial beam temperature
is similar to that in the CARM 4 5].
36
50
E
'11
co
0
W
40
1
fr
-r
3: '30
0
0
Cr
co 20
zLi
z
1 r1l
%W
0.1
0
0.2
.
0.3
0. 4
AXIAL MOMENTUM SPREAD(V
Figure 24: Intensity growth rate as a function of the fractional axial momentum
spread uplp,
=
'p,11P,-,l =
'p2/P2
37
Chapter 3
The 33 GHz Relativistic Klystron
Amplifier Experiment
The MIT 33 GHz relativistic klystron amplifier has a compact and flexible design.
The central component of this experiment is a novel compact input cavity in which
an explosive emission electron gun is embedded. The electron gun generates a
kA,
350 kV annular electron beam with 19 cm radius and 2 mm thickness. The beam
is confined by a 10-17 kG solenoidal magnetic field produced by a superconducting
magnet. Up to
of beam current modulation has been observed. Magnetic and
electric probes placed along the drift tube are used to study beam bunching. Theoretical calculations and MAGIC simulations show good agreement with experimental
results.
3.1
Introduction
The relativistic klystron amplifier (RKA) generates high-intensity microwave power
and has a variety of potential applications in areas such as high power radar and directed energy applications. it has attracted wide attention in recent years and the
subject has been under extensive studies both experimentally and theoretically by
38
many groups 7 8 9 Their studies have been encouraging in terms of high output
power level and high efficiency at a frequency of 13 GHz.
The physical mechanism of the RKA had been well studied. The radio frequency
(rf) input power at the cavity gap excites a slow wave and a fast wave on the beam (see
figure 13). The two space charge waves beat and the envelop of the beat wave reaches
a maximum at a distance of approximately 14 of the beat wavelength from the cavity
(see figure 31).
In the linear regime, and in the long wavelength approximation
(wavelength large compared with the waveguide radius), the dispersion equation for
the desired azimuthally symmetric TM01 like mode of the circular drift tube is 21]:
P - kV)2 = (k 2C2
_ W2
(3.1)
I
071IO
where k is the wave number and V the beam drift velocity, I is the beam current,
V/C) ' =
_32)-1/2,
and Io is defined as Io = 27rEomo C3 /e ln(b/a
= 8.5/ ln(b/a)kA,
where b and a are radii of the drift tube wall and the annular beam, respectively.
From dispersion relation 3.1), the beat wave length is readily found:
A
27re
02 _ E
+I _l
W
2)
(3.2)
where E is the coupling constant and is defined as E = 1'3_y3J0.
The objective of the MIT RKA experiment is to build a compact system that can
operate at high frequency
33 GHz), with high efficiency and high power output.
Figure 32 illustrates the design of the input section of the experiment. This design
reflects our emphasis on the compactness and tunability of the system. The re-entry
cavity length and gap size are adjustable for maximum flexibility. In designing the
rf power input scheme, we circumvent the problem of building sophisticated mode
converters widely used by other groups, and simply inject the power radially through
a single window in the side wall of the cavity. Despite this injection asymmetry, the
technique yields very good mode purity and azimuthal uniformity. Once the beam
is modulated, the current modulation signal is then diagnosed by two sets of probes
distributed along the drift tube. The probes are simple electric and magnetic dipoles
39
FIRST CAVITY
5kA D.C.
SECOND CAVITY PLACEMENT DETERMINED BY MODULATION
ENVELOPE.
Figure 31: Envelope of the beat wave resulting from the superposition of the slow
and fast space-charge waves at the same frequency.
40
I
I
O
0CQ
W
0M
I
Ld
M
I
0
(X
a.
a.
9"
U
LLJ
W
-i
CQ
I--
CL
I
U
U
Ix
I
z
U
W
-i
Li
<
LLJ
P
O
z4
X
-
L'i
I
U
-ZO
I
II
IT
D
z
CL
7N
C:'
I
-i
II
i>
)-
:3 iE
3<
EU
Ml
C3 (
CY .
LA-:
Figure 32: Experimental setup for the input section of the MIT 33 GHz relativistic
klystron amplifier showing the input cavity, electron gun, drift tube and diagnostic
probes.
41
embedded in the drift tube wall.
3.2
3.2.1
Experiment
Relativistic Klystron Amplifier Experiment Setup
Experimental Parameters
The parameters of the experiment are listed in Table 31.
The high voltage is
provided by the NEREUS accelerator composed of a Marx generator and a water
filled transmission line. The beam has a voltage of 350 kV and current of
150 ns square top duration as iustrated
kA with a
in figure 33. It is produced by an explosive
emission cathode with a annular knife edge of the radius of 19 cm, and a beam
thickness of - 2 mm. The beam is confined by a . - 17 Tesla superconducting
magnet whose axial magnetic field profile is shown in figure 34.
The input rf power source for the excitation cavity is a magnetron with a tunable
frequency range from 31 - 34 GHz, peak output power of 400 kW, and 0.5ps duration.
A typical pulse shape is plotted in figure 35.
Input Cavity Cold Test
I have performed extensive cold tests to study the characteristics of the input
cavity and the drift tube. A small dipole sensor is used to measure the axial electric
field at the cavity gap near the interaction region at a radial position corresponding
closely to the annular beam radius. By rotating the dipole around the symmetric
axis of the cavity, we are able to observe the azimuthal field pattern of the mode.
With some fine tuning, we have obtained a very good azimuthally symmetric cavity
mode for a cavity length equal to 3.5AOwhere AO= 27rclw. Figure 36 shows a plot
of the axial electric field as a function of the azimuthal angle . The deviations of the
field's angular profile from uniformity is less than 2.
42
The desired mode resonates at
CAVITY DESIGN
Drift tube radius
Cutoff frequency
0.935 in.
Cavity outer radius
3.5 in.
2 in.
12.9 in.
0.875 in.
60
3.7 GHz
Cavity inner radius
Cavity length
Gap size
Cavity shunt impedance
Unloaded resonance requency
382 GHz
Q 400
Measured unloaded
INPUT RF PONVER
Frequency range
3.1-3.5 GHz
Pulse width
0.5 pec
Power
50 kW
SUPERCONDUCTING
Field strength
AGNET
10-17 kG
ANNULAR BEAM
Radius
Pulse width
Voltage
150 nsec
350 kV
Current
5 kA
1.9 cm
Table 31: Experimental parameters for the MIT 33 GHz Relativistic Klystron Amplifier.
43
50
0
-50
11-11
-100
_i:1
11-1
Lo
0
-150
1-i
0
-200
-250
-330
-350
0
200
400
600
800
1000
TIME (ns)
Figure 33: Typicalvoltage pulse trace.
44
1200
1400
I
Bz
kG)
22.5 kG
22.1 kG
-10
-5
0
5
10
axial position (cm)
Figure 34: Axial profile of the magnetic field provided by the superconducting magnet.
45
.,-%
- oUU
-
3:
-C,
z0
Ir
zw
0
:2
Li
-r
I-:2
0
D
t_
w
3:
a_
0
:D
q-
550
5DO
450
400
350
300
250
200
150
100
50
0
i.L-
rif
-- sn
-500
0
500
1000
TIME (nsec)
Figure 35: Typical magnetron pulse trace.
46
1500
90
W
in
W
N
I
0
I
:i
Cc
0
Z
270
AZIMUTHAL ANGLE (DEGREE)
Figure 3-6: Azimuthal profile of the axial rf electric field E,, at the cavity gap at a
radial position corresponding to the annular electron beam radius. The rf electric
field E,., is measured with a small electric dipole probe.
47
a frequency of 3382 GHz. The measured cavity Q value is 400 and 90
of the input
power is coupled into the cavity. Figure 37 shows the measured results. The input
and reflected rf power is measured by using directional couplers in the waveguide
system as shown schematically in figure 38.
Current Oscillation Diagnostics
In order to measure the beam current oscillation
1
have designed a set of beam
diagnostic devices. The beam diagnostic system consists of two sets of probes installed
in the drift tube wall,
electric dipoles and
current modulation amplitude.
magnetic dipoles. They monitor the
The reason for having both electric and magnetic
probes in the system lies in the problem of plasma formation. Previous
KA research
shows that electric probes may cause plasma formation around the tip area because of
the electric field,breakdown. This can cause serious disturbance to the beam current as
well as false beam bunching signals. So it was the general consensus among the
KA
community to se magnetic probes, which have rounded circular antenna, instead of
using electric probes with sharp tips for beam diagnostic purposes. However, in my
design, the probes are deeply recessed into the waveguide so that the electric field
calculated at each probe is significantly less than the breakdown level. With this
design we believed that the use of electric probes will not cause plasma formation in
our experiment. In addition, we have the advantage of gaining much more information
on the beam bunching by employing both kind of probes. The following section shows
that results measured from these two kinds of probes agree with each other.
The electric and magnetic probes are positioned
inch apart along the beam axis.
The electric dipole tips are approximately 3 mm in length and the magnetic dipole
loops are approximately
mm in radius. The electric and magnetic probes are located
on opposite sides of the drift tube and their sensor tips are embedded in the drift
tube wall so that they cause minimal disturbance to the beam propagating inside the
tube. The electric probes are designed to pick up the high frequency radial electric
field E, induced by the beam electron oscillations and the magnetic probes sense the
48
Iuu 90
80
RF input = 24 mW
.-O
C
70 -
zW
60 -
CD
I--
0
0
LL
U-
W
00
z0
0W
-j
LL
W
rr
50
40 -
0
0 0
0
0
0
0
0
0
30
0
e
0
0
20 41
%41
10
7
3.35
1
i
3.36
3.37
1
1
1
3.38
3.39
3.40
3.41
FREQUENCY(GHz)
Figure 37: Measured power reflection as a function of frequency in the absence of
the electron beam (cold tests).
49
t::
U
M
zcc
I
C)
L Z
z
C) C)
gm
t.-
=z
t-4
E- LV)
= -:4
E-
e. 0-
W
C)
P.
IM
W
co
E--
W
E
CD
C1.
rC:1
N
C4
C--
r-
Figure 38: Schematic diagram for the directional coupler setup for
input and reflected rf power
50
easuring0 the
azimuthal magnetic field dBOldt. To separate the real beam modulation signal from
extraneous signals, the probe signals are passed through 31 - 35 GHz band pass
filters. The output from the filters is then measured by means of calibrated crystal
detectors.
During the cold test the probes are calibrated in situ in terms of the known rf
power of a traveling TMol waveguide mode propagating in the drift tube. Thus, the
probe response is obtained in terms of the E, and Bo fields at the drift tube radius.
This is achieved by means of microwave interferometry I use the rf power generated
from a TWT to calibrate the probes. The rf power is modulated with a low frequency
square wave signal and travels in a rectangular wave guide in TEO, mode. The wave
is then equally splitted into two in phase branches by using a magic T. The wave in
one branch is used as reference and the wave in the other branch is converted into
a cylindrical waveguide mode TMol by passing through a mode converter. While
the TMO, mode travels in the drift tube, its E, and Bo -fieldsare picked up by the
electric and magnetic probes, respectively. The probe signals are then compared with
the reference signal in a lock-amplifier. Thus, a probe coupling constant K relating
the probe signal and the waveguide field can be measured. Figure 39 illustrates a
schematic diagram for the calibration system. The measured coupling constants for
the electric and magnetic probes are shown in figure 310 and 311
The dashed lines
in the figures indicates the operating points chosen for the experiment.
simulation is also used to relate the rf current modulations
MAGIC
to the magnitudes of
E, and Bo at a fixed axial positions.
3.2.2 Relativistic Klystron Amplifier Experimental Results
The current modulation signals measured during a typical shot by one set of the
probes are presented in figure 312. AR of the pulses have the same time duration as
the current pulse (_ 150ns), and in real time, they coincide exactly with the current
pulse. This provides us with additional evidence that the rf signals are indeed caused
by the bunching in the electron beam.
51
0
0-j
0
LAJ
L)
I--
::E
z
LAJ
-i
0
LLJ
0-j
V)
.9
P4
Figure 39: Schematic diagram for the probe calibration system.
52
4
I
-UUIt
^
A
r4
E
I.-,
le
1-11
"N'.
t
1-1 1
e-005
M
z
U)
z0
C.)
I e-006
0z
-j
EL
-D
00
1
1 e-007
C)
E
Q
tLj
-i
LLJ
1 a-0118
-2.5
-2.o
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
PROBE DEPTH (mm)
Figure 310: Measured coupling constants for the electric probe as a function of probe
depth. Dashed lines show the operating point chosen for the experiment.
53
I
e-005
0,
E
11-1
1
I0
I
EXPERIMENTAL
3r
I
1 a-006
DATA
I
,I
I
BEST FIT CURVE
M
I--
z
Ln
z0
0
0z
I a-007
I
a.
...................
.......... .......................
........................ .. ........ .............
. ..........................................................................................................
D
00
0
1 C-008
Li
z0
M
in-nna
I
4
-3
-2
1
0
1
2
PROBE DEPTH (mm)
Figure 311: Measured .coupling constants for the magnetic probe as a function of
probe depth. Dashed lines show the operating point chosen for the experiment.
54
2
B1
I
1
0
0
-1
-':1
=3
LU
-2
W
O
0
-1
Cd
Cd
.2
O
_3
-3
11
0
-4
-4
-5
-5
-6
.6
0
250
500
750
1000
0
250
TIME (ns)
750
1000
750
1000
TIME (ns)
3
4
1
1
0
--
500
0
1
-i
6-2
-1
CIO
u -3
W
(D
<-4
(D
-2
-3
01-5
I
0
>-6
-4
-5
-7
_8
-6
0
250
500
750
1000
0
TIME (ns)
Figure 312:
B27 B3, and
and 500 kW
cm from the
250
500
TIME (ns)
Current modulation signals measured by successive magnetic probes B1,
B4, for a 150 nsec pulsed annular electron beam at kA and 350 kV,
input rf power. The probes are located 6 cm, 8.5 cm, 11 cm, and 13.5
center of the cavity gap, respectively.
55
The probe signals possess two distinctive features. One is that each pulse has a
relatively wide flat top. The flat portion represents about 90% of the pulse width.
The other feature is the sharp spike at the leading and trailing edges of the rf pulse.
A plot of the leading-edge spike amplitude as a function of frequency in figure 313
shows that the spike peaks at a frequency of 3382 GHz, the resonance frequency of
the unloaded cavity. The width of the resonance curve correspond to a cavity Q value
of 380. Recalling that the cold cavity Q value is about 400, we believe that the spikes
are a result of the strong fields induced in the largely unperturbed cold cavity, at a
time of initial crrent
burst. The initial electrons entering the gap have relatively low
density, and at that moment the cavity's Q value is not broadened by the beam, so
the rf field at the gap remains strong. These conditions result in a strong modulation
of the initial electrons entering the gap. Similar effects occur at the trailing edge of
the pulse.
Plotted in figure 314 are the probe amplitudes measured during the flat portion
of the current and voltage pulse. The bunching peaks at 338 GHz and reaches a
modulation amplitude of
of the
kA total current. As expected, the cavity's
loaded Q value is now reduced by one order of magnitude with Q -- 40, in agreement
with MAGIC simulations 22]. This reduction in Q is due to cavity beam loading.
Since the beam is modulated by the rf field at the gap, the current oscillation amplitude should scale linearly with the field strength at the gap, which is proportional
to the square root of the rf input power. The experimental data plotted in figure 315
demonstrates this relation.
An important parameter of the RKA is the knowledge of the optimal distance
between the first and the second cavities. The second cavity should be positioned
where the beam current modulation grows to its first maximum
modulation wavelength given by equation
about 14 of the
3.2)). Figure 316 shows the current os-
cillation amplitude as a function of axial distance from the center of the gap. It is
seen that the rf current bunching reaches a maximum at the distance of 13 cm, which
is about 14 of the beat wavelength.
The small differences between the magnetic
dipole measurements and the electric dipole measurements seen in figure 316 are as
56
9
8
3: 7
E
tu 6
0
zO
5
4
M
W
3
0-
2
Y
co
1
0
3.365
3.370
3.375
3.380
3.385
3.390
3.395
FREQUENCY(GHz)
4
(b)
.E 3 -
Unloaded
Uj
C)
=382
:3
I--
zO
2-
2
LU
YQ-
I
co
0 0
0
3.365
I
3.370
--
I
I
-I-
3.375
3.380
3.385
-
-
-
1
3.390
3.395
FREQUENCY(GHz)
Figure 313: Frequency spectrum of the magnitude of the leading spikes in the current
modulation signal pulses, and the corresponding unloaded cavity Q: (a) measured with
the third ma,,,netic probe; (b) measured with the third electric probe. Both probes
are 11 cm from the center of the gap.
57
z0
15
7
6-
................O...
Z)
0
20
5
zW
4-
1-
cc
CE:
:D
0
4!
3-
W
(D
2 -
zW
1
0ir
Loaded Cavity
=40
W
n- n
I
3.28
I
I
I
I
I
i
I
I
3.30
3.32
3.34
3.36
3.38
3.40
3.42
3.44
3.46
FREQUENCY (GHz)
Z
0
I
I
6 -
D
0
02
-
zUj
4-
(b)
fr
0
0
00
IC
0
3-
Jb 00
%
0
O*
0
W
O 2-
zW
0
0
0
0
0
Loaded Cavity
=40
0
1
IC
Uj
rL
0
3.28
I
i
3.30
3.32
--- -
I
1
3.34
3.36
3.38
I
I
1
3.40
3.42
3.44
3.46
FREQUENCY(GHz)
Figure 314: Frequency spectrum of the percentage current modulation in the flat
portion of the beam pulses, and the corresponding loaded Q: (a) measured with the
third magnetic probe; (b) measured with the third electric probe. Both probes are
11 cm from the center of the cavity gap.
58
8
z0
0
I-I
:D
6-
0
0
1
0
0
:E
1-
z
LLI
ir 4 M
Z)
0
Ld
0
I-zLLJ 2 C)
cc
LLJ
a_
0
I
I-
0
5
--
-
I
I
10
15
-- T
20
25
SQUARE ROOT OF THE RF NPUT POWER IN KILOWATTS
Figure 315: Percentage current modulation as a function of the square root of the
input rf power, measured with the third electric probe.
59
z
0
D
00
72
U
5
4 -
A
1 0 kG Guide Field
0
17 kG Guide Field
0
IL
0
0
6
9
I--
Z
W
= 3D
0
W
0
z
LU
0Er
I
2(a)
1
LU
n0-
0
I
I
I
I
I
I
I
I
2
4
6
8
10
2
14
6
18
DISTANCE FROM CENTER OF CAVITY GAP (cm)
6
0
5
1 0 kG Guide Field
0
17 kG Guide Field
2 4-
zW
cr 3
0
w 2-
z
W
W
(b)
1
0
0
2
4
6
8
10
12
14
16
18
DISTANCE FROM CENTER OF CAVITY GAP (cm)
Figure 316: Percentage current modulation growth as a function of beam propagation
length for a kA beam and with 10 kG and 17 kG guide field respectively: (a)
measured with magnetic probes; (b) measured with electric probes.
60
yet not fully understood). Since the beat wavelength decreases with higher total dc
beam current, as predicted by equation 3.2), we have taken another set of data by
decreasing the cathode-anode gap and thereby increasing the current. Figure 317
shows that indeed the bunching maximum now occurs at a shorter distance of 11 cm,
in agreement with expectations.
61
z0
5
Z)
4 -
0
0
2
zLU
I--
0
0
3 -
0
0
0
CC
CC
D 2W
0
zLIJ
1 0 kG Guide Field
0
1
C)
M
LLI
a_
n 0
I
I
I
I
I
I
I
2
4
6
8
10
12
14
I
6
18
DISTANCE FROM CENTER OF CAVITY GAP (cm)
Figure 317: Percentage current modulation growth as a function of beam propagation
length for a beam current higher than 5 kA and with 10 kG and 17 k guide field
respectively, measured with electric probes.
62
Chapter 4
Conclusions
The cyclotron two-stream instability presented in Chapter 2 is the first selfconsistent nonlinear model for studies of the proposed double-stream cyclotron maser
in particular and the cyclotron two-stream instability in general. The model was used
to analyze the linear and nonlinear coupling of the cyclotron space-charge waves on
two weekly relativistic electron beams copropagating along a uniform magnetic field
with an inverted population in the perpendicular momentum. The growth rate and
the saturation level of the cyclotron two-stream instability were determined from the
small signal theory and computer simulations for parameter regimes of experimental
interests. Good agreement was found between linear theory and simulations in the
small signal regime. When the Doppler upshift in frequency is much greater than the
cyclotron frequency, the instability growth rate was shown to decrease rapidly with
increasing axial velocity spread within each electron beam.
It was shown that the cyclotron two-stream instability is primarily electrostatic.
As a result, the electromagnetic energy flux through the waveguide cross section was
found to be negligibly small. This calls for further exploration of an efficient input
and output coupling scheme for the double-stream cyclotron maser. Such a need for
for identifying efficient coupling schemes is shared with other slow wave systems.
The numerical example given in chapter 2 assumed for simplicity two overlapping
annular relativistic electron beams. In practice, however , there may exist a small sep63
aration between the beams. The gain is expected to decrease as the beam separation
increases. To answer the question of how the gain varies as a function of beam separation, one must use the multimode formalism to solve the exact eigenvalue problem
for the small-signal regime.
For the MIT 33 GHz relativistic klystron amplifier, the input cavity stage has been
completed. We built a compact input cavity, designed a beam diagnostic system and
observed current modulation. With a 50 nsec pulsed annular beam at
kA and 350
kV and 50 kW rf input power, we observed up to 5% beam bunching. We found that
the current modulation scales linearly with the square root of the input rf power and
grows roughly sinusoidally as a function of axial distance. From the first maximum
of the bunching growth, we determined the optimal position for the second cavity,
which will be installed for a full RKA experiment. Theory and numerical results are
found to agree with the experiment.
64
Bibliography
[1] C. Bekefi, J. Appl. Phys. 71
9), page 1 2 and 3
May 1992.
[2] C. Chen, G. Bekefi, and W. Hu, Phys. Fluids B 5 12), page 1 2 and 4 December
1993.
[3] V. Granatstein and I. Alexeff, Eds., High Power Microwave Sources (Artech,
Boston, 1987).
[4] V. L. Bratman,
N. S. Ginzburg,
G. S. Nusinovich,
M. I. Petelin,
and P. S.
Strelkov, Int. J. Electron. 51 541 1981).
[5] G. Bekefi, A. DiRienzo, C. Leibovitch, and B. Danly, Appl. Phys. Lett. 54 1302
(1989); A. DiRienzo, G. Bekefi, C. Chen, and J. Wurtelc, Phys. Fluid B 3 1755
(1991).
[6] P. Catravas W Hu, C. Chen, 1. Mastovsky, and G. Bekefi, SPIE Conference
1994
[7] Y. Y. Lau, M. Friedman, J. Krall, and V. Serlin, IEEE Transactions on Plasma
Science, P-18,
553 1990).
18] D. G Colombant, Y. Y. Lau, M. Friedman, J. Krall, and V. Serlin, Proc. SPIE
1407) 22 1992).
[9] J. S. Levine, M. J. Cooskey, B. D. Hartneck, S. R. Pomeroy, M. J. Willey, M.
Friedman, and V. Serlin, SPIE 1629,19
f1O]A. Bers, private communication.
65
1992)
[11] A. Bers and C. E. Speck, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Report 78, 110 (1965).
[12] R. C. Davidson, Theory of Nonneutral Plasmas (Benjamin, New York, 1974).
[13] A. Bers and S. Gruber, Appl. Phys. Lett. 6 27 1965).
[14] C. Chen and J. Wurtele, Phys. Rev. Lett. 65, 3389 1990).
[15] Y. Y. Lau and D. Chernin, Phys. Fluids
[16] C. Chen and J. Wurtele, Phys. Fluids
4 3473 1992).
3
1991).
[17] C. Chen, P. Catravas, and G. Bekefi, Appl. Phys. Lett. 62, 1579 1993).
[18] H. S. Uhm, Phys. Fluids
5, 3388 1993).
[19] C. Chen, B. G. Danly, G. Shvets, and J. Wurtele, IEEE Trans. Plasma Sci.
PS-207 149 1992).
[20] A. Fruchtman, and L. Friedland, IEEE J. Quantum Electron. QE-19, 327 1983).
[21] R. J. Briggs, Phys. Fluids 19, 1257 1976).
[22] C. Chen, P. Catravas, and G. Bekefi, Proc. SPIE 1629, 29 1992).
66