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THEORY AND COMPUTER SIMULATION OF A NEW TYPE OF PLASMA CHERENKOV MASER (ELECTRON BEAM, MICROWAVE GENERATOR)

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8708428
Pointon, Timothy David
THEORY AND COMPUTER SIMULATION OF A NEW TYPE OF PLASMA
CHERENKOV MASER
University o f California, Davis
University
Microfilms
International
Ph.D.
1986
300 N. Zeeb Road, Ann Arbor, Ml 48106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
THEORY AND COMPUTER SIMULATION OF A
NEW TYPE OF PLASMA CHERENKOV MASER
Timothy David Pointon
B.S. (University of Nevada, Reno) 1980
M.S. (University of Nevada, Reno) 1982
DISSERTATION
Submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in
Applied Science
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
Committee in Charge
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ACKNOWLEDGEMENTS
I wish to thank my parents,
father,
for
their
continued
throughout my graduate studies.
friends,
whose
work easier.
members
who
support
and
in
support
particular
and
my
motivation
1 am also grateful to my
and willingness to help made my
Thanks
are
inspired
me
due
also
to
those
faculty
to put in the extra effort to
learn new physics.
A
special
DeGroot
for
acknowledgement
is
due
to
Dr.
John
his unique role as my major professor.
His
enthusiasm and wealth of ideas almost always presented me
with
new
approaches to seemingly intractable problems I
encountered in my research at Davis.
Finally I wish to give special thanks to Adam Bridge
and
Robert
Walraven
provide unlimited
resources
and
use
of
Multiware,
of
their
the
company's
technical
development of the CYLTMP code.
Inc.
They kindly
computational
assistance
This was
during
invaluable
che timely completion of this dissertation.
-
11 -
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to
TABLE OF CONTENTS
Acknowledgements
ii
List of symbols
v
List of figures
x
Abstract
xii
CHAPTER 1. INTRODUCTION
1
Background
1
Problem Statement and Outline of Dissertation
8
Instability Mechanism
11
Mathematical Model
15
CHAPTER 2.
LINEAR THEORY
20
Introduction
20
Dispersion Relation and Field Structure
Equations
21
Normal Modes without the Beam
24
The Effect of Adding the Beam
34
Discussion
40
CHAPTER 3.
THE CYLTMP PARTICLE SIMULATION CODE
41
Introduction
41
Field Solver
44
Particle Pusher and Particle/Grid Interpolation
46
The Full Computational Cycle
53
Diagnostics
56
iii
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CHAPTER 4. CYLTMP SIMULATIONS OF THE PLASMA
CHERENKOV MASER
61
Introduction
61
Choosing Input Parameters
62
Simulation Results
65
Saturation Mechanism
79
CHAPTER 5.
SUMMARY AND FUTUREDIRECTIONS
APPENDIX A.
RELATION
ANALYTIC TREATMENT OF THE DISPERSION
APPENDIX B.
ORTHOGONALITY RELATIONS
93
98
REFERENCES
105
110
iv
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LIST OF SYMBOLS
A
A/
Vector potential
Cross-sectional area of the beam
Cross-sectional area of the cells centered on
the ith radial row of the 2-D r-z grid
Cross-sectional area of the plasma
B
A/
Pn
Magnetic flux density.
a & component —
In thiswork,
B<£ at the (ifj)'th point of the 2-D r-z
in CYLTMP, at timelevel n+1/2
B has only
^
grid
c
Speed of light in vacuum
e
Magnitude of electronic charge (positive)
E
A
Electric field. In this work E has only r and
z components — Er and Efi
~
E2sj'EAj
E a and Er at the (i,j)'th point of the 2-D r-z
grid used in CYLTMP, at time level n
Linear driver field array for g'th group at
time level n
Total field energy in a CYLTMP simulation system
Electrostatic component of field energy in
CYLTMP
Electromagnetic component of field energy in
CYLTMP
Total kinetic energy of
'th species in CYLTMP
Ezk onfr"') Rad;*-al profile of Ea for the TM6a mode with
axial wavenumber ka
foi
One particle distribution function for species t<
g
Index for simulation particle groups
g'th group of simulation particles for species c<
i
Index in the r-direction on the 2-D r-z grid
used in CYLTMP
v
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, Minimum and maximum indices of cadial rows
’ °ver which particles of group g extend
Ift
Alfven Current —
Ik
Beam current
I0 (I^)
Modified Bessel functions of the first kind of
order zero (one)
j
Index in the z-direction on the 2-D r-z grid
used in CYLTMP
J0 (J^)
Bessel function of order zero (one)
J
Current density. In this work, J has only a
z-component — Ja
J- i\
J
z component of J at the (i,j)'th grid point at
time level n+1/2 in CYLTMP
K0(K^)
Modified Bessel functions of the second kind of
order zero (one)
kB
Boltzmann's constant
k4
axial wavenumber
ka(u
n'th discrete axial wavenumber or "mode" in
a CYLTMP system — a&k-^.
kjjuft.
Effective perpendicular wavenumber of the
TM an'th mode
L
Axial length of interaction region and the r-z
grid used by CYLTMP
m
Electron rest mass
N 0(Na)
Neumann function of order zero (one)
N-.
No. of groups of simulation particles for
species
N^
No. of simulation particles in group g
NR
No. of points in the r-direction for the 2-D
grid
NZ
No. of points in the z-direction for the 2-D
grid
nL
Unperturbed beam density
vi
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np
Unperturbed plasma density
n!L»'v
Axial linear particle density of g'th group of
particles at time level n
n^J
Particle density of species
at the (ifj)'th
point of the 2-D grid at time level n
pa
Electron's axial momentum component
Pr *■
Axial momentum of the r'th simulation particle
at time level n+1/2
R
Radius of waveguide and radial extent of the r-z
grid used by CYLTMP
rt
Inner radius of annular plasma
r^
Outer radius of plasma
r^
Radius of Beam
r^
Radius of i'th radial row of the r-z
grid on which E ^ is defined
r£ fr^
S.
Radii of the closest radial rows
to r^ — fi±A£
z.
Linear axial weighting functionfor gridparticle interpolation in CYLTMP
tgfr{
Time at saturation of the instability i.e.
when field energy is at its first maximum
V0
Unperturbed beam velocity
va
Electron's axial velocity component
vpv*
Wave's axial phase velocity
VjT4^*
Axial velocity of r'th simulation particle
at time level n+1/2
w* ;
3
Radial weights for grid-particle interpolation in CYLTMP
Zj
Axial location of the jth column of the
2-D r-z grid on which ^
is defined
Zp
Axial position of r'th particle at time
level n
oCon>
n'th zero of J0
vii
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Normalized beam velocity -- V0/c
■^0
Normalized wave phase velocity — v ^ / c
X “^2.
Normalized beam energy — (1- ^ ) **
£
Linear growth rate
Quantum unit of wavenumber in the finite
CYLTMP system —
Thickness of plasma —
ra - r4
Dimensions of unit cell in the 2-D r-z grid
Finite time step used in time integration of
field and particle equations in CYLTMP
Thermal velocity spread of beam
Thermal velocity spread of plasma
£
1-D dielectric function
£ a^.
z-z component of the 3-D dielectric tensor
V B e a m
to microwave energy conversion efficiency
©b
Beam temperature (in energy units)
0e
Plasma temperature (in energy units)
Charge density
(TV
w
<T
Charge density at the (i,j)'th point of the
2-D r-z grid at time level n
Plasma conductivity
Electrostatic potential
Electrostatic potential at the (i,j)'th point
of the 2-D r-z grid at time level n
3C
Perpendicular wavenumber in vacuum i.e.
radial profile of E % i n vacuum is a linear
combination of I0(ocr) and K0(yr)
Perpendicular wavenumber in plasma
y.±
Perpendicular wavenumber in the beam
Angular frequency of the TM6A'th mode
viii
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(Ote
Electron cyclotron frequency —
Plasma frequency of plasma electrons —
OJpt»
Plasma frequency of beam electrons
_ S.
Radial Laplacian operator —
w
J4.tr
TV~»
ix
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LIST OF FIGURES
£aga
Figure
1.1
The experimental plasma Cherenkov maser
6
1.2
Radial profile of the interaction region
of the experimental plasma Cherenkov maser
9
2.1
Plot of dispersion function versus M for
fixed k^.
27
2.2
Radial profile of Ea for selected TMOA> modes
29
2.3
Radial profile of E s and Er for a T M ^ mode
31
2.4
2.5
2.6
3.1
4.1
4.2
w versus k4diagram for the TM qo branch of
the dispersion relation
Effect of the beam on the TM00 branch
Growth rate of the TM
of plasma density
35
38
modes as a function
39
2-D r-z grid used for the field solver in
CYLTMP
43
Time history of EM field energy and beam and
plasma kinetic energies for a CYLTMP simula­
tion of the device at n. =2x10'
cm ,
© e = lOeV and © b= 100ev
68
4.5
Time history of most unstable TM0o mode, and
comparison of growth rates from the code
with linear theory for n~=2x1013 cm-3,
©e.= lOeV and @ t= lOOeV
-*3
As in Figure 4.1, but with np = SxlO1* cm
-•3
As in Figure 4.2, but with nP " 5x10 13 cm
As in Figure 4.1, but with np = 8xl013 cm"3
4.6
As in Figure 4.2, but with nP = 8X10*”5 cnf^
75
4.7
As in Figure 4.1, but with e b = lOkeV
76
4.3
4.4
X
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69
72
73
74
4.8
As in Figure 4.2, but with 0^ = lOkeV
77
4.9
Beam to microwave energy conversion
efficiency from CYLTMP simulations
78
Qualitative features of growth rate versus
k and time history of electrostatic field
energy for the 1-D two-stream instability
81
Phase space diagram of plasma electrons at
saturation from the CYLTMP simulation of
Figures 4.1 and 4.2
85
Phase space diagram of beam electrons at
saturation from the CYLTMP simulation of
Figures 4.1 and 4.2, together with axial
profile of Ea on axis
86
4.10
4.11
4.12
xi
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ABSTRACT
Theory and computer simulation of a new experimental
high-power
microwave
generator
is
presented.
In this
device, a circular waveguide is partially filled
dense
annular
plasma.
When
electron beam pulse passes
region,
microwaves
(;£ 20% ).
modes
in
are
an
intense
through
emitted
the
with
waveguide.
The
high
radiation
efficiency
mechanism is
waves
by
The linear theory is analyzed first.
A dispersion relation and field structure
the
vacuum
v ^ < c ) TM
stimulated Cherenkov emission of these slow TM
the beam electrons.
a
relativistic
central
The plasma creates slow (i.e.
the
with
equations
are
derived
for
azimuthally symmetric TM modes of this
system.
Numerical solutions demonstrate the existence of
the slow TM waves without the beam, and confirm that some
are unstable in the presence of the beam.
To analyze the
non-linear theory a new particle simulation code has been
developed.
of
This code is described in detail, and results
simulations of the experimental device are presented.
In these simulations, the
system
initially
evolves
in
good quantitative agreement with linear theory, while the
non-linear
saturation
experimentally
amplitudes
observed
are
efficiencies.
consistent
with
Saturation
linear instability is shown to be due to trapping of
of
the
beam electrons, and the saturation amplitudes agree quite
well with a simple trapping model.
xii
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1
CHAPTER 1. INTRODUCTION
Background
In recent years, great effort has
the
generation
been
devoted
to
high-power electromagnetic radiation
4" 5
using intense relativistic electron beams
.
Quite
apart
from
of
the
extremely
high-power
already been achieved, interest in
from
the
fact
that
conventional
these
shorter
microwave
in
than
tube
particular the gyrotron ( electron
operating
schemes
stems
they provide efficient, high-power
sources at wavelengths far
with
levels that have
is
attainable
technology.
cyclotron
In
maser
),
the millimeter to submillimeter range, and
at even shorter wavelength, the free electron laser, have
both proved to be practical efficient sources of tunable,
high-power radiation.
An
alternative
concept
that
has
received
less
A .O
attention is the so-called Cherenkov maser
.
device, a relativistic electron
through
waveguide
beam
passes
lined with a layer of dielectric.
alters the dispersion characteristics of
and
for
judiciously
chosen
in
vacuum.
These
a
This lining
the
waveguide,
parameters, creates normal
modes with phase velocities lower than c - the
light
In this
modes
speed
of
can then be excited by
Cherenkov emission from relativistic electrons satisfying
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V > V fV\. .
have
Experiments
shown
that
and
efficient
computer
operation
simulation"
is
possible
at
centimeter and millimeter wavelengths.
One of the principal limitations on the power levels
attainable
current.
charge
in
such
devices
is
the
If a beam is propagated in
of
the
beam
limit on the beam
vacuum?
this
space
itself creates a potential barrier
that the beam electrons must overcome.
that
the
It is easy to see
provides a limit on the beam current.
linearity of Maxwell's equations? this potential
By the
barrier
is directly proportional to the beam density - -e^>Boc
Thus for any fixed accelerating
ultimately
voltage
VQ ?
a beam density such that -e
there
is
> -eV0 . This
limitation can be overcome by injecting the beam
dense plasma (n^ >> n b ).
.
into
a
As the beam propagates through
the plasma? the plasma electrons are driven
out
of
the
beam volume? effectively neutralizing the space charge of
the beam (The plasma electrons are preferentially
out
in
since
the
driven
they are less massive than the beam electrons
lab
frame).
For
a
beam
propagating
in
a
conducting medium? the characteristic neutralization time
for
the
radial
ejection
of
the
excess
charge
is
= 1/4tt(T . However? for a typical laboratory plasma?
14.
-3
say
n p = 10
cm
and
Te =10eV?
this
time
is
-17
^3x10
sec.
Thus the conducting fluid model must by
modified to include electron inertia? leading instead
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to
3
the
characteristic
^b
is
the
neutralization time TT*/ ^ &b/c, where
thickness
establishment
of
a
of
the
beam-plasma
beam.
Since
the
equilibrium in a drift
tube of length L requires a time of at least
since typically L >>
and
, any such equilibrium must have
almost complete neutralization of the space charge of the
beam.
Having
neutralized
the
appears.
The attractive
limitation
space
charge,
a
new
magnetic
force
between the beam electrons is now unbalanced, and so
beam
shown
pinches
itself
12.
long
agothat
distribution
down.
the
For a slab geometry, it was
independent
of
the
radial
of the current density, the maximum current
that can propagate in a space charge neutralized beam
is
the so-called Alfven current:
IR =
where
-(DkPDpv
Cm*
^ = v/c, and ^ = ( 1 -
)
The maximum current in a drift tube is modified
geometric
factors
neutralized
of
order unity.
beam
can
by
applying
self-pinching
magnetic field.
only
by
The space charge
be stabilized
from
this
a sufficiently strong axial
However, it should be pointed
out
that
in applying the Alfven limit to a beam-plasma system, one
must use the total current, including any currents in the
plasma.
In
fact, large reverse currents are induced in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the plasma, that almost completely
current
behind
neutralize
the head of the beam.
the
beam
An observer fixed
in the plasma as the beam head passes by sees
a
rapidly
rising magnetic fi^ld due to the azimuthal magnetic field
of the beam passing by with the beam.
The electric field
associated with this time-varying magnetic field drives a
current in the
These
plasma
opposite
to
the
beam
current.
induced plasma currents persist behind the head of
the beam
current
for
a
decay
distance
characteristic
in the plasma.
of
resistive
In practice this distance
is so large that essentially all of
the
beam
except
a
thin layer at the head is fully current neutralized.
With the foregoing considerations, it
would
appear
that unlimited beam currents in plasmas are possible.
practice, beam currents
Since
these
are
limited
instabilities
by
result
In
instabilities.
in
growth
of
electromagnetic waves, the possibility of harnessing them
for
the generation of electromagnetic radiation presents
itself.
The
obvious.
possible
Since
neutralized beam
high
Thus
power
study
relativistic
the
radiation
currents,
levels
of
advantage
is
of
comes
scaling
to
schemes
is
from essentially
extraordinarily
possible, at least in principle.
electromagnetic
electron
such
beam-plasma
instabilities
in
systems has possible
application in high-power radiation generation.
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5
The pioneering work in this field has been
the
Soviet
Union
instabilities
has
(work
in
the
focussed
U.S.
mainly
done
in
on beam-plasma
on
electrostatic
instabilities and applications to plasma heating ).
In a
series of papers in the late 1970's and early 1980's, the
basic
theory
of
what
they
13*— 1
1
?
electronics" was presented
culminated
in
efficient
the
first
plasma
constructed,
the
termed
microwave
. This work has recently
experimental
microwave
basic
"plasma
device
generator
features
.
has
An
been
of which are shown in
Figure 1.1.
In this device, the dense annular plasma
created
pulsing the plasma cathode first.
by
is
Then, the
beam diode is fired and the intense relativistic electron
beam
passes
through
the
plasma.
High-power microwave
emission is observed to emerge from
waveguide.
This
device
maser described above.
dielectric
lining
is
from
output
coaxial
analogous to the Cherenkov
The annular plasma
replaces
the
as theslow wave structure creating
electromagnetic modes with
Radiation
the
the
phase
device
is
velocities
the
Vpt\
<
c.
result of coherent
Cherenkov emission of these waves by the beam
electrons.
Thus we call this device the plasma Cherenkov maser.
In their experiments, they ran
parameters
inner and
radius
fixed:
outer
rb=0.55cm,
Waveguide
radii
beam
the
following
radius R = 1.45cm, plasma
r,=0.67cm
energy
with
and
r;L=0.73cm,
beam
E^=480keV, beam current
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CoiIs
f or
ia I
S7I
FTI
IS
?7
!
rv7
|JE
x
te
rn
A A A ! A ! B* fieId
Cathode
Anode
HIgh-densIty pIasma ,
/
\
Col Iector
v/////y////y/y/7777///y//77s
^/Relativistic
e l e c t r o n b e a m y/y
V ///////////////////////////////2 /,
Cathode
inner coax i a
conductor
Circular waveguide
N , __________
REB
Diode
Interaction
region
Figure 1.1.
O u t p u t c o a x ia I
w a v e g u i de
The experimental plasma Cherenkov maser.
Ib=900A, and a pulse length of 50ns.
output
power
the
We will focus here on
results of varying the plasma density when the other
parameters were held
22.5kG,
fixed
resonator
( External
length
24cm,
scatter of the beam) . The
over
the
range
wavelength
3cm
varied
np«&2xl0
±
from
.
corresponding
to
&
1.8cm.
output
Peak power
of
Since the beam power is 430MW, this is
an efficiency of 21%.
Power output was observed to
below n p «#10
slowly for n p > 2x10
13
cm
13
-3
cm
drop
, while it dropped off
-3
Despite the success of the device, there
published
angular
20) MW was observed at a plasma density
cm
rapidly
minimum
field
density was varied
|3
»4
-3
n p ~ 10
- 10
cm
The
output of (90
and
magnetic
plasma
( n ^ / n p 0.02 - 0.002 ).
off
of
was observed for various configurations of
the other experimental parameters.
the
Dependence
is
little
theory on it ( compounded by the the fact that
apparently some refences are unavailable outside the USSR
).
The
theory
that
is
available
is mainly analytic
treatment of the linear theory, based on
approximations.
For
the conditions for
approximations
limiting
value
the parameters used in the device,
the
validity
are violated.
this system is required for
of
the
most
of
the
Thus numerical analysis of
a
truly
pertinent
theory.
This is the problem addressed by this dissertation.
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8
Broblem Statement and Outline of Dissertation
In this
physics
dissertation
we
will
of the interaction region.
address
of
the
This will be done by
considering the simplified problem of
problem
only
the
initial-value
an infinitely long waveguide with the radial
structure of the interaction region, in
which
the
beam
and plasma densities and velocities are initially uniform
everywhere.
Several
This
system
is
depicted
in
Figure
1.2.
important questions about the actual device will
remain unanswered.
address
First,
with
this
model
we
cannot
the problem of charge and current neutralization
when the beam and plasma are physically seperated.
is
a
transient
established.
net
This
process that depends on how the beam is
We simply assume here that the plasma has a
positive charge density to fully neutralize the beam
charge, and that the plasma has a return current equal to
the
beam
current.
Secondly,
oscillator ( rather than
a
the
actual device is an
single-pass
amplifier),
so
that in analyzing the actual device, the axial boundaries
cannot be ignored.
simplified
Nevertheless,
the
physics
of
this
problem is still interesting, non-trivial and
directly applicable to the most fundamental aspect of the
experimental
device
- the generation of electromagnetic
waves from beam-plasma instabilities in
region.
In
what
simplified model
the
interaction
follows, we will loosely refer to the
of
Figure
1.2
alone
as
the
plasma
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of
9
region
3
D>
Q>
>
Q_
JZ
O)
X
-Q
04
Figure 1.2.
Radial profile of the
the plasma Cherenkov maser.
interaction
O
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
Cherenkov
maser
or
PCM.
Any
references
to the full
experimental device will be explicitly noted.
The outline of this dissertation is as follows.
In
the remainder of this chapter, a brief description of the
basic instability mechanism
simplified
and
justification
mathematical model used is given.
2, the linear theory of the device is
the
dispersion
relation
is
for
In Chapter
presented.
derived.
Then
First
numerical
solutions of the dispersion relation and resulting
structure
of
the normal modes are given.
we describe the new
investigate
the
particle
non-linear
first present results of
simulation
problem.
simulations
compare the results with linear theory.
by a discussion of the non-linear effects
results
model.
are
field
In Chapter 3,
code
In
of
the
used
to
Chapter 4 we
the
PCMf
and
This is followed
in
which
the
interpreted with a simple semi-quantitative
Finally/ in Chapter 5, we summarize the
results/
and discuss future directions for this project.
This problem has
dissertation/
we
many
have
free
focussed
used in the experimental device.
results
in
this
work/
parameters.
In
this
only on the parameters
Thusf in all
numerical
the following parameters always
have the constant values listed below:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11
R
r
r
r
nb
p*
=
=
=
=
=
=
1.45cm
0.658cm
0.718cm
0.538cm
2.3x10!* cm
V6/c = 0.857
These
parameters
experiments.
are
Since
close
to
those
used
in
the
the particle simulation code uses a
finite radial grid size Ar, this
is
as
close
to
the
experiments as we can get.
Instability. Mechanism
Before plunging
useful
to
develop
ahead
a
simple
beam-plasma interaction.
that
although
with
Cherenkov
effects
analysis, it
physical
picture of the
emission
is a single particle
in
densities at
beam
are dominant.
Although
the
nature of Cherenkov emission is obscured with
this terminology, the basic
solution
which
Thus we talk of "beam
waves" rather than single beam electrons.
underlying
is
First, it should be pointed out
effect, we are interested
collective
the
of
the
result
equations
in
remains
the
-
detailed
collective
regime
predict instability only if the phase velocity of
these
unstable waves is less than the beam velocity.
The
linear
electromagnetic
instability
to
generates
the
waves in the plasma Cherenkov maser is a
two-stream instability.
similar
that
the
simple
In fact,
the
physicsis
one-dimensional
very
electrostatic
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two-stream instability.
geometic
Since the latter has none of the
complications of the PCM, let us first focus on
it for the moment.
In the
1-D
electrostatic
case,
we
consider the problem of an infinite homogeneous plasma of
density np penetrated everywhere by
with
and
density
plasma
dispersion
homogeneous
nb and velocity Va =
electrons
and
fixed
relation
for
the
E s
can
can be
be
.
beam
electron
For cold beam
plasma
ions,
the
electrostatic
waves
a O
1.1
io (fca-kVoY
and
Where
beam
written
Etw.k') = 1 - "E -
and
a
are
plasma
frequencies
the
plasma
respectively.
Without the beam term, the solution of equation 1.1 gives
the
normal modes of the plasma
= iWpjf independent of k
), while dropping the plasma term gives the normal
modes
of the beam
DO -
k Vo st
With both terms
equation
in
CO
included,
for
equation
fixed k.
are
with
part
unstable
positive
mode,
imaginary
which
gives
is
a
quartic
For sufficiently small k
( k^cOpt/Va ), a pair of roots
the
1.1
rise
complex.
The
corresponds
to
the
root
to
an
two-stream
instability.
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13
We can understand this instability as follows.
total
energy
( kinetic
energy
of
the
The
oscillating
electrons plus electrostatic energy of the field ) of the
beam waves is
Thus for &>> 0, the slow beam wave ( negative sign ) is a
"negative
energy
amplitude
E0,
the
electromagnetic
wave.
wave".
total
is,
energy
for a certain wave
of
the
beam
relationship
this
happens
between
perturbations.
In
are
the
the
180
because of
e
slow
out
the
density and
beam
of
the total velocity is low, and vice versa.
velocity
wave,
is
the
electrostatic
seperation, resulting in a
energy
of
the
wave.
net
these
high
field
lowering
where
The resulting
drop in the kinetic energy exceeds the positive
in
phase
phase. Thus the total
density ( equilibrium plus perturbation)
energy
and
field is actually lower than without the
Physically
perturbations
That
due
of
definite
to
charge
the
total
By energy conservation, if such a
wave is excited, the lost energy must go into the
source
that established the wave.
The two-stream instability results from coupling
of
the plasma waves with the resonant slow beam waves ( i.e.
those
beam
waves
with
, which
correspond
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to
14
wavenumbers
k
for
L6p\,«
COpe,).The plasma wave
excites the beam wave which reduces
beam.
the
energy
the
This lost energy goes into the plasma wave, whose
amplitude therefore increases, increasing
of
of
the
amplitude
the beam wave, which drops the beam energy even more,
and so on in a positive feedback loop.
The
waves
grow
together, transferring energy from the beam to the plasma
waves.
In the PCM, the interaction is
beam
wave
still
a
is
similar.
The
slow
no longer purely electrostatic, but it is
negative
energy
wave.
The
plasma
replaced by an electromagnetic wave with an E s
wave
is
component
to drive the beam electrons into exciting the beam
wave.
Unlike the 1-D case however, the electromagnetic wave has
transverse components coupled to
Maxwell's
equations.
same rate as G 2
compared
to
PCM,
get
we
electrostatic
the
oscillation
conversion
and
Eg,
component
of
growth
rate
frequency ).
beam
electromagnetic
energy
field
plasma kinetic energy ) from an essentially
interaction
of
by
Consequently they too grow at the
( assuming
the
its
is
small
Thus in the
into
both
energy
( and
longitudinal
the E g. component of the electromagnetic
wave with collective oscillations of the beam electrons.
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15
Mathemat i c.al...Model
In
general,
the
analytic
treatment
of
electromagnetic instabilities in relativistic beam-plasma
systems is extremely complicated.
the
plasma
However
analysis
Cherenkov maser is greatly simplified by the
fact that the transverse velocity components of the
electrons
of
do
not
play
beam
a key role in the interaction (
unlike, for example, the gyrotron or
FEL
).
In
fact,
large transverse velocity components degrade performance,
and in a well designed PCM they are
reasonable
of
neglecting
altogether.
l&ce./
condition
plasma electrons.
the
transverse
Physically,
realized with a strong axial
the
Thus
as
a
approximation, we can make the very important
simplification
components
small.
this
magnetic
&<* ^pc<
velocity
condition is
field
satisfying
>> 1 for both the beam and
Since transverse
velocity
components
are neglected, it follows that only azimuthally symmetric
perturbations can be excited.
of
Maxwell's
equations
to
Thus we restrict
this case.
In general, the
normal modes of the electromagnetic field in a
inhomogeneously
filled
with
a
analysis
waveguide
dielectric have all six
field components, and cannot be seperated into purely
and
TE
modes
20
. . However,
the
azimuthally symmetric
modes of an inhomogeneously filled circular waveguide
which
the
dielectric
special case in which
function
such
a
TM
depends
seperation
in
only on r is a
is
possible.
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16
Since
the
interaction
electric field, we
retain
electromagnetic
field.
have only
Ep
Ea ,
requires an E a component of the
only
the
TM
Azimuthally
and
modes
of
the
symmetric TM modes
components.
With
these
assumptions, Maxwell's equations reduce to
5E r
_
Si
3 E*
=
3
x -3
Sr
| ^Er
H T W
—
j£
. .
1 ‘4
1.3
where Ea, Ep,
To
complete
particles.
waves
and
the
Since
are all functions of r, z and t.
model,
we
are
U> *'10^ » we treat
neutralizing
must
dealing
the
background.
we
the
with high frequency
plasma
For
describe
the
ions
as
a
fixed
beam and the plasma
electrons, collisional effects are completely negligible,
but
kinetic
non-linear
present
electron
effects
behaviour
are
in
critical
the
strong
as the instability grows.
species
f * (r,z,pa,t).
with
a
1-D
in
determining the
electric
fields
Thus we describe each
distribution
function
Since the transverse momentum components
are zero, the distribution
function
satisfies
the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1-D
17
Vlasov equation
where va = pg/^ m, and ^
J 1 + p^/m**c*
The electron current is
9*
J«(r,z,t) = -ef
2. -foi (
1.7
o.
Equations 1.3-1.7 are the basic equations used
the
mathematical
model of the PCM.
here
for
These equations can
only be handled numerically.
In the early stages of
amplitudes
are
unimportant.
small,
the
and
instability,
kinetic
the
wave
effects
are
In this case, it is appropriate to use
linearized
fluid
electrons.
As a further assumption, we will assume
the
velocity
equations
the
beam
and
plasma
that
spread of the beam and plasma electrons is
negligible ( J^v^ << V0 and
beam
for
the
velocity).
&v^ << ^ , where
is
the
The linearized fluid equations for the
density and velocity perturbations of the plasma are
= o
1.8
=
- eEa
at
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18
while
the
linearized
fluid
equations
for
the
beam
perturbations are
^
+ O
W
+ Vo
= 0
1.9
^W
In
+
these
Vo ^5 V?tb
equations,
equilibrium
radius
=
it
quantities
( where
they
has
been
- e £ -a
assumed
n^, n p and
are
that
the
are independent of
non-zero).
The
linearized
current density is
1.10
Now assume
fields
have
that
the
perturbation
the form f (r) e
quantities
and
. g0 iving for the
perturbation current as a function of E ^ , we can rewrite
Maxwell's equations as
i£L
= i V , E r -|I*-
Er
l.u
«
1-12
1.13
where
is the z-z component
of
the
linear
dielectric
tensor, defined by
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19
r
f
£„(Wrk*,r) =<
"z*
4
o«r$rk
-
1
1
rVa< r < r l ] .14
-
n ^ r ^ f 2.
V. 1
Equations 1.11-1.14 are the starting point for the linear
theory.
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20
CHAPTER 2. LINEAR THEORY
In this chapter? we analyze the linear theory of the
We first derive the dispersion relation relating to
PCM.
and k2 ? and the equations for the field structure of
modes.
Analysis
stages.
First? we find the solution
relation
these
equations
is
to
the
That is we look for the normal
infinitely
plasma.
long
waveguide
containing
For brevity? we call
waveguide”.
beam
in two
dispersion
this
modes
the
only the annular
system
(
recall
fluid
that
we
are
the
using
"plasma
starting
point
the
These modes are
for the analysis of the system with
the beam? both for the initial-value problem
axial
cold
model for the plasma and the beam)?
both 03 and ka are real for these modes.
without
of
Since there are no losses in the absence of
collisionless
the
done
and resulting field structure in the absence of
the beam.
the
of
the
boundaries)?
and
the
( with
or
steady-state
boundary-value problem.
However as mentioned in
1?
the initial-value problem of the
we
focus
only
on
infinitely long waveguide.
real?
independent
Thus
variable?
we
and
consider
use
the
chapter
k3
roots
the effect of adding the beam.
u) (k%) with
positive
imaginary
a
dispersion
relation to find the allowed frequencies co(k9 ).
consider
as
Then we
We find complex
parts
which
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21
correspond
to
unstable
waves.
Finally the results of
linear theory are summarized.
Dispersion Relation and Field Stjcucture eauations
From Maxwell's equations - equations
1.11-1.13,
we
easily derive an equation for E &
2.1
where
■=*
The transverse field components are easily obtained
from
E « using Maxwell's equations
ELr “
Ca£
c-z.—
2.2
\JZ'
2.3
Writing out equation 2.1 for each region
explicitly,
have
vacuum:
plasma:
Vr
beam
Vr*
:
+
0Cp
« O
2.4
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we
22
where the constants
, OCp and
are given by
-i)
2.5
^ i f c s s * - 0
We choose the signs this way because we are interested in
the
and
frequency
ye? are
range to< ck2 , £»><
The equations for E s are
both real.
equations of order zero.
y.
In this range
Bessel
Since E s must be finite at the
origin, we can write the general form of the solution as
~
AJ0 (orkr)
BId (xr)
E a (r) =
+ CKft(xr)
rt <
r < r,
<
2.6
rt4
DJ0 (a^r) + ENtt(«pr)
r * r^
V^, FI0 (9Cr) + GKa (jcr)
Ja and N s are Bessel and Neumann functions of order zeror
while
zero.
I e and
The
unknown
K0 are modified Bessel functions of order
radial
boundary
conditions determine
The
requirement
quantities.
E ^ (r=R) = 0.
Secondly, since E%
component
at
interfaces,
itmust
interfaces.
component of
first
the
be
Finally,
D, E r
beam/vacuum
by
the
that
field
plasma/vacuum
continuous across
and hence
across the interfaces.
is
is a tangential
and
the
these
continuity of the normal
must be
continuous
Using the relations
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23
J o (z)
=
N*<z> = “» ±(*)
K*<z> = “K^ 2)
the boundary conditions lead
to
a
set
of
homogeneous
linear equations for the coefficients
AJ0 (*jrt )
= B I d (3crb )
+ C K 0 ( a r fc)
BIe (7cr, )
+ CK„(oc r, )
= DJ0 (5Cfr, )
+ E N 0 (3C|lr|)
D J o t e pr * .)
+ E N ^ C X p r^ )
= F l^ jc r^ )
+ GK 0 ( ^ c r z
PI0 (JCR)
+ GK 0 (3CR)
=0
l*brh*
)
2.7
= 7C(”BI^ (# rfc) + CK^jcr^))
3C(-BIA(5cr| ) + CK4 (*r4 ))
=
) + ENA (^r4 ))
+ ENA (s^r2 )) = 3C(-FI4 (*ra ) + G K ^ r ^ ) )
We get non-trivial solutions if the
coefficients
is
zero.
This
determinant
leads
to
of
the
the dispersion
relation
Q.
2.8
1o(3cr,>) [ l 5(3cr,') + Q K 0 Cxni]
In this equation,
Dp - yt-n^fotp
VTNafapri}
is the dispersion function without the
29
X d(7fr0
beam,
while
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
24
other undefined quantities are
"I* —
-t PJftQfpO)
+■ PNdC^ffaT)
f
% (
«
4- l « ( y R ^ Ka.(^r^\
\ I » (X *i) K atxR ) — lofjCR?} Ko C^CHt) /
Q
2 .10
3C IjLtyQ 4-S lo txq,^
s
-Sk,(xO
£
ss
This
X b *3a.(Xh^
Jo^b^
form of thedispersion relation
21.In
deriving
this
is
given
in
Ref.
form, we have used the relation
W(R 0 (z), IQ (z)) = 1/z, where W is the Wronskian
operator.
We
by
begin
the
analysis of
equation
2.8
first
considering solutions without the beam.
Normal Modes without the Beam
Without the beam, the general form for E
, equation
2 .6 , simplifies to
Ea
=i
BIa(*r)
0 ^
r < r,
° J o ^ p r^ + EN0 (jCjj)
r, <
r<
PIe(5Cr) + GKB(xr)
r2 < r ^
r^ 2.11
R
While the dispersion relation - equation 2.8, becomes
Dp = 0
2.12
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25
Analytic treatment of
relation
can
be
done
even
this
only
simplified
in certain limiting cases.
Rather than tackle this immediately,
look
at
dispersion
it
is
clearer
to
the numerical solutions to this equation first.
Plots of the numerical solutions
provide
a
qualitative
understanding of the nature of the normal modes, and help
to motivate the analytic approximations.
Two programs have been developed for
solution of the linear problem.
the
numerical
ANALYZE_DSPFN solves the
dispersion relation, while GET_FIELDS used these
to
find
the
corresponding
field
results
structure.
Both
ANALYZE_DSPFN and GET_FIELDS contain a switch to turn the
beam
on
or
off, allowing these programs to investigate
either the plasma waveguide alone, or the full linear PCM
problem.
ANALYZ EJDSPFN
uses
iteratively find the complex roots
from the dispersion relation.
Muller's
to (k4 )
ANALYZ E__DSPFN
is
an
for
99
real
to
ka
As with all iterative root
solvers, it performs only as well as the
Thus
method
initial
interactive
requires some careful input from the user.
guess.
program
that
It allows the
user to graphically locate the neighborhood of a zero and
reset convergence parameters before actually calling
root
solver.
the
By contrast the calculations performed in
GET_FIELDS are straightforward, and can be
done
without
any user supervision.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
Before
solving
the
dispersion
relation
for
the
plasma waveguider it useful to recall the results for the
azimuthally symmetric TM
waveguide.
modes
of
the
empty
circular
These modes satisfy the dispersion relation
2.13
and have radial structure
2.14
r
where
n = 1,
2,
...).
being
Note
the
nth
zero
at
Thus
the
high
on
In the plasma
the
dispersion
characteristics.
frequency roots to equation 2.12 must be
given approximately by
equation
2.14.
At
frequencies
the plasma will have a strong effect.
k 2 and go
Dp
over
, Dp
a
is also real.
range
of
Figure 2.1 shows a
frequency
for
the
parameters at a plasma density of n p = 2 x10
wavelength
of
(
frequencies t*> >>00p«. , the plasma has
high
very little effect
Ja
that for any GO and k^, the phase
velocity of these waves is greater than c.
waveguide
of
X» =
3cm.
This
density
For real
plot
of
experimental
-3
cm
and a
is roughly the
density at which the efficiency of the device is maximumf
while
this
wavelength is that of the most unstable wave
in the presence of the beam.
We see that there are roots
to equation 2.12 corresponding to modes with Vp^ < c.
In
fact there are an infinite number of low frequency roots.
However
the
very
low frequency roots are not physical.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
100
50
CL
o
-50
-100
-2
1
o
«/«pe
Figure 2.1. The dispersion function without the
beam at plasma density np = 2 x10 '* cm
, plotted
over a range of frequency for fixed k^= .25wfft/c (
wavelength 3cm).
= .25 is the freqency at
which vpi^ = c.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
They arise from using the cold fluid model for the plasma
beyond
its
range
of
validity.
In reality, waves with
phase velocities comparable to the
the
plasma
are
strongly
thermal
Landau damped.
plasma model is valid only for W >>kay ^ .
since
v^
<<
velocity
of
Thus the cold
Nevertheless,
c, some of the higher phase velocity slow
waves are physically significant.
When the field structure corresponding to the
of
the
dispersion
relation
these modes becomes clear.
profile
of
Ea
for
roots
is computed, the nature of
Figure 2.2 shows
the
radial
the first three modes with Vpy* > c
(a-c), and the first three modes ( in order of decreasing
frequency)
with
essentially the
slightly
by
< c (d-f).
vacuum
waveguide
the plasma.
TMon modes, for n = 1, 2,
exist only
Modes with Vp^ > c are
TM
modes,
perturbed
We therefore label these modes
... .
Modes
with
v^
<
c
because of the plasma. It is natural to label
these modes in order of
decreasing
frequency,
counting
downwards.
Thus we call the highest frequency slow wave
the TM QO
mode, the next highest frequency slow wave the
™o,-i or™ o T
mode
an<* so on*
Higher TM0^ modes have
successively more wiggles in the plasma while
the
field
structure in the vacuum region is almost the same.
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29
.5
1
0
0
1
0.0
0
1
(d)
(a)
1
2
0
0
1
0
1
2
0
1
(e)
1
i
0
1
Vl!
!>
1I
I
I}!
I
<f
0
k
r /R
1
r /R
1
(c)
Figure 2.2. Radial profile of E g
that E a(r=0) = 1) corresponding
roots in Figure 2.1:
(a) to = .366
(b) (0 = .520
(c) to = .798
( normalized so
to the following
(d) w = .191
(e) tO = .039
(f) to = .020
The plasma occupies the shaded region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
The TMoo mode is of particular interest in the
Not
only
does
it
satisfy
the condition for Cherenkov
emissionr v ^ < c, but it has the simplest
the
structure
to excite by a
relativistic
electron
beam.
The
full
structure of T M ^ wave from Figure 2.2 is shown in
Figure 2.3.
large
Note
outside
that
the
the
transverse
plasma.
Thus
components
the
plasma.
outside
It is this characteristic that allows good
coupling of the TMW wave with a TEM wave in
waveguide
are
electromagnetic
energy of the TM M mode is predominantly located
the
in
of all the slow waves, making it the easiest
plasma
field
PCM.
of
the
experimental
device,
the
while
component is largest on axis to maximize the
output
the Ea
interaction
with the beam.
The question of completeness of the TMork modes is
technical
problem
beyond
simply assume here
field
at
represented
time
by
t=0
a
the
scope
of this work.
that
any
azimuthally
in
the
plasma
linear
symmetric
waveguide
combination
of
the
can
J-Oo
We
TM
be
plasma
waveguide TMen modes
Ea(r,a.It-o>)A <Jk2e *^2 .
a
2.15
Ws-fc
Subsequent time evolution of the field is then
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31
0.0
0.2
r/R
(a)
4
3
2
1
0
1 «-*
0.0
0.2
0.4
0.6
0.8
1 .0
r/R
(b)
Figure 2.3. Radial profile of (a) Ea and (b) Er,
for
the
TMoo
wave
with
k*= .25u)f£/c at
n^ = 2x10
cm"3 . The fields are normalized so that
E% (r=0) =1. The plasma occupies the shaded region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
2.16
_i. ^ot\(
To
find
the
expansion
orthogonality
coefficients,
we
need
an
condition for inodes with the same k8.
The
simple orthogonality condition for TM modes of the vacuum
waveguide
comes from Sturm-Liouiville theory.
is
not
a
Sturm-Liouiville
However equation 2.1
equation,
because
is a
function of both CO and r ( the plasma is dispersive, and
only
partially
fills
the waveguide).
It is clear just
from looking at the radial profile of the T M e^ and
modes
that
they
do
not satisfy jrAc
Nevertheless, there exists a more
condition
that
rather than
general
comes
special
relation
general
*
orthogonality
directly from Maxwell's equations
mathematical
structure
'
.
The
and its derivation are in Appendix B.
For the special case here in which we consider two
m
TMeo
modes
and <\. having the same k4' the orthogonality condition
can be written
r~iv\
2.18
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33
Physically, this condition ensures that the total
in
the
field
energy
given by equation 2.17 and its associated
transverse components, is equal to the sum of the
energy
in its component modes.
Finally in this section, we
consider
the
analytic
treatment of the dispersion relation, equation 2.12.
For
the TMon modes with n ^ 1, equations 2.13
are
reasonable
approximations.
However,
principal interest is to derive
and
for
analytic
the PCM, the
approximations
for the slow TM modes, especially the T M m mode.
done in ref.
A.
The
21, and this work is contained in
principal
result
2.14
This is
Appendix
is an approximate dispersion
relation for the slow TM modes
2.19
where n = 0 , "1 , "2 , ... and
r
n = 0
n < 0
Where
A p = r^ - r^.
Equation
A f<<C4 ,
2.19
and
assumes
In
that
£5 1.
JCR «
1,
>:>
These conditions can only be
satisfied for a low frequency wave and a very dense
plasma.
Figure
2.4
compares
the
TM 00
thin
branch of the
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34
dispersion relation computed
1*3 -3
2.19
at n p = 2x10
cm .
ra
*+4,
equation
2.19
numerically
Even
is
though
still
approximation of the TM ^ branch.
with
a
equation
9CR
2
fairly
and
good
The field structure of
the TM o0 mode/ is given to lowest order by
0 ^
r ^ r-
Ea =<
2.20
\xS^
'A ker
Er
=
ra><
r ^ R
0 ^
r£ r
<
2.21
r- < r < R
sc 8* % r
This concludes the discussion of the normal modes of
the
plasma
waveguide.
We
now
consider the effect of
the
slow
adding the beam.
Ths ..Effsct
.Adding the.-Sg.am
When the beam is added,
become
can
unstable, due to collective Cherenkov emission by
the beam electrons.
the
T M on modes
TM0q
mode
We focus only on the instability
with
a
of
beam of the appropriate energy,
since this is the mode of interest in the PCM.
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35
0.4
0.3
0.2
0. 1
0. 1
0.2
0.3
0.4
0.5
^ ^ z/^pe
Figure 2.4.
The T M 0o branch of the dispersion
relation at n. = 2 x 1 0 cm"3 .
Solid curve is the
numerical solution and the dashed curve is equation
2.19.
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36
The standard approach to the analytic
beam-plasma
weak ( n ^
the
interactions
«
n^
is
treatment
of
to assume that the beam is
), and treat it as
a
perturbation
normal modes of the beamless system.
of
The details of
this calculation for the PCM can be found in Appendix
The
A.
principal results are that the frequency of the most
unstable TM qq wave is given by
W** «
Wpe, —
2.22
where the cutoff frequency is defined by
COe.
2.23
85
The wavenumber of this wave is given by W =
kaV0 , while
its growth rate is
\
a
Equation 2.24 is
implies
a
\ 2 i V y 6?
valid
minimum
2.24
if
<<
1.
Equation
2.22
plasma density for which instability
occurs.
In fact this is observed in the experiments. As
1$
-3
the plasma density is reduced below 10
cm , the power
output drops
off
sharply.
However,
substituting
the
experimental parameters into equation 2.23, the predicted
cutoff density is n ^ 2 .8 x10
cm
.
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37
The analytic results, equations 2.22 - 2.24
fact
good
are
in
approximations only for beam densities so low
as to be of
little
investigate
the
practical
linear
interest.
effects
To
carefully
of the beam, numerical
solution of the full dispersion relation, equation 2.8 is
required.
The numerical solution of equation 2.8 follows
the same procedure as that
described
for
the
beamless
case.
The
effect
of
adding
the
beam
used
in
the
experiments to the TM00 branch of the dispersion relation
13
_3
at a plasma density of n p « 2x10
cm
is shown in
Figure
2.5.
The
unstable over the
rate
plot
range
shows
04
r U
that
tv-
^
the
0.35.
TM^
mode is
The
growth
is
maximum
at
k-g.® .25tOp(6/c.
Since
ii
..a.
2.5x10 sec ,
the
maximum
growth
rate
is
-a
3.5x10 sec .
Thus, this wave increases by an order of
magnitude in 0.66nsec!
to
This is extremely
fast
compared
the 50nsec pulse length used in the experiments.
growth rate of the TM ^ branch with plasma density
parameter
is
shown
in Figure 2.6.
which
as
a
This plot is in CGS
units to compare the "absolute" growth
that
The
We find
IZ -3
the TMO0 wave is unstable only for np1^ 6x10
cm ,
is
in
good
agreement
with
rates.
the
experimentally
observed values.
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38
0. 3
3
©
QC
0.0
0.3
0.4
0. 5
(a)
0.016
0.0 1 2
©
Q.
3
3
0.008
v_/
E
0. 00 4
0.000
0.4
0.0
(b)
Figure 2.5. (a) Real and (b) imaginary parts of the
TMfto branch of the dispersion relation with the beam
at n p = 2x10
cm"3 . The dashed curve in (a) is the
(purely real) TMm branch without the beam.
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39
xl 0®
6
5
4
3
2
1
0
0
2
4
6
8
10
1 .6
2.0
k z (c m " 1)
(a)
x10s
16
12
8
*o
4
0.4
0.8
1 .2
k z (c m " 1)
(b)
Figure 2.6. Growth rate of the TMoo branch of the
dispersion relation
with plasma density as a
parameter. Each curve is labelled with its plasma
density in units of 10*^ cm'3 .
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40
D iscussion
The TM QO mode
central
of
the
role in the PCM.
waveguide
plays
a
In the presence of a beam with
the appropriate energy, it is
rapidly
plasma
unstable
and
compared to typical pulse lengths.
it couples very efficiently with
the
TEM
grows
very
Furthermore,
mode
of
the
for
the
output waveguide in the experimental device.
The linear
lower
limit
for
experiments.
exponential
theory
successfully
operation
However linear
of
accounts
the
theory
PCM
predicts
growth of the unstable waves.
that slow and ultimately terminate the
waves
attainable
determining
non-linear
non-linear
wave
growth
theory.
problem
We
is
unbounded
of
these
Since the
critical
in
the efficiency of any actual device, we need
assumptions, and even
handle.
amplitude
in
The processes
are outside the realm of linear theory.
maximum
a
observed
Analytic
must
then
therefore turn
treatment
make
is
to
some
extremely
of
the
simplifying
difficult
to
computer simulation to
investigate the non-linear effects.
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41
CHAPTER 3. THE CYLTMP PARTICLE SIMULATION CODE
Introduction
Particle
analyze
simulation
methods
are
widely
used
to
.
Of
non-linear problems in plasma physics
particular interest to plasma microwave
multi-dimensional
(2-D,
2-1/2D
fully electromagnetic codes
2i’2TJ
electronics
are
and 3-D), relativistic,
.
These
codes
retain
the most physics, and require tremendous computing power.
Since our computational resources are
VAX
limited
to
computers (VAX 780, MicroVAX II ), we cannot perform
such calculations in a realistic amount of time.
the
small
time
used
steps required
Lorentz
in
to
force.
such codes is spent on the intricate
push
the
particles
with
for
-eE^.
the
electrons
is
of
the
Since
with
of
the
essential
called
CYLTMP
TM waves and Plasma) to analyze this model.
CYLTMP can be used not only
system
equation
PCM is contained in these equations, we
have developed a particle simulation code
(CYLindrical
full
the much simpler
There is no magnetic force.
physics
the
However, if we restrict ourselves to the
model described by equations 1.3 - 1.7, the
motion
Most of
azimuthal
for
the
PCM,
but
on
any
symmetry in a circular waveguide
with a strong magnetic field constraining
the
to
this
general
in
computer
only
move
axially.
applicability, and the
Despite
tremendous
savings
particles
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42
time
compared
with
a
full
2-D
relativistic
electromagnetic code, CYLTMP appears to be the first code
of its kind.
We therefore discuss it in detail here.
CYLTMP self-consistently advances
the
value
of
forward
in
time
the electromagnetic field components on a
two-dimensional r-z grid, and the the one-dimensional (z)
positions
and
momenta
of
simulation
representing the plasma and beam.
are
advanced
using
The
field
densities
interpolated
positions and velocities.
turn
advanced
using
components
finite-difference approximations of
Maxwells equations (equations 1.3-1.5), with
current
particles
from
The particle
charge
the
and
particle
momenta
are
in
electric fields interpolated fromm
the grid, and the particle
resulting new momenta.
positions
updated
with
the
The field solver is essentially a
simplified version of the one used in the NRL code CYLRAD
9.1
as
described
by
Boris
interpolation between the
novel
features
of CYLTMP.
.
The
grid
particle
and
pusher
particles
and
are
the
In this chapter, we consider
in detail these two halves of the calculation,
how
they
are coupled together in the complete computational cycle,
and finally some of the diagnostics used in CYLTMP.
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43
z=0
Z= L
-O — S3— O — S3
o — S3— O — S3
I
I I I
NR D — • —
I
O—
I
S3—
I
I
I
I
I I
I
I I I
I
I I I
r=R
-O — S3— O — S3
O— »
1 1 1 !
NR-1 □— •— □— 6
-□— © — □ — 9
I
1
B
I
h
I
zl
2 O — S3— O — S3— 7K 2 □-
I
O — S3
I
• □I — ©I AAr
-©—
I
1 O1— S3—
1, O — S3—
O
1 □ - © — □ — ©1 1 2
2
NZ
II
0
-
S3
I ©I
I I
$3— O — S3
-
r=0
NZ+1
Jz
m
i
□ - " EI f
S3 —
I
□— ©
-
0 -
9 - - p,
§
Figure 3.1. The 2-D r-z grid used for the field
solver in CYLTMP. The "computational box" 0 « z < L ,
0 < r £ R is shown with the thick outline.
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44
F ie ld Sply.sx
The two dimensional r-z
grid
on
which
components are defined is shown in Figure 3.1.
covers a computational box O ^ z ^ L , O ^ r ^ R .
row
of
the
field
This grid
The
extra
points in each direction is included to simplify
the boundary conditions and particle pusher.
quantities
are
The
field
defined at the following grid points and
times
Eaij
=
E*((i-YUr.(i-V)Aa,a*)
Ertj = Ec( tt-OAr, -1)A*.,C&)
{U-£> Af, 0 - V)A5L
H D
$)
=
&t)
3-1
.<i-i) ae , I'-frt')
Note that these quantities are staggered in both time and
space.
The electromagnetic field components are advanced
with the following finite
difference
approximations
to
equations 1.3-1.5
3.2
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45
where
T\ = fo-
V) &r
= fi d:
and
c^-
x
Because of
the
staggering
of
the
SL
field
definitions/
equations 3.2-3.4 are second order in both space and time
( i.e.
finite difference truncation errors are
Equations 3.2-3.4 hold in the interior mesh/ but
compute
the
conditions.
boundary
points, we need specific boundary
boundary
CYLTMP
currently
conditions
-
is
runs
periodic
conducting boundary at r=R.
r=0
to
The
handled by symmetry.
in
with
the
z, and
simplest
a perfectly
computational
boundary
Explicitly/ these boundary
conditions are:
1.
Periodic in z.
For all quantities/ the j=NZ+l column
is equal to the j=l column.
2.
Symmetry at r=0
3.5
n-4*1
3.
|
i
Conductor at r=R
3.6
Equations 3.2-3.6 form a complete set of equations/
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once
46
we have a prescription for computing
Particle Pusher and Particle/Grid Interpolation
Since the particle motion
the
force
law
is
is
one-dimensional,
simply £, - qE^e%, the basic particle
handling facilities in CYLTMP are essentially those
in
a
one-dimensional
review the 1-D
and
electrostatic code.
non-relativistic
Let us first
electrostatic
case
clarify the discussion and define some notation.
rest of this dissertation "particle"
mean "simulation particle".
will
used
be
to
For the
taken
to
Actual plasma particles will
be referred to as "real particles".
In a 1-D electrostatic code, the
considered
to
be
particles
can
infinite slabs of charge with uniform
charge density normal to the direction of
inhomogeneity.
We will take the z-axis to this direction.
The particles
move in a computational box ( here a line ) from
z=L,
on which a grid Z*^ = j A z is defined.
step, a charge density
based
on
the
particle
self-consistent
With this field we
particle,
repeated
and
z=0
to
At each time
is assigned to the grid points
positions.
Then
difference approximation to Gauss' law is
the
be
used
a
finite
to
find
electrostatic field Ej on the grid.
compute
the
force
acting
advance it forward in time.
continuously,
simulating
the
on
each
The cycle is
self-consistent
collective behaviour of this multi-particle system.
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47
Central to the operation of the code is
function
S(Zj-zr)
that
to or from a particle at z r
species
c& at
density
3.S.
density
a
force
particle
of
q^vrS(Zj-zr)/£z
q,«,S(Zj-zr}/Azr and
to
grid point Zj .
turn, the field at grid point Zj acts
with
A
with velocity vr contributes particle
density S(Z^-zr)/&z, charge
current
weight
interpolates a quantity at grid
point
zr
the
q^EjS(Zj-zr).
on
this
In
particle
The most common choice of
weighting function is linear weighting
j * jo
j = je + 1
st
3.7
j = j© »jo + 1
where Z? £ z_ < Z. . , and
4®
r Ja-HL
With
this
function
a
&z = ftr - Z;
1
j
particle
contributes
particle
density only to the two grid points it lies between.
relative
contribution
to
directly
proportional
to the distance between the these
points and the particle.
simple
This
two
grid
weighting
points
scheme
is
has
a
physical interpretation in terms of "finite-size"
particles.
their
these
The
Since the
grid
points
neighbors by a distance
Az,
considered to be at the center of a
are
seperated
from
each grid point can be
cell
of
length A z
extending outAz/2 from the grid point in each direction.
If we now imagine the particles to have a
finite
length
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48
equal
to
the
grid
spacing
bzt S^(Zj-zr)
fraction of the particle in the jth
this
linear
weighting
in
the
is simply the
cell.
CYLTMP
uses
axial directionr so its
particles can be considered to have a length of Az in the
z-direction.
We can summarize the computational cycle
electrostatic
code as follows.
ft
(umh
and velocities z r and vf
at
of
a
1-D
Given particle positions
time
step
n , we
first
compute a charge density on the grid by
"
A a i r l ’* ?
S l a i -a?')
3.8
Next solve the field equations to get the electric
Ej* on the grid.
field
Then advance the particle positions and
velocities to the new time step ( correct to
by
the staggering in time between vr, zr and Er )
»
-eg?**
^
3.9
where the electric field at particle r is
Er"
-
2 EjS(zr
a nr )
3.10
3
With the particle positions and velocities at
time
step
n+1, we can then repeat this cycle.
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49
In
CYLTMP,
although
one-dimensional,
the
the
particle
particles
assign
points on one or more radial rows.
given
species o<
are
divided
motion
charge
to grid
The particles
up
into
Particles from group
for
to
by
the user at t=0.
the
for radial rows
t0 imax,ou^ ' these radial boundaries being
specified
a
groups G ^ ,
contribute
charge and current densities of species
is
inputs
Physically each particle
in group G*^ can be visualized as an annulus of thickness
Az
and
inner
and
outer
radii
r7Wtoi oj and rtIfllWX.O*}
respectively, representing a collection of real particles
in
this
volume,
all moving with the same velocity.
reality, since E % varies with r, the
real
the
eventually
volume
of
the
particle
different velocities, and
However
on
a
grid
with
the
will
annulus
in
CYLTMP
is
particles in the
weighted
that
particles
will
in
acquire
break
many radial rows,
slowly from one row to the next.
In
up.
can vary
The approximation
made
we use the same Es for all the real
annulus,
this
E a being
a
carefully
average of E^ at the "near" grid points.
Given
that there is no radial motion of the real particles,
they
all
maintain
if
the same axial velocity, the annulus
moves as a single rigid super-particle.
Furthermore,
if
the real particles in the annulus start with a prescribed
radial density profile, this profile is also preserved in
time.
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50
Although
several
the
radial
particles
rows,
we
assign
can
their
model
arbitrary
profiles by radially weighting the charge
different rows spanned by the particle.
group
have
the
particle
density
same
on
density
the
charge
to
radial
assignment
to
All particles in
profile.
Thus,
the
grid between rows i a n d
i-wvm.oi^ can be written in the factored form
3.11
where
^
=
3‘12
is the axial linear particle density density of particles
in
group G ^.
Note that CYLTMP uses linear weighting in
the axial direction, and that
*s
dimensionless.
It is convenient to put the 1/length3
dimension in nQ
-
the "background" density that defines the electron plasma
frequency
used
as the unit of time.
Particle densities
for each species are then defined relative to
w«*cy,
are
the
radial
weights for group
above discussion, they remain
uniquely
defined
constant
in
na .
The
From the
time,
being
by the initial radial density profile.
At t=0, the unperturbed density profile is
1 U (f\ ^
* YUo
( O Vtfte)
3-13
where g^ and h ^ are dimensionless functions specified
the
user.
The
axial
positions
of
by
the particles are
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51
defined so that
3*14
where n <,*^o = N ^ / N g
density of group
group
).
is
the
average
linear
particle
( N,*^ is the number of particles in
Comparison of
equations
3.11,
3.12
and
3.14 show that the radial weights are defined by
U o /
w^ l
Thus
n u .0
= ^
3-15
in CYLTMP, the full charge density on the2-D
grid
from all the particles is given by
^
“
HoZ. ^
3.16
Similarly, the current density on the grid is
°
3-17
where
(?VO u ^
*
21
V r Si( 5 ^ -
Z.^)
3.18
This completes the particle to grid interpolation scheme.
We
also
need
to
interpolate
particles for the particle
picture
E g from the grid to the
pusher.
From
our
physical
of the simulation particle, the average force on
the real particles within its annular volume is
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52
Thus in CYLTMP we compute the following radially weighted
average driver field for group
/71.
where A^ = TT ( r£
radial cell.
- r^
) i s the endface area of the ith
Note that the radial weight factors for the
driver field can also be computed once
t=0.
3.19
and
We then move the particles in group
a 1-D electrostatic code.
for
all
at
just as in
The field at the rth
particle
in group G«u^ is
3.20
We update the particle momentum with this field
tv+K.
n
^
”* ^
=
^ E r At
3.21
With the new momentum, we then update the particle energy
and position
*
3.22
n * 'i
z ^ - z "
,
jv
At
3-23
;i ? v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
The F.ull.Computation Cyglfi
With the above definitions, we can now
full
computational
cycle used in CYLTMP.
examine
the
Starting with
ri
a
the quantities at time level n - Eayj ,
fk
A**%
zc ,
- w e perform the following steps:
»
1.
Compute B^.>
2.
Advance the particle positions and momenta
and
pr
from equation 3.2.
.
For
each
driver field using
particle
using
in
the
equation
o+l
zr
.
group, we compute the linear
equation 3.19.
group
3.21
and
to
Then
for each
we obtain the new momentum
the
new
position
using
equation 3.23.
t\*k from
3. Compute J2 -
calculation
is
equations 3.17
actually
and
3.18. This
performed as the particles
are being moved.
4. Advance
and
E rij
t0 time
level
n+1 using
equations 3.3 and 3.4.
nfrl
0+1
We now have Ea-j
'2 , Eri^ ,
,
o+l
zr
. o+V
, and pjand can
,
repeat the cycle.
As emphasized above, all the radial weights
can
be
computed once and stored for use at each later time step.
Thus the particle mover is almost as fast as a
electrostatic
mover.
The
trick
of
pure
1-D
spreading
the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
particles out over more than one radial row results in
tremendous
saving
a
in
the
number
of particlesrequired (
and hence CPU time per
run
).
theexperimental
In
PCM,
the plasma thickness is about l/25th of the radius of the
guide, while the beam radius is about 9 times the
thickness.
plasma
Typical CYLTMP simulations require about 4096
particles ( 256 grid points and 16 particles/cell
thermal
velocity
distribution).
scheme in which the
radial
particles
If
were
we
tied
a
used a simple
to
a
single
row, the minimal calculation would need 25 radial
rows, with one row for
the
plasma
We would be forced to used (l+9)x4096
andnine forthe
beam.
= 40 960 particles.
However, examining the radial profile of the
Ea
for
TMd0 wave,
is slowly varying in the beam region, while E r and
vary rapidly
simulation
near
we
the plasma.
use
spread over 2 rows
particles
a
typical
CYLTMP
50 radial rows, 1 group of particles
for the
spread
In
over 6
plasma,
rows
for
and
3
groups
the beam.
double the radial resolution of the above
of
Thus we
example
while
using only 16 384 particles. The time required to do the
grid calculations on a 256x50 grid is about the
the
time
required
to
push
this
same
as
number of particles.
Using the simple scheme would halve the time spent on the
grid
but
increase
factor of 2.5.
better
results!
the time spent on the particles by a
CYLTMP runs
33%
faster
and
produces
The radial weighting and particle group
radial row assignments are run-time user inputs.
Thus if
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
a
series
of
runs
are
matter to first find
radial
row
to be performed, it is a simple
optimum
assignments
particle
by
simply
thicknesses
trying
and
different
combinations on a single run.
Having shown that CYLTMP is an efficient solution to
its
restricted
model,
we
compare
relativistic electromagnetic code.
require
it
with a full 2-D
Such
a
code
would
far more particles than CYLTMP - at least 50 000
particles for a 256x50 grid.
Furthermore
each
particle
must go through a complex electromagnetic particle pusher
to advance its position and momentum at each
Finally,
a
2-D
"divergence
continuity
At
code
cleaning"
is
must
step
preserved
go
to
to
through
an
ensure
that
used
in
CYLTMP
step.
elaborate
charge
second order in &T,A% and
It is easy to show that the charge
densities
time
-
and
current
equations 3.16 and 3.17 -
already satisfy the continuity equation to second
order,
and so CYLTMP does not have to do this step.
With
these
conservative
considerations,
estimate
it
also
probably
a
that CYLTMP uses only 2-5% of the
CPU time required by a full 2-D code.
must
is
Furthermore,
one
consider that the added complexity of the 2-D
code makes it more difficult to maintain
and
use.
The
price paid for these savings is that CYLTMP is restricted
to the simplified model given by equations 1.3-1.7, while
this is just a special case for a full 2-D code.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
Diagnostics
Since we will
CYLTMP
be
simulations,
history
energy.
of
the
results
of
the
it is appropriate to define some of
the diagnostics used.
time
discussing
the
The principal diagnostics are
components
of
the
the
total system
The total field energy is
3.24
where tVv= tT(r*^-r^) and ^ “ TTfr^
This energy is the sum of the electrostatic energy due to
charge
seperation
( plasma
waves
) and the energy in
purely electromagnetic waves (V*E = 0 ).
We
make
this
distinction by writing the electric field as
S, * Ee« + Ses
3-25
Only the purely electromagnetic component of
the
energy
can be radiated out of the system as microwaves, so it is
important to make this distinction quantitative.
easily
This is
done by using the scalar and vector potentials in
the Coulomb gauge
3.26
Eei - - V §
We then
define
3.27
the
electrostatic
and
electromagnetic
components of the field energy in the system as
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57
£es
=
^EtA
*
£esc,v)^|
3.28
+
3.29
Note that at a given point
E •E = E-_
+ E—
2E •E
E5
ElA ^ES v
a* a *
and that in our geometry Egs*E5M ^ 0.
^F£ =
^6S +
Thus
+ 2 J < W Eps *
With our choice of boundary conditions
and
gauge
to
condition
on
integral vanishes.
A,
it
is
easy
the
Coulomb
show that the
Thus definitions 3.24 and
3.25
lead
to the consistent condition
Efe -
^es
In CYLTMP,
Once
+^'E<v\
is
3.30
computed
directly
from
the
fields.
we have the charge density from the particle mover,
we can then solve for
the
scalar
potential
using
the
finite difference approximation to Poisson's equation
r? S iu t,\
- Z<-i 1
+ <~i 5 i . - i , S
be'*'
Si.yH -
3.31
*
- 4nrf ii
This is a set of linear equations that can be
efficient
matrix
zLtH
methods
that
exploit
the
solved
by
sparseness of the
. The electrostatic energy is
then
computed
from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We then compute ^ SMfrom Bpg and
This
using equation 3.30.
division of 18re consumes considerably more CPU time
than the computation of
itself, because the
solution
to equation 3.31 is non-trivial.
The total kinetic energy for species
=
2 1 Z I ft'iVWv 2-
is
■
3.33
where we have used the identity
V
.!)««■ - T a j * * '
-
This is useful to avoid the roundoff
computing
error
inherent
^ - 1 for non-relativistic particles.
In addition to these energies, CYLTMP can also
the
time
save
history of Fourier modes of 1-D quantities and
Fourier-Bessel modes of 2-D quantities.
is
in
periodic
Because the grid
in the z-direction and has N 2 grid points,
there are N a distinct axial wavenumbers in the system
!>«■ " A k *
*
+
3.35
where
A k a = 2Tt
N-a.ks
3.36
For a function defined on the grid points f^ = f (Zj),
define the finite Fourier transform
r
. 1 T £
£» _ s ; r * » e
3 - 37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
we
59
Its inverse is
°
e
n
3.38
The most useful FFT mode histories saved
by
CYLTMP
are
for the linear particle density of a specified group
\ ^
£
^0
These are used to observe the behaviour
of
longitudinal
waves in the selected species.
To analyze the electromagnetic field
CYLTMP
can
save
Fourier-Bessel
modes
on
the
grid,
of the magnetic
field
%ij
3.40
Where NWPk is an arbitrary normalization constant, and <Aoin
is
the
mth
zero
of
JQ .
These
mode amplitudes are
directly proportional to the amplitude of the
with
the
same
ka ,
larger amplitude than
TM
mode
once
the
latter has grown to much
all
the
other
plasma
waveguide
modes with the same ka .
In addition to time
histories,
CYLTMP
also
saves
"snapshots”
of selected quantities on the grid at chosen
time steps.
1-D quantities and their FFT's
versus axial grid point Z
These quantities include
CYLTMP
can
also
or wavenumber k
n ^
,
are
plotted
respectively.
'
and
•
generate contour maps of B^.* , and the
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60
electrostatic potential
of
.
, together with vector
maps
Finally, to display maximum information about
the particles, CYLTMP can generate snapshots of z-v^
and
z-p^ phase space for arbitrarily chosen particle species.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
CHAPTER 4
CYLTMP SIMULATIONS OF THE PLASMA CHERENKOV MASER
injttpflwgEifln
In
this
simulation
chapter
we
apply
code to the PCM.
the
CYLTMP
We begin with a description
of how the principal parameters in the
are determined.
of
the
PCM
parameters
are
presented.
corresponding
i3
These
to
-3
cm
density
are
used
varying
use
in
the
over
the
. Results of these simulations are in
In addition,
the
able to carefully examine the effect of
beam temperature on the efficiency.
linear
simulations
simulations
those
good agreement with these experiments.
simulations
code
Next, results of a series of simulations
experiments, with the plasma
range 2-8x10
particle
Saturation
of
the
instability is found to be due to trapping of the
beam electrons in the Enfield of the most unstable wave.
We
present
substantiate
simulation
this
results
assertion.
to
Finally,
qualitatively
we
find
fair
quantitative agreement between the maximum energy loss of
the beam in the simulations and those
predicted
from
simple semi-empirical trapping model.
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a
62
Choosing Input Parameters
Since some care is needed
parameters
for
procedure
used
CYLTMP,
for
the
we
PCM
to
determine
begin
by
the
input
outlining
simulations.
the
All
PCM
simulations used the following parameters:
grid size:
NR = 50
NZ = 256
radial
parameters:
/^r = 0.0299cm
rb = 18 A r = 0.538cm
r4 = 22 & r = 0.658cm
r^ = 24 Ar = 0.718cm
The beam was split into three groups, each particle being
6 Ar
wide,
while a single group was used for the plasma
with particles 2 Ar thick.
While Ar
value
of
and the radial parameters were
A z
simulated.
From linear
there
technical
want
as
are
point
many
the
used depended on the plasmadensitybeing
theory,
number of the most unstable TM
Az,
fixed,
two
is
factors
also
we
computed
wave - kao.
to
consider
discussed below ).
points/wavelength
as
requires A z to be as small as possible.
the
wave
In choosing
( a
third
First, we
possible.
This
Secondly, since
we are trying to model a continuous range of wavenumbers,
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63
we
want
&
to
be
as
small
as possible.
From
equation 3.36, we see that this is accomplished by
as
many
points
as
possible,
making
possible, and making k ^ one of the
latter
two
requirements
requirements
arbitrary
for
spatial
we
ka&= 10 A k % .
points/wavelength
ran
The
the
an
with the maximum
= 256, and
This
for
modes.
resolution. As
always
number of axial grid points
setting
higher
as large as
are completely opposite to the
maximum
compromise,
Az
using
Az
fixed
by
gives Ak^/k^ = 0.1 and 25
most
unstable
wave.
This
choice appears to have worked quite well.
Once Ar, Az and n^
cannot
be
arbitrarily
are determined the
chosen.
The
time
step
finite difference
equations 3.2 - 3.4 are numerically
unstable
too large.
limit can befound by
In principle this upper
deriving the modifications that finite Az,
make
to
Ar
At
and
known
first
appear
in
frequencies
to be stable in the continuous limit.
is a very difficult calculation, and all we need
At
is
the dispersion relation, and finding the At at
which imaginary parts
modes
if A t
to be less than this upper bound.
of
This
is
for
Thus in practice,
we simply use the limit for a homogeneous plasma
in
2-D
Cartesian geometry
At*
X
AX
Ay%
multiplied by a safety factor,
4.1
4
typically
in
the
range
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64
.7-.8.
This has been perfectly satisfactory.
The particle momenta of species ot were
to
approximate
initialized
a Maxwellian distribution in the species
rest frame
« N
where
o
= kgT^, and N ^
such
that
fa(p'a')
C
is
a
— 4.')j
4*2
normalization
constant
= 1 (
where
is the modified Bessel function of the second kind
order
one).
of
The particle momenta were then transformed
into the lab frame
Pa
=
where
the
>
( ^'2- +■
4,3
|&0 = Vd /c is the drift velocity of
lab
frame.
Numerical
instabilities can result if
either the beam or plasma temperatures
In
practice,
since
the cold plasma is
plasma
species o*. in
are
too
low
n^ , only the instability with
a
problem.
The
for
stability
temperature
condition
in
on
the
the
1-D
non-relativistic electrostatic case is
^
where
the
*
P
X p= <
\
3
|
*
st^ie electron Debye
plasma.Fortunately, this
length
non-physical instability
is self-stabilizing in the sense that if equation 4.4
not
satisfied
initially,
of
is
the instability heats up the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
plasma until it is, at which point the instability
off.
turns
In CYLTMP, a plasma simulation particle represents
many more electrons than a beam particle, so keeping
the
plasma temperature low greatly reduces the noise level of
the entire simulation.
a
low
plasma
Thus our simulations started with
temperature
( ©e=10eV,
and
simply let the non-physical instability stabilize
while
the
evidence
of
simulations,
smaller
physical
the
non-physical
grew.
One
instability
can
see
of
the
but it saturates much earlier and at a much
amplitude
Consequently
instability
itself
it
than
the
physical
instability.
was assumed to be harmless, and all the
simulations were done with the lOeV plasma temperature.
Simulation Results
Two series of simulations were performed.
series
The first
was designed to test CYLTMP and the linear theory
codes ANALYZE_DSPFN and GET_FIELDS against
one
another.
In these simulations, a single TM qq wave was excited in a
system with extremely low beam and plasma temperatures
even
here
the
the
problem
of the cold plasma did not
affect have a major effect on the physical
The
instability).
fields on the grid and perturbations in position and
momentum for each species were initialized
from
(
the
components
linear
at
codes.
selected
The
using
electromagnetic
points,
the
output
field
appropriate
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
Fourier-Bessel mode of the electromagnetic field, and the
appropriate linear particle density FFT
observed
performed,
and
in
About
every
a
simulations
the
oscillation
case,
theory.
Further, total
&
1% ) with
energy was conserved to
better than .5%, despite the violence of the
-
all
dozen
frequency and growth rate agreed very well (
linear
were
to oscillate sinusoidally with an exponentially
growing amplitude envelope.
were
modes
instability
converting more than 30% of the beam energy into field
energy.
CYLTMP
This was taken to be overwhelming evidence
that
( and incidentally the linear theory codes ) were
working well.
Confident that CYLTMP was working well, we began the
second
series
of
simulations.
This series focussed on
the realistic case of excitation of the TM00 waves
thermal
noise
in the system.
Experimentally, it is not
difficult to obtain a relatively low-temperature
<< c
) plasma,
but
it
is
much
( i.e.
more difficult to
produce a relativistic electron beam with a small
in
axial
momentum.
We
from
therefore
spread
focussed on the PCM
efficiency as a function of both plasma density and
temperature,
keeping
the plasma temperature at lOeV and
other parameters equal to those used in the
In
these
beam
simulations,
the
plasma
experiments.
particles
were
initialized with a small negative drift velocity so
that
the net plasma current was equal and opposite to the beam
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
current.
Furthermore a fixed ion background
was
placed
in
the
same region as the plasma so that the net charge
in
the
system
was
zero
( consistent
with
periodic
boundary conditions ).
Figures 4.1 and
results
of
B^lOOeV.
much
a
4.2
show
simulation
some
of
principal
13
-3
= 2x10
cm
and
in which
With these temperatures
the
W p and
A,Vb are
both
less than the beam velocity V0 = .857c, so that the
cold fluid theory should still be valid.
configuration,
93.5%
of
the
total
In the
system
kinetic energy of the beam, 1.6% kinetic
plasma,
2.8%
is
initial
energy
energy
of
magnetostatic
currents.
total
field
of
Figure 4.1a shows
field
energy,
over
and
component
in
beam and plasma drift
time
its
history
of
electrostatic
the
and
The total field energy rises
30% of the energy in the system.
electromagnetic
part
the
the
and
electromagnetic components.
to
the
electrostatic energy due to the radial
charge seperation, and 2.1% is electromagnetic energy
the
is
overtakes
the
Note that the
electrostatic
rises to over 20% of the total energy.
Figure
4.1b shows the time history of the kinetic energy of both
the
beam
and
the
plasma.
At saturation, the beam has
lost over 40% of its initial kinetic energy.
of
this
energy
has
Roughly 50%
gone into electrostatic waves ( of
which roughly half is in electrostatic field energy,
the
other
half
and
is in kinetic energy of the oscillating
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
TOTAL
ES
EM
200
400
600
BOO
1000
1200
800
1000
1200
(a)
1. 0
BEAM
0.8
0.6
0 .4
0.2
PLASMA
0.0
200
400
600
Q
4I
(b)
Figure 4.1. CYLTMP time history of (a) total field
energy and its electrostatic and electromagnetic
components, and (b) kinetic energy of the beam and
plasma.
n f = 2xlOra cm
and
= lOOeV in this
simulation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
0"*
0
200
400
600
800
1000
1200
«pef
(a)
0.0 1 5
o
a.
0.010
3
0 .005
0.000
0 .4
(b)
Figure
4.2.
Fourier-Begsel
modes
of
the
electromagnetic field - from the CYLTMP
simulation at n„ = 2x1013 cm"3
and
©»,= lOOeV.
(a) Time history of mode 9 ( k^= .234wfe/c ) - the
most unstable mode.
(b) Linear growth rate of S&n,
for n=7 to 12. The curve is the numerical solution
of the TM«o branch with the beam at this plasma
density.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
plasma
electrons
electromagnetic
).
The
waves.
other
We
50%
define
is
the
in
purely
efficiency
of
conversion of beam energy into microwaves as
* =
4 .5
-do
For this simulation, we get an efficiency of 18%.
Figure 4.2 provides evidence that the instability is
indeed
the
growth
of
TM 00 waves
linear theory of Chapter 2.
history
of
Figure 4.2a shows
the
time
the most unstable Fourier-Bessel mode of the
EM field { m=l, n=9 ).
rather
as predicted by the
than
Mode
9
was
mode 10 because the
the
most
unstable
used to compute Az
was calculated assuming the plasma is stationary, whereas
the
plasma
is
actually
drifting opposite to the beam.
After some erratic behaviour
amplitude,
this
mode
at
grows
early
exponentially
distinct growth rate until it saturates
The
growth
m=l,n=7
to
time
m=l,n=12
Fourier-Bessel
field
theory
drift velocity).
the
with a very
quite
are
modes
plotted
in
together with the growth rate of the TM
linear
small
abruptly.
rate obtained from the linear regime for the
electromagnetic
from
and
simulation
of
Figure
the
4.2b,
branch computed
( with compensation for the plasma
The agreement between linear theory and
is
excellent.
We
will
return to the
non-linear behaviour later.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71
Figures 4.3 to 4.6 show the effect
plasma
density.
Figures
of
varying
the
4.3 and 4.4 correspond to the
results displayed in Figures 4.1 and 4.2, but with
1*3 -3
np = 5x10
cm , while Figures 4.5 and 4.6 are results
at n
r
= 8x10
quite
good,
1^
-3
cm
but
plasma density.
and
4.8
density
as
from
the
to get poorer with increasing
This is discussed
the
These
in
lOkeV.
The agreement with linear theory is
appears
show
temperature.
.
effect
are
Figures
the
1
linear
results
and
theory
Figures
of varying
the
using
4.7
plasma
the
same
2, but a temperature of
At this temperature,
cold
below.
significant
deviations
can be seen, although we
still have an efficiency of 6%.
The efficiencies obtained from the second series
simulations
are
shown in Figure 4.9.
seen to decrease as the plasma density
is
consistent
efficiency
increased
with
the
of
The efficiency is
increases.
experiments,
in
This
which
the
was
observed to increase rapidly as np
13
-3
above 10
cm , reach a maximum value of of
*'**20% at np= 2x10
cm
and then decrease as the
plasma
density is increased still further. Unfortunately, it is
very
difficult
13 -3
np«2xl0
cm .
to
do
However
the
we
simulations
already
below
know from linear
theory
that
the
TM DO
waves are
stable
below
\z cm_x , which explains the low density cutoff on
np ~ 6x10
the
efficiency.
Finally,
note
that
the
efficiency
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72
0°
TOTAL
,ES
EM
200
400
600
800
1000
1200
800
1000
1200
^pe ^
(a)
BEAM
0 .4
PLASMA
200
400
600
(b)
Figure 4.3.
As
n p = 5x10 ^ cm"3 .
in
Figure
4.1,
but
with
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
1(T
1-7
200
400
600
800
1000
1200
(a)
0.015
0.010
a.
3
0.005
0.000
(b)
Figure 4.4.
nf = 5x10
cm
ka= .33w^,/c).
As
in
Figure
4.2,
but
with
.
(a) Time history of mode 10 (
(b) Modes n=5 to 12 are shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
10"1
TOTAL
ES___
EM
10"'
200
400
600
800
1000
1200
800
1000
1200
(a)
BEAM
0.8
0 .4
0.2
PLASMA
0.0
200
400
600
6>pe*
(b)
Figure 4.5.
As
np = 8x10 13 cm”3 .
in
Figure
4.1,
but
with
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
200
400
600
800
1000
0.2
0 .4
1200
CJn.t
(a)
0.010
0.0 0 8
Ol 0.006
4)
3
0.004
0.002
0.000
0.2
c k z/<ype
(b)
Figure 4.6.
As in
Figure
4.2,
but
with
np = 8x10V3 cm"3 .
(a) Time history of mode 10 (
kft= .SSw^/c ). (b) Modes n=5 to 12 are shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
10°
TOTAL
Ed
200
400
600
BOO
1000
1200
800
1000
1200
(a)
BEAM
0.8
0.6
0 .4
0.2
PLASMA
0.0
200
400
600
(b)
Figure 4.7.
As in Figure 4.1, but with
0^= lOkeV.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
10"*
200
400
600
800
1000
1200
(a)
0.015
o
a
3
0.010
0.005
0.000
(b)
Figure 4.8. As in Figure 4.2, but with ©v» = lOkeV.
(a) Time history of mode 10 ( k^= .26(0*, /c ).
(b) Modes n=5 to 12 are shown.
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78
_ T n r . T..
0.20
▲
^b«om
▲ 100aV
■ IksV
♦ lOkeV
0.15
■
P
■
0.10 -
*
A
-
m
—
♦
-
♦
▲
0.05
♦
0.00
■1
0
2
4
6
8
10
np ( x 1 0 13crrf3)
Figure 4.9.
CYLTMP simulation results of beam
energy to microwave conversion efficiency as a
function of plasma density with beam temperature as
a parameter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
decreases
with increasing beam temperature.
This too is
consistent with experiment, although they could not
temperature
directly.
Instead,
over the anode of the beam
scatter,
and
the beam.
they
diode
consequently
to
vary
used a thin foil
introduce
angular
an axial velocity spread, in
It was observed that the efficiency dropped as
the foil thickness increased.
Saturation Mechanism
As the linear instability evolves and
waves
grow,
the
wave
amplitude
the
unstable
spectrum
becomes
increasing peaked at the wavenumber of the most
wave,
since
these
waves
Given a small initial
amplitudes
magnitude.
can
grow
noise
grow
faster than the others.
level,
linearly
unstable
the. unstable
by
several
wave
orders
In this case, by the time non-linear
of
effects
are important, the wave amplitude spectrum is essentially
a delta function at the k^of
Thus
the
early
non-linear
dominated by the most
saturates
by
the
of
the
wave.
wave.
The
instability
the beam electrons in the Ez
field of the most unstable TMm
for
unstable
stage of the instability is
unstable
trapping
most
wave.
The energy
source
instability is the kinetic energy of the freely
streaming beam electrons.
Once the TM 00 wave
becomes
electrons
too
large,
the
are
troughs of the wave and no longer freely
amplitude
trapped in the
stream
through
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80
the system.
Thus the energy supply is turned off and the
instability saturates.
To
clarify
non-linear
these
behaviour
ideas,
of
electrostatic two-stream
we
the
first
consider
the
non-relativistic
1-D
instability.
It
has
already
been pointed out that the linear behaviour of this system
and the PCM are qualitatively
behaviour
similar.
The
non-linear
is also similar, and the simpler case has been
z ° i - 3Z
extensively
studied
.
Figure
qualitative
features of the growth rate versus k , which
is similar to Figure 2.5b.
the
growth
Wf*/V6.
rate
We
peaks
consider
dominated by this wave.
In
the
4.10a
shows
electrostatic
k
= k 0 , which
is
the
non-linear
effects
the
case,
close
to
to
be
The general features of the time
history of the electrostatic field energy
are
shown
in
Figure 4.10b.
Consider an electron with velocity Vc; =V0-vpv» in the
wave
frame (i.e.
in the lab
the frame travelling with velocity v ^
frame)
? -S.sinking1,
through
where
for
an
electrostatic
now,
Since the potential energy V = -e$
the electron energy is conserved.
potential
is constant in time.
is constant in
time,
Clearly an electron is
most easily trapped if it has this velocity at the bottom
of
a
potential
well.
The energy equation for such an
electron is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
6
(a)
(b)
Figure 4.10.
Qualitative features of (a) growth
rate
versus
k,
and (b) time history of the
electrostatic field energy, for the non-relativistic
1-D electrostatic two-stream instability.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
mv
+ eJ
f'l.
\\<W<>
=
- e.
The electron is just trapped when its velocity is zero at
the
crest
of
a
potential
barrier.
Thus the minimum
potential amplitude for trapping is
|L =
4.6
Z-e-
For
potentials
oscillate
larger
back
)
in
this*
the
electron
and forth in the potential well.
trapped electrons
minimum
than
( those
starting
near
a
Deeply
potential
a strong potential well
stray far from the potential minimum.
will
cannot
In this case,
the
electrons oscillate in a roughly parabolic potential
.
2
Thus,
they
execute
simple
harmonic
motion
at
the
so-called trapping frequency
(
a).
*c
4.7
iw
With these considerations, we can explain the qualitative
features of Figure 4.10b.
As the two-stream
grow,
and
close
evolves,
the
waves
we can soon neglect the effect of all but the
most unstable wave.
is
instability
to
The phase velocity of this wave
V9 ,
electrons
travel
electrons
and
so
much
are
that
more
trapped
v^
in the wave frame, the beam
slowly
first.
than
In
the
plasma
fact, the wave
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83
amplitude never comes close to being large enough to trap
the
plasma
electrons, which behave linearly right up to
saturation of the instability.
the
trapping
amplitude
As the wave
,
the
fluctuations evolve non-linearly
n ^ 10r*^
at
the
regions
decelerates the electrons.
end up in these bunches.
in
sinusoidal
into
dense
above
density
bunches
(
space where the Ea field
Most of
the
beam
electrons
These electrons thus decelerate
and supply energy to the growing wave.
the
grows
In the lab frame,
electrons continue to lose kinetic energy until they
are completely turned around in the
have
wave
velocity -V^ in the wave frame ).
maximum
field
continue
their
the
occurs.
oscillation
t their
frequency
hence
amplitude
in
velocity
the
its
( i.e.
This is when the
the
electrons
potential
in the
lab frame, increases.
draw energy from the wave and
Half
As
frame
wave
well at
frame,
and
Thus the electrons now
amplitude
decreases.
a trapping period later, the electrons attain their
maximum velocity and the wave amplitude is at a
minimum.
A short time later, the wave amplitude is increasing once
again.
as
the
wells.
that
Thus we get slow oscillations in the field energy
electrons
slosh back and forth in the potential
However, equation 4.7 is an only an approximation
works
best
for
deeply
trapped
Furthermore equation 4.7 is strictly valid
wave
amplitude is constant.
electrons.
only
if
the
These effects randomize the
relative phases of the electrons in their oscillations in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
the
wells, and
potential
distinct.
the
bunches
become
less
Thus the amplitude of the slow oscillations in
the field energy decays.
This simple physical picture also accounts very well
for
the qualitative features of the non-linear behaviour
of the field
energy
in
the
PCM
simulations.
Direct
evidence that trapping of the beam electrons has occurred
in the simulations is seen by examining
the
z-va
phase
space of theelectrons.Figure 4.11 shows the z-va phase
space of theplasmaelectrons
simulation
after
of
Figures
saturation
electrons
of
exhibit
at
top*,t = 600
4.1 and 4.2.
the
The
The
plasma
only mildly non-linear behaviour.
electrons
at
By
this
vortices in phase space are a characteristic
signature of trapping.
velocity
the
This time is right
instability.
contrast Figure 4.12a shows the beam
time.
for
axis
at
They are roughly centered on
V = Vpv» (
asymmetric
because
the
of
relativistic effects)/ and the space axis where E^ssO and
dE^/dz > 0 ( so that -eE^, is negative to the right of the
center of a vortex and positive to its left).
on
the
perimeter
of a vortex are oscillating back and
forth in theregion
of space
Recall
are
that
there
nine
occupied
by
wavelengths
unstable wave in this system/ and were
lack
Electrons
it
that
vortex.
of
the most
not
for
the
of a vortex at z^«135, we would have nine vortices.
The missing vortex is due to destructive interference
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of
Figure 4.11. Phase space density of the plasma
electrons ( every second particle shown) in the
simulation of Figure 4.1 at uy,t = 600 (just after
saturation).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
0.8
<n
0 .4
0.2
50
200
100
250
Z
(a)
0 .0 4
o
a.
0.02
3
o
£N
Ui
®
-0 .0 4
150
100
200
250
Z
(b)
Figure 4.12. (a) As in Figure 4.11, but for the
beam electrons, and (b) axial profile of E a along
the cylinder axis at this time.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
the
field
of
the
mode 9 TM0ll wave with the other TM00
waves (from Figure 4.2b, their amplitudes are not totally
negligible).
Thus the vortices represent beam electrons
trapped in the troughs of the most unstable wave.
The qualitative features
accounted
for
by trapping.
in efficiency with
interesting
of
that
Figure
4.9
are
also
First consider the decrease
increasing
although
plasma
the
density.
linear
It
growth
is
rate
determines how fast an unstable wave grows, it has almost
no
effect
on
the
saturation
amplitude.
Instead, the
trapping amplitude depends strongly on Ve- v ^ .
the
larger V ^ - v^
is, the larger the trapping amplitude
is (we will make this quantitative later
shows
the
density
).
normalized phase velocity
growth rate ^
for
Clearly,
for
the
most
which
the
simulations
Table
4.1
VPV\/C and the
unstable
waves
were
at
each
performed
(
obtained from ANALYZE_DSPFN)
nP
(xlO® cm"3 )
.782
.795
.804
.824
2
3
5
8
.0127
.0137
.0141
.0094
Table 4.1. Phase velocity and growth rate of the most
unstable TM00 wave as a function of plasma density.
Recall
that
= .857.
Thus
monotonically
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88
decreases
with
increasing
np .
The
trapping
model
predicts that the efficiency
should
decrease
with
13
(at least for np> 2x10
cm
). This is
increasing n p
consistent with experiment
that
and
the
simulations.
Note
there is no apparent correlation between the growth
rate and saturation amplitude.
The decrease
temperature
is
easy to see.
velocity
in
also
efficiency
with
increasing
consistent with trapping.
With a velocity spread Av*,,
difference
parameter
is
not
V8 - A v b-vpK.
Furthermore, since vpV%atV0 ,
»
so
A v v>/V0 ,
the
trapping
chapter
2.
This is
the
critical
V6-vpVs, but
A v b/(Ve -vpyj
amplitude
sensitive to velocity spread than the
is
linear
far
more
theory
of
This is also seen in the simulation results.
The oscillation frequencies and growth rates at
are
beam
© b =ikeV
almost identical to those at © b=100eV, both in good
agreement with the "cold" linear
efficiencies
at
A v fa,
the
However,
the
© b=lkeV are typically down 25-30% from
the ©j^lOOeV values.
spread
theory.
Also,
amplitude
with
an
initial
thermal
of the oscillations in the
field energy after saturation should decrease because
of
the
reduced coherence of the electron motion in the wave
13
-S
troughs. At n f>= 2x10
cm
we see oscillations at
e ^ l O O e V as shown in Figure 4.1a, but they are absent at
0^=10keV in Figure 4.7a.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
Having seen that
accounts
for
non-linear
many
trapping
of
the
behaviour,
we
predictions.
The
beam
the
beam
electrons
qualitative features of the
need
simplest
so-called rigid-rotor
cold
of
23
model .
some
trapping
We
assume
f(z,pfi,t=0) = n ^ (p_g. “Po )•
quantitative
model
an
In
is
the
initially
the
rigid
rotor model we simply assume that at saturation, all
the
electrons
The
have
turned
distribution
f (z,pa ,t=t
around
at
) = n^
(p-£ -p.f ).
in
the wave frame.
saturation
Because
is
of the strong
bunching of the beam electrons at saturation, this is not
as
crude
an
approximation as one might at first think.
This model works quite well in the non-relativistic case,
6 33
but must be modified for a relativistic beam '
Consider an electron with velocity
frame.
in
the
lab
In the wave frame ( moving with velocity
in
the lab frame), the electron moves with velocity
pi
P° ~~
4.8
Inverting
ffpW -t
4.9
*0
If the electron turns around in the wave frame, its final
_/
/
velocity is yj-= - £»& • Thus in the lab frame, the final
velocity is
P°
-f
t,
-
1
4.10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
The kinetic energy lost by the beam in the lab
( V - ' k Jmc2*,
’
&r= ( 1 " z )-'s.•
where
frame
= ( 1 -^o ) *
T^e initial
electron
energy
is
and
in
the
wave frame is
^6
=
( 1 - $ 0f>fY^)
-V
i.
where
= (1 -
4.H
f>^)
Inverting this relation, we get
"io m
f® Pp*)
4.12
From this it follows trivially that
'fo *
4.13
Thus the energy lost in the lab frame can be written
Uo'-fc
We follow
=
the
4-i4
notation
of
Ref.
6
and
introduce
a
normalized strength parameter
S
=a
£ ^
A
4.15
where A = 1 The energy lost by the electron
relative
to
its
total
(including rest mass ) energy is then given approximately
by the dimensionless relation
'>i»
^ --+ S
4.16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
This is the fractional energy loss of a
as
single
electron
measured in the lab frame when it turns around in the
wave frame.
If the rigid-rotor model were usedf equation
4.16 would therefore also give the fractional energy loss
of the beam.
contained
It is shown
therein,
that
in
a
Ref.
6,
and
references
much better estimate of the
fractional energy loss of the beam is
-
1.5S ( 1 +
4.17
This a semi-empirical expression based on
the
foregoing
analysis together with examination of computer simulation
results.
The factor
increase
in
while
of
1.5
comes
from
an
effective
S due to a decrease in ff* near saturation,
-3/*_
additional (1+S)
factor is due to
a
the
relativistic
effect
in
which
only
a
fraction of the
electrons rotate coherently in phase space.
The results of applying equation
simulations
are
shown
in
Table
4.17
to
the
4.2 ( Note that while
&E/^mc
can be simply expressed solely in terms of
the
meaningful quantity is AE/(])o“l)mc
more
PCM
S,
and this
is the quantity listed in the table ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
(xlOftfcm~3 )
S
.484
.400
.342
.213
2
3
5
8
theory
simulation
.38
.38
.36
.32
.41
.36
.29
.19
Table 4.2. Comparison of beam energy loss at saturation
from theory and simulation.
This model's prediction
fair
agreement
of the beam
with
temperatures used
loss is
the simulations.
agreement at higher density is
beam
energy
in
The poorer
probably due to the finite
in the simulations.
Since V0-v^
is much smaller at the higher densities ( see Table 4.1),
even
running
at
lOOeV,
the thermal spread of the
beam electrons significantly reduces the efficiency
the
"cold"
case.
Despite the success of this model in
predicting the maximum
efficiency
is
exactly what
microwaves.
over
still
fraction
energy
unknown
of
the
loss
of
the
because
we
lost
energy
beam,
do
the
not know
goes
into
This question requires further study.
gummaiy
Computer simulations of the PCM have been
using
CYLTMP.
These
simulations
have
performed
confirmed
linear theory of the instability presented in chapter
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
2.
93
More
importantly, the saturation amplitudes predicted by
the simulations are in good agreement with experimentally
observed
instability
efficiencies.
and
subsequent
Saturation
non-linear
of
the
linear
behaviour
is
qualitatively explained by trapping of the beam electrons
in the Es field of the most unstable TM00 wave.
the
magnitude
Finally,
of the beam energy lost at saturation can
be explained fairly well with a simple trapping model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
CHAPTER 5. SUMMARY AND FUTURE DIRECTIONS
In this dissertation, we have investigated the basic
physics of the generation of electromagnetic radiation in
the interaction region of the plasma Cherenkov maser.
In
chapter 1, a relatively simple mathematical model of this
system is created by neglecting the
components
of
theory
this
of
principal
the
exact
has
In chapter 2, the linear
been
particle
device
the
plasma
of
the
particle
waveguide,
solved.
The
together
with
effect of the beam on these
simulation
code,
with
described
the
using this code are presented in chapter 4.
field
amplitude
the
linear
theory.
They
electromagnetic
is created by rapid growth of the slow T M ^
in good quantitative agreement with
a
in
Simulations of the interaction region of
have confirmed that the large
numerical
waves
solutions
The saturation amplitude of the
field has been shown to be limited
beam
fully
The non-linear theory has been addressed
chapter 3.
of
velocity
is the exact numerical solution of the
calculations
modes.
new
model
result
slow TM waves in
electrons.
transverse
by
trapping
of
the
electrons in the troughs of the most unstable wave,
in fair agreement with a simple trapping model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
Both the linear
lead
to
results
and
non-linear
consistent
with
calculations
experiment.
theory calculations have shown that the growth
the
TMoa
waves
have
Linear
rates
of
decrease
significantly as the plasma
13 -3
density is reduced below n^ ~ 10
cm , and in fact these
12, .5
waves are stable for np^6xl0
cm . This is in good
agreement with the sharp drop in power output observed in
the
experiments as the plasma density is dropped below
13
-X
np~ 10
cm .
More
importantly,
the
efficiencies
computed
with
experiment.
CYLTMP
are in fairly good agreement with
In particular, the simulation results show a
peak
efficiency of about 20% when using a "cold" beam at
13
-3
n^= 2x10
cm
, just as is seen in the experiments.
However, despite these successes, much work remains to be
done on the theory of this device.
First, no attempt to
been
made.
optimize
the
efficiency
Clearly, to optimize the efficiency, we need
to choose parameters that maximize Va - vpV>
having an appreciable growth rate.
experimental
device.
operation
nf = 2x10
13
-3
cm
For
(
and
these
that even greater efficiency can
other
current
parameters
efficiency
A 0 = 3cm.
while
still
We have used only the
radial parameters, beam energy and beam
optimal
has
ea
of
the
we
find
20%)
at
However, it is possible
be
achieved
by
values of R, r4 ,rz , rt , Va
and nb .
Of course
we can expect that
considerable
effort
went
using
into
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
96
choice
of
these
parameters for the Soviet experiments.
However, it is not clear to what degree
the
approximations,
analytic
they
when
to
relied
exact
fully
on
numerical
solutions are clearly
required
optimize
the
choice of parameters.
This search for optimum parameters
could be automated with a program incorporating parts
ANALYZE_DSPFN.
These
optimum
parameters
can
of
then be
tested with CYLTMP.
Perhaps the single most important step to
with
this
project
is
to
conditions used in CYLTMP.
actual
device
can
be
reflecting
propagating
to
fill
boundary
at
A
before
and
a coaxial
scheme
to
eliminate
back into the system from the coaxial
The simplest
approach
would
the end of the coaxial guide region with an
if reaches the end
zsjn.
would be injected at z=0r and those
reinjected
a
z=0,
artificial absorbing resistive current to
wave
boundary
For the fields, we would have
waveguide would be needed.
be
axial
A more realistic model of the
waveguide extending beyond z=L.
waves
the
taken
constructed with only a moderate
increase in complexity.
perfectly
modify
be
at
z=0.
damp
out
the
The beam particles
"collected"
at
z=L
Finally the plasma particles can be
treated with self-consistent finite conductivity boundary
walls
at
both ends
CYLTMP simulations to
axial
boundaries.
.
These modifications would allow
investigate
Furthermore
the
we
effects
could
of
the
also examine
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97
transient effects as the beam
is
initially
established
such as current neutralization of the plasma.
To complement this version of CYLTMP, a new
of
ANALYZE_DSPFN
would be needed for the linear theory.
Simple axial boundary conditions for
are
a
version
the
linear
theory
perfectly conducting wall at z=0, and a partially
|8,u
reflecting boundary at z=L
. While to and k^must
still
satisfy equation 2.8, the axial boundary conditions force
both u> and k^to be complex.
used
to
determine
This modified program can be
optimum
parameters
efficiency with the axial boundaries.
to maximize the
These
parameters
can then be tested with CYLTMP.
Finally, in a real design of an
actual
device,
we
must examine the effects of not neglecting the transverse
velocity components.
The
complicated,
but
still
theory,
need
a
we
electromagnetic
linear
theory
tractable.
much
more
For the non-linear
multi-dimensional
particle code.
is
relativistic
Although this code would
be expensive to run, we could use it
sparingly.
For
a
given problem, we could use several runs of CYLTMP ( at a
fraction of the cost of a single run of the big
to
code
),
determine most of the "interesting" input parameters,
thereby minimizing the number of runs needed with the big
code.
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98
In conclusion, the plasma Cherenkov maser is a
interesting
addition
electron generators
efficiency
and
to
of
output
the
coherent
power
experimental device are most
that
growing
family
radiation.
of
the
impressive.
of
free
The
high
very
It
very
is
first
hoped
further work with this device will continue, to see
if it really can achieve
the
great
expectations
these
early results promise.
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99
APPENDIX A
ANALYTIC TREATMENT OF THE DISPERSION RELATION
In this appendix,
solutions
to
the
we
derive
dispersion
approximate
relation
for the slow TM
waves without the beam, and then derive an
for
the
(complex)
analytic
approximation
frequency shift for the TM00 wave in
the presence of the beam.
Given the
complexity
dispersion
is
that
relation,
it
clear
of
the
such analysis
requires many simplifying assumptions, which is why it is
left
to
the
appendices.
first presented in Ref.
The major results below were
21.
However,
this
article
is
extremely terse, and so it seems appropriate present this
work here also, with
additional
steps
to
clarify
the
derivations.
We
start
dispersion
by
relation
»
3cR<< 1 and
satisfied
wave
w <<
considering
for
a
•
without
equation
the
beam.
2.12
-
the
We assume that
1.
Physically, these conditions are
very
dense plasma and a low frequency
Since
ocpr^ >>1,
it
OCp rt >> 1, while JCR <<1 implies that
follows
«
1.
that
We use
the asymptotic expansions for the Bessel functions
A. 1
and
the
lowest
order
small
argument
expansions
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for
100
|z| «
1
JoCa) ^ 1
3iW>
%.[£)& \
— 1
KoU') **
^
a. 2
-k.
Keeping only the appropriate lowest order terms, we
find
the quantities in equations 2.9 and 2.10 simplify to
0CJ.C, (
"Tool? A
A.3
\ ca>if + T si\i? /
T
Xt)
fis
y P suv ^<3*
4- Crt(u)-VX jl\
Xp 6efo \H •+■ Xu) — Psu*(ip+\i')
A. 4
A. 5
where (J =
0fprA - ^ , \x= 3(p^ , and A p= r4 - r^.
From these
equations,
it
follows
that
equation
2.12
simplifies to
X
lW
X
j.
=
*•«
Equation A. 6 has an infinite number of
which
implicitly
defines
U>
for
roots
a fixed
for
Xj, (
k2 ).
For
A f<<^k and In ■Sfe*1, the right hand side of equation A.6
is very small.
the
zeroes
of
In this case, the roots must lie close to
the
tangent
function.
Expanding
tangent function equation A.6 is approximately
a -7
where n = 0, 1, 2, ... ( i.e. Xj_> 0 )
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the
101
Since
we assume that
but
neglect
it
we keep it for n = 0,
for
n > 0.
Thus
the
roots are
approximately
l~A^ — j— T"
V
rk » O
J
/
A. 8
OTT
n>o
In keeping with the nomenclature of Chapter 2, we define
**■
k iof,
C
i
----- rrzr
^ * 0
-<
*.»
I
(p£f
n < °
Then, squaring both sides, equation A.8 can be written
^9Cp h^)
® IK-Io* ^p)
or
-
o
-
which leads to the dispersion relation
Won
®
\
**
VCs■
l C*
----r— ;
A. 10
It should be emphasized once again that equation A.10
derived under the assumptions .xR <<1,
and In ^ ^
plasma.
small.
Ap<< rz
1.
We next consider the field
wave.
3Cj,r4 >>l,
is
structure
of
the
TM 00
measures the change in phase of E s across the
Clearly for the T M 00 wave this phase
change
is
Thus the radial structure of this field component
is approximately
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102
Alo(xr')
o
Blofor^
+ C K 0( ^ 0
Ea must vanish at r = R
Using
approximations
Sr^Ct.
and
A.2,
be
and
Aml1
continuous
at
r = r^.
keeping only the lowest
order terms
o ^ r<
E *
—
1 InrM
A. 12
Finally from equations 2.2 and A.12
( noting
that
the
next term in Es for r < r4 in the expansion of I0(5cr) is
Mjxr) ), we find
*<
(^ r
Er ~
o
A. 13
We now consider the effect of the beam on
wave.
TM ^
To make progress, we assume that the beam density
is low ( W^»\»<*w pe.), and treat the beam as a
of
the
the
T M 00 wave.
perturbation
As mentioned in the introduction we
consider the interaction to be a resonant coupling of the
the
T M 00 wave
the beam
with the slow beam wave.
density
is
so
low
that
at
We assume that
frequencies
of
interest the dispersion relation of the slow beam wave is
simply 60 = kaV0
satisfying
(i.e.
k2V0 >> Wpb
this relation,
).
For to and
ka
The condition for
resonant coupling at wavenumber ka is that
to =
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an<^
103
to = U)$o •
The dispersion relation for the TMO0 wave is
Applying the resonance condition, the resonant
frequency
is found to be
A. 14
We see that resonant interaction cannot occur for
densities so low that
plasma
U)f<< $ekj.ooV0 .
To find the growth rate at the
resonant
frequency,
we return to the full dispersion relation - equation 2.8.
From equations A.3 - A.5, noting that for the TM
wave,
tan )sA esXi.» we have
A.15
For
the
beam
assumptions
terms,
that
we
already
3C rb <<1.
know
from
Although
previous
C0» k ^ , we
assume that the beam density is so low that
In this
case
approximations
19Cb)a
e.\oc\
A.2,
the
and
beam
so JXbr^j << 1.
terms
in
Using
equation 2.10
become
A. 16
Q
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104
Finally, to lowest order, equation 2.8 is simply
Thus
the
Dp = Q.
dispersion relation for the T M 00 wave with the
beam is approximately
0* -
J
jOpb
IT
* r,
v *
^ ( w -k,vy
With the right hand side equal to zero this is
dispersion
equation
A - 17
just
the
for the TM06 wave without the beam,
which the resonant TM ^ wave satisfies with the frequency
given
approximately
= oW
.
by
equation
A.14
and
wavenumber
Treating the beam term as a perturbation,
we look for solutions to equation A.17 of the form
00 * Woo +
where
)£wl <<
After some lengthy algebra, we find
5 (2. 6O0o +
A. 18
where
5
A. 19
o
Then in the limit
J
«
_Sw
I «■.!
*00
A. 20
The frequency shift is given simply by
[iW-S*
\ W 60/
tfs
\
2.
Taking the cube root, the unstable root - i.e.
A. 21
the
with positive imaginary part is
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root
From equation A. 22 the real
shifted
downward.
part
of
the
frequency
is
This ensures that the phase velocity
of this wave is less than the beam velocity.
The
growth
rate is
Finally, plugging back the solution A.21
into
condition
A.19, we see that the frequency shift A.22 is valid if
«
k.
A -24
This concludes the analytic treatment of
relation.
It
is
the
dispersion
clear that many assumptions have been
made to derive these results.
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106
APPENDIX B
ORTHOGONALITY RELATIONS
In the analysis of the electromagnetic fields
general
cylindrical
waveguide
(
i.e.
in
arbitrary
transverse profile, but uniform along the cylinder
with
a
axis)
the cylinder axis taken to be the z-axis, we find a
discrete
set
of
wave-like
solutions
to
Maxwell's
equations
r
1
8,1
j3lx
We are using the notation of Jackson
is
a
general
indices. E\
the
*34»
v
, section 8.11. a
mode index which may stand for one or two
and
are the transverse, and E^and
longitudinal
label indicates
field components respectively.
the
direction
in
which
the
The (±)
wave
is
travelling.
In this appendix we focus only on the
of
two such modes
X
an<3
• We assume the waveguide to
be filled with a non-permeable
arbitrary
dielectic
both dispersive
restriction
and
placed
tensor
a
the medium is lossless.
medium
( JV =1)
£ (x,y,w,k»)
function
on r%i
£
orthogonality
of
with
an
which can be
space.
The
only
is that it be Hermitian, i.e.
In this case, both
00
and
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k%
107
are
real.
If
^
is
a scalar constant, the modes B.l
break up into TM and TE modes.
longitudinal
component
), and it is
easy
to
Each mode
has
a
single
( E a (B^ ) for TM (TE) modes
2.0
show
that these longitudinal
components satisfy the simple orthogonality relation
B.2
where S is the transverse surface of the waveguide.
the
general
problem under consideration here, the modes
cannot be seperated into purely
equation
B.2
no
longer
TM
and
applies.
conservation.
This
complex
conjugate
of
a
and
general
tied
to
relation and its derivation
23
which
only
=ky.
Maxwell's equations for mode
Dotting equation B.3 for
modes,
closely
are modifications of those given in Ref.
considered modes in which
TE
However
orthogonality relation exists which is
energy
For
X*
X
take the form
with Bp,
equation
B.3
and
for
adding the
t+)
0 (+)
with d x
gives
B.5
From similar manipulations on
Hermiticity of £
equation
B.4,
using
we obtain
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the
108
fcjr.vxar*§r-™r*
=
B 6
-i £ ^
* LW > £
"
w ^|.lw **»W ) ] .g ^
Subtracting equation B.6 from equation B.5, and using the
vector identity
B «Vxk - ft-Vxg ^ V-£xB
we get
v-(sr»sr*6?*»s;v
tr
B. 7
(*>
X
We write V
X
where ^
as
- &
+ **la
is the transverse gradient operator.
Using the
explicit form of the fields - equation B.l, the left hand
side of equation B.7 can be written
7.(s?*sT * s
* &X
)
jn-MfeV
+ * tav*
*£?*§>)
Note that in the last term, we only have
fields
the
transverse
because the longitudinal components cannot give a
z-component
in
the
cross
product
terms.
Using
divergence theorem
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the
where C is the boundary curve of S and n is
pointing
normal
on
C.
But
vanishes.
outward
= nxE. *6
= 0 on the surface of a conductor.
integral
the
Thus
the
and
line
It therefore follows from equations
B.7 and B.8 that
Equation B.9
is
First#
that
note
the
if
general
we
variable# and multiply both
result
we
let
sides
are
seeking.
as a continuous
of
equation
B.9
This is the simple statement that the energy of the
flows
at
the
group
velocity.
by
wave
For different modes in
equation B.9 there are two cases of special interest.
In the first case# we consider
look
to
be
fixed
and
at the orthogonality of modes with different ks(&).
Physically, this applies to the boundary-value problem in
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110
which a localized source or boundary wall is being driven
at a single frequency w*.
away
from
allowed
the
source
Then
have
wavenumbers ^(We).
all
waves
frequency
propagating
but with the
Assuming
that
is
independent of k2 , equation B.9 reduces to
+ E ^
°
B.ll
s
Now in the derivation
- (O
b |n
and
_ to
d
p.
ofB.ll,
had we originallyreplaced
e /->
with
bj*and
Oja
the result wouldhave
been
For
ifc follows that
[
*
&
=°
b-12
Equation B.12 states that the total power
sum
flowing
in
a
of such waves is equal to the sum of the total power
flowing in each wave.
All overlap integrals of different
modes in the integral for the total power vanish.
In the second case, we consider the problem in which
the
modes
have
frequencies W (k4).
initial-value
a
common
ka,
but
with
different
This situation is applicable to
problem
studied
in
this
the
dissertation.
Equation B.9 then reduces to
KFor
iS»0
•& -v
the integral vanishes.
From
b .13
equation
B.13
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Ill
it
follows
that the total field energy in a sum of such
waves is equal to the sum of the
total
energy
in
each
individual wave.
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112
References
1.
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Plasma Sci., Vol.
PS-13, no.
6
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