State of the Art Simulations of Magnicon
Vyacheslav P. Yakovlev*, Oleg A. Nezhevenko*,
Oleg V. Danilov*?t, Dmitry G. Myakishev^ and Michael A. Tiunov1^
*Omega-P, Inc., New Haven, CT 06511, USA
Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia
f
Abstract. The magnicon is a highly attractive candidate to be the RF source for a future multi-Tev
linear collider. Physical models and computer codes have been developed which can provide startto-end self-consistent simulations of a magnicon, including precise simulations of the highconvergence electron gun, RF-system, magnetic system, and beam collector. The 3-D beam
dynamics simulations include realistic fields, finite beam size and transverse space charge effects.
The codes allow one to provide steady-state simulations of the entire tube, so as to evaluate transient
process of magnicon excitation, parasitic mode self-excitation, stability analysis, and tolerance
analysis. The results of the simulations are found to be in good agreement with magnicon
experiments. A brief description of the physical models and simulation codes employed will be
given.
INTRODUCTION
One of the most attractive candidate for the RF source for future particle accelerator
applications is the magnicon - a microwave amplifier with circular deflection of an
electron beam [1]. Magnicons have demonstrated high power at high efficiency, good
phase stability and stable operation into a resonance load [1,2]. Magnicons provide the
best parameters in the decimeter, centimeter, and millimeter wavelength domains, which
are used in accelerators [3-6]. These characteristics render the magnicon not only
attractive for collider applications, but also for a variety of other accelerator applications.
Experimental investigations have confirmed the unique properties of magnicon. The first
magnicon experiment at the Budker Institute of Nuclear Physics (ESP) yielded an
efficiency of 73% at 915 MHz with output power of 2.6 MW [3]. A second harmonic
pulsed magnicon amplifier at 7 GHz has demonstrated 56% efficiency with 55 MW
output power and pulse duration of 1 jis [4]. These encouraging results prompted
Omega-P, in close collaboration with the Naval Research Laboratory (NRL), to
undertake design, construction and evaluation of an 11.424 GHz, 60 MW pulsed
magnicon [5]. At present, the tube is conditioned up to power levels of 15 and 25 MW
for 1.0 and 0.2 jisec pulse widths respectively, and already has been used for high power
tests of active RF pulse compressors and dielectric-lined accelerating structures [7].
Additionally, Omega-P built a third harmonic magnicon at 34 GHz [6].
The magnicon is a conceptually new RF source (see Figure 1, which displays a
schematic of an 11.4 GHz frequency-doubling magnicon amplifier [5]) and has a number
of novel characteristics that require special treatment in numerical simulations. In this
device, an initially linear beam is deflected in a series of deflection
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
421
WR-90 WAVEGUIDE
VACION PUMP
ELECTRON GUN
FIGURE 1. Schematic of 11.4 GHz magnicon amplifier. Inserts at the left show RF field pattern for
deflecting cavities (TMno mode) and for the output cavity (TM2io mode) .
cavities containing synchronously rotating TMno modes in presence of DC magnetic
field. The deflection cavities include a drive cavity, the gain cavities, and finally the
penultimate cavities, that contain RF high fields and do the bulk of the beam deflection.
For effective deflection, the DC magnetic field in the deflection system should be such
that \£2/co\ ~ 1.5-1.7, where £2 is the cyclotron frequency and co is the drive frequency [1].
The output cavity contains a synchronously rotating TMni0 mode that extracts principally
the transverse beam momentum as microwave power at the drive frequency or one of its
harmonics (n is harmonic number). In the output cavity for efficient extraction of energy,
the DC magnetic field should be chosen such that |I2//itf)| ~ 1. The electron dynamics in
the magnicon are essentially three dimensional in nature. In the ideal case of a thin beam,
all of the electrons entering the drive cavity at a single point in time experience identical
forces during their transit through the entire device. Once steady-state operation is
achieved, the operation of the device is time-invariant when viewed in a frame corotating at the RF-drive frequency. This synchronism makes possible extremely high
efficiencies, since all the electrons will experience identical RF fields. For this reason,
the experimental parameter that most directly impacts magnicon efficiency is the
transverse size of the electron beam, since the larger the beam, the more that radial
gradients in the deflection cavity RF fields create a spread in the electron momentum and
phase at the entry to the output cavity. In fact, a near-Brillouin beam is needed to
maximize the magnicon efficiency. Techniques for the generation and transport of
electron beams near the Brillouin limit have been developed and demonstrated [8]. For
parameters of current interest the beam radius rb is indeed small compared to the RF
422
wavelength wavelength A: rb/h= 0.01-0.03 [1-5]. Moreover, it has been amply
demonstrated—both numerically and experimentally—that the fringing fields associated
with beam tunnels and cavity openings affect the beam dynamics and the efficiency [9].
Thus it is imperative to include accurate realistic modeling of the cavities, beam tunnels
and fringing fields in simulation codes. In addition, it is necessary to have the means to
investigate transient processes and non-linear regimes of magnicon operation.
THE MAGNICON CODE
The magnicon simulation program [10] was developed in parallel to the magnicon
experimental program. Recently it allows provide "start-to-end" (from the cathode to the
beam collector) self-consistent simulation of the entire tube. The code is an integrated
system that includes high-precision evaluation of the electron gun; accurate calculation of
the static magnetic field; high-precision calculation of RF fields; 3D beam dynamics in
realistic fields including beam loading and space charge; steady-state modeling of the
magnicon; transient process calculations; stability analysis; tolerance analysis; collector
simulation; graphical user interface. The code works both on PC under Windows and on
NERSC Cray (in this case PC is used as remote X-terminal). The code block diagram is
shown in Fig. 2.
FIGURE 2. Block diagram of the magnicon code.
The Magnicon Gun Simulations
High power RF sources in the centimeter and millimeter wavelength domain for the
next generation of linear colliders require utilization of beams having powers of 100's of
423
MW's. Because of cathode loading limitations and danger of the voltage breakdown, it is
necessary to develop guns beam compression exceeding 1000:1 [8]. In order to achieve
the beam having radii close to Brillouin limit for such compression, one should have the
emittance of the order of thermal one [8,11]. Actually, it determines the general
requirement for the gun simulation code: numerical noise should give negligible
contribution to the beam emittance, i.e., this contribution should be much smaller than
thermal emittance. A special code has been developed for simulation of the electron guns
with high transverse area compression [12]. This code is based on current pipe model.
Relativistic effects are taken into account. Finite-element method is used for potential
approximation based on the second-order "serendipic" elements that allow achieve
accurate geometry and field description and, thus, decrease numerical noise. In addition,
special low-noise emission model is applied [12], which allows achieve smooth cathode
current density distribution. The code provides an accurate aberration calculations
including cathode edge effects analysis, because the mesh cell size may vary by two-three
orders in magnitude (see Fig. 4). The realistic magnetic field distribution is used in order
to describe influence of stray magnetic field (which may limit the beam radius in the
magnetic system [8]) and to simulate the gun matching with the magnetic system.
Thermal effects are taken into account as well in order to simulate the beam halo. Three
guns have been designed and built using this code demonstrating parameters close to
calculated values (see Table I).
Gun
7 GHz magnicon (Budker INP)
11.4 GHz magnicon (Omega-P/NRL)
34.3 GHz magnicon (Omega-P)
Table I
Beam
Power
100 MW
108 MW
98 MW
Beam Area
Compression
2300:1
1400:1
>2500:1
Status
Reference
In operation
In operation
In operation
[8]
[13]
[14]
11.4 GHz magnicon gun layout, electron trajectories and equipotential lines are shown
in Fig. 3.
608 R t c n i
3.4
0o2095
3.6095
FIGURE 3. 11.4 GHz magnicon gun layout in (r-z) plane, electron trajectories and equipotential lines.
Measured transverse beam area compression is 1400:1. The electron trajectories and
equipotential lines near the cathode edge of this gun are shown in Fig.4a. The mesh near
the edge is shown in Fig.4b.
424
SupErSAM U2.03, U n r i a n t i t l
SuperSflH U2.03, U n r i a n t s t l
RICH i
RICH:
1.25
ZICM)
FIGURE
lines near
near the
the cathode
cathodeedge
edge(left),
(left),and
andthe
the
FIGURE4.4. Some
Someelectron
electron trajectories
trajectories and
and equipotential
equipotential lines
finite-element
~5 µ(I near
near the
the edge
edge up
upto
to ~5
~5mm
mmin
inthe
thegun
gungap.
gap.
finite-elementmesh
mesh(right).
(right). The
The mesh
mesh size
size varies
varies from
from ~5
The
11.4 GHz
GHz magnicon
magnicon (calculated
(calculated and
and
The beam
beam envelopes
envelopes in
in magnetic
magnetic system of 11.4
measured)
are
shown
in
Fig.
5.
The
calculations
take
into
account
thermal
effects,
the
measured) are shown in Fig.
calculations take into account thermal effects, the
cathode
aberrations caused
caused by
by the
the gun
gun electrodes
electrodes
cathode edge
edge effects,
effects, stray
stray magnetic
magnetic field and aberrations
and
andmagnetic
magnetic system.
system.
1.5
1.25
R,mm
1
0.75
0.5
0.25
0
11
22
3
3
4
4
5
5
66
77
88
99
Z,cm
Z,cm
Measured —•—Calculated
Calculated
—•—Measured
FIGURE5.5.Beam
Beamenvelope
envelope in
in magnetic
magnetic system
FIGURE
system containing
containing 95%
95% of
ofthe
thebeam
beamcurrent,
current,calculated
calculatedand
and
measured.
measured.
The DC
DC Magnetic
Magnetic and
and Electrostatic
The
Electrostatic Field
Field Calculations
Calculations
Thefield
field in
in magnicon
magnicon magnetic
magnetic system,
The
system, which
which usually
usually has
has axial
axial symmetry
symmetry(with
(withsmall
small
perturbation
far
from
the
axis),
is
calculated
using
2D
SAM
code
[15]
based
on
perturbation far from the axis), is calculated using 2D SAM code [15] based onBoundary
Boundary
Equation Method.
Method. ItIt is
is possible,
possible, because
because the
Equation
the magnet
magnet yoke
yoke typically
typically works
worksininthe
theregime
regime
far
of
saturation,
and,
thus,
the
relevant
magneto-static
problem
is
linear.
Third-order
far of saturation, and, thus, the relevant magneto-static problem is linear. Third-order
approximation isis used
used for
for the
the secondary
secondary sources
approximation
sources distribution.
distribution. Special
Special regularization
regularization
procedure
is
used
to
avoid
mathematical
incorrectness
of
the
problem.
procedure is used to avoid mathematical incorrectness of the problem. ItIt gives
gives high
high
accuracy
of
the
field
calculations,
which
is
necessary
to
describe
stray
fields,
accuracy of the field calculations, which is necessary to describe stray fields, that
thatare
are
typically by three-four orders smaller than the field in a magnetic system. In addition,
typically
by three-four orders smaller than the field in a magnetic system. In addition,
SAM code provides an accurate field calculation in the magnetic system allowing avoid
SAM code provides an accurate field calculation in the magnetic system allowing avoid
425
the noise in calculations of electron dynamics. An example of the magnetic system
evaluation is shown in Fig. 6. The same code is used for electrostatic field calculations.
In Fig. 7 there is the result of optimization of the 34.3 GHz magnicon gun insulator [14].
2" 1250°
—— Calculated field
• Measured field
FIGURE 6. Calculated and measured axial field distribution in the magnetic system of 34.3 GHz
magnicon.
R(cm)
-36
-27
-18
-9
0
z(cm)
FIGURE 7. Equipotential lines in the gun insulator.
The Magnicon RF System Calculations
Deflecting of cavities of a magnicon are typically axisymmetric, and field symmetry
perturbation in a drive and output cavity is small in the area of the beam propagation.
2'/zD SuperLANS code [16] is used for RF resonant field calculations. This code is based
on the second-order field approximation and allows provide an accurate calculations of
426
Date:06/06/102
RICH)
________FREQUENCY (MHZ 1=34431.996
6.14076
Z(CH)
6
35
4
30
25
2
H,kA/m
En, MV/m
a)
0
-2
20
15
10
-4
5
-6
0
1
2
3
4
5
6
7
1
l,cm
l,cm
2
3
4
5
6
7
l,cm
b)
b)
c)
c)
FIGURE8.8.Field
Fieldpattern
pattern of
of the
the operating
operating mode
mode in
in 34.3
34.3 GHz
FIGURE
GHz magnicon
magnicon output
output cavity
cavity (a);
(a);surface
surface
electric field
field (b)
(b) and
and surface
surface magnetic
magnetic field
electric
field (c).
(c).
allthe
the six
six field
field components
components in
in the
the cavity
cavity for
for operating
operating (or
all
(or parasitic)
parasitic) multipole
multipole and
and
monopole
modes.
The
code
provides
also
an
accurate
calculation
of
the
surface
monopole modes. The code provides also an accurate calculation of the surface fields,
fields,
essential for
for high-power
high-power RF
RF amplifiers.
amplifiers. An
An example
example of
itit isis essential
of the
the field
field pattern
pattern and
and
surfacefield
fieldcalculations
calculations isis shown
shown in
in Fig.
Fig. 8.
8.
surface
TheBeam
Beam Dynamics
Dynamics Simulations
Simulations in
The
in aa magnicon
magnicon RF
RF System
System
Thebasic
basicidea
ideaunderlying
underlying the
the physical
physical model
model of
of the
The
the beam
beam dynamics
dynamics in
in magnicon
magniconisis
that
the
fields
in
a
magnicon
RF
system
can
be
separated
into
two
parts.
The
that the fields in a magnicon RF system can be separated into two parts. The first
first part
part
is
resonant
(i.e.,
oscillates
primarily
near
a
multiple
of
the
operating
frequency
ω0 ,
is resonant (i.e., oscillates primarily near a multiple of the operating frequency nncoo,
where nn isis aa natural
natural number)
number) with
with aa slowly
slowly varying
varying envelope.
where
envelope. The
The remainder
remainder isis
nonresonant and represents the self-fields. In a magnicon the transverse dimension of
nonresonant and represents the self-fields. In a magnicon the transverse dimension of
the electron beam is small compared with the wavelength and thus retardation effects
the
electron beam is small compared with the wavelength and thus retardation effects
may be neglected near the source, i.e., the electron beam. That is, the instantaneous
may be neglected near the source, i.e., the electron beam. That is, the instantaneous
location of the electrons determines the self-fields inside the beam. Thus, neglecting
location of the electrons determines the self-fields inside the beam. Thus, neglecting
longitudinal effects of space charge, at every time step the self electric field may be
longitudinal
effects of space charge, at every time step the self electric field may be
determined using instant 2D space charge distribution in a beam transverse cross
determined using instant 2D space charge distribution in a beam transverse cross
427
section (note that full 3D simulations [17] show that longitudinal space charge effects
are not important for the beam power at least up to -150 - 200 MW). Similarly, the
self-magnetic field is determined by the instantaneous current distribution and its
evaluation can also be simplified. It should be recalled that the self-magnetic
transverse field force for a relativistic beam tends to cancel the self electric field force
to within a factor I//, where 7 is the relativistic factor. This model is exact in the
absence of electron beam rotation. However, in a magnicon the beam rotates as a
whole around the cavity axis. Due to this rotation additional, azimuthal eddy current
electric fields are induced. In practice these fields are small since the rotation velocity
of the electron beam is small compared to the speed of light. Due to the stroboscopic
nature of the electron beam motion in the rotating RF fields the dynamics of any single
beam slice are to similar to any other since electrons undergo identical motion but for
a displacement in phase. Thus the instantaneous distribution of charge can be
calculated from the motion of a single slice. Further, a two-dimensional subdivision
of the electron beam can be usefully employed, in lieu of a three dimensional one,
allowing considerable savings in the required computer memory and time. An
example of beam dynamics simulations in 34.3 GHz magnicon is shown in Fig. 9.
z=21.7 cm
z=33.7 cm
z=24.7 cm
z=36.7 cm
z=27.7 cm
z=39.7 cm
2.7
26.2305
38.761
35.2915
39.822
ZICN)
FIGURE 9. Charge distributions in the beam transverse cross-section for different longitudinal
coordinates for 34GHz magnicon (left) and beam trajectories in (r-z) plane (right). The cavity chain is
shown.
Transient processes are analyzed by using the slow-time-scale approximation. The
model takes into account excitation of the drive cavity by an external RF source [18]
and mode coupling through the output ports. Time integration of slow-time-scale
equations is combined with electron trajectory integrations to determine the correct
steady state self-consistent solution—if any—during transient processes, or to identify
428
parasitic mode self-excitation. The complex power loss of the electron beam, which
determines the transient process, is calculated simultaneously with the particle
trajectory integration. The model is also developed which gives the possibility to
analyze the magnicon operation when the drive cavity is excited through one port. It
is possible because of the gyro-tropic effect specific to magnicon [3]. In this case the
RF field in the cavities has elliptical polarization, the electron dynamics is different for
different RF phase, and it is necessary to consider a number of beam slices in order to
35.8
37.34
36.87
40.41 41.94 Z(cm)
\ 35.8
c)
Hr(kA/m)
35.6
37.34
37.34
38.67
40.41 41.94 Z(cm)
36.67
40.41 41.94 Z(cm)
Hr(kA/m
38.87
40..41
41.94 Z(cm)
J \ 35.8
37.34
FIGURE 10. An example of the parasitic mode self-excitation in the magnicon cavity (a): transient
process for the operational mode (top) and the parasitic mode (bottom). Phase slippage of the parasitic
mode is also shown. Field map (b), electric field (c), and magnetic field (d) along the cavity for
operational (left) and parasitic (right) modes.
429
describe the beam dynamics and transient process. This takes, of course more
computer time, but allows make tolerance analysis, i.e., to calculate the tube operation
changes for small transverse and longitudinal misalignments of the RF cavities and
magnetic system. When the self-consistent solution is found, a stability test is
performed to check how far is the operation regime of the instability threshold [10].
Lyapunov's method is used for stability analysis by linearizing slow-time-scale
equations near the stationary point. The derivatives of the right hand sides of these
equations are numerically determined by solving the steady state problem with
perturbed RF amplitudes and phases. The instability growth rates and parasitic
oscillation frequencies are given by the eighenvalues of the matrix of the linear
system. An example of the simulation of parasitic mode self-excitation in 34.3 GHz
magnicon is shown in Fig. 10.
The Beam Collector Simulations
For the beam collector simulations Mote-Carlo method is used in order to describe
secondary emission and backward reflection of the electrons. Experimentally
measured data [19] is implemented for secondary emission and reflection coefficient
dependence vs. energy and angle of an incident electron as well as for energy and
angle distribution of the secondary and reflected electrons vs. an incident electron
parameters. The code calculates the distribution of the power deposition over the
collector surface and possible electron reflection into the RF system. An example of a
collector optimization is shown in Fig. 11.
FIGURE 11. a) Particle trajectories in (r-z) plane in the beam collector of 34 GHz magnicon in the
absence of drive power (space charge is taken into account), b) Particle trajectories in the beam
collector for full power operating condition.
430
SUMMARY
Physical models and computer codes have been developed which can provide startto-end self-consistent simulations of a magnicon, including precise simulations of the
high-convergence electron gun, RF-system, magnetic system, and beam collector.
The 3-D beam dynamics simulations include realistic fields, finite beam size and
transverse space charge effects. The codes allow one to provide steady-state
simulations of the entire tube, so as to evaluate transient process of magnicon
excitation, parasitic mode self-excitation, stability analysis, and tolerance analysis.
ACKNOWLEDGMENTS
This work was supported by the Division of High Energy Physics, US Department
of Energy. The authors wish to thank Prof. J.L. Hirshfield for careful reading of the
paper.
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