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Formation and transport of high-perveance electron beams for high-power, high-frequency microwave devices

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A dissertation entitled
Form ation and T r a n s p o rt o f H igh-P erveance
E le c tro n Beams f o r H igh-Pow er, H igh-Frequency
Microwave D ev ic e s
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Mark A. B asten
Degree to be awarded: December 19 96
May 19___
August 19------
Approved by Dissertation Readers:
October 31, 1996
Date of Examination
Dean, JUraduate School
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
FORM ATION AND TRANSPO RT OF
HIGH-PERVEANCE ELECTRON BEAMS
FOR HIGH-POW ER, HIGH-FREQUENCY
MICROWAVE DEVICES
By
M a rk A. B a ste n
A
D I S S E R T A T I O N S U B M I T T E D IN PARTI AL F U L F I L L M E N T O F T H E
R E Q U I R E M E N T S F OR T H E D E G R E E O F
D
(E
o c t o r
o f
l e c t r ic a l
P
E
h il o s o p h y
n g in e e r in g
)
at the
U N IV E R S IT Y O F W IS C O N S IN - M A D IS O N
1996
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1
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INTENTIONALLY
LEFT BLANK
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ii
Form ation and Transport o f H igh-Perveance Electron
B eam s for High-Power, High-Frequency M icrowave
D evices
Maxk A. Basten
Under the supervision of Professor J. H. Booske
At the University of Wisconsin - Madison
One of the problems in the scaling of high-power vacuum microwave sources to
higher frequencies is the need to transport beams with high space-charge density,
since the rf circuit transverse dimensions tend to decrease with wavelength. The
use of sheet electron beams can alleviate this difficulty since large amounts of
current can be transported, at reduced space-charge density, by increasing the
width of the beam. Thus, focusing problems and space-charge debunching effects
axe avoided and, in addition, the rf power density is reduced due to the extended
nature of a rectangular or planar rf interaction circuit.
Historically, the main
disadvantage of using sheet beams is the susceptibility to beam distortion and
break-up due to the diocotron instability, at least in solenoidal focusing schemes.
In this dissertation it is shown with analytical and numerical studies that sheet
electron beams can be stabilized through the use of periodically-varying magnet
arrays.
In particular, the offset-pole periodically cusped m agnet (PCM ) array
is shown to provide stable beam transport and side-focusing of elliptical sheet
electron beams w ith remarkably high space-charge density (> 1 0 0 A /cm 2) at low
beam voltages ( < 4 0 kV) compatible with compact microwave devices.
Logically, the well-known stability problem with sheet beams has also largely
discouraged the development of electron sheet beam sources. We have identified a
novel sheet beam forming system which uses magnetic quadupoles and a conven­
tional Pierce gun to transform an initially cylindrical beam into a high aspect-ratio
elliptical sheet beam . This m ethod has the added advantages for proof-of-principle
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sheet beam transport experiments in th at the components are readily available or
easily fabricated, and th at changes in the beam size can be m ade ’on-the-fly’ in an
experiment through adjustm ents of the quadrupole gradients. Both 2-d and 3-d
envelope simulations and 3-d particle-in-cell simulations establish the potential of
this technique, and indicate th at a high-quality 5.4 cm x 0.2 cm, 2 A, 10 kV ellip­
tical beam can be produced with modest quadrupole gradients. An experimental
investigation verified the basic principle of this m ethod, however the beam size was
limited to 4 ± 0.5 cm x 0.4 ± 0.1 cm due to beam rotation induced by leakage
flux from a solenoid m atching section between the gun and the quadrupole lattice.
Another m ethod to increase the transportable beam current, and thus mi­
crowave output power, is through the use of periodic perm anent quadrupole magnet
(PPQM ) arrays for focusing high-perveance round electron beams. The focusing
force from PPQM arrays can be an order of m agnitude greater than th a t of con­
ventional periodic perm anent magnet (PPM ) arrays for beam voltages from 10 40 kV. Analytic and numerical studies dem onstrate the potentiad of this configu­
ration but also show th at the magnitude of the beam ripple is often more severe
in PPQM systems than with PPM focusing. An experim ental investigation using
a rectagular-block PPQ M array with a period of 2 cm and an adjustable gradient
from 750 - 2000 g /cm and a 0.27 A, 10 kV electron beam was conducted. Mea­
surements of the beam radius at different positions in the array axe in qualitative
agreement with the predictions of envelope and particle-in-cell simulations.
I certify th at I have read this thesis and certify th at in
my opinion it is fully adequate, in scope and in quality,
as a dissertation for the degree of Doctor of Philosophy.
John>H. Booske
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iv
Acknowledgem ent
There axe a number of people and organizations which supplied support and as­
sistance at various stages of this dissertation th at the author would like to ac­
knowledge.
Funding for this work was provided in p a rt by the DoD Vacuum
Electronics Initiative as managed by the Air Force Office of Scientific Reseaxch
(AFOSR) and the Naval Research Laboratory. Com puter resources were provided
by National Energy Research Supercom puter Center and the Computer-Aided En­
gineering Center of the UW-Madison. The WARP PIC code and assistance was
provided by the Heavy Ion Fusion Group of LLNL. Drs. Steve Lund and D. Grote
deserve special mention for their add in operating this particle sim ulation code.
The TRACE3D, POISSON/PANDIRA, and EGUN sim ulation codes were pro­
vided by the Los Alamos Accelerator Code Groups. Use of the MAGIC PIC code
was provided thanks to the Mission Research Corporation (MRC) and the MAGIC
User’s Group as administered by AFOSR. Deserving of special thanks for advice
and tim ely modifications to the MAGIC code are Drs. David Sm ithe and Larry
Ludeking of MRC. Useful consultations are acknowledged with Drs. J. Schaxer of
the UW-Madison, R. True of Litton Industries, S. Lund of LLNL, N. Dionne of
Raytheon Industries, and C. Arm strong of Northrop-Grum man.
Assistance from a number of undergraduate students is also gratefully acknowl­
edged. Vaxious sub-components of the experiment were constructed with the help
of L. R auth, M. Gemelke, R. Thompson, J. Anderson, and T. Wadzinski. J. An­
derson and Z. Smith also provided help on some of the sim ulation studies.
A large debt is owed to my reseaxch advisor, Dr. John Booske, for his encour­
agement, advice, and wealth of ideas. Finally, I would like to acknowledge the
support and patient understanding of my wife and daughter, Sharon and Emily
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Basten, and my parents, Aaron and Loretta Basten.
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vi
Contents
A b s tr a c t
ii
A c k n o w led g e m en t
1
In tr o d u c tio n
1
1 .1
High-power Sources of Microwave and Millimeter Wave Radiation .
3
1.2
Applications of High-Power Microwaves and Millimeter Waves . . .
7
1.2.1
Enhanced Radar Systems
7
1.2.2
Linear RF-driven A ccelerators....................................................
11
1.2.3
Microwave Processing of Materials
..........................................
12
1.2.4
RF Sources for F u s i o n .................................................................
12
1.3
1.4
2
iv
Extended Beam Configurations
...........................................................
.............................................................
13
1.3.1
Historical R e v ie w ...........................................................................
14
1.3.2
Study of High-Perveance Sheet Electron B e a m s ...................
17
Periodic Permanent Quadrupole Magnet F o c u sin g .....................
19
T r a n s p o rt o f S h e e t E le c tro n B eam s
2.1
Semi-Infinite Sheet Beam Stability
21
.......................................................
2.1.1
Beam Envelope Trajectory S im u la tio n s.............................
2.1.2
PIC Simulations of the Semi-infinite Sheet Beam
22
25
.................
27
2.2 Offset-Pole PCM for Beam E d g e-F o cu sin g ............................................
28
2.2.1
PIC Simulations with E d g e-F o cu sin g .......................................
32
2.2.2
3D Magnet Simulations of the Offset-Pole P C M ..................
36
2.2.3
Hybrid PCM-Quadrupole Sheet Beam F o c u s in g ...................
37
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vii
3
Formation of Sheet Electron Beams
3.1
40
2-D Theory of Electron T r a n s p o r t...........................................................
42
3.1.1
Physical Quadrupole Magnetic F ie ld s .......................................
48
3.1.2
Variations in Longitudinal V e lo c ity ...........................................
50
3.1.3 Elliptical Sheet Beam Design Case
............................................
53
3.2 3-D Analysis of Electron Transport in the Quadrupole Beam-Forming
S y stem ............................................................................................................
4
E x p e rim e n ta l In v e s tig a tio n o f S h e e t B eam F o rm a tio n
4.1
60
4.1.1 Pierce Electron Beam Source
......................................................
61
............................................
62
...............................................................
66
4.1.4 Vacuum S y s te m ................................................................................
69
4.1.5 Electron Beam Pulsing C irc u it.....................................................
70
......................................................................................
71
4.2 Experimental R e s u lts ...................................................................................
72
4.1.3 Quadrupole Lens Array
4.1.6 Diagnostics
P e rio d ic P e rm a n e n t M a g n e t Q u a d ru p o le F o cu sin g o f R o u n d E lec­
tr o n B eam s
6
60
Description of the E x p e rim e n t.................................................................
4.1.2 Solenoid Magnet M atching Circuit
5
53
75
5.1
Analytical Study of PPQM F o c u s in g .....................................................
76
5.2
Coupled-Mode Analytic Theory of Beam O scillation...........................
82
5.3
Numerical Solution of the PPQM Envelope E q u a tio n s.......................
86
5.4
PIC Simulations of PPQM Focusing........................................................
SS
E x p e rim e n ta l In v e s tig a tio n o f B e a m F ocusing in P P Q M A rra y s 91
6.1
The Pierce Electron Source and Matching Optics
..............................
91
6.2
The PPQM A rra y .........................................................................................
93
Design Equations for a PPQM A r r a y ........................................
94
6.2.1
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viii
7
6.2.2
Mechanical D e s i g n ........................................................................
96
6.2.3
Magnetic Field M easu rem en ts.....................................................
97
6.3 Experimented Measurements of Beam T r a n s p o r t ...................................
99
Summary
104
7.1 Formation and Focusing of Sheet Electron B e a m s ................................. 104
7.2 PPQM Focusing S t u d y .................................................................................. 107
Bibliography
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110
1
Chapter 1
Introduction
This dissertation reports on the theoretical and experimental investigation of novel
magnetic focusing schemes of high-perveance electron beams which may allow for
the development of compact, high-power, vacuum-based microwave and m illimeterwave sources. The beam perveance is defined as the quantity
where /*, is
the beam current and Vj is the beam voltage. To avoid bulky voltage supplies
compactness generally constrains the beam voltage below 40 kV. Since rf o u t­
put power increases with beam current we see that larger beam perveance values
correspond to higher rf output power in compact microwave sources. Two highperveance configurations are examined: the use of extended beams (sheet beams
or m ultiple electron beamlets) and the use of strong-focusing periodic quadrupole
magnets in transporting high current-densitv round electron beams. This report
begins by describing the present-day and potential future applications of highpower microwave sources then discusses the limitations faced by state-of-the-art
conventional microwave tubes.
The use of sheet electron beams is one attractive solution for overcoming powergeneration and power-handling deficiencies of conventional sources, since high cur­
rent can be propagated in close proximity to waveguide walls or rf cicuits w ithout
the problems associated with high space-charge density or excessive rf power den­
sity. A review of past work on sheet electron beams reveals that, for the m ost
part, few recent sheet beam studies have been undertaken. A main reason for this
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2
lack of interest is due to the well-known susceptibility of sheet beams to the diocotron instability, at least in solenoidal focusing configurations. Chapter 2 presents
an analysis of sheet electron beam transport through a periodic magnet focusing
array.
In contrast to solenoidal focusing, it is found that periodically focused
beams are stable.
A suitable magnet geometry, providing for stable transport
and side-focusing on the beam edge is identified. Analytic analysis, beam envelope
equations, and 2 | particle-in-cell (PIC) codes are used to dem onstrate sheet beam
stability and side-focusing.
In Chapter 3 a novel m ethod for the formation of sheet electron beams using
a round-beam electron source and magnetic quadrupoles is presented. This con­
figuration is especially suited for laboratory experiments since it provides a cheap,
easily constructed, and param etrically flexible method to generate sheet beams.
The results of 2-D and 3-D beam envelope analyses, and 3-D PIC transport codes
dem onstrate the feasibility of quadrupole formation of sheet beams. An experi­
m ental investigation of this technique is presented in Chap. 4. Features of this
experimental design include the use of a Pierce diode electron source and four
low-gradient quadrupoles to form a 5.4 x 0.2 cm elliptical sheer electron beam.
In Chap. 5 another focusing technique using periodic permanent quadrupole
magnets (PPQM s) in close-packed arrays is examined. This scheme has the po­
tential to allow for the focusing of higher space-charge density beams than con­
ventional periodic permanent magnets (PPM s) due to the strong focusing effect of
magnetic quadrupoles. Substantial benefits may be realized even for low-voltage
beams compatible with compact, high-power microwave tubes. A detailed look at
the beam dynamics in these arrays is made with analytic design equations, en­
velope equations and particle simulations. In Chap.
6
the design, construction,
and operation of an experiment to test PPQM focusing is discussed. Chapter 7
is an overview and summary of both the sheet beam and PPQM configuration in
which potential problems with either scheme are addressed and the prospects for
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3
utilization in future microwave devices is evaluated. Recommendations for further
study are also given.
1.1
High-power Sources of Microwave and Mil­
limeter Wave Radiation
Radar has long been one of the prime forces driving the development of sources of
high-power (> 100 W average ) coherent radiation in the microwave (1-30 GHz) and
millimeter-wave (30-300 GHz) regions of the electromagnetic spectrum. Although
the first operational radar set (the 1938 British Chain Home system) operated
at 25 MHz (within the HF band), the 12 m eter wavelength severely lim ited the
detection of the relatively small aircraft used in the early days of W W II. This
system was soon upgraded to the VHF band and. by the end of the war. into the
lower frequency microwave bands. W ith the advent of 'stealth' technology and
the desire to acquire other targets of small cross-section, this push towards higher
frequency radar systems, along with higher output powers for long-range detection,
is still very evident today [1 ].
Indeed, the progress made in the development high-frequency, high-power sources
is impressive. Figure
1
shows the state-of-the-art for microwave tubes today [2].
The eaxly development of the klystron and the conventional magnetron (not shown
in Fig. 1 but comparable to the curve for klystrons) allowed for the replacement
of the gridded triodes and pentodes used in the early radar systems. Although
incoherent (lacking in phase stability) and having narrow bandwidth, th e mag­
netron could be used in radar systems by providing feedback for coherency in the
interm ediate frequency (IF) of the receiver [3]. In addition, the development of
phase-locking between separate oscillators, or between multiple oscillators and a
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4
reference source, allows for the combining of tube o utput for phase-stable, highpower operation. While the instantaneous bandwidth is still extemely narrow, me­
chanical tuning techniques can be used to provide frequency agility over a broader
bandwidth. A factor of two in frequency tuning and a factor of 100 increase in
peak power has recently been reported on a system of four mechanically tuned
magnetrons [4]. This system provides continously tunable output (in both output
frequency and power) from 1-3 GHz and 10 kW-400 MW with the use of a novel
combiner-attenuator scheme.
Another workhorse of modern-day radar systems is the klystron amplifier,
shown schematically in Fig. 2.
In contrast to the m agnetron, the position at
which the electron beam is created (the cathode) is spatially separated from the
position of the interaction of the beam with the rf wave. Aside from advantages
gained from a modeling standpoint, the separation of the interaction region and
cathode/anode regions also reduces the problem of 'pulse jitter* (defined as the
variation from pulse-to-pulse of the time-delav between the turn-on time of the rf
and voltage pulses) associated with magnetrons and other crossed-field devices.
Figure 2 also serves to illustrate features which are common to other classes
of conventional microwave tubes such as the traveling wave tube (TW T) and the
backward-wave oscillator (BW O). The electron beam is formed, focused, and ac­
celerated in a cylindrical diode (a Pierce diode, or gun) and propagates through an
aperture in the anode under magnetic focusing provided by solenoids or permanent
magnets. The beam enters the interaction region where the kinetic energy of the
electrons is converted into electromagnetic energy of the rf wave. Spatial modu­
lation of the beam density excites space-charge waves on the beam which, when
in an appropriate phase with respect to the wave, causes coherent amplification of
the rf power. After the interaction region, the spent beam propagates into a col­
lector region - which may have a depressed potential applied for added efficiency
enhancement. Because most of the beam ’s energy is in the longitudinal direction
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5
(along the axis), and very little is in the transverse direction, these tubes are often
labeled linear beam tubes to distinguish them from crossed-field devices, such as
the magnetron, and from devices utilizing beams with high transverse energy, such
as the gyrotron.
Klystrons generally have the best noise characteristics at high-power of any
class of microwave tube - making them ideal sources for long-range tracking and
acquisition radars. Signal-to-noise levels of -110 dB/M H z (measured from the cen­
ter frequency) can be achieved - far in excess of the -70 dB/M Hz requirements
for most tracking and acquisition systems [5], Several GW of output power at 1.3
GHz and 1.7 GW at 3.5 GHz have been reported [6 ] for short pulses (< 100 nsec.)
High-power, phase-stabilitv. and low-noise also make klystron amplifiers attractive
for high-gradient linear accelerators, and some of the most impressive recent work
on these devices has been undertaken at the Stanford Linear Accelerator Cen­
ter. Recent results include the fabrication and testing of a 52 MW. 1.5 ^second,
solenoidallv-focused tube at 11.4 GHz [7]. SLAC has also undertaken the design
and fabrication of a tube of similar output parameters using periodic permanent
magnets (PPM s). This concept, attem pted in the Soviet Union and Germany,
has the advantage of replacing expensive and bulky solenoid magnets (sometimes
superconducting) with a relatively compact and lightweight PPM stack.
Another class of devices, the traveling wave tube amplifier and backward-wave
oscillator, are also mainstay components of today's radar and communication sys­
tems. These two devices are often similar in geometry (with the exception of an
input rf coupler present in the amplifier) but in the TWrT the interaction proceeds
along the length of the device in the same direction as the drifting electron beam,
while in the BWO the interaction builds in the direction opposite to the electron
drift motion. Both devices are capable of large bandwidths. but suffer from rela­
tively low average-power lim itations at high frequency due to the fragile nature of
their slow-wave circuits. BW O’s providing at least 20 m W of output power to 170
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6
GHz,
1
mW to 280 GHz, and 0.1 mW to 1500 GHz are commercially available [8 ].
Even though the original theory of the helix-TW T was developed over four
decades ago [9], the helix-TW T is still one of the most successful microwave tubes
for generating phase-stable, large bandwidth, m oderate average-power radiation.
In addition, the relatively small size of high-frequency helix-TW Ts makes them
vital to airborne radar and electronic warfare (EW) systems, communication sys­
tems, and they are often used as input drivers for high-power microwave tubes.
Commercially packaged TW Ts generating a kilowatt of average power at frequen­
cies up to 15 GHz are readily available. Of special note is the recent development
of microwave power modules (MPMs) [10]. MPMs are completely integrated rf
sources (including power conditioner, solid-state rf input, and a m iniature helixTW T) capable of several hundred watts of wideband microwave power package in
a volume the size of a thin paperback book.
Other variations of TW Ts have also enjoyed much success. Tunnel-ladder (or
Karp) circuits axe the basis for space-based millimeter wave tubes including a 400
W, 28 dB gain tube with an instantaneous bandwidth of 2.3 % centered around 28.2
GHz that was recently developed for NASA [1 1 ], The coupled-cavity TW T (often
employing a ferruled circuit or ladder, with coupling apertures between segments)
improves on output power over the helix, but at the cost of lower bandwidth. A
ladder circuit coupled-cavity TW T has been developed that produces 100 W of
continous-wave (CW) power with a 20 % bandwidth, centered at 90 GHz [12].
Another class of tube that has been the subject of a considerable amount of
recent attention are gyrotron-type devices. In the most basic configuration these
devices utilize just smooth waveguide as the interaction circuit, hence they are
capable of handling much higher beam and rf power densities at high frequen­
cies than their slow-wave counterparts. Gvrotron oscillators for electron cyclotron
heating and current drive of fusion plasmas have been developed which provide
MW of CW power at 110 GHz [13]. and over 500 kW at 140 GHz [14].
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1
7
U nfortunately for airborne radar systems, the application of gyrotron-like de­
vices as amplifier configurations requires large magnetic fields provided by bulky
solenoids or superconducting magnets. Of recent note is the operation of a gyrotrontraveling wave tube amplifier experiment demonstrating over 20 dB of gain, 10 %
overall efficiency.
6
kW of output power over a 33 % bandwidth region from 27-38
GHz [15]. This experiment used a tapered interaction circuit and profiled mag­
netic fields provided by superconducting magnets operating at 1.45 T. However,
precise control and optimization of the field profile was required in order to provide
large bandwidth. In addition, scaling this experiment to 95 GHz would require an
increase of the magnet strength by a factor of three. Magnet requirements are
relieved somewhat by operating in harmonics of the fundamental, at reduced gain,
but these devices are mainly still in the research stage at this point. For example,
a research program is currently in progress to develop a 100 kW peak power, 45
dB, 4th-harm onic gvrotron-klystron amplifier operating at 10 kW average power
at 95 GHz [16].
1.2
Applications of High-Power Microwaves and
Millimeter Waves
1.2.1
Enhanced Radar System s
Certainly one of the main driving forces behind the development of high-power,
high-frequency vacuum tubes is the development of m ilitary radar systems. Con­
sider the m axim um range equation for radar [17]:
^
= [PTA 2( 7 / 4 n \ 2( k T / T ) ^
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( 1)
8
where PT is the peak power of the source, <r is the effective cross-sectional area of
the target, A is the effective axea of the antenna, A is the free-space wavelength, k T
is the tem perature of the receiver, and r is the radar pulse duration. Obviously,
the range increases proportionally to the source signal output power and, noting
th at X = c / f where c is the speed of light and / is the frequency, the range also
increases with frequency for a fixed effective antenna area and target size.
Furthermore, the resolution of the radar increases with higher frequency - an
im portant consideration for m ilitary radars looking for small cross-section missiles
and stealth-clad targets. A rather new use for high-resolution radars is the tracking
of space debris in orbit around the earth. T he accumulation of decades of debris
poses a serious problem to present and future space missions and especially to the
space station. Even objects on the order of a centim eter in cross-section (such as
a nut or bolt) can cause critical damage to the sensitive areas of a satellite. It has
been estim ated that, due to the long-ranges and the resolution required, a radar
source at 35 GHz would require about 20 MW of output power over 100 fi seconds
- about a factor of 30 larger than any existing source [18].
For some radar applications bandwidth is also an im portant factor. The range
resolution for a radar system with instantaneous bandwidth A f is given by A R =
c /2 A / [19].
Thus larger bandwidths imply a sharper resolution in the target
range. Operation over a wide frequency range is also desirable to avoid electronic
countermeasures such as jam m ing and false echoing. As an alternative to widebandwidth long-pulse radar systems, equivalent range resolution can be obtained
through the use of extremely short-pulse (a few nanoseconds) systems in a time-offlight configuration. In this case, the same expression for the range resolution may
be used if we make the identification of A f = r -1 . However, since the operation of
the radax depends on the average power, this type of application requires extremely
large peak powers (on the order of a GW) at very high repetition rates (> 100 Hz)
to achieve comparable range.
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9
Having identified some param eters of m erit for sources for future radar systems
(average power, peak power, bandwidth, frequency) we can further identify other
param eters which are especially im portant for airborne, spaceborne. and shipborne
radar sets. Two obvious considerations are compactness and efficiency. A main
contribution to the weight and size of the rf source is the associated voltage supply
and focusing magnets. Although recent advances have made available 200 kV,
200 MW [20] sources of pulsed power, these supplies are probably still an order of
magnitude too heavy and large for use on anything but the biggest of airborne plat­
forms. Hence, this particular niche of radar sytems will require sources operating
in a low voltage (< 40 kV) regime [21].
An advantage of highly relativistic devices is that large amounts of current can
be transported due to the partial self-focusing provided by the azim uthal self mag­
netic field. Mildly relativistic devices must rely on external m agnetic focusing to
provide confinement and. in the case of gyro-devices. for the interaction with the rf
wave. Again, size and weight restrictions prevent the use of conventional solenoid
magnets ( 2-10 kG) or superconducting solenoid magnets ( 1 0 - 1 0 0 0 kG) on any­
thing but the largest platforms. It is encouraging to note that some investigation
of perm anent magnet solenoidal focusing structures has been conducted, in part
spurred by the development of high-remanence permanent magnet m aterials. A
perm anent magnet structure providing a 2.5 kG uniform longitudinal field over 23
cm for focusing in linear beam tubes has recently been reported [22]. The magnet
structure has a total mass of 15 kg (less than a third of the weight of an equivalent
solenoid), including cladding on the exterior to prevent flux leakage to neighboring
components, and flux shielding to accomodate a flux-free cathode and a collector
region. While extremely promising, this magnet still yields in size and weight to
periodic permanent magnet systems used commonly in helix-TWTs and BWO de­
vices. Commercially available T W T ’s, utilizing cylindrical PPM stacks, have been
fabricated with magnet periods of less than
1
cm with on-axis field am plitudes of
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10
over 2 kG. More recently, the size and weight advantages provided by PPM stacks
have led to their use in high-peak power klystrons for linear accelerators [7].
Since it is unlikely that high-peak power sources will be developed for compact
platforms in the forseeable future for time-of-flight radar systems, we m ust also be
careful to consider the noise of our source. Since velocity information is obtained
through the Doppler shift of the return in conventional systems, the power level
must drop off as sharply as possible away from the center frequency, and side-lobes
and spurious sources of noise must be limited. While most radar system s require
— 40 dB/M Hz and battery fire-control systems require -50 dB /M H z. acquisition
and tracking radars require at least -70 dB/MHz in the noise reduction from the
center frequency [5]. Narrow line-widths. elimination of competing modes, and a
quiescent electron beam are necessary in a low-noise source.
The beam contributes to noise in a variety of ways. The ideal rf source would
be one in which a cold, drifting electron beam interacts with the rf wave such that
all electrons give up kinetic energy at the same rate and with identical phase svnchronicity. Obviously, any realistic electron beam will have a finite tem perature, or
spread, in the velocity distribution which prevents exact synchronicity. Contribu­
tions to velocity spread are multiple: finite tem perature of the (therm ionic) cathode
(typically 1100° K. or about 0.1 eV), cathode surface roughness, gun fabrication
errors, the introduction of tranverse electron velocity due to the m agnetic focusing
structure, and focusing mismatches. Instabilities within the electron beam, or due
to the presence of ions, can lead to noise or loss of the beam to the tube walls.
Another m ajor contributor to tube noise is the so-called partition noise caused by
beam interception within the interaction region.
Based on the preceding discussions we can now summarize the requirements
for future m ilitary and civilian radar systems. First of all, there is an imm ediate
need for wide-bandwidth tubes at 35 GHz and 95 GHz tubes to take advantage of
the atmospheric windows around those frequencies. At 35 GHz. tubes having less
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11
than an octave of bandwidth must have the product of the frequency (in GHz) and
the power (in MW) be larger than 0.5 [1]. The requirement for tubes with over
an octave axe relaxed to less than 0.5, but frequency-power factors approaching
this number Eire desired. As an example, a moderate-bandwidth tube operating at
35 GHz must attain at least 14 kW of output power. At 95 GHz the bandwidth
constraints axe lowered (somewhere in the neighborhood of 5-10 % is desired),
but the same formula requires sources capable of at least 5 kW of power. Phasestable tubes for Doppler radars m ust also have noise figures below -40 dB/MHz.
In addition to all of these requirements, a candidate source must be packaged
(tubes, magnets, power supply, cooling, etc.) to meet stringent size and weight
requirements.
1.2.2
Linear RF-driven Accelerators
A m ajor reason for the demise of the Superconducting Super-Collider was due
to spiraling costs of m ajor components, including the large number of rf sources
required to drive the particle beam. Future high-gradient linear accelerators are
envisioned which will reach TeV energies in a fraction of the length and cost of the
SSC - provided that suitable rf sources can be developed. The design frequency
of these accelerators vary from a G erm an/FSU collaboration near 10 GHz to the
American concept at 11.4 GHz. As has been recently discussed [7]. the energv-perpulse goal of
1
kJoule for accelerators is the same as for rf sources for nanosecond
radar and electronic warfare applications. To date only a fraction of this energy
level has been reached - tubes have operated at 100 MW for 1.5 fts or at
for
10
ns - and substantial improvement will need to be realized.
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1
GW
12
1.2.3
Microwave P rocessing o f M aterials
An axea that has been receiving considerable interest recently is the use of mi­
crowaves to process materials - especially for sintering ceramics (see, for exam­
ple, Refs. [23, 24, 25] and the references therein). Ceramic m aterials processed
with microwaves have been shown to posess superior qualities to those processed
in conventional ovens. In general, the absorption of ceramics increases with fre­
quency [23], so inexpensive, moderate-power ( 2-100 kW) sources of microwaves
and millimeter-waves will need to be developed for rf sintering to be m ade costeffective.
1.2.4
RF Sources for Fusion
Electron cyclotron heating (ECRH) and current drive of fusion plasmas is an at­
tractive method of driving fusion reactors. The small wavelength simplifies the
launching scheme and allows for the possibility of controlling disruptions through
the relatively precise deposition of energy within the plasma. At least a hundred
megawatts of average power could be required at millimeter-wave frequencies from
110 GHz to 280 GHz - depending on the confinement field of the tokam ak. Longpulse (essentially continous-wave. or CW) gvrotrons have been developed within
the FSU and the US that can provide 0.5 MW for 10 seconds or
1
MW for
1
second. The free-electron laser (FEL) is an alternative source which could provide
high average power (through high repetition rates of short pulses) with the added
benefit of frequency tunability. However, this in turn will require the development
of moderate-power, frequency-agile sources to be used as rf drivers for the FEL.
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13
1.3
Extended Beam Configurations
In general the output power of a microwave tube may be written as P0 =
where
17
7 7
V*,
is the tube efficiency, /& is the total beam current and Vj is the beam
voltage (where the beam has total kinetic energy given by eVj,). Increases in tube
output power must therefore come through corresponding increases in either the
tube efficiency, beam current, or beam energy. Efficiencies of 60 % are achieved
in some devices through the use of energy recovery in depressed collectors. It is
unlikely that future advances in tube efficiency alone will be enough to significantly
advance the state-of-the-art. Also, for reasons discussed previously, requirements
on tube compactness make it desirable to operate below 40 kV in beam voltage.
Thus, increases in the amount of beam current are the only realistic option for
increasing the output power of individual tubes.
Especially problematic for the development of high-power. high-frequency rf
sources are limitations on beam space-charge density and rf power density within
the rf circuit and output. Generally speaking, as the frequency increases the crosssectional area of the interaction region decreases as the inverse-square of the fre­
quency. Constraining the same output power and beam fill-factor (the ratio of
beam-to-wall radii) in our hypothetical device as we increase frequency implies
th at the same current will need to be pumped through a shrinking waveguide. Cor­
responding increases in current density (and space-charge for fixed beam voltage)
require increased magnetic focusing. Furthermore, decreases in device efficiency
occur due to enhanced velocity spread acquired in the drift region and the effects
of space-charge debunching within the interaction region. Likewise, our device will
need to accomodate higher rf power density within the output waveguide as the
circuit dimensions decrease - leading to problems with rf breakdown and cooling.
Sources utilizing an extended beam configuration - either m ultiple electron
beamlets or sheet electron beams - are an attractive alternative to round beams
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14
for microwave devices.
Figure 3 show the progression of a single round-beam
device into a m ulti-beam or sheet beam configuration. High current, at reduced
current density, can be transported through thin clearance apertures and in close
proximity to walls or rf structures.
Size and weight reductions axe realized in
extended beam configurations over a comparable num ber of single-beam devices
due to the shared focusing structure and vacuum tube. A general property in slowwave devices is that the the rf electric field decays exponentially away from the
surface of the interaction structure (as in, for example, grating, helix, or rippled
wall circuits). Devices utilizing sheet-beams. therefore, may carry high-current
and yet be thin enough to m aintain good coupling to an appropriate rf circuit. In
addition, restrictions on total rf power are eased due to the wide output guide.
1.3.1
H istorical R eview
The advantages of using sheet electron beams for the transport of high-current
through microwave circuits were appreciated at least as early as the mid-1950s.
Soon thereafter experimental investigations of the transport of sheet beams in
uniform axial magnetic fields showed the existence of an instability which led to
beam kink and filamentation [26. 27. 28]. Figure 4 shows one such example of
beam breakup. Labeled the Webster instability at first, it was recognized that the
driving mechanism behind it was linked to the coupling of the radial motion of
the beam, induced by space-charge. with the focusing provided by the longitudinal
field [27, 29]. In later work it was shown that the driving mechanism was indeed
due to an E x B velocity shear across the top and bottom of the electron beam [30,
31, 32], where E is the transverse space-charge electric field. Now refered to as the
diocotron instability, the phenomena largely discouraged the use of sheet beams
for several decades.
Periodic focusing, rather than a unform guide field, was proposed as a possible
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15
m ethod to avoid filamentation. Early concepts investigated electrostatic periodic
focusing of sheet beams [33, 34], but Sturrock pointed out that magnetic periodic
focusing had the potential to focus higher current beams for reasonable magnet
field strengths [35]. Although Sturrock considers beam ”stability'’ in this work (and
in a companion article on a general theory of sheet beams in periodic fields [36]),
the stability criterion thus developed is actually a condition for effective transport
of the beam through the focusing stack. Although later analyses, m ainly based on
the plasma fluid-equations [32, 37, 38], have investigated the issue of sheet beams
in uniform focusing under a variety of conditions, few investigations of sheet beam
stability in periodic fields have been undertaken.
One notable exception is recent work done at the University of M aryland as
part of a sheet-beam free electron laser experiment [39]. This work investigated
the stability of sheet beam transport through a 1 cm. 5-period. S0O-16OO g planax
wiggler array. The 'sheet* beam actually begins as a planar multiple beam let array
produced by seventeen 1.0 mm holes drilled into an anode plate appoxim atelv 3.0
mm apart. However, by the end of the transport channel (5 cm ) the beam lets merge
into a continous sheet approximately 3 m m x 30 mm in extent. The beam used in
this experiment was relativistic - 100 and 400 kV beam voltages - and produced
by a field-emission ("cold**) cathode for short (< 100 ns) pulses. About 26 A of
current was present in the the beam after scraping at the anode plate. Hence, the
experimental parameters used in this study are in the relativistic (high-voltage),
low-space charge regime - a fact bolstered by the relatively good agreement of the
actual beam transport with single-particle codes mentioned in the article.
As an indication of the increasing interest in and appreciation of sheet beams
for use in microwave devices, a recent paper design study compares a sheet beam
klystron to round beam klystrons at 11.4 GHz for accelerator applications [40]. Due
to the high beam current, only moderate beam voltages (400 kV) are necessary
to provide the same output power as highly relativistic (3000 kV) klystrons. In
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16
addition, an efficiency enhancem ent of a factor of 2-4 is deemed possible through
the reduced space-charge in the sheet beam. Solenoidal focusing is envisioned
for this device. This work was based on an earlier paper design study [41] using
a planar magnetic wiggler for beam focusing. Although these designs appear to
provide marked improvement in the state-of-the-art of high-power klystrons, the
experimental dem onstration has yet to be made and. furtherm ore, basic questions
of sheet-beam stability and transport need to be addressed.
Finally, for completeness we note that sheet beams in their own right have
received increasing interest for use in ion accelerators [42] and for heating of fusion
plasmas for the same reason they are attractive for microwave devices - namely the
ability for transport of high current. A relativistic. solenoid-focused sheet electron
beam of dimensions 0.6 x 13 cm was produced for injection into a fusion mirror
machine [43]. Although 100 % transport was dem onstrated through the transport
channel for very high current density (over 1 kA /cm 2). the beam did suffer from
edge distortion associated with the diocotron instability.
Multiple Electron Beam s
The idea of using m ultiple electron beamlets to increase the power output of vac­
uum tubes was first proposed in the 1950‘s [44. 45]. and experimental investiga­
tions soon followed. Boyd et al. constructed and operated a 10 beamlet. 750 MHz
klystron with a single m agnetic focusing circuit which demonstrated a ten-fold
increase in output power and enhanced bandwidth. [46] A twelve beam travelingwave klystron (wherein m ultiple cavities are tied together by waveguide and the
wave is amplified along the longitudinal and tranverse extent of the tube) demon­
strated over 5 MW, 23 dB gain, and 16 % bandwidth at S-band. [47]
For obscure reasons, these early promising results did not lead to further in­
vestigations of m ultibeam devices in the US. However, as recently reported in the
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17
Soviet Union developed a wide range of multi-beam tubes over the span of several
decades. These included a
1
MW, 40 dB S-band klystron dem onstrating 5-10 %
bandwidth and a 36-beam disk-loaded TW T at 10 GHz exhibiting roughly a factor
of 1.5 more output power than a comparable single-beam device. [48] Details are
sketchy, but generally these tubes employed multiple beams spaced axound the
axis in a round configuration (versus planar as in [47]). Solenoidal focusing was
probably used to confine the beams in these devices.
1.3.2
Study o f H igh-Perveance Sheet Electron B eam s
Although sheet beams systems were initially proposed over three decades ago. and
the advantages to be gained by extended beam systems are well-known, it is only
recently that renewed interest has resurfaced. However, these recent works on
sheet-beams are. without exception, limited to the high-voltage, low space-charge
regime which is inconsistent with compact microwave devices for mobile systems.
A major reason for the lack of effort to pursue space-charge dom inated designs
is the ubiquitous presence of the diocotron instability, at least in conventional
solenoidal focusing.
Stable, focused propagation of space-charge dominated sheet beams over long
transport distances is one of the main objectives of study of this thesis.
This
includes the elimination of the diocotron instability, low-ripple focusing in the thin
transverse dimension of the sheet beam, and edge-focusing for confinement in the
wide dimension. A recent article [49] proposed an offset-pole. periodically cusped
magnet (PCM) configuration which has the potential to provide stable, smoothflow confinement of sheet electron beams. This thesis continues and expands the
theoretical analysis from [49] using period averaged analyses of electron stability,
electron trajectory equations, and particle-in-cell numerical simulations presented
in Chap. 2.
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18
The formation of sheet beams is another area th a t has not received much past
attention. The sheet-beam FEL [39] experiment used a anode aperture plate to
mask, or scrape, an initially round beam from a low-vacuum compatible cold cath­
ode to form a sheet. Beside being highly inefficient, this method suffers from addi­
tional disadvantages for use in low-voltage devices, which often employ therm ionic
cathodes operating in a high-vacuum environment. Gas evolution and the liber­
ation of m etal ions from the aperture plate can lead to cathode poisoning and
degradation.
Furthermore, the presence of gas near the beam can lead to the
formation of ions within the electron beam which can spoil the beam velocity
distribution and focusing - an effect that is enhanced by the nearly unavoidable
presence of secondary electrons created at the plate. Finally, although appropri­
ate for low-pressure experiments where up-to-air and pumpdown cycles are short,
any modification of the sheet-beam size by replacement of the aperture requires
an extended bakeout procedure that can take days to complete in high-vacuum
systems.
A few design studies of sheet-beam thermionic electron guns have been under­
taken. Virtually all studies have used 2-D codes that are incapable of modeling
the focusing at the edge of the em itter. In any case, no such gun will be readily
available for use in sheet-beam transport experiments in the near future. Hence,
we have identified a novel method of forming large aspect-ratio elliptical sheet
electron beams using magnetic quadrupoles and a round-beam electron gun. This
method is especially suited for laboratory experiments because it is relatively inex­
pensive, easily fabricated, and has the added advantage of flexibility in the beam
size through external adjustm ents of quadrupole param eters. Chapter 3 describes
the analytic models, two-dimensional and three-dimensional envelope equations
and particle-in-cell transport codes used to evaluate beam formation.
An experim ental system for investigating sheet-beam formation is presented
in Chap.4. Details of the magnetic quadrupole sheet-beam forming system are
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19
provided and experim ental results are discussed.
1.4
Periodic Permanent Quadrupole M agnet Fo­
cusing
Another novel configuration which holds promise for increased microwave output
power is the use of periodic perm anent quadrupole magnets (PPQ M s) to focus
high density round electron beams. Figure o illustrates the basic configuration. A
round electron beam propagating along the e-axis with a drift velocity vzo is focused
inward in the x — z plane by the v:o x B y Lorentz term. In the other transverse
plane the beam is defocused during the first half-period of the array. The field
polarity and the direction of the focusing reverses during the next half-period of
the magnet stack and, provided the magnetic field gradient is strong enough, then
a net inward focusing counteracting the beam space-charge is achieved.
In PPM stacks beam focusing is achieved through the interaction of the trans­
verse oscillatory motion induced by the periodic radial component of the magnet
stack and the oscillatory B. component. The net inward focusing that occurs can
be understood by period-averaging the transverse wiggle motion and the periodic
axial field. In contrast, focusing in the PPQM configuration occurs through the
’strong’ axial component of the velocity, v:o. interacting with the 'strong* trans­
verse component of the quadrupolar magnet field. Hence, the enhanced focusing
strength of PPQM s over their PPM counterparts should make possible the focus­
ing of higher perveance electron beams for increased microwave output power in a
single device.
In order to quantify this statem ent consider the magnitude of the focusing
force of a PPQM array with a magnetic field gradient. B^q - The m agnitude of
the focusing force at the beam edge may be written as. e v ^ B ^ ' r ^ . where e is
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20
the electron charge and r &0 is the beam radius. In PPM focusing the focusing
term may be written as, (evgBz0 cos(kmz )) where vg is the oscillatory rotational
velocity induced by the PPM stack. Here B zo is the on-axis PPM field m agnitude,
km = 27r/Am where Am is the magnet period, s is the drift coordinate, and (...)
denotes a spatial average over magnet periods.
Assuming a flux-less cathode,
conservation of magnetic flux through the beam may be invoked to write the
rotational wiggle velocity as, vg = 0r&o = ( eB zo/2m) cos(kmz)ri,0. Noting that
the period-average of cos2(..) = 1/2 we can write the ratio of PPQ M -to-PPM
focusing force term s as:
F™™
qBz0 /Am
V
’
Here we have used the non-relativistic expression. vzo
B j(kG )
K
y j ( 2 e \ \ / m ) , where V& is
the beam voltage.
As will be discussed later, field gradients of 4 kG /cm are easily achievable with
SmxCoy or NdFeBo permanent magnet m aterials in quadrupole arrays. Typical
on-axis B z0 m agnitudes range from 1-2 kG for PPM stacks used in helix-TW Ts.
Figure
6
shows a plot of Eq. 2 against the beam voltage Vj, for a range of different
field param eters. Using the more conservative 2 kG number for the PPM field we
find that the ratio of the available field strength ranges from 1.4-2.7 from 10-40
kV in favor of the quadrupole focusing configuration. Using a
1
kG field value the
ratio becomes 5.4-10.S from 10-40 kV. Assuming a corresponding increase in the
amount of space-charge which can be confined in a PPQM array then an increase of
40% to an order of magnitude in the beam perveance can be realized for low-voltage
beams. It should also be noted that the advantage of PPQM focusing continues
to scale with \/H> which may prove particularly useful for typically high-voltage
devices such as klystrons.
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21
Chapter 2
Transport of Sheet Electron
Beam s
The offset-pole PCM configuration is shown in Fig. 7. In contrast to planar wigglers, the PCM has the same polarity across the midplane and no transverse mag­
netic field on-axis. Thus, for short magnet periods, confinement is produced with
small transverse components of the electron velocity - a necessary condition for
efficient operation in linear beam tubes. The poles are offset with respect to one
another to provide a static side field for edge focusing. For permanent magnet
materials, where linear superposition of fields approxim ately holds, the fields can
be thought of as comprised of a purely varying periodic field provided by the center
section of magnets (—x m > x < x m in Fig. 7) and static side-focusing fields near
the transition at ± x m % 1 . Another configuration which is investigated utilizes
periodic quadrupolar edge fields to provide sheet beam side focusing. This scheme,
although more complex to fabricate, allows for the strength of the side fields to
be set independently of the central PCM focusing array. Hence, beam matching
to reduce tranverse oscillations can be more readily achieved in the quadrupolar
PCM array.
In this chapter a theoretical investigation of sheet beam focusing is presented
for both magnet configurations. The beam dynamics are examined using analytic
methods, numerical solutions of the transverse beam envelope equations, and de­
tailed particle simulations. Where appropriate, comparisons to beam focusing in
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22
conventional PPM stacks are m ade in order to highlight the fact th a t in m any ways
the use of PCM arrays on sheet beams is an extension of existing technology to
a new regime, rather than a radical departure from conventional microwave tube
design. In either magnet array robust side-focusing is achieved for remarkably
high space-charge beams. It is also shown w ith analytical expressions and m agnet
design codes that the fields required for sheet beam focusing are well within the
capabilities of modern perm anent magnet technology.
2.1
Semi-Infinite Sheet Beam Stability
Consider the offset-pole PCM configuration in Fig. 7. Near the center of the array
the magnetic fields can be approximately represented by [49]:
By,m{y.=) % - B0s\nh(kmy) cos(km=)
(3)
B..m( y .z ) % + B 0 cosh(fcmj/)sin(fcmr)
(4)
where Bc is the on-axis m agnitude of the field. km =
and Am is the spatial
period of the magnet array. As a first step towards investigating beam stability,
we consider the case of a semi-infinite sheet electron beam of half-thickness yb
propagating through a semi-infinite PCM array.
Equations 3 and
4 are now
valid everywhere along the beam width. Furthermore, werestrict ourselves to thin
beams with y* <§; ymwhere ym is the half-height of the magnetstack.
YVe can now
simplify the above equations:
By,m(y' -) ~
B0kmy cos{kmz )
S.-.m (-) ~
+ B a s\n{km:)
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(5)
(6)
23
For short PCM periods, where the transverse velocities i . y are much smaller
them the drift velocity i , we can map the z coordinate into time through the
transform ation z -+ u0t, where u0 is a constant w ith i ss u0. This so-called
paraxial approximation is commonly made in ray tracing and envelope analyses of
electron optics systems. The transformation of the m agnetic focusing fields into
the time domain takes the form:
t) » - B 0kmycos(u;mt)
(7)
Bz.m(y-t) % +£„sin(u;mf)
(S)
where u:m = Armu 0 is the frequency associated with the beam drift through the
periodic magnet array. The time-dependent equations of motion for an electron
of charge —e, mass m. under the influence of the above magnetic fields and a
space-charge electric field
E y ,s
are:
•r =
y
Here
=
- u j c z ( t ) y + u.iey(< )u „
(9 )
+ * ’c : { t ) x
(10)
-
—
m
E y .s
= e B y{t)/m. u.'c: = e B z( t ) /m . and the self electric field in the x direction
is taken to be zero for the semi-infinite beam.
We now proceed by separating time-scales under the assumption that motion
due to the self-electric field evolves on a slow time-scale relative to the motion
due to the oscillating m agnetic field in our short period array. This is equivalent
to the condition u pe
u;m for the regime of interest, where u:pe = n0e2/e0m is
the plasma frequency for our beam of charge density nae. The transverse motion
may be separated into fast and slow time-scale oscillatory behavior according to
x = x / + x 3 and y = yj + ys. Substituting these expressions into Eq. 9 under the
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24
assumption th at y <C u a, and separating the fast and slow time-scales gives:
x j « —kmy3u>C0u0 sin(wmf )
(11)
idco
X/ ~ kmys-— u0 cos(umt)
(1 2 )
Or, integrating gives:
for the fast time-scale motion in x where u;co = eB0/ m .
Applying the same method to Eq. 10. except equating slow time-scale term s
by averaging over fast time-scales. gives:
y, =
m
E y.s + u;cr (t ) x f
(13)
where the bar denotes the time-averaging of a quantity. Substituting Eq. 12 into
this equation yields:
y3 = — Ey>s +
m
* ’co2
cos2 (wmt)ys
(14)
which, when averaged over the fast time-scale, u;m_ l. leads to:
y, = ~ — E y.3 + ^ - y s
m
1
(15)
Proceeding as in Ref. [50]. we further assume that the slow time-scale oscillation
is of sufficiently low frequency that electrostatics apply. Under this quasistatic
assumption we may look for solutions to Eq. 15 of the form el{kl~wtK For a thin
beam, such that 2ybk <§C 1 . the space-charge electric field can be written as E y%s %
( —m/2e)2ybkujpe2ys Substituting this into Eq. 15 and using ^ —> —fa; gives norm al
mode solutions of the form:
(16)
which correspond to stable, bounded, periodic betatron oscillations. The same
analysis in a uniform guide field B = B 0z leads to solutions of the form:
2
u; % ± i —^ —2kyb
(17)
which is a purely growing perturbation corresponding to the diocotron instability.
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25
2.1.1
Beam Envelope Trajectory Simulations
The equations of motion for a non-relativistic electron in an external periodic
magnetic field and the self-electric fields due to a semi-infinite sheet beam given
by
9
and
10
were solved using a ray equation approach for the particle trajectory.
Again we employ the short-period assumptions x,y C : a u 0 to use the paraxial
ray approximation £ —> u0^ and the transform ation t —>• z / u 0. Under these sim­
plifications the coupled electron equations of motion were numerically solved. A
plot of the resulting electron motion is shown in Fig. 8 , and serves to illustrate some
general behavior of electron motion in periodic fields. This simulation is for a 10
kV beam with a current density of
12
A /cm 2, focused by a PCM having a 3.75 mm
period and a field am plitude of 700 G. Note that the z-axis has been normalized
to the magnet period (Z „=z/A m) and that the transverse coordinates have been
normalized to the initial beam half-thickness (Yn=y/y&. Xn= x /y 6 ). The motion
in the X-Z plane is a combination of a linear E x B drift with the high-frequency
wiggle motion due to the PCM. Also occuring are slow time-scale betatron oscil­
lations. These oscillations are more readily apparent in the Y-Z plane and can be
seen to have a period of roughly
12
magnet periods.
Betatron oscillations in the sheet beam lead to a breathing, or ripple, effect
of the beam thickness, since for a cold beam or nearly-cold beam the electrons
oscillate in phase. It is im portant to minimize this effect through m atching’ of
the beam betatron oscillations with plasma oscillations due to space-charge. For
a semi-infinite sheet beam, the condition for beam matching is given by u.'pe zz
co/ V 2 = u>0 where u)q is the betatron frequency. For the beam param eters of
Fig.
8
this criteria gives B 0= 507 G for a matched condition: Fig. 9 shows the
resulting elimination of the betatron motion. The remaining oscillation is solely
due to the PCM fields and this represents the minimum ripple condition on the
beam.
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26
As the magnet period increases the beating of the bi-harmonic motion of the
beam comprised of betatron and PCM oscillations can lead to a condition where
the PCM oscillation period is comparable to the betatron period, or harmonics
of the betatron period.
Constructive interference between the two oscillations
creates large-amplitude. or completely unconfined, orbits which lead to beam loss
to the transport channel. This situation is similar to the case for round beams
in permanent periodic magnet (PPM ) focusing treated in Mendel et al [51]. In
this article the radial equation of motion for the electron in a round beam was
recast in the form of a M atthieu's equation. Tabulated solutions for M atthieu’s
equation showed the existence of a series of passbands and stopbands as the PPM
period was increased corresponding to transport of the beam through the focusing
stack or loss of the beam to the channel walls. Similarly, for the semi-infinite sheet
beam case, the numerical solutions for the trajectory of electrons at the beam edge
(envelope calculations) also show the existence of higher order passbands which
correlate very well with the M atthieu’s solutions.
Mathematically, this can be
seen if we recast Eq. 14 using the relation cos2 (0) = 1/2[1 —cos(20)]:
2
y 's
=
-
— E y .s
m
+ ^ [1
1
-
cos(2a;in0 ]y j
(IS)
which is of the same form as the M atthieu's equation given in Ref. [51].
The agreement between the trajectory simulations and the M atthieu's equation,
as tabulated by Mendel, for the position of the pass- and stopbands is very good.
For the beam param eters used in Fig. S the first stopband occurs at Am= 35 mm
and the second pass band occurs between 60-S0 mm. The relatively poor focus­
ing available in the higher passbands, leading to larger am plitude electron orbits,
probably negates any advantage to be gained by using longer magnet periods.
We also note th at, besides the aforementioned similarity between planar PCM
and PPM focusing pass- and stopbands, the analogy between the two magnet
systems also carries over into the basic scalings required to achieve beam focusing.
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27
In general the magnet period. Am, must be smaller than the scale length associated
with plasm a oscillations, Ap = 2jru 0 /u/p, in order to focus the beam against spacecharge. Typically, in PPM systems, a ratio of Am/Ap < 0.33 is desired [52] and a
comparable scaling is required in PCM focusing arrays. In addition, th e magnet
period should be smaller than the betatron scale length. Ag = 2~u0/ujg, in order
to avoid large amplitude oscillation and beam current loss. Consideration of the
point at which the first stopband occurs in Ref. [51] gives the scaling Am/A^ < 0.25.
2.1.2
PIC Simulations o f th e Sem i-infinite Sheet B eam
The 2 -A-d PIC code MAGIC [53] was used to test the predictions of the analytic and
trajectory models from Sec. 2.1 and Sec. 2.1.2. MAGIC numerically solves plasma
and electromagnetic problems through integration of Maxwell's equations and the
Lorentz force equation in two-dimensional space and three-dimensional velocity
coordinates. As in Secs. 2.1 and 2.1.2. the paraxial transformation c —►u0t is used
to map the drift coordinate r into the tim e coordinate so a 2 -d code can be used
to model a 3-d situation.
The following simulations (see Fig. 10 and Fig.
11)
load a semi-infinite sheet
beam with charge-densitv pmin = 2.06 x 10“ 3 C /m 3, a thickness of 2.0 mm. and
an initial z-velocitv of u0 = 5.9 x 10‘ m /s corresponding to a beam energy of 10
keV. The beam propagates through a planar transport channel with perfectly con­
ducting top and bottom walls spaced 20 mm apart. Periodic boundary conditions
on the side walls are used to model an infinitely wide beam. An initial density
bump was used to prime the simulation for the lowest-order mode of the diocotron
instability - thereby shortening the simulation run-tim e by a considerable amount.
Figure 12 illustrates the geometry of the density bump. In these simulations, the
thickness of the bump, zl was chosen to be the same as the thickness of the beam
(2 mm), and the ratio of the maximum-to-minimum charge density was pmax/Pmin
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28
= 1.6. The same simulations w ithout the density bump verified the sam e final
result.
Figure 10 shows the resulting transport of a semi-infinite beam in a periodic
magnet array having a period Am= 3.75 mm and a field magnitude of 700 G. The
tim e (and distances normalized to the magnet period) for plots a-d are: t =
ns (z/Am = 0.0), 0.5 ns (8 .8 ). 0.7 ns (12.3), 1.2 ns (21.2),
1 .8
0 .0
ns (31.7). and 12.4
ns (218.7 periods), respectively. Although the beam suffers from a m ism atch, and
corresponding breathing in beam thickness, the beam is confined and no evidence
of filamentation is extant. The breathing of the beam thickness is in qualitative
agreement with the trajectory sim ulation using the same beam and m agnet pa­
rameters shown in Fig. 8 . This sim ulation demonstrates that sheet beams focused
by PCM m agnetic fields are robustly stable against the diocotron instability.
In marked contrast, the sim ulation for the same beam loaded into a uniform
solenoidal focusing field exhibits large scale filamentation corresponding to the
lowest-order diocotron mode. The same field magnitude (B0 = 700 G) is used in
the simulation shown in Fig.
11.
Here the times of plots a-d are: t=0.0 ns. 0.5.
1.4. 2.8. 4.6. and 11.5 ns. respectively. The evolution of the instability, driven by
the E x B 0 velocity shear across the top and bottom of the beam, is clearly seen.
A kink has begun to form by 0.5 ns and by the end of the run a large density
bunch and a pulling away of the beam from the side walls is clearly visible.
2.2
Offset-Pole PCM for Beam Edge-Focusing
Consider the offset-pole PCM configuration in Fig. 7. Near the center of the array
the magnetic fields can be approxim ately represented by:
B
y
^
+ B 0sinh(kmy) cos(km:)
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(19)
29
B z,m(y, z) % - B 0 cosh(kmy) sin(kmz)
(20)
where B 0 is the on-axis m agnitude of the field, km = 27r/Am, and Am is the spatial
period of the magnet array. As shown in the previous sections these field compo­
nents provide stable focusing of semi-infinite sheet-electron beams for short PCM
periods.
Side-focusing in the wide transverse dimension of the beam in this configuration
is provided by the fringe fields of the array. Analytic expressions for the fringe fields
may be obtained by considering the magnetic equivalent of Fig. 7 as pictured in
Fig. 13. For field solutions near the interface of the 'pure' PCM array and the
side magnets, we can further replace the side magnets with semi-infinite sheets of
’surface magnetic charge' as shown in Fig. 14. The side fields are obtained from
Coulomb’s law by integrating over the surface charge:
(21)
B y j x . y ) = C „ J f , m [(y _
(22)
where Cb is a dimensioning constant. The magnetic surface charge density. pm,
can be w ritten as:
-Po[$(y'- y m ) - $ { y ' + y m )] for x > o
+po[S(y' - y m ) ~ S(yt + ym)] for x <
0
If we recognize that far from the interface the contribution of the sheets produces
a constant magnetic field. B so. then we may replace Cbp0 —■
►Bso. Substituting
Eq. 23 into Eqs. 21 and 22. and integrating over all four magnetic sheets gives:
B x,3 =
—
^ 7
{ln[(x - x m )2 + (y + ym )2] - ln[(x - x m )2 + (y - ym )2]
ln[(x
+
x m )2 +
(y
—! / m ) 2 ] +
in[(x
+
x m )‘
+
(y
+
y m )2] |
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(24)
Plots of Eqs. 24 and 25 are shown in Fig. 15 for several magnet aspect ratios
(ym/x m). These plots and a consideration of the beam cross-section as sketched
in Fig. 14 illustrate the dynamics of the side focusing in the offset-pole PCM
configuration. Focusing of the beam in the wide transverse dimension (in x) is
provided by the vzoB VtS force where v~0 is the beam drift velocity. The linear
region of BVtJ near the x = 0 midplane makes possible a force balance against the
linear space-charge electric field and. as will be discussed below, thereby allows for
beam matching in the wide transverse direction. The Bx,s component is zero along
the midplane but off the midplane the polarity of BItS is such that particles are
pushed away (defocused) along the y direction. However, the adverse effects due to
the BX'S field can be overcome for very thin beams (y&
x<, < xm) where particles
axe close to the axis all along the midplane, and especially for highly elliptic sheet
beams where the tapering of the beam at the edge reduces defocusing.
Sheet Beam Matching
As noted, the B y,s side field is approximately linear around the point x = 0. Along
the midplane (y = 0) we can write Eq. 25 as:
(26»
Assuming a very wide magnet array with ym/ x m
1 and using the expansion
tan - 1 (t/> 4 - e) % f — -£ (1 — e/v) we can write:
By-J(x.O) ss
7T
\x„- m 2
)x
(27)
Based on the above discussion of the BIiS fields, if we consider elliptical beams
then the electric space-charge field of an elliptical beam of m ajor radius X6 , minor
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31
radius y&, and space-charge density ne may be written as [54]:
£ r (4) =
ne j/6
-------------- x
4“ J/6
to
(28)
-y
to
%b
4“ J/6
Comparing the defocusing force due to beam space-charge and the inwaxd focusing
force due to the By<3 side fields we can use Eq. 27 and the y component of Eq. 28
to write a force-balance condition:
=
2ec <3Z0 \ y m/ x mJ
(29)
where wpe (= ne 2 /e 0 m) is the electron plasm a frequency. From the above equation it
is evident that the required side magnetic field amplitude can be held to reasonably
small values provided that the beam aspect ratio yb/*b is made small. Fortunately,
this condition is the same as required to avoid defocusing effects from the Br,s
magnetic field term, implying that high current densities can be transported in
highly elliptic beams using m oderate side field magnitudes.
Sheet Beam Tilt
One other adverse effect of the addition of static side focusing fields to the periodic
center fields is a tilt in the beam m idplane from the magnet midplane, as illustrated
in Figure 16. This can be understood by noting that the addition of side fields
to the center fields causes a rotation in the axis along which the By,t component
of the total magnetic field is zero. In other words, since the side field changes
polarity across the x = 0 plane, yet the center PCM fields are sym m etric, the
beam will settle into an equilibrium around the null field axis. Using the linear
approximation to the side By,s field in Eq. 27 and noting that the period average
of Eq.
19 is kmy B 0/ \ / 2 near the m idplane, we have the following expression for
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32
lines of constant magnetic field:
1
2Vm
B y A x , y ~ 0)|conatant ~ —j=B0kmy - B so----- r i
V2
7TX m
(30)
The null field axis occurs at an angle given by:
0o = ta n - 1 (y/x) = ta n - 1
(31)
\Trkmx m x m B 0 }
For most beams of interest the small amount of tilt incurred is not problem atic.
For example, anticipating beam simulations in the following sections, if we have
Vm/xm
= 1/16.
Am/xm = 0.3/4. and B so/ B 0
= 1/2. then Eq. 31gives60« 0.33
mrads.
A highly elliptic sheet electron beam with an aspect ratio of x^/t/t
would thereby have its effective width increased by 80X(,/yi,. or about
2 %.
= 56
Only
in cases of only highly elliptic beams having extreme space-charge. requiring laxge
side magnetic fields for focusing, would beam tilt be problematic in pushing the
beam edges far enough off axis into the defocusing Br,s field.
2.2.1
PIC Sim ulations w ith Edge-Focusing
The 2 -d PIC code MAGIC was used to study sheet beam focusing in offset-pole
PCM
fields.As
before,the x drift coordinate is transformed into the tim e frame
using
z —*uat
according to the paraxial assumption discussed previously. The
static side magnetic fields given in Eqs. 24 and 25 are superimposed upon the
central PCM fields in Eqs. 19 and 20. Under the paraxial assumption and for thin
beams with fcmj/6
1 we can rewrite Eqs. 19 and 20 as:
By.miy.t) % Bo{kmy)cos{^mt)
(32)
B z.m{ y J ) « —B0 sin(u,’mf )
(33)
Here we define uim = kmii0 as the magnetic oscillation frequency corresponding to
a beam drifting along a spatially varying magnet array with km = 2^/Am where
Am is the magnet period, with a drift velocity u0.
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33
The first beam considered here is a highly elliptic sheet beam with m ajor and
minor radii of
= 1.4 cm and
= 0.025 cm (note Xb/yt = 56) with a total
beam current of /*, = 10 A and beam energy of eV& = 10 keV. The current, spacecharge, and paxticle density of the beam corresponds to 90.9 A /cm 2, 15.5 m C /m 3,
and 9.7 xlO 16 m - 3 respectively. The beam is initially loaded with a Gaussian
velocity distribution in all three velocity coordinates. The rms velocity spread
(ratio of vth,: / u 0 where vth,z is the therm al velocity in r) is taken to be 1 %, which is
consistent with, but somewhat larger than, the velocity spread normally produced
in Pierce-type linear beam sources. Equipartition of the velocity distribution is
assumed and the rms therm al spread in the transverse components is also taken
to be vth . x / u 0 = vth.,y/ u 0 = 1%. Approximately 1000 particles are loaded on to a
200x50 rectangular grid with conducting walls located at i = ± 2 cm and y = ±
0.25 cm. Figure 17 shows the evolution of the beam cross-section subject to the
magnetic focusing fields of Eqs. 24. 25. 32. and 33 with Am = 3 mm. B a = 1200 G.
B so = 534 G, x m = 4.0 cm, and ym = 0.5 cm. Both the x and y axes are in units of
centimeters. The beam cross-section is shown for simulation times corresponding
to (top-to-bottom ) r = 0.0. .238. .477. .715. .954. 1.19. and 2.3S nsec. Using the
z — t transformation discussed previously, these times correspond to 0.0. 4.64. 9.28.
13.93, IS.57, 23.21. and 46.42 magnet periods, respectively.
The particle plots of Fig. 17 dem onstrate several things. First, relatively high
current density beams can be focused without instability with modest, and readily
achievable, magnetic field magnitudes. Thus the prospect for microwave tubes uti­
lizing sheet beams in PCM-focused configurations would appear to be realizable.
Here we note th at the total beam perveance is h / V 3*2 = 10 //per vs. For compar­
ison we note that the PPM-focused 50 MW klystron of Ref. [7] has a total beam
perveance 350 A/(440 kV )3^2 = 1.2 //pervs. Furthermore, since the perveance of
the sheet beam configuration is obtained for low beam voltages, the possibility
of constructing compact high-power systems without bulky high-voltage supplies
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34
is distinct. Second, although beam m atching is more difficult to achieve in both
tranverse dimensions simultaneously, the beam of Fig. 17 is relatively well-matched
in the thin transverse dimension. In th e other transverse plane the beam oscillates
inward and outward as it encounters space-charge and side-field forces on a timescale which is slow compared to u;m-1 . Another noteworthy point is th a t the
long-term confinement (> 40 magnet periods) establishes the robustness of the
beam focusing in offset-pole PCM arrays.
For comparison purposes. Fig. 18 shows the same beam in a static, uniform
solenoidal magnetic field oriented along the z-axis (pointing out of the page). The
field magnitude, B zo is taken to be the peak PCM magnetic field used in Fig. 17, or
B zo = 1200 G. Recalling that the growth rate of the diocotron instability scales as
(jjpe2f%jce, or as the ratio of the charge density, ne. to the magnetic field am plitude.
B zo, we note that by using the peak PCM field rather than the rms value of the
sinusoidal PCM field (i.e. B0/ \ / 2) we are. in effect, taking the 'best case' condition
of solenoidal focusing for this comparison. Still. Fig. 18 shows that the beam begins
to shear and tilt early on in the simulation and eventually filaments and scrapes
the transport channel walls near the end of the simulation.
To further illustrate the attractiveness of the offset-pole PCM array for focusing
high-perveance sheet beams we scale the beam current by a factor of five (to 50
A) for the same size beam as in Figs. 17 and IS. Constraining ourselves within
achievable limits of PPM /PC M magnet technology we take B 0 = 4000 G as the
upper limit of the PCM field. Since the space-charge field is linear with ne (or /&)
we also scale the side-field am plitude by approximately a factor of five to B so = 2515
G. All other magnet and beam param eters remain the same. Figure 19 shows the
evolution of this 455 A /cm 2 beam for simulation times of (top-to-bottom ): r = 0.0,
.238, .477, .715, .954, and 1.19 nsec. corresponding to 0.0, 4.64. 9.2S, 13.93, 18.57,
and 23.21 magnet periods, respectively. Again we see stable focusing is achieved
in both transverse planes leaving us with the pleasant problem of just how such a
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35
high current-density beam might be generated in the first place. Figure 20 shows
the evolution of this beam in a solenoidal field with B zo = B 0j \ J 2 = 2828 G. As
before, the presence of beam instability causes filam entation and eventual particle
loss to the channel walls.
Though these cases are instructive, such a short period array (Am = 3 mm)
poses a fabrication challenge. Figure 21 shows the evolution of the same beam as
in Fig. 17 in a magnet array having Am =
1
cm. As before, we have Ba = 1200
G, B so = 534 G, xm = 4.0 cm. and ym = 0.5 cm, and the beam remains robustly
confined by the magnet array. From top-to-bottom . these plots correspond to 0.0.
1.39, 2.78, 4.18, 5.57. 6.96 and 13.93 magnet periods, respectively. One notable
difference is an enhanced tilt in the beam m idplane which, recalling the discussion
of beam tilt and the scaling of 0o ~ Am. is to be expected if only the magnet period
is increased. In this simulation the tilt angle is less than a few degrees, but even
this could be reduced by either increasing B 0 or by widening the magnet array.
Finally, the simulation of Fig. 22 is included, for reasons that will be made
more clearly in Chap. 3. since this beam corresponds to a sheet beam which can
be produced using a novel formation system for laboratory experiments. The beam
param eters here are /*, = 2 A. xi, = 2.7 cm. y* = 0.1 cm. and Vj, = 10 kV. An rms
velocity spread (defined previously) is assumed to be
1%
in all three momenta
coordinates. Magnet array parameters are Am = 1.0 cm. Ba = 250 G. Bso = 157
G, x m = 5.0 cm, and ym = 0.5 cm. Approxim ately 2000 particles are loaded on
to a 400x40 square grid extending from ±5.0 cm in x. and ±0.5 cm along y. The
plots of the beam cross-section are taken at r = 0.0. 0.94. 1.89. and 2.S2 nsec,
which corresponds to 0.0. 5.5. 11.0. 16.5 magnet periods.
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36
2.2.2
3D M agnet Sim ulations o f the O ffset-Pole P C M
The 3-D finite-element m agnetostatic code TOSCA [55] was used to model the
offset-pole PCM and check the analytic expressions for the side fields.. The threedimensional model of the PCM. the 2-D baseplane layouts of the polepieces. and
the x-y mesh from the simulations are shown in Fig. 23. Some difficulty was found
in using periodic boundary conditions at the endplanes (along z) to model an
infinite number of PCM periods. Instead an extra half-period was added to the
end of the ’period of interest*. For small gap-to-period ratios of the PCM stack
(0.5 in this case), including only one extra half-period was found to be sufficient
and simulations showed th at the effect of additional half-periods was slight.
The available version of TOSCA cannot model the perm anent magnet PCM
assumed in the analytic calculation of the side-fields leading to Eqs. 24 and 25. In­
stead, iron-core and air-core electromagnet models were used as an approximation
to a magnet array constructed from high-remanance perm anent magnet materials.
The B-H curve for iron polepieces were taken from the 1010 steel table used in the
POISSON/PANDIRA [56] set of 2-D magnet design codes. Here the array width
and height are x m = 2.0 cm and ym = 0.75 cm. and the m agnet period is Am
= 3.0 cm. Individual magnet cores have dimensions (wxhxt) of 5.0x1.75x0.5 cm.
The inner spacing between cores along r is
2 .0
cm. Each magnet coil is offset 0.75
cm from the array m idplane and has a rectangular cross-section with dimensions
(wxh) of 0.75 cm x 1.25 cm. The inner transverse dimensions of the coil are the
same as the width and thickness of the cores.
The calculated B y side-field from this simulations are plotted in Fig. 24 along
with the analytic results of Eqn. 25. The top figure shows the normalized B y,t
total magnetic field component, which can be visualized as the superposition of
the oscillatory PCM component and the B VtS side-field. The lower plot shows the
only the side-field component, which is found by subtracting the oscillatory PCM
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37
component. Also plotted as the solid curve is the analytic expression of Eq. 25,
which assumes a constant magnetization across the magnet face.
Excellent agreement is obtained for the air-core case, but only modest agree­
ment is obtained for the steel-core electromagnets. The reason for this is due to the
fact th at the within the air-core coil the flux is more evenly distributed for a long,
thin coil than is the case for iron. Hence, the air-core case more closely resembles
the perm anent magnet configuration and the assumptions used in the derivation of
Eq. 25 are better modeled with an air-core m agnet. Since magnet tunabilitv may
be desirable for proof-of-principle laboratory experiments, and the presence of a
high-^i material such as iron greatly enhances the achieved field for a given applied
current, it still may be beneficial to use iron-core arrays. However, it should be
noted th at, recalling the previous discussion, beam matching will be difficult to
achieve without a linear By,s field region to balance the linear space-charge forces
along the x-axis of the beam and magnet array.
2.2.3
Hybrid PCM -Q uadrupole Sheet Beam Focusing
A more flexible focusing scheme for sheet electron beams is a hybrid structure
consisting of a PCM array in the center and canted, periodically-varying side arrays
for beam edge-focusing. This hybrid structure is illustrated in Figs. 25 and 26. The
basic mechanism for edge-focusing is similar to the focusing mechanism in periodic
perm anent quadrupole magnet arrays, as discussed in Sec. 1.4. where a gradient
in the side fields is required to focus against beam space-charge. The flat array of
Fig. 25 in itself does not provide adequate edge-focusing but if the side magnets
axe canted, as shown in Fig. 26 then the gradient required for edge-focusing can
be achieved.
The main advantages of this configuration over the offset-pole configuration of
Fig. 7 are twofold. First, beam matching is more easily achieved in both transverse
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38
planes since adjustm ent of the side field gradient can be m ade through the side
m agnet cant-angle or spacing independently of the central PCM array. The second
advantage is the elimination of the beam tilt problem, which is especially im portant
for relatively ’thick’ beams where the combination of beam tilt and static B x%3
defocusing fields can lead to loss of the beam. In the hybrid array the side-fields
are also periodically-varying so the period-averaged magnet axis remains along the
array midplane.
Assuming the usual linear superposition property of perm anent magnets we
can simply write down the side magnetic fields of the PC M /quadrupole hybrid as:
5 r.i(y .-) = B q{ — ) sin(A:m- + ©,)
(34)
m
By,3( x , z ) = B q{ — )s\n{km: + o3)
(35)
ym
where, as before, km =
where Am is the magnet period. x m and ym are the
PCM magnet array width and height. The quadrupole field at the center of the
canted poles surface is B q and o 3 is the phase difference between the PCM and
quadrupole stacks.
The MAGIC PIC code was again used to study the focusing in the hybrid array
under the paraxial transform ation km: —>ujt/u0. Figure 27 shows an example of
sheet beam evolution in a hybrid array with magnetic focusing fields given by
Eqs. 35, and the PCM fields in Eqs. 32 and 33. In addition a two-period magnetic
field taper is applied to the side-fields to reduce the intitial tilt experienced by
the laminar input beam at the entrance to the array. The tap er is implemented
by increasing the side-field am plitude by Bq/4 over each half-period until the full
m agnitude is reached at the beginning of the third magnet period.
Such field
tapering is commonly done at the entrance of FEL wigglers for the same reason.
Approximately 1000 particles are loaded into a
200
x 50 grid with conducting walls
located at x = ± 2 cm and y = ± 0.25 cm. Here the magnet array has Am = 3
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39
m m , B 0 = 1200 G, B q/ x m = 3500 G /cm , x m = 4.0 cm, and ym = 0.5 cm. As
in Fig. 17 the sheet beam has a m ajor and minor radii of Xb = 1.4 cm and j/6 =
0.025 cm (note Xb/yb = 56) with a total beam current of lb = 10 A and beam
energy of eVb = 10 keV. All other simulation and beam param eters, including the
initial transverse and longitudinal velocity spread, are as in Fig. 17. The beam
cross-section is shown for simulation times corresponding to (top-to-bottom ) r =
0.0, .238, .477, .715, .954. 1.19, and 2.38 nsec. Using the z — t transform ation
discussed previously, these times correspond to 0.0. 4.64. 9.2S. 13.93. 18.57. 23.21,
and 46.42 m agnet periods, respectively.
The particle plots of Fig. 27 dem onstrate that stable, long-term focusing of
high-density sheet electron beams can be achieved in the hybrid magnet array.
Furtherm ore, the beam is more readily m atched in the wide dimension and the
am plitude of radial oscillations is reduced. Another point of note is that the beam
tilt with respect to the midplane has been eliminated.
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40
Chapter 3
Formation o f Sheet Electron
Beam s
Logically, the problems associated with the stable transport of solenoidallv focused
sheet beams have also discouraged the development of electron beam sources capa­
ble of producing large aspect-ratio beams with low em ittance. Few design studies of
thermionic electron guns producing linear sheet-beams have been reported. One of
the difficulties with such approaches is the inherent three-dimensional (3-d) nature
of the problem. Accurate modeling at the em itter edge is particularly troublesome
and would require a 3-d gun design code. The development of such a gun would
likely involve a considerable outlay of tim e and money - neither of which are often
available in sufficient quantity for proof-of-principle experiments.
A recent sheet-beam free-electron laser experiment [57. 39. 5S] used an anode
plate to mask, or scrape, an initially round beam from a field-emission (cold) cath­
ode to form a sheet beam. Besides losing a fraction of the beam, this m ethod
suffers additional disadvantages for low-volt age devices utilizing therm ionic cath­
odes operating in high-vacuum. Gas evolution and the liberation of m etal ions
from the aperture plate can lead to cathode poisoning and degradation. Further­
more, the presence of gas near the beam can lead to the formation of ions which
can degrade the beam focusing and quality. These effects can be further enhanced
by the presence of secondary electrons created at the aperture plate. Finally, al­
though appropriate for low-pressure experiments where up-to-air and pumpdown
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41
cycles axe short, any modification of the sheet-beam size by replacement of the
aperture requires an extended bakeout procedure that may take days to complete
in high-vacuum systems.
In view of the current unavailability of thermionic sheet-beam guns and the lim­
itations with masking, we have developed a practical method of forming elliptical
sheet electron beams with large aspect-ratios using a matching section consisting
of magnetic quadrupoles. Figure 28 illustrates the basic idea with a simple two
quadrupole lattice. The first quadrupole focuses the initially round beam in one
plane and defocus the beam in the other plane. The beam space-charge field con­
tinues to defocus the beam in the drift region between the quadrupoles. and a
second quadrupole with a gradient oriented opposite to the first and an appropri­
ately chosen strength and placement corrects the electron trajectories to produce
a highly-elliptic paraxial output beam. This method and simple variants with ad­
ditional focusing quadrupoles is especially suited for laboratory experim ents since
it is relatively inexpensive and is easily achieved with conventional hardware. In
addition, for adjustable electromagnet quadrupoles. it has the added advantage
of flexibility by allowing variation of beam parameters through adjustm ents of
quadrupole strengths external to the vacuum envelope by varying the currents of
the electromagnet field coils.
This chapter discusses the theoretical and practical aspects of a quadrupole
lens system and is organized in the following manner: first, the single-particle
transport of an electron through an ideal quadrupole pair is discussed and it is
shown th at, without space-charge, it is not possible to obtain highly elliptic parax­
ial output with an initially lam inar round beam. Next, the effect of space-charge
on the beam envelope is investigated. The coupled beam envelope equations are
numerically solved in two-dimensions and demonstrate that the presence of spacecharge helps to correct the slope of the envelope in the focusing plane so that
highly elliptical, paraxial output becomes possible. The model is then modified to
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42
include the effects of finite beam emittance, variations in longitudinal velocity to
third-order, and third-order magnet fringe fields based on a bell-shaped fall-off of
the quadrupole strength with axial distance from the center of the magnet. The
analysis indicated th a t the quadrupole method could produce highly elliptic output
beams with m odest quadrupole gradients, but the issue of degradation of beam
quality (e.g. em ittance growth) through the quadrupole lattice is not addressed.
Moreover, considering the rapid variation of the transverse beam envelope from a
round to a highly elliptic cross-section, the 2 -d paraxial assumptions made in the
previous studies are of limited validity.
The analysis is then extended to three-dimensions through the use of more de­
tailed envelope simulations and particle-in-cell (PIC) simulations. Results indicate
that significant axial self-electric fields are present and can lead to the evolution
of non-uniform space-charge and a subsequent over-focusing of the beam in the
space-charge depleted regions. Both the 3-d envelope and PIC simulations agree
well for the beam envelope. Simulations indicate that a 10 kV. 2 A round beam
with an initial radius of 0.3 cm can be transformed into a sheet beam of dim en­
sions 5.1 x 0.2 cm through the use of four idealized magnetic quadrupoles. The
required field gradients of the quadrupoles are easily achievable (< 60 G /cm ). and
the beam exhibits acceptable increases in em ittance. Simulations with non-ideal
quadrupole fringe field profiles based on iron-free perm anent magnet quadrupoles
(PM Q’s) indicate th at the same-size beam can be obtained through adjustm ent of
the magnetic field gradients to account for the axial fringe field.
3.1
2-D Theory of Electron Transport
Consider the transport of a single electron of charge —e and mass m through a
quadrupole m agnetic field as shown in Fig. 2S. To first order, assuming hyperbolic
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43
m agnet pole faces, the magnetic fields axe given by [59]:
n
B0
B* = — 5 -y
*f>0
(36)
R —
- - S i rI
By
ilfl
Here B a and R a are the quadrupole field and radius at the pole tip. T he quantity
B a/ R 0 is often referred to as the quadrupole gradient.
Substituting these fields into the non-relativistic equations of m otion for the
paxticle gives:
dax
— =
at1
e
e B0
— vzB y = ----- u,0— x
m
m R0
L0
(37)
cPy
-171
ati =
e
m
e B0
= + ~m V:o~B~y
ti0
where the longitudinal velocity is assumed to be approximately constant. Replac­
ing
27
—> vzojz gives the following ray equations describing the particle motion:
„
x
y
Here ka2 (=
<Px
= T
T =
az*
„=
(Py
t
d zzi
e B0 1
d v-0x = ~ k^ x
m R0
2
(38)
e B0 1
= m R
5—
0 vzo y = k° y
is called the quadrupole excitation. Primed quantities denote
derivatives with respect to r. The solutions to these equations represent converg­
ing (in the focusing plane) and diverging (in the defocusing plane) motion of the
particle and are given by [60]:
x(z) =
x '(r)
=
j 0 cos kaz +
sin k0z
K0
—k0x 0sin k0z + x 0'cos kQz
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(39)
44
y{z) =
y0 cosh k0z + yd sinh k0z
y'{z) =
k0y0 sinh kQz +
cosh k0z
k0
(40)
where x0, xd, y0, yd refer to the position and slope of the incident particle.
Consider the case of an electron moving through two identical quadrupole mag­
nets with the same excitation {kx = k2 = ka), length L, and separated by a drift
distance of length d.
The second quadrupole is rotated through an azimuthal
angle of 7r/2 with respect
to the first m agnet in order to
form a "sym m etric
quadrupole pair”. Thus the motion in one plane (the x-z plane) is focusing-driftdefocusing and in the other plane (v-z) the motion is defocusing-drift-focusing.
The quadrupole fields are first-order and assumed constant over L. and the drift
region is assumed to be field-free. If the input trajectory of the electron is assumed
to be paraxial {e.g. x d = 0,yd =
0
) then the slope of the particle as it leaves the
second quadrupole is given by:
x / = fc0 x0[cos k0L sinh k0L —k0d sin k0L sinh kaL —sin kaL cosh k0L\
(41)
yj' = fcDy0[cos k0L sinh k0L — k0ds'm k0L sinh kaL —sin k0L cosh kaL\
If paraxial output from the quadrupole lens system is desired, then the term in
brackets in Eqs. 41 must be zero. Furthermore, we physically require d > 0. which
leads to the condition: tan h (k0L) > taxi(k0L). This condition is strictly only true
when k0L = 0, but is approximately satisfied in the limit k0L —¥ 0 . This thinlens limit is only valid for the case where the transverse displacement of the ray
through the lens system is sufficiently slight such that the force restoring the tra­
jectory to paraxial flow is approximately equal to the original bending force. Thus,
highly elliptic paraxial output flow for a m atched input beam is not possible in the
sym m etric quadrupole lens pair without beam space-charge. Although the algebra
is more involved and not presented here, it may be more generally dem onstrated
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45
th at, without space-charge, highly elliptic paraxial flow is not possible in either the
nonsymmetric pair (fct 7^ k2) or a symmetric quadrupole triplet (ki = k2 = k2/ y j 2 ).
W ith the addition of beam space-charge in the model the situation is con­
siderably altered, and highly elliptical, near-paraxial flow is possible. In the fo­
cusing plane, the relatively thin dimension of the beam induces relatively large
space-charge electric field forces which act to correct the trajectory in the drift
region. Additionally, space-charge pushes out the envelope in the defocused direc­
tion. Careful adjustm ent of the second quadrupole position and gradient allows
for highly elliptic paraxial flow in both planes.
After Lapostolle [61] . the electric self-field for an elliptical beam with constant
charge density (n e ) may be approximated by:
m 2 V
Et {3) = - - U!p
e
X +Y
(42)
F
L y
<*>
-
-
e ^
,
2
^
x
,,
+ y y
Here X and Y correspond to the minor and major radii of the elliptical beam crosssection, respectively, and u.’p( = ne2/e0m) is the electron plasma frequency. In the
limit of a circular beam (A' = Y = Rb0)• we see that Eqs. 43 give the correct form
of the beam self-field. Similarly, in the limit }' » A', the field component in the x
direction approaches that of an infinitely long charged sheet.
In the transformation from round to elliptical cross-section, the total beam
area is not conserved and the charge density is a function of r. Assuming that
the initially uniform-densitv beam remains uniform in the transform ation, we may
rewrite the plasma frequency as a:p2{z) = u;po2 .4*0//M r ) = u,'po2^ p - , where Ab is
the beam cross-sectional area and the subscript (o) refers to that param eter at the
quadrupole entrance. As we are interested in the beam envelope trajectory, we let
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46
x -¥ X and y -* Y and recast Eqs. 43 as:
v (*) _
~
2
e
Rbo2
X +Y
(43)
E (*) by ~
e
a; 2
Rb° 2
X +Y
It is appropriate at this point to include an em ittance term in th e equations
of motion to model the case of an finite em ittance beam. After Lawson [62], the
effect of electron transverse kinetic pressure on the beam envelope may be included
as additional defocusing terms of the form:
V3where
(44)
y'3
and t ny are the normalized beam emittances in the x and y directions.
The envelope equations for the beam with space-charge and finite em ittance now
become:
V"
2 \ r , u'’po
A = -k0 A H
- ..
Y" = k S Y +
,
enx‘
, ^
+ -TXT
(4o)
r + Tk
U;0J A + }
V3
<46>
For ease of calculation, we normalize the ray equations with r n = z / L . X n =
X / X Q = A /Rbo* Yn = Y / Y 0 = Y/Rbo and ^
■ We may now write Eqs. 45
and 46 as:
X„" = - ( W ) 2.Y„ + lk,L f y - l - r r + (
*‘ n
K " = + l k „ i ) 2r , +
i
*n
+ r„
)2( - ^ )2A j
“ 60
*^60
+ (-£-)’(£ !.)’
Itbo
tlb o
(47)
-*-n
*
(48)
In
where primed quantities now denote the derivative with respect to cn and we have
defined kp = 2 tt/Ap =
Equations 47 and 48 are coupled through the space-
charge term and are valid in the limit A'n\ Yn'
1.
To illustrate the effect of space-charge on ellipse formation we numerically
solve the above equations for a zero-emittance laminar beam entering an ideal
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47
nonsymmetric quadrupole pair.
The quadrupoles are separated by a field-free
drift region of length d. Figure 29 shows the trajectory of the beam envelope in
the focusing and defocusing planes. The beam has an initial current density of
30 A /cm 2 and energy of 10 keV. The first and second quadrupoles have gradients
of 492 G /cm and 118 G /cm , respectively.
Each has a length of 1.62 cm and
they axe separated by a drift distance of 0.49 cm. The beam at the end of the
second quadrupole is nearly paraxial in both planes and the ellipticity (Yj / X j ) is
108. This extremely elliptic example may be of limited value for research purposes
since the large aspect ratio would make spatial resolution in the thin dimension
very challenging. Assuming a incident beam of radius R(,0 = 1 mm (corresponding
to an existing commercial gun: Litton: model M707) the resulting beam would
be an ellipse with major radius 1.57 cm and a minor radius of 144 fim. However,
this example does serve to illustrate several important features.
A comparison
in Fig. 29 of the beam trajectories with space-charge to the case where spacecharge is neglected shows dram atic differences, even within the first quadrupole,
dem onstrating that it is a very im portant effect in beams with either large current
density or low longitudinal velocity. In the drift region, space-charge affects the
focused dimension strongly and bends the trajectory back to near-paraxial flow.
In the absence of space-charge the focused ray ends up crossing the axis. and. after
further defocusing by the second quadrupole. a magnified negative beam image is
formed in that plane.
Due to the large changes in transverse position in both planes, there exists a
near decoupling of force effects at the second quadrupole. Since the magnetic field
scales with displacement, the second quadrupole bends the defocused ray back
into paraxiality but has very little effect on the focused ray. Hence, in very elliptic
cases only the first quadrupole gradient, drift length and the space-charge density
strongly affect the trajectory of the focused ray. For a beam of a given energy and
current density, changes in beam ellipticity while maintaining paraxial flow, may
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48
be made through careful adjustm ent of the quadrupole gradients and separation.
3.1.1
Physical Quadrupole M agnetic Fields
So far the quadrupole has been assumed to be ideal. The quadrupole magnetic field
abruptly rises from zero, is constant over a length, L, and then abruptly returns
to zero. Provided this length is recognized as an effective length of the physical
quadrupole, this model is relatively useful for understanding the physics of the
focusing. However, in an experimental design one has to consider realistic profiles
where the effect of the fringe fields is modeled more carefully. After Septier [63],
we choose a bell-shaped field model as shown in Fig. 30.
Other field models
representing experimentally measured profiles may be easily substituted. The bell­
shaped model has a length of constant field from —r 0 to
with a Lorenztian
fringing field on either side. The function describing this field distribution is given
by:
/(-) =
i
for —oc
U + t-^ )2)2
1
for - r 0 < r < r ,
l
for
( i + ( ^ ) 2)2
The constants
+ :0
°
(49)
< : < oc
and 6 prescribe the width of the flat-top and rate of fall-off of
the profile, and, in general, will be determ ined from an experimentally measured
magnetic profile. For the purposes of this paper, we choose 2z 0 = Lj'l and b is
found by constraining equal areas under the curve described by Eq. 49 with the
ideal field profile of length L. This condition gives 6 = L/tt. The two field profiles
are compared in Fig. 30.
In the flat-top region the fields rem ain the ideal linear fields treated earlier.
However, in the fringe region higher-order fields may be necessary in the field
expansion. If the condition .V, Y -C R 0 is strictly satisfied throughout the particle
trajectory, then the first-order fields given by Eq. 37. m odulated by the profile
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49
function in Eq. 49 axe sufficiently accurate. For highly elliptic flow this condition
m ay not be completely satisfied and higher-order field term s m ust be included. To
third order these fields are given by [63]:
Br =
~ G f ( z ) y + - ^ /" ( - ) ( 3 x 2y + y3)
By =
- G t t z ) x + ^ f " ( z ) ( 3 y 2x + x 3)
Bs =
-G f'{z)xy
(50)
where G = B 0 / R 0 is the quadrupole gradient discussed previously. It is straight­
forward to show that these fields satisfy V • B = 0. Note th at B z is considered
third-order since in the equations of motion B : is cross-multiplied with x' or y'.
We also allow for overlap and cancellation in the magnetic field profiles of
the two quadrupoles and the drift region between the quadrupoles is no longer
assumed to be field-free. If the quadrupoles are sufficently close, significant field
cancellation can occur. In addition, we assume that magnetic shielding would be
necessary to limit stray flux at the ends of the quadrupole system from affecting
other components, such as the electron gun and the sheet-beam focusing system.
As a reasonable assumption, we assume that shielding can be designed and placed
such th at the field profile given in Eq. 49 abruptly rises from zero at a position
L /2
from the center of the first magnet. Similarly, on the output end of the pair,
we assume that the field is shielded such that the profile falls back to zero at L/2
from the center of the second magnet. For the field profile considered here, the
field amplitude is only about 40% of the peak, and the magnetic effects of the
tail beyond this point are slight. The model considered here is easily changed to
conform to any particular experimentally measured quadrupole system profile.
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50
3.1.2
Variations in Longitudinal Velocity
We investigate the effect of longitudinal velocity variation for the case where
X ',
1 may no longer be satisfied, but X f2, Y ' 2 axe small enough th a t fourth-
order terms in transverse velocity may be neglected. To a reasonable approxim a­
tion we may neglect space-charge depression and write v z o 2 a; vx 2 + v y 2 -\-v:2. Using
£ = v z± we have:
M ~ ) % ---------- ^ ------—(1 + X '2 + T '2)?
Noting that
(51)
we have for the X equation of motion:
X " = {— )2(X ,X /,(l)+ y ',V ',(1)) A " + ( ^ ^ ) 2—
— H jx B ) r+ ^
(52)
Jzo
To simplify Eq. 52 we note that X " (or V") appear on both sides of the equation.
Hence, the procedure is to factor and substitute so that .V" (or V") only appears on
the left-hand side, and then expand the denominators containing the small term s
X ' and Y ' . In the course of these calculations, use will be made of the expansions:
[ 1 + 1 ( A " J + V"2)]
2
(53)
1
1
^ r = ^ [ l + (A"2 + V'2)]
l’:'
V,0-
(54)
I'zo
Substituting these expressions in Eq. 52. we proceed to expand and multiply, re­
taining only terms to third order in transverse velocity. As a short-cut. we note
th at the first term on the right of Eq. 52 already involves the second-order term s
of X
'2
and X ' Y ' . Hence, we use the superscript (1) on X " and Y " to denote that
we need only consider the first-order components of these terms and the result will
be fully third-order as intended. Assuming that the em ittance term is small, we
can, by inspection of Eq. 52. write down expressions for .Y"(1) and
th at are
accurate to first-order:
A"'“ > = ----{Y'B; - By) + ( ^ ) 2_ i —
mv.0
vz0
A + >
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(55)
51
Y»( D = -----! _ ( 5 r _ x ,Br) + ( ^ ^ ^ ) 2—i —
m v zo
vzo
A+ r
(56)
Combining Eqs. 50 and 52-56.normalizing as in Sec. 3.1 and keeping only terms
to third-order in transverse velocity, we obtain:
A„" = -(fc„i)2[l +
xA„(3Vn2 + A„2)]
+ l ^ r [ l + ( ^ ) 2(2A„'2 + i ; 12+ AVK')]
An “r
^
+ (l-,i)2( ^ ) 2A'„'>;'[/(z)A„ - - ^ /" ( r ) A 'n(3V;2 + A„2)]
2
o
t.- / n
r//
(k 0 L) 21^R b X
nYnYn’Rbof
'( z )\ +, ^~ -;y - -3
"
'A.n
+
(57)
The K ray equation is obtained from Eq. 57 by replacing A'n with K and sub­
stituting k
2
—> —k
2
and enI —> eny. While not explicitly shown in Eq. 57. we
note th at the derivatives / ' and f " contain factors of j and -p- and Eq. 57 is fully
normalized.
The normalization param eter ^
is a measure of the strength of the higher-
order field terms and the fringe field in the quadrupole. This is explicitly seen if
we rewrite this param eter as
Thus, the effect of the fringe field is
small for quadrupoles with a small ratio of radius-to-length. ( ^ -C 1). provided
that the transverse displacement of the envelope is small ( R^ Xn.
< 1). In
highly elliptic beams, one m ust be careful that the quadrupole radius is sufficently
large to justify the third-order magnetic field expansion in Eq. 50.
W ith the above discussion in mind, we further simplify Eq. 57 by dropping
term s that scale as ( ^ “)4. The equation of motion for A'„ becomes:
X n" =
1 , R b o >2/ 0 f t V' f t \
~ ( k 0 L)2[1 + ^ ( - ^ ) 2( 3 .w 2 + K )]/(- )A'n + - ( k 0 L f f " ( =)
( W
Rb
X
A„/?6o2(3V;2 + An2) +
+
^bc
(k0 L) 2 ( ^ ) \ \ \ r n' f ( z ) X n + ( k oL) 2 ^ X
L
An
[1 + ( ^ ) 2(2AY2 + K * + AV K')]
+ v; 1
1 LL
nYnYn'f'(z)
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52
r2
+
2
-D- f4 rv- -3
(58)7
v
This equation (and the associated Yn equation) describe, to third-order, how the
envelope of the beam evolves through the quadrupole lens system . Information
about non-elliptical distortions (aberrations) in the beam cross-section can be
gained from the solution of Eq. 58 for those envelope components which are not
in the x z or yz planes.
An experimental study of quadrupole aberrations [63]
showed small distortions in the beam shape which, in principle, can be corrected
in a careful lens system design. For the present purposes of this paper we examine
only the envelope radius in the focusing plane (xr) and the defocusing plane (yx).
Coupling terms for the ellipse elements in the these planes are zero except for in
the denominator of the space-charge term s, and the ray equations are considerably
simplified to give:
X . " = -(l-„ £ )2[l +
A'„3
A„
T
In
L
rlbo An
(59)
K" = +(i-„£)2[i + | ( ^ ) 2v;,2]/(--)>; - (i.i)2-^ /" (-)v ; 3
. .(j t - )2-.fi
A'n + V ;1
L
+ i 2 tny2
RboAYnZ
(60)
[ ]
Finally, it is acknowledged that the em ittance term was handled in an ap­
proximate manner in the above analysis. Its inclusion illustrates that the effect
of nonzero em ittance on the beam envelope is not negligible, and should be in­
cluded in a careful design. Strictly speaking, however, there is difficulty in relating
the em ittance as defined above with an experimentally-measurable rms em ittance.
Several authors [61, 64] have derived envelope equations for the effective beam
width and em ittance which are easily related to measurable param eters in the lab­
oratory. The derivations of these rms envelope equations require linear focusing
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53
forces (and elliptical beam sym m etry). Hence, the rms equations would not rigor­
ously apply in the case where higher-order field and velocity variation effects are
included in the ray equations. However, in the spirit of the sm all-term corrections
applied in the above equations, an rms approach may only represent a reasonable
error in the analysis.
3.1.3 Elliptical Sheet B eam D esign Case
In order to illustrate the potential of quadrupole formation of elliptical sheet beams,
we give a design example using an existing commercial round-beam electron gun.
Equations 60 and 60 were solved numerically for a 10 keV beam with a current
density of 2 A /cm 2. The resulting beam envelope in the focusing and defocusing
planes is shown in Fig. 31. Assuming a beam radius of 0.6 cm at the quadrupole
entrance, the output beam has a m ajor radius of 3.61 cm, minor radius of 0.16 cm
and an ellipticity of 22.5. The first quadrupole has a gradient of 63.8 G /cm and the
second has a gradient of 17.6 G /cm . A plot of the normalized m agnetic field profile
used in this simulation is shown in Fig. 32. The magnets have a flat-top region
of length 1.41 cm, and the separation between the centers of the the quadrupoles
is 3.1 cm. Assuming a quadrupole radius of 6 cm. the magnetic fields at the pole
tips for the two quadrupoles are a modest 383 G and 106 G.
3.2
3-D Analysis of Electron Transport in the
Quadrupole Beam-Forming System
The results of the previous section indicate that it is possible within 2d paraxial
approximations to construct a simple matching section lattice consisting of two
non-ideal quadrupoles to form an elliptical sheet electron beam. In this section
we now compare the 2d analysis to more complete 3d calculations for an ideal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
m agnetic quadrupole lattice to determ ine the validity of the 2d solution and to
gain insight on the beam em ittance through the matching section.
Equations 45 and 46 were solved for a ( z ) = -Y(z) and b(z) = Y ( z ) by numerical
integration from a specified initial envelope state with radii a and 6 and divergence
angles a' = da/dz and b' = db/dz specified at z = 0 for given beam energy,
current, and emittance. Here, a and 6 are the principal radii of the elliptical
beam envelope which are assumed aligned to the x- and y-coordinate axes. This
integration was carried out under the assumption of preserved em ittance (i.e.,
cx =const and ey =const). Fig. 33 shows one such 2-d solution for beam transport
through a matching section formed from two ideal quadrupoles. The initial and
final beam energy, current, envelope radii, and divergences are given in Table 1,
and the parameters of the magnetic quadrupole lattice are given in Table 2. The
initial em ittance was taken to be zero, consistent with an initially cold paraxial
input beam.
Detailed evaluations of the idealized sheet beam matching section described
above and other alternative designs were carried out by performing simulations
with the WARP code[65]. The WARP code is a 3-d electrostatic and nonrelativistic PIC code that was originally developed for the simulation of space-chargedom inated beams in heavv-ion fusion applications. The code employs a cartesian
mesh th at allows for transverse bends, a capability from which the nam e WARP
is derived. In these simulations all self-magnetic fields are neglected, correspond­
ing to a neglect of self-field term s of order
order term .
~ 0.04 times the retained leading
A so-called semi-Gaussian beam with uniformly distributed space
charge and a Gaussian velocity distribution is injected at the r = 0 plane. The
injected beam has a round K-V type envelope of specified circular cross section
and zero divergence angle (i.e.. a = b. and a' = b' = 0). The Gaussian distributed
transverse x- and y-velocitv spreads are set from the specified beam em ittances
and radii, and the longitudinal velocity spread about the axial velocity specified
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55
by the injected beam energy (£/,) is chosen to be equal to the transverse velocity
spread, corresponding to an equipartitioned beam emerging from the electron gun.
Particle weights are set to obtain the specified beam current. For com putational
efficiency, the simulations are carried out in a steady-state mode where particles
are injected at every tim estep and field-solves for the beam self-electric field are
carried out at specified m ultiples of the particle advance tim estep until the beam
fills the system and settles into a steady, time-invariant state. Such a solution will
correspond to the mid-pulse structure of a long pulse. Also, for com putational
efficiency, the fieldsolve for the self-electric field employs an FFT m ethod on a
rectangular parallelpiped mesh with assumed 4-fold symmetry. In this m ethod,
all particles are gathered in one transverse quadrant of the beam using sym m etry
operations and the fieldsolve is carried out over this single quadrant. This leads
to a 4-fold enhancement of the particle statistics for a given transverse grid size,
thereby allowing use of a higher resolution grid.
WARP simulations were carried out to evaluate the validity of the previous
sheet beam matching section formed from two ideal quadrupole magnets. These
simulations were employed using a 64 x 64 transverse grid in one quadrant of the
beam, 128 axial grid points, and approximately 200.000 particles. Results from
these simulations are presented in Table 1 and Fig. 34. and beam and lattice
parameters axe listed in Tables 1 and 2. The top plot in Fig. 34 shows a crosssection of the beam distribution in the xy plane and the bottom plot is a crosssection in the yz plane. Evidently, the results obtained are substantially different
than those predicted by the ideal 2-d envelope simulations. The reason for this can
be understood as follows. The axial variation of the beam envelope is significant
in this system. This causes the axial self-electric field of the beam to push the
front edge of the beam forward along the axis, thereby causing a local depletion of
space-charge at the front edge of the beam. In turn, this leads to an overfocusing
of the beam by the applied quadrupole field in the in the yz plane and then an
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56
inflection through the y = 0 axis. Instead of recollimating a converging beam, the
second quadrupole causes further divergence. Thus, it appears the neglect of axial
gradients in the usual 2-d envelope formulation invalidates the simple envelope
equations when applied to short sheet beam matching sections with significant
axial variation. In addition to the disagreement in envelope evolution, WARP also
predicts a significant rms x-emittance growth (Table 1.) th at is associated with
the development of space-charge nonuniformity.
Further verification of the importance of 3-d self-field effects was carried out
with the TRACE3-D beam envelope code. TRACE3-D is a 3-d beam envelope
code written for accelerator applications[66]. The code can handle a variety of
different magnets and includes space-charge fields under the assumption that they
are linear and can be represented by a transfer m atrix. This linear space-charge
approximation corresponds to a uniform density profile within transverse cross sec­
tions of the beam. TRACE3-D also includes an optim ization routine which solves
for magnet param eters (i.e.. field strengths, axial lengths and separations) given
a desired beam size and emittance. Moreover, because TRACE3-D is based on
a reduced moment description of beam behavior, it is much faster to run than
the 3-d WARP PIC code, and is therefore appropriate for rapid iteration of de­
sign parameters.
However, approximations inherent in the reduced description
also render the code inappropriate for evaluations of possible degradation of beam
quality (em ittance growth) due to nonlinear self-field forces. In the absence of non­
linear applied fields, the TRACE3-D model will predict constant rms emittance
throughout an ideal quadrupole based matching section. Due to the detailed beam
manipulations in the m atching section, it is possible that the real beam can suffer
significant em ittance growth. Such possible em ittance growth can be evaluated
with the WARP code.
TRACE3-D simulations of the previous sheet beam m atching section formed
from two ideal quadrupole magnets were carried out and compared with results
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57
from the WARP PIC code. Results obtained and beam param eters are presented in
Fig. 35 and in Tables 1 and 2. As expected, TRACE3-D predicts constant beam
em ittance in contrast to th e significant emittance growth predicted by WARP.
However, as might be expected for such a strongly space-charge dom inated beam
[1
(er 2 + ey2)/2fcrb2; where n> is the average beam radius] this growth does not
influence the envelope structure significantly and there is close agreement between
TRACE3-D and WARP for the evolution of the beam envelope. This good agree­
m ent in envelope structure suggests that TRACE3-D contains sufficient physics
for rapid conceptual design work. Then WARP can be employed to check if the
resulting design is acceptable from the standpoint of beam quality.
Motivated by the previous analyses, an alternative sheet beam matching section
was designed using TRACE3-D to iterate a lattice of four ideal quadrupoles lenses
rather than two. This approach reduces the axial gradients of the system, thereby
reducing axial self-field effects and rendering design iterations less complicated.
T he four quadrupole system has the added practical advantage of allowing more
flexible adjustm ent of the final beam parameters by adjusting the gradients of the
electromagnet quadrupoles via the electromagnet currents without repositioning
the quadrupole lattice. The beam envelope obtained by analogous TRACE3-D
simulations of a four quadrupole sheet beam matching system is illustrated in
Fig. 36 . Corresponding beam and lattice parameters are listed in Tables 3 and
4. Note that the final sheet beam state is highly elliptical with an aspect ratio of
a/b ~ 27.
Similar WARP simulations to those carried out for the two quadrupole m atch­
ing section were done for the four quadrupole system to verify the TRACE3-D
design and determine if sufficient beam quality is m aintained. These simulations
employed a 64 x 64 transverse grid in one quadrant of the beam. 256 axial grid
points, and approximately 200.000 particles. Results from these simulations and
beam parameters are presented in Tables 3 and 4 and Figs. 37 and 38. Initial
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58
transverse beam em ittances cx = cy = 9.2ir mm-mrad were chosen to be consistent
with electron gun simulations and existing sources[52]. T he axial velocity spread
of the assumed equipartitioned beam corresponds to 0.15%. The x z and yz planar
cross-sections of the beam obtained by WARP (Fig. 37 ) are in good qualitative
agreement with the corresponding TRACE3-D envelope (Fig. 36 ). In Fig. 38, the
highly elliptical (aspect ratio a/b ~ 24) sheet beam cross section in the xy plane
is presented at the exit plane {z = 18.5 cm) of the m atching section.
The predicted rms em ittance growth of the beam in the x-x' and y-v' phase
spaces are plotted in Figs. 39 and 40. Although WARP predicts significant xem ittance growth (Table 3), the final value is still small enough to be acceptable
for planned sheet beam transport experiments[67]. This em ittance growth is as­
sociated with the development of nonlinear space-charge fields during the beam
evolution.
The rapid oscillation observed in the x-em ittance near r = 15 cm
(Fig. 39) is a consequence of the rms em ittance measure being employed and is
not a true measure of the phase-space area. Nevertheless, it is anticipated that the
distribution distortions will become thermalized at a level near the maximum indi­
cated increase in rms x-em ittance. In contrast, negligible rms y-em ittance growth
is observed in Fig. 40.
In all simulations described above, the quadrupole fields were assumed ideal
with constant gradient inside the effective axial length of the magnet and zero
outside, corresponding to zero axial fringe field. Substantial fringing will occur
in magnets where the ratio of quadrupole radius to axial magnet length (i.e., the
magnet aspect ratio) becomes comparable to unity. For the quadrupoles in the
sheet beam matching sections examined here, this will be the case because the
axial lengths of the quadrupoles are short and the radii will need to be greater
than the largest transverse beam extent to insure that the beam only samples the
linear portion of the field gradient. Thus it is expected th at the axial fringe field of
the magnets will have a significant effect on beam focusing in sheet beam matching
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59
sections.
A fringe-field model based on iron-free perm anent magnet quadrupoles (PM Q ’s)
[68, 69] was used in the TRACE3-D envelope code to examine this issue. Since the
fringe extent is largely based on the magnet aspect ratio, it is expected that the
PMQ model will be qualitatively sim ilar to th at of the electromagnets likely to be
employed. For sheet beam m atching sections it is found that th e effective length
of the quadrupole can be substantially larger than the physical axial length of the
m agnet, so the total quadrupole field gradient. GT( z ). is a sum over contributions
from neighboring magnets. Figure 41 is an example of the beam envelope evolution
through a four-lens system of the same total axial length as in Fig.
36. The
total field gradient is also plotted for reference. The axial length of the individual
magnets have been shortened to 1.9 cm from 2.S cm to reflect the increased effective
length of the magnets. Although adjustm ent of the quadrupole magnet strengths
axe necessary, substantially the sam e final beam param eters as the ideal quadrupole
case can be obtained (compare to Fig. 36). Table 5 summarizes the maximum field
strengths at the axial magnet midpoints, physical lengths, the radii of the clear
bore aperture and magnet assembly, and the axial positions of the permanent
m agnet quadrupoles used in this example.
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60
Chapter 4
Experim ental Investigation o f
Sheet Beam Formation
An experimental investigation of sheet electron beam formation using magnetic
quadrupoles has been designed, constructed, and operated based on the theoret­
ical analysis presented in Chap 3. Section 4.1 describes the main components of
this experiment and features of their design and fabrication. Experimental mea­
surements are presented in Sec. 4.2.
4.1
Description of the Experiment
Based on the theory and results presented in Chap. 3 we have designed an experi­
mental configuration to test the formation and transport of sheet electron beams.
Figure 42 is a schematic of the experiment. The electron beam is formed at the
M690 electron gun and is focused through a magnetic matching section consisting
of three electromagnetic solenoids and a iron flux shield around the gun. Another
flux shield between the solenoids and the quadrupole lattice serves to lim it solenoid
magnetic flux at the lattice. This matching section was necessary to propagate the
beam away from the gun-cathode region in order to accomodate a vacuum valve
in the system to protect the gun. The sheet beam is formed using the magnetic
quadrupole lattice. Vacuum ports at the end of the tube allow for various di­
agnostics to be used to measure beam characteristics. In the following sections
a description of these main components of the experiment - the electron source,
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61
magnetic m atching section, quadm pole lattice, vacuum system, voltage pulsing
circuit, and diagnostics - will be discussed in more detail.
4.1.1
Pierce E lectron B eam Source
The electron beam source used in this experiment is a gridded Pierce gun (model
M690) manufactured by Litton Industries. This gun is capable of being operated
at up to 20 kV and an adjustable beam perveance (/*, = aj,V&. where /*, and Vb
are the beam current and voltage, respectively, and qj is a factor know as the
gun perveance) ranging from 1.S6 - 2.3 jzpervs. The gun perveance. normally a
fixed value in non-gridded guns and depending only on the gun-anode geometry,
may be adjusted through changing the grid-to-cathode potential difference in the
pulsing circuit, as will be discussed further in Sec. 4.1.5. Nominally, the grid was
driven at 200 V with respect to the cathode voltage (at -10 kV) to give a perveance
value of 1.90(± 0.05) /zpervs. Figure 43 shows the measured beam current as a
function of the applied cathode voltage. The solid line indicates a perveance of 2.00
/xpervs and the dashed line is the best-fit curve of l.90(±0.05) /zpervs. Nominal
gun operating param eters are listed in Table 6.
Typically, off-the-shelf electron guns from m anufacturers of microwave tubes
are built to be directly welded on to the vacuum envelope. In this experiment
it was necessary to accomodate an in-line valve between the gun and the rest of
the vacuum tube to allow for up-to-air changes in the vacuum system (including
the insertion of various beam diagnostics) downstream from the electron source.
The valve serves to keep the sensitive cathode in a high-vacuum environment for
the short time the rest of the tube is at atmospheric pressure. A high-vacuum
2.75 inch flange was welded to the anode which, in turn, was bolted to an in-line
metal-seal vacuum valve. The vacuum valve and gun assembly is then bolted to
the rest of the tube.
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62
The dispenser cathode is a BaO impregnated Tungsten m atrix which is heated
to approximately 1150° K by means of a filament seated ju st under the cathode
surface. A custom -built filament heater supply produces the required 6.3 Vac, 5
Aac power to drive the cathode surface to the tem perature necessary for efficient
electron emission. Several interlocks are used to prevent cathode heating in the
case of a vacuum leak and to prevent an instantaneous over-voltage condition when
the cathode supply is first turned on.
Electon emission and trajectory simulations were performed by the manufac­
turer using the DEMEOS electron gun simulation code [52]. These simulations
assume that no external magnetic fields are present at the cathode surface. The
predicted beam radius at the focus is 0.3 cm for a 10 kY. 2A beam. A highquality electron beam is also predicted with a low value of the beam transverse
emittance (e£ = 16 ((x2) ( ( d x / d : ) 2) — {x d x / d z )2) is the rms x-em ittance squared,
with a similar definition for ejj). If we assume an equipartitioned Gaussian beam
emerges from the cathode then the thermal axial velocity spread is given by
8 v:/ v zo
= ex/2rbo = ey/2r(,0 = 0.0048 [70]. These predicted beam parameters are
the same as used in the theoretical investigation of sheet beam formation presented
in Sec. 3.2 (see Figs. 36. 37. and 3S).
4.1.2
Solenoid M agnet M atching Circuit
As has been previously mentioned, an accomodation for a vacuum valve is required
between the gun assembly and the rest of the vacuum tube.
Since the beam
focus occurs only 2.44 cm away from the cathode center (see Table 6) a magnetic
matching section is required to transport the beam away from the cathode and into
the quadrupole lattice. The simplest configuration for achieving this is a matching
section comprised of electromagnetic solenoids with appropriate flux shields to limit
the magnetic field at the cathode and within the quadrupole lattice, as shown in
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63
Fig. 42. The design of this magnetic matching section is discussed in the following
sections.
Design
The solenoid matching section design was based upon several criteria. First, the
cathode must sit in a flux-free, or nearly flux-free, region to preserve the electro­
static gun optics. The field must then rise quickly at the beam focus to a uniform
flat-field region with a field magnitude, B0, which is large enough to provide ade­
quate low-ripple focusing of the beam. The flat-top region of the field profile m ust
also be long enough to allow for the gun vacuum valve, vacuum tube, and sideports for the vacuum pumps and beam diagnostics, and the solenoids should be
space far enough to allow for access to the tube. Field uniformity is also desired to
prevent large magnitude radial beam oscillations. Also, the field must be clamped
off quickly, through means of a flux shield, at the solenoid-quadrupole transition
to eliminate stray flux in the quadrupole lattice. In addition, all of this m ust be
achieved using available solenoids and power supplies.
Assuming no flux threads to the cathode, the best match of the beam into the
solenoid occurs when the magnetic field at the beam focus is equal to l / \ / 2 of the
full field, B 0 [70]. Hence it is im portant that the magnetic field ramp up quickly
at the beam focus. As the axially streaming beam encounters radial magnetic field
components at the field ramp the beam acquires a rotation velocity proportional
to the strength of the axial magnetic field. The field magnitude at which the
magnetic focusing forces exactly cancel the forces due to radial self-electric field and
the rotational motion is known as the Brillouin field. B(,r■ Theoretically, a beam
focused at the Brillouin field undergoes smooth flow without radial oscillations,
or beam ripple. In practice, however, a Brillouin-focused beam is susceptible to
perturbations which upset the force balance and can lead to large beam oscillations
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64
- it is an unstable focusing equilibrium. Generally, low-ripple focusing is achieved
for magnetic fields a factor of 2-5 above Bhr- For a non-relativistic electron beam
the Brillouin field may be written as:
Bbr « j ( —
)(£ )( — )
v 7Tt 0ec /?, rbo
where m, e, and
and Ib and
0
(61)
Z are the electron mass, charge, and normalized drift velocity
are the total beam current and radius. Using the beam param eters
from Table 6 we find th at Bbr ~ 394 G.
The 2-d m agnet design code PANDIRA [71] was used to design the m agnetic
circuit according to above constraints. PANDIRA can model electrom agnet coils
and magnetic m aterials with internal or user-defined B-H curves under the as­
sumption of azim uthal symmetry. The iron flux shields used in this experim ent
were m anufactured from cold-rolled low carbon (1008-1010) steel. The internal
B-H curve for this material was used in the following simulations.
The six available magnet coils have an inner bore radius of 10.16 cm. an outer
radius of 35.56 cm. and a thickness of 3.SI cm. The coils are capable of handling
over 300 A of direct current at field magnitudes of 4 kG and are constructed from
hollow rectangular copper tubing imbedded in a fiberglass mold. Filtered water
flows through the copper tubing at a rate of 8 1/s in order to provide cooling for
the coils. Based on the magnet simulations only three coils are necessary in the
magnetic circuit. These three coils are coupled in series and are driven by means
of a Sorenson DCR40-500A (40 V, 500 A) power supply.
Figure 44 shows the three elecromagnet pancake coils, the gun flux can. the
quadrupole flux shields, and the magnetic field profile predicted by the design code
for one such simulation. The gun flux shield consists of an iron can (i.d. = 5.0 cm,
o.d. = 8.0 cm, L = 16 cm) with a front plate having a thickness of 0.5 cm. The
plate is split into halves to facilitate the m ounting of the electron gun and has an
inner aperature with a 1 cm radius for the gun-anode tubing.
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65
The B 0 / v/2 point occurs at approxim ately 36 cm and the cathode of the M690
gun sits in less than 10 G leakage flux at z = 33 cm. The average peak magnetic
field is 840 G and the field ripple is less than ±4% . If we assume adiabatic electron
dynamics, then a 4% field variation corresponds an acceptable 2% variation in
the average beam radius (£r&/r£, ss 15Ba/ B 0). Using the Brillouin field calculated
previously we find that the average focusing field is over a factor of two above the
Brillouin flow value.
The flux shield between the solenoids and the quadrupole lattice consists of
two separate soft-iron annuli. The inner ring is split into halves to accomodate
the installation of the vacuum tube. The thickness of each plate is 1.3 cm and the
inner aperature radius of 1 cm allows for the beam drift tube. T he amount of flux
leakage into the quadrupole region is less than 20 G.
Field Measurements
Field measurements of the m agnetic field profile were made to verify the design
prediction. Field measurements were made with a Bell 750 Gaussm eter and a
calibrated 10 kG axial Hall probe also manufactured by F.VY. Bell. A 24 Vdc
stepping motor attached to a linear motion drive was used to uniformly sweep
the probe along the axis of the circuit and the field was read directly to a digital
oscilliscope through the use of the voltage outputs on the Gaussm eter. Voltage­
tim e traces were then downloaded to a PC and the drive sweep rate was used to
convert the data into field and distance values.
Figure 45 shows the measured axial magnetic field profile for three different coil
currents. Also shown is the predicted profile for a coil current of 160 A. Note that
the quadrupole flux shield has been pushed away from the solenoids to allow more
room for the vacuum tube than in the design of Fig. 44. However, the PANDIRA
predicted field is in good agreement with the measured field. Some variation near
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66
the gun and quadrupole flux shields is attributable to manufacturing differences
between the actual soft-iron used and that measured for the internal permeability
table of the PANDIRA code.
The measured d ata indicates that a 1500 G m agnetic focusing field can be
achieved for coil currrents of 200 A. This corresponds to a factor of 3.75 times
the Brillouin field and adequate beam focusing is expected.
The approximate
Gauss/amp calibration factor is 7.5 G /A ± 0.1 G /A .
4.1.3
Q uadrupole Lens Array
Contained in this section are a detailed description of the design, construction, and
testing of the quadrupole lattice used to transform the round electron beam into
an elliptical sheet beam as described in Sec. 3.
Magnet Design
As with the solenoid matching circuit the PANDIRA magnetic design code was
used to model th e iron-core quadrupoles in the sheet beam forming system. The
type of iron used in the quadrupole construction was selected to conform closely to
the internal soft-iron B-H table of PANDIRA. Figure 46 shows the cross-section
of one of the quadrupoles.
Each magnet consists of four hyperbolic polefaces
surrounded by a rectangular current-carrying coil.
We can take advantage of the octal symmetry of each magnet by modeling
just one-half of each pole (splitting the pole through the center) and by using the
appropriate boundary conditions at each symmetry plane. Figure 47 shows the
base model used in the PANDIRA simulations including the outline of 1/8 th of
the quadrupole cross-section. The coil outline and the calculated magnetic field
gradient are also plotted. Assuming 26 AWG wire is used in the coil (dia.
=
0.0159 in.) and 80 turns/coil then only 3.5 A is necessary to obtain a quadrupole
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67
field gradient of 40 G /cm . Hence only 420 am p-turns of current are needed to
drive the quadrupoles to the required 60 G /cm gradient (see Sec. 3.2) for sheet
beam formation. As can be seen in Fig. 48 the iron poles are very fax away from
saturation and linear behavior of the m aterial can be expected.
Based on the previous modeling and the 3-d beam modeling of Sec. 3.2 we have
constructed a four quadrupole m atching section with the specifications shown in
Table 7. The length of the quadrupoles has been shortened to 1.905 cm to reflect
the longer effective magnet length including the fringe fields. The core of each
m agnet is constructed of annealed 1008-1010 steel and the hyperbolic polefaces
were cut with computer-controlled electric discharge machining for precise toler­
ances. Each quadrupole has four coils with 140 turns of 26 AWG high-tem perature
magnet wire. The required field gradient of 60 G /cm can be achieved using less
than 5 A of current which is provided by low-voltage dc supplies. At equilibrium
the maximum coil tem perature is 150° C and within the specifications of the wire
without auxiliary cooling.
Field Measurements
Field measurements of the quadrupole magnets were made to verify the field struc­
ture before mounting them to the rest of the system. Measurements were taken
with both transverse and axial Hall probes which were swept using a linear motion
drive. D ata was recorded on a digital oscilliscope and downloaded to a PC.
Figure 49 shows the measured tranverse field Br as a function of the axial length
for several different tranverse positions along the t/-axis. The current through each
coil is 1.0 A in this case and the quadrupole m agnet coils are hooked in series for
this measurement to ensure that the current per magnet is uniform. Also shown
are the outlines of the iron quadrupole cores so the axial position of the magnets
can be seen for comparison. Figure 50 shows a similar measurement for a magnet
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68
current of 1.5 A.
Based on the quadrupole field profile measurements at 1 A, 1.5 A, and 0.5 A
(not shown here) it is possible to determine the average quadrupole gradient profile
for the case when the lattice magnets axe driven with the same current. Figures 51
and 52 show the result of dividing the measured magnet profile by the position of
the probe for the 1.0 A and 1.5 A cases. Consistent quadrupole gradients axe seen
for the y = 1.91 cm and 3.82 cm positions for each current. This indicates th a t
the magnets are indeed not near saturation. The smaller gradient found for the
5.72 cm transverse position is attributable to the expected roll-off of the field for
large tranverse positions (see Fig. 47).
Figure 53 shows the measurement of the quadrupole fields for the case when
the magnets are driven with currents th at give a close m atch to the gradient profile
used in the theoretical model of Fig. 41. The current values and axial positions
of the magnets are listed in Table 8. The gradient profile is again determ ined by
dividing the measured field by the transverse position and is shown in Fig. 54. Also
plotted in Fig. 54 is the theoretical gradient for a lattice of permanent magnets
having the properties in Table 5 for comparison. Good agreement is obtained
between the permanent magnet model and the actual measured field of the ironcore electromagnet quadrupoles except near the fringe fields on either end of the
lattice. The reason for the slight disagreement at the ends is attributable to the fact
that the flux within the iron is not uniform while the perm anent magnet model
assumes a uniform m agnetization across the magnet m aterial in the derivation.
Hence, since the ’source* of the field lines is stronger near the magnet edge in the
permanent magnet model, a slower fall-off can be expected.
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69
Solenoid-Quadrupole Interface
The interface between the solenoidal matching section and the quadrupole lattice
is im portant for several reasons. First, enough overlap between the end of the
solenoidal field profile and the beginning of the quadrupole profile must exist in
order to transport the electron beam into the quadrupole lattice. On the other
hand, too much overlap will interfere with the transverse quadrupole fields and the
beam dynamics in the quadrupole lattice. The design of a flux shield between the
solenoids and quadrupole lattice has already been discussed in Sec. 4.1.2. However,
this shield was designed without reference to its effect on the quadrupole lattice
since this would require a fully 3-d design code and very substantial com puter
resources.
Therefore, as an approxim ation to the expected field overlap, we compare the
two profiles in Fig. 55. Subsequent transport experiments indicated that the inner
pole-piece of Fig. 47 was too thick (0.1S75r ) and this was replaced by a high
perm eability/i-m etal with a thickness of 0.060*.
Another difficulty is the reduction of effective quadrupole flux due to the pres­
ence of the flux shield. T he quadrupole gradient was measured in the presence of
the flux shield in order to gauge the reduction of quadrupole tail-field. and these
measurements are shown in Figs. 56 and 57. The normalized field gradient in the
presence of the plate (dashed line) is compared to the case without iron (solid line).
As can be seen, the gradient is reduced by nearly a factor of two at the flux shield.
4.1.4
Vacuum System
The vacuum system used in the sheet beam forming experiment is described in
detail in this Section. Some of the details from the mounting of the electron gun
have been presented in Sec. 4.1.1 and a schematic of the system is shown in Fig. 42.
The entire vacuum envelope is comprised of stainless steel (type 304) tubing and
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70
flanges which are m ated together with m etal (copper) gaskets. A novel feature of
the system is the use of rectangular tubing. The inner dimensions of the rectangular
tubing are 10.16 cm x 2.54 cm - sufficient for expansion of the sheet electron beam.
A custom rectangular flange at the end of the tube has three 3.34 cm mini-flanges
for the insertion of various beam diagnostics to be described later.
The entire system is evacuated by a 120 1/s turbo-molecular pump backed by
a oil-free diaphragm pump. The absence of oil in the pump assembly is a desired
feature to prevent backstreaming into the rest of the system. The pump assembly is
separated from the rest of the tube by a pneum atic gate valve which is interlocked
to close the valve and protect the electron gun if a power failure occurs. Nominal
base pressures of 1.0e-9 Torr are regularly achieved in the system. Pressures of
o.e-8 Torr are typical while the cathode is hot and the beam is being pulsed into
the tube.
4.1.5
Electron Beam Pulsing Circuit
The cathode voltage pulse is supplied by a Velonex 570 hard-tube pulser. While
this supply is rated to provide 10 kV. 2 A of output in practice operation is
typically limited to around 1.5 A. Hence the gun is typically operated around 8
kV (see Fig. 43) with a maximum pulse length of 100 n sec.
The grid is biased simultaneously with the gun cathode by means of the voltage
dividing circuit shown schematically in Fig. 58. It was previously discussed that
approxim ately a 200 V bias is required at the grid with respect to the cathode
potential to achieve the 2 /zperv nominal gun perveance. The values of R 2 and Ri
used in this circuit are 6 kfl and 250 kQ, respectively. A slightly higher value of R 2
is required to overcome grid charge-up problems. In addition, a shunt capacitance
across the cathode-grid (Cgk = 200 nF) is used to damp out reflexive oscillations
between the cathode and grid.
As shown in Fig. 43 a final gun perveance of
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71
1.90(±0.05) ^pervs is achieved with this pulsing circuit.
4.1.6
D iagnostics
Beam Current and Voltage Measurements
The total electron beam current and the beam current deposited downstream on a
collector were monitored by Rogowski coils manufactured by Pearson Electronics.
Both current transformers were calibrated for currents up to 1 A by using a square
voltage pulse across a precision 50
resistor. Each were determ ined to be accurate
to ± 1 % for the range of current across a 100 \is pulse width.
The beam voltage (the voltage applied to the cathode) was measured with a
Tektronix high frequency resistive voltage probe. The probe was calibrated to
within 0.1 % using a 100 V. 100 fis voltage pulse and was checked against two
other Tektronix probes to 10 kV for accuracy.
Beam Profile Measurments
The beam cross section was measured using Cerium-doped quartz glass scintilla­
tors. Past experience has shown that accurate beam images are created with a
minimum of blooming and diffusiveness. In this respect they are far superior to
phosphors in the beam power density range of interest here.
The 7.5x1.27x0.4 cm ( wx h x t ) rectangular glass plates were coated on one
side with a thin, plasma-deposited molybdenum metal layer. The purpose of the
molybdenum is twofold: first to bleed beam current from the glass surface to an
external lead and. second, to block out back-light from the hot cathode surface in
order to give a clear bluish beam image. Molybdenum was chosen as the surface
m aterial since thermionic cathodes are relatively unaffected in the event molybde­
num atoms flow backstream and deposit on the cathode surface.
The thickness of the molybdenum layer must be chosen carefully to be thick
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72
enough to allow for efficient charge bleeding and for good lifetime over a large
num ber of beam current pulses. Yet, it m ust be sufficiently thin th a t some electrons
penetrate to the Cerium-doped glass below. The surface thickness of 250 ±30A
was chosen to satisfy both of these criteria and is based on particle stopping power
calculations. This thickness was verified using profilometer m easurem ents.
One difficulty encountered in the actual measurement of collected current on
the molybdenum-covered glass sam ple is a reduction in the apparent current due
to reflected primary and secondary electrons. From [72] we find th a t the secondary
electron yield coefficient (secondary electrons per primary electron) ranges from
0.35 - 0.25 from 5 - 1 0 kV for molybdenum. Similarly, the reflected primary
electron yield (reflected primary electrons per primary electron) ranges from 0.36
- 0.38 from 5 - 1 0 kV. Since the energy of the secondary electrons is low - in
the range of a few hundred volts - we expect the brunt of the current loss to the
vacuum walls will be carried off by the reflected primary electrons. A calibration
of the amount of current reduction for the case when the entire beam is hitting the
glass samples shows that a calibration factor of Ic sr 0.66/6 accurately accounts for
the apparent current loss from 5 - 9 kV.
4.2
Experimental Results
In this section we discuss the experim ental results of sheet beam form ation through
the quadrupole lattice. Beam images were photographed and Figure 59 shows a
representation of the beam image of a quadrupole-formed sheet electron beam at
z = 10.6 cm from the outside edge of the beginning of the rectangular vacuum
tubing. Using this point as our reference the axial positions of the quadrupole
cores are at z \ 1 = 1.0 cm. : 2l = 2.9 cm, r i 2 = 5.3 cm. z 2 2 = 7.2 cm.
z 2 = 11.5 cm,
= 9.6 cm,
= 13.9 cm. and z2A = 15.S cm, where the superscript corresponds
to the quadrupole number and the subscript 1.2 correspond to the first and last
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73
edge as z increases. The quadrupole-side of the flux shield is at a position z = -1.0
cm. Thus, the beam images are taken approximately midway through the third
quadrupole. The beam has V& = 4.06 kV, h = 547 mA and the collected current is
approxim ately Ic = 421 mA. The quadrupole currents are Iql = -1.6 A. I q2 = 1.8
A, 7,3 = -4.2 A. and 7,4 = 0.0 A and the quadrupole separations are as listed in
Table 8. The solenoid field in this case is Bs = 1160 g. Figure 60 shows the beam
voltage, em itted current, and collected current for this case. An obvious feature
of this beam is that the beam midplane is tilted with respect to the glass plate
and, hence, with the vacuum tube and quadrupole midplanes. Nevertheless, an
elliptical sheet electron beam with a » 4 ± 1.0 cm and b % 0.4 ± 0 .1 cm has been
produced.
As noted in Sec. 4.1.3 the flux shield between the quadrupole lattice and the
solenoid m atching magnets was reduced to 0.060 in. from the original value of
0.187 in. This was done in order to increase the overlap between the quadupole
field and the fringing field of the solenoids. Care must be taken to balance this
condition with the need to limit the amount of leakage flux into the quadrupole
lattice since the leakage field can be of the same m agnitude as the transverse
quadrupolar fields. Several different thicknesses of the quadrupole flux shield were
used in an effort to balance these two requirements but. unfortunately, the tilted
4 cm x 0.4 cm beam is the best case.
The main problems with solenoidal leakage flux is two-fold.
causes a beam rotation due to
v± x
First of all, it
B zs. The magnetic quadrupoles themselves
have a B z field component but the effect of this component on the beam dynamics
is small, for paraxial beams, since the integral:
/•Am/2
/
Jo
vLBz(z)d=
is approximately zero by the sym m etry of the quadrupolar B z.
(62)
However, the
introduction of a static, unidirectional solenoidal component leads to B z effects
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74
which to not integrate to zero over the lattice length. The added field causes the
beam to rotate. This is a severe problem for a sheet beam forming system as
proposed here since the rotation leads to beam loss along the thin dimension of
the vacuum tubing. The second problem is the loss of predictive capability since
many of the codes used here can only include a uniform B z component if at all,
and not the realistic tapering evident in this experiment.
An illustration of the effect of solenoid leakage flux on the beam dynamics can
be made using the WARP code with a uniform Bz field all across the length of
the lattice. Figure 61 shows the beam focusing with B z = 100 g for the same
quadrupole and beam param eters as used in Tables 3 and 4 and Figs. 37 and 38.
Figure 62 shows the beam cross section at r = IS.5 cm. R ather dram atic differences
in the beam focusing are evident, as is the tilt of the beam in Fig. 62. Similar runs
with Bz = 50 g and B z = 10 g show a decreasing amount of beam distortion and
tilt as the field is decreased.
Although these experim ents dem onstrate proof-of-principle generation of large
aspect ratio elliptical sheet electron beams with the quadrupole beam forming
method, the use of the solenoid m atching section severely affected the results. One
way around this is to elim inate the solenoidal section altogether and to place the
rectangular vacuum tubing and quadrupole lattice near the electron gun. This
implies the removal of the vacuum valve protecting the gun. Provided that the
up-to-air cycles are of short duration and that the gun is opened to a dry nitrogen
or similar water-free gas the gun can be reactivated and used. This configuration
would more closely resemble the use of the quadrupole forming m ethod in an actual
device.
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75
Chapter 5
Periodic Perm anent M agnet
Quadrupole Focusing of R ound
Electron Beam s
A potential focusing geometry for high-perveance round electron beams involves
the use of permanent quadrupole magnets in a close-packed alternating gradient
array as shown in Fig. 5. As we have seen, such an array, utilizing the 'strong'
focusing components of the quadrupolar field acting on the beam drift velocity,
can provide an enhanced focusing force of
2 -1 0
times the force of conventional
PPM arrays for voltages from 10-40 kV and field gradients of 2-4 kG /cm . Further
enhancement is realized as the beam voltage is increased.
The cross-section of a PPQM array shown in Fig. 63 serves to illustrate the basic
focusing mechanism. A round electron beam propagating along the c-axis with a
drift velocity vi0 is focused inward in the x —z plane by the v:o x B y Lorentz term.
In the other transverse plane the beam is defocused during the first half-period of
the array. The field polarity and the direction of the focusing reverses during the
next half-period of the magnet stack and, provided the magnetic field gradient is
strong enough, then a net inward focusing counteracting the beam space-charge is
achieved.
In this chapter we examine the problem of a round electron beam focused in
a PPQ M array through the use of analytical methods, beam envelope equations,
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76
and numerical simulations. In Sec. 5.3 scaling considerations for the design of a
close-packed PPQ M array are discussed and scaling param eters are identified for
magnet designs. Where appropriate, comment is made on the relative m erits of
PPQM vs. conventional PPM focusing.
5.1
Analytical Study of PPQ M Focusing
As previously discussed, the basic focusing mechanism in the PPQM array is pro­
vided through the vz x B± Lorentz force term . The magnetic field of such an array
can be written as [73]:
Br{y. -)
=
| y + ^ - ( 3 x 2 i/-f t/3) | cos(A:mr)
B y{x.z)
=
| x + ^ - ( 3 y 2x + r 3) | cos(A:mr)
£ ,( x ,y ,r )
=
(63)
km— x ys\ n( kmz)
r„
Here, as before, we have km = 2 » / \ m where Am is the periodicity of the magnet
array, and B q is the magnetic field at the quadrupole radius. r ?. Note that these
equations are equivalent to the expansion in Eq. 50 with the axial profile function
f ( z ) = cos (A:m;) and that V • B =
0
is satisfied.
For an elliptical beam with m ajor and minor radii a and b we have for the
space-charge field (Eq. 43):
E
t
(
x
)
=
e
m
Ey ( y) = —
e
u.'p e 2 — ^ —
x
a+ o
2 a
— ~ry
a+b
(64)
where wpe = ne 2 / c0m is the electron plasma frequency, m and —e are the electon
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77
mass and charge, and n is the beam electron density. Provided th at the beam en­
velope excursions are small (a «
6)
then the above equations can be approxim ated
by the linear expressions:
Ex(x) = ~ ^ u j pe2x
(65)
E y {y) = “
(66)
Wpe2y
Following the development of Sec. 3.1 we drop the non-linear field term s in
Eqs. 64 and use Eqs. 65 in the x. y electron equations of motion to obtain:
at*
+ — vg cos (kmz)x - l~u,'pe2x = 0
rq
1
( Px
1
ujc
2
u. cos {kmz)x - -u :pe x =
rq
1
—
at*
(67)
0
As in Sec. 3.1 and following the notation in Ref. [73] we make the paraxial ap­
proximation,
—y vz-jz —*
where q = kmz. We may now write:
fx
dq2
+
COS
7 -7
+
cos (kmz)y - ^ z ^ y =
. . .
1
[kmz)x - 7 - 7
7 -r
= 0
(6 8 )
0
If we define A = us^/'luZm2 and B = (u;c/ r 7 )r-/u ,-m 2 then the above equations may
be recast as:
+ B cos(q)x — Ax = 0
dq*
(py
— - B cos{q)y - Ay = 0
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(69)
78
Now, letting q = 2u these equations can be explicitly written in the form of
specialized Matthieu-Hill differential equations [73]:
cPx
-p— + 4B cos(2tt)x —4 A x = 0
au 2
(70)
< py
—— —4B cos(2u)y —4Ay = 0
au 2
The general form of the specialized M atthieu-Hill equation is:
^
+ (a + 2 /? c o s 2 7 > = 0
(71)
Solutions to this equation are oscillatory with bounded (stable) or unbounded,
exponentially-growing (unstable) envelopes depending on the coefficients a and
0. Figure 64 shows the position of the first instability band as a function of the
param eters a and 0. Stable solutions exist for a . 0 below and to the left of the
lower dashed line. Similarly, stable solutions exist above the upper dashed curve.
Between the two dashed lines the solutions are unbounded and correspond to
unstable electron flow. We also note that higher order regions of stable flow occur
for larger values of a.
3
than plotted in Fig. 64. but the stable solutions in these
regions have large-amplitude envelope oscillations and are therefore im practical for
use in a real system.
Comparison of Eqs. 70 and 71 lead to the identifications o = 4.4 and
3
=
2
B.
To good accuracy the edge of the first instability band in Fig. 64 is given by
a = 1 —1.140. W ithout space-charge the edge of the instability band is determ ined
by finding the intersection of the band edge with the 'beam ' line a = 0. This occurs
for 0 S =
0 .8 8 .
or in terms of beam and quadrupole parameters:
B. = |
—
= {U7 / ^
rCm
21
V
= 0.44
(72)
The meaning of the above equation becomes clear if we recognize that (uic/rf,)vz =
k 0 2 v z 2 and note th at k 0 is the quadrupole strength param eter defined in Sec. 3.1,
k 02 = e B q/ ( m r qv: ). In the thin lens approximation, the focal length, f q. of a single
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79
m agnetic quadrupole with strength ka is given by 1 //, = k 0 2 lej f . Here lcj j is the
effective length of the quadrupole defined by:
f Xm/2 B 2 (z)dz = B 2 leff
(73)
Jo
For the sinusoidal magnet profile used here, the effective length of a single quadrupole
is Am/4. Substituting the focal length into Eq. 72 gives:
B’ = T h = T
T r - s* 0-1¥Jq = OAA
47T2 Jqlcff
<7 4 >
Recalling that the focal length is defined from the center of the lens (the principal
planes coincide by sym m etry) as shown in Fig. 65. then we see that the first
unstable bandoccurs for / , ss Am/4, e.g. the magnet focusingcauses
through the center axis
in asingle half-period.
a deflection
Therefore, if the electron was
initially focused in the first half-period as shown in Fig. 65. then it enters the next
half-period with a diverging trajectory which diverges further due to the defocusing
action of the second quadrupole. For an array of alternating-gradient quadrupoles
as in the PPQM geometry this means the beam envelope will continue to grow
with successive periods and eventually be lost to the transport channel walls.
We now consider the solutions to Eq. 70 with space-charge. A ^ 0. The beam
line in this case is given by a =
= (!2 A / B ) 3 . as shown in Fig. 64. Solving for
the intersection of the beam line and the first instability band gives:
1 - 4.4
■= " W
l?5)
Recalling the definition of .4 we have that:
4.4 =
2-
± f _ = 2^ !
”*m ^z
(76)
As an example, for a beam with A = 0.27 A. Vj = 10 kV. r /, = 0.1 cm. and with
Am = 2 cm, we have that Am/Ap = 0.3 and the edge of the first stopband now
occurs at B, = 0.36. Hence, the critical value of magnetic field gradient at which
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80
instability occurs has been reduced to 0.64C?,,, where Ga is the critical field gradient
w ithout space-charge.
It is instructive at this point to examine the envelope equation analogous to
Eq. 70 for PPM focusing. From Ref. [51] we have:
S
+ 5 ^ ( 1 + c o s 2 u ) r ~ 5 u'> V =
(77)
0
where the linearized space-charge electric field approximation is again used. Mak­
ing the definition of .4 as before and B' =
/u : 2 where
uji
= u.’c/2 is the electron
Larmor frequency, we can rewrite the above equation as:
dPr
-r—r + B
du1
(1
+ cos 2 u )r — .4r = 0
(78)
W ithout space-charge we have a = B r and 3 = a / 2 in the Matthieu's equation map
of Fig. 64 (the solid line representing a = 33). and the first instability band occurs
at a = 0.66. Including space-charge we have for the beam line a = B —.4 = 23 —.4.
Solving for the critical value of
3
we find that 20c = B's = 0.66(1 + .4).
So, in general, the addition of space-charge in PPM focusing leads to the intu­
itive result that a higher PPM focusing field can be used without driving the beam
into instability. Since the magnetic focusing force is azimuthally symmetric and.
on a period-averaged basis, oriented inward, the increased magnetic field pressure
serves to counteract the outward pressure on the beam envelope due to spacecharge. This physics is incorporated into Eq. 77 as the constant ^ 3 " m ultiplying
unity in the middle term. Upon comparison we see that there is no analogous
term in the envelope equations for PPQM focusing. This is a reflection of the
difference between PPM and PPQM focusing. The bi-planar. non-svrnmetric na­
ture of PPQ M focusing (focusing-defocusing in each transverse plane) means th at
the beam envelope cannot be constricted simultaneously in both transverse planes.
If the field gradient is increased to attem pt to counteract space-charge the beam
envelope expands even more in one plane - explaining why the gradient m ust be
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81
decreased to avoid beam instability as the space-charge density increases. This
point will be explored further by numerically solving the beam envelope equations
in the following sections.
Focusing systems are typically designed to operate well away from the stability
edge in order to avoid problems with the extra defocusing forces due to spacecharge, finite beam em ittance, longitudinal velocity variations, and magnet errors
in a real system. Thus the required reduction of the field gradient for finite spacecharge in PPQM system s is not necessarily a severe liability.
Away from the
stability edge the potential exists for beam focusing with either smaller size/weight
magnet packages than equivalent PPM systems or for the focusing of beams in highperveance regime for which PPM systems are inadequate. Both properties become
especially im portant as the trend towards high power mi Hi meter-wave devices in
smaller packages continues.
It is desirable to com pare the required field gradient of the PPQM system to
the field am plitude in PPM focusing to transport a given electron beam. One way
to accomplish this is assum e that each focusing array is the same 'distance' away
in param eter space from the stability edge of Fig. 64. Then we can take a direct
ratio of the critical
3
values, neglecting space-charge. for our test beam:
3mom.
'ppqm |
vO.SS
. u u ___ ^
x r lcr,‘ ■ 0 3 ~ ~
where
k i
=
=
ujc / 2 v z .
For comparison purposes we assume both focusing
systems have the same period and we can cancel out J 2. Thus, for a test beam
a similar distance away from the stability edge in each focusing system, we have
that ko / k i 2 « 1/3. Using the definitions of k 0 and k i we find:
G[glcm\ ~ 0.025-y^=L
Figure
66
(SO)
shows a plot of the quadrupole field gradient as a function of the equiv­
alent PPM field strength. The meaning of this plot can be understood through
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82
an example by considering the curve for a 10 kV electron beam. If, at a partic­
ular magnet period, the beam is focused at a 2000 g on-axis PPM field strength
then the equivalent quadrupole field gradient is only 1000 g/cm . Since perm anent
magnet quadrupoles can easily generate field gradients to 4000 g/cm while PPM
stacks axe usually limited to less than 4000 g the possibility exists that the relaxed
field requirements of quadrupole focusing may allow for a smaller overall m agnet
assembly. We also note that the potential benefits to be gained are enhanced as
the beam voltage increases.
5.2
Coupled-Mode Analytic Theory of Beam Os­
cillation
It is instructive at this point to consider the approxim ate analytic solutions to
Eqs. 69 as developed in Ref. [74]. Here it is postulated that the radial beam motion
can be expanded for small oscillations into coupled oscillation modes according to:
Vr(9. z) = ^ 2 \ 'rm( : ) e x p ( - i 2 ni 6 )
(81)
m
S r(9. r) = ^ 5rm( r ) exp( —i'lmO)
m
where the sum over m is taken from zero to cc. 5 r represents the radial displace­
ment and ly is the normalized radial velocity of the beam.Furthermore,
the radial
beam oscillations lead to a space-charge electric field which may be decomposed
into a static (m =
0
) term and an oscillatory contribution (for m ^
£ ro
=
Erm =
0
):
- r p - ( l - S ro)
2 c0
~ r^ ~
~^-o
exp(—i 2 m 0 )
m
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(82)
83
Substitution of these equations into the beam envelope equations of motion
(Eqs. 69) leads to a solution for the dom inant radial motion of the beam (details of
this analysis may be found in Ref. [74]). It is found that the S r0 and S r2 oscillations
dom inate the electron motion, where S ro corresponds to an azim uthally symmetric
breathing of the beam radius and Sr2 is the elliptical oscillatory motion with period
Am due to the quadrupole focusing field. The S r 2 motion can be solved for:
k 2
Sr 2 = *'Z K, jji 2° rZp
. 2 COS k™Z
(83)
A glance at the resonant denom inator shows that for Am s: v/2Ap then largeam plitude beam oscillation can be expected. The meaning of this is straightfor­
ward. If the magnet period increases (or, conversely, if the space-charge density
increases) such that Am approaches Ap then the effect of space-charge is to kick
the beam envelope outward over successive magnet periods. However, this is a
well-known phenomenon in conventional PPM systems and typically a focusing
structure is designed for which Am < qAp. where a is of the order of 1/3.
Using Eq. 83 the equation of motion for the Sro radial component is found:
z. 2
,72 c
dz 2
+
-£ -5
2
-u- 2
/. 2
=2 L ------------ ^ ----- _ [ l + cos(2A-mr)]
2
2 ( 2 km 2 - k p2y
J
(84)
The solution to this equation consists of a homogeneous part (oscillating at the
magnet scale-length. Am) and a inhomogeneous contribution which consists of a
static term and a term which oscillates at half the magnet period:
S r
2nh
r%' h
=
, ------ r - C0S(2k „ z ) +
----------- , -------------------
2(2Am2 - V ) ( S k j - k „ 2)
”
1 -------------- ^
--------
2 (2 k „ 2 - k p2)
As pointed out by VVessel-berg [74] an equilibrium point for
5 r2
( 85 )
occurs when:
( ^ 7 ) =
(86)
Substitution of this into Eq. 85 leads to the equilibrium solution for the inhomo­
geneous part of Srok
Sro =
2
2
oACm
2
cos( 2 kmz)
Kp
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(87)
84
Suppose th at the equilibrium condition Eq.
86
is not m et. but rather the left-
hand side of the equation is equal to orfcp2, where a is some arbitrary constant.
Noting that a =
when a ^
1.
1
gives us the equilibrium solution of Eq. 87, we examine the case
Then equation Eq. 85 becomes:
ak
2
Sr2 = Yul 2 p L 2 x cos( 2 kmz) + 1 - Q
(oKm
(88)
Kp )
Averaging this over magnet periods leads to the simple asymptotic expression
(Sri) = l —o:. Hence, for a focusing strength k 0 2 greater than the critical value
defined in Eq.
86
we have that a > 1 and the average beam radius is compressed
from the initial value ((.Sr2) <
0
). Conversely, an under focused beam with a <
1
leads to a steady-state beam with a radius larger than the initial value.
However intuitive this situation is (larger focusing strength implies smaller
beam radius) it must be kept in mind that the equilibrium condition of Eq.
86
and
the equilibrium solution Eq. S7 must be compared to the other ripple component
S r2 - In view of the previous discussion regarding the S ri ripple component and the
edge of the stability band, consider the scenario for worst-case (largest) Sro ripple
when k 02 = QA4k m 2 % km2/2 near the instability edge. Since, also at worst-case,
we require k m 2 / 2 ~ k p 2 or larger, we have k
2
% k m 2 / 2 ss kp2. A comparison of
the two amplitude term s of Eq. S3 and Eq. S7 leads to the conclusion that the Sr 2
amplitude dominates by the factor 7/2 = 3.5 for this case. Elsewhere. S r 2 is even
larger in magnitude than S ro■ Consequently, we can restrict evaluation of beam
confinement criteria to a discussion of the asym metric ripple of 5r2.
Following Ref. [74] we take Eq.
86
and the m agnitude of 5 r2 to obtain the
following conditions:
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85
where e = |Sr2 |- Recalling previous discussions we see th at the first equation
represents a condition on the focusing strength necessary for an equilibrium radius
equal to the initial beam radius. Focusing strengths above the equilibrium value in
Eq. 89 imply a period-averaged beam radius which is smaller than the initial value.
The second condition says th at, for a particular ripple value e. we require a field
strength below that of Eq. 89 in order to avoid the resonance in the denominator
of Eq. 83.
A plot of Eqs. 89 is shown in Fig. 67 for beam param eters corresponding to
an actual commercial Pierce electron gun. Here the beam current is A = 0.27 A,
H = 10 kV, rjo =
= 0.27 /zpervs.
1
m m. and the corresponding beam perveance is .4^ = h/Vb 3 ^ 2
The lower dashed lines correspond to the first ('equilibrium ’)
equation of Eq. S9 for two ripple values S r 2 =
0 .1
and
0 .2 .
Similarly the two
upper dashed curves are from the second ('ripple” ) equation in Eq. 89. According
to the above analysis a focusing array should be designed with a gradient and
period such that we are below and to the left of the ripple constraint and above
the focusing constraint for a beam with a particular entrance ripple. By dividing
the two equations in Eq. S9 we can eliminate the ripple param eter to obtain:
C = ifcmV
(90)
The solid line in Fig. 67 shows the required quadrupole field strength for a partic­
ular beam density and magnet period to satisfy both equations simultaneously for
a given beam ripple.
Figure
68
shows the focusing conditions for another electron source having per­
veance of 2.0 /ipervs and a beam diameter of 6.0 mm. Comparing the beam current
density for both 10 kV beams we have that this beam has a current density which
has been reduced by a factor (2/0.27)(2/6)2 = 0.S2. Since the ripple condition
does not depend explicitly on the beam density we see that the biggest effect on
the focusing curves is to relax the required field gradient to satisfy the equilibrium
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86
condition for this beam.
5.3
Numerical Solution of the PPQ M Envelope
Equations
As we shall see, the conditions of Eqs. 89 provide a guideline of how to choose a field
gradient and magnet period in order to focus a beam in a PPQM array, but they
underestim ate the amount of ripple produced on the beam. The problem with the
development of Eqs. 89 from the envelope equations of motion lies with the smallsignal expansion of radial modes. Another type of radial beam oscillation which is
present in periodic focusing systems is known as betatron motion. Since we typically
have Am < Ap, the oscillation on the beam envelope is bi-periodic with a highfrequency (Am) component and a slowly varying am plitude modulation (betatron
motion). In the azimuthallv symmetric PPM focusing the amplitude m odulation
can be kept smaller than the initial beam radius (t*(-) < r0) by increasing the
magnetic focusing strength. However, the situation is more complex in the case of
PPQM focusing. The bi-planar sym m etry of quadrupolar focusing means that in
one plane the beam is initially defocused while being focused in the other plane.
The maximum radius of the beam ( including ripple) is an im portant factor in tube
design - we must insure that current is not lost to the rf structure or transport
channel. Since the maximum beam radius cannot be confined to the initial radius
then beam ripple may be a more troublesome issue in PPQM focusing.
In order to investigate the effect of betatron oscillation on the effective beam
diam eter we numerically solve the electron envelope equations. Eqs. 69, using a
forward finite-difference method. An example of the resulting envelope motion is
shown for both transverse planes in Figs. 69 and 70. The beam parameters used
here axe the same as in Fig. 67 and the magnet period and quadrupole gradient are
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87
1.0 cm and 1870 g/cm , respectively. The solid lines represent the predicted motion
of the beam envelope as a function of the normalized axial distance, z / \ m. The
transverse components x and y have been normalized to the initial beam radius.
We find that the beam motion is comprised of the fast-scale oscillations due to the
periodic focusing/defocusing of the quadrupole array and the slow-scale betatron
motion. If we take the definition of the beam ripple as the average displacement
from the average beam radius, or:
r
_ 2 ^rmax r m t n )
r “ I2 /Vr1 m a x +» r' m t n /)
m n
then we find that the fast-scale motion has a ripple of Sr = 13 %. This is in
good agreement with the predicted value from Eq. 89 (see Fig. 67). However, in
term s of designing an actual device it is more critical to know the maximum beam
excursion from the initial beam radius. We define A r = (rmax — ro)jr 0 as the
parameter which tells us what the maximum beam extent is in the PPQM array.
Including both periodic oscillations in Fig. 69 we find th at A r = 0.30. e.g. the
beam edge will exceed the initial radius by 30%.
Figures 71 and 72 show the beam envelope for the same beam in a magnet array
having Am = 2 cm and G = 936 G /cm . This corresponds to a point to the lower
right andacross
the line S r 2 = 0.2 in Fig. 67. Thepredicted Sr isapproximately
0.26 - again ingoodagreement with Eq. 89 -
but the beam A r is over 0.70. So
the beam envelope in this case will be 70% larger than the initial radius.
Figure 73 shows the ripple A r as a function of the field gradient for the same
beam as in Figs. 67. 71, and 72 for magnet periods of Am = 1.0 cm and 2.0 cm. In
general, the smaller the magnet period is compared to the plasm a wavelength the
wider the range over which the ripple is a minimum. Also, sm aller magnet periods
tend to lead to smaller ripple values. As can be seen in Fig. 73 the minimum ripple
band occurs across a small range of G (800 - 1000 G /cm 2) for Am = 2.0 cm. For
Am = 1.0 cm the band is considerably wider (1S00 - 3000 G) and the minimum
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88
ripple within this band is Ar = 0.4 - 0.6.
For these beam param eters the condition Eq. 90 gives the equilibrium gradient
values of 1106 G /cm and 553 G /cm for Am =
1
cm and 2 cm, respectively. Compar­
ison with the ripple predicted by the numerical solution of the envelope equations
in Fig. 73 shows th a t the total ripple is severely underestim ated by Eqs. 89. In
general, the m inumum ripple point is found to be a factor of two or more higher
than the gradient value predicted by Eq. 90, and the true total ripple is at least a
factor of two higher when betatron motion is included in the analysis.
It should also be pointed out that the beam focusing becomes unstable above
the minimum ripple point as the gradient is increased in Fig. 73. In the case of
Am = 2.0 cm the critical gradient occurs when k 0 2 / k m 2
0.32 (for G % 1000
g/cm ) which is good agreement with value of ka2 / k m 2 % 0.36 calculated in the
example of Sec. 5.1. T he difference between these two values can be traced to a
breakdown of the small-ripple approximation in solving the Matthieu-Hill equation
with space-charge effects included.
5.4
PIC Simulations of PPQM Focusing
In order to check the accuracy of the analytic and envelope treatm ents of PPQM
focusing described in the previous sections we use the particle-in-cell simulation
code, MAGIC. Again we make the paraxial approxim ation described in Sec.
2 .2 .1
and to transform the drift coordinate c into the tim e coordinate according to
2
—»• u0t, where u 0 is the beam drift velocity. Under this approximation we have
for the time-dependent quadrupolar fields:
Bx(y*t)
=
~ ~
+ -y^-(3x2y + 1/3) | cos(kmu 0 t)
By(x.t)
=
—^
j x + ^ - ( 3 y 2x + x3) j cos(kmu 0 t)
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(92)
89
B z[x, y, t)
= km— xysin(kmu 0 t)
r?
Figure 74 shows the evolution of beam cross-section as a function of the nor­
malized drift length down a quadrupole stack. An initially cold, uniform density
beam is simulated on a
1 0 0 x 100
grid with uniform grid spacing of dx = dy = 0.016
cm. Approximately 600 particles are initially loaded in the sim ulation (only 1/4
of these are shown in Figure 74). The beam param eters used in this simulation
are the same as used in Fig. 69: /& = 0.27 A. Vj, = 10 kV, r& = 1.0 m m , and the
magnet period is 1.0 cm. VVe include the B: magnetic field term here but neglect
the higher-order term s in B x and B y under the good approximation (rf,/r ? ) 3 <C 1
for this particular beam. Higher order magnetic field effects may be im portant de­
pending on the ratio ( r t / r , ) 3 for a particular beam (i.e. extremely 'thick' beams).
The magnet array has G = 1S71 g/cm which corresponds to a pole-tip field of 3200
G at r = r, = 1.71 cm. As can be seen, the beam undergoes elliptical motion as
it progresses down the array (corresponding to the S r 2 motion in Sec. 5.2), and
exhibits relatively low-ripple focusing. The cross-section is mainly elliptical, how­
ever, the diamond-shaped behavior seen in some plots is due to the effect of the
quadrupolar B z fields in Eq. 93. By the time the beam has traversed 21 magnet
periods it remains well-confined and no evidence of instability can be seen.
The ripple motion of the beam in Fig. 74 can be quantified by determ ining the
rms transverse position coordinates for the collection of particles in the beam as a
function of z j Am. Thus X rms = y/(x2) where (x 2) =
x t2 and similarly for the
y coordinate. The ripple calculated in this manner from the particle simulations
can be seen in Figs. 69 and 70 as the open circles connected by the dashed lines.
Here the rms quantities have been normalized to the initial rms value in order
to more conveniently see the fractional ripple A r . As can be seen the agreement
between the envelope and PIC simulations is excellent, and both m ethods show
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90
th at the ripple equations of Sec. 5.2 are inadequate to accurately determine the
beam radius along the m agnet array. Figures 71 and 72 show a sim ilar comparison
between the envelope and PIC results for a PPQM array having Am = 2 cm and
G = 936 G/cm.
Finally, Figs. 75 and 76 show the beam ripple in the x and y planes for the
same beam and magnet param eters as in Figs. 69 and 70 when the electron beam is
given an initial rms velocity spread of Svz / v z = 0.001. By equipartion we assume
th at Svx/ v z = Svy/ v z =
Sv:/ v : The normalized beam envelopes for the case
with an initially therm al beam (filled circles) can be compared to the case with
an initially cold beam (open circles) and the numerical envelope solution (solid
line). The therm al beam envelope undergoes large oscillations near the beginning
of the PPQM stack but eventually these oscillations settle down to resemble the
cold beam results, at least in peak-to-peak magnitude. Interestingly, the average
beam radius is reduced in both planes when thermal effects are included. This is
a rather counterintuitive result since the usual effect of transverse beam em ittance
is to act as another defocusing force - manifesting itself as an outward 'pressure'
on the beam envelope. In a subsequent simulation the grid size in each transverse
dimension was halved to dx = dy = 0.00S cm and the num ber of particles was
increased by a factor of 4 (approximately 2400 particles) to determ ine whether
non-physical em ittance growth due to grid noise was a factor. The results of this
simulation for the beam envelope were virtually identical to Figs. 75 and 76. A
complete explanation of this initial velocity-spread simulation result (i.e.. whether
it is an accurate physical prediction or a consequence of an incomplete physical
model) remains a topic for future study.
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91
Chapter 6
Experim ental Investigation o f
Beam Focusing in P P Q M Arrays
Based on the analytic and numerical investigations reported in Chap. 5 we have
conducted an experim ental investigation of PPQ M focusing of round electron
beams.
The basic experimental configuration is as shown in Fig. 42 with the
exception of a different electron gun and a rectangular-block PPQM array in place
of the quadrupole lattice in Fig. 42. In this chapter we describe the experim ental
design of the m ajor components including the electron gun, the solenoid m atching
section, the PPQ M array, and beam diagnostics. Discussion of the beam transport
experimental results is given in Sec. 6.3.
6.1
The Pierce Electron Source and Matching
Optics
The M690 electron gun in Fig. 42 was replaced with another commercially available
Pierce diode (M707) which produces a beam with .4^ = 0.27 /ipervs to 10 kV with
a beam radius of
%
=1
mm [52]. These beam parameters correspond to the
param eters used in the theoretical analyses shown in Figs. 67. 71. 72. and 74.
Since it is desirable to keep the gun under vacuum when changes to the vacuum
system are m ade we must accomodate for a vacuum valve to be place between the
gun and the rest of the system. This, in turn, makes it necessary to use a solenoid
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92
matching section to propagate the beam from the electrostatic focus of the gun,
past the valve and pum ping port, and into the PPQM array. As in the experim ent
described in Chap.3 it is also necessary to use flux shields to prevent solenoid flux
from disturbing the gun optics and to separate the solenoid from the PPQ M array.
All solenoid/flux shield components are as described in Sec. 3.3 with the exception
th at the gun flux shield has inner aperture which has been reduced from 0.75 in.
to 0.50 in.
The gun focusing with the measured solenoid field was checked using the EGUN
gun optics code. These simulations predict that a high-quality beam can be pro­
duced in this system with rms axial velocity spread below 1 % and beam radii from
0.5 - 1.0 mm. Beam measurements were made on the beam using a translatable
beam current probe and agree well with the simulated beam size. Velocity spread
measurements were also made, which indicated rms spread on the order of 2%-3%,
however these values are probably too high due to secondary electron effects in the
measurement. The actual velocity spread more likely lies between the theoretical
and measured value. Figure 77 shows the measured beam radius and rms velocity
spread as a function of the solenoid guide field.
We recall the discussion of Sec. 4.1.2 regarding the Brillouin num ber, or the
ratio above the magnetic field am plitude which exactly cancels space-charge forces.
Using Eq. 61 and assuming a beam radius of
1
mm we find that this gun is being
operated at a Brillouin number between 1.9 - 3.S for a solenoid field from 750 G 1500 G. Adequate beam focusing should result for this range of Brillouin numbers.
Details of the solenoid focusing system were previously discussed in Sec. 4.1.2
and the adverse effect of solenoid leakage in the quadrupole sheet beam focusing
system were noted in Sec. 4.2. We note here, as will be seen, the effect of leakage in
this experiment should be reduced for two reasons. First, the quadrupole gradients
here are on the order of 750 G /cm or higher so the relative effect of the solenoid
leakage compared to the strong gradients is very much reduced. Second, the PPQM
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93
array has a much smaller quadrupole radius than the quadrupoles used in Chap.4.
Thus the quadrupole field loss to the nearby flux shield for the first set of m agnets
will be relatively smaller for the PPQM array due to the fixed aperture size in the
flux shield.
6.2
The PPQM Array
In this section the design, fabrication, and testing of a rectangular-block PPQ M
array is discussed. The advantage of using rectangular blocks, as shown schem at­
ically in Fig. 78. is that the rectangular blocks are much cheaper and easier to
m anufacture since the curved surfaces of the round-aperture PPQM stack (see
Fig. 5) do not need to be machined. The cross-section of such an array is shown
in Fig. 79. The rectangular blocks minimize the amount of machining to be done
since the blocks of rare-earth magnet m aterial can be sintered very close to the
final dimensions. The caveat is that a reduced achievable field gradient is sm aller
for the rectangular-block array and careful consideration of this must be considered
in the design of an effective PPQM array.
The choice of permanent magnet material from several candidate m aterials was
made based on a couple of criteria. The unit cost of both SmiCos and S1112C 0 1 7
magnet materials was roughly twice the cost of NdFeBo materials. However, since
the NdFeBo m aterial is prone to corrosion it is often electroplated - increasing the
effective unit cost. The remnant fields. B r of the three materials are sim ilar with
B r ss 0.90, 1.0, and 1.1 T for SmiCo5. Sn^C otr. and NdFeBo. respectively. For this
application the magnet array will operate at room tem perature so the accelerated
corrosion of NdFeBo at elevated tem peratures is not an issue and electroplating is
not required. Therefore NdFeBo was the m aterial of choice for this experim ent.
In the following sections the calculation of the achievable field gradient for
rectangular block quadrupoles is presented and the actual physical design of an
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94
array having Am = 2 cm is discussed. Field measurements of the PPQ M array
were conducted and verify the design equations.
6.2.1
D esign Equations for a P P Q M Array
Interest in using perm anent magnet quadrupoles for the focusing of high-energy,
relativistic particle beams in accelerators has grown over the last decade-or-so with
the advent of some of the newer high-remanence perm anent magnet m aterials. As
a consequence, there is a growing body of literature concerning the calculation
of field gradients in perm anent magnet quadrupole systems. However, since most
accelerator transport lattices are comprised of high-gradient quadrupoles separated
by drift lengths on the order of. or often greater than, the quadrupole length the
overlap of nearby fringing fields is not considered. Also, consider the integrated
force effect due to the quadrupolar B: field:
/ •Am
/
Jo
/2
v±Bs(:)d;
(93)
where uj. is a perpendicular component of the particle velocity. In the paraxial
approximation t’j. does not change strongly as the beam traverses the quadrupole
and so the above integral is near-zero since B :(z) is sym m etric about the longitu­
dinal magnet midplane. Since fringing effects are negligible and B z is unim portant
to the focusing dynamics much of the analytical investigations have concentrated
on the simpler task of calculating the 2 -d quadrupolar fields (i.e. infinitely long
quadrupoles).
For the purposes of calculating beam focusing dynamics the ac­
tual quadrupole profile function Cj_(-) can be related to the idealistic hard-edged
quadrupole through an effective length by:
r m' 2 GL 2 {z)dz =
Jo
Gonefj
where G 0 is the 2 -d quadrupolar field gradient.
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(94)
95
A full-blown calculation of the 3-d fields of the PPQ M array is not analyti­
cally tractable. So, with the previous discussion in m ind, we can determ ine the
2-d quadrupolar fields and relate th at to the realistic 3-d quadrupolar field for
the PPQM array. Following Ref. [75] the magnetic ’scalar’ potential satisfies the
following equation:
V 2 $ m(x ) = 4tt(V- M)
(95)
where $ m(z ) is the m agnetic scalar potential and M is the m agnetization vector.
Consider the single block m agnet of Fig. SO. where the definitions used in specifying
the magnet geometry are the same as in Ref. [75]. Here r, and r„ are the inner
and outer radii of the magnet from the origin. lz is the magnet length along the
easy-axis, or direction of m agnetization, and ly is the magnet width. An angle r
is a function of the param eters ly and r, and defines the angle subtended by the
magnet element. For a constant, uni-directional magnetization which is oriented
parallel to the long-axis. and the inner and outer surface normals. Eq. 95 may be
solved as:
Here the sum is over the eight magnet surfaces comprising the magnet array in the
transverse xy plane (two surfaces per magnet). The quadrupolar magnetic field is
then determined from B { x ) = —V $ m( r ) .
The details of this calculation are presented in Ref. [75] and therefore we just
present the results as they pertain to the array design. The maximum quadrupolar
magnetic field magnitude at the inner radius (r = r,) is obtained for the case
when the inner corners of the m agnets touch (r, = ly/ 2 ) and when the outer radius
becomes large (r 0 —>■oo). We note that then r = 45°. The maximum possible field
is then Bq = 0.64f?r where B r is the remnant field of the m aterial. For comparison,
we note from Ref. [75] that the maximum B q for the round-aperture array of Fig. 5
is 0.83Br .
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96
O f course the condition r 0 -* oo cannot be realized in practice, and a realistic
magnet design for microwave tubes will try to minimize r 0 to save on cost and
weight. The interior field of the m agnet array can be written as:
g , = R .- s m ( 2 r ) ■
7r
s z + tan r
where, we recall, s = r 0 / r t- =
1
(97)
+ /r /r,-. By the geometry of Fig. 80 we have that:
t
= ta n - l (/y/ 2 r,-)
(98)
Figure 81 shows a plot of the scaling of this equation with the quadrupole radius
for lx = 3.6 cm and /y = 1.0 cm . For these particular magnet length param eters
the maximum reduction factor of 0 . 6 is obtained when the inner magnet edges are
touching, r,- = 0.5 cm.
As an example of a calculation of th e field gradient for a realistic PPQ M stack,
we take lx = 3.6 cm, ly = 1.0 cm. Am = 2 cm, and r, = 1.84 cm. Then by the results
of the above and Fig. 81 we find that the field strength should be approxim ately
Bq « 0.64 x 0.20 x Br - Dividing both sides by the quadrupole radius and using B r
= 11.7 kG, we find that the expected quadrupole gradient is G % 1500 G /cm .
6.2.2
Mechanical D esign
For the reasons discussed previously NdFeBo is the material of choice for the array
in this experiment due to its low cost since plating is not necessary. The array
will operate at room tem perature and any corrosion will be confined to the surface
and will have a small effect on the bulk magnet properties. The m anufacturer
(Magnetic Components. Inc.) m aterial designation is N34I\X01-NdFeBo with a
rated B r % 11.7 kG. This material has good tem perature stability and consistency
(rated to ± 3%).
The individual blocks were made with lx = 3.6 cm. ly =
1 .0
cm. and /- = 1.0 cm
with tolerances of ± 0.0127 cm. The m easured magnet-to-magnet field strength at
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97
the surface of individual magnets was found to be 4.7 ± 0 .0 8 kG for a pole-to-pole
variation of less than 2%. The individual magnets were placed in alum inum rails
in a quadrupolar configuration and clamped into place using 2-56 nylon-tipped
set screws. Each aluminum rail fits into a slotted plexiglass jig which holds the
magnets a t a fixed quadrupole radius and at a fixed 90° angle with respect to the
other quadrants of the array. This configuration allows for the quadrupole radius
to be varied from rq = 1.59 cm to 3.18 cm and thereby allows for adjustm ent of
the quadrupole gradient.
6.2.3
M agnetic Field M easurem ents
Field m easurements of the quadrupole magnets were made to verify the field struc­
ture before mounting them to the rest of the system. Measurements were taken
with both transverse and axial Hall probes which were swept using a linear motion
drive. D ata was recorded on a digital oscilliscope and downloaded to a PC.
Figure 82 shows the measured transverse (quadrupolar) magnetic field of the
array for severed different transverse positions. The radius of the quadrupole array
is r, = 1.84 cm. The average magnetic field gradient, determined by dividing by the
probe position, was found to be G = 1650 G/cm ± 125 G /cm . The relatively large
uncertainty is a consequence of the finite size of the Hall probe sensing element
(rs
0.254 cm across) and some lateral motion of the probe as it swept down the
array. However, the measured average gradient is in reasonable agreement with
the theoretical gradient of 1500 G /cm from Sec. 6.2.1 for a rectangular-block
PPQM array with this quadrupole radius. A more instantaneous m easurement
of the entire gradient profile could be obtained using a pulsed-wire type of field
measurement. A prominent feature of Fig. 82 is the presence of term ination effects
at the beginning and end of the array. However, by modifying the radius of the
first and last half-period of the array the end-effects. and their effect on the beam,
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98
can be reduced.
Figure 83 shows the measured longitudinal magnetic field of the quadrupolar
array along the x = 0 midplane. Since we expect this field to be zero here this
measurement gives and indication of th e ’error’ field of the axrav. The longitudinal
field errors are largely random - perhaps indicating randomly placed m agnets with
field variations and/or alignment errors. In any event the m agnitude of the axial
field is less than 30 g and the effect of the random variations is likely to average
out of the beam dynamics.
In Ref. [69] the amplitude function F(z) for a permanent magnet quadrupole
extending to :
—oo is calculated.
The magnetic fields of the semi-infinite
quadrupole are the given by:
# r ( y .- )
=
G(z)y = GaF(z)y
By(x.z)
=
G(z)x = G0F{z)x
B:(x.y.z)
=
0.0
(99)
where G0 = Bq/ r q is the constant quadrupole gradient. From [69] the am plitude
function for the semi-infinite quadrupole is given by:
2 2
+ iV + 4 + &/
I/, U0 ---------------------------------------
(
100 )
where r, and rQare the inner and outer radii of the permanent magnet. Here i/,,0
is defined by:
r
V i.o=
- i -1/2
l + ( — )2
(101)
The constant gradient G0 is defined as the uniform value well away from the magnet
edge at z = 0. Here
G0 = 2 BrCm( - - - ) = G( - oc )
r,
r,.
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(1 0 2 )
99
where th e factor Cm depends on the number of blocks M in the array, Cm =
sin(37r / M ) / ( 37 r/M ).
We now invoke the linear superposition property of perm anent magnet m ateri­
als. For a realistic magnet of length lm the profile function is found by calculating
Eq. 100 for two magnets of opposing gradients, one with an edge at z= 0 and the
other at z = /m, and summing the result. For an array of magnets this procedure is
used for each individual half-period of the array and the final result is obtained by
numerically summing subsequent evaluations of Eq. 100. Figure 84 shows an ex­
ample (dashed line) of this calculation for the magnet param eters of Fig. 82. Also
shown in Fig. 84 is the average measured field gradient (solid line). In practice, the
gradient am plitude G0 in Eq.
102
must be adjusted to obtain the correct normal­
ization for the rectangular-block array, however, excellent agreement is obtained
for the to tal am plitude function Fj(z). Some disparity exists for z > 15 cm but this
is a ttrib u ted to errors in the positioning of the Hall probe in the measurements.
6.3
Experimental Measurements of Beam Trans­
port
In this section we discuss an experim ental investigation of beam transport through
a PPQ M array. The beam voltage and current diagnostics are as discussed in Sec.
4.1.
T he m ain diagnostic for measurement of the beam cross-section is again
Cerium-doped glass. The glass was cut into disks with d = 12.7 mm and one
surface was coated with a thin molybdenum layer with r = 250 A ± 50 A for charge
bleed-off. The disk is placed in a isolating ceramic holder and a wire leading to
an electrical vacuum feedthrough gives a path to ground for the collected beam
current. A calibrated current transform er is used to measure the beam current.
As noted in Sec. 4.1.6 the molybdenum metal layer is a source of secondary
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100
and reflected prim ary electrons which are blown from the surface and collected
elsewhere on the stainless vacuum envelope.
As a consequence, the measured
collected current is less than the total current in the beam even under conditions
where the entire beam is clearly hitting the collector. A calibration of the m easured
collected current with the true collected current shows that approximately 33% of
the current is lost to secondaries in the m easurement. This ratio is relatively flat
for beam voltages from 2-8 kV and will be used to adjust all subsequent collected
current measurements.
It should be noted th at the reflected primary electron
coefficient (the ratio of reflected primary electrons to impacting primaries) in this
voltage range for molybdenum is approximately 0.30. which is in good agreement
with the observations.
The flux shield between the solenoid m atching section and the PPQM array
was reduced in thickness to 0.0625 in. Measurements of the beam diam eter ju st
ahead of the flux shield show’ that the image of the beam cross-section varies from
2 - 3 mm from solenoid fields ranging from 1575 G - 1200 G for l j = 5 kV. At
a fixed solenoid field of 1575 G the beam diam eter ranges from
2
- 3.5 mm from
H = 5 kV - 9.2 kV. Allowing for some blooming of the optical image of the glass
these measurement are in reasonable agreement with Fig. 77. Great care was taken
to center the beam with respect to the vacuum tube (and hence the quadrupole
array) by viewing the beam image as the gun position was adjusted to place the
beam on the magnetic solenoid axis.
The PPQM array was then mounted around a 0.75 in. diam eter vacuum tube
with the help of the plexiglass jigs dicussed in Sec. 6.2.2.
The radius of the
quadrupoles could be adjusted from 1.55 cm to 3.18 cm to vary the peak magnetic
field gradient of the array from 2017 g/cm to 733 g/cm in discrete steps. Table 9
lists the quadrupole gradients for the radii used in the experiment. A comparison
to the plot of Fig. 67 show that these gradients should take the beam through both
the S r 2 =
0 .1 , 0 .2
lines and the equilibrium lines for this electron beam with Am
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101
=
2
cm.
In order to m itigate the end-effects of the quadrupole gradient at the beginning
of the PPQM array on the beam dynamics the radius of the first half-period of
the array was increased by 0.635 cm. Also, the effect of the flux shield on the
gradient amplitude was noted in Sec. 4.1.3, where a 50 % reduction was measured
neax the shield. Here the quadrupole radius is roughly half of the m agnet radius
of Sec. 4.1.3 (2 cm versus 4.2 cm), but the distance between the first quadrupole
and the flux shield is also nearly half that of Sec. 4.1.3 (1 cm versus 2 cm). Thus,
the scaling of the first gradient peak in the PPQM array should be approximately
50% of that of Fig. 84, which reduces the peak gradient to nearly the same value
as in the center array.
The glass witness plate was placed at four subsequent positions relative to the
magnet array. These positions are zg =
6 .8
cm. ll.S cm. 13.8 cm. and 16.8 cm from
the front edge of the array, corresponding to 3.4. 5.9. 6.9. and S.4 magnet periods
down the stack. The last position is more than one period from the end of the
array to avoid problems with magnet array term ination effects. Figure 85 shows
the measured beam radii, a and 6 . as a function of distance down the magnet array.
The uncertainty in the measurement of the beam radii is approximately ± 0.5 mm,
however if significant image blooming is occuring then these measurements may be
too large. The axial distance is measured relative to the front edge of the PPQM
array, where z = 0 corresponds to the beginning edge of the first half-period. Here
the quadrupole gradient is G = 825 g/cm and the field magnitude of the solenoid
m atching section is 1575 g. The beam current and voltage are If, = 94 mA and VI
= 5 kV.
Figure
at
2
86
shows the beam radii as a function of the quadrupole magnet strength
= 16.8 cm. or 8.4 magnet periods down the stack. Again, the beam current
and voltage are /<, = 94 mA and V\ = 5 kV, and B s = 1575 g. This measurement
gives an indication of the amount of beam ripple through the magnet stack, with
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102
severe ellipticity and ripple occuring as the gradient is increased.
T he beam radius as a function of the beam voltage is shown in Fig. 87. Again,
these measurements are taken at z — 16.8 cm with B 3 = 1575 g and G = 825
g/cm . The general trend of increasing beam voltage is evident, though it is not
clear whether the increase is due to a larger input beam radius or due to some
effect of the periodic focusing. Figure
88
shows the measured beam radius at the
entrance of the array as a function of the beam voltage. The initial beam radius
increases by a factor of two from 5 - 9 kV. Based on this it may be surm ised th at
the larger input beam radius largely accounts for the larger beam radius observed
in Fig.
88
near the end of the quadrupole array. Finally, the initial beam radius
as a function of the solenoid m atching field is plotted in Fig. 89. As expected, a
tighter beam is obtained for larger solenoid field.
Much of the above data was collected at a beam voltage of 5 kV in order to
protect the molybdenum surface coating from being melted away by the electron
beam. It is instructive, therefore, to look at the predictions from the analytic
theory of Sec. 5.2. the envelope simulations of Sec. 5.3. and the PIC simulations
described in Sec. 5.4 for \ l = 5 kY\ Figure 90 shows a plot of the ripple and
equilibrium conditions of Eqs. S9 for the two S r 2 values of 0.1 and 0.2. Also plotted
as the solid line is the equlibrium gradient which satisfies both focusing conditions
simultaneously. According to this theory by adjusting the quadrupole gradient
from 400 - 1000 g the beam can be taken through the equilibrium gradient and
through the ripple condition for S r 2 =
0 .2 .
We also note that the M atthieu-Hill
solutions to the beam envelope without space-charge give Gc = 1040 g/cm as the
critical gradient for beam instability. Using the crude approxim ation of Eq. 75 to
include space-charge the instability band edge is reduced to Gc = 880 g/cm .
Figure 91 shows the predicted beam ripple. Ar, from the numerical solutions of
the envelope equations. For gradients from 700 - 900 g/cm the solutions predict
weak instability of the beam envelope - meaning that the beam envelope grows
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103
at a slow enough rate that it takes tens of magnet periods before an order of
m agnitude growth is evident. The envelope predicted by a PIC simulation for this
case is shown in Fig. 92. It is evident th at the average beam envelope is growing
in both planes, however, it grows at a slow-enough rate th at the beam envelope
increases approximately a factor of two in one plane and a factor of 2 - 4 in the
other transverse plane. These growth factors are in relatively good agreement with
the measured beam radii in Fig. 85.
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104
Chapter 7
Sum m ary
In this section a summary of the work described in this dissertation is provided,
with emphasis on the novel aspects of this research. Because the subject m atter
was largely divided into two distinct topics - sheet electron beam formation and
focusing, and PPQM focusing of round electron beams - this summary is divided
into two separate sections. Section 7.1 gives an overview of the theoretical investi­
gation of high-perveance sheet electron beam focusing in periodic arrays and, also,
a sum m ary of the theoretical and experimental investigation of a novel sheet beam
forming technique. Recommendations for further experimental work are given, es­
pecially on a new experimental configuration designed to elim inate problems with
the m atching solenoid section. In Section 7.2 a sum m ary of a theoretical and ex­
perim ental investigation of using PPQM focusing of high-perveance round electron
beams is provided. Again, the novel aspects of this work are emphasized and the
prospects for use of this type of focusing system are commented upon.
7.1
Formation and Focusing of Sheet Electron
Beams
The use of extended beam configurations, especially sheet electron beams, has the
potential to greatly advance the state-of-the-art for vacuum microwave devices.
Particularly as the current trend towards the high-frequency millimeter-wave band
continues, where device dimensions tend to decrease with A. there is a need to
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105
overcome some of the lim itations inherent in conventional round beam devices.
Sheet electron beams have the advantage th at the total amount of beam current
in a single device, and therefore the total output power, may be increased without
inordinate increases in the beam space-charge density or rf power density in the
microwave circuit. Therefore a sheet-beam microwave tube can largely avoid both
dc and ac defocusing problems due to beam space-charge and subsequent efficiency
loss. Similarly, the extended type of rf interaction circuit compatible with sheet
beams allows for increased total rf power capacity without electric breakdown.
As pointed out, the potential benefits of using sheet electron beams have been
appreciated for several decades, however, the presence of the diocotron instability,
at least in simple solenoidal focusing systems, largely discouraged the implemen­
tation of sheet beams in actual devices. It has been shown here, using analytical
and numerical investigations, that space-charge dominated sheet electron beams
can be stabilized in periodically varying focusing configurations. A particular type
of configuration - the offset-pole periodically cusped magnet (PCM) array - has
been identified which provides stable focusing in both transverse dimensions of the
beam and the low transverse velocities compatible with linear beam microwave
tubes. Analytic expressions for the magnetic fields of the PCM array were derived
and verified with 3-d m agnet simulations. The use of these fields in numerical PIC
simulations established the potential of offset-pole PCM for focusing very high
space-charge density electron beams. A particular attraction of this configuration
is that it leaves the sides open for input or output coupling of electromagnetic
radiation.
An alternative configuration, the hybrid quadrupole-PCM array, was also shown
to be effective in sheet beam focusing and has the added advantage of more easily
allowing for magnet tapering at the entrance of the array and eliminating the beam
tilt encountered in the PCM array. In any event, for most beams of interest the
amount of beam tilt in the PCM is not problematic.
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106
A legacy of the lack of modern investigations on using sheet beams is th at
very few studies of fabricating sheet beam electron guns for linear devices have
been conducted. Such a study would involve a considerable outlay of tim e and
resources for the design and fabrication of a sheet beam gun. neither of which
is often possible for proof-of-principle experim ental investigations of sheet beam
focusing. A novel, alternative method of sheet beam formation using an initially
round beam and magnetic quadrupole lenses has been identified. This m ethod has
the advantages of low-cost construction, since quadrupole magnet design is well
understood and round beam electron guns are readily available, and th at changes
in the beam shape can be made 'on-the-fly' by adjusting quadrupole strengths
without the need to break the vacuum system . Furthermore, since quadrupoles
have been used extensively by the accelerator com m unity for decades, there exists
a large variety of 2-d and 3-d design codes and an extensive knowledge base in the
literature.
A theoretical investigation of the quadrupole sheet-beam forming m ethod was
made based on 2-d and 3-d beam envelope formulations. 3-d PIC codes, and magnet
design simulations. Although the 2-d formulation demonstrated the general utility
of this method for sheet-beam formation, it proved inaccurate due to the neglect
of the self-E- space-charge fields. The 3-d envelope simulations and particle sim­
ulations, which include this term, agree extrem ely well on the beam envelope and
establish the potential of using a magnetic quadrupole lattice to form high-quality,
variable-size sheet beams for transport experim ents.
Based on these theoretical studies a four-quadrupole magnet lattice was de­
signed fabricated, and tested.
An ’off-the-shelf' round beam electron gun and
vacuum components were procured. Due to the desirability of having a vacuum
valve between the electron gun and the rest of the vacuum system, and the proxim­
ity of the electrostatic beam focus to the cathode, it was necessary to use a solenoid
matching section to propagate the beam from the gun into the quadrupole lattice.
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107
A carefully designed flux shield was used in an attem p t to separate the quadrupole
fields from the solenoid fields but, yet, allow for enough overlap to insure the
propagation of the beam into the quadrupole lattice. Although a 4 cm x 0.4 cm
sheet electron beam was generated - experimentally dem onstrating the ’proof-ofprinciple’ - the leakage of solenoidal flux into the quadrupole lattice caused tilt
in the ellipse which greatly hampered beam formation and made comparison with
theory difficult. An alternative configuration was proposed which eliminates the
solenoid m atching section altogether. It is this au th o r’s opinion that subsequent
experiments with the new configuration will produce m ore substantial results.
7.2
PPQM Focusing Study
Although magnetic quadrupole focusing of particle beam s has been used for several
decades in accelerator research, it has been only recently th at the development of
high-Br perm anent magnet materials has driven interest in their use in compact
permanent magnet quadrupoles in replacing bulky electrom agnets. The advantage
to using quadrupoles is the strong focusing provided by the v:
x
B ± force - which
are the ’strong’ components of the velocity and quadrupolar magnetic field. Rec­
ognizing this, there may be advantages to using periodic permanent quadrupole
magnets (PPQM s) in close-packed arrays for focusing high space-charge beams in
linear beam tubes. A comparison of the magnetic focusing forces between PPQM
arrays and conventional PPM focusing shows that a 2-10 increase in the focusing
force may be realized for low-voltage (1 0 -4 0 kV) beam s. Further increases occur
and the beam voltage is increased.
Early investigations of PPQM focusing concentrated on the beam envelope
equations without space-charge. but more recent treatm ents included space-charge
in an effort to determ ine the amount of ripple on the beam radius. It was shown
here that both treatm ents seriously underestimate the am ount of ripple due to the
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108
neglect of betatron motion in the approximations m ade in the analyses. Numerical
solutions to the envelope equations demonstrate th at th e m agnitude of the beam
excursion from the initial radius can be a factor of two or greater than that pre­
dicted by the analytic studies. Numerical PIC simulations agree with the beam
envelope motion predicted by envelope simulations. Though the position of the
instability bands is correctly predicted by the analytic theory, the enhanced ripple,
even away from the instability edge, can cause problems with beam loss in a real
device.
An experimental investigation of PPQM focusing was conducted in order to
check the validity of the analytic, envelope, and PIC treatm ents of the beam focus­
ing. The experimental configuration consisted of a commercially-available Pierce
electron gun, matching solenoid optics to propagate the beam into the PPQM
array, and a variable-gradient. 10-period, rectangular-block NdFeBo PPQM array
with a periodicity of 2 cm. To the author's knowledge, this is the only such ex­
perim ental investigation of electron beam focusing using the close-packed PPQM
array in a configuration compatible with linear-beam vacuum microwave devices.
The beam radius was measured near the end of the PPQM array under a variety
of beam and magnet param eters. The results of this investigation largely agree
with the numerical envelope solutions and PIC simulations and indicate that the
simple analytic theory is inadequate to accurately predict the magnitude of the
ripple on the beam.
Despite the enhanced focusing force available in PPQM focusing over conven­
tional PPM focusing, there are some fundamental difficulties with utilizing PPQM
focusing in compact, linear beam tubes. In azimuthally-sym m etric PPM focusing
the beam radius may be confined to a value equal to the initial envelope radius
through increasing the on-axis magnetic field magnitude. In quadrupole arrays the
focusing is not azimuthallv symmetric and the beam radius is initially defocused in
one tranverse plane while focused in the other tranverse plane. Thus, the maximum
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109
beam radius cannot be confined to the initial beam radius and, in fact, the beam
ripple is quite often larger than the ripple in a comparable PPM stack. This is a
severe problem for those devices th at may use tight beam fill-factors to optim ize
the coupling impedance between the beam and the rf wave. In some respects the
focusing in PPQM stacks is analogous to wiggler focusing, as in free-electron lasers,
and both systems share some of the same challenges in application. For example,
increases in beam space-charge, u;p, require smaller field gradients for focusing. As
with the transverse field component in wigglers a higher field gradient in PPM Q
focusing can lead to large am plitude oscillation, transverse beam velocity, and po­
tential beam reflection for high space charge beams. O ther technical difficulties
may exist in m atching the Pierce gun into the quadrupole stack (due to azimuthal
symmetry concerns) and with assembling and tuning a magnet array with twice
as many magnets as PPM stacks, however these particular issues are beyond the
scope of this study. Still, for those applications using very high space-charge den­
sity beams where higher ripple is not problematic then the PPQM method may
present an attractive alternative in regimes where PPM focusing is ineffective.
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110
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List o f Figures
1
Average power versus frequency for various microwave devices. . . .
122
2
Schematic of the klystron amplifier............................................................... 123
3
Progression to a high-perveance extended beam configuration. . . . 124
4
Evolution of the diocotron instability for a sheet beam ............................125
5
Schematic of the PPQ M a rra y .......................................................................126
6
Comparison of PPQ M and PPM focusing strengths..................................127
7
Schematic of the offset-pole PCM array.
8
Sheet beam envelope trajectories................................................................... 129
9
Sheet beam envelope trajectories under ’m atched’ conditions.................130
10
MAGIC simulation of a PCM-focused sheet beam .....................................131
11
MAGIC simulation of a solenoidaily-focused sheet beam ......................... 132
12
Initial density perturbation in the MAGIC simulations............................133
13
Magnetic equivalent of the offset-pole PCM ................................................134
14
Magnetic surface charge model for the offset-pole PCM array.
15
Normalized edge fields of the offset-pole PCM a rra y ................................136
16
Illustration of beam tilt in offset-pole PCM a r r a y . ..................................137
17
PIC simulations of an elliptical 90.9 A /cm 2 beam in a PCM array. . 138
18
Evolution of an 90.9 A /cm 2 sheet beam in a uniform magnetic field. 139
19
Evolution of an elliptical 455.0 A /cm 2 beam in a PCM array. . . . .
20
Evolution of an 455.0 A /cm 2 sheet beam in a uniform magnetic field. 141
21
Evolution of a 90.9 A /cm 2 beam in a long-period PCM array.
22
Evolution of am experimentally realizable beatm in a PCM array. . . 143
23
Model of the offset-pole PCM configuration used in TOSCA................. 144
24
Predicted side-fields for a offset-pole PCM arrray......................................145
.................................................. 128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
...
135
140
. . . 142
119
25
Schematic of the hybrid PCM /quadrupole configuration......................... 146
26
Tilt applied to the hybrid PC M /quadrupole configuration..................... 147
27
Evolution of a sheet beam in a hybrid PCM /quadrupole array. . . . 148
28 Schematic of the quadrupole sheet beam forming method................. 149
29 2-d beam envelope trajectories with and without space-charge.
. . . 150
30 Bell-shaped quadrupole field model.........................................................151
31
2-d envelope trajectories for the two quadrupole lattice.....................152
32
Magnetic field profile in the 2-d envelope simulation.......................... 153
33
2-d envelope solution of the two quadrupole matching section.
34
WARP simulation of the two quadrupole matching section............... 155
35
TRACE3D simulation of the two quadrupole matching section. . . . 156
36
TRACE3D simulation of the four quadrupole matching section.
37
WARP simulation of the four quadrupole matching section.............. 158
38
Simulated cross-section of the beam ........................................................159
39
Beam em ittance in in x-x' phase-space................................................. 160
40
Beam em ittance in in y-y' phase-space................................................. 161
41
TRACE3D simulation using the perm anent magnet model...............162
42
Experimental configuration for the sheet beam formation experiment. 163
43
Measured M690 perveance.........................................................................164
44
Calculated axial field and the magnetic circuit model........................ 165
45
Measured axial magnetic field of the solenoid matching section.
46
Cross-section of the magnetic quadrupole lens..................................... 167
47
Quadrupole model and the calculated magnetic field from PANDIRA.168
48
Saturation check of the magnetic quadrupole lens.................................... 169
49
Measured magnetic field of the quadrupole lattice for 1 A......................170
50
Measured magnetic field of the quadrupole lattice for 1.5 A...................171
51
Measured magnetic field gradient for 1 A drive current...........................172
52
Measured magnetic field gradient for 1.5 A drive current....................... 173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
. . . 154
. . 157
. . 166
120
53 Measured field of the quadrupoles for the currents listed in Table 8. 174
54 Comparison of the gradient to the permanent m agnet m odel................. 175
55 Overlap of the quadrupole and solenoidal magnetic fields........................176
56 Measured quadrupole gradient near the flux shield for y = 1.91 cm.
177
57 Measured quadrupole gradient near the flux shield for v = 2.82 cm.
178
58 Schematic of the M690 pulsing circuit.......................................................... 179
59 Image of the 4 cm x 0.4 cm sheet electron beam ....................................... 180
60 Beam voltage, current and collected current waveforms........................... 181
61
WARP simulation with a 100 g solenoid field..............................................182
62
WARP simulation with 100 g solenoid field - beam cross-section. . . 183
63
Cross-section of a round aperture permanent magnet quadrupole.
64
Parameter-space map of the specialized M atthieus equation..................185
65
Physical picture of beam instability in PPQM focusing........................... 186
66
Comparison of PPQM gradient to PPM field m agnitude.........................187
67
Focusing map for a beam with An = 0.27 and Db = 2 m m ....................188
68
Focusing map for a beam with Ay = 2.00 and Db = 6 m m ....................189
69
Envelope solutions in y
70
Envelope solutions in y for beam with Ay = 0.27 and Db= 2 mm. . 191
71
Envelope solutions in x for Ay = 0.27. Db = 2 m m. and Am = 2 cm. 192
72
Envelope solutions in y for Ay = 0.27, Db =
73
Predicted envelope ripple for a beam with \ y = 0.27. Db = 2 mm.
194
74
Cross-section of PPQM-focused beam with A y = 0.27. Db = 2 mm.
195
75
Beam envelope in x for
Am = 2 cm with velocity spread.................196
76
Beam envelope in y for
Am = 2 cm with velocity spread.................197
77
Measured beam radius and velocity spread for the M707 gun
78
The rectangular PPQM array.........................................................................199
79
Cross-section of the rectangular PPQM array............................................ 200
80
Geometry of the rectangular-block array.....................................................201
. 184
for beam with Ap = 0.27 and Dj = 2 mm. . 190
2
mm. and Am = 2 cm. 193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
198
121
81
Reduction factor for ly = 1 cm and lx = 3.6 cm .................................... 202
82
Measured transverse field of the PPQM a r r a y . .................................... 203
83
Measured longitudinal field of the PPQM a r r a y . .................................204
84
Average quadrupole gradient of the PPQM array versus theory.
85
Measured beam radius versus distance down the array........................ 206
86
Measured beam radius as a function the quadrupole gradient.
87
Measured beam radius as a function of the beam voltage................... 208
88
Measured initial beam radius as a function of the beam voltage. . . 209
89
Measured initial beam radius as a function of the solenoid field.
90
Ripple and equilibrium curves for Vj, = 5 kV ..........................................211
91
Ripple factor A r predicted by envelope simulations for V& = 5kV. . 212
92
Beam envelope predicted by PIC simulations for \'l = 5 kV..............213
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
. . 205
. . . 207
. . 210
122
10a
klystrons
gyrotrons
gridded
tubes
0
£
o
couple-cavity
TWTs
helix TWTs
Ql
0
O)
0
0
MPMs
>
<
X
10°
0.1
X
solid-state
devices
1
10
100
1000
Frequency (GHz)
Figure 1: Average power plotted against frequency for several types of vacuum
tubes. Corresponding output parameters for several type of solid-state sources are
also shown for reference.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
heater
beam
cathode
anode
input
cavity
output
cavity
collector
Figure 2: Schematic of the klystron amplifier, showing the electron source, trans­
port channel, input and output cavities, and the beam collector. In general, many
vacuum microwave devices share a similar configuration, with different types of
interaction circuits replacing the cavities in this figure.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
circuit or
waveguide
conventional round
beam device
beam
circuit or
waveguide
I
/
multi-beam
configuration
sheet beam
configuration
sneet electron
beam
circuit or
waveguide
Figure 3: The progression of a single round-beam device into an extended-beam
configuration, either multi-beams or sheet beams. Extended-beam devices are
capable of producing higher output powers in a compact source.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
(a)
Fic. 3. (a) to (d). Strip
beam instability for a 1.25cm wide beam V'—120 v,
B —410 gauss, Z.-0.4 m,
/• ■ 7, IS, 32, and 41 maf
respectively, (el Strip beam
purposely excited by elec­
trical deflection r —22 v,
B —850 gauss, 1 -0 .1 5 m,
/*—70 in.
(b)
(O
(d)
(«)
Figure 4: Evolution of the diocotron instability for a sheet beam focused by a
uniform axial guide field.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
Figure 5: Schematic of the PPQM array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
10
8
1.0 kG
6
LL
cr
1.5 kG
4
2
2.0 kG
0
20
40
Vb ( k V )
Figure 6: A comparison of the focusing strengths of PPQM and conventional PPM
arrays.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
top view
i__
-X m
X m
Figure 7: Schematic of the basic configuration of the offset-pole PCM array. In
contrast to planar wigglers used for FEL experiments, the PCM has the same
magnet polarity across the midplane. An offset in the magnets produces sidefocusing in the wide dimension of the beam.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
XZ p I a n e
C
•V
c
X
c
X
1
fl
1
4
I
Zn
«
t/
ft
?
I
1C
T
c
1C
I•
YZ p i a n e
1
I
C
1
2
1
4
ft
Zr* ■ c / I ■
Figure 8: Plots of the normalized electron position and velocity in the semi-infinite
PCM array. The high-frequency oscillations correspond to the periodicity of the
magnets, while the low-frequency oscillations correspond to betatron oscillations
for the unm atched beam.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
XZ
p i*ne
C
•u
I
2
T
I
Figure 9: M atched beam transport for B 0 = 507 G. Note the absence of betatron
oscillations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
IB
E
E 0
sn
-IB
8
IB
H(mm)
IB
IB
Cd).
B
.
►
•
-IB
H(mm)
ie
IB
(e)
E
E B
B
x(mm)
18
ib
cfl.
a
-IB
B
K(mm)
ta
B
x(mm)
ib
Figure 10: Simulation dem onstrating stable transport of a semi-infinite sheet beam
in a rapidly oscillating m agnetic field corresponding to a PCM array with a period
of 3.75 mm and peak field B Q = 0.07 T. Initial beam conditions were Pmax/Pmin
= 1.6, pmin = 2.06 x 10~3 C /m 3, a thickness of 2.0 m m, and an initial velocity of
u0 = 5.9 x 107 m /s, corresponding to a beam energy of 10 keV. The time (and
distances normalized to the magnet period) for plots a -f Me: t = 0.0 ns (z/Am =
0.0), 0.5 ns (8.8), 0.7 ns (12.3), 1.2 ns (21.2), 1.8 ns (31.7), and 12.4 ns (218.7
periods), respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
IB
E
E B
a
-IB
-1 8
B
IB
(c).
IB n
y
i
B
•
►
«
-IB
x(mm)
ie
IB
Ml
B
i
•
B
x(mm)
IB
10
IB
(e).
E
E B
-IB
0
x(mm)
10
IB
cn.
B
-18
-IB
x(mm)
10
0
K(mml
10
Figure 11: Evolution of the diocotron instability for a semi-infinite sheet beam in
a uniform axial magnetic field with am plitude B 0 = 0.07 T. The beam param eters
are the same as in the preceding plot. The tim e (and distances normalized to the
magnet period) for plots a -f are: t= 0.0 ns (z/Am = 0.0), 0.5 ns, 1.4 ns, 2.8 ns,
4.6 ns, and 11.5 ns, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
charge
density 11
POO
A
horizontal coordinate, x
Figure 12: Model of the density bump used to seed the MAGIC simulations for the
lowest-order diocotron mode. Here A was chosen to be 2.0 mm in the simulations.
Longer run-tim e simulations without the bump verified the final results.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
Figure 13: Magnetic equivalent of the offset-pole PCM.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
Figure 14: Magnetic surface-charge model for analytic solution of the side fields
near the transition region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
136
4
m
y.s
Y/X =-0.50
2
-0.25
0.00
0
025
0.50
2
-4
1
0
1
Figure 15: Plot of the normalized side fields B Xt3 and B y_s for several aspect ratios
2/m/^m- The horizontal axis in the x-coordinate normalized to the magnet width,
2m -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
137
Figure 16: Illustration of beam tilt in offset-pole PCM array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
-
2
-
1
0
1
2
Figure 17: PIC simulations of a high current density elliptical sheet electron beam
in an offset-pole PCM array with ij, = 1.4 cm, y* = 0.025 cm, A = 10 A , V& =
10 kV. The magnet array has Am = 3 mm, B 0 = 1200 G, B l0 = 534 G, x m = 4.0
cm, and ym = 0.5 cm. From top-to-bottom, these plots correspond to 0.0, 4.64,
9.28, 13.93, 18.57, 23.21, and 46.42 magnet periods, respectively. Both the x and
y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
139
-
0.2
0
0.2
-
0.2
0
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
''$%t:
V.SJV. ~.\i< •**»"«•
0
-
A
0.2
-
2
^
-
'
.HV*.
-
1
0
1
2
Figure 18: PIC simulations of a high current density elliptical sheet electron beam
in a uniform magnetic field with Bxo = 1200 G. The beam parameters and the
times at which the cross-section is plotted are the same as in the previous figure.
Both the x and y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
140
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
1
1
0
-
0.2
-
2
-
1
0
1
2
Figure 19: PIC simulations of a high current density elliptical sheet electron beam
in an offset-pole PCM array with x b = 1.4 cm, yb = 0.025 cm, Ib = 50 A , Vb =
10 kV. The m agnet array has Am = 3 mm, B a = 4000 G, B to = 2515 G, xm = 4.0
cm, and ym = 0.5 cm. From top-to-bottom, these plots correspond to 0.0, 4.64,
9.28, 13.93, 18.57, 23.21, and 46.42 magnet periods, respectively. Both the x and
y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
141
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-
0.2
-
2
-
1
0
1
2
Figure 20: PIC simulations of a high current density elliptical sheet electron beam
in a uniform magnetic field with B xo = 2828 G. T he beam parameters and the
times at which the cross-section is plotted are the same as in the previous figure.
Both the x and y axes axe in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
142
0.2
0
-
0.2
0.2
0
-
0.2
0.2
0
-0.2
*•1
0.2
0
-0.2
0.2
-
0
0.2
0.2
0
-0.2
0.2
0
-
0.2
-
2
-
1
0
1
2
Figure 21: PIC simulations of a elliptical sheet electron beam in an offset-pole
PCM array with Am = 1 cm. Here /& = 10 A, xj, = 1.4 cm, y& = 0.025 cm, Vb
= 10 kV, B0 = 1200 G, B ao = 534 G, xm = 4.0 cm, and ym = 0.5 cm. From
top-to-bottom, these plots correspond to 0.0, 1.39, 2.78, 4.18, 5.57, 6.96 and 13.93
magnet periods, respectively. Both the x and y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
143
0.5
_0 s—.—.—.—,—
-5.0
-2.5
0
-------------------2.5
5.0
Figure 22: Evolution of an elliptical sheet electron beam in an offset-pole PCM
array with x b = 2.7 cm, yb = 0.1 cm, Ib = 2 A , Vb = 10 kV. The magnet
array has Am = 1 cm, B 0 = 250 G, B t0 = 157 G, xm = 5.0 cm, and ym = 0.5
cm. From top-to-bottom , these plots correspond to 0.0, 5.5, 11.0, 16.5 magnet
periods, repsectively. B oth the x and y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
air
iron
Y (m)
1
I
-l.294e«ai
L
X
J
I
L
•l.84Be«ll
X
X
X
X
«.848£*flt
X
J.
X
X
I.254C»|1
X
fl.4?4£»|i
X (m)
Figure 23: Model of the offset-pole PCM configuration used in the 3-d magnet
design code TOSCA. The pole geometry is shown in the baseplane and the core
and coil dimensions are listed in the Sec. 2.2.2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
145
0.5
E
w_
o
c
x = 0.00
x = 1.95
x = 2.28
x = 2.60
x = 2.93
o
CD
-0.5
-1.0
1
0
3
2
Z (cm)
1.0
. analytic
result for
0.8 -perm , m agnets
o
c
C
O
CD
0.6
air core
0.4
0.2
0
1010 steel
core
0
1
2
3
4
5
X ( cm )
Figure 24: Predicted B VtS side-fields for the offset-pole PCM array as a function
of the transverse dimension x. Good agreement with the analytic expressions is
obtained for the air-core model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
/
./
Figure 25: Layout of the hybrid PCM /quadrupole configuration before canting the
side magnets to provide a linear field gradient.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
Figure 26: Quadrupolar side magnetic fields for beam edge-focusing are created
by tilting the side magnet array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.1
-0.1
Figure 27: Evolution of high current density elliptical sheet electron beam in an
hybrid PC M /quadrupole array with
= 1.4 cm, y& = 0.025 cm, h = 10 A , Vj
= 10 kV. T he m agnet array has Am = 3 m m , B 0 = 1200 G, B q/ x m = 3500 G/cm ,
x m = 4.0 cm, and ym = 0.5 cm. From top-to-bottom , these plots correspond to
0.0, 4.64, 9.28, 13.93, 18.57, 23.21, and 46.42 magnet periods, respectively. Both
the x and y axes are in units of centimeters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
Quadrupole
Magnet
Quadrupole
Magnet
Figure 28: Schematic of the quadrupole beam-forming system with transverse
x- (solid) and y- (dashed) beam envelopes indicated. An initially round input
beam is transform ed into a high aspect-ratio ellipse through the use of m agnetic
quadrupoles.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
2
1
0
1
-2
0
.5
1.0
1.5
2.0
2.5
Z„ = Z/L
Figure 29: Beam envelope trajectories through the magnetic quadrupole pair
in the presence of space-charge (solid curves) and without space-charge (dashed
curves). The envelope trajectory in the x z plane (Xn) is strongly defocused due
to space-charge and highly elliptic paraxial flow is possible.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
z =L/4
U2
o
0.8
go
*N
;x
X
CD
0.4
-
2
-
1
0
1
2
Z/L
Figure 30: The bell-shaped quadrupole model used in the 2-d envelope analysis.
The dashed line is the equivalent ideal quadrupole model for r 0 = L f 4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
1.5
1.0
>
X
c
c
0
2
1
3
Z„ = Z/L
Figure 31: Envelope trajectories for a sheet-beam with ellipticity of 22.5 using
round-beam param eters from an existing commercial gun. Third-order effects of
longitudinal velocity variations and a realistic magnetic field are included in the
model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
153
1.5
1.0
5
0
5
0
2
Z„ = Z / L
Figure 32: Magnetic field used in calculating the envelope.
quadrupole gradients of 64 G /cm and 18 G /cm are required.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Only modest
154
4.0
E
o
.Q
CO
Quad 2
Quad 1
2.0
60.7 G/cm
0.0
0.0
5.0
10.0
Z ( cm )
Figure 33: 2-d envelope solution of a two quadrupole matching section for the the
beam and lattice param eters listed in Tables 1 and 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
Y(cm) vs Z(cm)
4.0
0.0
-4.0
0.0
5.0
10.0
Figure 34: 3-d PIC simulation of the two quadrupole matching section for the
beam and lattice param eters in Tables 1 and 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
156
2.7
0
•2.7
0
5
10
Z ( cm )
Figure 35: 3-d beam envelope simulation of the two quadrupole m atching section
for the beam and lattice param eters in Tables 1 and 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
0
5
10
15
Z (cm)
Figure 36: 3-d beam envelope simulations of a four quadrupole m atching section
for the beam and lattice param eters in Tables 3 and 4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
158
X(cm) vs Z(cm)
40
0.0
10.0
20.0
Y(cm) vs Z(cm)
Figure 37: 3-d PIC simulation of the four quadrupole m atching section for the
beam and lattice param eters in Tables 3 and 4. At the extraction plane (z = 18.5
cm) the beam diam eters are approxim ately 2a = 4.8 and 26 = 0.2 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
1.0
E
0.0
>-
-
1.0
-4.0
-2.0
0.0
2.0
4.0
X(cm)
Figure 38: 3-d PIC simulation of the four quadrupole m atching section for the
beam and lattice param eters in Tables 3 and 4. The beam cross-section at the
extraction plane (z = 18.5 cm) is shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160
Ex (rc-mm-mrads)
100.0
50.0
0.0
0.0
10.0
20.0
Z (cm)
Figure 39: Evolution of rms beam em ittance in x-x' phase-space.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
10.0
(spejiu-ujuj-u) ><3
5.0
0.0
0.0
10.0
20.0
Z (cm)
Figure 40: Evolution of rms beam em ittance in y-v' phase-space.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
3.5
50
25
0
•25
0
5
10
50
15
Z (cm)
Figure 41: 3-d beam envelope simulation of the four quadrupole matching section
using a realistic fringe field model based on PM Q’s. The total quadrupole gradient,
GT(■?), is plotted on the right vertical axis for reference. Magnet parameters are
listed in Table 5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
Figure 42: Experim ental configuration to investigate sheet beam formation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
2.0
<
0.5
0
2
6
4
8
10
V ( kV)
Figure 43: Measured gun perveance for the M690 gridded electron source. The
solid line represents a perveance of 2.00 /xpervs and the dashed line is a best-fit
value to the m easured beam current. The fitted perveance of the gun is 1.90(±
0.05) /zpervs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
1000
40
800
600
E
o
cc
400
quad
flux
shields
gun
flux
shield
0
solenoid
matching
coils
30
60
200
90
120
Z (cm)
Figure 44: Predicted axial magnetic field and the geometry of the magnetic circuit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
166
1800
= 200 A
Pandira
(1010 table)
1200
R ( cm)
160
N
100
600
-30
0
30
60
90
120
Z ( cm)
Figure 45: Measured axial magnetic field of the solenoid m atching section. Good
agreement is found with the predicted field profile.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
167
tY
300 G ( 3 / R = 63 G /c m )
B ean
1010 ste e l
440 A m p - t u r n s
(5.5 A /tu rn )
0«oortm«nf of Eloctn'col
o
Coom
terisconsin/U
£nsin«nns
dmvne
orjity
fpuW
adison
----------_
ExraW8
JJ&-JTU
Quadrupole
, t (ref,ptLayout
»\ )
aneA
..
i— H a s t e n
'
r
i
S
i
~
-
'-! „-----------
Figure 46: Cross-section of the magnetic quadrupole lens and the transverse mag­
netic field predicted by PANDIRA. The coil. beam , and vacuum tube cross-section
are shown for reference.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
200
quadrupole outline
(1010 steel)'
B/R = 40 G/cm
150
icoil \
\(80 turns, 3.5 A)
0
100
E
o
m
reflection planes
(octal symmetry)
-*
0
2
4
6
8
10
X ( cm )
Figure 47: The quadrupole octant used in the 2-d PANDIRA magnet design sim­
ulations. The calculated magnetic field is shown as a function of the transverse
distance along the magnet.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
169
1200
- standard 1010 B-H table (Poisson)
900
o
DC
600
B / R = 0.146 I
o
00
300
0
2000
4000
6000
8000
Amp-turns (A)
Figure 48: Simulated saturation of the quadrupole lens does not occur until over
4000 amp-turns of current - well above the design range of the magnets.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
170
0.2
Y = 5.72 cm
3.82 cm
1.91 cm
0.00 cm
0
X
CO
-
0.1
-
0.2
0
5
10
15
20
25
Z ( cm )
Figure 49: Measured magnetic field of the quadrupole lattice for a 1 A drive
current. The transverse position of the probe is listed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.2
0.1
0
i
o
X
CD
-
0.1
-
0.2
0
5
10
15
20
25
Z ( cm )
Figure 50: Measured magnetic field of the quadrupole lattice for a 1.5 A drive
current. The transverse positions of the probe are as in the previous figure.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
172
40
20
E
o
O
0
O
-20
-40
0
5
10
15
20
25
Z ( cm )
Figure 51: Measured magnetic field gradient of the quadrupole lattice for equal
drive currents of 1.0 A.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
40
20
E
o
O
0
O
-20
-40
0
5
10
15
20
25
Z ( cm )
Figure 52: Measured magnetic field gradient of the quadrupole lattice for equal
drive currents of 1.5 A.
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
174
0.4
Y = 5.72 cm 3.81 cm .
1.91 cm
0.00 cm
0.2
O
X
-
0.2
-0.4
5
10
15
20
25
Z ( cm )
Figure 53: Measured m agnetic field of the quadrupole lattice for the drive currents
listed in Table 8.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
permanent
magnet
model
50
E
o
0
0
O
measured
gradient
profile
-50
0
5
10
15
20
25
Z ( cm )
Figure 54: Measured magnetic field gradient of the quadrupole lattice for the
drive currents listed in Table S. This is the final experimental configuration. A
comparison to the fringe field model used in the 3-d beam envelope analysis of Sec.
3.2 shows very good agreement.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B ( G auss/10 )
176
200
200
150
150
100
100
-50
-50
60
80
100
120
Z ( cm )
Figure 55: Overlap of the quadrupole and solenoidal magnetic fields.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
177
y = 1.91 cm _
Gt ( G / c m )
1
0
1
0
4
2
Z ( cm )
Figure 56: Measured quadrupole gradient near the flux shield for y = 1.91 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
178
2
y = 3.82 cm
Gt ( G / c m )
1
0
1
2
0
4
2
Z (cm)
Figure 57: Measured quadrupole gradient near the flux shield for y = 2.82 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
*179
Anode
Cathode
g*
Heater
Velonex 570
Pulsar
(10 kV, 2A, 100ns)
To Cathode
Heater Supply (6.3Vac, 5A)
Figure 58: Schematic of the M690 pulsing circuit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
180
v<ss«.l
uj«JI
O .H * C . \ Cm
%
Figure 59: Reproduction of the beam image of the 4 cm x 0.4 cm sheet electron
beam.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
r
r
T
T
T
T
T
T
T
Figure 60: Beam voltage, current and collected current waveforms.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
182
X vs
. 04
Z
-------------------------------------------------------------------------------------
.02
Figure 61: Plot of the beam cross-section in x z and yz when a 100 g solenoidal
field is transposed on the quadrupole lattice. The beam dynamics are severely
affected.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
183
vs
02
01
-
r# g f r j L
s.jnsn' f>£.'
-
. 01
-
. 02
s
s
s
s
Figure 62: Plot of the beam cross-section in it/ at : = IS.5 cm when a 100 g
solenoidal field is transposed on the quadrupole lattice. The axes are in units of
meters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
184
Figure 63: Cross-section of a round aperture perm anent magnet quadrupole.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
185
3
a=2p [PPM]
2
STABLE
UNSTABLE
a=Ap
1
PPQM (space-charge)
0
a=0 [PPQM]
STABLE
P=0.88
1
0
0.5
1.0
1.5
P
Figure 64: Parameter-space map of the specialized M atthieu's equation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.0
186
Xm/2
Figure 65: Focusing instability in PPQM systems occurs when the thin-lens focal
length approaches the physical length of the magnet half-period.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
187
4000
3000
10 kV
2
O
CL
CL
E
2000
2 0 kV
o
O)
CD
1000
= 100 kV
0
1000
2000
3000
4000
B0 ( g ) [PPM]
Figure 66: Comparison of PPQM gradient versus the equivalent PPM field mag­
nitude.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
188
10000
ripple
condition
A = 0.27, D. = 2 mm-
5000
E
2000
o
03
°
1000
- equilibrium
- condition
500
V
200
0
1
2
Xm ( cm )
Figure 67: Focusing map for a beam with Afj. — 0.27 and Db = 2 m m.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
189
10000
ripple
condition
A = 2.0, Dh = 6 mm
G ( g/cm )
5000
2000
1000
"equilibrium
500 -condition
200
0
1
3
2
m (v Cm )7
Figure 68: Focusing map for a beam with A f i = 2.00 and
Df , =
6 mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
190
2.0
1—
i— i— i
|— i i
i
i
|
i
i
i
T 1----1----1---
r
= 1.0 cm _
n
1.5
XC
1.0
0.5
0
J
I
L
I
I
I
l
I
I
10
I
1
j
L
i
15
i
i__
20
ZA,m
Figure 69: Envelope solutions for a beam with Afj. = 0.27 and Db = 2 mm. Here
the magnet period and gradient is 1 cm and 1S70 g/cm .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
2 .0
1.5
1.0
0.5
0
0
5
10
15
20
ZA,m
Figure 70: Envelope solutions for a beam with Afi = 0.27 and Df, = 2 mm. Here
the magnet period and gradient is 1 cm and 1S70 g/cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
192
2 .0
X = 2.0 cm
1.5
1.0
0.5
0
0
2
4
6
8
10
Z /A .m
m
Figure 71: Envelope solutions for a beam with Afj. = 0.27 and Dj = 2 mm. Here
the magnet period and gradient is 2 cm and 936 g/cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
193
2 .0
2.0 cm
.5
.0
0.5
0
0
2
6
4
8
10
Z/km
Figure 72: Envelope solutions for a beam with A/z = 0.27 and Db = 2 mm. Here
the m agnet period and gradient is 2 cm and 936 g/cm .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
194
6
= 0.27 jipervs, Db = 2 mm
= 2 cm
4
2
0
0
1000
2000
3000
4000
G ( g/cm )
Figure 73: Predicted ripple from envelope solutions on a beam with Afi = 0.27,
Db = 2 mm, and Am = 1 and 2 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
195
0.50
10.5
0.25
E
o
>*
-0.25
-0.50
14.0
E
o
0.50
21.0
7.0
§
E
u
-0.25
-0.50
-0.50
0.25
-0.25
x ( cm)
0.50
-0.50 -0.25
0
0.25
0.50
x (cm)
Figure 74: Cross-sectional plots of a PPQNl-focused beam with
2 mm. Here Xm = 1 cm and G = 1871 G /cm 2.
A./J.
= 0.27, Db
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
196
2.0
.0 cm
^ vz/vz * 0 . 1 %
1.5
1.0
0.5
0
0
5
10
Z IX
15
20
m
Figure 75: The beam envelope in x predicted by PIC simulations for a beam with
Afi = 0.27 and Db = 2 m m and an initially therm al beam with a rms velocity
spread of 8vx/ v z = 0.001. Here the magnet period and gradient is 2 cm and 936
g/cm . Here the magnet period and gradient is 2 cm and 936 g/cm . The filled
circles show the normalized envelope as a funtion of r when an initially therm al
beam is simulated. This can be compared to the result for an initially cold beam
(open circles) and the result from beam envelope simulations (solid line).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
I
2 .0
I
I
r
I m = 1.0 cm
1.5
> C 1.0
i •
/ <: • j
0.5
•
u
•
i\
t
i:\■•/ \t•i \i
w i >
•
\*\
;
*' *
1 ••
5
J
0
0
!
I
L
I
5
I
I
I
I
10
I
I
= 0 .1 % J
I
15
I
I
20
m
Figure 76: T he beam envelope in y predicted by PIC simulations for a beam with
Ay. = 0.27 and Db = 2 mm and an initially therm al beam with a rms velocity
spread of 8vs/ v z = 0.001. Here the magnet period and gradient is 2 cm and 936
g/cm. The filled circles show the normalized envelope as a funtion of r when an
initially therm al beam is simulated. This can be compared to the result for an
initially cold beam (open circles) and the result from beam envelope simulations
(solid line).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Sv^v., (% )
198
0.8
0.6
fit-to-point
rms velocity spread
E
E
b e st fit
0.2
0
500
1000
1500
2000
Guide Field Bz ( G )
Figure 77: iVIeasured beam radius and velocity spread for the M707 gun.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7S: The rectangular PPQM array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 79: Cross-section of the rectangular PPQM array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
201
Figure 80: Geometry of the rectangular-block array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
202
0.7 5
q.max = 0 .6 4 B
0.6 0
0.45
c
0
0.30
0.15
1
2
Tj ( cm )
Figure 81: Reduction factor for ly = 1 cm and lT = 3.6 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
203
10
~i
i
|
i
i
i
i
|
i
i
i
i
|
i
i
i
i
[
i
i
i
5
CD
-5
j
-10
0
15
10
i
20
i
i
i
25
Z ( cm )
Figure 82: Measured Bx transverse magnetic field of the PPQM array for v = 0.0,
0.10, 0.26, 0.57, 0.73, and 0.S9 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
204
0.010
0.005
N
CO
-0.005
-
0.010
0
5
10
15
20
25
Z( cm)
Figure 83: Measured B z transverse magnetic field of the PPQM array for y = 0.0,
0.10, 0.26, 0.57, 0.73, and 0.89 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
V
205
10
5
E
o
O
0
N
o
5
-10
0
5
15
10
20
25
Z (c m )
Figure 84: Average measured quadrupole gradient of the PPQM array (dashed
line) versus the calulated value (solid line).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
206
5
Vb = 5 kV, lb = 94 mA
G = 825 g/cm, Bs = 1575 g
4
3
2
1
0
0
5
10
15
20
z ( cm )
Figure 85: Measured beam radius as a function of distance down the PPQ M array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
207
10
v b = 5 kV, lb = 94 mA
z = 16.8 cm, Bs = 1575g
6
,
b ( m m )
8
4
2
0
500
1000
1500
2000
G ( g / c m)
Figure 86: Measured beam radius as a function of the quadrupole gradient.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
208
5
z = 16.8 cm
G = 825 g, Bs = 1575 g
4
( UJIU ) q ‘
3
2
1
0
4
6
8
vb ( kV)
Figure 87: Measured beam radius as a function of the beam voltage.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
209
3
z = -1.0 cm
Bs = 1575 g
rb ( m m )
2
1
0
4
6
8
Vb ( k V )
Figure 88: Measured initial beam radius as a function of the beam voltage.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
210
3
_ z = -1.0 cm
- V. = 5 kV, L = 94 mA
2
( LUIU ) QJ
1
0 *—
1000
1200
1400
1600
1800
Bs ( 0 )
Figure 89: Measured initial beam radius as a function of the solenoid field.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
211
10000
A, = 0.27, D, = 2 mmripple
condition
5000
E
o
= 5 kV
2000
O)
°
1000
500 -equilibrium
condition
200
0
2
1
3
L m (v c m ')
Figure 90: Ripple and equilibrium curves for A „ = 0.27. Db = 2 m m . and Vb = 5
kV.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
212
5
Vh = 5 kV, L = 0.095 A
4
3
2
1
0
0
200
400
600
800
1000
G ( g / c m)
Figure 91: Ripple factor A r predicted by envelope simulations for Vj = 5 kV, A^
= 0.27, and Db = 2 mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
213
6
Vb =
5
kV, lb = 0.095 A
Xm= 2 cm, G = 848 g/cm
4
2
0
0
5
10
15
Z ( cm )
Figure 92: Beam envelope predicted by PIC simulations for Vj = 5 kV.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
214
List o f Tables
1
Initial and final beam parameters for a two-quadrupole matching
section................................................................................................................. 215
2
Lattice param eters for a two quadrupole m atching section......................216
3
Initial and final beam parameters for a four quadrupole sheet beam
m atching section...............................................................................................217
4
Lattice param eters for a four quadrupole sheet beam matching section.218
5
Perm anent magnet quadrupole param eters for a four quadrupole
m atching section with fringe fields............................................................... 219
6
Nominal beam and operating param eters for the M690 electon gun.
7
Specifications for the quadrupole magnet array..........................................221
8
Quadrupole magnet parameters for the experim ental configuration.
9
Quadrupole field gradients used in the experim ent................................... 223
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
220
222
215
B eam
P a r a m e te r
Energy (£b)
Current (I)
x- Radius (a)
x- Div. (a')
y- Radius (b)
y- Div. (b')
x-emit. (cr )
y-emit. (ey)
In itia l
10 keV
-2.0 A
0.60 cm
0.0
0.60 cm
0.0
4.6rr
4.6-
2-d
10 keV
-2.0 A
3.9 cm
0.0
0.1 cm
0.0
4.6 tt
4.6-
F in al
T R A C E 3 -D
10 keV
-2.0 A
2.1 cm
0.11
1.4 cm
0.27
4.64.6-
W ARP
10 keV
-2.0 A
2.0 cm
0.10
1.2 cm
0.2S
9.0 tt
10.6-
Table 1: Initial and final beam param eters for a two-quadrupole matching sec­
tion. The initial and final beam states are measured at z = 0 and at z = 10 cm.
Em ittance values are in units of m m — mrads.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
M agnet
P a r a m e te r
Field Gradient (B0/R<,)
Effective Axial Length (L)
Axial Center Position
Q u ad 1
60.7 G /cm
2.S cm
2.9 cm
Q uad 2
-26.3 G/cm
2.S cm
7.4 cm
Table 2: Lattice param eters for a two quadrupole matching section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
217
B eam
P a r a m e te r
Energy (Sb)
Current (I)
x- Radius (a)
x- Divergence (a')
y- Radius (b)
y- Divergence (b')
x-em ittance (ex)
y-em ittance (ey)
In itia l
10 keV
-2.0 A
0.30 cm
0.0
0.30 cm
0.0
9.27rmm-mrad
9.27rmm-mrad
F in a l
W ARP
T R A C E 3 -D
10 keV
10 keV
-2.0 A
-2.0 A
2.7 cm
2.4 cm
0.0
0.0
0.1 cm
0.1 cm
0.0
0.0
9.27rmm-mrad 100~mm-mrad
9.2:rmm-mrad 9.2:rmm-mrad
Table 3: Initial and final beam parameters for a four quadrupole sheet beam
matching section. The initial and final beam sta te are measured at z = 0 and at
z = 18.5 cm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
218
M agnet
P a r a m e te r
Field Grad. (B 0/ R 0)
Eff. Axial Length (L)
Axial Center Pos.
Q uad 1
30.8 G /cm
2.8 cm
2.9 cm
Q uad 2
-17.3 G/cm
2.S cm
7.2 cm
Q uad 3
59.5 G /cm
2.8 cm
11.6 cm
Q uad 4
-22.5 G /cm
2.8 cm
15.9 cm
Table 4: Lattice param eters for a four quadrupole sheet beam m atching section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
219
M agnet
P a r a m e te r
Max. Field Grad.
Physical Length
Clear Bore Rad.
Outer Assembly Rad.
Axial Center Pos.
Q uad 1
90.0 G /cm
1.9 cm
4.0 cm
7.5 cm
2.9 cm
Q uad 2
-117.6
1.9
4.0
7.5
7.2
G /cm
cm
cm
cm
cm
Q uad 3
180.4 G /cm
I.9 cm
4.0 cm
7.5 cm
II.6 cm
Q u ad 4
-84.9 G /cm
1.9 cm
4.0 cm
7.5 cm
15.9 cm
Table 5: Perm anent magnet quadrupole parameters for a four quadrupole matching
section with fringe fields.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
220
P a r a m e te r
S p e c ific a tio n
Beam Voltage ( Vj)
Beam Current (/<,)
Perveance (Qfc)
Cathode Radius (rc)
Focal Length ( / c)
Beam Radius at Focus (r;,0)
Tranverse x-emittance (er )
Tran verse v-emittance (ey)
Axial Velocity Spread {Sv:/ v :o)
Cathode Heater Voltage (Y\)
Cathode Heater Current (/*)
Cathode Tem perature (Tc)
10 kV
2.0 A
2.00 ^pervs
0.762 cm
2.44 cm
0.30 cm
9.2;r-mm-mrads
9.2~-mm-mrads
0.004S
6.3 Vac
4.95 Aac
1150° C
Table 6: Nominal beam and operating param eters for the M690 electon gun.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
221
M a g n e t P a r a m e te r
S p e c ific a tio n
Inner Radius (Rq)
Axial Length (/?,)
Coil Width (Ii'c)
Coil Height (Hc)
Turns /Coil ( Nc)
Nom. Peak Gradient (G)
Max. Current
Max. Temperature
4.2 cm
1.91 cm
l.S cm
1.8 cm
140
60 G /cm
6 A
150° C
Table 7: Physical and magnetic specifications for the constructed quadrupole lens
array.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
222
M agnet
P a r a m e te r
Nom. Field Grad.
Physical Length
Clear Bore Rad.
Nom. Current
Center Pos.
Q u ad 1
Q uad 2
30 G/cm
1.9 cm
4.2 cm
1.6 A
2.9 cm
-30 G /cm
1.9 cm
4.2 cm
l.S A
7.2 cm
Q uad 3
60 G/cm
I.9 cm
4.2 cm
3.1 A
II.6 cm
Q uad 4
-30 G /cm
1.9 cm
4.2 cm
l.S A
15.9 cm
Table 8: Quadrupole magnet parameters for the experimental configuration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
223
Q u a d ru p o le R a d iu s
1.55
1.S4
2.22
2.4S
2.86
3.10
cm
cm
cm
cm
cm
cm
Q u a d ru p o le G ra d ie n t
2017 g/cm
1650 g/cm
1283 g/cm
1054 g/cm
825 g/cm
733 g/cm
Table 9: Quadrupole field gradients used in the experiment.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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