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Design and operation of an inverse free -electron -laser accelerator in the microwave regime

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Design and Operation of an Inverse Free-Electron-Laser Accelerator
in the Microwave Regime
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
Rodney Bruce Yoder
Dissertation Director: Prof. Jay L. Hirshfield
December 2000
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UMI Number 9991248
Copyright 2001 by
Yoder, Rodney Bruce
All rights reserved.
UMI
UMI Microform9991248
Copyright 2001 by Bell & Howell Information and Learning Company.
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©
2001
by Rodney Bruce Yoder
All rights reserved.
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Contents
Acknowledgm ents
1
Introduction
1
1.1 Historical sketch.............................................................................................
2
....................................................................
6
1.3 IFEL th e o ry ...................................................................................................
9
1.2 Essential physics of the IFEL
2
vii
1.3.1
Simplified r e s u lt s ............................................................................
10
1.3.2
Three-dimensional results
IT
............................................................
Simulation Param eters and R esults
24
2.1 MIFELA p a ra m e te rs ...................................................................................
26
2.1.1
Entry reg io n.....................................................................................
26
2.1.2
Acceleration re g io n .........................................................................
27
2.1.3
Extraction re g io n ............................................................................
29
2.2 Beam p a ra m e te rs ..........................................................................................
29
2.3 Acceleration re s u lts ......................................................................................
31
2.4 Accelerated beam p ro p erties.......................................................................
33
3 Experim ental D esign and Construction
37
3.1 Design overview .............................................................................................
37
3.2 MIFELA com ponents....................................................................................
38
3.2.1
Acceleration s tr u c t u r e ...................................................................
38
3.2.2
RF input and output co u p lin g ......................................................
41
ii
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iii
4
3.2.3
Axial magnetic field
.......................................................................
43
3.2.4
Wiggler magnetic f i e l d ....................................................................
44
3.3 Auxiliary s y ste m s.........................................................................................
55
3.3.1
RF power d is trib u tio n ....................................................................
55
3.3.2
RF gun.
MeV beamline and diagnostics....................................
57
3.3.3
S p e c tro m e te r....................................................................................
58
3.3.4 Vacuum and mechanical s u p p o rts .................................................
63
3.3.5 Radiation shielding...........................................................................
66
6
D ata C ollection M ethods and R esults
69
4.1 Experimental m ethod....................................................................................
69
4.1.1
T im in g ................................................................................................
69
4.1.2 P ro c e d u re ..........................................................................................
70
4.1.3 Data a n a ly s is ....................................................................................
72
4.2 Acceleration re su lts .......................................................................................
73
4.2.1 Experimental d a t a ...........................................................................
74
4.2.2 Comparisons with theory and co m p u tatio n .................................
78
5 Scaling Possibilities and C onclusion
84
5.1 Scaling the M IF E L A ....................................................................................
84
5.2 Summary and co nclusions...........................................................................
87
B ibliography
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90
List of Figures
1.1
Sketch illustrating the IFEL mechanism
.................................................
1.2 Graph of the IFEL potential function.............................................
18
1.3 <&-AE phase-space orbits for the IFEL accelerating p o te n tia l...
18
8
1.4 Graph of 3. vs. axial field for steady-state orbits in the F E L .........
23
2.1 Calculated ideal wiggler and axial magnetic fields in MIFELA
....
28
2.2 Average computed particle energy in MIFELA vs. axial distance . . .
32
2.3 Average computed transverse and axial velocity components of parti­
cles in MIFELA. vs. axial distance
..........................................................
32
2.4 Calculated electron energy vs. distance for various injected ernittances
34
2.5 Computed longitudinal phase space at exit of M IF E L A .............
36
2.6
Computed beam spot at exit of M IFE LA ...............................................
36
3.1
Schematic outline of the MIFELA e x p e rim e n t.....................................
39
3.2 Scaled drawing of the M IF E L A ......................................................
39
3.3 Drawing of RF input and output couplers for M IF E L A .............
42
3.4 Detail of wiggler magnet construction.............................................
45
3.5 Schematic of wiggler shunt c irc u it...................................................
47
3.6 Wiggler shunt positions and calculated field t a p e r .......................
47
3.7 Measured DC wiggler f i e l d ................................................................
49
3.8 Circuit diagram for the wiggler magnet pulsed-current supply
3.9
....
49
Typical current pulse shape for the wiggler magnets u p p l y .................
50
3.10 Comparison of wiggler field vs. time inside and outside ofMIFELA .
53
iv
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3.11 Drawing of the RF distribution system for M IF E L A ...........................
56
3.12 Sealed drawing of the BPL
59
6
MeV b e a m lin e ........................................
3.13 Electron beam spectra on entrance to MIFELA for several slit positions 60
3.14 Sample current traces in the beam line.....................................................
61
3.15 Beamline arrangement and spectrometer at MIFELA e x it..................
64
3.16 Computed beam positions at the spectrom eter.....................................
65
3.17 Layout of the B P L .....................................................................................
67
4.1
Comparison of null spectra with and without wiggler and RF fields
4.2
Measured energy spectra for two input phases at
4.3
Measured output energy for an input beam energy of 5.24 MeV. versus
6
.
75
MW input power .
77
p h ase..............................................................................................................
79
4.4
Comparison of simulation with experiment for a 5.24 MeV injected beam 80
4.5
Simulation results compared with data for a 5.62 MeV injected beam,
accelerating phase.........................................................................................
4.6
82
Simulation results compared with data for a 5.62 MeV injected beam,
decelerating phase.........................................................................................
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List of Tables
2.1
Structure parameters for simulation of MIFELA
vi
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Acknowledgments
When I began tins thesis. I had no idea how complicated and time-consuming it
would turn out to be. I owe a huge debt of gratitude to many people who contributed
their expertise and time to what became a large and multifaceted project.
Dr. Jay Hirshfield. my advisor, deserves tremendous credit for guiding me
through the stages of the experiment, inspiring me when I felt that it was all im­
possible. and lending the benefit of his experience when problems arose. Thank you.
Jay, for trustworthy advice on a huge variety of subjects: for your moral, professional,
and financial support: for always having my interests at heart: and for fostering an
atmosphere of collaboration and collegiality in the Beam Lab and Omega-P. The rest
of the folks around the lab deserve lots of thanks as well. Mike LaPointe. who wears
many hats around the BPL but manages to run the lab in between everything else,
was generous with time, assistance, and advice. When I needed to accomplish any­
thing in the lab. Mike always knew the simplest method and generally helped me get
it going with a smile. Thanks for the mentoring. Mike—you taught me most of what
I know about laboratory skills. Mei Wang, fellow grad student, set up and perfected
the beamline that delivered electrons to the MIFELA, without which I couldn't have
done a thing. Saveliy Finkelshteyn. our invaluable technician, was perennially cheer­
ful even when repairing the magnet power supplies for the n-th time, and produced
reliable electronics units for anything I needed. Jimmy Fang joined us towards the
end of the project, and helped out with several unpleasant tasks (moving the lead
wall, anyone?). Thank you all for your help— for pitching in with major van mm
work, giving me the benefit of your ideas, and generally helping to make the BPL a
friendly place to work.
vii
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v iii
I relied upon the foundational work of a number of people when I joined the
MIFELA effort. The design of a microwave IFEL was begun in the early 1990s by a
collaboration consisting of Jay Hirshfield at Yale/Omega-P. Prof. Thomas Marshall
and Dr. Tingbin Zhang at Columbia, and Drs. Achintya Ganguly and Phillip Sprangle
at the Naval Research Lab. The early designs were worked out by Tingbin Zhang and
Tom Marshall in 1995-96. and I also used Tingbin’s work on tapering the wiggler field
via resistive shunts. Prof. Marshall remained with the MIFELA project throughout
its duration, visiting Yale frequently and keeping in touch via phone and email. He
kindly made his own data available as well his expertise in the FEL field. I thank
him for his interest and support, his ideas and good advice, and his generous loan
of equipment.
Achintya Ganguly gave me his code, with which I performed the
simulations in Chapter 2. and patiently taught me how to use it.
When things
weren't working. I was glad to know that support was only a phone call away.
Other scientific collaborators who deserve thanks include Dr. Soo Yong Park, of
the POSTECH Department of Physics and Pohang Light Source in Pohang, Korea.
On his visits with the Yale Beam Physics group, he gave Mei and me the benefit
of his long experience with accelerators, beamlines. and cryo-systems. and helped us
improve our current output considerably. Dr. Michael Shapiro, of the MIT Plasma
Group, designed the microwave input and output couplers for the MIFELA, and Dr.
Steven Gold of NRL loaned me a high-voltage charging supply when ours failed. I
also thank the Klystron Group at SLAC. particularly George Caryotakis and Ron
Koontz. for making available the XK-5 klystron which powers the BPL. Finally, the
MIFELA experiment was initially supported by the U.S. Department of Energy, High
Energy Physics Division, through a Small Business Innovation Research grant.
Not least, I must mention the talented Gibbs Machine Shop crew at Yale, who
built much of the experiment—particularly Vinnie Bernardo, who usually managed
to “squeeze in" my jobs. David Johnson, master welder, and Tom Hurteau. Thanks
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for being skilled machinists, for constantly coming over to help us out of some mess,
for gamely trying to understand my drawings, and for being there when we needed
you.
I thank the members of my thesis committee— Profs. Jack Sandweiss. Peter
Parker. Charles Sommerfield. and Colin Gay—for their helpful comments and critique,
and my outside reader. Dr. Arie van Steenbergen.
I have saved some essential contributions to my graduate work until last: the
personal support and goodwill of my family and friends. To my various housemates,
particularly Bala, Bari§, Manik. Holly, Carly, and Michelle, goes my heartfelt appre­
ciation for many years of happy home life, family-style bonding, and close friendships.
Several of my fellow graduate students in the physics department were staunch com­
panions and morale-boosters. beginning with those infamous problem-set sessions and
continuing through exams and dissertations: thanks to Ron. Scott. Peter, and espe­
cially Mitch and Nigel. I was involved in many New Haven musical organizations,
which provided refreshing creative outlets to help me keep a mental balance— I think
in particular of the Yale Camerata and its members: the Christ Church choir and
Rob Lehman: and the PiP organization and Sara Laimon. My family, especially my
parents, gave me a great deal of moral support over what turned out to be a long
haul. Thank you. Mom and Dad. for your love and confidence in me.
Finally, my wife. Juliette Wells, deserves her own paragraph, if not her own
entire section. Not only did she make my work possible in many practical ways, such
as feeding me during late-night experimental runs and carefully proofreading and
improving the clarity of the final text, but I could hardly have gotten through these
years of intermittent frustration without her constant encouragement and assurance
that the project would eventually finish. Her faith in my ability and prospects never
wavered, even when mine did, and though it was clear what she was getting into, she
married me anyway. To you, Juliette, I give my deepest gratitude and all my love.
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Unmindful of the glories which await
His progress, he doth not accelerate
His current, with unseemly haste. ...
—Henry Ellison. "On the Thames" (1851)
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Chapter 1
Introduction
The free-electron laser or FEL. a device which can produce or amplify coherent
radiation by means of an electron beam modulated by magnetic fields, has become
widely known since its development in the early 1970s [1 . 2]. Less well known, but
relying on the same physics, is the inverse FEL (IFEL). which is a particle accelerator
rather than a radiation source, so that radiative energy is absorbed (rather than
produced) by a modulated electron beam. While an IFEL was first described in
1972 [3]. experimental proof of the principle was not carried out for twenty years [4],
and no one to date has demonstrated significant energy gain or high efficiency in an
IFEL.
The development of extremely high-powered pulsed lasers in the 1980s and
the resulting interest in laser acceleration have encouraged new experimental efforts
involving the IFEL. The device described in this thesis (the Microwave Inverse FreeElectron Laser Accelerator, or MIFELA) is a proof-of-principle electron accelerator
using the IFEL mechanism but scaled to the microwave regime. Operation at long
wavelength simplifies diagnostics and allows one to work with parameters, such as
phase, that are difficult to access in laser experiments, so that detailed theoretical
predictions can be verified. The work presented will show strongly phase-dependent
acceleration of a magnitude not previously reported for IFELs. with minimal spread­
ing in energy and phase, verifying the predictions of numerical simulation. Scaling of
this proof-of-principle experiment to higher gradients is also discussed.
1
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1.1
Historical sketch
The idea of energy exchange between an undulating charged-particle beam and
an external electromagnetic wave, now often called the free-electron laser interaction,
has its earliest roots in 1933. when Kapitza and Dirac [5] introduced the stimulated
Couiptou effect and proposed an experiment to observe it. predicting that momentum
would be transferred between electrons and photons during scattering collisions and
furthermore th at the probability of such transfer would be increased by the presence of
scattered radiation. The symmetry of such an interaction—particles may gain energy
and be accelerated or lose energy to radiation and undergo deceleration—is clear in
this first work and in most later theoretical treatments, but for a long while the idea
was only developed experimentally through stimulated-radiation devices, leading up
to today's free-electron lasers (FELs). Work on accelerators was to come much later,
when high-power lasers made them more attractive.
The earliest ancestors of the FEL were born out of a need for high-frequency
microwave tubes in the 1950s [6 ]: the idea of using stimulated synchrotron radiation
to build a microwave source began to take shape at Stanford as a result of the experi­
ments of Motz and coworkers in 1951-53. who calculated and measured the spectrum
of spontaneous radiation from an electron traversing an undulator. or wiggler. for
the first time, with some experimental results in both the optical and millimeter
regimes [7. 8 ]. Later in that decade. Motz and Nakamura calculated the power radi­
ated by an undulating beam into a waveguide [9] and discussed the amplification of
external waves by the same mechanism [10]. In 1960. Phillips operated a low-power,
low-energy device called the "ubitron," which could produce
1 0 -cm
microwave out­
put [1 1 ]; he also speculated that the same principle might be useful for an accelerator.
The ubitron s timing was not fortunate, since competing devices were better under­
stood and had higher efficiency, and the project temporarily faded into obscurity
due to lack of interest within the microwave community. The idea remained alive at
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Stanford, however: in 1968. Pantell proposed a microwave cavity device ("Compton
laser") based on Motz’s idea [12]. and by 1970 John Madey. then a Stanford grad­
uate student, had begun work on what is usually called the first true free-electron
laser, a machine that worked via a stimulated resonant interaction in which highly
relativistic electrons in an undulator produced optical amplification. His first paper
describing the theory of the device (treated quantum mechanically as a stimulatedbremsstralilung problem) was published in 1971 [13]: an apparatus was soon under
construction [1 ], and in 1976 the successful observation of gain in a
1 0 .6
f.im FEL was
reported by his group at Stanford [2]. A year later, a 3.4 ^m FEL oscillator with a
mirror resonator was demonstrated as well [14].
Theoretical and computational effort strengthened considerably in the wake
of these results, which in many cases were more complex than predicted. In the
beginning, much of the analytical emphasis was on the phenomena surrounding the
growth of the laser wave, its spectrum and harmonics [15. 16]. and calculations of gain
in a large number of regimes [17. 18. 19. 20.
2 1 ].
Of more direct interest to accelerator
theorists were beam-dynamical results that began to appear slightly later and that
used classical electrodynamics, it having been shown that quantum treatments were
unnecessary in orbit calculations [22. 23]. By 1979. Kwan and Dawson had proposed
an axial guiding magnetic field to increase stability [24]. which soon became common
and was the subject of much analysis [25]; it was also shown that a very strong axial
field could increase gain by exploiting the cyclotron resonance [26. 27]. The 1980s
saw full nonlinear calculations of beam dynamics in three dimensions, using tapered
undulators with physically realizable wiggler fields as well as axial fields [28, 29. 30],
and observed FEL behavior was more often able to be modeled in detail by numerical
codes.
Meanwhile, formal similarities between particle dynamics in FELs and in
accelerators were pointed out by Kroll, Morton, and colleagues [31. 32. 33]. By the
end of the decade, the subject was judged sufficiently mature for several textbooks
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and monographs to appear [6 . 34. 35. 36].
None of the early workers had much interest in the acceleration possibilities of
the FEL mechanism. There was a little Soviet work in the early 1960s on accelera­
tion using an axial field alone to maintain a gyrating beam [37. 38] (the inverse of the
cyclotron resonance maser [39. 40. 41. 42]). but the fields needed were unrealistically
large. Palmer, at roughly the same time as Madey's original work, made the first
direct connection to accelerator physics when he rediscovered the FEL interaction
while looking for ways to use newly available terawatt lasers to accelerate electrons.
Palmer’s 1972 paper [3], generally cited as the first description of an inverse FEL ac­
celerator (IFEL). also contains descriptions of "electromagnetic generators” which he
called "laser-like" and is probably the first work to emphasize the inverse relationship
of FELs and IFELs. Like Madey. Palmer used a helical wiggler magnet and circularly
polarized radiation in his designs; he also speculated with some prescience that an
FEL would be self-bunching.
While FEL development quickly gained speed and propagated through the
physics and engineering communities, little experimental work was done on IFEL
acceleration for some time, although quite a bit of analytical and computational work
was carried out: these results mostly date from the 1980s. when higher available laser
powers sparked interest in all kinds of laser acceleration schemes, including the IFEL.
Some of the first computational work was done by a group associated with the Naval
Research Laboratory [43. 44], and. by the middle of the decade, high-gradient, highenergy laser IFELs had been studied and discussed in a number of conferences and
papers [45, 46. 47], where a variety of numerical results had taken into account not
only variable wiggler periods and axial fields but also higher-order effects, including
emittance growth and radiation damping [48].
On the experimental side, several experimental groups in the early 1980s had
made careful measurements of energy spectra on electrons that had undergone the
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FEL interaction and demonstrated a mixture of energy gains and losses roughly in
keeping with theoretical expectations. Warren et al. at Los Alamos saw an average
energy gain th a t was greater than zero (with large spread) if the electron energy was
below resonance [49], and groups under Edighoffer [50] and Slater [51] observed small
amounts of energy gain in experiments at TRW and Boeing/M.S.N.. respectively. The
first definitive experimental proof of the IFEL principle, however, came in 1992. nearly
a decade later, when Wernick and Marshall at Columbia put two wigglers in series
and demonstrated that a beam decelerated in the first wiggler could be re-accelerated
in the second, using a helical wiggler with 1.6 mm radiation [4]. (A group at the
Yerevan Physics Institute, in what was then the Soviet Union, reported acceleration
at 10 /im in 1989 [52. 53. 54. 55], but their results were only preliminary and appear
doubtful.) The Columbia finding was a relatively low-energy proof-of-principle result,
with about 9% of available electrons accelerated from 750 keV to
1
MeV.
At about the same time, an experimental program began to build a 10-/im
IFEL at Brookhaven National Laboratory's Accelerator Test Facility [56]. The BNL
group published their first results in 1996 [57]. finding that the high-energy portion
of their 40 MeV electron beam was accelerated by up to 2.5% in a gigawatt guided
laser beam. The result that was of most direct interest to the accelerator commu­
nity was the degree of bunching of the electron beam at the laser wavelength that
occurred during the interaction, and extensions of the work focused on that aspect
of the process [58]. Current experimental efforts at BNL-ATF include an IFEL as
a microbunching pre-injector for other advanced accelerator prototypes, such as an
inverse Cerenkov accelerator [59].
The device described in this thesis, the MIFELA, is an intellectual descendent
of the Columbia experiment and was first planned around 1993 as a research col­
laboration between Omega-P, Inc.. Yale University, Columbia University, and the
Naval Research Laboratory [60, 61], with the accelerator itself located in Yale's Beam
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6
Physics Laboratory. Intermediate stages in the design and simulation processes of the
MIFELA were described at several Advanced Accelerator Concepts workshops [62. 63]
and Free Electron Laser Conferences [64. 65].
1.2
Essential physics of the IFEL
Electromagnetic (EM) waves have always fascinated of accelerator designers,
since their electric field components are by far the strongest available in nature and
are easily generated and controlled. Using them for acceleration, however, is not so
straightforward. All of the acceleration methods and structures based on EM waves
have been designed to sidestep their universal limitation: the strongest electric fields
in a wave are transverse to its propagation direction and therefore difficult to use in
accelerators. In fact, unbounded EM waves in a field-free vacuum cannot impart any
net acceleration to a charged particle [6 6 ]. a result which can be proven in a variety
of ways and is often known as Palmers Theorem. For acceleration, then, the waves
must be modified or augmented with other fields or structures: broadly speaking,
practical acceleration methods are either structure-based, using traveling waveguide
modes or cavity excitations, both of which have longitudinal field components, or
(more experimentally) laser-based, in which focused beams are combined with ex­
ternal magnetic fields or a medium such as a gas or plasma. By using a laser in a
medium, longitudinal fields can be created by displacement of charged particles [67]
or the inverse Cerenkov effect [6 8 ]; magnetic fields impart transverse motion to a
particle, which can then be acted on by the transverse electric fields of a wave. The
IFEL is essentially a mechanism for coupling the electron to the wave by imparting
transverse motion to the electron via magnetic fields.
The IFEL process can be thought of as having two stages. Transverse mag­
netic fields that either vary periodically in space in a single plane or are "circularly
polarized" about an axis move electrons off-axis through the v x B force, where the
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 .
fields cause them to spiral around the axis or wiggle transversely. Once the electrons
have non-negligible transverse motion, they have velocity components parallel to the
electric field and can gain kinetic energy. Because the acceleration rate initially de­
pends on the electron velocity, the process is usually called second-order. The trick,
so to speak, is to tailor the magnetic fields so that the electron’s spiralling or wiggling
motion and the EM wave are always in the same relative phase. While the electron
indeed falls behind the wave, whose phase velocity is greater than c, the phase slip­
page is such that the electron transverse velocity changes sign synchronously with the
RF wave field. This resonance is pictured in schematic form in Figure 1.1.
Several different but equivalent physical pictures illuminate the way this res­
onance occurs. We will briefly describe one which invokes a quantum-mechanical
understanding of single-particle dynamics in an undulator. Imagine a single electron
traveling relativistically along the axis of the wiggler. initially without any driving EM
field present. When the periodic magnetic field of the wiggler is Lorentz-transformed
to the electron's frame, it gains electric-field components and becomes an electromag­
netic wave, since the electron's speed is very close to c. The electron therefore “sees"
an EM wave propagating in the waveguide, which on the single-particle level we vi­
sualize as a single virtual photon that approaches and scatters off the electron. The
photon is absorbed, and another is emitted: a Compton scattering process. The emit­
ted photon can be detected in the laboratory. The direction of emission can be either
parallel or anti-parallel to the electron's motion; the corresponding electron recoil is
thus backward or forward. (The backward case corresponds to ordinary undulator
radiation.)
So far we have described emission of a single photon by a single electron with
no driving field present. If we now include a driving wave, we add a photon to
the process. The electron is now approached by two photons, moving in opposite
directions: with two photons present, stimulated Compton scattering takes place, in
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
8
C
V
t
Figure 1 . 1 : A sketch illustrating the IFEL resonance, by which the electron slips
behind the EM wave at a rate such that the electron transverse velocity and EM field
stay always in the same direction. The drawing shows a transverse wiggler. for clarity,
but the same principle applies in the helical case. The dashed line shows the electron
trajectory', while the solid high-frequency figures represent the wave's electric field
at several locations along the electron's path. Note that the EM wave is gradually
moving ahead of the electron.
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
9
which the electron absorbs the photon from one beam and is stimulated to emit it
again by the existence of the other beam. Because of the stimulated nature of the
process, the efficiency is much higher than in the single-photon case. The recoil is
again positive or negative, depending on which beam is absorbed: consideration of the
dynamics shows that there must be a slight frequency difference between absorbed
and emitted photons, so that by carefully choosing the exact driving frequency, the
device can be "tuned" between acceleration and deceleration.
This quantum picture oversimplifies the case, however, by implying that FEL
gain is an inherently quantum phenomenon, since classical approaches yield the same
results in the small-signal regime. Because the classical pictures are more quantitative
and less intuitive in nature, their discussion is relegated to the next section, where
two classical approaches are used to build a simple theory of the IFEL.
1.3
IFEL theory
While early FEL theorists analyzed the physics of the interaction in a quantum
mechanical framework [13], it was shown soon afterwards that a classical description
was equally valid for any realistic experiment [2 2 ], an approach which is now stan­
dard. Likewise. Palmer's IFEL paper [3] made use of a simple classical framework
for estimating the accelerator parameters, and practically all of the IFEL theoretical
work since then has been in a classical pendulum-equation formulation, thus show­
ing its similarity to other branches of accelerator physics. This observation was first
made formally explicit by Kroll. Morton, et al. [33. 31, 32]. who used the idea of
the synchronous or resonant particle, as in storage-ring theory, to simplify the FEL
dynamical equations.
The full dynamical problem of the helical IFEL is simple to state: a relativistic electron is acted upon both by the static magnetic field of a wiggler. which is
transverse to the axis and rotates around the axis as a function of axial distance.
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 .
10
and by the fields of a circularly polarized electromagnetic wave, propagating in a
circular metallic waveguide. This simplicity is somewhat deceptive, however—even
the fields themselves are a challenge to write down in full, and the coupling of the
electron’s velocity components through the fields leads to intractable equations unless
approximations are made. When further refinements to the experiment are added—
such as an axial magnetic field or a tapered wiggler period, both of which are used
in MIFELA—the problem can be treated with full accuracy only by computation.
As a result, numerical computations are relied upon heavily for practical IFEL de­
sign questions after theorists have done what is possible analytically; the situation is
similar in other subfields of accelerator physics.
VVe will begin our survey of IFEL theory with some quick estimations derived
from a highly simplified version of the dynamical equations in the absence of an
axial guiding field. The following sections present more accurate results, including
higher-order effects and stability results with a non-zero guiding field.
1.3.1
Simplified results
Our statement of the simplified problem begins by idealizing the fields present.
We consider the field of the wiggler to be purely transverse and to be periodic in 9
with period Aw = 2~/kw. so that it is described entirely by the expressions
BWr{z) = Bw cos kwz
( 1 .1 )
BWy{z) = Bw sin kwz
(1.2)
Note that this field does not satisfy Maxwell’s equations, because it has nonzero
divergence;leaving
out the 0-and r-components is equivalent to neglecting
of order k^p, asdiscussedbelow. We
terms
also consider the RF fields to bethose of a
circularly polarized plane wave:
Ex(z) = E0sin[ks{z - ct) + 0Q]
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
(1.3)
11
Ey{z) = E0cos[ks{z - ct) + oQ\
(1.4)
where ks = 2tt/As is the free-space wavenumber of the driving radiation, and
B= -:xE
c
(1.5)
using MKS units, as we will do throughout. In the expressions here, we have sup­
pressed radial dependence for both wiggler and RF fields, in part by ignoring the
existence of a conducting boundary supporting guided, not plane, waves. This as­
sumes that the electrons stay close to the axis—or. more precisely, that the radius of
electron orbits p is small compared to the size of the waveguide and that kwp
1.
This approximation is often reasonable for laser-driven FELs. but is not particularly
accurate for the microwave IFEL. Since the shortcuts taken here and later in this
section lead, as one can imagine, to fairly crude quantitative predictions, we will use
them for descriptive purposes only. The simulations in Chapter 2 present a more
accurate picture of MIFELA dynamics.
The electron orbits are determined, given the fields, from the Lorentz force
equation
F = e[E + v x B ]
(1.6)
where v is the electron velocity. We take the electrons to be relativistic. so that
the velocity ratio 3 = v /c ~
be
helical, which forhigh
1.
and we anticipate that the steady-state
orbits will
energy implies that the transverse velocitycomponents
3*. 3y <§; 3::. Under these conditions, the familiar near-cancellation of the transverse
electric and magnetic forces of the RF fields takes place:
i
where the relativistic factor
7
=
— [Ex - v s(B„ + B Ky)]
7771
(1.7)
=
- [ E x( l - 3 : ) - v : B Wli}
(1.8)
7771
is defined by
r
^
.
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 .
(i.9)
12
Since
1
—J ; ~ l
I / 2 7 2 for 3 —* 1 . the transverse equations of motion are
—
determined to order I / 7 2 solely by the wiggler fields, a rather robust approximation
valid even for MIFELA.
Considering only the wiggler fields, then, we have simple equations for the x
and y motion of the electron:
—e
x = — v-Bws m k wz
7m
£
y = — l k B w c o s kXL,z.
7m
Using the approximation r = u:t ~ ct to integrate
1 .1 0
(1-10)
(1. 1 1 )
and 1 . 1 1 . we find that the
total transverse velocity component J 2 = .J2 + J 2 satisfies
Uxl = ^ % = ym ckw
7
(U 2)
where aw is a dimensionless quantity, a normalization of the vector potential, which
is known as the wiggler strength parameter.
(u3)
/1 tC - r i f f r
Integration of eqns. 1.10.
1 .1 1
twice shows that the helical trajectory has a radius of
/1 1 . \
p = — .
( 1 *14)
To calculate the rate of energy change of the electron, we consider the electric
field, which is the only field that can do work on the particle:
dW
d~( ,
—— = — me = eE • cp
dt
dt
so that from 1.12 and 1.3. 1.4 we change variables as above to obtain
dy _
dz
eEodv
7771c-
^
+
CQS^
V
- cos[ks(z — ct) + 0Q]sm kwz'j
=
s i n ^ r - ks{z - ct) + o 0 j .
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 .
(1-15)
(1-16)
13
We will define
(I> = kwz — ks(z — ct) + <Dq
(1-17)
to be the "ponderomotive phase"—the phase of the electron relative to the beat wave
created by the wiggler and RF fields—and if we further use the dimensionless electric
field strength commonly taken in laser physics
_ eE0
p.Eq
a s = ---------- = — y r -
m ajs
(1.18
mc-ks
then the energy gain formula simplifies into the convenient form
d~/
dz
ksasa
sin$
7
(1.19)
which indicates that the rate of change of energy squared is constant:
d~r
— = 2ksasawsm<&.
dz
( 1 .2 0 )
Two things become obvious in eqn. 1.19. The first is the presence of the relative
phase term s i n . which shows the invertability ofthe FEL mechanism.
phase factorscorrespond toincreasing energy with
Positive
c(the IFELaccelerator), while
negative phase gives electron deceleration and the FEL. Secondly, the energy change
is nonlinear and decreases with increasing
7
if all other parameters are constant.
This is the reason for tapered wiggler fields, which are commonly used in FELs of all
kinds: usually a change in aw is introduced to compensate for the change in
7
in the
denominator.
In order for cumulative energy exchange to happen in this system, we require <D
to be approximately constant, that is. to vary only slowly on the scale of the device.
Let usconsider anelectron moving through the wiggler with axial velocity uz. If we
beginfrom z = t = 0, the value of <5 after one undulation period, z = A^,. is then
given by
$ =
=
kwXw —ks ^Au, —c—
2 - -
2 t t^
f
j
-
1^ +
-F <p0
0o
R e p r o d u c e d with p e r m i s s io n of t h e cop 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 rm is s io n .
(L'-l)
( 1 .2 2 )
14
and so for constant lf> we require
1~3Z
3.
(1.23)
Now we write 3- as
3Z = (J- - J l’ )
and. substituting eqn.
1 .1 2
for
11.24)
we have
(1.25)
so that finally 1.23 becomes
(1.26)
which is a resonance condition for the electron energy relative to wiggler and RF
parameters and is equivalent to maintaining constant phase <f>. Note once more that,
as we saw above, we must increase \ w or am to maintain resonance as
7
increases.
Another derivation of this resonance condition, usually seen in FEL contexts,
illuminates an alternate physical meaning of eqn. 1.26. If the electron is relativistic.
the wiggler magnetic field seen in the electron frame, as discussed in Section 1.2 above,
becomes an electromagnetic wave under Lorentz transformation, with a Dopplershifted wavelength
(1.27)
When this "'virtual photon" scatters off the electron at angle 9. assuming minimal
recoil, the wavelength seen in the laboratory frame for the scattered photon is Doppler
shifted again:
Alab =
7 ^ ,(1
- 3Zcos (9).
(1.28)
Combining these two. we obtain
Alab = Aa,(l —3z cos 9)
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 er. 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 .
(1 .2 9 )
15
and making the approximation 1.25 for J: we have
(1.30)
For small
6.1
— cos 6 « A#2. and the other cosine factor can be taken as equal to
1.
with the result that
(1-31)
Setting A|ab = As for a resonant process, we arrive at a generalization of 1.26 that
applies to FEL radiation, where
6
c-axis. (Thus in the IFEL process.
is the angle of the radiation with respect to the
6=
0.)
We can expand this simple account of IFEL dynamics by including non-resonant
particles in the picture. As we said earlier. 1.26 can be viewed as a resonance condition
on 7 . such that for a given configuration of fields, a resonant electron will have energy
7r
given by
(1.32)
and be associated with a (constant) resonant phase ^>r by
dz
sin<I>r .
(1.33)
7r
Now if electrons are allowed to have energy
7
^
7 r.
equation 1.19 is unchanged in
form, but the phase $ can no longer be taken as constant in r. Taking the derivative
of 1.17.
(1-34)
so that we can write (compare eqn. 1.23)
(1.35)
(1.36)
(1 .3 7 )
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16
which combines with eqn. 1.19 to form a pair of coupled nonlinear equations for the
evolution of 7 .
If we now investigate the regime of small deviations from resonance, such as
might be
expected to apply in an actual installation, we can linearize the equations
around the resonant energy in order to solve them. Following [47], weintroduce the
small quantity
^ = 7 ^ 7r < 1
(L3g)
7r
and linearize with respect to q. assuming that, as 7 r increases, the accelerator pa­
rameters are varied according to 1.32 to maintain synchronism. Considering 1.19 and
1.33. we can write
^“ (7 ‘ -
7p )
= 2asa u,A*I1.[sin<&(c) - sin<&P]
(1.39)
and if wefurther assume that 7* varies slowly compared to q. so that
1 dq
1 (h';
7q 7dz: » 3-T
T*
7 - dz
( L4Q)
then linearizing and substituting gives the equations
dq
dz
&
— «
q.sQ-u*kw
7r
[sin‘I>(c) - sin<I>r]
2kwq{z)
(1-41)
(1.42)
which are a form of the familiar differential equation for a driven pendulum.
6
=
.4 sin 9 + B. commonly used in the physics of standard linear accelerators to describe
phase oscillations and stability [69. 70]. Eqns. 1.41. 1.42 depict a regime of stable
oscillations about an ideal trajectory, or alternatively a moving potential well or
"bucket" which traps electrons that are near resonance in energy and phase. The
buckets here have increasing energy with increasing <f>: it is known (see [69]) that
such an equation is derivable from a Hamiltonian with a potential function of the
form V'(<&) = —i(Q/u,'o)2 (cos$ -t-<Fsin$0)- Such a potential is graphed in Figure 1.2:
R e p r o d u c e d with p e r m i s s io n of t h e cop 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 rm is s io n .
17
Figure 1.3 shows a number of particle orbits in phase space, including both trapped
and untrapped particles.
As the "stationary phase"
$0
is increased in Figure 1.3.
the size of the trapping bucket decreases, and more of the orbits are unbounded (i.e..
more particles are unaccelerated).
1.3.2
Three-dim ensional results
While a complete calculation of particle motion and energy gain in an IFEL is
far beyond the reach of analytic methods and susceptible only to numerical study,
some results of interest can be obtained by considering the correct wiggler fields and
waveguide modes. In this section, we will outline a three-dimensional electron orbit
theory and its implications for particle stability.
A planar wiggler field—one that changes direction in a single plane—is known to
consist of a series of harmonics at the magnet periodicity. Since an array of permanent
magnets can produce such a field, it is somewhat simpler to construct than a helical
field, as well as being somewhat simpler to analyze, because the variations transverse
to the axis are relatively easy to model. A helical wiggler of the sort we are interested
in discussing can be very complicated in comparison. The present experiment, like
most helical-wiggler designs, involves a bifilar current winding, in which two current
helices running in opposite directions are interspersed symmetrically. While the field
on the axis of such a configuration has long been known (see [71]), calculations off-axis
are much more involved. An exact analytical formula for the correct magnetic field,
with all components, inside and outside of a bifilar helical wiggler has been found by
Park et al. [72]; it is not, however, very usable in its full-blown form, since it consists
of an infinite sum over harmonics of the wiggler period, each term of which is itself
an infinite sum involving modified Bessel functions of increasing order. The field
exactly on axis, however, does reduce to the ideal case of Eqns.
1 .1 ,
1.2: the 9 and z
components vanish, leaving only a radial component that rotates with increasing 2 .
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 er. 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 .
18
5
15
10
20
25
30
Figure 1.2: Graph of the potential function V'(<&) ~ cos <5 + <£sin<I?o. showing a series
of "buckets'" for particle trapping; <l>o = tt/ 8 .
9
0
9
0
5
2.5
7.5
10
12.5
15
17.5
Figure 1.3: Orbits in A E-$> phase space for particles moving according to equations
1.41 and 1.42. with
$ 0
= 7tt/8. Several trapped (i.e. closed) orbits are visible, as well
as many unbounded ones.
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 er. 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 .
19
The magnitude of the axial field for a wiggler of wavenumber kw = 2tt/X w and radius
rm is given by
B l {0) = - ^ l h i L [ kwrwKo(kwrw) + K\(k wrw)\
7T
(1.43)
where Ki{x) is the modified Bessel function of the second kind, of order z. and I is
the current in each winding.
Without using the full machinery of [72], however, the field dependence on radius
alone may be deduced for a given axial field, due to the symmetry of the magnetic
potential [73]. Following this approach, we obtain the following compact expression
for the field near the axis, which we will use in the analysis to come:
B (r)
=
2B L[I[{kwr) cos(d - k1L.z)er - ( i/A:1t-r)/i(A:u.r) sin(0 - kwz)e 0
+ A {kux ) sin(0 —A-'u.-Je-] -f- B qG:
(1.44)
where /„ and I'n are the modified Bessel function of the first kind of order n and its
derivative with respect to argument, and where we have included a constant axial
guiding field of strength B q.
We can now write down the full equations of motion for the electrons in the
absence of an accelerating field—i.e. the unperturbed orbits assuming constant
7
.
It is most easily done in a coordinate basis that is helical and rotating with the
wiggler [19], which is obtainable from Cartesian and cylindrical coordinates by the
transformations (with \ = 9 —kwz)
e!
= cos \ er —sin \ eg = cos(\ —6)ex —sin (\ —0)ey
(1.45)
e2
= sin \ ^ + cos \eg = sin (\ —9)ex + cos(\ —9)ey
(1-46)
63
= er
(1-47)
so that the three velocity components become v\, v2, V3 (note t/3 = u: ). This comes
at a slight calculational cost: since the axes are changing in time, the time derivative
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 .
20
on the left side of the Lorentz equation with
7
constant.
~ v = — (v x B)
at
(1.48)
7m
must include the time derivatives of the basis vectors. Some calculation shows that
e t — kw~&>
e2 =
(1.49)
—kwz e i
(1.50)
(e 3 is of course fixed). To simplify the final expressions further, we let
A = kwr
(1.51)
and in order to use A as an independent variable, we formally set
A = kwr = kw[i'[ cos \
4 - c2 sin
\}.
(1-52)
We define \ similarly in terms of tq and r2. Expanding the Lorentz equation compo­
nentwise and simplifying, the equations of motion become
i>i
= - c 2[fio - kwv3 + 2 n wI y{\) sin \] + VinwI2{\) sin 2 \
(1.53)
c2
= —^ ^ [/^ ( A ) c o s2 \ + /o(A)] +
(1-54)
03
= - t ; 1n u;s in 2 \/ 2 (A)
—/t'u-C3 + 2f2u./i(A) sin \]
4 - c2 flu;[/2 (A)cos2\ 4 - / 0 (A)]
(1.55)
with two additional equations, due to our definitions above:
A =
X =
kw(u{ cos \ 4- v-2sin \ )
A
sin \ 4- c2 cos \) - kwv3
(1.56)
(1.57)
where Q0w = eB 0_w/ 7m.
A complete solution to these coupled equations clearly cannot be obtained, but
if we restrict ourselves to steady-state solutions which correspond to the physical
situation experienced by an electron beam, considerable simplification is possible.
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 er. 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 .
21
For steady-state motion (no acceleration) in our helical coordinates, we have t>t =
ih = t'.j = A = y = 0. and we also require that i/-> = 0. which means that the electron
orbit is in the same sense as the wiggler field, rather than its opposite. Under these
conditions. Eqn. 1.56 becomes \ = ± tt/2 . and hence 1.57 simplifies to
\ —
_ —
l’1 _ — L’w
f
— T*
Uj
L'z
where we haverecognized tq tobe the transverse
simplificationand
il -O'
(l.U O j
orwigglevelocity
uw. Algebraic
use ofthe Bessel-function identities on the firstthree
equations
yield a single relation:
_______2 v 3Q
L’1
w I i ( \ )
_______
\[Q 0 - k wv3 ± 2 Q M X ) Y
1
1
Eqns. 1.58 and 1.59 are sufficient to determine t’i = cw. v3 = u:. and A if any one of
them is fixed; we choose to make A the independent variable and define
^O.tu
Oo.ur = ---kwc
, . ..n >
(1.60)
so that
2a wI l (\)
a 0 - v :/ c ± 2 a wh ( \ Y
(-
]
We use the energy constraint
to rewrite the velocities in terms of 7 . finally obtaining a single expression that relates
the orbit parameters to the beam energy and fields necessary for steady-state motion:
L/2
A
7
= A2ttQ ± (1 + A2 )2qiu/ 1(A).
(1.63)
Note that this definition 1.60 implies
Qtt'
from 1.13. with q 0 normalized similarly.
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 rm is s io n .
( 1 .6 4 )
A plot of the solutions of this equation for given values of 7 and B w is shown in
Figure 1.4. where axial velocity vz/c is plotted against axial field strength B0. The
two curves represent the two solutions implied by the signs in eqn. 1.58, with the
left-hand curve corresponding to the upper sign. In the FEL literature, these are
generally referred to as Group I and Group II orbits, respectively [30].
The stability of these solutions can be calculated by examining small deviations
from the steady state. Carrying out a perturbative expansion in which tq = vw -f 5rq.
u-y = Sv-1, t’.i = v. +Sv3. \ = ± 7t / 2 + 6\ . and A = ^ v w/ v z + S\. we obtain a system of
five first-order equations, which can then be simplified to give a pair of fourth-order
equations of the form
(1.65)
where Q[ 2 are defined in terms of the parameters in 1.59 and play the role of fre­
quencies in the solution to the perturbed equations. The perturbation and solution
process is carried out in [29], where equations are found for the trajectories that re­
duce to the one-dimensional expressions in the limit A —* 0. B 0 —<>0. In the interests
of avoiding needless clutter, we will not exhibit these solutions here: for our purposes
it is sufficient to note that the presence of squared frequency-like terms in the dif­
ferential equation 1.65 implies regions of instability for f l j 2 < 0. This is indeed the
case, and the dashed portions of the curve in Figure 1.4 are those in which the orbits
are unstable.
The theoretical operating point of MIFELA is marked in the same figure. It is
very near the curve but does not fie precisely on it. showing that we have somewhat
exceeded the capability of the small-gradient, constant-wiggler model. At this point,
we turn to numerical simulation to complete the dynamical analysis, which is the
subject of the next chapter.
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23
1
0.8
0.6
CO.'*
0.4
0.2
5
10
B o (kG)
15
20
Figure 1.4: Graph of the normalized axial velocity 3Z = vz/c of steady-state orbits
as a function of axial field B q, measured in kilogauss. from equation 1.63. For these
curves.
7
= 12.74 and Bw = 2.2 kG. The left-hand curve contains Group I orbits
(vw = —Xv: ) and the right-hand curve Group II orbits {vw = Ac.): dashed portions
represent unstable orbits. The diamond (<>) shows the computed operating point of
MIFELA.
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Chapter 2
Simulation Param eters and
Results
This chapter describes the final parameters for the MIFELA. which were chosen
and optimized through numerical simulation, and ideal acceleration results calculated
using these parameters. The reader should bear in mind that the MIFELA experi­
ment. as eventually constructed, deviates from this ideal design in several ways, due
to various practical problems and constraints in the laboratory setup. This design
is therefore included here not so much to anticipate the experimental results in later
chapters as to outline the theoretical best-case scenario for a MIFELA of this en­
ergy and frequency; the results below served as a guide during construction of the
experiment.
An approximate one-dimensional result for an electron accelerator, which had
been obtained by Zhang and Marshall at Columbia [74]. was used as a starting point.
In order to model the IFEL interaction more accurately, a fully three-dimensional
FEL simulation code was chosen. The code, written by H. Freund and A. Ganguly
of the Naval Research Laboratory, is known as ARACHNE and contains a nonlinear,
slow-time-scale integration of the equations of motion for each individual particle [75.
76. 77], including the full spatial dependence of the wiggler and radiation fields [72].
In ARACHNE. the radiation fields are expanded in normal modes of a cylindrical
waveguide, space charge is included, and the orbit equations are integrated in full
without averaging. The code has been benchmarked successfully against numerous
FEL experiments [78. 79]. Since the code merely keeps track of the energy carried by
24
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25
the particles and fields, it can be used for either FEL or IFEL calculations by varying
the relative phase at which particles are injected into the RF fields.
In order to successfully accelerate in the IFEL. one must first impart transverse
velocity to the electrons. This is often done via what is known as adiabatic entry, in
which the wiggler field as a function of axial distance is brought up from zero over
a distance much longer than one wiggler period. A similar scheme can be used for
beam extraction. In earlier work of Zhang and Marshall [64]. a form of this principle
was found to effectively generate the proper axis-encircling orbits, and with some
refinement it was retained in the design of the MIFELA. In this scheme, an electron
passing through the device crosses three distinct regions: entry, acceleration, and
extraction. During entry, the electron is “spun up" from an initial velocity which
is purely axial by slowly increasing the wiggler and guiding fields. By the end of
the entry region, the electron has attained significant transverse velocity, and. as it
crosses into the acceleration region, it is spiralling around the axis. This is. of course,
necessary for acceleration. During the acceleration process, because the transverse
velocity component is the only component that can be increased, the guiding field is
tapered to keep the orbit size from growing excessively. When the electron exits the
acceleration region, the increased kinetic energy, located in the transverse velocity
component, must be transferred back to the parallel component before it can be
useful. The beam is thus spun back dowm again until most of its transverse velocity
is gone.
The following section describes the details of accomplishing this design for the
MIFELA with the equipment available at Yale. Readers who are interested in the
amount of acceleration that could be achieved with this type of design if significantly
higher power (perhaps at a much higher frequency) were to be used are referred to
Chapter 5. where the expected scaling of MIFELA results with increased power and
frequency is discussed.
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26
2.1
MIFELA parameters
The most important parameters of the MIFELA were imposed by what was
available in the Yale Beam Physics Laboratory, where the experiment was built. The
source of microwave power in the BPL is an S-band klystron, operating at 2.856 GHz.
which immediately hxes the scale uf th e accelerator. T h e aame k lystron powers th e
electron beam source, an RF gun that produces a bunched beam with an energy of
6
MeV. Power is split between the gun and the accelerator, with a variable phase
shifter to make sure the correct phase relationship is maintained. Since the nominal
maximum power output of the klystron is 25 MW. and beam production from the gun
requires about
8
MW of that power, the simulations used an RF power in MIFELA
of 15 MW.
2.1.1
Entry region
Much effort was devoted to finding the optimal configuration for the input
region, since setting up axis-encircling orbits is probably the most important single
requirement for accelerator operation. The ideal entry section would increase the
transverse velocity component of injected electrons at the expense of axial velocity,
with no net change in kinetic energy: meanwhile, the center of the spiralling orbits
would remain on the axis. In the final design for MIFELA, the wiggler and guiding
fields are increased from zero to their acceleration-strength values over a distance of
five wiggler periods. RF power is introduced into the accelerator from the beginning
(i.e.. at the same time as the particles), but since the orbits are far from resonance,
very little energy change takes place for most of the entry distance. The wiggler
period is kept constant during this region at 11.75 cm.
It was discovered th at the most stable orbits are achieved by increasing the
magnitude of the wiggler field using a “sine-squared profile." that is. a nonlinear
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profile in which the field magnitude obeys
B w(:) = B ua sin"
where B w0 =
2.2
for
0
< r < 5A,,,
(2.1)
kG is the initial value of the wiggler field in the acceleration region
and Xw = 11.75 cm is the wiggler period. The axial guiding field is increased linearly
from zero to its full value (1.7 kG) over the same distance. Plots of these wiggler and
guiding fields are shown in Figure 2.1.
2.1.2
A cceleration region
Once the electrons have been spun up until their transverse velocity is approxi­
mately correct for resonance (see Fig. 1.4). acceleration begins. In order to maintain
the correct resonance relation (eqn. 1.26). the wiggler period in this region is tapered
linearly, which means that the rate of increase of the normalized field aw is equal
to the calculated acceleration rate of the electrons. The magnitude of the guiding
field is less prescribed: designs at higher gradient have used a two-segment field, with
different slopes, but for the MIFELA at current power levels it proved best for the
beam spot size to slightly down-taper the field.
The tapers are not large for MIFELA. because the input power level and the­
oretical gradient are both relatively small. The wiggler period, initially 11.75 cm,
is tapered linearly to 12.32 cm over the 92.75 cm acceleration length. This gives
rise to an increase in the wiggler field from 2.2 to 2.3 kG. or equivalently a wiggler
strength parameter aw (from eqn. 1.13) that varies from 2.4 to 2.75. Meanwhile, the
guiding field is decreased from 1.7 kG to 1.61 kG over the same length. (Refer again
to Fig.
2.1
for a plot.) The field decrease was unexpected, but simulation shows that
a negative taper leads to better energy spreads at output.
Injected microwave power travels in the MIFELA in the circularly polarized
T E tl mode. The waveguide dimensions were chosen to maximize field strength near
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28
2.5
0.5
co
-0.5
0
50
100
150
200
z (cm)
Figure 2.1: Plot of ideal wiggler (solid line) and axial (dashed line) magnetic fields
vs. axial position for MIFELA. as found through simulation. The entry, acceleration,
and exit regions correspond to 0 < r < 59 cm. 59 < z < 151 cm. and c > 151 cm
respectively.
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29
the axis; this necessitated using as small a cylinder as possible, that is. operating it
near cutoff. The chosen radius of 3.14 cm leads to a waveguide index n = uj/ck = 0 .2 .
and with an input power level of 15 MW. the normalized field strength as is equal to
0.16.
2.1.3
Extraction region
The same early work of Zhang and Marshall [64] concluded that the beam could
efficiently be spun back down onto the axis—the inverse of the spin-up process in the
entry region—by linearly tapering the fields back down to zero over a distance of
roughly five wiggler periods. In the design adopted here, both fields were tapered
linearly from their final values to zero over a distance of 58.75 cm. with the wiggler
period once more remaining constant.
The structure parameters in the simulation are summarized in Table 2.1.
2.2
Beam parameters
The properties of the injected electron beam were chosen to correspond realisti­
cally to the expected output of the RF gun. The gun design (a 2 ^-cell version of one
in use at the Stanford Synchrotron Radiation Laboratory) is optimized for output at
6
MeV [80]; the beam itself consists of microbunches 5 ps in duration, each containing
up to 109 particles. By the time the beam is injected into MIFELA, the use of an
achromatic beamline and energy selection allows energy spreads of less than 1 %. The
finite bunch length corresponds to a phase spread relative to the RF power of about
tt/30 ~ 0.1 radians. In the simulation, a conservative phase spread of tt/10 rad was
used, centered at an RF phase angle of +0.65tt rad (corresponding to
= tt/2 in
eqn. 1.26).
Current is reduced during the energy selection, and in practice only about
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10'
30
Table 2.1: Structure parameters for simulation of acceleration in MIFELA. by region.
Entry region
Length
Wiggler field
L i = 58.75 cm
B w = 0 - 2 . 2 kG. nonlinear ramp
Axial magnetic field
Wiggler period
Wiggler radius
B 0 = 0-1.7 kG. linear ramp
Au, = 11.75 cm
rw = 3.84 cm
A cceleration region
Length
Waveguide radius
Free-space RF wavelength
Waveguide index, n = uj/ck
Input RF power level
L -2= 92.75 cm
R = 3.14 cm
Xs = 10.5 cm
n = 0.2
Pin = 15 MW
Normalized RF field strength. as = eE 0/mcajs as = 0.14
Circular
Polarization
Waveguide mode
TE„
A,t. = 11.75-12.32 cm. linear taper
Wiggler period
Wiggler radius
rw = 3.84 cm
Wiggler field strength
Wiggler strength parameter. au, == eB w/mckw
Axial magnetic field
B w = 2.2-2.3 kG
aw = 2.4-2.75
Bo = 1.7-1.61 kG. linear taper
E xtraction region
Length
Wiggler field strength
Z/3 = 58.75 cm
Axial field strength
B q = 1.61-0 kG, Unear ramp
B w = 2.3-0 kG, linear ramp
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31
particles are likely to be injected per microbunch: this corresponds to a peak current
of 0.32 A. with an average macrobunch current of 4.5 mA. At a repetition rate of
1
Hz. the overall average current is 9 x 10- 9 A. Since beam loading was not expected to
be significant, the value of the beam current made little difference to the simulation,
and it was set at 0.1 A.
The simulated beam was injected on axis and given an initial diameter of 0.7
mm. with particles evenly distributed within the beam. Beam energy spread and
emittance could be varied: some results presented here assume a cold beam, while the
effect of beam quality is included in Section 2.3 below.
2.3
Acceleration results
When one examines the calculated evolution of electron energy over the length of
the MIFELA. the three regions of the device are clearly visible, as seen in Figure 2.2,
which shows the average value of particle energy as a function of axial distance.
During injection, as the beam is spun up. energy is transferred from axial to azimuthal
velocity components, with only minor change in the total energy. The slight decrease
observable in the figure may be due to a momentary decelerating resonance created
as the fields increase. Figure 2.3 shows that, with initial cold-beam injection, the
transverse velocity ratio Jj. = tq_/c rises from
velocity
0
to about 0.26. at the expense of axial
which falls from an initial value of 0.997 to about 0.96.
During acceleration, despite the nonlinearity of the interaction, the beam energy
rises at a nearly constant rate due to the compensating wiggler field taper. The
effective gradient in the acceleration region is 0.81 MeV/m.
In the extraction region, particles quickly fall out of resonance and energy change
is nearly zero: at the same time. Fig. 2.3 shows that energy is being transferred from
transverse to axial velocity components. At the end of the extraction region, the
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32
6.8 r
t
6.6
6.4
6.2
6
5.8
0
50
100
150
200
z (cm)
Figure 2.2: Plot of the average calculated electron energy in MeV. as a function of
axial distance r. in the MIFELA. The acceleration region consists of c-values between
59 cm and 151 cm: the effective gradient is approximately constant at 0.81 MeV/m
in this region.
0.8
0.6
0.4
0.2
0
50
100
150
200
z (cm)
Figure 2.3: Plot of the average electron velocity components in the MIFELA. as a
function of axial distance. The upper (dashed) line shows the axial velocity ratio
J: = u./c, and the lower (solid) line shows the transverse ratio
= v±/c. Transfer
of kinetic energy between components takes place in the injection and extraction
regions at either end of the device.
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33
transverse velocity has nearly vanished (d i = 0.03). and the beam may be steered or
focused in a spectrometer for energy analysis or further use.
If a physical beam is considered, the property that has the greatest effect on the
acceleration is the beam's angular divergence. The effects of divergence are normally
represented by the so-called transverse einittance £, defined in a given direction x by
£, = y V X M 2) -
(xx-y -
(2.2)
where x. x' are. respectively, the position and divergence angle relative to the beam
axis, and the angle brackets denote averages. Including a transverse rms emittance
for the injected beam of 0 .57T mm mrad results in increased emittance growth and a
final average energy that is 50 keV less than that of the cold beam case: if the injected
emittance rises to tt mm mrad. the average energy gain is reduced by 150 keV. For
an initial emittance of more than about 1.5tt mm mrad. beam loss during "spin-up”
(i.e. in the injection region) becomes significant, and the exiting beam is severely
degraded in energy. Comparisons of output beams for various initial emittances are
made in Figure 2.4. A low-divergence injected beam is clearly of critical importance
to a high-performance device.
2.4
Accelerated beam properties
For the cold beam result described in the previous section, phase and energy
spread were minimal, as shown in Figure 2.5. while beam spot size was on the same
order as that of the entering beam, as shown in Figure 2.6. Fig. 2.5 displays the
calculated longitudinal phase space of the exiting beam for a computation using
1000
particles which were given spreads in initial phase, azimuthal angle, and radius. That
each dark line is in fact a group of
100
particles with the same initial phase shows
that the acceleration is much more sensitive to phase (via equation 1.26) than to the
other variables. Consequently, particles injected near each other in phase remain so
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34
6.8
6.6
6.4
6.2
5.8
5.6
0
50
100
150
200
z (cm)
Figure 2.4: Plot of average calculated electron energy as a function of axial distance in
the MIFELA. assuming a range of injected emittances. The top curve is the cold beam
case: the lower curves have, from top to bottom, transverse emittances at injection of
0.3". 0.5rr. l 7r. 2.8tt. and 4" mm mrad. Since in the last two cases the most divergent
electrons hit the accelerator walls and are lost, the increased emittance in the last
case makes little difference to the output, except to decrease the current.
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35
throughout, with an overall phase spread that increased only from tt/ 10 or 0.31 rad to
0.53 rad. This strong phase trapping is a characteristic feature of IFEL acceleration.
The energy spread in Fig. 2.5 is dependent almost entirely on the entering phase
spread: in this case the overall spread in energy is 0.15 MeV. or 2%. for all particles
in the simulation.
The beam spot at the exit of MIFELA. shown in Fig. 2.6. has been some­
what compressed in the azimuthal direction and extended in the radial direction, as
compared to the entering spot. The arrows on the axes in the figure show the lim­
its containing 80% of the particle distribution in each direction. The initial beam
profile—circular, with diameter 0.7 mm—is replaced by a more rectangular distribu­
tion. with sides of 1 mm in the x direction and 0.5 mm in the y direction, for an
increase in area of approximately a factor of two. Note that the beam as a whole has
been slightly displaced from the axis as well: in this case steering would be necessary
to center the beam for further experiments. Calculated rms transverse emittance
at the exit is 0.43" mm mrad. although since we have used an initially cold beam
having di. =
0.
exhibiting the transverse phase space on exit is not very physically
meaningful, as emittance growth cannot be calculated.
As discussed in the previous section, beam divergence has a significant effect
on the interaction in MIFELA. Nonzero divergence at injection leads to increased
divergence throughout the device, with emittance growth occuring largely in the
injection region rather than during the acceleration process. Future designs may lead
to a more sophisticated adiabatic entry region which is less disruptive to beam quality.
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36
6.75
>
2
>>
6.7
-
so
6.65
6.6
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
phase angle (rad)
Figure 2.5: The calculated longitudinal (£ -$ ) phase space at the exit of MIFELA.
for a cold beam of 1000 particles. Each dark smear contains 100 particles of initially
identical phase.
0.24
'J
C
>»
-
0.45
-
0.4
-
0.35
-
0.3
-
0.25
x (cm)
Figure 2.6: The calculated beam spot at the exit of MIFELA, for a cold beam of 1000
particles. Coordinates are measured from the axis. The arrows on the graph axes
indicate 80% of the particle distribution in each coordinate. Note the different x and
y scales.
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Chapter 3
Experimental Design and
Construction
3.1
Design overview
The MIFELA structure may be described simply as a smooth-walled circular
waveguide. with externally applied magnetic fields and filled with RF power, into
which a beam is injected at one end and extracted at the other. The details of
construction and operation for each of these components are described below; the
whole is summarized here.
The injected MIFELA beam is produced by an RF gun. which operates at the
same microwave frequency as the accelerator itself. Up to 23 MW of RF power from
the laboratory klystron is thus divided between gun and accelerator, with appropriate
phase delays, while a specially designed input coupler ensures the correct travelingwave mode in the accelerator structure. The beam, meanwhile, travels through an
1 1 -element
achromatic beamline which performs energy selection and focusing before
injection into MIFELA. Current and position monitors allow fine-tuning of the dipole
and quadrupole fields.
After its exit from MIFELA. the energy of the beam is analyzed by a magnetic
dipole spectrometer, in which the beam is dumped into a Faraday cup and the return
current measured.
The axial guiding field is generated by a series of 18 independently-controlled
solenoids, in order that any field profile may be produced. The wiggler consists
37
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38
of a pair of helical filaments, which are wound directly on the exterior wall of the
waveguide and pulsed at high current via a capacitor discharge bank. Up-tapering
the wiggler period accomplishes the increase in the wiggler field over the length of the
structure, while the nonlinear field shape in the beam entry region is brought about
by reducing the current using resistive shunts between the two windings.
A schematic diagram of the entire experiment is shown in Figure 3.1. and a
scaled drawing in more detail of the MIFELA itself is presented in Figure 3.2.
3.2
MIFELA com ponents
3.2.1
Acceleration structure
The structure itself is the least complicated aspect of the MIFELA. consist­
ing entirely of a single smooth-walled cylindrical waveguide, which, by enforcing RF
propagation in the correct waveguide mode, keeps the velocity of the driving radiation
resonant with the particle beam and magnetic fields. The structure also serves as the
vacuum vessel and supports the windings for the wiggler magnet, which are wound
on its exterior wall.
As indicated in Table 2.1. the waveguide was designed to operate in the T E u
mode, with circular polarization. The RF coupling scheme which sets up this prop­
agating mode is described in the next section. The only remaining design question
is the choice of guide radius, which, when combined with the operating frequency,
sets the waveguide index and hence the overall scale of the machine. For the highest
possible acceleration gradient, it is clearly preferable th at the electron beam should
experience high fields as much of the time as possible. Given that the radius of beam
orbits (see equation 1.14) is usually small compared to the guide radius, it is to our
advantage to keep the guide as small as possible, in order to maximize the pow'er den­
sity and hence the RF field magnitudes. This reasoning leads to a radius near cutoff
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39
Klystron
RF gun
Power
splitter/
phase
Spectrometer
and d ia g n o s t ic s
Dipoles
nergy s e l e c t o r
Accelerator
SI li I LCI
□
RF
abs or be r
Figure 3.1: Schematic outline of the MIFELA experiment, showing the connections
between the major components. While this drawing is not to scale, it correctly shows
the layout in the BPL.
R esistive s h u n ts
^ f o r w i g g l e r fi el d t a p e r )
RF I n p u t p o r t
(from klystron
(to
RF o u t
absorbers)
Solenoids
XXXa a X A A A A X /
Injection port
(from beam line)
ux
W iggler
directly
ca< 3 e
wound
o n t o pipe
Steering
B e a m exit p o rt
(to d iag n o stics)
coi
50 cm
Figure 3.2: Scaled drawing of the MIFELA itself, showing all components. The
resistive shunts for the downtaper of the wiggler field at entry are shown schematically.
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40
for the waveguide. For a microwave frequency of 2.856 GHz. the cutoff radius in the
T E u mode is 3.075 cm; the radius of the MIFELA waveguide was set at 3.14 cm,
giving a so-called waveguide refractive index n = jj/ck of 0.2. This number affects all
the other parameter choices, since the periodic wiggler field must be combined with
the traveling RF wave to produce a traveling potential well which is resonant with
the electron energies. As the BPL's klystron is theoretically capable of producing up
to about 23 MW. using 8 MW to power the gun would leave roughly 15 MW available
for acceleration.
With a refractive index so close to zero, the construction tolerance on the inner
radius was considerably tightened, since small changes in radius have a large effect
on RF propagation velocity. Furthermore, the waveguide needed to be machined
in a single piece in order to avoid the issue of RF reflection and beam disruption at
waveguide joints. Our challenge was therefore to locate a facility which could machine
as long a pipe as possible with as close a tolerance as possible. The final product,
which became the MIFELA. was about six feet in length (183 cm) with the inner
radius machined to a tolerance of ±0.001 inch, or roughly 0.1%. After installation,
measurement of the waveguide cutoff showed an effective radius of 3.136 cm. The
construction material was stainless steel, for the sake of higher wall resistance, with
an initial wall thickness (the minimum required for machining) of 0.25 in or 0.64 cm;
this value was later reduced for the sake of pulsed-field penetration (see Section 3.2.4
below).
It would be possible to maintain much higher effective RF power levels—and
hence higher acceleration gradients—in the MIFELA by using an excited cavity rather
than a traveling-wave structure. Because this is significantly less straightforward to
engineer, it was not attem pted in this proof-of-principle experiment. The underlying
physics, however, would be identical, and the fabrication tolerances could potentially
be relaxed. See Chapter 5 for a discussion of this idea, together with other possible
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
extensions of the experiment.
3.2.2
RF input and output coupling
In order to set up the required circularly polarized T E U wave in MIFELA. a welltested input coupler that had already been used for other Beam Physics Laboratory
experiments was chosen. With this design. RF power is fed simultaneously into two
WR-284 waveguide arms. 90° apart, attached to the side of the structure. When
the input power in the two arms is equal, the RF inside the structure is correctly
polarized to high accuracy, and matching stubs in the arms reduce reflected power
to less than 1%. Once inside the coupler, the radiation is cut off in the direction of
the beam port, and thus propagates entirely into the MIFELA waveguide. A scaled
drawing of an input coupler is shown in Figure 3.3.
This coupler design was initially made for a device with a slightly different inner
radius than MIFELA. Rather than redesigning the matching stubs for the new size,
it proved simpler to lengthen the coupler and introduce a radius transition region in
which a stepwise taper was used for best reflection performance, as seen in Fig. 3.3.
The middle "step" dimension is one quarter of a guide wavelength, with the result
that waves reflected from the front and rear edges of the step interfere destructively,
and the amplitude of reflected waves approaches zero.
In this traveling-wave device, unused RF power must be absorbed at the down­
stream end of the MIFELA. (With a beam energy gain of roughly 1 MeV for a mac­
robunch of 1011 particles, about 0.1% of input RF energy is given up to the beam,
making beam loading negligible.) The input coupler design was used in reverse as an
output coupler on the far end, each arm terminated with a matched load containing
wedges of lossy ceramic composed of 60% aluminum nitride and 40% silicon carbide
(see Fig. 3.3). The beam exit port, like the entry port, is well below cutoff for the
RF. Measured overall reflection at the klystron was always less than 5%.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
Uatcning
stub —.
0
30 cm
Figure 3.3: The top figures are scaled drawings of the coupler design used for RF
power input and output for MIFELA (side and end views). The two waveguide
arms at 90° are shown correctly positioned in the end view drawing, although one is
invisible in the side view. The step-taper in radius can be seen. The lower figure, at
the same scale, shows the design of the matched loads used for RF absorption at the
output couplers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
3.2.3
A xial m agnetic field
For optimal adjustability, the axial magnetic field, necessary for adding stability
to the transverse electron orbits, is provided by a series of water-cooled solenoidal coils
arranged side by side. Each of the solenoids (built by Ogallala Electronics) consists
of a pair of current windings with square cross-section, and can provide a peak axial
field of about 1.6 kilogauss at the maximum rated current of 90 A. When at least four
units are adjacent to each other and operated at once, the maximum field becomes
about
2
kG with negligible field ripple.
The MIFELA experiment uses 18 of these coils arranged in series, as shown in
Fig. 3.2. An individual coil has a large effective field width: each solenoid is
1 0 .8
cm
long and produces a field with a full-width at half maximum of 20.1 cm. By varying
the current in each coil, the field profile along the axis can be made to follow- any
slowly-varying function. When each coil is powered by a separate computer-controlled
current supply, an arbitrary field profile can be calculated and applied using a single
computer program. A "flux cage” of iron bars enclosing the solenoids provides a flux
return path that both prevents field leakage and strengthens the axial field.
Because the variation in field described in Table 2.1 is quite gentle, the MIFELA
interaction was in practice rather insensitive to variations in axial field slope: thus a
constant rather than down-tapered field was used w-ithout apparent degradation of
the beam output. A current of 57 A in each coil located in the acceleration region
produces a flat field of 1.15 kG. w-hich became the operating range for the experiment
(see Chapter 4).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
3.2.4
W iggler magnetic field
Wiggler m agnet construction
The wiggler field is created by a pulsed-current electromagnet consisting of a
bifilar helix: two helical filaments, interspersed symmetrically and carrying current
in opposite directions, wound with equal pitch and running the length of the device.
These filaments are connected at the far end. so that the outgoing current in one side
returns through the second side in the opposite direction. The combination creates a
strong transverse field in the accelerator which rotates around the axis with changing
r (and is thus, in a sense, circularly polarized). A detailed diagram is given in Figure
3.4. As mentioned briefly above, the magnet filaments are wound directly onto the
exterior of the accelerator tube, although they are kept electrically isolated. This
provides for maximum field on axis, since field increases with decreasing winding
radius. To taper the wiggler field, the winding period was increased along the device:
the downtaper to zero at the beam entry end was accomplished with a series of
resistive shunts, discussed below.
For generation of fields on the order of
2
kG. as required by the design, the
filaments had to be able to comfortably conduct pulsed currents of up to 50 kA with­
out needing cooling. A current pulse of this magnitude also requires good structural
rigidity. To accomplish this, the filaments were made of AWG 4 solid bare copper wire
(diameter = 5.2 mm) wrhieh was wound tightly onto the accelerator tube and fixed
at the ends. Prior to winding the wiggler. the tube was covered with several layers of
Kapton insulating tape, providing 20 kV insulation with a thickness on the order of
100 /zm. In order to keep the winding period accurate, an insulating rubberized-foam
spacer of the correct width (as shown in Fig. 3.4) was wound onto the tube at the
same time. By varying the spacer width at the correct rate, the winding period was
varied smoothly and accurately over the length of the device. 15 periods were used
in all. of which five make up the entry region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
Wiggler
filament
7 /A V /7 )
Insulating
spacer
Current shunt
Insulating
spacer
Wiggler
filament
Figure 3.4: Schematic drawings showing details of the wiggler magnet construction.
The upper drawing shows one end of the accelerator structure, with the wiggler
filaments wound on the outside of the waveguide; the lower drawing is a cross-section
of the structure, showing the wiggler winding and spacers as well as an example of a
resistive shunt like those used in the beam entry region.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
The shaping of the initial field uptaper is performed by a series of 23 resistive
shunts connecting the two filaments of the wiggler winding, with the position and
number of shunts dependent on the relative resistances of shunt and wiggier filament.
As long as each shunt is azimuthallv symmetric, then no unwanted field distortion is
introduced: in practice, this is accomplished by shunts that are semicircles of AWG 4
solid Nichrome wire (cross-sectional diameter 4.7 mm) attached in parallel between
the two filaments in order to cancel each other's field (see Fig. 3.4). The resistance of
each shunt is 5.0 mQ. For the sake of durability, the shunts are fastened with copper
brackets that are themselves bolted directly to the wiggler filament using drilled and
tapped holes.
Because the resistance per unit length of the Nichrome wire is roughly 100 times
greater than that of the copper filament, the current loss from the filament at each
shunt is relatively small, and the overall field taper is quite smooth. A fitting program
developed by Dr. T. B. Zhang of Columbia University [81] was used to optimize the
shunt locations in order to produce the field taper shape prescribed by equation 2 . 1 :
the computational problem is that of a resistor network with variable resistances
(Figure 3.5). The resulting prescription for shunt locations is shown in Figure 3.6;
given the locations, it is easy to calculate all currents (also shown in Fig. 3.6) and the
net resistance using KirchhofTs Law. The shunt arrangement leads to a somewhat
increased overall inductance for the wiggler. which could significantly increase the
high-frequency impedance if the circuit otherwise had very low inductance: this has
been described in detail with reference to an FEL experiment at CEA/CESTA in
France, where a similar shunt method was used to build adiabatic entry and exit
fields with a helical wiggler [82, 83. 84]. In this case, however, since the capacitor
bank powering the wiggler has a large inductance of its own. any change due to the
wiggler shunts is negligible.
Benchtop measurements of the wiggler magnetic field using DC current are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
PcuL o
•R«
P cu L -i
AAA^
■Re
'- ^ W V
P cA q
-V A -
P cu^I
Pcu^-2
-A A A -----
Pcu^-N
A V W
R«
R.
-A A A ---------
-A A A — 0
PCii^N
P cu^-2
tot
tot
Figure 3.5: Schematic diagram of the wiggler shunt circuit, where R s is the resistance
of a shunt. pCu is the resistance/unit length of the copper, and Lx is the length of
copper filament between two shunt locations. The total current / tot is divided among
N shunts, so that less current passes through each successive wiggler segment.
30 ------------ ---------------------------------------------------25 r
<
20
C
u
b3
U
W)
00
15 tb
I10 b
5
L
L
4.
o
O
o o
0
0
10
20
30
40
50
z (cm)
Figure 3.6: Wiggler shunt positions and currents: each shunt is represented by a
diamond.
The --position of the shunt is shown on the horizontal axis, with the
computed current flowing in the segment of wiggler filament downstream of each
shunt on the vertical axis, exhibiting the shape of the field taper.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
presented in Figure 3.7. The field, which is shown in one projection in order to make
the period visible, follows the specified sine-squared uptaper shape in the injection
region well. The measured field magnitude within the acceleration region is 0.08 ±
0.005 gauss per ampere. DC. (A pulsed field complicates this situation: see below.)
Powering th e wiggler
In order to produce the required pulsed wiggler current of tens of kiloainperes, a
five-element capacitor discharge bank was built, consisting of five 10.6-jiF capacitors
rated at 15 kV each. Initially, the bank was conceived as a pulse-forming network
which would be able to generate a flat-topped current pulse of width at least 30 //s:
however, the large internal inductance of the capacitors rendered this unworkable.
In the final model, the capacitors are discharged into the wiggler without any added
inductance, using an ignitron switch. A second "crowbar" ignitron is used to short
out the bank after one half-cycle, preventing ringing of the capacitors. (Figure 3.8
shows a diagram of the circuit.) This design produces a rounded current pulse having
a half-width (FWHM) of about 41 /.is (shown in Figure 3.9). Maximum current for
a given initial voltage depends strongly on the impedance of the load: using AWG 2
welding cable in a twisted-pair for the wiggler leads, as well as clean brass connection
hardware, the peak current in amperes is about four times the charging voltage in
volts.
Finite p u lse length and field penetration
While using a pulsed-current magnet for the wiggler field made a helical magnet
easy to build and avoided the expense and tolerance demands of a permanent magnet
array, it had one quite troublesome consequence regarding penetration of the wiggler
field into the MIFELA waveguide, due to the screening of a rapidly changing magnetic
field by a conducting wall. It was initially anticipated th at eddy currents in the
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 rm is s io n .
49
„ -O——- N- —
4
' S A
f
h
r
"
4
9 ' > i ♦ ii
-
:.... / ......t X .?..♦....
_
/ 44
®
o
*
9
\*
4
&
v
*
4ij
.i .
V
❖ ... *
4
*
!
'
4
'
.7
♦•
■•?•■•♦
r• r-
-
9
* !
_ .......... ....................... < b . - v 4 - -
*
%
it
-6
40
20
0
60
«!
- *-* -■■ *■;-- -4 i-
80
%
100
;/
*
120
z (cm)
Figure 3.7: Plot of the x-projection of the wiggler magnetic field, as measured on the
benchtop using DC current. The solid line is interpolated from the data points; the
dashed line shows the total field amplitude.
*0-15 *V
C N IT h O N
t
SL 770J
jcnarqina
’ Pecrson* current
supply
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monitor
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o/ooo
wwim fti
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(5C22 T>iyrotron)
Figure 3.8: Schematic circuit diagram for the capacitor bank and discharge circuit
used to power the wiggler magnet.
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
Figure 3.9: Typical current waveform from the wiggler pulse, monitored by a current
transformer on the positive lead. The shape is constant regardless of amplitude; this
trace was obtained with an initial charging voltage of
1
kV.
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. 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 .
51
waveguide wall might somewhat lower the peak field seen by the electron beam. For
the sake of the higher wall resistance, the structure was machined from stainless steel
rather than copper, as mentioned in section 3.2.1 above; the wall thickness was also
made as small as possible. Despite these precautions, bench tests of field penetration
gave very poor results: the peak field within the pipe was less than 35% of the
field without the pipe present, and the measured field pulse was greatly delayed and
widened in time. The situation is shown qualitatively in Figure 3.10. which shows
the best case finally achieved, as described below.
A simple mathematical model was useful in predicting the relative effect of dif­
ferent parameters on the field penetration. The propagation of magnetic field through
a conductor can be treated as a diffusion problem, described by the usual diffusion
equation with a coefficient equal to 1//icr (where fi and a are the permeability and
conductivity, respectively, of the conductor). Using a one-dimensional approach for
simplicity, integration of the Green's function for that equation yields the expression
(3.1)
for the field a distance x inside a conductor at time t. where the field at the exterior
surface is suddenly increased from 0 to B0 at time t = 0. and where a = ^la . Note that
this behaves correctly in all limits, since B{0. t) = Bo, B (x, 0) = 0. B —* 0 as x —* oc.
and B —►B0 as f —• oc. Using this equation and breaking a time-dependent external
field into many discrete "time slices-’ of constant field, it was possible to numerically
approximate the situation inside MIFELA. albeit without taking the field polarization
or the curved geometry into account.
Since eddy current shielding is a transient effect, it would seem that increasing
the wridth of the current pulse would lead to higher internal fields. Analysis showed,
how'ever. that this method was very inefficient, since even doubling the width (wrhich
would be impossible to achieve in practice) resulted in only an additional
10%
pene­
tration. The final approach involved machining the exterior in order to reduce the wall
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 .
52
thickness to about 4 mm—the absolute minimum required for structural rigidity—
and the addition of eight longitudinal grooves cut into the exterior wall and equally
spaced in azimuth, designed to increase the effective resistance for axis-encircling eddy
currents.
The final measured penetration value with these changes was 52 ± 3%. leading
to a net wiggler field (as seen by the electrons) of 0.037 ± 0.003 gauss per ampere.
This just suffices to generate a field of 2.2 kG with the maximum available current of
60 kA. A plot of the measured internal field, compared with the field in the absence
of a shielding wall, is shown in Figure 3.10.
Steering error and correction
The alert reader may have noticed a discrepancy between the overall design
length of the MIFELA specified in Chapter 2 (221.9 cm) and the final length of the
constructed acceleration structure (183 cm. as mentioned above, the maximum that
could be machined with precision). To compensate for this shortfall without further
reducing the already small energy gain, it was decided to abbreviate the exit region
by terminating the wiggler at the end of the acceleration region, then leaving the
fields to diminish naturally beyond the wiggler's end. Calculation indicated that this
arrangement would loosely approximate the rate of decrease of the design fields; the
guide field could be maintained through the region if orbit stability required it.
While it was certainly expected that the smooth decrease of the transverse
velocity component would be less well shaped with this arrangement, the problem
proved on installation to be much worse, apparently due to exacerbation of the already
non-ideal field shape by magnetic shielding in the region. Essentially, the output
beam dropped out of resonance almost instantaneously upon exiting the wiggler itself,
without spiraling back to the axis as it was meant to do. The beam at the end of
the MIFELA interaction, having gained a transverse velocity component of 3±_ =
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 er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
53
0.2
0.15
0.05
U
1)
c
00
a
2
*---- -------- ---------0.05 1---------:--------- 1-------------0
40
80
12 0
160
200
240
280
Time (ps)
Figure 3.10: Comparison of the measured wiggler field inside MIFELA vs. time (solid
line), with the field in the absence of the conducting wall (dashed line).
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 rm is s io n .
54
0.2-0.3. left the MIFELA structure with a divergence angle of up to 10°. which
generally prevented it from entering the exit beam pipe at all. Measurement of beam
trajectories through the wiggler in the absence of RF power confirmed a transverse
deflection at exit, which was proportional to wiggler field and constant in direction.
Since the direction of the transverse velocity was constant, it could in principle
be corrected by a simple steering magnet at the appropriate location. Several versions
of such a magnet were tried: the successful version was a pair of transverse coils giving
a vertical field, located in the transition region of the output RF coupler (see Fig. 3.2).
Limited space between the coupler and the axial-field solenoids constrained the design
considerably; the coil radius was perforce somewhat less than the coil separation
distance. For the final design, the field on axis had a half-width (FWHM) of 10 cm
with a peak value of 4.8 gauss per amp. Since the field necessary to counteract the
transverse velocity component was on the order of 300-350 gauss, currents up to 75 A
would need to run in the coil. With a total coil resistance of 0.13 Q. however, nearly
700 W would be dissipated in such a coil if it were run continuously. After some
attem pts to build coils with smaller resistances ended in conflagration, it became
clear that pulsed current would again have to be used in order to avoid overheating.
However, the coil’s location on the output coupler, where the copper vacuum wall is
11
mm thick, would lead to an extreme shielding effect on a pulsed field, making very
long pulses (> 0.5 sec) necessary. The magnet was therefore electronically controlled
to make the power supply output nonzero for two seconds, timed so that the field
was maximal during beam transit—in effect, giving a somewhat rounded pulsed field
lasting for 1.5 seconds. Further details of steering procedures are given in C hapter 4.
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
55
3.3
Auxiliary system s
3.3.1
RF power distribution
The Beam Physics Laboratory has as its source of microwave power an XK5 klystron operating at 2.856 GHz with a nominal peak power output of 25 MW.
Although the klystron can sustain a repetition rate of up to 10 Hz. operation at
one pulse per second is the norm for BPL experiments. From the klystron, which
is powered by a custom-built modulator giving a pulse duration (flat-top) of
2
/is.
power is distributed through the laboratory through WR-284 rectangular waveguide.
As summarized in Section 3.1 above. RF power from this klystron must be
divided between the electron gun and the acceleration structure during operation of
the MIFELA. This is accomplished by a combination power splitter and phase shifter
which feeds RF power to the gun and/or the structure, in any ratio, with the insertion
of an adjustable phase delay. This unit, together with a second, earlier splitter, also
provides a measure of isolation protecting the klystron from reflections at the gun.
There is a further 3-dB splitter on the accelerator section in order to divide power
evenly between the two input waveguides (see Section 3.2.2). The RF power system
is shown in Figure 3.11.
Power measurements were made using crystal detectors mounted on directional
couplers located just after the klystron, just before the gun. and at the MIFELA. The
crystals were calibrated with known power sources at a particular voltage output, and
attenuation on the couplers varied until the same voltage was reached; this method
avoids difficulties with nonlinear crystal response.
The use of a single RF power source immediately solves any potential syn­
chronization problems, while operating the gun and MIFELA at the same frequency
causes bunches to be injected into every RF cycle in the accelerator.
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
56
From
klystron
Vacuum
windows
Matched load
To RF gun
Ion pump
Ion pump
3-dB hybrid
coupler
Matched
load
Power splitter/
phase shifter
Beginning of
MIFELA
1 meter
Figure 3.11: Drawing showing an overhead view of the RF distribution system in
MIFELA. to scale. Locations of ion pumps are also marked.
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 er. F u r th e r re p r o d u c tio n pro hibited w ith o u t p e r m is s io n .
57
3.3.2
R F gun, 6 M eV beam line and diagnostics
The RF gun installed in the BPL is a 2^-cell design made by AET Associates
which generates beam for several BPL experiments.
Such a gun is in essence a
miniature linear accelerator consisting of several cavities, connected by irises, which
are filled with RF power. The first cavity is terminated at the halfway position hv a
hot cathode (forming the "half-ceir); since the electric field at the cathode is then
maximal, during the peak of every RF cycle electrons are ejected into the cavity and
rapidly accelerate away. The subsequent two full cavities serve both to accelerate and
to focus the bunched beam. Because the mechanism inhibits space-charge buildup at
the cathode, this design can produce large beam currents. The BPL's gun requires
roughly 9 MW of microwave power at 2.856 GHz to produce a
6
MeV beam (the
energy at which beam quality is optimal), but can be run from 4 to 7 MeV by varying
the input microwave power and cathode temperature.
In normal operation, the beam consists of a series of five-picosecond micro­
bunches. containing up to
10 °
particles each; the microbunches are separated by 350
ps (i.e.. one RF period) within a macrobunch that lasts the length of the RF pulse—
in this case. 2 /is. The gun was designed (see [80]) to produce a well-focused beam
(normalized transverse emittance of 3tt mm mrad) with energy spread of order 1%:
however, some energy selection in the beamline is necessary to remove a low-energy
"tail" in the beam spectrum. This process reduces the total particle number; in
practice, with no need for a high current, bunch sizes closer to
107
particles were
used.
In order to be achromatic, the beamline consists of three legs connected by
90° dipole magnets, as shown in Figure 3.12. A half-slit located in the center of the
second leg, where energy dispersion has been introduced, is used to narrow the energy
spectrum. Figure 3.13 shows the beam energy spectrum, measured well downstream,
for three different slit positions, showing that the low-energy tail is indeed scraped
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 er. 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 .
58
off by the slit.
To produce a high-quality beam for injection into MIFELA. the beamline con­
tains a total of nine quadrupoles and tliree steering magnets, as well as the two dipoles
just mentioned. The beamline design was carried out by M. Wang from a preliminary
design by M. Borland, with magnet locations and strengths optimized using the
INFINITY
COSY
numerical code [85]. The best computed rms transverse emittance at the
entrance to MIFELA was less than 0.3tt mm mrad in both directions. Actual rms
beamline emittances at the MIFELA entrance ranged from 0 .67 T to 1 .57T mm mrad.
measured using a single quadrupole scan method in which values of beam spot size
versus quadrupole strength were fitted to the standard beam envelope equation. (For
normalized emittance. multiply these values by
7
= 12.7.)
Several types of beam diagnostic devices were installed and are shown in Fig. 3.12.
Each leg of the beamline contains a toroidal current monitor, which uses a current
transformer to generate a time-dependent voltage signal which is proportional to the
beam current. Phosphor screens, located at the slit and at the entrance to MIFELA.
could be viewed with a CCD camera connected to a frame grabber and computer,
enabling beam images to be digitized and analyzed. (Analysis was carried out using
the ‘‘NTH Image" program [8 6 ].) An additional current monitor and screen are lo­
cated at the output of MIFELA, before the spectrometer (discussed below). Sample
current traces from each of the current monitors are shown in Figure 3.14.
3.3.3
Spectrom eter
A magnetic dipole spectrometer, located at the output of the accelerator, was
used to measure beam energy spectra. A vertical magnetic field deflects the electrons,
causing those of the selected energy to pass through a beam pipe and vertical slit be­
fore being collected in a Faraday cup. The return current must pass through a 10 Cl
resistor, giving a voltage signal proportional to the beam current. A scaled drawing
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
59
S te e rin g
C u rren t m o n ito r
j m agnet
RF gun
Dipole
valve
Q u a d ru o o le s
tu rb o -p u m p
V
50 cm
H a lf-s lit
Q uaarupoles
C u rren t m o n ito r
P h o sph o r
sc re e n
C u a d ru p o ie s
/ I\
valve
Beginning
of MIFELA
S teerin g
m agnet
Figure 3.12: Scaled drawing, showing an overhead view of the
6
MeV beamline in
the BPL. The RF gun is shown at one end. with 90° dipole magnets separating the
three legs. Quadrupole and steering magnets, as well as current monitors and viewing
screens, are marked and labeled.
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 er. 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 .
60
250
200
3
3
eu
£
U
3
150
100
5.9
6
6.1
6.2
6.3
6.4
E n erg y (M eV )
Figure 3.13: Electron beam spectra at the entrance to MIFELA for three locations
of the energy selection half-slit in the beamline. showing that the low-energy tail on
the spectrum is removed by the slit. The solid line is measured with the slit removed
(i.e.. no energy selection); the other two traces show spectra with the slit inserted
0.5 inches (long dashes) and 0.6 inches (short dashes). The variation in peak energy
shows the accuracy limit of the spectrometer. Spectra have been scaled to the same
peak current value.
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. 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 .
61
0 .5
<
eu
t
u
e:
cC3
0 .4
0 .3
0.2
U
4
6
10
12
T im e (us)
Figure 3.14: Sample current traces for beamline operation, measured with the toroidal
monitors at (from top to bottom) the middle of the beamline; injection into MIFELA:
after MIFELA: and the current signal from the Faraday cup at the spectrometer.
They have been displaced relative to each other for easy viewing. Some current loss
at the second dipole is visible.
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 re p r o d u c tio n proh ibited w ith o u t p e r m is s io n .
62
of the arrangement is shown in Figure 3.15. which also includes a cross-section of
the detector and slit arrangement. The slit was designed to allow maximum vacuum
pumping at the detector, and the Faraday cup was shaped to minimize secondary
electron emission. Again, the
cosy
in f in it y
code was used to design a device that
would maximize both energy resolution and signal-to-noise ratio. The results were
calibrated by comparisons with theory, both analytical and numerical, and with sim­
ilar data obtained at the dipole magnets.
A toroidal electromagnet with cylindrical iron pole pieces, taken from an earlier
experiment, was used to build the spectrometer. The field on axis was flat-topped
with soft edges, and had an effective pole radius measured at 8.7 cm. With a field
per unit current of 155 gauss per ampere, up to 1.4 kG could be generated without
overheating. For a relativistic beam, the beam rigidity Bp in a bending magnet is
proportional to the particle momentum, and in practical units this becomes [87]
£[T]p[m] = ^ d £ [G e V ]
(3.2)
«J
(where p = Rta.n.6 is the bending radius. R is the effective field radius and 6 is the
bending angle). In this case, we find that fields near 950 gauss will be required to
select a beam of 6.0 MeV. with the energy vs. current graph having a slope very near
to 1 . for R = 8.7 cm and 9 = 67.5o.
The smallest measurable energy change in the spectrometer, intended to be
at least 0.1 MeV. proved to be more critical than expected, since the final energy
changes obtained were on the order of hundreds of keV (see Chapter 4). Figure 3.16
show's computed transverse locations at the slit versus energy for a number of beams,
varied both in energy and in x-position at the MIFELA exit. A 2% difference between
particle energies (equivalent to 0.1 MeV when the MIFELA is operated near
6
MeV)
leads to a spatial separation of several millimeters, so that with a slit wridth of 3 mm
centered at zero we exclude all particles whose energies differ from that selected by
more than 0.1 MeV. This design is also relatively insensitive to position spreads.
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
63
An experimental test of spectrometer sensitivity can be made by comparing
the slit scraping results in Fig. 3.13 with the spatial dispersion expected from the
beamline bending magnets. That is. the decreased spectral width in the maximal
insertion ease can be compared with calculations of scraping loss in the beamline.
yielding a reasonable estim ate of the spectrometer's resolving power. Comparing the
second and third spectra in Fig. 3.13. wre see that the spectrum narrows by 40 keV
on the low-energy side when the slit is extended from 0.5 to 0.6 in (a change of
0.254 cm). Simulation of the beamline shows that these two slit positions will scrape
particles whose energies are too low by 3% and 4%, respectively. The change in width
for these two spectra should hence be roughly 1% of the peak energy, or 60 keV in
this case. The rough agreement between these estimation methods implies that final
energy changes of 0.1 MeV can certainly be distinguished by this spectrometer, with
a smallest discernable change between shots of perhaps 70-80 keV.
3.3.4
Vacuum and mechanical supports
The entire MIFELA experiment is kept at high vacuum: during operation,
measured pressure in the accelerator cavity itself was kept in the 1-5 x 10“ ' torr range,
falling to about 4 x 10- 8 torr when unused. Two 180 l/s turbo-molecular pumps were
operated continuously to maintain these pressures, with one on the beamline prior
to injection into MIFELA and the other just after the output. (These locations are
shown in Figs. 3.12 and 3.15 respectively.) Additional pumping on distant portions of
the vacuum in the RF waveguide was provided by two 20 1/s ion pumps at locations
marked in Fig. 3.11.
The entire experiment is mounted on aluminum frames, which support the
solenoid magnets on whose inner bore the accelerator rests. Precise spacers keep
the accelerator level and centered, and the waveguide arms are supported from the
ceiling. The beamline before and after the accelerator is fully adjustable in height
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
64
Phosphor
screen
Current
monitor
Spectrometer
Steering
mcgnet
50 cm
Slit
•)
Figure 3.15: Scaled drawings of the beamline and spectrometer arrangement at the
exit of MIFELA. with an enlargement showing the dimensions of the collector and
collimator attached to the spectrometer.
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 er. 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 .
-20
-15
-10
-5
0
x (mm)
5
10
15
20
Figure 3.16: Calculated transverse positions for a variety of sample beams at the
location of the spectrometer slit, showing spatial separation as a function of deviation
from the design energy. W ith a slit width of 3 mm. centered at the origin, an energy
width of ±2% is obtained, or about 0.1 MeV for a
6
MeV beam.
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 er. 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 .
66
and position, so that the accelerator and magnets, the heaviest elements, determine
the alignment of the whole system. For an experiment at these wavelengths and with
macroscopic beam sizes, high alignment precision is not required; the beam paths are
currently aligned to within several millimeters.
3.3.5
Radiation shielding
Most electron accelerator experiments create penetrating radiation principally
through x-ray emission by the particles involved. This occurs via both synchrotron
radiation, emitted whenever charged particles travel on a curved path (such as inside
a bending magnet in a beamline). and bremsstrahlung. in which particles being slowed
or stopped in a beam dump give up their excess kinetic energy in the form of highenergy photons. Other, more involved, means of radiation production can operate at
high electron energies, due mainly to secondary interactions undergone by emitted
gamma-rays.
Since the Beam Physics Laboratory operates inside a shielded vault isolated from
the operator's area by 40 feet of tunnel and 3 feet of concrete, shielding requirements
for MIFELA were less strict than those for a less protected facility: in practice,
only specific high-intensity locations required extra shielding. Radiation levels were
monitored during operation by an ion chamber mounted on the laboratory ceiling,
and dose rates were measured by
20
photographic film badges placed throughout the
vault. While the radiation level in the vault usually averaged between 50 and 100
mRem/hr while the beam was running, the operator’s area received no measurable
dose. A plan of the laboratory is shown in Figure 3.17. showing shielding walls and
dosimeter locations.
During the experiment’s operation, the MIFELA acts as a radiation source in
three distinct regions: the RF gun. the beamline. and the beam dump/spectrometer.
The gun itself is a potent x-ray source, at a level of tens of rad/hr. depending on
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 er. 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 .
67
Control Area
MIFELA
Klystron
gate
Offices
Work Areo
Shut cown switch
•
R a d ia tio n
3
High voltage sites
sites
a
A rea monitors ( f ilm
o
interlocked access points
badges)
Figure 3.17: Floor plan of the BPL. showing shielding walls and the location of the
MIFELA. Locations of dosimeters, access points, and interlocks are also marked.
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 rm is s io n .
68
the beam current: it produces a spectrum of energetic photons peaked at roughly
2/3 of the beam energy. The beamline bending magnets also are a source of x-rays
due to synchrotron radiation, which is directed in the familiar "searchlight" cone
of a relativistic electron and hence points mainly into the tunnel walls. X-rays are
produced at the beam dump through bremsstrahlung, with a spectrum ranging up to
the full electron energy.
A lead wall
8
inches thick was constructed between the tunnel exit and the
MIFELA experiment in order to shield the gun and beamline.
The wall, which
crosses the width of the vault, extends upward for several feet from below the vertical
position of the beamline in order to block any line-of-sight photon travel out of the
tunnel.
A lead housing 12 inches thick in all dimensions was built around the beam
dump itself: this shielding adequately absorbs x-ray photons up to an energy of 6.7
MeV. Since the measured beam energies in this thesis never reached more than 6.1
MeV, such a barrier was sufficient. Because photons above 6.7 MeV have the potential
to create secondary neutrons in lead (by collision with one common isotope of Pb),
however, upgraded experiments would need to address many more shielding questions.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Chapter 4
Data Collection Methods and
Results
4.1
Experimental m ethod
4.1.1
Timing
Synchronization of the many pulsed elements of MIFELA was crucial to the
experiment. The beam pulse of roughly 1 f.is duration was. of course, always within
the 2 fjs RF pulse in MIFELA. since they came from the same source. However, the
pulsed wiggler and steering magnet fields (with 40 fis and 1.5 sec widths, respectively)
had to be timed correctly to be maximal during beam transit.
While the klystron and RF gun were pulsed at a rate of 1 pps (assuming normal
operation of the klystron modulator), the capacitor bank generating the wiggler cur­
rent pulse took up to 15 seconds to charge, making wiggler operation much slower. A
single experimental “shot:: was therefore generated by initiating capacitor charging
for the wiggler. When the capacitor bank had reached a preset voltage level, it was
discharged so as to be contemporaneous with the next available beam pulse. This
required that the discharge trigger signal be given some milliseconds in advance; the
discrepancy is due to intrinsic switching times in the wiggler current unit, the width
of the wiggler pulse itself, and the finite diffusion time necessary for the wiggler field
to enter the acceleration structure.
Because the steering coil had the longest pulse time of all. it had to be triggered
69
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
70
even further in advance. W ith steering correction, then, the wiggler signal is not
given until nearly
1
second after the steering trigger, so that the first klystron pulse
after the capacitors are charged serves to start the trigger sequence, and the MIFELA
interaction does not take place until the second pulse.
4.1.2
Procedure
Due to the many subsystems involved, measuring energy change in the MIFELA
was a multi-step process. The measurement procedure is described here.
The beam energy depends sensitively on the RF power level in the gun and.
through the current level, on the gun cathode temperature. Thus the first steps in
every experimental run were to find a suitable operating point for the gun and then
to optimize the beamline magnets for the beam energy obtained.
An initial measurement of beam energy, giving a null spectrum, was obtained
using a beam passing through the MIFELA in the absence of RF power and wig­
gler field (although the axial field was retained in order to stabilize the beam as it
traveled through the acceleration tube). Since changing phase settings on the power
splitter/phase shifter unit had a small effect on the RF reflections from the gun.
and hence on the beam energy, this initial spectrum was valid only at a given phase
setting.
As described in Section 3.2.4. the wiggler exit region was ineffective at removing
transverse velocity from the exiting beam, and a steering magnet had to be used for
trajectory correction. Because the only way to find the necessary steering field was
to vary the magnet strength until beam appeared after the MIFELA output, the
steering magnet had to be calibrated on an unaccelerated beam. i.e.. without RF in
the accelerator. Thus the spectrometer setting obtained for the initial unchangedbeam measurement was left fixed, the wiggler magnet was operated, and the steering
magnet current was varied until a spectrometer signal again appeared. For exactness,
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
71
the steering current for each run was "balanced" against the wiggler field that was to
be used in the measurement. Once the steering current had been found, both steering
and wiggler currents were left unchanged for the rest of the run.
Note that this adjustment process depends on the impossibility of acceleration
using magnetic fields alone: more specifically, the energy of the spectral peak must
be unchanged from its value without wiggler fields. In practice, this means that while
spectral shapes may be compared with and without wiggler fields in the absence of
RF. the location of the peak is forced to be identical.
In certain cases in which
an unoptimal beam leads to large beam loss in MIFELA—for example, if the input
beam energy is far from
6
MeV—it is conceivable that one part of the spectrum
could suffer greater loss than another, so that the peak would actually be shifted. We
argue, however, that such cases would be recognizable by a significant mismatch of
the spectral shape, and in tiny case numerical analysis indicates that pan&Ifcs spread
in energy' do not have dissimilar trajectories.
All of the setup to this point in the experiment was carried out with no power in
the MIFELA at all. The fields in the accelerator, nevertheless, were correct only for
a specific beam energy set by the amount of RF in the gun. The RF levels in the gun
and accelerator could not be varied independently; only their ratio was adjustable
at the splitter. Thus it was somewhat cumbersome to "add RFMto the MIFELA in
order to take acceleration measurements: the overall RF level had to be increased
so that, with the splitter set to some appropriate ratio, the gun received the same
power level that it had before. Furthermore, since the splitter settings were manually
controlled by knobs on the splitter/shifter unit, located in the experimental vault,
they could not be adjusted during operation due to the radiation hazard. Hence, the
only possibility was to reset the splitter for the desired power level in the MIFELA,
increase the total RF power, and attem pt to reproduce the gun energy observed in
the initial setup. In practice, the beamline fields were sufficiently sensitive to beam
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
energy that if the fields were left unchanged, it was clearly visible when the correct
gun operating point had been reached; the maximum error in operating energy was
on the order of 10-20 keV. below the spectrometer sensitivity.
At this point, data was taken: that is. with the wiggler energized, and with all
of the beamline. axial, wiggler and steering currents held to their earlier values, the
spectrometer current was varied and the collected current measured. As described
in the previous section, the MIFELA was operated in single-shot mode and a beam
spectrum was built up from a series of shots as the spectrometer was scanned through
the energy range. The energy spectrum with RF in MIFELA in the absence of wiggler
field proved to be unchanged from the null result, so that an unaccelerated spectrum
could be obtained by scanning the spectrometer without pulsing the wiggler. This
provided a check against slow drift in the beam energy, which occurs because of
temperature effects in the gun and the klystron modulator, and a complete spectral
comparison for every scan.
4.1.3
D ata analysis
Because of the pulsed, single-shot nature of the experiment, each individual
energy spectrum consisted of a series of digitized collected current signals vs. time,
each obtained at a different spectrometer setting. A spectrum was assembled by
assigning a single peak value to each current trace, after baseline subtraction to
eliminate voltage offset and noise in the electronics. (The wiggler pulse, because of
its high current, was particularly difficult to isolate from the diagnostics, so a null
(no-beam) trace was generally taken as a baseline.)
Sources of uncertainty in this reduction come both from indefinite peak heights
and from the uncertainty in the spectrometer current/energy calibration. In the
selection of peak heights, an average baseline (i.e., a straight line representing the
average of any noise fluctuations) was compared with the largest signal value. On the
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
73
occasions when the current trace was noisy or showed structure, this process itself
was inexact.
To compensate for shot-to-shot variation in injected beam current, the peak
current measured in the current monitor nearest to the gun was also recorded for
each shot, and the collected currents were scaled to correspond to an identical in­
jected beam current. While this process never significantly changed the energy of the
spectral peak, it did generally change the spectral shape somewhat.
Shot-to-shot variation in the wiggler current was monitored throughout, but
during usable runs it was negligible compared to other sources of uncertainty.
4.2
Acceleration results
When the data collection stage of this experiment was finally reached, an unex­
pected problem appeared which affected any possible demonstration of the MIFELA's
performance. While the designs presented in Chapter 2 made use of the full output
capacity of the Beam Physics Laboratory's klystron, deterioration of the klystron
cathode caused the peak available power to fall to roughly 13 MW by the time that
data collection was possible. After reflections were taken into account, the usable RF
power for both gun and accelerator combined was measured at 12.7 MW. of which
only about
20%
could be fed to the accelerator if a usable beam was to be generated in
the gun. Even with this compromise, the full
6
MeV of beam energy was unavailable,
and results presented here use injected beams ranging from 5.2 to 5.6 MeV. further
decreasing the beam quality and acceleration gradient.
While this limit on available power leads to results which are somewhat less
dramatic than expected, the energy changes measured can nonetheless be compared
with numerical results obtained with the same (non-optimal) initial conditions: thus,
the computer code can still be benchmarked against the data. Of particular inter­
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
74
est is the response of the accelerator output to variation in the RF phase at which
the beam is injected. Phase response measurements are discussed in Section 4.2.2
below, showing that the actual MIFELA performance is comparable to calculated
predictions.
4.2.1
Experim ental data
The data presented here are in the form of spectrum comparisons for beam
energies with and without accelerator operation. Energy change can occur only if RF
power and wiggler fields are both present: no energy deviation was seen for operation
with RF alone, for example.
Spectra for these three null cases are compared in
Figure 4.1. where it can be seen that operating the wiggler causes beam loss which is
not entirely uniform in energy, but nevertheless preserves the overall spectral shape.
Spectra presented here generally compare the energy output during full MIFELA
operation with that of the beam under RF power alone, since the latter was the
easiest to obtain.
Since the expected energy change in MIFELA is dependent on the initial phase
through equation 1.19. repeated here:
d-1
ksa,saw
— =
sin<P
a:
7
(4.1)
and since the long RF wavelengths allow for precise phase control on the injected
beam, the phase value is the most important experimental parameter. While the
reported '‘value" of the phase here is the phase difference between output arms of
the shifter unit, it is understood that such a number represents an arbitrary choice
of zero-point, as energy change variation under differing phases is the real physical
quantity of importance.
Figure 4.2 shows output spectra obtained at two different phase values. 155°
apart, taken with an input beam energy of 5.62 MeV and RF power of about
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6
MW.
75
x
10
8
6
4
0
5.55
5.60
5.65
5.70
5.75
5.80
5.85
5.90
Electron energy (M eV)
Figure 4.1: Comparison of measured spectra without RF or wiggler fields (solid line,
square data points), with RF only (short dashes, diamonds), and with wiggler only
(long dashes, circles). Peak heights have been scaled for the same initial current,
and spectra have been shifted so that peak energies coincide. Spectral shapes are
consistent between traces, with evidence of beam loss on the wiggler-alone spectrum
leading to a narrower peak.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
76
The wiggler current was 32 kA. leading to an initial field of 1.3 ± 0.1 kG. and the
axial field was flat at 1.58 kG. Beam current at injection was equal to 80 mA.
The plot for
6°
shows an acceleration of 0.34 MeV. with an uncertainty of
roughly 0.4 MeV; there is clear separation between the unaccelerated and accelerated
spectra, since both of them have widths of 0.15 MeV. There are no measurable unac­
celerated particles. While there is beam loss on the order of 30%. the spectral shape
is nevertheless quite consistent between the two. with a suggestion of structure on
the high-energy side appearing in both curves. The effective acceleration gradient is
0.43 MV/m.
The plot for 161°. with a distorted spectral shape in which the average and
peak energies differ, is harder to interpret. While the simple equation 4.1 implies
that energy loss at this phase ought to be equal to energy gain at the opposite phase,
the inherent asymmetry of the tapered wiggler design will modify this relationship, so
it is no surprise that the two spectra are not equal and opposite. The peak is shifted
in the positive direction, however, by 0.15 MeV. approximately equal to the width of
the original spectrum; if average energies rather than peak energies are compared, the
energy gain is 0.07 MeV. which falls at the limit of the spectrometer sensitivity. We
propose that the unoptimal conditions of injection into the decelerating phase lead
to greater beam instability and sensitive dependence on conditions in MIFELA: see
the following subsection, in which both of these cases are compared with simulation.
We emphasize at this point that no one to date has reported IFEL acceleration
results of this kind: unambiguous acceleration of the entire exiting beam, with no
untrapped particles, and with energy gain more than twice the width of the spectral
peak. The percentage energy change of 6 % for all exiting particles currently exceeds
others in the literature.
Figure 4.3 shows output beam energy as a function of phase for a different set
of initial conditions. For this data set. input power was about 3 MW. with wiggler
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
iI
Accelerating phase (6°)
00
7
rZ
-/<
r t;
6
■I ..........V-
c
u.
5
I
“Z
3)
u
zj
4
<
U
AE = 350 keV
:
>
\
/
3
—
J
L
2
-i*
1
0
5.40
5.50
5.60
5.70
5.80
5.90
6.00
6.10
Beam energy (MeV)
Decelerating phase (161°)
AE = 150 keV
10
8
e
g
“3u
6
4
0
5.50
5.60
5.70
5.80
5.90
6.00
Beam energy (MeV)
Figure 4.2: Experimental energy spectra for two phases, separated by 155°. showing
results for accelerating and decelerating phases. In each case, the exiting beam spec­
trum under MIFELA operation (solid line, circles) is compared with the spectrum in
the absence of wiggler fields (dashed line, squares).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
78
current of 30 kA producing an initial field of
1.1
±
0 .1
kG and axial field of 1.58 kG.
The beam energy at input was 5.24 MeV. with a peak injected current near 40 mA.
The maximum energy gain in this case is 0.20 ± 0.02 MeV. occurring at a phase
value of 40°. Again, we see an asymmetric result for the energy change vs. phase,
with results for "decelerating" phases being both mostly positive (although near zero)
and less clean than results for accelerating phases.
The error bars on this graph give an indication of the uncertainty involved in
assigning a single number to the energy in each case. Since each number represents a
complete spectrum, results differ depending on whether one takes peak values, average
values, average values with background subtraction, or median values for the two
energies. Exit current was very low for phases opposite the accelerating maximum,
thus decreasing signal-to-noise ratios to the lowest allowable value. Furthermore,
since beams became widely spread in energy, identifying “the" beam energy or even
the peak energy became somewhat subjective. The increased error bar size on the
"decelerating phase" portion of the plot reflects both of these effects.
4.2.2
Comparisons w ith theory and com putation
The phase response shown previously in Fig. 4.3 is compared with simulation
in Figure 4.4 below. Simulation parameters in this case were 3.0 MW of input RF
power, a beam radius of
0 .1
cm. initial wiggler field magnitude of 1.15 kG, and a
constant axial field of 1.58 kG. The beam was monoenergetic and included a Gaussian
distribution in divergence angle, with vz/v having a =
0 .0 1 .
The figure shows an excellent fit between modeling and theory for acceleration
phases.
The only free parameter in use was an overall additive constant on the
energy data, representing additive uncertainty in the spectrometer calibration with
the steering correction included: this constant has a best-fit value of -70 keV. D ata
in the deceleration phases is much less clean: agreement is imprecise, and one of the
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
79
5.5
5.45
5.4
5.35
5.3
u
S
5.25
Injection energy = 5.24 MeV
5.2
-50
50
100
150
200
250
300
phase angle (dcg)
Figure 4.3: Experimental plot of output beam energy for an input beam energy of
5.24 MeV. as a function of injection phase. Error bars denote the uncertainty in the
identification of output energy values, due to wide energy spread or multiply-peaked
spectra.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Injection energy =
5 .2 4 M eV
\
5.2 j►
5.15
•50
0
50
100
150
200
250
300
350
phase angle (deg)
Figure 4.4: Simulation of output beam energies (solid line) compared with measured
output energy data as a function of injection phase for a monoenergetic initial beam
of energy 5.24 MeV. and with 3.0 MW of RF power. 1.15 kG of wiggler field, and
1.58 kG of axial field.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
81
d a ta points appears to be well above the expected energy. We examine simulation
results for decelerating phases more closely below, where this discrepancy is partially
resolved.
Figures 4.5 and 4.6 superimpose the acceleration spectra from Fig. 4.2 with the
results of model calculations for those cases. Here, the model parameters were the
same as those used above, but with increases in the RF power and wiggler field (6.0
MW of RF power and an initial wiggler field of 1.25 kG); also, the best-fit energy ad­
justm ent of —70 keV taken from the previous example was included. Use of a physical
beam with energy spread on the order of the measured spectral width results in an
accelerated-beam spectrum that very closely matches the experimental data. Again,
the decelerating phase match was not as good, with both spectral shape and peak lo­
cation differing from their experimental values, although part of the distribution does
overlap with that of the injected beam. Note, however, that the simulated spectral
width is increased by 40% from the accelerated case, with a flattened peak: also, with
4672 injected particles in the simulation, the accelerated beam contains 4088 parti­
cles. whereas the decelerated beam contains only 3505. In other words, simulation
does predict a higher degree of beam loss, a greater beam spread, and a somewhat
distorted spectral shape for the decelerating phase as compared to the accelerating
phase: asymmetry in the gain curve is also present. It is quite probable that these
characteristics indicate overall instability in the decelerating-phase case—certainly
intuition expects that the beam should be less well behaved when the wiggler taper
no longer matches its energy gain—and that the resulting sensitivity of the beam to
exact field and injection conditions would make its simulation less reliable.
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
82
10
0
_ 2
1~
-
5.4
■ ■-
■-
-
-------------------------------------------------------------■------------------------------ i--------------------------------------------------------------
5.5
5.6
5.7
5.8
5.9
6
:
6.1
Electron energy (MeV)
Figure 4.5: Simulation result in the accelerating phase for an input beam with finite
energy spread, compared with experimental results from Fig. 4.2. The injected beam
is centered at 5.62 MeV. with
6
MW of RF power and 1.25 kG of wiggler field.
Injection spectra are on the left: the simulated beam is the solid histogram, overlaid
with the experimental injection spectrum (dashed line with circles). On the right are
the simulated (solid line) and experimental (dashed line and diamonds) accelerated
spectra.
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 er. 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 .
83
10
S
— ..................................................... *............................-O r . ........
<
°
- .................
0
A
|
6 :..............................rr....^...A* A ....
O
r
=
4 ’.
^
L
^ I
I
=
3
2 :
0
_
5.3
0 ;^ f
© ...!......
-
“
x<y.0 ,'® ................
9
7 \> «
'•
^
o .....
! ■.
I °°
j
-
-
«
—
5.4
5.5
5.6
5.7
5.8
5.9
Electron energy (MeV)
Figure 4.6: As the previous figure, but for the decelerating phase. The injected
spectra are in the center: the simulated beam is the solid histogram, overlaid with
the experimental injection spectrum (dashed line with circles). The simulated out­
put spectrum (long dashes) and experimental output spectrum (short dashes and
diamonds) overlap but do not coincide.
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 er. 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 .
Chapter 5
Scaling Possibilities and
Conclusion
5.1
Scaling the MIFELA
The low acceleration gradient of the present structure, limited as it is by avail­
able power resources, raises the question of what one might expect from a MIFELA
that was scaled upward in power and/or frequency.
Equation 1.20. quoted again here
dr
—r - = 2ksasaw sin <P
dz
(5.1)
implies that the rate of energy gain squared is constant for constant wiggler period;
the rate of energy gain, d ^ jd z, is thus maximum at r =
0
and decreases over the
length of the experiment. The net gradient is therefore dependent on the length
of the structure.
However, tapering the fields allows the gradient to stay roughly
constant at something close to its initial (maximum)value.
obtained from Eqn. 1.19 for the simulations in Chapter
MV/m when
7
In fact, the rough value
2
gives a gradient of 0.79
is equal to its initial value of 12.7. a result which is surprisingly close
to the numerical value of 0.81 MV/m for the same situation. We may therefore
estimate the likely MIFELA gradient using eqn. 1.19.
d'y
ksasaw .
— =
sin<P.
dz
7
with
7
set to its injection value.
84
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 rm is s io n .
(5.2)
85
To simplify matters, we assume that the waveguide refractive index n and the
ratio of wiggler radius to pitch rw/ \ w remain constant under frequency scaling—in
effect, that the MIFELA dimensions scale inversely with frequency. The RF power
level is proportional to the electric field amplitude, which in a circular waveguide
scales with power and waveguide dimensions via
(5.3)
so that
eE0
mujc
>/ P r f
Rl>jti1/'2
(5.4)
and with n constant, the product Rui is constant as well, and the normalized RF field
strength scales entirely as the square root of input power.
The wiggler field B w is proportional to
(5.5)
and hence with rw/ \ w constant, we observe that
aw ~ ~r~
(5-6)
is constant under frequency scaling. The overall gradient, then, scales with the square
root of power (through as) and with the RF frequency (ks).
Using this reasoning, a MIFELA operated with 150 MW of input power at a
frequency of 34 GHz (parameters which have been proposed for next-generation RF
accelerators [8 8 . 89]) would have a gradient 12\/T0 ~ 36 times greater than the device
outlined in Chapter 2. and thus possible gradients of 30-35 M V/m. Such a device
would be reasonably straightforward to construct, with a waveguide radius of perhaps
4 to 5 mm. and would be free from some of the fabrication difficulties posed by RF
cavities at high frequencies.
We also note that the use of an RF cavity rather than a traveling waveguide
would substantially increase the effective RF fields present in the structure. If one
R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
86
calculates the traveling-wave power needed to generate the same peak field strengths
as found in a cavity T E u P mode, one finds that the result is proportional both to n,
the waveguide index of refraction, and Q , the cavity quality factor. With Q-values
highly dependent on frequency, a cavity at 2.8 GHz with a Q of 18.000 gives field
strengths 15 times greater than a traveling wave alone, whereas a cavity at 11.4 GHz
with Q = 34.000 increases peak fields by a factor of 28. The considerable engineering
problems of such a cavity would include development of a feedback system to provide
frequency stability, and these ideal Q-values are higher than what has been achieved
to date using normal-conducting cavities, which currently reach Q's in the low tens of
thousands [90]. Nevertheless, the current experiment with unchanged frequency and
power levels could plausibly give acceleration gradients of 1.7 M eV/m with a cavity.
Since beam loading is minimal for the cases considered here, it would be conceiv­
able to design a "recirculating-power" MIFELA. in which unused microwave power
would be collected at the MIFELA output and re-injected into the waveguide again.
Such a circular resonator could build up large power levels while avoiding some of the
difficulties of constructing and tuning a cavity with very high Q. and would have the
advantage th at the entire wave could be available for acceleration. Modern recircu­
lating systems have exhibited power gains near 10. and continue to improve [91, 92].
Starting from Eqn. 1.16 without substituting as gives an alternate version of
the same scaling rule, i.e.
j : ~ E0aw
(5.7)
which for a guided wave gives identical results to those above, since waveguide radius
R scales inversely with frequency for fixed n (see 5.3 above), and aw is roughly con­
stant. This version, however, is also useful for laser-driven IFELs [57. 47], in which
the spot size acts as a kind of waveguide radius. The upgraded
10
fj.m CO2 laser
at Brookhaven ATF has been calculated to have an electric field amplitude of 21.9
GV/m at the focus, with a Rayleigh length of 30 cm. An IFEL at this frequency.
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 .
87
even one with an unchanged wiggler parameter, could have gradients thousands of
times larger than those in this experiment, giving an energy gain of more than
in one Rayleigh length for a
6
1
GeV
MeV beam. Although synchrotron radiation losses will
begin to be significant at these energies, the electron transverse velocity will decrease
with increasing 7 . so that high drive power and large energy could ameliorate this
effect. One study has found [46] that the practical energy limit on a high-energy
IFEL is at least several hundred GeV. although the beam emittance could begin to
deteriorate at lower energies. While such an accelerator would make large demands
on field and phase precision and might require several tesla of axial field, the potential
utility could be great.
5.2
Summary and conclusions
In this thesis, we have explored the prospects of the inverse FEL mechanism
for electron acceleration using microwave radiation as a driving field. Since previous
experimental efforts to demonstrate IFEL acceleration used laser or millimeter-wave
beams, this work represents a new and distinct proof of the IFEL principle, including
some capabilities not available to earlier experimenters. Beginning with feasibility
studies using numerical simulation to predict the abilities of a microwave IFEL. we
have taken our nominal laboratory capacity of 15 MW of microwave power at 2.856
GHz and. using this value, designed a "best-case" accelerator. This device, which
utilizes a tapered helical wiggler and an axial guiding field, both near
2
kilogauss,
exhibits an almost-constant acceleration gradient of 0.81 MeV/m. Simulation results
show minimal energy spread and modest emittance growth, with sensitive dependence
on initial beam injection phase, as expected.
The simulations were used as a guide in engineering and constructing a MIFELA
in the Beam Physics Laboratory, designed to accelerate a
6
MeV electron beam
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 er. 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 .
88
produced by an RF gun. Engineering solutions and features include a pulsed bifilar
helical winding to create the wiggler field, shunt resistors to ensure adiabatic beam
entry, and continuously variable injection phase, with a very small effective "phase
window.” Automatic synchronization between RF fields and electron beam is enforced
by splitting a single RF pulse between the RF gun and the accelerator structure.
While limitations in the available equipment reduced the available beam energy
and RF power to less than optimal levels, it was still possible to observe acceleration of
an amount never previously demonstrated. When operated at the appropriate phase,
the MIFELA exhibited energy gain of 0.35 MeV, with virtually unchanged energy
spread, and an accelerated fraction of nearly 70%. The importance of injection phase
was demonstrated repeatedly, as the small phase window permitted phase spreads
that were near zero on the scale of the RF wavelength; we have reported on the
degraded beam quality, energy spread, and insignificant energy change that result
from off-phase injection.
For accelerating-phase values, simulation results agree well with experiment.
The amount of measured energy change and the width of the region of accelerating
phase coincide strongly with those calculated. Results are less exact for decelerating
phases, suggesting that increased sensitivity to beam quality and magnetic field con­
ditions makes exact simulation harder in those cases; in any case, the asymmetry of
the tapered-wiggler design leads to quite different behavior for accelerated and decel­
erated beams. These findings imply that if an IFEL were to be used as a pre-injector
for some larger machine, energy spreads created by large phase width at injection
could be a major concern, and efficient bunching would then become a priority.
The simulation results for accelerating phases having been validated by exper­
iment. we report on further prospects for this version of the IFEL when scaled to
different frequency regimes and power levels. Simple estimates for the change of out­
put energy with changing parameters show that, using 150 MW of 34 GHz radiation.
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 er. 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 .
89
gradients over 30 MeV/m appear likely: such a MIFELA could thus be useful as a
pre-injector for other high-energy machines. Laser IFELs could conceivably compete
with other high-gradient laser acceleration mechanisms. Continuation of this work
may show to what extent this potential can be realized.
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 .
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R e p r o d u c e d with p e r m is s io n of t h e cop 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 .
95
[91] C. Nantista et al., in Proceedings of the 1993 IEEE Particle Accelerator Con­
ference, Washington. DC. edited by S. T. Corneliussen and L. Carlton (IEEE.
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[92] S. G. Tantawi. R. D. Ruth, and P. B. Wilson, in Proceedings of the 1999 IEEE
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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 .
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