Twisted waveguides for particle accelerator applications
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Citation
Wilson, Joshua L., Yoon W. Kang, and Aly E. Fathy. “Twisted
Waveguides for Particle Accelerator Applications.” IEEE, 2009.
129–132. © Copyright 2012 IEEE
As Published
http://dx.doi.org/10.1109/MWSYM.2009.5165649
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Institute of Electrical and Electronics Engineers (IEEE)
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Thu May 26 23:57:18 EDT 2016
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http://hdl.handle.net/1721.1/71807
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Detailed Terms
Twisted Waveguides for Particle Accelerator Applications
Joshua L. Wilson # , Yoon W. Kang ∗ , and Aly E. Fathy ∗∗
#
University of Tennessee, Knoxville, TN (Now with MIT Lincoln Laboratory, Lexington, MA)
∗
Spallation Neutron Source, ORNL, Oak Ridge, TN
∗∗
University of Tennessee, Knoxville, TN
designed to support slow wave accelerating modes with good
figures of merit. Here, we discuss the twisted guide from a
microwave engineering perspective. First, the geometry of the
twisted structures is described mathematically. Secondly, the
eigenmodes and dispersion curves are compared to corrugated
RF structures. Finally, two twisted prototypes are discussed,
with excellent agreement obtained between experiment and
prediction.
Abstract— A novel microwave device for accelerating charged
particles based on twisted waveguide is presented. Twisted guides
support slow-wave TM modes whose phase velocity could reach
the speed of light c. The axial electric field in these structures can
travel synchronously with the particles to achieve nearly uniform
acceleration in a traveling-wave topology. The advantages of using
twisted guides over conventional RF accelerating cavities are discussed. We present two types of twisted accelerating structures,
one analogous to the well-known disk-loaded accelerating structure, and the other analogous to the popular elliptical (or TESLAtype) accelerating geometry. The propagation characteristics of
these two structures are considered, and prototypes are made to
experimentally validate our theoretical results.
Index Terms— Slow wave structures, accelerating structures,
guided waves
II. T WISTED S TRUCTURE G EOMETRY
A simple example of a twisted structure is a twisted
rectangular waveguide, originally analyzed by Lewin [4]. In
this case, the waveguide is essentially a straight rectangular
guide twisted uniformly about its longitudinal axis. Although
this simple case is instructive, it is not necessarily good
for accelerating particles. Therefore, we wish to consider
other cross sectional geometries other than a rectangle. Our
strategy is to pick a cross section that generates a 3D twisted
structure whose longitudinal cross section matches some given
rotationally symmetric accelerating structure. This can always
be done as follows:
Start with some rotationally symmetric structure defined by
I. I NTRODUCTION
RF cavities have long been used to accelerate charged
particles to achieve high electric field strengths and because
of the wide availability of high power RF sources. However,
special care must always be taken to match the phase velocity
of the wave in the RF structure (whether it be traveling or
standing) to the velocity of the particle to be accelerated. This
constraint prohibits empty straight waveguides and cavities
from being used, because the phase velocity in such structures
is always greater than c. Dielectrics can sometimes be used to
circumvent this problem, but dielectrics introduce difficulties
of their own, such as outgassing and dielectric breakdown.
In practice, corrugated structures are almost always used to
achieve slow wave operation, since these structures tend to
have high accelerating electric field and can also be used in a
superconducting regime.
On the other hand, corrugated structures present problems
such as more expensive manufacturing, more difficult tuning
procedures, and the potential for “trapped modes”; these are
modes whose frequency resides in a stop band and, as a
result, are not able to propagate out of the structure [1]. Such
modes are particularly problematic in high Q superconducting
structures, where they may be permitted to persist for a long
time with little decay and act detrimentally on the particles.
In this paper, a twisted waveguide accelerating structure is
discussed that seeks to combine the advantages of a straight
uniform waveguide with those of an empty corrugated structure. The twisted guide was originally proposed by Kang
[2], with a more detailed analysis carried out in [3]. It
was shown [3] that the twisted waveguide structure can be
978-1-4244-2804-5/09/$25.00 © 2009 IEEE
ρ < g(z),
(1)
where g is a periodic function with period ∆z. Next, a 2D
transverse cross section is defined in polar coordinates (ρ, φ)
φ(x, y)∆z
.
(2)
ρ(x, y) < g
π
We then take this cross section and twist it about the longitudinal axis at a twist rate
π
p=
.
(3)
∆z
(p is in Rad
m ). The 3D geometry produced by this method
will always yield a longitudinal cross section identical to the
original rotationally symmetric accelerating structure.
For example, assume we began with a regular disk-loaded
accelerating structure (a cylindrical waveguide periodically
loaded with metal irises.) Applying the procedure described
above will yield the notched cross section of Fig. 1(a). This
cross section is then extruded while twisting at a constant rate
to yield the shape (b), whose longitudinal cross section is the
same as the disk-loaded structure we began with.
129
IMS 2009
Fig. 1.
b
a
A twisted analog to the disk-loaded slow-wave structure
Fig. 3.
Twisted analog of an elliptical accelerating structure.
Stanford Linear Accelerator (SLAC) [5], designed to accelerate relativistic electrons, and constructed its helical analog.
It was discovered upon CST [6] simulation that the fields
were very similar to the fields of a disk-loaded structure, with
the exception that the resonant frequency had shifted up by
roughly 30%. As a result, the phase velocity was too high
to accelerate the electrons. To compensate for this, the outer
radius was increased until the phase velocity was brought back
down to the speed of light c. The dimensions for the final
twisted structure are shown in Table I, while the simulated
fields are shown in Fig. 4. A periodic boundary condition
was enforced on each end of the structure, with 0 degrees of
phase shift between the two ends. This yielded a wave with the
same propagation constant as the orginal SLAC accelerating
structure.
z
Fig. 2.
Geometry of an elliptical or TESLA-type accelerating
structure.
Because this twisted structure has such structural similarity
to the disk-loaded accelerating structure, it is expected that
it may be a good candidate for particle acceleration. This is
indeed the case, as will be demonstrated in the next section.
Before we discuss the electromagnetic properties of this structure, we will introduce another class of structures formed using
the same procedure just discussed. In particular, we consider
an elliptical cavity, also widely used in the accelerating field.
The longitudinal cross section of this rotationally symmetric
structure can be described in cylindrical coordinates by joining
ellipses and tangent lines as shown in Fig. 2.
Following the procedure for creating the twisted analog, we
twist the cross section of Fig. 3a, and form the 3D structure
of (b).
TABLE I
PARAMETERS FOR TWISTED ANALOG OF DISK - LOADED
ACCELERATING CAVITY
Parameter
Frequency
Inner radius
Outer radius
Twist rate
Notch angle
Phase advance per cell
Phase velocity
Value
2.84
4.13
5.493
89.76
1.048
2π
3
2.98 × 108
Unit
GHz
cm
cm
Radians/m
Radians
Radians
m/s
The dispersion curves for this structure are shown in Fig. 5
for several values of the twist rate p. These simulations were
carried out using the straight guide equivalent model of [7].
Unlike the dispersion curves of corrugated structures, it
may be noticed that there are no stop bands present for
this twisted structure; this is an important feature which we
shall return to later in this section. Instead, the dispersion
curves resemble more closely the dispersion characteristics of
a straight waveguide.
The second structure considered, the elliptical guide, was
designed following one of the superconducting cavity designs
at the Spallation Neutron Source (SNS), operated at 805 MHz
III. E LECTROMAGNETIC CHARACTERISTICS OF T WISTED
G UIDES
In this section, we begin by presenting some specific
characteristics of the twisted analogs of the disk-loaded and
elliptical accelerating structures. We then generalize our results
by reviewing some of the salient features of the twisted guide
in contrast to corrugated accelerating structures.
A. Two twisted cavity examples
First, consider the twisted “notched” structure of Fig. 1.
We began with a disk-loaded accelerator design from the
130
Fig. 6. CST simulated electric field distribution in twisted elliptical
analog.
Fig. 4. CST simulated electric field distribution in twisted diskloaded analog.
the operating point on the dispersion curve, allowing it to
operate at a different phase velocity. On the other hand,
corrugated accelerating cavities that operate in a standing wave
mode must be tuned cell by cell, adding to the manufacturing
difficulty.
4.5
TEM limit
4
frequency(GHz)
3.5
p=67.3R/m
It should be mentioned in this discussion that the dispersion
curves predicted for an infinite twisted waveguide may be
different than the curves for a waveguide terminated by metal
end walls, as discussed above. This is because of the perturbation introduced by the metal walls, whose imposed boundary
conditions do not conform to the periodic field solutions.
To develop an accurate model of the structure accounting
for all end effects, a 3D numerical method must be used.
However, a simplification to 2D can be made if the waveguide
is assumed to be infinite [7], an assumption that often does not
significantly alter the dispersion characteristic, but can cause
large discrepancies in the electric field strength close to the
metal walls.
3
2.5
p=337R/m
2
1.5
1
20
30
40
50
60
Beta(R/m)
70
80
90
Fig. 5. Dispersion curves for the twisted analog of a disk-loaded
structure, shown for several different twist rates.
B. Discussion
A second important feature of the twisted guide is that
the fields along the center axis of the twisted guide (where
the particles ideally reside) can be designed to be purely
axial. This is because for the designs considered, each electromagnetic mode solution is necessarily degenerate due to the
rotational symmetry of the structure. In particular, any field
solution rotated 180 degrees about the center axis must also
be a solution, i.e. the physical structure maps back onto itself
under this rotation. Adding these degenerate field solutions
together, one obtains both electric and magnetic fields that
are purely axial at the center, since any transverse component
of the original field solution is annihilated by the rotated
field solution. This is important because any transverse field
components along the particle axis can cause defocusing of
the particle beam and result in instability.
There are several prominent electromagnetic features of
twisted accelerating structures. First, we notice that the phase
velocity can be adjusted to well below the speed of light.
Also, twisted structures can be easily tuned if operating in
a standing wave mode simply by shifting the position of the
terminal end walls, effectively changing the allowed values
of the propagation constant. This, in turn, will slightly shift
Finally, as mentioned previously, the twisted guide has
dispersion characteristics resembling a straight waveguide,
rather than possessing the stop bands normally associated
with periodic structures. This is because there exists an exact
mathematical analog that relates the twisted guide to a straight
waveguide [8], [7]. As a result, the problem of “trapped
modes” can be effectively circumvented.
and used to accelerate non-relativistic ions at 61% the speed
of light. Once again, the twisted analog to this structure had a
somewhat higher resonant frequency than the original structure
(by roughly 30 %), so the outer dimension of the twisted
structure was increased in order to keep the phase velocity
constant. The final design was scaled to operate at 2.8 GHz,
and had an inner radius of 1.24 cm, an outer radius of 6.37
cm, and a twist rate 96.2 Rad/m. The twisted structure admits
field solutions of the type shown in Fig. 6.
The dispersion curves are similar in appearance to Fig.
5, and will be presented in the next section along with
experimental data.
131
TABLE II
C OMPARISON OF MEASURED AND PREDICTED RESONANT
FREQUENCIES FOR THE TWISTED NOTCHED STRUCTURE
Mode
TE-like
TE-like
TE-like
TM-like
TM-like
TM-like
Fig. 7.
Fabricated 2.8 GHz twisted prototypes.
a
Measured Frequency (GHz)
1.874
1.994
2.378
2.639
2.814
2.913
Predicted Frequency (GHz)
1.876
1.978
2.334
2.636
2.822
2.953
have less problem with trapped modes compared to normal
corrugated accelerating structures. A technique was presented
to find twisted analogs to rotationally symmetric structures.
Two specific examples were considered: one analogous to the
disk-loaded accelerating structure, and the other analogous to
the elliptical cavity. Two prototypes were fabricated and measured experimentally, and excellent agreement was obtained
between the predicted and measured results.
This twisted guide accelerating structure has great promise
for use in future accelerators. The wide range of achievable
phase velocities indicates that twisted structures can be used
for both relativistic and non-relativistic particle accelerators.
Since no dielectrics are employed, the twisted guide is also
a good candidate for superconducting applications. Finally,
twisted guides could possibly be mass produced more cheaply
than conventional corrugated structures. The advantages of
twisted guides and cavities should not be overlooked in the
design of the next generation of particle accelerators.
b
Fig. 8. Dispersion curves for twisted prototypes. Solid lines are
from simulation, and x’s represent experimental data points. (a) is
for notched shape (the top curve is the TM-like accelerating mode,
and the bottom is a TE-like mode). (b) is for elliptical shape.
IV. E XPERIMENTAL R ESULTS
ACKNOWLEDGMENT
Two prototypes were fabricated at S-band; one had a
notched shape, and the other an elliptical shape. These prototypes are shown in Fig. 7. They were manufactured by
QuickParts using 3D printing, and then electroplated on the
inner surfaces and flanges. Copper end plates were affixed
to each end, and probes were placed in the end walls. The
resonant frequencies were measured using a network analyzer,
and a beadpull measurement [9] was performed to measure the
electric field distribution along the center axis of the twisted
structure.
Since the resonant frequencies are known from measurement and the propagation constant can be obtained from the
beadpull measurements, it is possible to construct dispersion
diagrams for the TM modes of interest. These curves are
shown in Fig. 8. Despite the perturbations to the periodic
fields introduced by the metal end walls, the experimentally
measured dispersion curves were close to those predicted by
theory. The measured resonant frequencies for the notched
structure are also compared to our theoretical results in Table
II.
This work has been sponsored by ORNL-SNS. The Spallation Neutron Source is managed by UT-Battelle, LLC, under
contract DE-AC05-00OR22725 for the U.S. Department of
Energy.
R EFERENCES
[1] F. Marhauser, H.-W. Glock, P. Hulsmann, M. Kurz, and H. Klein, “Search
for trapped modes in tesla cavities,” in Proceedings of the 1997 Particle
Accelerator Conference, vol. 1, May 1997, pp. 527–529.
[2] Y. W. Kang, “Twisted waveguide accelerating structure,” in 9th Workshop
on Advanced Accelerator Concepts, Aug. 2000.
[3] J. Wilson, Y. Kang, and A. Fathy, “Twisted structures and their application
as accelerating structures,” in Proceedings of the 2008 Linear Accelerator
Conference, Sept. 2008.
[4] L. Lewin, Theory of Waveguides. London, Newness-Butherworths, 1975.
[5] R. P. Borghi, A. L. Eldredge, G. A. Loew, and R. B. Neal, Design
and Fabrication of the Accelerating Structure for the Stanford Two-Mile
Accelerator. Academic Press, New York, N. Y., vol. 1.
[6] CST Microwave Studio 2006 Users Manual.
CST Ltd., Darmstadt,
Germany.
[7] J. Wilson, C. Wang, A. Fathy, and Y. Kang, “Analysis of rapidly
twisted hollow waveguides,” IEEE Transactions on Microwave Theory
and Techniques, vol. 57, pp. 130–139, Jan. 2009.
[8] D. Shyroki, “Exact equivalent straight waveguide model for bent and
twisted waveguides,” IEEE Transactions on Microwave Theory and
Techniques, vol. 56, no. 2, pp. 414–419, 2008.
[9] D. Goldberg and R. Rimmer, “Measurement and identification of HOMs
in RF cavities,” in Proceedings of the 1997 Particle Accelerator Conference, vol. 3, May 12–16, 1997, pp. 3001–3003.
V. C ONCLUSION
A novel twisted waveguide based accelerating structure has
been presented. These structures support TM modes with an
axial electric field that can be used to accelerate charged
particles. These twisted structures can be easier to tune and
132