PEG Structures for Multi-Beam Devices
David Yu, David Newsham, and Alexei Smirnov
DULY Research Inc., Rancho Palos Verdes, CA 90275.
Abstract. Photonic band gap structures with single or multiple defects show potential for use in
single-beam and multi-beam klystrons and particle accelerators. The primary concerns are the
coupling between the modes at each individual defect site and the damping of unwanted higher
order modes. A conceptual design of a PEG based, multi-beam klystron and methods to damp
HOMs and to cool and tune the structure are presented.
INTRODUCTION
Photonic band gap (PEG) structures [1-5] are well known in solid-state physics,
and recently have received attention among accelerator researchers for possible
applications as rf power extractor and/or accelerating cavities. The PEG cavity
generally consists of a periodic lattice of (vertical) conducting cylinders bounded on
top and bottom by a conducting plate. The primary feature of the PEG is that the
periodic lattice creates an electromagnetic band structure analogous to electron band
gap formation in semiconductors. The traditional rf cavity structure has a spectrum of
discreet modes beginning with the fundamental and continuing upward. An infinite
PEG structure exhibits frequency bands of allowed modes ("pass bands") interspersed
with frequency bands where no modes exist ("stop bands"). In general, the modes that
exist in the pass band have fields that extend throughout the lattice. If a "defect" is
created in the band structure by altering or removing a single rod or rod pattern, then
one or more "defect modes" may exist in the stop bands. In this case, the field of the
defect mode is localized to the region near the defect and decays away exponentially
in all directions. The band structure for the TM and TE modes of an infinite triangular
lattice are shown in Figure 1 along with the global band gap diagram for the same
lattice parameters.
The PEG cavity may be used with a single defect, i.e. a center rod removed from
the lattice, for application to an rf generator (e.g. klystron or gyrotron), or to an
accelerator. It may also be used with multiple defects for application to a multi-beam
klystron or a multi-beam accelerator. The PEG cavity has unique properties that can
facilitate simple coupling of input and output power in the fundamental mode. A main
feature of the PEG structure is that it has effective damping of higher order modes. It
has potential to significantly simplify the design and improve the performance of high
power multi-beam klystrons and accelerators. Figure 2 illustrates two PEG structures,
one with a single-defect and another with a six-defect lattice.
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
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12.0
16.0
TE modes
TM modes |
12.0
.
8.0
£ 8.0
4.0
4.0
0.0
0.0
r
15.0
25.0
TM modes
20.0
10.0
015.0
8
10.0
5.0
5.0
TE modes
0
.0
0.1
0.2
0.3
0.4
0.0
0.0
0.5
a/b
0.1
0.2
0.3
0.4
0.5
a/b
FIGURE 1. Band structure and global band gaps for an infinite triangular-lattice PEG structure with
lattice filling fraction a/b = 0.2, where a is the rod radius and b is the distance between two adjacent
rods, a) TM modes, b) TE modes.
Single Defect
6 Defects
FIGURE 2. Schematic drawing of a) a single-defect, triangular-lattice PEG structure, b) a six-defect,
triangular-lattice PEG structure. Open circles represent missing rods.
The PEG structure has several advantages that make it attractive for multi-beam
operation. They are simple to manufacture, and can be constructed using simple rods,
plates, and cavity walls. The oversized structure results in increased power handling
capabilities; and for a PEG cavity oversize does not always imply overmodes. The
fundamental mode is confined to the vicinity of a local defect site, and the "lattice"
modes that occur in the pass band of the overall structure extend beyond the defect
where they may be efficiently suppressed. Finally, it is possible to have several
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strongly coupled modes in the same structure while collectively suppressing unwanted
modes.
SINGLE-DEFECT PBG
Figure 3a shows the fundamental single-defect mode of a triangular PBG cavity.
Figures 3b-d show higher order modes that appear to extend throughout the cavity
indicating that they may be modes of the lattice that are outside the stop band, and
hopefully easily damped at the periphery of the structure. In particular, Figure 3c
shows the dipole mode that was presented in the wakefield study performed by
Shapiro [2]. Issues of the singe-defect PBG cavity include cooling, tuning, and
minimizing overvoltage. A single-mode, single-defect PBG cavity was constructed
using a careful choice of the rod radius and lattice constant [6].
FIGURE 3. Electric field contours for single-defect modes: a) fundamental (accelerating) mode,
b-d) higher order (lattice) modes.
MULTI-DEFECT PBG
We have extended the concept of a single-defect PBG cavity to a multi-defect
cavity. Preliminary analysis of a six-defect cavity was performed. Because each of
the six defects in the lattice represents an individual oscillator, the coupling of these
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oscillators should result in six normal modes, five of which are not the accelerating
mode. These six normal modes are shown in Figure 4.
FIGURE 4. Six normal modes that result from the coupling of 6 oscillators in a 6-defect PEG cavity.
The "+" and "-" indicate the relative axial field directions in the defect, and "0" indicates that the defect
site is split by the null line and a local dipole-like mode should exist in that defect site.
Figure 5 shows the first two of the 6-defect modes. The left-hand figure in Figure 5
is the fundamental accelerating mode. One of the five coupled oscillator modes is
shown in the right-hand side of Figure 5. Unlike the single-defect case where higher
modes extend throughout the lattice, this mode appears to be bound to the defect sites.
Further design work will be needed to identify each of the modes, determine the mode
separation, and identify techniques to increase the mode separation.
FIGURE 5. Examples of the 6-defect modes, a) is the 6-defect accelerating mode and b) shows one of
several higher order coupled oscillator modes.
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Calculations using SUPERFISH indicate that of these 6 modes, only 4 are
independent. This is consistent with the results of a theoretical analysis of six coupled
1-D harmonic oscillators. From the SUPERFISH results, the separation between the
fundamental mode and the next higher mode is larger than 200 MHz, for a 17 GHz
PEG cavity with a=0.79 mm and b=6.40 mm.
To increase the coupling between the six defect sites, seven additional rods were
removed near the center of the cavity. Some of the modes of this cavity appear similar
to those of a single-defect cavity, except that the larger size lowers the fundamental
frequency. Figure 6a shows the fundamental mode in the larger cavity, and Figure 6b
shows an overmoded field pattern that exploits the basic 6-defect pattern of the cavity.
Figure 7 plots the axial electric field versus the distance from the center of the PEG
cavity along the vertical line. The localized spots of null field values are a result of
passing through the conducting rod.
a)
b)
FIGURE 6. Two modes of the 6-defect lattice with 7 central rods removed.
1.2
1.0
0.8
_ 0.6
< 0.4
<0.8
~ 0.2
f
0.0
1-0.2
Li.
1 -°-4
"* -0.6
-0.8
-1.0
Distance from the Cavity Center (cm)
Distance from the Cavity Center (cm)
FIGURE 7. Axial electric field versus distance from the cavity center for two modes of the 6-defect
PEG cavity with the 7 central rods removed.
It is seen from Figure 6 and 7 that the fields in the defects are strongly coupled to
each other, as well as to the large opening at the center. These two modes may play a
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role in the design of a power input/output coupler based on overmoding. In addition
to a six-defect cavity, we are also studying other multiple-defect PEG cavities,
including linear arrays of multiple defects which may be useful for sheet beams.
MULTI-BEAM RF STRUCTURES
Conventional klystrons generally consist of a series of cylindrical resonant cavities
separated by drift tubes with predetermined lengths for optimal bunching. A gun
produces an electron beam that is first velocity modulated by the rf field in the input
cavity, then subsequently bunched by one or more intermediate cavities, and finally,
the beam power is converted into rf energy in the output cavity before the beam is
dumped into a collector. Multi-beam klystrons generally use an array of specially
designed cavities coupled in an appropriate manner to synchronize the beams. By
replacing several of the metal rods with beam holes, a single multi-defect PEG
structure can be used to simultaneously interact with multiple electron beams, each
passing through an individual beam hole in the structure. A simplified schematic of
such a device is shown in Figure 8. In the multi-beam klystron concept currently
under development, six beams enter the first PEG cavity, and are velocity modulated
by a TMOl-like mode set up in the defect site around the beam holes by an external
drive signal coupled to the center (as shown) or via waveguides to the periphery of the
structure (not shown). The beams are then bunched by a series of intermediate
resonant PEG structures. Finally, the beams are energy modulated at the last PEG
structure, which also has an output waveguide coupler, and the spent electron beams
enter a common collector after the beam energy has been extracted. The two modes
shown in Figures 6 and 7 may play an important role in achieving efficient coupling.
input gain
cavity cavity
bunching
cavity
output
cavity
FIGURE 8. Schematic layout of a PEG based multi-beam klystron. A multi-beam accelerator would
have no space between successive cavities.
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A similar scheme could be used for a multi-beam traveling-wave accelerator. Other
than extracting energy from an applied rf field rather than supplying the field, the
major differences are that there would be no drift space between successive cavities,
and at the end, the electron beam would leave the system intact rather than being
dumped in the collector. The cavity design for the accelerating structure would have
similar properties and function with the inter-mediate cavities in the klystron design.
HIGHER ORDER MODES
Figure 9 shows the first higher order mode that does not appear to be bound to the
multiple defect sites. As seen in the figure, the cavity TM02 mode results in a dipolelike mode at each defect region where the radial null of the TM02 mode would occur.
It is important that this mode be suppressed for successful operation of a multi-beam
device. Possible methods to damp this mode include the introduction of absorbing
material around the structure and coupling to waveguides, either located radially at the
periphery of the cavity via rectangular waveguides or axially coupled to the center of
the cavity via a round waveguide. Currently, we are studying the effect of coupling
this mode to a rectangular waveguide using SUPERFISH and GdfidL to calculate the
value of the loaded Q and estimate the damping effectiveness.
Figure 9. Electric field contours for a higher order mode of the 6-defect PEG structure.
COOLING AND TUNING
There are two aspects of cavity tuning. The first is the designed tuning to set the
frequency of the fundamental defect mode; it is integral to the design process. The
second class of tuning is used to make up for slight numerical and manufacturing
errors (tolerance) that cause the part to deviate from the ideal design and cannot be
avoided in practice. It should be possible to provide some shift in the frequency of the
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fundamental defect mode by altering the shape and/or the position of the innermost set
of rods surrounding each defect site, and by altering the diameter of the coupling hole
at the defect site. Care must be taken when altering the basic lattice structure to insure
that the band gap properties of the PEG device are not lost due to altering the rod
design.
To make up for manufacturing tolerance and to achieve proper frequency resonance
(coupling) at (between) each defect site, individual site tuning will probably be
required. One proposed method of tuning each individual defect site is by slightly
dimpling or bending one or more of the rods that surround that defect. Other than
issues of sensitivity and asymmetric rod bending that will naturally arise, the defect
lattice has a particular feature that each defect site shares 2 rods with neighboring
defect sites. If one of these shared rods was bent in order to tune one defect site, it
would most likely require the opposite bend to provide the equivalent tune to its
shared defect site and a classic case of frustration occurs. This frustration would
become extreme in the case of a large number of defects regularly spaced in the
lattice. For the 6-defect case, limiting the choice of rods to be bent can solve this
frustration. If only 3 rods are bent, as shown in Figure 10, then the altered rods would
be next-nearest neighbors to the defect site and would likely have less influence upon
them. If the dimple method were chosen, then the alteration to the rod would only be
present on the portion of the rod facing the defect, and thus, should not interfere with
the adjacent defect sites, but may reduce the effectiveness of the cooling by reducing
the area of water flow.
The rods that are inherent in the PEG geometry provide a convenient method for
cooling the structure. By replacing select copper rods with copper tubing, they can be
internally cooled using water. Analysis by MIT [2] of single defect PEG structure
indicated that the local temperature rise on the interior surface of the rod was
approximately twice that of a standard pillbox cavity. In the 6-defect structure (shown
in Figure 10) the problem may be greater for the rods that border on 2 defect sites
because of the nominally double heat flow.
Differences between the heat load on the rods that share defect sites with the rods
adjacent to only a single site may result in a temperature difference between rods.
This temperature variation, and the resulting difference in expansion may play a role
in the tuning of the cavity. It may be possible to tune individual defect sites by
carefully controlling the temperature of individual rods that border that cavity.
Although the external plumbing may be complicated for such a scheme, the dynamic
nature of thermal cooling may provide an elegant solution to the tuning problem.
In addition to the rods that make up the PEG lattice, the cooling requirements of the
cavity end plates and iris openings may present a problem. It is hoped that conductive
cooling of these end plates will be sufficient, otherwise an active cooling channel
would need to be introduced.
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FIGURE 10. Six-defect triangular lattice showing the water filled cooling tubes surrounding each
defect site, and indicating the 3-rod tuning scheme in the upper left defect site.
ACKNOWLEDGMENTS
The authors would like to thank M. Shapiro, C. Chen, R. Temkin, E. Smirnova, S.
Schultz and N. Kroll for useful discussions and comments. This work is supported by
DOE SBIR Grant No. DE-FG03-02ER83400.
REFERENCES
1. M.A. Shapiro, et al., PAC'99, p. 833; M.A. Shapiro, et al., Phys. Rev. Special Topics -Accelerators
and Beams, Vol 4, 042001 (2001).
2. M.A. Shapiro, et al., PAC 2001 p. 930.
3. D.R. Smith, N. Kroll, and S. Schultz, AAC'94, p. 761.; D.R. Smith, L. Derun, D.C. Vier, N. Kroll, S. Schultz,
and H. Wang, AAC'96, p. 518.
4. S. Schultz, D.R. Smith, and N. Kroll, PAC93, p. 2559.
5. E.I. Smirnova, et al., "Simulation of Photonic Band Gaps in Metal Rod Lattices for Microwave Applications, to
be published in Journ. of Applied Physics, vol.91, Feb. 1, 2001; E.I. Smirnova, et al., PAC2001, p. 933.
6. E.I. Smirnova, this publication.
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