Proceedings of the 16th Annual Meeting of Particle Accelerator Society of Japan
July 31 - August 3, 2019, Kyoto, Japan

PASJ2019 FRPI007

WAKEFIELD DAMPING IN THE DIELECTRIC ASSIST ACCELERATINGSTRUCTURE
Shingo Mori∗ , Mitsuhiro Yoshida†, KEK, Tsukuba, Japan
Daisuke Satoh‡, National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

Abstract
The dielectric-assist accelerating (DAA) structure is adielectric-inserted normal-conducting cavity, which provides high Q value at room temperature. This cavity has thepotential to apply for a long-pulse and high-duty accelerator
which can provide high average current. In the high current
accelerator, the wakeﬁeld from the transverse beam oﬀset
causes the emittance growth. We consider several options todump the wakeﬁeld by modifying a radius of the beam holeand introducing a microwave absorber in the choke structure.

INTRODUCTION
A dielectric-assist accelerating (DAA) structure [1, 2] ispromising in power eﬃciency and can be a new candidateof accelerating cavity for high duty operation because it provides high eﬃciency Q ∼ O(105 ) at room temperature andhigher Q value at low temperature. Due to the structuresof the dielectrics and the use of the low-loss dielectric material [2], the ﬁve cell DAA cavity provides the Q-value of
1.2 × 105 at room temperature.
In our previous work, we introduced the choke structure
to the DAA cavity [3]. The motivation of this structure wasto obtain better thermal contact between the copper structure
and the dielectrics without brazing them inside the cavity.
Due to the high Q value of the DAA cavity, this cavity
is expected to enable us the longer-pulse operation at roomtemperature and provide us the larger average current. However, the high-Q nature of the DAA structure would cause
severe long-range wakeﬁeld. The multi-bunch instability
caused by the higher-order mode or the lower-order mode
would limit the maximum current from the requirement of
the beam breakup, the emittance growth, and the energy
spread.
Since the eﬀect of the wakeﬁeld of the normal DAA structure has not been considered, we consider the long-range
wake function by combining the analytical calculation andthe numerical eigenmode simulation. Then we consider introducing the wakeﬁeld damping structure to the new design
proposed in our recent work [3] and compare the long-range
wake function in the normal and extended DAA structure
and the C-band choke-mode cavity [4].
This paper is organized as follows. In Sec. 2, we will
introduce the formulae of the longitudinal and transverse
component of the long-range wake fuction and explain theeigenmode analysis we use. In Sec. 3, we will show the resultof the eigenmode analysis and the obtained wake function
in both normal DAA cavity and the new design. In Sec. 4,
we will summarize our results.

WAKEFUCTION CALCULATION
THROUGH THE EIGENMODE ANALYSIS
We consider the equivalent circuit model of the RF cavity, where the circuit resistance corresponds to the shunt
impedance of the cavity Rz (x, y) in the axial direction withoﬀset x in the x-direction and y in the y-direction. Thecircuit impedance for a driving frequency ω is written asZz (x, y, ω) =∞Rn (x, y),ωnω
n=0 1 + iQ n ω − ωn(1)where n is the label of all modes in the cavity without beamhole. Rn (x, y) is the shunt impedance of the n-th mode. Q n
is the Q value of the n-th mode. ωn is the eigenfrequency ofthe n-th mode. Since the Panofsky-Wenzel theorem relates
the momentum change in the transverse direction with thetransverse derivative of that in the axial direction, we canobtainωZ⊥ (x, y, ω) = ∇⊥ Zz (x, y, ω).
c(2)Then the wake function in the axial direction and transverse
direction can be obtained from the Eq. (1) as follows:∗ smori@post.kek.jp† mitsuhiro.yoshida@kek.jp‡ dai-satou@aist.go.jp- 1034 -wz (x, y, s)∫ ∞ωs1dωZz (x, y, ω)ei c ,=2π −∞
∞

ωn Rn
ωn s
ωn s 2=cos 1 − 1/4Q n
exp −Qn
2Q n ccn=0ωn s
1, (3)
sin 1 − 1/4Q2n−2c
2Q n 1 − 1/4Q n

Proceedings of the 16th Annual Meeting of Particle Accelerator Society of Japan
July 31 - August 3, 2019, Kyoto, JapanPASJ2019 FRPI007

 ⊥ (x, y, s)w∫ ∞ωsi=dω Z⊥ (x, y, ω)ei c ,2π −∞∫ ∞ic
i ωs=dω ∇⊥ Zz (x, y, ω)e c ,2π −∞ω∫ ∞∞ ⊥ Rn (x, y)
ωsi
∇c 
 ei c ,=dωωn2π −∞
ω n=0 1 + iQωn ωn − ω∞ ⊥ Rn (x, y)c∇ωn s
2= −sin 1 − 1/4Q n2cn=0 Q n 1 − 1/4Q nωn sexp −.
(4)
2Q n cDue to the dielectric structure in the DAA cavity, theeigenmodes inside the DAA structure is not a pure TM orTE modes. It is convenient to distinguish each eigenmode byusing the waveform, e.g. number of nodes in the radial direction or number of the poles in the azimuthal direction in thecross-section of the transverse plane between neighboring
dielectric disks.
In order to calculate the eigenmode of the DAA cavity, wecalculate the unit cell of the DAA structure with one layer ofthe dielectric disk in the center of the unit cell and periodic
boundary condition in both ends. Considering the 1 [m]accelerating cavity of the DAA structure with 37 unit cell,each eigenmode has a family of the 38 normal modes withphase advance nπ/37 (n = 0, · · · , 37). The normal modesof n = 0 are called as zero mode and those of n = 37 arecalled as π mode. The DAA structure is designed so that
accelerating TM02-like mode becomes π mode.
Due to the beam-cavity interaction, the beam bunch losses
energy inside the cavity and excites the eigenmodes which
can decelerate the bunch. When beam has a ﬁnite oﬀset,
the deﬂecting modes also acquire energy from the bunch.
In the family of the normal modes, the transit-time eﬀectsuppresses the non-synchronizing normal modes. Then wewill consider the synchronizing normal modes for TM0m,TM1m, TE1m (m = 1, 2).

RESULT
Normal DAA cavity
Fig. 1 shows the schematic ﬁgure of the DAA test cavity. [2]Figure 1: The normal DAA cavity of the test cavity with 6cell.

Fig. 2 shows the dispersion curve of the normal DAA
structure. Here we omit the TE0m-like modes.

Figure 2: The dispersion curves in the normal DAA cavity.
The orange curves denotes the TM1m-like mode. The blue
curves denotes the TE1m-like modes. The red curve denotes
the TM0m-like modes. The black line denotes the phase of
the modes synchronizing with the relativistic electrons.

New design with the choke structure and the microwave absorverIn our previous work [3] we proposed to apply choke
structure to the DAA structure in order to easily remove heatfrom the dielectric disks. Here we introduce the microwave
absorber around the low-loss dielectric disks as shown inFig. 3. The SiC ring around the dielectric disk can damp theFigure 3: The new design of the DAA cavity. The white component shows the low-loss ceramics and the black component
around the dielectric disk is the SiC ring as a microwave
absorber.
higher- or lower-order modes in the cavity propagating inside
the transmission line. The decay length of the microwave
can be estimated as
d=c
ω

2.
tan δ(ω)(5)

Fig. 4 shows the frequency dependence of the decay lengthof the microwave, where we use the loss tangent tan δ(ω)
of the SiC measured by Y. Takeuchi et al. [5]. This curve
tells us the radial thickness of SiC of 10 [mm] is enough toreduce the modes with frequency over 3 [GHz] transmittingin the radial transmission line.
Fig. 5 shows the result of the longitudinal wake-function
for both normal and the new DAA cavities. Since the longitudinal wake-function is dominated by the accelerating- 1035 -Proceedings of the 16th Annual Meeting of Particle Accelerator Society of Japan
July 31 - August 3, 2019, Kyoto, Japan

PASJ2019 FRPI007

Figure 4: The frequency dependence of the decay length ofthe microwave in side the SiC with parameter in Ref. [5]mode, the result does not change in both cavity. Fig. 6 shows
the magnitude of the longitudinal wake-function at s = 0 foreach eigenmodes in both cavity.

Figure 5: The longitudinal wake-function per unit length
obtained from the eigenmode analysis. The blue curve denotes the result of the normal DAA cavity. The orange curve
denotes the result of the new cavity.

Figure 7: The transverse wake function per unit length andper unit oﬀset calculated from the eigenmode analysis. The
blue curve denotes the result of the normal DAA cavity. Theorange curve denotes the result of the new DAA cavity. The
red curve denotes the wake function of the normal conducting C-band structure. [4]Figure 8: The contribution from each eigenmode to thetransverse wakefunction at the origin s = 0.01 [m].

DISCUSSION AND CONCLUSION

Figure 6: The contribution from each eigenmode to thelongitudinal wakefunction at the origin s = 0.

We investigate the long-range wakeﬁeld in the DAA structure using the eigenmode analysis of both the higher ordermodes and the lower order modes. Since the DAA structure
has about one order of magnitude higher Q value comparedto the normal conducting cavity, the wakeﬁeld excited inside the DAA structure stands for a longer time than thenormal conducting one. We ﬁnd the wake function in thenormal DAA cavity has the decay constant of 4 [μs], while
the C-band choke mode cavity has the decay constant of
0.5 [ns] [4]. The new design with the choke structure andTable 1: The Result of the Eigenmode Analysis

Fig. 7 shows the result of the transverse wake function forboth normal and the new DAA cavities. Here we also plot
the transverse wake function obtained in the literature [4].
Fig. 8 shows the magnitude of the transverse wake-function
at s = 0.01 [m] for each eigenmodes in both cavity.
Table 1 shows the frequency, the phase advance of v p = cand Q value of each eigenmodes. The frequency and thephase advance is the result of the normal DAA cavity. Notethat the result of Q value does not contain the eﬀect of theendplates, because of the models used in the eigenmode
calculation have the periodic boundary.

modef [GHz]37φπTM11(+)
TM11(−)
TM12
TE11
TE12TM01
TM02Acc.
TM022.931
4.093
7.077
5.4266.622
5.265
5.719
6.558

1927−2835353434-32- 1036 -QNormal
3.6 × 104
1.2 × 105
8.6 × 104
4.9 × 104
9.5 × 104
6.0 × 104
1.7 × 105
6.2 × 104

Choke
303
1.0 × 105
89
95
218
98
1.7 × 105105

Proceedings of the 16th Annual Meeting of Particle Accelerator Society of Japan
July 31 - August 3, 2019, Kyoto, Japan

PASJ2019 FRPI007

the SiC ring shown in Fig. 3 reduces the transverse wakeﬁeld
well and we obtain the decay constant of 4 [ns]. Then thetransverse wake eﬀect in the new structure is improved by athousand times better than the normal DAA structure.
This means that in the linac application for FEL the same
current can be achieved by the 8 times longer ﬂat-top length.
This is promising since the high Q nature of the DAA structure can omit the pulse compressor and would provide us thelonger pulse length compared with the conventional normal
conducting cavity.

ACKNOWLEDGEMENTS
This work is supported by JSPS KAKENHI Grant
No. 16H02134. The authors would like to thank N. Shigeoka from the Mitsubishi Heavy Industries Mechatronics
Systems Ltd. for their continued support of the developmentof DAA structures. We thank the helpful discussion withH. Ego in KEK.

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[2] D Satoh, M Yoshida, and N Hayashizaki. Fabricationand cold test of dielectric assist accelerating structure.
Physical Review Accelerators and Beams, 20(9):091302,
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[3] Shingo Mori, Daisuke Sato, and Mitsuhiro Yoshida. The
design optimization of the dielectric assist acceleratingstructure for better heat and gas transfer. In 10th Int. Partile Accelerator Conf.(IPAC’19), Melbourne, Australia,
19-24 May 2019, pages 1179–1181. JACOW Publishing,Geneva, Switzerland, 2019.
[4] N Akasaka, T Shintake, and H Matsumoto. Optimization on wakeﬁeld damping in c-band accelerating structure. Technical report, SCAN-9904050, 1998.
[5] Yasunao Takeuchi, Tetsuo Abe, Tatsuya Kageyama, Hiroshi Sakai, Kazuo Yoshino, and Masahiro Ando. Control of rf dielectric properties of sic ceramics. 2010.

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