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67th ICFA Adv. Beam Dyn. Workshop Future Light Sources
ISBN: 978-3-95450-224-0
ISSN: 2673-7035

FLS2023, Luzern, Switzerland

JACoW Publishingdoi:10.18429/JACoW-FLS2023-MO3B2

BEAM DYNAMICS USING SUPERCONDUCTING PASSIVE HARMONICCAVITIES WITH HIGH CURRENT PER BUNCH
A. Gamelin∗ , V. Gubaidulin, A. Loulergue, P. Marchand, L. S. Nadolski, R. Nagaoka
Synchrotron SOLEIL, Gif-sur-Yvette, France
N. Yamamoto, KEK, Ibaraki, Japan
Abstract
In 4th generation synchrotron light sources, harmonic cav-

ities (HCs) are critical components needed to achieve therequired performance. They provide longer bunches, whichhelps to reduce statistical effects (intra-beam scattering andTouschek effect). In "timing" modes, where the bunch spacing is larger than in conventional modes and the numberof particles per bunch is higher, this need is even greater.
In this article, we present the beam dynamics in the highcurrent per bunch regime and how it interacts with the single
bunch collective effects. In particular, a dipole-quadrupole
instability is observed above the microwave threshold and acoupling between the dipole and cavity modes is shown tolimit bunch lengthening at low current. The effective gainfrom the use of HCs in terms of lifetime, emittance, andenergy spread is also discussed.

resonance, one can set a lower limit on the needed shunt
impedance:
𝑉1
𝑅𝑠 ≫
.
(4)
2𝑘 𝐼0
For 𝑉1 = 1.8 MV, a 4th HC and the lowest beam current of
20 mA, the total shunt impedance of the HCs should be muchlarger than 11.25 MW. As the system should also be stable athigher currents, the total 𝑅𝑠 /𝑄 should not exceed 100 W toavoid instabilities [5, 6]. Both conditions are only possible
with superconducting (SC) cavities, which typically have 𝑄 0around 108 , while normal conducting (NC) cavities typicallyhave 𝑄 0 of a few 104 . For these reasons, we consider in thisarticle a double RF system using SC HCs as detailed in Table
1 and for such a SC passive system there is 𝜓 ≈ 𝜋/2 andthus sin 𝜓 ≈ 1 in Eq. (3).
Table 1: RF Cavity Parameters

INTRODUCTION
In this paper, we explore the beam dynamics of a double
RF system with harmonic cavities (HCs) in the context ofhigh currents per bunch. The parameters of the SOLEIL II
project [1–3], which aims to replace the existing SOLEIL
storage ring by a 4th generation synchrotron light source areused throughout the article.

Double RF System
For a passive unloaded cavity at the kth harmonic of the
RF frequency 𝑓 𝑅𝐹 , the only knob is the cavity tuning angle
𝜓 or equivalently the cavity detuning Δ 𝑓 :Δ 𝑓 = 𝑓𝑟 − 𝑘 𝑓 𝑅𝐹 ,𝑓𝑟
𝑘 𝑓 𝑅𝐹Δ𝑓tan 𝜓 = 𝑄−
≈ 2𝑄,
𝑘 𝑓 𝑅𝐹
𝑓𝑟
𝑓𝑟

(1)
(2)

where 𝑓𝑟 is the cavity resonance frequency and 𝑄 is thecavity quality factor (equal to the unloaded quality factor 𝑄 0
in that case). The beam-induced voltage 𝑉2 in the harmonic
cavity (neglecting form factors) is given by [4]:𝑉2 = 2𝐼0 𝑅𝑠 cos 𝜓 ≈ 𝐼0

𝑅𝑠 𝑘 𝑓 𝑅𝐹
sin 𝜓 ,𝑄 Δ𝑓(3)

where 𝐼0 is the total current, 𝑅𝑠 is the cavity shunt impedance(using the circuit ohm definition). For a main cavity voltage 𝑉1 , an approximate voltage of 𝑉2 ≈ 𝑉1 /𝑘 is needed
to get close to the near flat potential (NFP) conditions [5]and longer bunches. Using the beam voltage induced at∗ alexis.gamelin@synchrotron-soleil.fr

Parameter

NC MC

SC HC

Harmonic number 𝑘
Shunt impedance 𝑅𝑠 (per cavity)
Unloaded quality factor 𝑄
𝑅𝑠 /𝑄 (per cavity)
Loaded quality factor
Cavity number

1
5 MW
35 700
140 W
6 000
4

3 or 4
4.5 GW
108
45 W
108
2

BEAM DYNAMICS
In this part, the beam dynamics in 8-bunch mode at
100 mA and in single bunch mode at 20 mA is studied bytracking using mbtrack2 [7]. Results for SOLEIL II main operation mode, uniform filling at 500 mA, using the same RF
system but without short-range wake, were already discussed
in Ref. [5].

Short Range Wake
To be able to estimate accurately the beam dynamics at ahigh charge per bunch, a preliminary impedance model including the most important features of SOLEIL II (resistive
wall with NEG coating, tapers, BPMs, ...) is included in thesimulations [8]. Figure 1 shows the relative variation of thebunch length and energy spread when the single bunch current is varied from 0 mA to 20 mA. It can be observed thatthe micro-wave instability (MWI) threshold, characterized
by an energy spread increase, is reached at 3 mA.
Even in uniform filling mode, where the current per bunchis limited to 1.2 mA, the short-range wake effect is important

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JACoW Publishingdoi:10.18429/JACoW-FLS2023-MO3B2

model is included. In addition, very fast beam losses areobserved, even without short-range wake, which are driven
by the mode coupling between the dipole Robinson modeoriginating from 𝑓 = 𝑓𝑠0 and the cavity mode from 𝑓 = Δ 𝑓 .
The cavity (Robinson) mode has been introduced inRef. [11], and recently studied in Refs. [12, 13], and is theconsequence of the beam modulation at 𝑓 𝑅𝐹 ± Δ 𝑓 feedingback into the main cavity. Here, this effect only originatesfrom the HC, i.e. not from the double RF system, and thus
can be described by the point bunch Robinson theory applied
to the HC [9]:Figure 1: Relative variation of the bunch length 𝜎𝑧 , energy
spread 𝜎𝛿 and R factor (Eq. (6)) versus single bunch current.
Zero current RMS values: 𝜎𝑧0 = 8.6 ps, 𝜎𝛿0 = 0.09 %.
and allows to increase the maximum bunch length by 30 %compared to what was shown in Ref. [5].

“Weak” Dipole-quadrupole Instability
In the 8-bunch mode at 100 mA, a “weak” dipolequadrupole instability is observed in a given detuning range
only when the short-range wake is included in the simulation.
The instability is characterized by an in-phase oscillation of
the beam center of mass and bunch length which does notlead to beam loss. The typical amplitude of the oscillations
is about 20 ps for both the center of mass and the bunch
length.
The coherent bunch spectrum, defined as the absolutevalue of the Fourier transform of the mean position of thebunch, is shown in Fig. 2 with and without the short range
wake while the 3rd HC detuning is varied. In the left plot,
one can see the dipole Robinson mode frequency decreasingfrom the unperturbed synchrotron frequency 𝑓 = 𝑓𝑠0 when
the HC detuning decreases and the system is getting closer
to the NFP conditions. In the right plot, in addition, thereis the dipole synchrotron mode, from the inclusion of theshort range wake, which is constant because of the (negative) incoherent tune shift compensated by the dynamic
coherent frequency shift for mode 𝑚 = 1 [9]. When the
dipole synchrotron mode is reached by what we believe tobe the quadrupole Robinson mode around Δ 𝑓 ≈ 20 kHz, this“weak” dipole-quadrupole instability appears.
This instability seems remarkably similar to the fast mode
coupling Robinson instability described in Ref. [10] but isdriven by the dipole synchrotron mode excited by the MWI
instead of the dipole Robinson mode. The instability issimilarly observed for the 4th HC but closer to the NFP andin both cases it is possible to go beyond the detuning at which
the modes couple and get back a stable beam.

Mode Coupling Between The Cavity and Dipole
Robinson Mode
The same “weak” dipole-quadrupole instability is observed in single bunch mode at 20 mA when the impedance∞∑︁
𝑒𝐼0 𝛼𝑐𝑝𝜔 𝑅𝐹 𝑍 ( 𝑝𝜔 𝑅𝐹 )
×
𝐸 0𝑇0
𝑝=−∞

− ( 𝑝𝜔 𝑅𝐹 + Ω)𝑍 ( 𝑝𝜔 𝑅𝐹 + Ω) ,

Ω2 = 𝜔2𝑠0 + 𝑗(5)where Ω is the coherent mode complex frequency, 𝜔 𝑠0 =2𝜋 𝑓𝑠0 , 𝜔 𝑅𝐹 = 2𝜋 𝑓 𝑅𝐹 , 𝛼𝑐 is the momentum compaction
factor, 𝐸 0 is the reference energy, 𝑇0 is the revolution period,and 𝑍 is the HC impedance.
Figure 3 shows the coherent bunch spectrum superimposed with the dipole Robinson mode and cavity modecomputed numerically from Eq. (5). When the short-range
wake is not included, there is a perfect agreement between
the tracking and the point bunch theory. The addition of
the impedance model strengthens the coupling between themodes, which lowers the instability threshold. In our case,the mode coupling is fully independent of the synchrotron
damping time, differently from what is stated in Ref. [13].
This instability is a strong limitation for HC operation at
low total current, the threshold is lower for lower harmonic
HCs as the detuning to get the required voltage to reach NFP
is lower. It explains why the 3rd HC can provide much lessbunch lengthening than the 4th HC at 20 mA as shown in
Table 2.

IBS AND TOUSCHEKThe reference Touschek lifetime for the SOLEIL II TDR
lattice [2] is 3.0 h without errors, calculated for 1.8 MV,
1.2 mA and the zero current RMS values: bunch length 𝜎𝑧0 =8.6 ps, energy spread 𝜎𝛿0 = 0.09 %, horizontal emittance𝜖 𝑥 = 84 pm rad and vertical emittance 𝜖 𝑦 = 25 pm rad exitedby white noise.
To compute the Touschek lifetime 𝜏 for non-Gaussianbunches including the IBS, we use the following method.
Firstly, the equilibrium bunch length and energy spread are
extracted from mbtrack2 tracking results including the HC
and the impedance model. They are used to compute analytically, using the ibsEmittance command from elegant [14],new values for the equilibrium energy spread and emittances.
Then the Touschek lifetime 𝜏Piwinski is computed using the
Piwinski formula [15] with the IBS equilibrium values andan arbitrary bunch length 𝜎𝑧0 . Finally the bunch profile 𝜌from tracking is used to compute the Touschek lifetime 𝜏

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Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

67th ICFA Adv. Beam Dyn. Workshop Future Light Sources
ISBN: 978-3-95450-224-0
ISSN: 2673-7035

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

67th ICFA Adv. Beam Dyn. Workshop Future Light Sources
ISBN: 978-3-95450-224-0
ISSN: 2673-7035

FLS2023, Luzern, Switzerland

JACoW Publishingdoi:10.18429/JACoW-FLS2023-MO3B2

Figure 2: Coherent bunch spectrum from tracking data while the 3rd HC detuning is varied for the 8-bunch mode at 100 mA
without (left) and with (right) short-range wake.

Figure 3: Coherent bunch spectrum from tracking data while the 3rd HC detuning is varied for the single bunch mode at
20 mA without (left) and with (right) short-range wake. The dipole and cavity modes shown in the dashed lines are thenumerical solutions of Eq. (5).
using a scaling law [16]:∫𝜏 = 𝜏Piwinski × ∫𝜌02 (𝑧)𝑑𝑧
𝜌 2 (𝑧)𝑑𝑧= 𝜏Piwinski × 𝑅 ,(6)

where 𝜌0 is the longitudinal line density corresponding to a
Gaussian bunch of 𝜎 = 𝜎𝑧0 and 𝜌 is an arbitrary longitudinalline density.
This approach is not fully self-consistent, as the elegant
IBS calculation assumes Gaussian bunches, and any further
bunch lengthening from the IBS is neglected in the 𝑅 factor.
The IBS would need to be included in the tracking simulation
without assuming a Gaussian shape for a fully self-consistentcalculation, but the result should give a good approximationof the combined effects.
The energy spread increase from the MWI plays a significant role for the IBS and Touschek at high bunch charge.
Without HC, the Touschek lifetime is quasi-constant in therange from 10 mA to 20 mA as the current per bunch increase is compensated by the energy spread blow-up (responsible for a 60 % increase at 20 mA), the emittance increase
from IBS and the bunch lengthening.
The performances, including Touschek and IBS effects,
resulting from the maximum bunch lengthening found by
tracking are shown in Table 2. In uniform mode at 500 mA,
the 3rd HC is more effective in lengthening the bunches than
the 4th HC. In 8-bunch mode at 100 mA, both harmonics
give similar results while for single bunch at 20 mA the 4th
HC provides better performances than the 3rd one.
Overall, the HC provides a lower lifetime gain for the highcurrent per bunch modes (factor 2) compared to what is
obtained for the uniform mode (factor 4). A first reason isthat the bunches are already quite long due to the impedance
without HC, and the effective lifetime gain is only the ratio
𝑅HC /𝑅noHC . The HC bunch lengthening also reduces theenergy spread, if above the MWI threshold, and the emit-

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67th ICFA Adv. Beam Dyn. Workshop Future Light Sources
ISBN: 978-3-95450-224-0
ISSN: 2673-7035

FLS2023, Luzern, Switzerland

JACoW Publishingdoi:10.18429/JACoW-FLS2023-MO3B2

Operation
Mode

HC

Bunch
LengthCurrent
per Bunch

Emittance
H/V

R

Lifetime

EnergySpreadUniform
Uniform
Uniform
8 bunches
8 bunches
1 bunch1 bunch1 bunch

Off
3HC
4HC
Off
3/4HC
Off
3HC
4HC

13 ps
64 ps
46 ps
27 ps
77 ps
33 ps
41 ps
69 ps

1.2 mA
1.2 mA
1.2 mA
12.5 mA
12.5 mA
20 mA
20 mA
20 mA

103/30 pm.rad
90/27 pm.rad
91/27 pm.rad
126/38 pm.rad
112/34 pm.rad
130/39 pm.rad
132/40 pm.rad
122/36 pm.rad

1.6
7
5
3.4
8.7
4
5
8.3

5.1 h
22.4 h
17.0 h
1.5 h
3.6 h
1.3 h
1.7 h
2.5 h

1.06 × 10−30.95 × 10−3
0.97 × 10−3
1.86 × 10−3
1.21 × 10−3
2.31 × 10−3
1.88 × 10−3
1.55 × 10−3tance blow-up from IBS which tends to reduce the lifetime
improvement.

PERSPECTIVESAs shown in this paper, high current per bunch modes inultra-low emittance rings are challenging for a double RFsystem, and the resulting Touschek lifetime can still be quite
low despite large bunch lengthening factors. In addition,
beamlines may require a limit on the maximum bunch length
to maintain good resolution for “timing” mode experiments.
A possible way to improve further the Touschek lifetime
would be to go to a round beam, i.e. 𝜖 𝑥 = 𝜖 𝑦 , but it brings
additional difficulties for lattice design and operation.
For the SOLEIL II project, it was decided to focus on
the uniform mode at 500 mA and on a new 32-bunch mode,the other modes being kept but with lower priorities. As aconsequence, we are now investigating an NC passive HC
solution based on ESRF 4th HC design [17].

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Table 2: Performance in different operation modes with and without SC HC (RMS values for 𝜎𝑧 and 𝜎𝛿 ).