Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 WEP059

FEASIBILITY STUDY OF AN ACTIVE HARMONIC CAVITY FOR BUNCHLENGTHENING IN AN ELECTRON STORAGE RING
N. Yamamoto∗ , S. Sakanaka, KEK, Tsukuba, Ibaraki, Japan
P. Marchand, A. Gamelin, R. Nagaoka, Synchrotron SOLEIL, Gif-sur-Yvette, France
Abstract
Bunch lengthening from a double radio-frequency (rf)system combining fundamental and harmonic cavities (HCs)is essential to achieve the extremely low emittance which is
required for ring-based fourth generation synchrotron lightsources in low-to-medium energy range. The relatively low
required rf voltage in these machines as compared to third
generation ones, makes the use of normal conducting (NC)cavities more attractive. We report here some results of thefeasibility study of such a double rf system using NC active HCs for the SOLEIL Upgrade case. Particle tracking
simulations have been performed, taking into account the
instabilities related to the fundamental impedance of the cavities, and the results suggest that relevant bunch lengthening
factors can be achieved for all the operating modes, evenat the lowest stored current of 10-20 mA in a single bunch.
The use of NC instead of superconducting (SC) HCs should
result in significant cost savings for comparable performancein terms of bunch lengthening at the same HC frequency.

INTRODUCTION
Double rf systems combining fundamental and passive
HCs have been used for over 20 years in a number of third
generation light sources to lengthen the electron bunches
in order to improve the beam lifetime and stability [1–4].
Such systems are even more relevant in fourth and future
generation synchrotron light rings since bunch lengthening
is essential to preserve the extremely low emittance, especially for the storage rings of low-to-medium energy range.
Besides, the relatively low required rf voltage in the latter
makes the use of NC cavities more attractive.
For this purpose, a powered HC, so-called ”active HC”,
has some advantages as compared to the more commonlyused passive HC:• The external generator of an active HC allows to provide
sufficient voltage to lengthen the bunches even at low
stored current (operation with a single or few bunches)• A proper control of the rf external generator may circumvent unstable beam motions• An adaptive feedforward technique on both main andharmonic cavity generators may mitigate voltage fluctuations due to transient beam loading (TBL) [5].
The first point is important to maintain in the next generation
light sources a wide exploitation of the synchrotron radiation by preserving all the operation modes existing in thethird generation light sources. New concepts of synchrotron
radiation sources that emphasize flexibility are proposed by
KEK [6] and SOLEIL [7].

As for the second and third points, it has been reported
that these phenomena limit the bunch lengthening duringmulti-bunch operation [2–4, 8, 9], and overcoming them is an
important milestone in achieving extremely low emittance.
For active HCs, the TM020 cavity has been proposed because of its intrinsic high Q and low R/Q values [5]. Two different NC-TM020 cavities are being developed at KEK [10]and ESRF [11] with resonant frequencies of 1.50 GHz and1.41 GHz respectively.
In this note, the feasibility study of active HCs, assuming
both the SOLEIL Upgrade v0356 lattice [7] and ESRF-type
NC-TM020 HCs, is numerically carried out. The tracking
code mbtrack [12] is used to evaluate the beam stability andbunch lengthening performance.

ACTIVE HC SYSTEM
The total rf voltage 𝑉 (𝜙) seen by the beam is given by
𝑉 (𝜙) = 𝑉𝑐,1 cos(𝜙 + 𝜙1 ) + 𝑉𝑐,𝑛 cos(𝑛𝜙 + 𝜙𝑛 ),(1)

𝑉 (0) = 𝑈0 /e,(2)where 𝜙 is the phase angle of the beam with respect tothe main rf wave, 𝜙1 and 𝜙𝑛 are the synchronous phaseswith respect to the main and 𝑛th harmonic rf waves; 𝑉𝑐,1and 𝑉𝑐,𝑛 are the main and harmonic rf voltage amplitudes,
respectively; 𝑈0 is the energy loss per turn.
Here, we introduce a variable 𝜉,𝜉=−𝑛𝑉𝑐,𝑛 sin 𝜙𝑛.
𝑉𝑐,1 sin 𝜙1(3)

𝑉 ′ (0) and 𝑉 ″ (0), the first and second derivative of the total
rf voltage, can be expressed as functions of 𝜉 as follows:
𝑉 ′ (0) = −(1 − 𝜉)𝑉𝑐,1 sin 𝜙1 ,sin 𝜙1
𝑉 ″ (0) = −𝑉𝑐,1 (cos 𝜙1 − 𝜉𝑛).
tan 𝜙𝑛(4)(5)

The bunch lengthening is ”optimal” at the so-called flat
potential (FP) condition, when 𝑉 ′ (0) = 𝑉 ″ (0) = 0, whichis met by setting the parameters such that 𝜉 = 1 and 𝜙𝑛 =arctan(𝑛 tan 𝜙1 ). Once 𝑉𝑐,1 , 𝑈0 /e and 𝑛 are fixed, all thesystem parameters are determined from the above equations,(1) to (5), regardless of the stored current or operating mode.
In general, the complex cavity voltage 𝑉𝑐̃ can be considered as the superposition of a generator-induced component
𝑉𝑔̃ and a beam-induced component 𝑉𝑏̃ : 𝑉𝑐̃ = 𝑉𝑔̃ + 𝑉𝑏̃ . Ignoring the bunch form factor (assumed to be equal to 1), theHC beam-induced voltage is given by
𝑉𝑏,𝑛 =∗ naotoy@post.kek.jp- 698 -

2𝐼0 𝑅𝑠,𝑛cos(𝜓𝑛 ) cos(𝜋 + 𝜓𝑛 + 𝑛𝜙),(1 + 𝛽𝑛 )(6)Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 WEP059

where 𝐼0 is the stored current, 𝑅𝑠,𝑛 , 𝛽𝑛 and 𝜓𝑛 are the HC
shunt impedance, coupling factor and detuning angle (-𝜋/2< 𝜓𝑛 < 𝜋/2). Its detuning frequency Δ𝑓 is calculated fromits resonant frequency 𝑓𝑟,𝑛 and loaded-Q 𝑄𝐿,𝑛 :Δ𝑓 = 𝑓𝑟,𝑛 /(2𝑄𝐿,𝑛 ) tan 𝜓𝑛 ,(7)

𝑄𝐿,𝑛 = 𝑄0,𝑛 /(1 + 𝛽𝑛 ).

(8)̃ = 0, the HC voltage is only
In the passive HC case, 𝑉𝑔,𝑛the beam-induced component, and 𝜓𝑛 (or Δ𝑓) becomes the
only operational ”knob” for adjusting the HC voltage andphase. Then the FP condition can be only met for a single
value of the product 𝐼0 .𝑅𝑠,𝑛 and the power lost by the beamin the HC must be fully supplied by the MC generator.
In the active HC case, since the HC generator amplitude
and phase are also used as operational ”knobs”, the FP condition can be fulfilled regardless of 𝐼0 or 𝑅𝑠,𝑛 as far as instabilities do not occur.
In both cases, active or passive, approaching the FP condition means that the longitudinal restoring force is reduced,
and it is necessary to be careful about the occurrence ofinstabilities, especially at high beam current.

INVESTIGATION SETUP
The investigation has been performed for the SOLEIL
Upgrade, whose parameters are listed in Table 1. A combination of four EU-type 351.6 MHz main cavities (MCs) [13,14] and one to three ESRF-type 2-cell HCs [11] at the harmonic number 𝑛 = 4, is considered. The cavity parameters
are listed in Table 2. It is assumed that both MCs and HCs
are ”HOM free”. For the SOLEIL Upgrade, not only 500 mA
in multi-bunch but also 20 mA in a single bunch and 100 mA
in 8 bunches are requested. The double rf system should betherefore compatible with all these operation modes.
As a result of the double rf potential analysis, based on
Ref. [15] with the parameters listed in Table 1, the FP condition is met when 𝜙1 =1.296, 𝑉𝑐,4 =434 kV and 𝜙4 =-1.641.
In this case, the bunch length and lengthening factor (BLF)
are expected to be 40 ps and 4.7, respectively.
Using a single ESRF-type HC with a voltage of 434 kV
corresponding to the FP condition leads to a power dissipation 𝑃𝑐 of about 40 kW, which is close to the limit of what
Table 1: SOLEIL Upgrade Parameters (v0356)
Parameter

Unit

Value

Energy, 𝐸0
RF frequency
Energy loss per turn (no ID), 𝑈0
Main RF voltage, 𝑉𝑐,1
Energy spread
Momentum compaction factor, 𝛼Longitudinal damping time, 𝜏𝑒Synchrotron frequency w/o. HC
Natural rms bunch length

GeV
MHz
keV
MV

2.75
351.6
458
1.80
8.9×10−4
1.1×10−4
12.2
1.78
8.5

ms
kHz
ps

Table 2: Cavity Parameters of the Double Rf System
Parameter
Harmonic number, 𝑛Shunt Impedance, 𝑅𝑠 = 𝑉𝑐2 /2𝑃𝑐
Unloaded-Q, 𝑄0
Cavity coupling coefficient, 𝛽Cavity numberMC

HC (2 cell)

1
5.0 MΩ
35,0005.0
4

4
2.4 MΩ
27,0001.0
1∼3

could be handled [16]. That would be relaxed in using a second HC, however in return it will double the total R/Q hence
increasing the parasitic effects of the TBL and limiting thebunch lengthening [5, 17]. Therefore we have investigatedand compared the following three cases: one HC (1-HC),
two HCs (2-HC) and three HCs (3-HC) but one of them
detuned in the opposite direction for bunch shortening (BS)instead of bunch lengthening (BL).
The mbtrack simulations are performed in one-dimension(longitudinal) and both MC and HC impedance’s are considered as instability sources. A broad band impedance of
0.069 Ω/GHz/m (pure copper vacuum pipe) is assumed andtaken into account with the calculation technique describedin the Ref [18]. In the tracking simulation, the cavity voltage
and phase vary as the bunch distribution changes until thesteady-state condition is reached. Therefore, it is difficult toanticipate the setting that will lead to the target voltage andphase.
For the active HC case, the parameter set is determinedin advance to provide the specified cavity voltage and phase
under the predicted bunch distribution (a Gaussian distribution with the rms bunch length of 30 ps), which minimize
the required generator power for both MC and HC, and they
are treated as constant during each tracking simulation.
In the mean time, for both cases, the setting of the MC isslowly changed to achieve the specified voltage and phasewith the accuracy of 1×10−3 .
The tracking results for the various parameter sets weresystematically compared after several series of simulations.
Typically, 0.5 to 1 million turns were required to converge
towards a steady solution. The deviations from the target
values were about 4% in amplitude, 0.1% in phase and thestatistical accuracy of the calculated bunch length was estimated to be 0.1 ps.

HIGH CURRENT MULTI BUNCH MODE
In the bunch lengthening operation of a double rf system,
the resonant frequency of the HC is higher than the 𝑛th
harmonic of the rf frequency, that is positive detuning. In
this case, one needs to be careful as regards to the occurrenceof coupled-bunch instabilities and in particular the +1 mode,
when the detuning is large. It has been well studied for thecase of a single rf system. In the Ref. [19], the analyticalequation of the growth rate GRl for each mode number 𝑙
is derived with the assumptions of a point charge, equally
spaced 𝑀 bunches and a coherent frequency close to the- 699 -

Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 WEP059

3 SOLEIL_v0356, fs=1.78kHz, I0=500mA, τe=12.2ms

0.38

50

mode = 0

2

-1
1

10

mode = +1
Simulation points

010

FP detune(144 kHz)1.406GHz-1

10-200 -100

0

Rs of one HC(2.4 MΩ)

100

200

300

400

500

600

Figure 1: HC 𝑅𝑠 threshold for coupled-bunch instabilities (1HC case), taking into account the MC 𝑅𝑠 and the parametersin Table. 1. The threshold plots for coupled-bunch modes :-1 (red), 0 (green) and 1 (red). Unstable beam motions are
expected in the grey-colored area.
incoherent one (synchrotron frequency), as follows:
GRl =

𝛼𝐼0
2𝜈𝑠 𝐸0∞⎧⎫{∞ ++ )] − ∑ 𝑓 − Re [𝑍(𝑓 − )]} (9)∑ 𝑓𝑙,𝜇 Re [𝑍(𝑓𝑙,𝜇
𝑙,𝜇
𝑙,𝜇 ⎬
⎨
{
}
𝜇=1
⎩𝜇=0
⎭
±
𝑓𝑙,𝜇
= {𝜇𝑀 ± (𝑙 + 𝜈𝑠 )} 𝑓0

𝑍(𝑓 ) =

40

Bunch profile
Passive(β=1)(𝑅/𝑄) 𝑄𝐿
1 + 𝑖𝑄𝐿 (𝑓𝑟 /𝑓 − 𝑓 /𝑓𝑟 )where 𝜈𝑠 and 𝑓0 are the synchrotron tune and revolution
frequency.
By extending the Eq. (9) to the case of multiple cavities,
the thresholds of the HC shunt impedance 𝑅𝑠 are calculated
for the coupled-bunch modes +1, 0 and -1, and plotted inFig. 1 as a function of the HC detuning frequency. In thecalculation, the synchrotron frequency without HC is used
and the contributions from the MC and the longitudinalradiation damping are taken into account. Although, theseassumptions become no longer valid when 𝜉 approaches
unity, due to the reduction and spread of the synchrotron
frequency, we can use this type of plot as a guideline for thefollowing complicated tracking simulations.

1-HC case
Considering the 1-HC case with a shunt impedance of
2.4 MΩ, the detuning frequency that satisfies the FP condition is 144 kHz, indicated as ”FP detune” in Fig. 1. So,
the stable domain is expected within the detuning frequency
range from 350 kHz to the vicinity of ”FP detune.”
The mbtrack simulations have been carried out along theparameter set shown as ”Simulation points” in Fig. 1 andthe obtained rms bunch lengths are plotted versus 𝜉 in Fig. 2
for three 1-HC cases : active (in red), passive with 𝛽4 =1(generator ”OFF”, in blue) and passive with 𝛽4 =0 (in green).

0.42

0.43

Passive(β=0)
Passive(β=1)
ActiveActivePassive

30

(β=0)

20
0.85

Detuning Frequency [kHz]HC voltage [MV]
0.39
0.4
0.41

SOLEIL−U v0356
Vc:1.8MV, 2-cell HC x 1

Unstable10

Rms bunch length [ps]Total Rs,CBI threshold [M Ω]

10

0.9

ξ

0.95

1

Figure 2: Rms bunch length as a function of 𝜉, obtained
from the mbtrack simulations, for the active and passive (𝛽4= 0 and 1) 1-HC cases. In the inset, the bunch profiles at
maximum bunch length are compared for the three cases.
𝜉 is calculated from Eq. (3) using the steady values of eachsimulation result.
In the active case with the generator ”ON”, the maximum
bunch length of 33.6 ps (BLF of 4) is obtained for 𝜉=0.97 anda detuning of 142 kHz. With the generator ”OFF”, although
it is reduced, one can still achieve a bunch length up to
29.9 ps (BLF of 3.4) for 𝜉=0.98 and a detuning of 132 kHz.
It is confirmed that the longer bunches are achieved with the
HC generator ”ON” despite its slightly smaller 𝜉. Moreover,comparing the bunch profiles in the inset of Fig. 2, one cansee that the active case has a more symmetric shape. That is
an advantage of the active HC, which allows to cancel not
only 𝑉 ′ but also 𝑉 ″ .
The bunch lengthening and profile for the more usual
passive case with a coupling 𝛽4 =0 are also shown in Fig.2.
The achieved performance is rather similar to that of theactive case : maximum bunch length of 34.2 ps (BLF of
4.0) for 𝜉=0.98 with a detuning of 138 kHz and an almost
symmetric profile. However, it should be mentioned that thepassive 1-HC case is not compatible with bunch lengthening
in the operation modes at lower current.
In all the above studied cases, unstable beam motion withbunch length fluctuations along the bunch train, prevents
further bunch lengthening at the vicinity of ”FP detune.”
The analysis of the bunch mass center motion suggests theoccurrence of mode +1 oscillations, however it is foundthat the growth rate is much larger than that given by theextension of the Eq. (9). This instability is considered to beof the same nature as the one reported in Ref. [8] and [9]under the name, “coupled bunch mode 𝑙 = 1 instability” and“periodic transient beam loading instability”, respectively.

2-HC case
The instability thresholds for the 2-HC case are plotted asdashed lines in Fig. 3. Considering the detuning frequencyof 287 kHz that satisfies the FP condition, there is no stable
operating point to perform bunch lengthening for this case.

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Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 WEP059

3 SOLEIL_v0356, fs=1.78kHz, I0=500mA, τe=12.2ms

3 SOLEIL_v0356, fs=1.78kHz, I0=500mA, τe=12.2ms

mode = 02

Unstable

10

mode = +1-1

w. +1 mode damper

1

10

10

Total Rs ,CBI threshold [M Ω]

Total Rs ,CBI threshold [M Ω]

10

Rs of two HC(4.8 MΩ)

Simulation points

010

1.406GHzFP detune(287 kHz)w/o. damper-110-200 -100

0

100

200

300

400

500

mode = 02mode = +1

1

10

Rs of two BL HC(4.8 MΩ)

Simulation points
010

FP detune(200 kHz)1.406GHz-1

10-200 -100

600

Unstable

10

Detuning Frequency [kHz]

0

100

200

mode = -1
300

400

500

600

Detuning Frequency [kHz]Figure 3: HC 𝑅𝑠 thresholds for coupled-bunch instability(2-HCs case). The dashed and solid lines are obtained withand without +1 mode damper, respectively.

Figure 4: HC 𝑅𝑠 thresholds for coupled-bunch instability(2BL & 1BS HCs case).
Table 3: Summary of Bunch Lengthening Performance

Then the thresholds were re-calculated after including a +1
mode damper [20] with a center frequency of 352.4 MHz
and a bandwidth of 881 Hz. The result is shown as solid lines
in Fig. 3. A maximum damping rate of 666 s−1 is expected
for the coherent frequency at 1 kHz, where the feeding rf
amplitude of 240 V is technically feasible.
The results of the mbtrack simulations show that the maximum bunch length is limited at 19.6 ps (BLF of 2.3) for𝜉 = 0.81 and a detuning of 350 kHz. Beyond this 𝜉 value,
one observes an unstable beam motion whose property issimilar to that of 1-HC case. The unstable motion cannot
be suppressed by increasing the damper voltage within apractical range. The 𝜉 limit is lower than that of 1-HC case,which is likely due to the 2-fold increase of the HC 𝑅/𝑄.

3-HC (2BL & 1BS) caseAccording to the previous investigation, it is found that theunstable beam motion having the couple-bunch mode +1 andlarge growth rate reduces the bunch lengthening performancein the 2-HC case. Then we have introduced a third HC withnegative detuning as a counter force to the instability, aconfiguration that we called 2BL & 1BS.
In this case, the detuning of the BS HC is set to -400 kHz
leading to a beam induced voltage around 150 kV. Therefore,
in order to meet the FP condition, the 2BL HCs must provide
about 580 kV, which corresponds to a detuning of 200 kHz.
Although the presence of the BS HC tends to decrease
the damping from the 2BL HCs on the coupled-bunch mode
-1, the global contribution remains a damping effect andtherefore the stability of mode -1 should be insured from390 kHz to the FP tune, as shown in Fig. 4.
The results of mbtrack simulations show that the maximum bunch length of 28.4 ps (BLF of 3.3) is obtained at
𝜉=0.95 and a detuning of 219 kHz. The cavity voltage forthe 2BL HCs is 558 kV; the dissipation is then 16.2 kW foreach BL HC and 4.7 kW for the BS HC, which is quite relaxed as compared to the 1-HC case. The bunch lengthening
performance is in between the 1-HC and 2-HC cases and

HC setup
Multi bunch 500 mA
one HCone Passive HC (𝛽 = 0)
one Passive HC (𝛽 = 1)
two HCs
2BL & 1BS HCs
Single bunch 20 mA
one HC
8-bunch 100 mA
one HC

Bunch length

BLF

33.6 ps34.2 ps
29.9 ps
19.6 ps
28.4 ps

4.0
4.0
3.4
2.3
3.3

38.8 ps

4.6

30.0 ps

3.5

it is confirmed that the introduction of the BS cavity has asignificant effect on suppressing the mode +1 instability that
occurs near the FP.

DISCUSSION
From the comparison of the active HC results for the
SOLEIL Upgrade multi-bunch 500 mA mode, one finds thatthe best performance is obtained with a double rf systemthat consists of four EU-type 352MHz-MCs and one active
ESRF-type 1.41GHz-HC. The obtained bunch length of
33.6 ps (BLF of 4) is comparable to that of the SC passive
HC case for the same HC frequency [21].
Although it is not discussed in this paper, the bunch
lengths in single-bunch and 8-bunch modes have been also
evaluated for the 1-HC active configuration. The obtained
BLF is 4.6 in single bunch-20 mA and 3.5 in 8-bunch100 mA. In single bunch mode, it is considered that the
FP condition is met without the occurrence of the mode +1
instability, as 𝜉 values higher than unity can be reached (double hump bunch shape). On the other hand, the bunch lengthin 8-bunch 100 mA mode is limited owing to the mode +1
instability at 𝜉=0.94.
From these results, summarised in Table 3, it is numerically confirmed that BLFs > 3.5 can be achieved for all re-- 701 -

Proceedings of the 19th Annual Meeting of Particle Accelerator Society of Japan
October 18 - 21, 2022, Online (Kyushu University)

PASJ2022 WEP059

quested operation modes, using one active HC with a power
dissipation of about 40 kW, which is rather close to the estimated hardware limit. It is worthwhile mentioning that alarger number of passive HCs would be needed to achievebunch lengthening in the lower current modes : three (1BS & 2 BL) at 100 mA in 8 bunches and more than five at
20 mA in a single bunch.
Furthermore, the impact of the TBL should be investigated
if empty buckets are introduced in the bunch train to avoid
ion trapping.
Finally, the authors continue to investigate the impact of
rf techniques such as mode damper, direct rf feedback andfeedforward TBL compensation [5, 22]. As pointed outin ”INTRODUCTION”, the possibility of applying thesetechniques is another advantage of the active HC option as
they may allow for further enhancement of the BLF.

CONCLUSION
The feasibility of an active HC system for bunch lengthening is numerically investigated for the SOLEIL Upgrade
storage ring. The threshold of the coupled-bunch instabilities
driven by the cavity fundamental impedance’s is analyticallycalculated for each possible parameter set. Then the maximum stable bunch length is evaluated by simulations using
the particle tracking code mbtrack.
The maximum stable bunch lengths are obtained by using
a single 2-cell 1.41GHz-HC having a shunt impedance of
2.4 MΩ, combined with four EU-type 352MHz-MCs. Relevant BLFs are anticipated for all the requested operating
modes: 4.0 at 500 mA in 416 bunches, 3.5 at 100 mA in 8
bunches and 4.6 at 20 mA in a single bunch. The HC power
dissipation will be around 40 kW, which is close to the estimated hardware limit. R&D’s are ongoing at the ESRF to
demonstrate its feasibility in terms of power dissipation andrequired HOM damping for the SOLEIL Upgrade case [16].
In the 3-HC (2BL & 1BS) case, the power dissipation is relaxed however the maximum BLF is smaller (3.3 at 500 mA),
which is nevertheless better than the factor of 2.3 in the 2-HC
case, since the third HC with opposite detuning can reduce
the effective shunt impedance that causes instability andlimits the bunch lengthening.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant Number JP20H04459. It has been accomplished as part of a
RF collaboration between different labs (KEK, PSI, ESRF,
BESSY and SOLEIL). We thank our partners for the usefulexchanges.

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