FLS 2023

Lucerne, 27th August - 1st September 2023

Beam Dynamics Using Superconducting Passive
Harmonic Cavities with High Current per Bunch
Alexis Gamelin on behalf of SOLEIL II Project Team
1Overview

1. Introduction
1. SOLEIL II project
2. Passive superconducting harmonic cavities
2. Beam dynamics simulations with harmonic cavities considering short range wakes
1. PTBL instability
2. Dipole-Quadrupole instability
3. Coupling between cavity mode and dipole mode
3. Expected performances (emittance & lifetime) in high current per bunch modes

2SOLEIL II TDR lattice
12 m

22.5°7m

2BA type

8.0 m

22.5°

7BA HOA type𝛽x ∼ 1 𝑚𝛽z ∼ 1.5 𝑚

Present lattice at 4 nm.rad [32 dipoles]
46 % of straigth length11.25° 3.8 m

Section replaced
by a 3T superbend

4.2 m

11.25°

Upgrade at 84 pm.rad [126 dipoles]+reverse-bend to lower the emittance
25 % of straight length

3.11 m

4BA HOA type

Major features of the TDR lattice:• Fit “at best” the beamlines positioning• Permanent magnets for dipoles, quadrupoles
and reverse bends• Standard beam pipe inner diameter of 12 mm• 95 % NEG coated ring
• Off axis injection (using MIK)
• Beam lifetime of ~ 3 hrs w/o HC3

Harmonic cavities and timing modes
In (most) 4th generation low emittance storage rings, harmonic cavities (HCs) are critical components needed to reach design
performances.
They are mainly used to lengthen the bunches which provide:• Reduced intra-beam scattering (IBS)• Increased Touschek lifetime
Harmonic cavities used in synchrotron light sources are quasi-exclusively passive, i.e. powered by the beam, so they mostly
have been used in filling modes with high average current.
But it is in the “timing” modes where the current per
bunch is high that both the IBS and Touschek effects are
the strongest.
So, if we want to keep this type of operation modes inultra-low emittance rings, the HCs should provide alarge reduction of the IBS and Touschek effect.

Is it possible to have ultra-low emittancehigh current bunches ?

Current SOLEIL operation modes:
Operationmode

Total current

Current per
bunch

Charge perbunch

Uniform

500 mA

1.2 mA

1.4 nC

Hybrid (3/4)

450 mA

1.44 mA

1.7 nC

8 bunch

100 mA

12.5 mA

14.8 nC

1 bunch

20 mA

20 mA

23.6 nC
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Passive harmonic cavity
For a passive cavity at the 𝑚𝑡ℎ harmonic of the RF frequency 𝑓𝑅𝐹 , the only knob is the cavity tuning angle 𝜓or equivalently the cavity detuning Δf:

𝑓𝑟
𝑚𝑓𝑅𝐹
Δ𝑓tan 𝜓 = 𝑄−
≈ 2𝑄
𝑚𝑓𝑅𝐹
𝑓𝑟
𝑓𝑟

Δ𝑓 = 𝑓𝑟 − 𝑚𝑓𝑅𝐹

Cavity shunt impedance

The voltage in such a passive cavity is given by:(neglecting form factors)

Cavity quality factor

𝑅𝑠 𝑚𝑓𝑅𝐹𝑉2 = 2𝐼0 𝑅𝑠 𝑐𝑜𝑠(𝜓) ≈ 𝐼0sin(𝜓)𝑄0 Δ𝑓Total beam current

To get long bunches, by flattening the total RF voltage, the voltage in the harmonic cavity needs to be able to reach:

𝑉1 sin(𝜙1 )𝑉1𝑉2 = −𝜉≈𝜉𝑚 sin(𝜙2 )𝑚where𝜉=−𝑚𝑉2 sin 𝜙2=𝑉1 sin 𝜙1••= 0 without HC= 1 at flat potential conditions

For a main RF voltage 𝑉1 = 1.8 𝑀𝑉 and a 4th harmonic cavity, to cover a range from 20 mA to 500 mA, it needs:
1.8𝑀𝑉➢ 𝑅𝑠 ≫≈ 11.25 𝑀Ω with a total 𝑅𝑠 /𝑄0 < 100 Ω for stability at high current (PTBL, …).
2∗4∗20 𝑚𝐴➢ Only possible with Super Conducting (SC) systems, typically 𝑄0 ≈ 108 , while Normal Conducting typicallyhave 𝑄0 ≈ 104 .
5RF System
𝑉𝑡𝑜𝑡 (𝑡) = 𝑉1 cos 𝜔𝑅𝐹 𝑡 + 𝜙1 + 𝑉2 cos(𝑚𝜔𝑅𝐹 𝑡 + 𝜙2 )RF system (for this presentation):••

Main NC RF : ESRF - EBS fundamental cavity––––––Rs (per cavity) = 5 𝑀Ω𝑄0 = 35 000
𝑁𝑐𝑎𝑣 = 4
𝛽 = 5 (𝑄𝐿 = 5 833)
𝑉𝑐 = 1.8 MV
𝑓𝑅𝐹 = 352 𝑀𝐻𝑧

Passive SC HC : Super3HC–––––m = 3 or m = 4Rs (per cavity) = 4.5 𝐺Ω
𝑄0 = 108
𝑅/𝑄 = 45 Ω𝑁𝑐𝑎𝑣 = 2Because it is SC, 𝜓 is fixed to 𝜋/2 and 𝜙2 to −𝜋/2, theconditions to lengthen the bunches are quite simple:

cos 𝜙1 ≈𝑈𝑙𝑜𝑠𝑠𝑒𝑉1

𝑉2 ≈ 𝜉

𝑉1 sin 𝜙1
𝑅 𝑚𝑓𝑅𝐹≈ 𝐼0𝑚
𝑄 Δ𝑓

Which gives the (near) flat potential conditions for 𝜉 = 1:

𝑑𝑉𝑡𝑜𝑡
0 = 𝜉𝜔𝑅𝐹 sin 𝜙1
𝑑𝑡

11−𝜉

𝑑 2 𝑉𝑡𝑜𝑡
𝜔𝑅𝐹 𝑈𝑙𝑜𝑠𝑠
0=−𝑑𝑡 2𝑒

For a SC passive HC:• The MC phase 𝜙1 is the same as the one usedwithout HC.
• The second derivative of the RF voltage cannever be cancelled and is fixed by the losses.
• The MC and HC are totally independent systems.
6Overview

1. Introduction
1. SOLEIL II project
2. Passive superconducting harmonic cavities
2. Beam dynamics simulations with harmonic cavities considering short range wakes
1. PTBL instability
2. Dipole-Quadrupole instability
3. Coupling between cavity mode and dipole mode
3. Expected performances (emittance & lifetime) in high current per bunch modes

7

Uniform filling at 500 mA
Here are the stable settings found for a 3rd HC and a 4th HC for the SOLEIL II using multi-bunch tracking taking into account thebeam loading in the main and harmonic cavity (mbtrack2[1]):At high current, 500 mA in uniform filling,the 3rd HC allows to get a bit past the(near) flat potential conditions while the 4th
HC is limited before 𝜉 = 1.

At a given turn during the instability

Double bump

The limitation, in both cases, is the 𝑙 = 1
instability [2] / PTBL[3] but it is happening atdifferent distance from 𝜉 = 1.

𝜎𝜏0 = 9 𝑝𝑠

𝜉=−𝑚𝑉2 sin 𝜙2
𝑉1 sin 𝜙1

is the ratio of the harmonic“force” to the main cavity one

The PTBL threshold increases with thebunch length. It then explains why the 3rdHC threshold is higher than the 4th HC as it
naturally produce longer bunches.

How bunch lengthening from additional impedances impacts
these results ?

[1] Gamelin, A., Foosang, W., & Nagaoka, R. mbtrack2, a Collective Effect Library in Python. IPAC’21[2] Venturini, M. (2018). Passive higher-harmonic rf cavities with general settings and multibunch instabilities in electron storage rings. PRAB, 21(11), 114404.
[3] He, T., Li, W., Bai, Z., & Wang, L. (2022). Periodic transient beam loading effect with passive harmonic cavities in electron storage rings. PRAB, 25(2), 024401.

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Tracking vs single bunch current
Single bunch collective effects:• Considering a preliminary TDR impedance model (RW + geom. impedance, +20 components including NEG, tapers, IDs, …).
• Treated as wake function in the tracking (so no broad band fit).

RMS Bunch length

Energy spread ratio𝜎 ≈ 33 𝑝𝑠 for thesingle bunch mode𝜎 ≈ 27 𝑝𝑠 for 8 bunchmode𝜎 ≈ 14 𝑝𝑠 for the uniform
filling mode

Convergence from 5e5
macro particle in tracking
Microwave Instability(MWI) threshold in
between 2 mA and 3 mA

10

Tracking vs single bunch current
20 mA

20mA
0 mA

0 mA

Go from a sharp peak at zero current to a spread out
bump at higher current (MWI regime) at lower frequency(incoherent tune shift).

The synchrotron mode m=1 is constant because the (negative)
incoherent tune shift is compensated by the dynamic coherent
frequency shift for mode m=1 (Ng book p.68 & p.206)
The synchrotron mode m=2 (slightly shifted downward) is visible
in the MWI regime.
11Uniform filling at 500 mA (with SB collective effects)➢ The turning points corresponds to thethreshold of PTBL instability.
➢ The instability leads to a decrease of
the average harmonic voltage and thus
a decrease of the average 𝜉 for lower
HC detuning.

➢ The added bunch lengthening from theimpedance push the threshold towardhigher values.

1,2 mA/bunch
0 mA/bunch

Harmonic cavity detuning decreases

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8 bunch mode at 100 mA (with SB collective effects)
Mean center of mass

HC onHC off

Mean RMS bunch length

12,5 mA/bunch

HC off
0 mA/bunch

HC on➢ A dipole-quadrupole instability is observed in a given tuning range when the single
bunch collective effects are considered (MWI regime).
➢ This instability is observed at lower 𝜉 (and lower bunch length) for the 3HC.
➢ It seems like a “weak” instability: excited beam but not beam loss

13

8 bunch mode at 100 mA (with SB collective effects)
Dipole (Robinson) mode: starts at 𝑓𝑠0 and then decreases
when getting close to the (near) flat potential conditions.

Dipole-Quadrupole instability

Dipole (synchrotron) mode

3HC only

Quadrupole (Robinson) mode

3HC + impedance modelInteraction between the dipole synchrotron mode excited in the MWI regime and the Robinson
quadrupole mode.
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1 bunch mode at 20 mA (with SB collective effects)➢ The same kind of dipole-quadrupole
oscillations as in 8 bunch mode isobserved when impedance is added.
➢ In addition, fast beam losses
observed (w/ and w/o impedance).

20 mA/bunchare➢ The fast beam loss happens sooner (in 𝜉and in detuning) when the impedance istaken into account.

0 mA/bunch

Harmonic cavity detuning decreases

Center of mass15

Coupling between the cavity mode and dipole (Robinson) mode
The fast beam losses are driven by the mode coupling between the dipole (Robinson) mode originating from𝑓 = 𝑓𝑠0 and the cavity mode from 𝑓 = Δ𝑓:“The cavity mode can be understood as follows. When thebunch passes through the cavity, it excites, in addition to theequilibrium voltage with frequency 𝑓𝑅𝐹 , a transient voltage offrequency 𝑓𝑟 […]. When these two components of thevoltage act back on the bunch, the revolution harmonics ofthe beam are modulated by 𝚫𝐟 = 𝐟𝐑𝐅 − 𝐟𝐫 . The modulated
beam oscillation of frequency 𝒇𝑹𝑭 ± 𝚫𝐟 then feeds back on
the cavity leading to the cavity Robinson mode.”
Towne, N., & Wang, J. M. (1998).
Spectrum of single bunch longitudinal dipole modes. PhysicalReview E, 57(3), 3461.

The dipole (Robinson) and cavity modes are foundnumerically by solving the equation[1,2]:

3HC only (no impedance model)[1] Yamaguchi, T., Sakanaka, S., Yamamoto, N., Naito, D., & Takahashi, T. (2023). Systematic study on thestatic Robinson instability in an electron storage ring. Physical Review Accelerators and Beams, 26(4), 044401.
[2] He, T., Li, W., Bai, Z., & Li, W. (2023). Mode-zero Robinson instability in the presence of passive
superconducting harmonic cavities. Physical Review Accelerators and Beams, 26(6), 064403.

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1 bunch mode at 20 mA (with SB collective effects)
3HC only

3HC + impedance model

4HC only

4HC + impedance model➢ The addition of the impedancemodel strengthens the coupling
between the modes, which lowers
the instability threshold.
➢ Without SB collective effects, the 4th
HC allows to lengthen the bunch all
the way to double bump bunch in astable way.
➢ For the 3rd harmonic, the bunch
lengthening is very rapidly limited by
the merging of these two modes.

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Overview

1. Introduction
1. SOLEIL II project
2. Passive superconducting harmonic cavities
2. Beam dynamics simulations with harmonic cavities considering short range wakes
1. PTBL instability
2. Dipole-Quadrupole instability
3. Coupling between cavity mode and dipole mode
3. Expected performances (emittance & lifetime) in high current per bunch modes

18

Workflow IBS + Touschek + HC + Impedance
tracking(mbtrack2)
Impedance model + HC

Bunch length +energy spread vsbunch current

To compute the lifetime for non-gaussian bunches :𝜌0 is the longitudinal line density corresponding
to the gaussian bunch of 𝜎 = 𝜎𝑧0∫ 𝜌02 𝑧 𝑑𝑧
𝜏 = 𝜏𝑃𝑖𝑤𝑖𝑛𝑠𝑘𝑖 × 2= 𝜏𝑃𝑖𝑤𝑖𝑛𝑠𝑘𝑖 × 𝑅∫ 𝜌 𝑧 𝑑𝑧
Piwinski formula using the correct inputs
but a fixed bunch length 𝜎𝑧0

𝜎𝑧0 = 8.6 ps (𝑉𝑅𝐹 = 1.8 MV)𝜎𝛿0 = 0.09 %𝜖𝑥 = 84 pm.rad𝜖𝑦 = 25 pm.rad
𝐼𝑏𝑢𝑛𝑐ℎ = 1.2 mA

Bunch length +energy spread+ emittances𝜌 is an arbitrary longitudinal line density

Reference value for the uniform mode at 500 mA(for zero current bunch length):
••
•
•
•

IBS computation(analytic)

Touschek
lifetime

Piwinski(pyAT)𝝉 = 3.0 h19

Bunch lengthening w/o HC
Bunch length and Touschek lifetime without harmonic cavity (and IBS neglected):➢ The energy spread increase due to the Micro-Wave Instability (MWI) has a big impact on lifetime(+ 60% at 20 mA compared to 𝜎𝛿 = 𝜎𝛿0 )𝜏 = 𝜏𝑃𝑖𝑤𝑖𝑛𝑠𝑘𝑖 (𝜎𝛿 ) × 𝑅

SOLEIL II MAC#3 - 15-16 May 2023

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IBS + Touschek
For a 4th SC HC (best performances at low current but not at 500 mA):
Operation
mode

Harmonic cavity

Bunch length(RMS)

Current perbunch

Emittance H/V

Lifetime

Energy spread

Uniform

OFF

13 ps

1,2 mA

103/30 pm.rad

5.1 h

1.06E-3

UniformON

46 ps

1,2 mA

91/27 pm.rad

17.0 h

0.97E-3

8 bunches

OFF27 ps

12,5 mA

126/38 pm.rad

1.5 h

1.86E-3

8 bunches

ON77 ps

12,5 mA

112/34 pm.rad

3.6 h

1.21E-3

1 bunchOFF33 ps

20 mA

130/39 pm.rad

1.3 h

2.31E-3

1 bunch

ON

69 ps

20 mA

122/36 pm.rad

2,5 h

1.55E-3

8 bunch
100 mA

1 bunch
20 mA

21

IBS + Touschek
For a 4th SC HC (best performances at low current but not at 500 mA):
Operation
mode

Harmonic cavity

Bunch length(RMS)

Current perbunch

Emittance H/V

Lifetime

Energy spread

Uniform

OFF

13 ps

1,2 mA

103/30 pm.rad

5.1 h

1.06E-3

Uniform

ON46 ps

1,2 mA

91/27 pm.rad

17.0 h

0.97E-3

8 bunches

OFF27 ps

12,5 mA

126/38 pm.rad

1.5 h

1.86E-3

8 bunches

ON77 ps

12,5 mA

112/34 pm.rad

3.6 h

1.21E-3

1 bunchOFF33 ps

20 mA

130/39 pm.rad

1.3 h

2.31E-3

1 bunch

ON

69 ps

20 mA

122/36 pm.rad

2,5 h

1.55E-3➢ HC only provides marginal lifetime gain for high current per bunch modes because:
1. The bunches are already quite long due to the impedance (R = 3.4 at 12.5 mA and R = 4 at 20 mA).
2. HC bunch lengthening also reduces the energy spread and emittance blow-up.
➢ The vertical emittance is the only knob to increase further the lifetime but once past the round beam
condition, the curve is mostly flat (due to the “frozen beam” Touschek effect).
22

Conclusions and perspectives
Lessons learned using HCs in high current per bunch modes➢ Adding an impedance model to the HC tracking can considerably change the results (in particular for the high current perbunch regime).
➢ The PTBL instability threshold is increased for higher harmonic and more generally for longer bunches.
➢ Operation with HCs above the MWI threshold can lead to a dipole-quadrupole instability➢ The coupling between the cavity mode (detuning) and dipole mode can limit the HC performances at low total current.
➢ The effectiveness of the HC bunch lengthening in reducing Touscheck/IBS effects is only the ratio 𝑅𝐻𝐶 /𝑅𝑛𝑜𝐻𝐶

Other leads to achieve high current per bunch:➢ Active NC HCs▪ Should be effective a low current▪ More complex than passive HCs▪ Beam stability is also an important issue▪ Very little experimental experience but developing
fast (ESRF, ALBA/BESSY/DESY, …)➢ Round beams▪ Reduce Touscheck/IBS effects▪ More complex lattice design & operation

After this study, some objectives are now changingfor SOLEIL II project:➢ Only two “major” operational modes:▪ Uniform filling at 500 mA (1,2 mA/bunch)▪ 32 bunches at 200 mA (6,25 mA/bunch)▪ Others (8 bunches and single bunch) kept aslower priority modes➢ Now considering NC passive HC (ESRF-4th HCdesign) as baseline solution:▪ Similar performances at high current▪ Much cheaper than the SC passive option
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Thank you for your attention!

Also … a post-doc positions is open at SOLEIL on beam dynamics:
https://www.synchrotron-soleil.fr/en/job-offers/post-doctoral-position-beam-dynamics

Contact: nagaoka@synchrotron-soleil.fr

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