-1 mode instability at
SuperKEKB LER
K. Ohmi (KEK)
Accelerator Review Committee of SuperKEKB
Dec. 13-14, 2022
Thanks to H. Fukuma, T. Ishibashi, M. Migliorati, S. Terui, M. Tobiyama, D. Zhou

Beam size blow up in positron ring, LER• LER beam size blow up has been observed since early stage ofcommissioning in 2021.
• A series of measurement has been done in 2021-2022.
• The beam size blowup is single-beam and single-bunch effect.
Independent of Number of bunch=33,66,99,1567.
• It disappear when collimator open (reduced impedance).
• It appears at ny<0.6 and is serious at ny <0.58.
• The idea, in which a localized impedance contributes, was rejected, because of
Feed Back response.

• It seemed to be related to the sideband of x-y coupling (nx - ny - ns =n).
• The idea was rejected, see next.

• -1 (ny-ns) mode signal was seen at the blowup.
• The blow-up was disappeared at BxB feedback OFF.

Tune shiftΔ𝜈𝛽 = 2.00 × 10−19 ෍ 𝛽𝑖 𝐾⊥𝑖 𝐼(𝑚𝐴)𝑖

Collimator set in the beam size
measurement is Dny=0.01-0.013/mA

Tune Survey
No collision
Set kb=36.1x1015 V/Cνy – νx + νsνy – νx + 2νsνy – ν x••••

νy – νx + 2νs?
νy – νx + νs

The stop-band is remarkably spread when the instability occurs.
The vertical emittance is getting smaller for higher vertical tunes.
We can see a small stop-band around 0.595 with the middle bunch current of 0.72 mA/bunch (νx = 0.5310).
It had been difficult to inject in νy = 0.6 or above, and the vertical emittance decreases at the higher vertical tune. 4

Tune Survey
Troubles on beam
injection, irrelevant tobeam instability.

• The structure of the stop-bands in the lower and higher νx is exactly same.
• This probably indicates that the chromatic coupling (νy–νx+2νs=N line) is not related to thisinstability.
5Bunch Oscillation Recorder spectra• Tune 0.524,0.590, 100bunch, I=0.3-1.1mA/b.
• Emittance growth and tune peak ~0.57 appear simultaneously.
ey=30pm

ey=300pm
ey=120pm

FB gain (0.0,2.0)
ey =40pm

BOR and Pilot bunch spectra (0 mode)
Pilot bunch: Tail bunch, is shaked by frequencies
scanned, BxB feedback inactive

0.573 0.586
0.578

Courtesy H. Fukuma

0.579

Gated tune and BOR

ny0 =0.60

Ib (mA)ny

ny• Gated tune of the pilot bunch, tune ny and sideband ny-ns.
• BOR data of whole bunches• Peak seen in BOR is -1 mode ny-ns
Courtesy H. Fukuma

ny0 =0.595

Ib (mA)

Vertical Emittance w/wo D06V1
D06V1 open
close

open

close

side-bandside-band• When we fully opened the aperture of D06V1, the vertical emittance blow-up didn’t occur up to ~1.5 mA/bunch.
➢ (D06V1 aperture) close: ±2.9 mm, open: ±8 mm• The background level derived from the storage beam increased when we opened it. We‘ve used D06V1 as a primary
collimator to cut off the injection backgrounds, but these observations indicate this collimator contribute to suppress thestorage backgrounds too.
9Vertical Emittance w/wo BxB FB (Mar. 1st) 33-bunch operation• We observed the vertical emittance with turning on/off the feedback (FB) with small number of the bunches toavoid multi-bunch instabilities.
• When we turned on the FB, the blow-up occurred around 0.85 mA/bunch.
• When we turned off the bunch-by-bunch FB, the vertical emittance blow-up didn’t occur up to around 1.06
mA/bunch (poor injection rate above than this current).
• After the tuning of the FB to suppress the “-1 mode instability”, the blow-up didn’t occur up to ~1.44 mA/bunch(design bunch current in LER).
10

Vertical Emittance w/wo BxB FB[M. Tobiyama]

betatron frequency synchrotron frequency (like)betatron frequency• Two FB loops have been tuned to suppress fβ line (99.8-41 kHz) with resistive kick, but the FB becomes reactive for the fβ-fs line at
99.8-43 kHz. This enhances the -1 mode instability. The instability has occurred whether we turn on the FB or not (but with different
thresholds).
• M. Tobiyama tuned the phase of one loop by changing the FB filter to suppress the fβ-fs line, and this suppresses the instability with
1dB of the FB gain. However, the FB becomes reactive for betatron frequency line (0 mode).
11

Mechanism of the beam size blowup• Localized impedance at D6V1 is used.
• Beam size increases when -1 mode appears at ~0.9mA.
• The 0 and -1 modes are not coupled at the current. The
threshold of TMCI is ~2mA.

• Studies considering both the impedance and bunch-bybunch feedback are necessary.

Wake force used in simulations (prepared by D. Zhou, T. Ishibashi)• Wake field is calculated by ECHO3D (collimators), GDFidl (cavity and othervacuum components) and analytic (resistive wall).
• Kick Factor byKy, byKqy (V/C)• byKy+byKqy is -3.3x1016 V/C for collimators, and -1.8x1016 V/C for others. 5.1x1016 V/C for total.
• Tune shift is Dny=0.01/mA. The threshold of TMCI is around 2mA (ns=0.023)Δ𝜈𝛽 = 2.00 × 10−19 ෍ 𝛽𝑖 𝐾⊥𝑖 𝐼(𝑚𝐴)𝑖

Simulation for the impedance• I=1mA, Dny=0.01• No emittance increase.

0 mode is seen. Tune shift is 0.01/mA

Mode analysis for simple BxB Feedback• Resistive and reactive component of the feedback• The feedback works only in l=0 mode.

• Considering betatron phase difference between the feedback
𝐷𝑝,𝑦 ≡ 4𝜋𝑑𝑝,𝑦 𝛿𝑙0 𝛿𝑙 0 𝜓(𝐽′ )Δ𝐽
kicker and the wake source′Mode analysis for zero chromaticity• At zero chromaticity, imaginary part of tune apprear blow the
TMCI threshold as is demonstrated by E. Metral.
• The strength seems weak.
GrowthDamp

Mode analysis for xy=1.5• Effects of chromaticity is dominant compare with the resistive feedback.
GrowthDamp
Dn=0.01/(2p)~0.0017• Simulation using the simple FB system did not show the -1 mode
instability (see next page).
• -1 mode instability can not be explained by the simple feedback system.

1 tap resistive feedback• The simple feedback model• Use 2nd feedback loop, FB gain 0.1. No growth at G=0.15.
• Choose f(K2)= f(M1)+0.25-ny(0.585).
• No emittance increase. -1 mode is seen.

Bunch by Bunch feedback system in SuperKEKB-LER• 2 feed back loop working independently, TFBK1-FBMON1
and TFBK2-FBMON2• Max 10 tap
AX
BX
NX
Element Length.67792 19.5518 21.9781 PFZTFBK1 .00000.69845 19.9303 21.9759 FZTFBKP1 .55000.56341 17.6476 21.9913 PFZTFBK2 .00000.58394 17.9631 21.9888 FZTFBKP2 .55000.49527 23.9306 22.0432 PFBMON1 .00000-.49527 23.9306 22.1906 PFBMON2 .00000-.60424 18.2863 22.2474 PFZLFBK1 .00000-.70726 20.0962 22.2588 PFZLFBK2 .00000
1.71319 14.6430 27.4756 PMD06V1 .00000
0.0000 .08000 44.5250
IP.00000

s(m)
AY
BY
NY
#
EX EPX1489.30431 -1.1407 6.31089 22.7721 4006
1489.02931 -1.0404 5.71108 22.7648 4007
1490.83831 -1.7000 10.6686 22.8020 4009
1490.56331 -1.5997 9.76118 22.7977 4010
1499.90944 .84750 19.4097 22.9117 4034
1519.05569 -.84750 19.4097 23.1355 4046
1528.67382 1.50061 8.91787 23.2541 4070
1530.05382 .99740 5.47061 23.2857 4073
1870.26828 -10.133 67.3498 28.8574 4660 .516 -.0728
3016.30649 0.0000 .00100 46.5870 8097

model for SuperKEKB• Betatron phase difference• fy(M1)=22.9117, fy(K1)= 22.7721• Dfy(M1->K1)= 46.4474• fy(M2)=23.1355, fy(K2)= 22.8020• Dfy(M2->K2)= 46.2535

Kicker2

Monitor 1
Monitor 2
Collimator D6V1

Kicker1

FIR digital filter(𝑁𝑡𝑎𝑝 −1)

Δ𝑃𝐾 (𝑛) =෍

𝑐𝑜𝑒𝑓 (𝑘)𝑋(𝑛 − 𝑘)𝑘=0

Example of actual setting of the filter coefficients• Filter coefficients (~Mar. 11, 2022)• coef1={21623,-5530,-11430, 25925,-32767, 31362,-20832, 5288, 12317,25956};• coef2={26781,-26182, 7149, 2479,-22777, 25564,-32767, 19752};• Filter coefficients (~Mar. 12, 2022)• coef1={29144,-32767,-16328, 19950};• coef2={10883,-32767, 28452,-20750,-7342, 21524};

Resistive and reactive components for FIR filter
𝑁𝑡𝑎𝑝 −1

Δ𝑃𝐾 𝑛 =

෍𝑐𝑜𝑒𝑓 𝑘 𝑋 𝑛 − 𝑘 = −2𝑑𝑃 𝑃𝐾 𝑛 − 2𝑑𝑋 𝑋𝐾 (𝑛)𝑘=0• Relation of X(n-k) and XK(n), PK(n) are associated through thebetatron motion with the tune m=2pn.
𝑋 𝑛 − 𝑘 = 𝑅𝑒 (𝑋𝐾 𝑛 + 𝑖𝑃𝐾 (𝑛)) exp(𝑖𝑘𝜇 + 𝑖∆𝜙) = 𝑋𝐾 𝑛 cos(𝑘𝜇 + ∆𝜙) − 𝑃𝐾 (𝑛) sin(𝑘𝜇 + ∆𝜙)• Resistive and reactive components for FIR filter
1
𝑑𝑃 =
2

(𝑁𝑡𝑎𝑝 −1)෍
𝑘=0𝑐𝑜𝑒𝑓 (𝑘) sin(𝑘𝜇 + ∆𝜙)1
𝑑𝑋 = −
2(𝑁𝑡𝑎𝑝 −1)෍𝑐𝑜𝑒𝑓 (𝑘) cos(𝑘𝜇 + ∆𝜙)

𝑘=0• For Ntap=1, Df=p/2 is pure resistive, Df=p is pure reactive.
• In general, resistive and reactive components are mixed.
• For tune scan, m=2p(n0+dn), Df=2p(Df0+dn), where betatron phase ischanged at the section from the monitor to the kicker.

Resistive and reactive components• Filter coefficients (~Mar. 11, 2022)• coef1={21623,-5530,-11430, 25925,-32767, 31362,-20832, 5288, 12317,-25956};• coef2={26781,-26182, 7149, 2479,-22777, 25564,-32767, 19752};
D

U
D

U
Gain is weak at low tune.

• ( Mar.12, 2022)• coefU={29144,-32767,-16328, 19950};• coefD={10883,-32767, 28452,-20750,-7342, 21524};
U

D

U

D

Simulation BxB feedback ON• The simulation considers the betatron phase phases ofmonitors, kickers and D6V1 collimator.

Vertical emittance and FFT y at 1mA/b• First FBcoef (Mar. 11)
UD• GFB=0.05+0.05• GFB=0.02+0.02

Vertical emittance and FFTy at I=0.5mA• First FBcoef (Mar. 11)• GFB=0.05+0.05• No emittance increase at low current

Vertical emittance and FFT of <y>• Second FBcoef (Mar. 12)• I=1mA/b• Emittance increase at low ny is suppressed.

Effects of collimator offset – BxB FB OFF• Emittance increases 30->40pm
at 1mA/b for collimator offset
1mm at D6V1.
• This may explain small
emittance increase for thebunch current.
• Equilibrium orbit distortion asfunction of z. <y>=0.3sy,<yz>=0.1sysz.
• No large emittance growth forthe tune scan, 0.565<ny<0.585.

Effects of collimator offset – BxB FB ON• Filter coefficients (~Mar. 11, 2022)• FB gain(damping rate) 0.05(FB1)+0.05(FB2)• Emittance growth is seen at low ny.
• -1 mode is seen at every tune.

• No remarkable change from the case without collimator offset.

Monitor errors or digitized effects?
• Model with the monitor resolution and/or digitized 0.1-0.2sy.

• No remarkable effect or suppression of instability was seen.
• -1 mode instability occurs at high FB damping rate 0.1 in thesimulations at present.

Kicker noise• Equilibrium power of the BxB feedback P=0.2 W, R=10kW.
• P=V2/R, V~40 V. dpy/p0~10-8. (M. Tobiyama)• <x’>=(ey/by)1/2~1.7x10-6.
• 1%sy noise is introduced in the simulation.

G=0.1

No remarkable change.
Slightly enhanced.
No instability at G~0.06.

Summary• -1 mode instability, which is single beam and single bunch
phenomenon, occurs high bunch current, narrow collimator
aperture (Dny>0.01) and BxB feed back ON.
• The instability can be suppressed by tuning of the feedback.
• The instability strength is changed by condition of BxB feedback
and impedance damage.
• The -1 mode instability can be reproduced by simulation usingtransverse wake and high gain (G~0.1) multi-tap feedback.
• Higher ny>0.58 is preferred for the instability, but injection isworse in the high vertical tune.
• This instability sometimes causes troubles in the physics
operation due to condition of feedback and collimator andcareless tune change.

Thank you for your attention

Summary for feedback• Multitap (1st coef) and high gain feedback system (G=0.1)
cause -1 mode instability.
• Multitap FB with 2nd coef. and 1 tap FB (G=0.1) does notcause -1 mode instability.
• These explain the experimental results.
• The High gain G~0.1 may be controversial.
• Feedback may kick a bunch stronger than dpy=-Gy at a small amp.,y.

• Effect of monitor resolution and kicker error/noise, see later.

• Collision condition becomes worse when V tune go down carelessly.

Vertical Beam size vs ny at by*=1mm (Dec. 2021)
No collision

kb=36.1x1015 V/C
nx =0.529
I=0.10mA/b

nx-ny+2ns=N

Broad peak around thesynchro-beta resonance

I=0.90mA/b

ey~20pm@ny =0.61

Vertical Emittance w/wo FB (Mar. 28, Apr. 5)• In this study, two FB loops were tuned to suppress fβ line, but the number of taps was reduced so thatthey didn’t became reactive on fβ-fs line.
• When we turned off the FB on Mar. 28th, we were able to accumulate up to ~1.45 mA/bunch for 31-bunch.
• When we turned on the FB on Mar. 28th, the threshold of the -1 mode instability was increased to ~1.3
mA/bunch for 31-bunch.
✓ It was ~0.8 mA/bunch on Mar. 1st.

• When we turned off the FB on April. 5th, the threshold was ~1.3 mA/bunch 61-bunch (derived from multibunch instability?).
• When we turned on the FB on April. 5th, the threshold was ~0.95 mA/bunch 61-bunch.
Mar. 1st✓ Two FB loops were tuned to suppress the betatronfrequency and the number of taps was reduced.

✓ In the green dots, one of them was tuned tosuppress a frequency around (betatron –37
synchrotron).

Tune Survey• We scanned the vertical tune again after a tuning of the vertical bunch-by-bunch FB.

• In this survey, two FB loops were tuned to suppress the betatron frequency.
• The number of taps was reduced so that it would not be reactive as much as possible for a frequency
around (betatron – synchrotron).

• The vertical emittance blow-up didn’t occur around 0.9 mA/bunch on Mar. 28th.
• When we compare the lower bunch currents (~0.3 mA/bunch) w/wo the FB on Mar. 28th, itslightly suppresses the vertical emittance in some regions for the vertical tune.
2022-03-28 (31-bunch)2022-03-01 (97-bunch)νy-νx-νs=N

νy-νx-2νs=N
νy-νx-νs=N

νy-νx=Nνy-νx-2νs=N

νy-νx=N38

Vertical
Beam
size
vs
n
at
n
=0.538yx
No collision
kb=36.1x1015 V/C

No 2ny-mns =N

nx-ny+ns=N

I=0.30mA/b
nx-ny+2ns=N=36.1 ey=150pm

V Emittance depends on the
impedance.

kb =28.8 ey=78pm

I=0.90mA/b

Injection difficult
0.7mA/b at ny=0.6

Feedback system for a naive idea
• Betatron oscillation, (X+iP)n=e-inm(X+iP)0
1.
2.
3.
4.

𝑋=

𝑦
𝑃=

𝛽𝑦

𝛼𝑦 𝑦 + 𝛽𝑦 𝑦′

𝛽𝑦Position data of Tap number is measured.
Fourier amplitude and phase at a timing are determined. (X+iP)M=Aexp(-ifM)
Phase at Kicker fK=fM+Df. Betatron coordinate at kicker (X+iP)K=Aexp(-i fK)
Kick the beam proportional to PK(resistive) or XK(reactive), DP=-2aPK-2bXK
𝑁−1

𝐴 exp −𝑖𝜙𝑀

12
= ෍ (𝑋 + 𝑖𝑃)𝑛 exp(−𝑖𝑛𝜇) ≈
𝑁
𝑁𝑡𝑎𝑝
𝑛=0

2
(𝑋 + 𝑖𝑃)𝐾 = 𝐴 exp −𝑖𝜙𝑀 − 𝑖∆𝜙 ≈𝑁𝑡𝑎𝑝

Pure resistive feedback
2
𝑃𝐾 = −
𝑁𝑡𝑎𝑝(𝑁𝑡𝑎𝑝 −1)

∆𝑃 = −2𝑎𝑃𝐾෍
𝑛=0

𝑋𝑛 sin(𝑛𝜇 + ∆𝜙)(𝑁𝑡𝑎𝑝 −1)෍(𝑁𝑡𝑎𝑝 −1)෍𝑋𝑛 exp(−𝑖𝑛𝜇)𝑛=0

𝑋𝑛 exp(−𝑖𝑛𝜇 − 𝑖∆𝜙)𝑛=0

Reactive feedback component
2
𝑋𝐾 =
𝑁𝑡𝑎𝑝

(𝑁𝑡𝑎𝑝 −1)෍

𝑋𝑛 cos(𝑛𝜇 + ∆𝜙)𝑛=0∆𝑃 = −2𝑏𝑋𝐾

[K. Ohmi]

41

[K. Ohmi]

42

[K. Ohmi]

43

FB gain and collimator offset
FB Gain scan for yw=1mm

tune scan for yw=3mm

ny=0.565,
ny=0.565,
FB gain=0.05, 0.05

1 tap resistive feedback• Simplest feedback model• Use 2nd feedback loop, FB gain 0.1. No growth at G=0.15.
• Choose f(K2)= f(M1)+0.25-ny(0.585).
• -1 mode is seen, but no emittance increase.

Activities• 7th meeting (https://kds.kek.jp/event/41962/)• Mode analysis with BxB Feedback, K. Ohmi
• LER single bunch blow-up measurement at Apr. 5., K. Ohmi
• Machine study reports and PyHEADTAIL simulations using new wake, T. Ishibashi• 6th meeting (https://kds.kek.jp/event/41322/)• -1 mode and BxB FB, K. Ohmi
• Machine study report and impedance model updates, T. Ishibashi• 5th meeting (https://kds.kek.jp/event/40778/)• Study of Head-tail instability, K. Ohmi
• Report of machine studies, T. Ishibashi• PyHEADTAIL simulations for a situation of a machine study on Oct. 26th, 2021., T.
Ishibashi• PyHEADTAIL simulations: concentrated or distributed wakefield, M. Migliorati

Members (29):Alexei Blednykh
Demin Zhou
Emanuela Carideo
Frank Zimmermann
Hiroyuki Nakayama
Hitomi Ikeda
Katsunobu Oide
Kazuhito Ohmi
Kazuro Furukawa
Keisuke Yoshihara
Makoto Tobiyama
Mauro Migliorati
Mika Masuzawa
Mikhail Zobov
Andrii Natochii
Nicolas Mounet
Takeshi Nakamura
Rogelio Tomas
Sven Vahsen
Shinji Terui
Tadashi Koseki
Masaru Takao
Takuya Ishibashi
Tom Browder
Tor Raubenheimer
Na Wang
Yong-Chul Chae
Yoshihiro Funakosh
Yusuke Suetsugu
46

Activities• 4th (https://kds.kek.jp/event/40154/)• ECHO3D and its application, I. Zagorodnov• Computation of the impedance of collimators in the LHC, N. Mounet• Impedance Model Updates, T. Ishibashi• Analysis of the Bunch Oscillation Recorder, K. Ohmi
• Convergence studies and wakes for vertical collimators in ECHO3D, T. Ishibashi• Convergence study of PyHEADTAIL, T. Ishibashi• 3rd (https://kds.kek.jp/event/39972/)• Impedance model for SuperKEKB LER, D. Zhou
• Update on machine studies, T. Ishibashi• Beam dynamics simulations with the updated wake, M. Migliorati• Laslett tune shift in SuperKEKB and J-PARC MR, K. Ohmi
• Synchro-beta resonance chromatic coupling and wake force, K. Ohmi
• Convergence study of vertical collimators with GdfidL, T. Ishibashi• 2nd (https://kds.kek.jp/event/39472/)• Machine study items and so on, T. Ishibashi• Impedance calculations of collimators with simple geometries, D. Zhou
• 1st (https://kds.kek.jp/event/39138/)• Welcome, introduction and presentation of the subgroup, M. Migliorati• Introduction of TMCI members, collimators, tune shift and instability measurements, T.
Ishibashi• Impedance and wakefield model, D. Zhou
47
• TMCI and localized impedance, K. Ohmi

Effects of collimator offset• Dipole kick depends on the longitudinal distribution• Turn-by-turn change of y-z distribution for collimator offset
3mm at D6V1 is shown.
• An equilibrium distribution with a banana shape is formed
after several radiation damping time.