1. STABILISING INTENSE BEAMS
BY LINEAR COUPLING
Elias METRAL
Introduction, observations and motivation
Theory
Experiments
Conclusion
Elias Metral, CERN-PS seminar, 12/04/2000

2. Single-particle trajectory
INTRODUCTION
Single-particle trajectory
One particle
Circular design orbit
Low intensity  Single-particle phenomena
High intensity  Collective effects
2 stabilising mechanisms against transverse coherent instabilities :
Landau damping by non-linearities (space-charge and octupoles)
Non-linearities  Perturbations of the single-particle motion (resonances) !
Feedback systems
Elias Metral, CERN-PS seminar, 12/04/2000

3. THE IDEA (from R. CAPPI and D. MOHL) WAS TO :
OBSERVATIONS
In 1989, a coherent instability of the quadrupolar mode type driven by ions from the residual gas has been observed by D. Mohl et al. in the CERN-AA and successfully cured by adjusting both tunes close to 2.25
In 1993, a single-bunch instability of the dipolar mode type driven by the resistive wall impedance has been observed by R. Cappi in the CERN-PS and “sometimes cured” by adjusting both tunes close to 6.24
THE IDEA (from R. CAPPI and D. MOHL) WAS TO :
USE LINEAR COUPLING TO “TRANSFER DAMPING” FROM THE STABLE TO THE UNSTABLE PLANE, IN ORDER TO REDUCE THE EXTERNAL NON-LINEARITIES
Elias Metral, CERN-PS seminar, 12/04/2000

4. A general formula for the transverse coherent instabilities with
THEORY (1/16)
A general formula for the transverse coherent instabilities with
Frequency spreads (due to octupoles)
Linear coupling (due to skew quadrupoles)
Mode
coupling term
Linear
coupling term
x-dispersion
integral
x-Sacherer’s
formula
Elias Metral, CERN-PS seminar, 12/04/2000

5. Uncorrelated distribution functions (Averaging method)
THEORY (2/16)
Coherent frequency
to be determined
Uncorrelated distribution
functions (Averaging method)
Near the coupling resonance
is the lth Fourier coefficient of the normalized skew gradient
Elias Metral, CERN-PS seminar, 12/04/2000

6. Sacherer’s formula (single- and coupled-bunch instabilities)
THEORY (3/16)
Sacherer’s formula (single-
and coupled-bunch instabilities)
=> “low intensity” case
Head-tail modes
Coupled-bunch modes
Power spectrum
Pick-up (Beam Position Monitor) signal
-signal
-signal
Time
Time
One particular turn
Elias Metral, CERN-PS seminar, 12/04/2000

7. Let’s recover the 1D results
THEORY (4/16)
Let’s recover the 1D results
In the absence of - Linear coupling
- Mode coupling
In the absence of frequency spreads
Real coherent betatron frequency shift
Instability growth rate
=> Sacherer’s formula is recovered
These are the Laslett, Neil
and Sessler (LNS) coefficients
for coasting beams
Motions
=>
Instability
Elias Metral, CERN-PS seminar, 12/04/2000

8. THEORY (5/16) 1D criterion 1D criterion
In the presence of frequency spreads
(1) Lorentzian distribution
(2) Elliptical distribution
Overestimates Landau damping (infinite tails)
Underestimates Landau damping (sharp edges)
1D criterion
1D criterion
Keil-Zotter’s
stability criterion
Elias Metral, CERN-PS seminar, 12/04/2000

9. In the absence of frequency spreads
THEORY (6/16)
In the absence of linear coupling but in the presence of mode coupling => “high intensity” case
In the absence of frequency spreads
=> Kohaupt’s stability criterion against Transverse Mode Coupling Instability (TMCI) is recovered
In the presence of frequency spreads
=> A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping
Elias Metral, CERN-PS seminar, 12/04/2000

10. In the absence of mode coupling only for coupled-bunch modes
THEORY (7/16)
New 2D results
Stability criterion
(for each mode m)
In the absence of mode coupling only
for coupled-bunch modes
(and coasting
beams)
In the absence of frequency spreads
Necessary condition for stability
Transfer of
growth rates
Stability criteria :
Stable region
Full coupling?
No coupling
Full coupling
Elias Metral, CERN-PS seminar, 12/04/2000

11. => Normalised coupling (or sharing) function
THEORY (8/16)
for full coupling
=> Normalised coupling (or sharing) function
Elias Metral, CERN-PS seminar, 12/04/2000

12. In the presence of frequency spreads
THEORY (9/16)
In the presence of frequency spreads
(1) Lorentzian distribution
=> Same results with replaced by
Stability criteria :
No coupling
Full coupling
Transfer of both instability growth rates and frequency
spreads (Landau damping)
Elias Metral, CERN-PS seminar, 12/04/2000

13. (2) Elliptical distribution
THEORY (10/16)
(2) Elliptical distribution
A particular case : No horizontal tune spread and no vertical wake field
Stable region
Elias Metral, CERN-PS seminar, 12/04/2000

14. Approximate general stability criterion
THEORY (11/16)
Approximate general stability criterion
1) “far from”
2) “near”
=> Transfer of growth rates only
Necessary condition
THE TUNE SEPARATION SHOULD BE SMALLER THAN
THE ORDER OF MAGNITUDE OF IN ORDER
TO HAVE THE TRANSFER OF LANDAU DAMPING
Elias Metral, CERN-PS seminar, 12/04/2000

15. On the coupling resonance THEORY (12/16)
H-plane
Transfer of frequency spread (to Landau damp )
V-plane
“One plane is stabilised by Landau damping and the other
one is stabilised by coupling” 
Same result obtained considering both non-linear space-charge forces and octupoles for coasting beams => D. Mohl and H. Schonauer’s 1D stability criterion (gain of factor ~2)
Elias Metral, CERN-PS seminar, 12/04/2000

16. => Necessary condition for stability
THEORY (13/16)
In the presence of both mode coupling and linear coupling, neglecting frequency spreads
=> Necessary condition for stability
Elias Metral, CERN-PS seminar, 12/04/2000

17. THEORY (14/16)
Example :
=> Computed gain in intensity of about 50% for the classical ratio of factor 2 between the transverse sizes of the vacuum chamber
Elias Metral, CERN-PS seminar, 12/04/2000

18. SHARING OF DAMPING BY FEEDBACKS
THEORY (15/16)
SHARING OF DAMPING BY FEEDBACKS
An electronic feedback system can be used to damp transverse coherent instabilities. Its action on the beam can be described in terms of an impedance, which depends on the distance between pick-up and kicker, and the electronic gain and time delays
Electronics
Pick-up
Kicker
Beam
The stabilising effect of feedbacks can be introduced in the coefficient 
Its damping effect in one plane, can also be transferred to the other plane using coupling
Elias Metral, CERN-PS seminar, 12/04/2000

19. Frequency spreads (due to octupoles)
THEORY (16/16)
SUMMARY OF THEORY
1 general formula for transverse coherent instabilities in the presence of
Frequency spreads (due to octupoles)
Linear coupling (due to skew quadrupoles)
In the absence of coupling the well-known 1D results are recovered as expected
Effects of linear coupling (skew quadrupoles and/or tune distance from coupling resonances) :
Transfer of growth rates for “any” coupling
Transfer of Landau damping for “optimum” coupling
“Chromaticity sharing” (for Sacherer’s formula)
=>
Linear coupling is an additional (3rd) method that can be used to damp transverse coherent instabilities
Elias Metral, CERN-PS seminar, 12/04/2000

20. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9)
Experimental conditions
High intensity bunched proton beam
1.2 s long flat bottom at injection kinetic energy
Sacherer’s formula
=> coupled-bunch instabilities
Coupled-bunch modes
Most critical head-tail mode
number
for the horizontal plane
121 s-1
- 40 s-1
 Landau damping is needed
Elias Metral, CERN-PS seminar, 12/04/2000

21. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9)
See next slides
Observations
1D case 
Spectrum Analyzer
(zero frequency span)
Beam-Position Monitor
(20 revolutions superimposed)
10 dB/div
R signal
Center 360 kHz
Time
Time
(20 ns/div)
RES BW 10 kHz
VBW 3 kHz
SWP 1.2 s
One particular turn
Elias Metral, CERN-PS seminar, 12/04/2000

22. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9)
MEASUREMENT OF THE CERN-PS LINEAR COUPLING
Coupling resonance
No solenoid
In the PS
In the presence of linear coupling between the transverse planes, the difference from the tunes of the 2 normal modes is given by
Guignard’s coupling coefficient
It is obtained from the general formula (in the smooth
approximation used to study instabilities)
Measurement method : For different skew quadrupole currents, we increase and decrease in the vicinity of the coupling resonance and we measure the 2 normal mode frequencies using a vertical kicker, a vertical pick-up and a FFT analyzer
Elias Metral, CERN-PS seminar, 12/04/2000

23. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9)
Coupling measurements from mode frequencies by FFT analysis
Low intensity bunched proton beam
1.2 s long flat bottom at injection kinetic energy
“Mountain range” display for the “natural” coupling
FFT Analyzer
Time
Frequency
Elias Metral, CERN-PS seminar, 12/04/2000

24. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9)
=> Modulus of the normalised skew gradient vs. skew quadrupole current
Elias Metral, CERN-PS seminar, 12/04/2000

25. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9)
Stabilisation by Landau damping (1D case)
Simplified (elliptical) stability criterion : Keil-Zotter’s criterion
Theoretical frequency spread required
Experimental frequency spread required 
This is less than required by the theory by a factor 3 (without taking into account space-charge non-linearities...)
Elias Metral, CERN-PS seminar, 12/04/2000

26. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9)
Stabilisation by coupled Landau damping (2D)
Constant tune separation
Measurement
Theory
(Lorentzian vertical distribution)
Elias Metral, CERN-PS seminar, 12/04/2000

27. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9)
Constant octupole strength
Measurement
Theory
(Lorentzian vertical distribution)
Elias Metral, CERN-PS seminar, 12/04/2000

28. EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9)
CONCLUSIONS OF EXPERIMENT-1
The experimental results confirm the predicted beneficial effect of coupling on Landau damping
Using coupling, a factor 7 has been gained in the octupole current (for this particular case) => Less non-linearities
Difference between theoretical predictions and experiments  Space-charge non-linearities, impedance and tune spread models…
Further theoretical work => More precise treatment of the non-linearities in the normal modes
Elias Metral, CERN-PS seminar, 12/04/2000

29. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6)
Single bunch of protons with nominal intensity
1.2 s long flat bottom at injection kinetic energy
Bunch length
Transverse tunes
Transverse chromaticities
Growth rates [s-1]
Sacherer’s formula =>
Head-tail mode number m
Elias Metral, CERN-PS seminar, 12/04/2000

30. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6)
Observations
1D case 
Spectrum Analyzer
(zero frequency span)
Beam-Position Monitor
(20 revolutions superimposed)
10 dB/div
R signal
Center 355 kHz
Time
Time
(20 ns/div)
RES BW 10 kHz
VBW 3 kHz
SWP 1.2 s
Elias Metral, CERN-PS seminar, 12/04/2000

31. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6)
Stabilisation by linear coupling only
EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6)
since
 ~ no emittance blow-up
(limit)
The ~ same results are obtained for the ultimate beam
 ~ no emittance blow-up
but ~ no blow-up in the PS
Elias Metral, CERN-PS seminar, 12/04/2000

32. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6)
Voir le file presentation 1
Elias Metral, CERN-PS seminar, 12/04/2000

33. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6)
8 bunches of protons with nominal intensity
Theoretical stabilising
skew gradient
 coupled-bunch instabilities
Growth rates [s-1]  or
Head-tail mode number m
The ~ same results are obtained for the ultimate beam
Elias Metral, CERN-PS seminar, 12/04/2000

34. EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6)
CONCLUSIONS OF EXPERIMENT-2
The stability criterion for the damping of transverse head-tail instabilities in the presence of linear coupling only has been verified experimentally and compared to theory, leading to a good agreement (to within a factor smaller than 2)
The CERN-PS beam for LHC (nominal or ultimate intensity) CAN BE STABILISED using linear coupling only* (skew quadrupoles and/or tune separation). Furthermore, this result should be valid for “any” intensity (as concerns pure head-tail instabilities)...
* i.e. with neither octupoles
nor feedbacks
Elias Metral, CERN-PS seminar, 12/04/2000

35. LANL-PSR (from B. Macek)
OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING IN OTHER MACHINES
LANL-PSR (from B. Macek)
“Operating at or near the coupling resonance with a skew quad is one of the most effective means to damp our 'e-p' instability”
BNL-AGS (from T. Roser)
“The injection setup at AGS is a tradeoff between a 'highly coupled' situation, associated with slow loss, and a 'lightly coupled' situation where the beam is unstable (coupled-bunch instability)”
CERN-SPS (from G. Arduini)
“A TMCI in the vertical plane with lepton beams at 16 GeV is observed. Using skew quads ('just turning the knobs'), gains in intensity of about 20-30%, and a more stable beam, have been obtained”
=> MDs are foreseen to examine these preliminary results in detail
CERN-LEP (from A. Verdier)
“The TMCI in the vertical plane at 20 GeV sets the limit to the intensity per bunch. The operation people said that it's better to accumulate with tunes close to each other”
Elias Metral, CERN-PS seminar, 12/04/2000

36. The CERN-PS beam for LHC can be stabilised by linear coupling only
CONCLUSION
These results explain why many high intensity accelerators and colliders work best close to a coupling resonance blablablabla and/or using skew quadrupoles. They can be used to find optimum values for the transverse tunes, the skew quadrupole and octupole currents, and the chromaticities (=> sextupoles)
The CERN-PS beam for LHC can be stabilised by linear coupling only
Linear coupling is also used at BNL and LANL, and seems to be helpful in SPS and LEP => See future MDs
Using this “simple” formalism, the following results are also obtained:
Coherent beam-beam modes => Decoupling the 2 beams by making the tune difference much larger than the beam-beam parameter (A. Hofmann)
2-stream instabilities => Same stability criterion with negative coupling (Laslett, Mohl and Sessler)
THEIR IDEA !
ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF
Elias Metral, CERN-PS seminar, 12/04/2000