OBSERVATION OF COHERENT SYNCHRO-BETATRON
BEAM-BEAM MODES AT VEPP-2M
I.N. Nesterenko, E.A. Perevedentsev and A.A. Valishev
Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia

An experimental evidence is presented of the coherent
synchro-betatron beam-beam modes in the spectrum of coherent dipole oscillations observed at the VEPP-2M collider. The measured current-dependent behavior of themode spectrum is in good agreement with predictions ofour theoretical models, taking full account of the finite
length of colliding bunches. Potential effect of the synchrobetatron modes on the beam-beam blowup is discussed.

1INTRODUCTION

Among the beam-beam phenomena there is a large groupof effects concerning the coherent motion of colliding
bunches [1]-[4]. It has been recently proposed that a finite length of colliding bunches can lead to the coupling ofsynchro-betatron modes in the beam-beam system (Fig. 1)
and affect the beam stability [5]. However, there was noexperimental evidence of existence of such modes at thepresent colliders. This paper gives an outline of the theory
and presents the experimental investigation of the synchrobetatron beam-beam modes at the VEPP-2M collider, comparing the measured data with theoretical predictions.
IPIP

IP0,σ−1,σ−2,σ0,π−1,π−2,πFigure 1: Naming convention for the synchro-betatron
beam-beam modes.

2

THEORY: MODE SPECTRUM

The dipole motion of bunches, coupled via the beambeam force can be expressed in terms of coherent synchrobetatron modes. Usually the betatron coupling between thetransverse degrees of freedom is small, and therefore a separate treatment of horizontal and vertical synchro-betatronoscillations is a good approximation. Since the rise time ofthe head-tail instability is much shorter than the radiation
damping time in e + e machines, we can drop the radiative
effects in what follows.
For the case of small betatron oscillation amplitudes it
is often allowable to linearize the transverse force exerted
by the beams on each other. This linearized beam-beaminteraction is considered here.

We study the dipole moments as functions of the longitudinal position in the bunch. Discrete approximation of
the appropriate eigen-functions allows to reduce the task tolinear algebra.
We use the so-called “hollow beam” model. It assumes
that all particles of the bunch have equal synchrotron amplitudes and are evenly spread over the synchrotron phase,
forming a circle in the synchrotron phase space. This circle
is divided into N mesh elements, each characterized by its
transverse dipole moment (2 variables) and its number corresponding to its synchrotron phase. In the arcs synchrobetatron oscillations of the elements forming a bunch are
C B , whererepresented by the N  N matrix M
denotes the outer product, B is the betatron oscillation
matrix of dimension  , C is the N  N circulant matrix [6], which transports the dipole moment around the circle formed by the mesh elements with fixed synchrotron
phases and thus performs a synchro-betatron mapping of
N variables for each of the colliding bunches.
The linearized beam-beam interaction is described by aN  N matrix Mbb consisting of consecutive short kicks
and drifts between interactions of macroparticles sitting ineach mesh, and assumed to be transversely-rigid Gaussian
disks [2].
The complete one-turn matrix is the product of the arc
matrix M and the beam-beam matrix M bb . Its  -dependent
eigenvalues and eigenvectors completely characterize thesynchro-betatron modes of the beam-beam system and canbe obtained numerically using a computer algebra system[7], see an example in Fig. 2.

222 2=24

40.30

synchro-betatron mode tunes

Abstract

matrix model
tracking+1,π0.25+1,σ0,π0.200,σ−1,π0.15−1,σ−2,π−2,σ0.10νβ +νsνβνβ −νs0.05
00.04

0.080.12
ξ

0.160.20

0.24

Figure 2: Synchro-betatron mode tunes vs. the beam-beam
parameter  . Comparison of the circulant matrix model andtracking. Equal bunch intensities, : , s
: ,and the bunch length is :  .

07= 0 11= 0 03

3OBSERVATION TECHNIQUES

Vertical coherent oscillations of the bunches were observed
using the beam synchrotron radiation from the dipoles. The
optical image of the beam was focused into the movablescreen plane (Fig. 3). The screen was cutting off a portion of the light in the beam image plane. For a fixed edge
position, a displacement of the beam centroid resulted inmodulation of the light flux.
ySReedgeen

4 EXPERIMENTAL RESULTS

Figure 3: Edge detector scheme.

The light which passed through the optical system thenfell on the PMT (Fig. 4). The PMT signal, with modulation proportional to the beam displacement, was fed tothe fast ADC input. In our system we used the CAMACstandard 8-bit ADC with the 8k read buffer and minimumtransform time of 10 ns. The PMT bandwidth was adjusted
to observe separate turns of the bunch in the storage ring.
The ADC clock rate was exactly equal to the beam revolution frequency and the phase was locked to the RF phase ofthe bunch. Timing of the ADC start with the high voltage
beam excitation pulse was performed using the multichannel time interval generator (TIG in Fig. 4): the TIG trigger
signals were passed to both the ADC and the HV generator.

f0DLkicker

HV Pulse

TIG

Generatorout #1out #2

1:100

...
HV

Oscilloscope

SPG

input

SupplyCrate
Controller

PCFigure 4: Block diagram of the experimental setup.

C A M A Ctimingstart20 0 +1

ADC
input
RFtiming

VEPP-2M

The center of mass positions of the colliding bunches weresampled turn-by-turn. The Fourier transform of the collected data gave the coherent mode spectrum, where theproposed synchro-betatron modes of the beam-beam system were experimentally detected, and their spectrum wasmeasured as a function of the beam-beam parameter at different synchrotron tunes.
The complete results of the coherent beam-beam mode
spectra calculation with the account of the finite bunch
length are presented in [5, 7]. Since VEPP-2M had a negligible transverse impedance, we compare these experimental data with the simulation results for the case where thecollective interaction is completely due to beam-beam. Fig.
5 shows the dependence of the measured and calculated
synchro-betatron mode tunes on the beam current for equal
electron and positron bunch intensities.
In perfect agreement with the theoretical model, themeasurement has shown that besides the leading  and 
modes a number of synchro-betatron modes coupled via
the beam-beam force exist in the dipole mode spectrum.
These modes show up and disappear with the beam current change due to  -dependence of the beam-beam mode
eigen-states. For the  value less than the synchrotron
tune s , the state excited with the kick consists of only
two beam-beam modes,  and  , with the synchrotron
wavenumber m. In the range  s <  < s the initial condition is the combination of four eigenmodes:, ,  ,  . Here the first index labels the synchrotron
wavenumber, the second labels the coherent beam-beameigenmodes with even and odd symmetry between the twocolliding bunches, respectively. With larger  the dipole
moment passes on to ,  and later to  ,  modes. Because of small coupling of modes with large
synchrotron wavenumber these transitions do not show anapparent tune split.
An important parameter in the coherent beam-beam effect is the ratio Y between the coherent beam-beam  and modes tune split  and  . The rigid Gaussian beammodel [2] predicts the ratio =: . With non-rigid
beams, solution of the Vlasov equation [4] gives for our: .
case of flat beam and vertical oscillations =In our experiment  I was evaluated by fitting thetheoretical mode spectrum to the whole dataset of the measured spectrum, using a single fitting parameter, Y  . Then=0edge
screenPMTf002

scr

PMTSR

The similar observation channel was implemented forthe positron beam. For synchronization of the electron andpositron channels the clock pulse splitter was used with thedelay correction tuned by means of the additional cable in
the positron channel.
The HV pulse generator gave a one-turn kick to thebunch, with an adjustable amplitude of the excited oscillations. A minimum amplitude was equal to :  ,  being
the Gaussian vertical beam size. The kicker plate terminated in a matched load, to kick only the electron bunch.

2 +2()13 +3= 10= 1 21

0.17
0.16
0.15
0.14
0.13
0.12
ν0+νs

0.11

ν0

0.1
0.09
0

ν0-νs

5

10

15

20

25

30

Figure 5: Measured (circles) and calculated (lines) synchro-betatron mode tunes vs beam current I (mA). VEPP-2M was: , s:, cm, and the bunch
operated with one e + and one e bunches colliding at two IPs, length was 3.5 cm.

= 0 101 (I ) was evaluated from the luminosity measurement:
2ere y L :y =I
The main inaccuracy in  thus evaluated is due to statisticalstraggling of on-line
p L data and equals to  10%. A good
linearity of L vs I evidenced for current-independent
beam sizes at the IP, i.e. we had  / I without saturation,in the whole range of currents used. From our experiment
=: .
we have found Y
This result can be related to our experimental procedure.
Indeed, after the kick the transverse beam density distribution does not change during the measurement period of
8192 turns (or effectively less, limited by decoherence).
This is much less than the radiation damping time. For thisreason the situation appears to be equivalent to interactionof two transversely-rigid Gaussian bunches.

== 1 05

5 CONCLUSION
The observation system allowed to discover the synchrobetatron modes in the spectrum of coherent oscillationsof colliding bunches at the VEPP-2M collider. The measured spectra dependence on the beam current is in excellent agreement with analytical and numerical models[5, 7]. The measured coherent beam-beam mode tune split
is :  . This is close to the value expected from the rigid
Gaussian model which seems to be adequate to the used
experimental technique.
The above presented experimental evidence of the synchro-betatron beam-beam modes adds confidence to theconclusions of their theory. One of them is that for the
negligible transverse impedance the mode system remains
stable unless some of the mode tunes reach a half-integer
resonance.

1 05= 0 0069=6On the other hand, calculations involving the machine
impedance predict a coherent beam-beam instability without a threshold. Some, though not all, of the synchrobetatron modes can be damped by optimizing the betatrontune chromaticity. Since the theoretical models [5, 7] used
the linearized beam-beam interaction, their prediction ofinstability is not as conclusive as the above prediction ofstability. In a realistic nonlinear beam-beam system one
can expect saturation of such an instability at amplitudes ofthe order of the vertical beam size. However, this mechanism can cause a vertical emittance blowup detrimental tothe high performance of the flat-beam colliders.

6REFERENCES[1] H. Koiso et al., Particle Accelerators 27, 83 (1990).
[2] K. Hirata, Nucl. Instr. Meth. A 269, 7 (1988).
[3] A.W. Chao and R.D.Ruth, Particle Accelerators 16, 201(1985).
[4] K. Yokoya and H. Koiso, Particle Accelerators 27, 181(1990).
[5] E.A. Perevedentsev, in Proc. 1999 Particle Accelerator Conf.,
New York (1999), vol. 3, p.1521;
E.A. Perevedentsev, in Proc. Int. Workshop on ElectronPositron Factories, Sept. 21-24 1999, Tsukuba, KEK Proceedings 99-24, Feb. 2000.
[6] V.V. Danilov, E.A. Perevedentsev, Nucl. Instr. Meth. A 391,77 (1997).
[7] E.A. Perevedentsev and A.A. Valishev, Phys. Rev. ST-AB 4,024403 (2001);
E.A. Perevedentsev and A.A. Valishev, in Proc. 2000 European Particle Acc. Conf., Vienna (2000), p. 1223;
E.A. Perevedentsev and A.A. Valishev, in Proc. 2000 Int.
Computational Acc. Phys. Conf., Darmstadt (2000).