TRANSVERSE BEAM BUNCHING METHODUSING A DEFLECTING MODEK. Miyata and M. Nishi
Energy Research Laboratory, Hitachi, Ltd.
1168 Moriyama-cho, Hitachi-shi, Ibaraki-ken 316 Japan
Abstract
A transverse bunching method is presented for an electron
beam using a deflecting mode in a non-axisymmetric RFcavity. The method's principle is based on an interaction
between synchrotron and betatron oscillations. A damping ofthe betatron oscillation leads to a reduction of beam sizeat the expense of growth in energy spread. The damping time
can be controlled by varying the amplitude of the deflecting
mode or by detuning of the deflecting mode frequency from anacceleration frequency.
Introduction
In an electron synchrotron or storage ring, transverse
beam bunching usually depends only on radiation damping or
adiabatic damping. Here we present a transverse beam
bunching method by electromagnetic fields of a deflecting

y=;ax+P'x',(1)where a and fl are Twiss parameters.
The accelerating and deflecting mode cavities
aredesignated as Ace and TBB, respectively, and the actions of
the RF fields are treated like a delta-function. We use thebetatron phase space ( x , y ) and the synchrotron phase space( B , 5) where B is an RF phase and 6 is a energy deviation defined as(2)with E as electron energy and Eo as synchronous electronenergy.
The basic equations are described as follows 1 ' 2(1) At Ace and TBB
Llx=-7JL1oP,

mode.

The electric and magnetic fields affect betatron andsynchrotron oscillations. The electric field affects thesynchrotron oscillation through an energy change, and it
affects the betatron oscillation through a change of aclosed orbit. The magnetic field affects the betatronoscillation through momentum kick, which in turn affects thesynchrotron oscillation through a change of revolution time.
In the method, a betatron oscillation amplitude is damped
through a coupling of synchrotron and betatron oscillations.
This method can be applied to injection of a low energy
beam so that the beam is compressed within a good fieldregion to reduce beam loss. Another application is found in
reduction of betatron emittance in a high energy electronring.
Synchro-Betatron Oscillation
The synchrotron and betatronoscillations haveinteractions due to the following factors,(1) energy dispersion of a closed orbit at RF cavities,(2) chromaticity,(3) magnetic fields in RF cavities,(4) transverse dependence of electric fields in RF cavities,(5) change of revolution time due to the betatron oscillation, and(6) wake fields (transient electromagnetic field caused by
the beam-cavity interaction).
The synchrotron oscillation affects the betatron oscillation
through factors (1), (2) and (3), while the betatronoscillation affects the synchrotron oscillation throughfactors (4) and (5). Through factor (6), the synchrotron andbetatron oscillations affect themselves and interact witheach other. This is the main cause of beam instability. The
chromatid ty in factor (2) is usually corrected to zero tosuppress the head-tail instability of the betatronoscillation, so this factor can be neglected here. Furthermore the wake fields in factor (6) are assumed to beneglected.

L1 y=

(3)

1

[(0-;,- o~ )X

1 + o-:.

(4)x{ o~a7J-(l+o-:.+a~H+Cax.:-~)}+ft¢H],(5)

L1 0= 0 'Llo= eV(x+7Jo, 0)(6)

Eowhere5p=P- PoPo

(7)

with p as electron momentum andelectron momentum.
(2) From Ace to TBB

p o as synchronous

L1 0= 0'(9)

.t1 o= o .

(lo)(3) From TBB to ACC
(11)
(12)

(13)
wherefo

k 0 = 2 7r-Basic Equations

c 'In order to analyse synchro-betatron motion, basicequations are presented for numerical simulations. The
betatron motion is assumed to be linear, except for RF
cavities. An accelerating cavity and a deflecting mode
cavity are installed in a straight section of the ring. Forconvenience, a normalized phase space ( x , y) is used forthe betatron motion

f 0 : acceleration frequency,c : velocity of light.
In these equations, the following parameters are used.
,8A: betatron function at Ace,
P'T : betatron function at TBB,
fl. AT: betatron phase advance from Ace to TBB,
P.TA: betatron phase advance from TBB to ACC,h : harmonic number,

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(14)

am : momentum compaction factor,
a=TJTsinp.-(T(l-cosP.),(15)

b= TJT(l-cos f1.) +(T sin tt,(16)

f1. = f1. AT+ llTA :betatron phase advance per turn, (17)Tj; .

=(TaT7JT +ftT
(18)
The symbol LJ denotes a change of the following val~e •. The
subscripts - and + in Eqs.(4),(8),(11) and (12) Indicate
values just before and after each cavity. Equation (12) was
derived by Piwinski and Wrulich 1' 3 • In Eq.(l2), the firstterm including 6 is a phase shift by the change of thecircumference due to energy dispersion of the closed orbit,
while the second term, including x and y , is due to thebetatron oscillation.

The synchro-betatron motion was simulated by using thedata of the deflecting mode described above. Table 1 lists
basic parameters used here.
The basic equations described above satisfy the symplectic
conditions for a conservative systea, so that the number ofturns can be set on the order of one million in the
double-precision calculation. However, higher order terms ofx and yare neglected in the above basic equations.

Deflecting Mode
The electromagnetic effects of the deflecting mode aredescribed by using an electric voltage V and a "magneticvoltage" VH defined asV=/E.ds(19)

VH=Zo /H.ds(20)····~···-!
ir----r··--1'-1E s: longitudinal electric field,
H.: vertical magnetic field,
Zo: characteristic impedance in a vacu1111.
The change of the energy deviation Ll (} and the mo~~entum kick¢HbY the RF magnetic field are written as

Ll (}= e V cos 8( 21)¢H= e Vn sin 8(22)rp 1

rp 6 6 0

EoThe Maxwell equation relates the electric and ,magneticvolatages V and VH as follows,

avax= kVH

(23)

= 2rr-f(24)wherekc 'and f is an operating frequency of the deflecting mode.

Therefore the two voltages are written in the following form,
VH (X)= VHo (1+ gX 2 ) ,g
3J ,V(X)=kVHo (X+3X

Fig.l Structure of the deflecting mode cavity.

(25)
(26)

where X is the horizontal displacement from a central axisof the deflecting mode cavity. "Magnetic impedance" Z H isconveniently defined as
2Zn= Vno
Pw(27)

where Pw is power loss of the deflecting mode on the wall.
The deflecting mode cavity having a non-axisymmetricstructure was numerically analysed by a three-dimensional
computer code for electromagnetic field analysis based on
the boundary element method. Figure 1 shows the structure of
the deflecting mode cavity. In Fig. 2, the RF magnetic
fields of the TM11o mode are indicated by arrows for onefourth of the cavity. The resonant frequency is 150KHz, andwe obtained the magnetic impedance Zu of 6.9MQ withcopper wall, and the coefficient g of 38 m- 2 in Eq. (25).
Fig.2 RF magnetic field of TM11o mode in onefourth of the deflecting mode cavity. The numerical result was obtained by the computer code
based on the boundary element method.

Numerical Simulation Results-205-Therefore, the number of turns was restricted to twomillions.
Figure 3 shows the RF phase relation of accelerating anddeflecting modes for f = f 0 where the synchronous phaseis 180oFigure 4 shows the synchrotron and betatron motions andbeam size evolution with initial RF phase between 140° and
220°, initial energy deviation of zero, and initial betatron
amplitude of 3cm.
When the energy gain by the electric voltage is positive
for a positive horizontal displacement around the synchronous phase 180° , the betatron oscillation decays, while thesynchroton oscillation gradually grows. The damping time of
the betatron osciUation can be controlled by varying theamplitude of the deflecting mode or by detuning of thedeflecting mode frequency from the acceleration frequency.
Qualitatively we can say that the electric field of thedeflecting mode amplifies the synchrotron oscillation andTable 1.

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0221
N (Xl0 6 )Turn Number N (X 10 6 )

7J'

1.01

lOkV /m150 MHzReferences

5 kV1) A. Piwinski, "Beam Instabilities and Computer Simulations", DESY M-83-21 (1983).
2) D.P. Barber, G. Ripken and F. Schmidt, "A Non-Linear Canonical Formalism for the Coupled Synchro-Betatron Motionof Protons with Arbitrary Energy", DESY 87-036 (1987).
3) A. Piwinski and A. Wrulich, "Excitation of BetatronSynchrotron Resonances by a Dispersion in the Cavities",
DESY 76/07 (1976).

0.8\0.0~----/7,~----~----~----~/.

'\ /\'~ -0.8 __/ , - vcos {) L1 0 \ ' ,....._...../;<~.
0.::-1.20~--~~--~~--~~--~

901802N (X 106 )We presented a transverse beam bunching method wherebetatron oscillation can be damped by a deflecting mode at
the expense of growth of synchrotron oscillation when theenergy gain by the electric voltage of the deflecting modeis positive for a positive horizontal displacement aroundthe synchronous phase.

(TBB)

/00

Conclusions

Accelerating Voltage (ACC)

-0.4

~

2.07m

Acceleration Frequency

0..

J7J

3.1m

Accelerating Voltage (ACC)~!
"e 1....,

1.01

-0.43

(TBB)

0.4

-1

u:l

7J'aVolt age Gradient

•rl

•rl
.....

~

1.06m

1.3m

(ACC)

(TBB)

~

"•rl

7J

10hFunction

....,

-1.3

0.138

(ACC)

b"0a

1.58

Momentum Compaction Factor

Energy Dispersion

$Q)

•rl

~

4'E!

damps the betatron oscillation, and that the magnetic fieldof the deflecting mode amplifies the betatron oscillation
which attracts the particle toward the synchronous phase todamp the synchrotron oscillation. The effect of electricfield on the betatron motion is stronger than that of themagnetic field. That is why the betatron oscillation decaysand the synchrotron oscillation grows.
The betatron oscillation grows when the RF phase of thedeflecting mode is shifted by 180° from the one shown inFig. 3. When the initial RF phase is far from the synchronous phase, the particle runs away from the RF-bucket
generated by the accelerating voltage.
The RF power of the deflecting mode is calculated by Eq.
(27) as 1.5W for the voltage gradient of 10kV/m, while theRF power of the accelerating mode is 25W for the accelerating voltage of 5kV with a shunt impedance of lMQ.

})

Harmonic Number~"""0....,"'>

4Fig.4 Synchrotron and betatron oscillation amplitude and beam size.

20MeV

Horizontal Betatron Tune'-'~~

555u,--.

Basic Parameters

EnergyTwiss Parameters

2270

360

0 (deg)Fig.3 RF phase relation of accelerating and deflecting modes for f=fo . The voltage -V cos (}of the deflecting mode is shown for a positive
horizontal displacement.

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