STUDY OF LINEAR COUPLING COMPENSATION FOR THE STORAGE RING OF SPRING-8
H. Tanaka, R. Nagaoka, K. Yoshida, K. Tsumaki and M.Hara
RIKEN-JAERI Synchrotron Radiation Facility Design Team
2-2S-S, Honkomagome, Bunkyou-ku, Tokyo, 113, Japan
Abstract
Linear coupling compensation for the storage ringof SPring-S under the practical error condition isdescribed. It is found that the coupling effect is so strongthat we can not achieve the ring design performance
without some coupling compensation scheme, and thatthe method with only two skew quadrupoles is useful tothe coupling compensation to some extent.
Introduction
The storage ring of SPring-S is a Chasman Green
type low emittance ring. In this kind of storage rings,
strong quadrupoles and sextupoles are usually used inorder to strongly focus the beams and to compensate
In the case that these magnets arelarge chromaticities.
tilted and misaligned, they make the circulating beams
feel the strong skew quadrupole error fields. Therefore,
the strong linear coupling of lateral beam oscillations issuspected in the storage ring of SPring-S.
On the other hand, it is required for the variousexperiments by the photon beam users that the coupling
ratio of vertical and horizontal emittance is adjustablefrom 1 % to about 100 % (full coupling condition). Inaddition, it is crucial to suppress the increase of verticalemittance lower than 1 mm for the stable beam injection
with high efficiency.
For these requirements, the linear coupling should
be compensated enough to obtain 1 % coupling ratio. In
respect to the increase of the coupling ratio, it can beeasily controlled with the adjustment of the workpoint.
Linear coupling theory
A coupling coefficient Kq, which is derived fromHamiltonian function to include a skew quadrupole
potential through a canonical transformation and Fourier
expansion, is used to estimate the strength of the linearcoupling. The coupling coefficient Kq is expressed as1 ,2)(1)

known statistic treatment3) of Eq. (1). In this calculation,a tilt error of a quadrupole and vertical offset errors of a
sextupole, which are the first order of the error
perturbation, are assumed as a source of a skew
quadrupole component. The offset errors mean amisalignment of a sextupole and closed orbit distortion(COD) at a sextupole. For the normal operation mode(hybrid mode ) 4 ) in which the nearest resonance to theworkpoint is Qx- Qy = 32, the expectation value of x32 iscalculated as,[ x 32 l tilt = 34.5S [ e J ,[ K32 ] offset = 20S.S8 [ l'.z ] ,

(3)

(4)

[ K l til?+ [ K l ofset2•[ K32 l total =--./(5)

where El and 1'. z are the tilt error angle and the ofset error,
respectively, and [p] denotes the root mean square valueof a parameter p. From above formulae, it is found thatthe contribution from sextupoles is dominant in thestorage ring of SPring-S. This is general for the low
emittance ring where lots of strong sextupoles areinstalled. It is suspected that the root mean square valueof the alignment error, the residual COD, and the tilt error
is 0.2 mm, 0.1 - 0.15 mm, and 0.5 · mrad, respectively. The
expectation value of K3 2 for the storage ring of SPring-S
is estimated to be 0.05 - 0.055.
The coefficient Kq can be also calculated with thevertical amplitude beating of the tracking particle whichis started with zero vertical emittance. The values of x3 2for 10 rings are distributed from 0.006 - 0.07 and theaverage value is about 0.03 which is slightly lower thanthe expectation value. This may be because the errors of
tracking simulation are cut off at the double root meansquare value. In this paper, it is presumed that theexpectation value of K32 exists in the range from 0.03 0.055.
With. Kq and l>, the coupling ratio is estimated by Eq.
(2). The coupling ratio is shown as a function of Kq and asa parameter of li in Fig. 1. The parameter l> is 0.06 at thenormal hybrid mode. The maximum value of l) is limited towhere '¥ i and j3 i (i = x : horizontal plane, y : verticalplane) are Floquet phase and the betatron function, Qiand .p q are the betatron tune and the phase of q
harmonics corresponding to the strength of sine andcosine terms of Kq, k 8 and L are the strength of skew
quadrupole components and the ring circumference,
respectively, and e = 2rts!L (s : azimuthal length, L : ringcircumference) denotes the time variable. The parameterq represents the differential resonance condition and theresonance to be compensated often satisfies the conditionthat l) (l> = Qx-Qy - q) is minimum. The parameter l> denotes
the distance from the resonance.
The horizontal and vertical emittance are related toKq through perturbation treatment of the Hamiltonian
function and the resonance approximation as follows.

100~~"·=c.""
u"
<>I80

D

8~0.06

•

o~o.w

DD
D60

DD
D40
D0•••
D20• bl00.00

D•

0 040••••0.08Coupling Coefficient

•0.12
Kq

(2)

where r is the coupling ratio of vertical and horizontal
emittance.
Aspect of linear coupling in the storage ring of SPring-S
The expectation value of Kq is calculated by well-299-

Fig. 1

Coupling ratio v .s. coupling coefficient Kq

[Js:!

0=0.06• 0=0.100.2;;:;[J...
'<l[J[J[J=.e=[J[J0.1••[J[JIii.
0.00.00•••••[J[J••TURN N0.:

Scola:0.04

0.08

Coupling CoefficientFig. 20.12

Kq

Fig. 4Beam height v .s. coupling coefficient Kq
Horizontal emittance is 5.3 nmrad and beam widthis 0.33 mm and ~y is IO m.

=

0.22

1'-<...."'
E~..
~...

-;
u'"'

0.20
0.18
0.16

Fig. 3500

10.0 mm
1.0 mrod

Phase space plot at injection simulation

Linear Coupling compensation scheme
In order to compensate K q for the arbitrary
operation mode, all terms (cosine and sine terms) in righthand of Eq. (1) should be vanished with skew quadrupole
magnets. The linear coupling phase part in right hand ofEq. (1) is represented here with ~q and it is assumed thata set of only two skew quadrupoles, S 1 and S2, are installedat ~ q 1 and ~ q 2, respectively. The compensation can beperfectly achieved with this set, if ~q1 and ~q2 satisfy
the following relationS),~v~q1 + ~q2

Nux0=51.22
Nuy0=19.16

0.14 +-----,--.,-~--r---l•20-1010200Horizontal

10.0 mm

1.0 mrod

0.24.-------------,...

PyPx

0.3

Amplitude (mm)Amplitude dependent tune shift diagram
Cross represents vertical tune and white square
represents horizontal tune. Vertical amplitude
ratio is I%.

about 0.1, because the operation range of the workpoint islimited by the first and third order resonances from theviewpoint of dynamic stability of the ring. With theparameter condition mentioned above, the expectation
value of coupling ratio becomes 20 - 50 % at the normal
workpoint, S = 0.06 and IO - 20 % at S = 0.1. That is , it is notforeseen that the linear coupling can be controlled with
only workpoint tuning. The beam height at the high f3
straight section is shown as a function of Kq in Fig. 2 andfrom that Figure, it is estimated that the expectation valueof beam height becomes 0.1 - 0.15 mm at. S = 0.06 and 0.06 0.11 at S = 0.1.
·The situation of the coupling at the large amplitude
can be investigated ·with the amplitude dependent tune
shift diagram shown in Fig. 3. It is observed that theincrease of the horizontal amplitude makes ll small. This
fact means that the coupling ratio of vertical andhorizontal emittance becomes larger at the beaminjection than at the steady state after a radiationdamping. An example of a phase space map at theinjection are shown in Fig. 4. In this simulation, the beam
are injected into the ring with small vertical emittancewhich is negligible compared to the horizontal one(horizontal initial amplitude = 8 mm), but the verticaloscillation is excited by the skew quadrupole fields in thering and vertical emittance becomes quite large after
dozens of revolutions. We can find that particle motion inFig. 4 is almost full coupling through the divergence ofvertical phase space. That fact is also confirmed by thecoincidence of both tunes measured by the Fourier
analysis. The coefficient of K3 2 of this ring is about 0.025,
which is not so large.

= ( 27t 14) x n,(6)

where the subscript 1 and 2 represent the parameter of S 1
and S2, n is arbitrary odd integer. But actually, quite large
numbers of skew quadrupoles should be introduced toobtain two degrees of freedom for the compensation
totally, because it is difficult to find the set of two skew
quadrupoles satisfying the above relation for all
operation modes.
On the other hand, the storage ring of SPring-S ischaracterized by its large cell. periodicity, maximum 48
periodicity. With this large periodicity and the easy
constraints to operation flexibility, we can find the set of
only two skew quadrupoles satisfying Eq. (6). The phasedifference between two skew quadrupoles separated at
n 12 cells are represented with the superperiodicity N as,~q1 ='~'xi+'Pyi - ( Qx- Qy· q) X 9J,
Qy· q) x 82,
M = 82 • 9 1 = 27t x ( n 12 1 N ),~~ =~q1- ~q2 = qx (92 -81)
= 27t x q x ( n 12 IN ).
~q2 = '~'x2 + '~'y2 " ( Qx-(7)
(8)

(9)
(10)

If the workpoint can be determined as the combination of

odd and even number in integer part of betatron tune, we
can always make q some odd number. Furthermore, if N of
all operation modes can be limited a multiple of 4, we
make n 12 I N equal to 1/4 for all operation modes by thearrangement where two skew quadrupoles are separated
with the quadrant of the ring. Consequently, two skew
quadrupoles correctly satisfy Eq. (6). In the operation of
thestorage
ringofSPring-8,
theminimum
superperiodicity is 4 at the operation mode with 4 longstraight sections. The constrain to superperiodicity is notse,ious. As concerns the constrain to the workpoint, it is
not either serious. The workpoint of normal hybrid mode
is now composed of two odd numbers in integer part, but
it is easily adjusted with vertical detuning without
reduction of the dynamic stability.
With the above criteria, the relative positions of twoskew quadrupoles can be determined. Next, the specifiedposition in a cell for the skew quadrupoles should be-300-

selected. On the selection, following two things should beconsidered. One is to select the position in the dispersion
free section. If the skew quadrupoles are located in thedispersive section, the vertical spurious dispersion isinduced by the linear coupling and vertical emittancebecomes' large. The other is to select the position withhigh ~ x and ~ y. From Eq. (1), it is found that JC q is
proportional to the square root of ~x and ~y· The strengthof the skew quadrupoles can be reduced with high ~x and~ y. Considering these requirements, the position at theharmonic sextupoles in dispersion free section and at theedge of dispersion free straight section are selected ascandidates.

0(A)100200300

400500700800900 1000

700800900 1000

600

SUCCESIVE TURN N

Computer simulation
0The normal hybrid mode is used for the simulation of
linear coupling compensation. As the workpoint of thismode does not satisfy the even-odd combination, we can
not use the scheme described here directly. In order toinvestigate the effect of two skew quadrupoles whichsatisfy Eq. (6) exactly, we rearrange two skew
quadrupoles for the normal hybrid mode, regardlessperiodicity. This is enough to know whether our scheme
is useful or not.
The subroutine for the compensation simulation isadded to the tracking program RACETRACK6) R IKE N
version. The simulation is performed according to thefollowing procedure. At first the COD is corrected at thelevel of the residual COD lower than 0.15 mm for the ring
including normal errors (the misalignment error 0.2 mm,
the tilt error 0.5 mrad, the field error 5x 10- 4 , the gradienterror 5x 1 0- 4 ). After the COD correction, the strength of
the skew quadrupoles are fitted to minimize the maximum
vertical amplitude of the tracking particle in 50
revolutions. As the COD and Twiss parameter are distorted
during the fitting, the COD and the Twiss parameter
calculation, the COD correction can be used in fitting loop
to cancel the fitting error. Mapping of the coupling ratio
as a function of skew quadrupole strength can be used todetermine the initial strength.
The result of the simulation is shown in Table 1 andAn example of improvement of the coupling is shown inFig. 5. The simulation is performed for 4 rings of which
K3 2 is relatively large among 10 rings mentioned before
and remarkable improvement can be found in thecoupling ratio. The coupling ratio is reduced to one third one fifteenth of the initial value; 4 - 10 % by theskewofstrength
maximum
Thecompensation.
quadrupoles is about 0.023 m· 1.
Although it is difficult to find the reason why thecoupling can not be compensated completely, we guessfollowing two things. One is local phase distortion by thering errors between two skew quadrupoles which breaks
the indispensable phase relation. The other is the
problem of fitting performance which means theprogram can not search an optimum strength from thearbitrary initial condition.
Table 1
simulation result of the coupling compensation
No

Mter compensation

Before compensation

1C

Coupling ratio(%)

1C

53

0.012

3.9

20.039

30

0.015

6.5

30.039

30

0.021

II

40.047390.020

10200

300

400

500

600

SUCCESIVE TURN N
0.60~----------------------------------------;

0.450.30
0.15~

o.oo~

-0.15
X -0.30
-0.45
-0.60

(B)

I I I
I II rll
I I I I IIIII
l'ir' !rill•' :'lr• I' rl,'ll''''" "llllr''lr,l, ulr,,l, 1•,1" I''''''l''lrlrl"•l'•I '11,111' ~· ll,llrI ' I' ' Ill' I' '''l:l',rl Iii'' lr II ,,'I, I,, I 11II ' III 'I' lr' lrrr,lI IIll1r,, Ill llrrI , I'Iril ,lr,lr'11il
0100

200

300

400

500

600

700

800

900 1000

700

800

900 1000

SUCCESIVE TURN N

0100200300

400500600

SUCCESIVE TURN NFig. 5

Horiwntal and vertical emittance beatThis is the result of linear coupling compensation
for No 1 in Table 1. X and Y represent horizontal
and vertical oscillation amplitude. (A) and (B)denote before and after compensation.

Summarywaslinear coupling
theofsituation
Theinvestigated. It is found that the expectation value oflinear coupling coefficient is 0.03 - 0.055 and that thecoupling ratio is 20 - 50 % at the normal workpoint. Thesevalues are too large to control the coupling ratio withworkpoint adjustment. In order to satisfy requirementthat the coupling ratio is adjustable in wide range ( 1 % about 100 % ), some compensation scheme should be used.
The compensation scheme with only two skew
quadrupoles was proposed and the computer simulation
was performed to study its effect. It is found that thecompensation scheme is effective to some extent, but can
not satisfy our requirement, 1% coupling ratio. Detailinvestigation of this imperfect compensation and anothercompensation scheme based on its result should bepursued.
References

Coupling ratio(%)

0.070

1001. F. Willeke et al., AlP Conference Proceedings 184, Vol.
p.758
2. G. Guignard et al., CERN 76-06, 1976
3. J.P. Koutchouk et al., Proc. of Particle Accelerator
Conf., Washington, D.C., U.S.A., March 16-19, 1987
4. R. Nagaoka et al., RIKEN Accel. Progress Rep., 227, 1988
5. A. Wrulich et al., ESRF-SR/LAT-87-02, 1987
6. A Wrulich et al., DESY Rep., 84-026, 1984-301-