The 13th Symposium on Accelerator Science and Technology, Suita, Osalm, Japan, October 2001

Beam Response Analysis for the SPring-S Storage Ring with 4 by 4
Model Calibration Method (MCM)
H. Tanaka *, K. Soutome, M. Takao, SPring-8/JA SRI, 1-1-1 Kouto, Mikazuki,Sayo, Hyogo 679-5198
J. Schimizu, The Japan Research Inst. Ltd.,
1-33-8 Shin-machi, Nishi-ku, Osaka-shi 550-0013, Japan

Abstract
Restoration of linear optics is crucial especially for third
generation synchrotron radiation sources to improve key
beam parameters such as emittance, momentumacceptance and so on. We have thus been analyzing theerror distribution along the ring by using model
calibration method (MCM) since 1998. A part of results
was already utilized to improve the beam performance.
The model used in MCM was recently changed from 2 by
2 to 4 by 4 formalism which can treat a complete
transverse oscillation mode mixing. In this paper, wepresent obtained results with 4 by 4 formalism comparedwith those with 2 by 2 one.

2 4 BY 4 RING MODEL
The ring model used here is based on linearlized
Hamiltonian with thin nonlinear kicks as nonlinear
magnets and error fields. As a nonlinearity, we only take asextupole magnet into account. The full transverse optics
parameter calculation obeys the formalism in Ref. [5].

2.1 Energy Shift Treatment
The ring model can not treat with coupling between thetransverse orbit and pathlength change. We therefore
consider this effect approximately. When j-th STM at thefinite linear dispersion T/z,j is changed by Oz.j. thefractional energy shift MjiE is expressed by

1 INTRODUCTION
From the beginning of 1990, W. J. Corbett and hiscolleagues developed a new technique to estimate beam
optics parameters via the ring model [1]. The orbit
response is defined as a set of orbit differences measured
by beam position monitors (BPM's) with changing
strength of a certain dipole steering magnet (STM) bysome amount. The different response can be obtained byusing the different STM as a perturbator. The idea of thismethod is that the realistic ring model should be obtainedby fitting key parameters so that the calculated responses
agree well with the corresponding measured responses. By
using the model all estimated parameters are consistent
with each other. Here, we do not touch the detail ofparameter fitting, which one can see in Ref. [2]. This
technique was successfully applied to restore the optics
distortion in Advanced Light Source (ALS) [3].
We started applying 2 by 2 MCM to SPring-8 storage
ring from 1998 in order to estimate the distortion ofbetatron functions [4]. On the other hand, top-up
operation of the storage ring has recently been planned toincrease the time-averaged brilliance for several bunch
filling. In this case, it is essential to prevent a horizontal
oscillation from coupling with a vertical one, because invacuum undulators limit vertical aperture to around 4 mm
in half height. To optimize operational condition andcontrol distributed linear error fields if necessary, wechanged the model from 2 by 2 to 4 by 4 formalism whereskew error fields and precise optics parameters for eachtransverse eigen mode can be estimated.
'corresponding Author; Email; tanaka@spring8.or.jp

AEe ·.n ·-1-""'~ z=x y' 'a.£ 'E(1)

where a and L are respectively the momentum
compaction factor and the circumference. The suffices xand y stand for respectively the horizontal and verticalplanes. By using Eq. (1), the model response calculated by
the 4 by 4 formalism, Rm,i,j is modified asAERm,i,j ~ Rm,i,j + - -1 T/z,i·(2)E

The suffices i andj represents respectively i-th BPM andjth STM.

2.2 Fitting Parameters
The fitting parameters in the ring model are the following:•••

Horizontal and vertical BPM gain factors Gx,i andGy,i and the BPM rotation error around the beamaxis qi for 280 BPM's ( i = 1 - 280 ).
STM gain factors Gj and the STM rotation error
around the beam axis Pj for maximum -2x260
STM's ( j = 1 - max. 520 ).
236 thin integrated normal quadrupole error fields
and 132 thin integrated skew quadrupole error fields.

The measured horizontal and vertical response Xmes i andYmes,i are related to the model response Rm,i,j by '[ COS qi.. _ [ Xm,i,j ]-sin m
y m,i,jRm,I,J-- 83-sin ~icos ml[Gx,iXmes,i. Gy,iYmes,il' (3)The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

where Xm,i,j and Ym,i,j are respectively the horizontal
and vertical model response at i-th BPM. The kick angleof j-th STM Oz,j is also related to the kick angle set in themeasurement through the gain factors Gj and the STM
rotation error Pj.
Five normal and three skew quadrupole error fields are
distributed in one Chasman Green (CG) cell and six
normal and four skew quadrupole error fields are in amatching section for long straight of about 30 meter(LSS). The number of errors and their positions wereoptimized from the viewpoint of fitting convergence [4].
The model error field arrangement is shown in Fig. 1.
CGCellconvergence versus cut-off level. The open squares andcircles stand for respectively the s.t.d. of two normal
quadrupole error distributions CJn and that of two skew
ones CJs. Two distributions are obtained by a different setsof beam responses. The crosses stand for the r.m.s. ofdifferences between the model and measured beam
responses X· The cut-off level Vmin represents the ratio
against the maximum eigen value. In Fig. 2 we see thatthe objective X decreases as Vmin decreases, but that thes.t.d. of the error distributions increases in the region
where Vmin becomes smaller than 0.001. This shows thatthe convergence improvement does not contribute to theaccuracy improvement of the error fields in the small cutoff region. Accordingly we set Vmin to 0.001 in thiscalculation.

,., .., ,.. ,.,LSS-~BendingI:Quadrupole I :Sextupole~ :Normal 4 :Skew

Figure 1: Arrangement of normal and skew error fields.

3 BEAM RESPONSE ANALYSIS
3.1 Beam Response Measurement
At the original operation point, cross talk between thehorizontal and vertical beam responses is quite small. Toenhance the cross talk, we set the operation point near by
the linear differential coupling resonance line Vx - Vy =22.
The distance from the resonance is 0.02 -0.03.
The step of strength change for each horizontal STM isset to 0.035 mrad and that for vertical STM is 0.05 mrad.
To suppress a hysteresis effect, the direction of strengthchange is determined as to increase the absolute value of
initial setting strength.

Cut-off Level ofEigen Value V .

mmFigure 2: Fitting convergence v.s. cut-off level.

3.3 Error Field Distribution
Figure 3 shows a quarter of the estimated normal
quadrupole error distributions. In the figure, thedistribution calculated by 2 by 2 MCM is shown together.
:::::;' 0. 004 ,-~,..-r-.--r--,-r--r--T---,---r-.-,-~,.....,........,-,....-.-.---,--..,
Total NQ Number=236..s:: 0.003 A Quadrant of the Ring: Section A..§.

3.2 Fitting Convergence
The linearlized sensitivities of each fitting parameter
described in subsection 2.2 against beam responses arecalculated with the ring model. By using the calculatedsensitivity matrix, the fitting parameters are determined as
to minimize the square of differences between the modeland the measured responses by the least square method.
Here, to solve this linear problem we used the algorithmof singular value decomposition (SVD). Since the ringmodel is not purely linear, the problem is iteratively
solved changing the initial values of the parameters. Three
iteration steps are sufficient for the good convergence.
In the SVD algorithm, the convergence also depends onthe cut-off level of the eigen values. Figure 2 shows theiIZl-g0.002
0.001&0~ -0.001
0z -0.002 --<>-4x4: 2001/04/16"'0~ -0.003 :::=1~t ~gggm~~§=~~:s -0.004 0-84-----><--- 2x2: 2000/ll/9_C

8162432
40
Fitting ID Number

4856

Figure 3: Integrated quadrupole error distributions.

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001-~~~~4x4: 2001104/16 ~~~~~~~2x2: 20001101191
1- - ~ 4x4: 2000/11129 -Design

As seen in Fig. 3 the estimated error distributions do notdepend on the model used in MCM.
Figure 4 shows a quarter of the estimated skew
quadrupole error distributions. In the figure, three
distributions obtained by the different sets of beam
responses are also shown. The slow harmonics of thedistribution are well reproduced in every measurement. We
see the magnitude of high harmonic components is lesscompared with that of the quadrupole error distribution.

P.

1:1::::;' 0. 004 ,..-,---ro--r-r---r-r-r-r-,.--,~--,--r~,.-~,---.-,---ro~--r-r-.-,Total NQ Number=132
8::;' 0.003 A Quadrant of the Ring: Section A..
i/v

I·~

e.!

i)

H-:1:.;.

50

100

250
200
150
Pathlength [m]

300

350

50

100

250
200
150
Pathlength [m]

300

350

on[i 0.002

tJtl)

"fJ

60.0010~ -0.001~tl)

13 -0.002----o-- 4x4: 2001104/16

C;i-o-- 4x4: 2000/ll/29_A~ -0.003----l>--4x4: 2000111!29_B:s -0.00~36 240 244 248 252 256 260 264 268
Fitting ID Number

Figure 4: Integrated skew quadrupole error distributions.

3.4 Betatron Functions
Figure 5 shows estimated betatron functions for a quadrantof the ring. The betatron functions were calculated under
the usual operation point (Vx=40.15, Vy=18.36),
assuming the small shift of a operation point does notaffect the error distribution seriously. In the figure, threekinds of distributions are shown together with the ideal
distribution. One is estimated with the 2 by 2 formalism
and others with the 4 by 4 one. In the 4 by 4 analysis,
two different measurement data-sets were used. We see
three kinds of distributions have a good agreement. The
r.m.s. horizontal and vertical deviations are respectively-8 % and - 6.5 %.
The agreement between the results obtained with 2 by 2and 4 by 4 formalisms shows the skew error fields arerelatively weak. This means 2 by 2 formalism, where theoff-diagonal parts of the one turn map are ignored, is good
approximation for the SPring-8 storage ring.

3.5 Coupling Resonance Excitation
By using the estimated skew quadrupole fields, we canpredict the excitation of linear coupling resonance lines
around the operation point. The calculated excitation
strength of the nearest resonance line, Vx - Vy =22 agrees
well with the value measured by the minimum tune
separation. As seen in Fig. 6, we find one differential andtwo sum coupling resonance lines near by the operation
point should be suppressed to achieve the small verticalemittance corresponding to the H- V coupling ratio of 0.1%.

Figure 5: Distribution of betatron functions. Upper andlower graphs stand for respectively horizontal and verticalbetatron functions.
c::0·.:::

0.01.s~o0.001

K~v =40.15---+-- Difference-o--SumXv y=18.36

0u

@
c::
00.0001"'0 ~ 10~ 50:::

4-<00+"'~ 10-6.a "'~""0~abllo;10-7::;s0>·.gd)

0:::

10- s
20
15
10
50-20 -15 -10 -5
I =V +V -58 orl = v - v -22
sumydiff

yFigure 6: Excitation of linear coupling resonance near by
the operation point.

4 REFERENCES[1] W.J. Corbett, M.J. Lee and V. Ziemann, SLAC-PUB6111 (1993).
[2] J. Safranek, N.I.M. A388 (1997) 27.
[3] D. Robin, G. Portmann, H. Nishimura and J. Safranek,Proc. 5th EPAC, Barcelona, 10-14 June 1996, p.971.
[4] G. Liu, K. Kumagai, N. Kumagai, H. Ohkuma, K.
Soutome, M. Takao and H. Tanaka, Proc. 18th PAC, New
York, 29 Mar.-2 Apr., 1999, p.2337.
[5] H. Koiso, OH0'91, p.I-1-p.I-22 (1991) in Japanese, K.
Ohmi, OH0'91, p.II-1-p.II-33 (1991) in Japanese.

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