Measurement and correction of coupling in BEPC

CPC(HEP & NP), 2011, 35(12): 1143–1147 Chinese Physics C Vol. 35, No. 12, Dec., 2011Measurement and correction of coupling in BEPC *ZHANG Yuan() 1) YU Cheng-Hui() JI Da-Heng()XU Gang() WEI Yuan-Yuan() QIN Qing()Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaAbstract: The existing linear coupling theory and representation method are introduced briefly. The socalledlocal and global coupling is discussed in more detail. The vertical orbit distortion excited by a horizontalcorrector is represented with the coupling parameters at the corrector and the observation point. The formulais used to measure the coupling in BEPC. In order to correct the coupling, vertical correctors are used tochange the vertical orbit through sextupoles by a least square method. We also introduce and review otherfrequently used coupling measurement/tuning methods used in our machine.Key words:BEPC, coupling, luminosity optimizationPACS: 29.20.db DOI: 10.1088/1674-1137/35/12/0121 IntroductionQuadrupole alignment errors and the non-zeroclosed orbit in the sextupoles will introduce couplingbetween horizontal and vertical directions in a storagering. The detector solenoid in a collider can also leadto very strong coupling. The solenoid field compensationis a very important problem to be consideredduring the design process, since transverse betatroncoupling can impact the machine performance of acollider seriously.The coupling “contributes” to the luminosity bylocal coupling at the IP and global coupling. The fourlocal coupling parameters can be represented by theEdwards-Teng-Billing parameters [1, 2]. Global couplingis often called emittance coupling ɛ y /ɛ x . In factthe vertical emittance comes not only from the betatroncoupling but also from the vertical dispersion inan electron circular machine where the impact of radiationcannot be ignored. That is to say we need tominimize the local coupling parameters and verticaldispersion along the ring in order to reduce the verticalemittance. It is much more important to reducethe vertical dispersion in a low emittance machinesuch as the ILC/SuperB damping ring [3, 4].It would be ideal if we could tune local and globalcoupling independently. However, the BEPC storagering is very compact and there is no space to installmagnets which could serve as the knobs. Thereare 4 skew quadrupoles in total in each ring, wheredispersion is nearly zero. It seems that we cannottune the local coupling, the dispersion at the IP andthe vertical emission separately. That is to say, themachine tuning is much more difficult.In the following, we first introduce the normalmode analysis of a 4D one-turn map and present thevertical closed orbit distortion excited by the horizontalcorrector in a coupled machine. Then we obtainthe local coupling parameters along the ring bymeasuring the horizontal/vertical orbit distortion excitedby the horizontal dipoles. The local couplingparameters can also be obtained using a turn-by-turnbeam position monitor. Since the number of LiberaBPM [5] is very limited and the data are not well calibrated,so far we can not compare the results fromthe two methods. The familiar global coupling formulaand measurement method are also introducedand discussed. The frequently used measurement andcorrection methods during machine tuning in dailyoperation are introduced in the last part of the paper.2 Theory and simulation2.1 Local couplingWe follow the parametrization of coupling inReceived 24 February 2011* Supported by National Natural Science Foundation of China (10805051, 10725525)1) E-mail: zhangy@ihep.ac.cn©2011 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Instituteof Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

1144 Chinese Physics C (HEP & NP) Vol. 35Refs. [1, 2, 6]. The one-turn transfer matrix of 4Dphase space vector x = (x,x ′ ,y,y ′ ) is represented byT , written in terms of 2×2 submatrices,( )M mT = = V UV −1 , (1)n Nwhere( ) ( )A 0γI CU = ;V =; γ 2 +|C| = 1, (2)0 B −C + γIA is a 2×2 normal mode transfer matrix, which canbe represented by Courant-Snyder parameters,( )cos2πνA +α A sin2πν A β A sin2πν AA=.−γ A sin2πν A cos2πν a −α A sin2πν A(3)It is similar for B. The symplectic conjugate is( )C22 −CC + 12=. (4)−C 21 C 11In a weak coupling machine, mode A/B is similar tohorizontal/vertical oscillation, respectively. ν A /ν B isthe so-called horizontal/vertical tune. The one-turncoupled transfer matrix can be rewritten in the followingformT = G −1 ¯V Ū ¯V −1 G, (5)G is the Courant-Snyder transformation matrix,( ) ( √ )GA 01/ βu 0G = ; G u =0 G B α u / √ √ , (6)β u βu¯V = GV G −1 =(Ū = GUG −1 =γI−G B C + G −1A)G A CG −1BγI( )γI ¯C=− ¯C, (7)+ γI(R(2πνA ) 00 R(2πν B )), (8)where R is the rotation matrix.We restrict ourselves to a weak coupling machine,which is true in most cases. Then γ ≈ 1 along thering is a good approximation. The exception maybe in the IR region, where the detector solenoid andanti-solid field is very strong. When a horizontal correctorprovides a deflection θ x , the horizontal orbitdisplacement is∆x cod =θ x2sinπν x√βb,x β c,x cos(∆φ x −πν x ), (9)where ∆φ x is the horizontal betatron phase advancefrom the corrector to the observation point, β b /β c isthe beta function at the BPM and the corrector, respectively.The deflection in the horizontal direction wouldalso lead to a vertical orbit displacement along thering due to the transverse coupling. The vertical displacementcomes from two parts,1) the local coupling at BPMθ x√∆y cod,1 =− βc,x β b,y ¯Cb,22 cos(∆φ x −πν x )2sinπν xθ x√− βc,x β b,y ¯Cb,12 sin(∆φ x −πν x ).2sinπν x(10)We can notice that the first term on the right-handside may come from the tilt error of the BPM, sinceit is in phase with that of the horizontal distortion.2) the local coupling at the horizontal correctorθ x√∆y cod,2 =+ βc,x β b,y ¯Cc,11 cos(∆φ y −πν y )2sinπν y+ θ x2sinπν y√βc,x β b,y ¯Cc,12 sin(∆φ y −πν y ).(11)We can also notice that the first term on the righthandside may come from the tilt error of the corrector,since it is in phase with that of the verticaldistortion excited by a vertical corrector at the positionof the horizontal one. As a matter of fact, wedo not like to use the ¯C c,12 distribution as the couplingparametrization along the ring due to the finitelength of magnets, since most of the horizontal correctorsare implemented using the redundant coils inthe main bending magnets.According to the previous analysis, we can concludethat due to a horizontal deflection, ∆y codbpm is∆x codat a∆y cod∆x cod= ¯C b,22 k 1 + ¯C b,12 k 2 + ¯C c,11 k 3 + ¯C c,12 k 4 , (12)where the four coefficients k 1,2,3,4 are only related tothe decoupled linear optics of the machine. Afterthe machine is well calibrated by using the LOCOmethod[7], it is expected that we could use the theorymodel to calculate the coefficients.We performed done some simulations usingSAD [8]. The rotation angle along the s-axis ofquadrupoles is set randomly, then we knob the horizontalcorrectors and measure the transverse orbitdistortion. By analyzing the orbit response information,we can get ¯C b,12 at the BPMs. Tuning thevertical orbit by vertical correctors could change thecoupling, where the skewed quadrupoles come from

No. 12 ZHANG Yuan et al: Measurement and correction of coupling in BEPC 1145the vertical displacement through sextupoles. Fig. 1shows the real value of ¯C b,12 before/after correction.In the simulation, the emittance coupling is 6% beforecorrection and reduces to 0.9% after correction.Fig. 1. Simulation study of local coupling measurementand correction. The horizontal axisis the BPM index and the vertical axis is the¯C b,12 . The figure shows the real value before(dot) and after (square dot) correction.The local coupling parameters could also be measuredbe using a turn-by-turn beam position monitor[9]. When only the A mode is excited, the transverseturn-by-turn beam position can be written asx(n) = ɛ A γ √ β A cos(2πnν A ),y(n) = ɛ A√βB√¯C 2 12 + ¯C 2 22 cos(2πnν A +∆φ A ), (13)where n is the turn number, andcos∆φ A ≡− ¯C 22√ ¯C2 12 + ¯C;222¯C 12sin∆φ A ≡ √ ¯C2 12 + ¯C.222Inverting these expressions gives√β(A y)¯C 12 = γsin∆φ A ,β B x A¯C 22 = −γ√β(A y)cos∆φ A ,β B x A(14)(15)where (y/x) A is the ratio of the y amplitude to the xamplitude for the A mode.Similarly if only the B mode is excited, we couldmeasure the two local coupling parameters ¯C 12 and¯C 11 . Since the horizontal beam size is much largerthan the vertical one in a normal electron/positronstorage ring, the small amplitude oscillation in thehorizontal direction is more difficult to measure. Generallyspeaking it is not a good idea to measure thelocal coupling by exciting the B mode.2.2 Global couplingIn the smooth approximation and assuming a uniformlydistributed skew quadrupolar field, analyticformulas are derived for a constant focusing latticewhose s-dependent coupling strength j(s) is replacedby a uniform coupling strength given by [10]C = − 12π∮dsj(s)√β x (s)β y (s)e −i[φx(s)−φy(s)]+i(s/R)∆ ,(16)where R is the machine radius, s is the longitudinalcoordinate, β and φ are the Twiss parameters of theuncoupled lattice, and ∆ the fractional distance fromthe difference resonance. Up to the first order, |C|is equivalent to the “tune difference on the couplingresonance”, ∆Q min (also known as the “closest tuneapproach”). Their transverse RMS emittances arecoupled according toɛ x = ɛ x0 −|C|2 ɛ x0 −ɛ y0, (17)∆ 2 +|C| 2 2ɛ y = ɛ y0 +|C|2 ɛ x0 −ɛ y0. (18)∆ 2 +|C| 2 2It should be mentioned that the equations are givenin the absence of synchrotron radiation. Recent work[11] finds that the emittance sharing and exchangedriven by linear betatron coupling without radiationis much more complicated. In many cases we wouldlike to use a more simplified equation for eliminatingvertical dispersion even with radiation [12],ɛ y |C| 2=ɛ x 2∆ 2 +|C| . (19)2The minimum tune split could also be represented bythe matrix formalizm[13],∆Q min =det|m+ +n|π(sinµ A +sinµ B ) . (20)The most common way to adjust the coupling isto globally decouple the machine. The strengths ofnormal quads are adjusted to bring the normal modetunes together. Typically the tunes cannot be madeequal. The minimum tune split is used as a measureof the coupling in the machine. Skew quads are thenused to minimize the tune split. Finally the normalquads are returned to their original values to bringthe machine back to its normal operating point.Bagley & Rubin have pointed out that globalcoupling is ill-defined [9], because the change in the(m + +n) as the tunes are brought together dependson the detailed changes in the phase advances betweenthe skew quads and the observation point. This

1146 Chinese Physics C (HEP & NP) Vol. 35in turn depends on which quads are used to bring thetunes together. Since most circular machines run nearthe coupling Q x = Q y area, where the resonance densityis low, the global coupling method helps to someextent. Detailed coupling correction needs more detailedwork and we have to correct the local couplingalong the ring.3 Daily operation in BEPCLuminosity performance strongly depends on themachine condition. The coupling correction is animportant knob for optimization during daily runs,which includes the vertical emittance reduction, thelocal coupling and even vertical dispersion parametertuning at the IP.There are four skew quadrupoles in the arc, wheredispersion is nearly zero. They could be used to tunethe so-called global coupling parameters. At first, thepower supplies are unidirectional and we seldom usethe tuning method. In that case, we use the verticalorbit bump near the sextupoles as an alternative.The 3-bump vertical orbit knob near S5 (which is themost strongest sextupole in the arc) could help ustune the luminosity. It is unfortunate that we do notknow for sure how the vertical emittance, local couplingor even dispersion distortion at the IP help us,we just know that they really help.Since the autumn of 2010, the 4 skew quadrupolesin each ring have commenced operation. Most oftenthey work as a knob of the local coupling parameterat the IP, however both the global and local couplingsare changed. Since the dispersion is nearly zero at theskew quadrupoles, we ignore the vertical dispersiondistortion caused by the knob.The minimum tune split method is used to estimatethe emittance coupling in BEPC, since thebeam profile monitor is not well calibrated before October,2010. One quadrupole in the injection regionis used to change the transverse tune. The minimumtune separation is usually in the range of 0.01–0.02,and the coupling ɛ y /ɛ x is estimated about 1.0%–2.0%.The local vertical orbit 3-bump near S5 is used tochange the tune separation and sometimes the minimumsplit can achieve 0.005. During the luminosityoptimization, we do not care about the tune separationvery much. In most cases we just changethe bump and try to increase the luminosity. Asmentioned before, the one-quadrupole knob methodcauses beta-beating along the ring and the minimizedcoupling may be enlarged when the quadrupole is resumed.When the machine runs near the half-integer region,where the luminosity is higher, the minimumtune split method does not work since it will have tocross the synchro-betatron resonance line 2ν x +ν s = nwhen the transverse tunes move close to each other.Fischer [14] has developed a method for global couplingmeasurement and correction, which is based onN-turn maps fitted from turn-by-turn beam positiondata, where tune change is not needed. We have performedrelated simulations in our machine. It seemsthat the method does not work well. When we try toreduce the the so-called tune-split(calculated by oneturnmap) at some points, it seems that the emittancecoupling does not drop. We did not find a“golden” point where we could tune the coupling byits det|m + +n| value. Maybe because the machine isfar from a smooth one. That may show that the minimumtune split method is not a good approximationfor us.The local coupling is obtained by analyzing thetransverse orbit response to the horizontal correctors.When the detector’s solenoid is off, the method istested in the BER (the electron ring of BEPC) andthe result is shown in Fig. 2.Fig. 2. Local coupling measurement and correctionin the BER with the solenoid off. Thehorizontal axis is the BPM index and the verticalaxis is the ¯C b,12 .With the first correction, the vertical beam size dropsfrom 945 to 790 (random unit), that is to say, the verticalemittance drops to 70% of the original. We alsomeasure the Touschek lifetime by measuring the lifetimeversus the bunch current and obtain that theinverse of Touschek lifetime ratio is 33:48:95 amongthe original, the 1st and 2nd corrections. Since theTouschek lifetime τ ∝ σ x σ y σ z , we could also obtainthat the vertical emittance drops by about 50% afterthe first correction with the lifetime data. Theminimum tune split is also measured, it drops from

No. 12 ZHANG Yuan et al: Measurement and correction of coupling in BEPC 114712 kHz to 4 kHz for the 1st and 1 kHz for the 2ndcorrection. It is estimated that the vertical emittancecontributed by coupling drops by about 90% after the1st correction where the Eq. (19) is used. In spiteof the bad coincidence between different estimationmethods, we could conclude that the coupling is significantlyreduced in the ring.When the detector solenoid and the correspondinganti-solenoid are on, the correction method has alsobeen used successfully in our machine. Fig. 3 showsa typical example, where the rms size of ¯c 12 has beenreduced by about 80% to 0.009.The 4 transverse coupling parameters at the IPcould be tuned with the 4 skew quadrupoles. An alternativemethod is used to change the vertical orbitin the sextupoles, where the vertical dispersion at theIP is also disturbed. That is to say, we could tune fourx-y coupling parameters and two vertical dispersionsat the IP by changing the vertical orbit in sextupoles.It is expected that the tuning would not change theorbit at the IP, injection region, RF region and northIP (where the two colliding beams are separated verticallyto avoid the parasitic beam-beam effect). Thepreliminary experiments show that the method is notfeasible since the vertical orbit distortion could notbe ignored.4 Summary & discussionFig. 3. Local coupling correction with thesolenoid on. The horizontal axis is the BPMindex and the vertical axis is the ¯C b,12 .The error of local coupling measurement comesfrom the beta beating from the theory model and theBPM resolution. The sextupole and vertical magnetstrength error would also contribute to the correctionerrors. Since the wiggler 4w1 is also running in theBER, the LOCO method has shown that the opticdistortion is more serious in the BER than that inthe BPR (the positron ring of BEPC).In BEPC the solenoid of the detector is wellcompensated by the anti-solenoid on both sides of theIP [15, 16]. It is believed that the coupling mainlycomes from alignment error and the vertical orbitthrough the sextupoles. Coupling tuning knobs areindispensable to the luminosity optimization.In this paper we present a formula by which wecould obtain the local coupling parameters at theBPM using the orbit response information insteadof the single-pass BPM. This method has been successfullyused to measure and correct the couplingin our machine. We also introduce other couplingknobs for normal operation. The local coupling atthe IP and the global coupling could not be tunedindependently in BEPC, which would bring moretrouble to the luminosity optimization. In our localcoupling correction, so far we have not considered thevertical dispersion correction.We wish to thank the members of the BEPCcommissioning group for their hard work.References1 Edwards D, Teng L, IEEE Trans. Nucl. Sci., 1973, 20: 32 Billing M. Cornell Report No. CBN 85-2, 19853 Panagiotidis K, Wolski A, Korostelev M. Proceedings ofIPAC10, Kyoto, Japan, 46024 Liuzzo S M, Biagini M E, Raimondi P, Donald M H. Proceedingsof IPAC10, Kyoto, Japan, 15305 Instrumentation Technologies, http://www.i-tech.si6 Sagan D, Rubin D. Phys. Rev. ST-AB, 1999, 2: 0740017 WEI Y Y. Analysis of BEPC Optics Using Orbit ResponseMatrix, Proceedings of PAC2007, Albuquerque,USA8 Hirata K. An Introduction to SAD, CERN 88-04, 19889 Bagley P, Rubin D. Proceedings of PAC 1989, 87410 Minty M, Zimmermann F. Measurement and Control ofCharged Particle Beams. Berlin: Springer, 2003. ISBN 3-540-44197-511 Franchi A, Metral E, Tomas R. Phys. Rev. ST-AB, 2007,10: 06400312 Guignard G. Phys. Rev. E, 1995, 51: 6104–611813 Schanchinger L, Talman R. Manual for the ProgramTEAPOT, Noninteractive FORTRAN Version, AppendixG, 199614 Fischer W. Phys. Rev. ST-AB, 2003, 6: 06280115 YU Cheng-Hui, WU Ying-Zhi, XU Gang. HEP & NP, 2005,29: 418–423 (in Chinese)16 YU Cheng-Hui et al. Nucl. Instrum. Methods A, 2009, 608:234–237