Through horizontal and vertical excitations, we have been able to make a precision measurement of linear geometric optics parameters with a Model-Independent Analysis (MIA). We have also been able to build up a computer model that matches the real accelerator in linear geometric optics with an SVD-enhanced Least-square fitting process. Recently, with the addition of longitudinal excitation, we are able to build up a computer virtual machine that matches the real accelerators in linear optics including dispersion without additional fitting variables. With this optics-matched virtual machine, we are able to find solutions that make changes of selected normal and skew quadrupoles for machine optics improvement. It has made major contributions to improve PEP-II optics and luminosity. Examples from application to PEP-II machines will be presented

Cai, Y.

Colocho, W.

Decker, F. -J.

Seeman, J.

Sullivan, M.

Turner, J.

Wienands, U.

Woodley, M.

Yan, Y. T.

Yocky, G.

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Work supported  in part by Department of Energy contract DE-AC02-76SF00515SLAC-PUB-11925PRECISION MEASUREMENT AND IMPROVEMENTOF e+, e- STORAGE RINGSYiton T. Yan, Y. Cai, W. Colocho, F-J. Decker, J. Seeman, M. Sullivan,J. Turner, U. Wienands, M. Woodley, G. YockyStanford Linear Accelerator Center, Stanford University, Stanford, CA 94309AbstractThrough horizontal and vertical excitations, we have been able to make a precisonmeasurement of linear geometric optics parameters with a Model-IndependentAnalysis (MIA). We have also been able to build up a computer model thatmatches the real accelerator in linear geometric optics with an SVD-enhancedLeast-square fitting process. Recently, with the addition of longitudinal excita-tion, we are able to build up a computer virtual machine that matches the realaccelerators in linear optics including dispersion without additional fitting vari-ables. With this optics-matched virtual machine, we are able to find solutionsthat make changes of selected normal and skew quadrupoles for machine opticsimprovement. It has made major contributions to improve PEP-II optics andluminosity. Examples from application to PEP-II machines will be presented.presented at 10th EUROPE Particle Accelerator ConferenceEPAC’06, 26-30 June 2006, Edinburgh, UKJune 2006PRECISION MEASUREMENT AND IMPROVEMENTOF e+, e- STORAGE RINGS ∗Y.T. Yan, Y. Cai, W. Colocho, F-J. Decker, J. Seeman, M. Sullivan,J. Turner, U. Wienands, M. Woodley, G. YockySLAC, Stanford, CA 94309, U.S.A.AbstractThrough horizontal and vertical excitations, we havebeen able to make a precision measurement of linear geo-metric optics parameters with a Model-Independent Anal-ysis (MIA) [1]. We have also been able to build up a com-puter model that matches the real accelerator in linear ge-ometric optics with an SVD-enhanced Least-square fittingprocess [2]. Recently, with the addition of longitudinal ex-citation, we are able to build up a computer virtual machinethat matches the real accelerators in linear optics includ-ing dispersion without additional fitting variables. Withthis optics-matched virtual machine, we are able to findsolutions that make changes of selected normal and skewquadrupoles for machine optics improvement. It has mademajor contributions to improve PEP-II optics and luminos-ity. Examples from application to PEP-II machines will bepresented.INTRODUCTIONTo improve the optics of e+, e- storages rings, it would bevery helpful if one has an accurate lattice model. In somecases, the designed ideal lattice may serve such purpose tosome extent. However, in most cases, real accelerator op-tics improvement requires accurate measurement of key orall optics parameters. In this paper we present a completelinear optics model that can match the real machine linearoptics. We call such a complete model a virtual machine.The virtual machine model starts with the ideal designlattice as its initial state. All quadrupole strengths, sex-tupole feed-downs, all BPM gains, and BPM cross-planecouplings are then considered as variables to fit a completelinear-optics set of well chosen quantities that are obtain-able from both the computer virtual machine and the realmachine measurement. To interpret the above in short, onecan have a simple mathematical formulae,~Y ( ~X) = ~Ym, (1)where all variables are represented by the array (a vec-tor) ~X ; the to-be-fitted quantities obtained from the vir-tual model and their corresponding quantities from mea-surement are represented by the array ~Y and the array ~Ymrespectively. One may call that ~Y is the response to the~X and therefore is a vector function of ~X as is explicitlyshown in the equation.The response quantities, we have chosen for geometricoptics, are the local Green’s functions and the phase ad-∗Work supported by DOE contract DE-AC02-76SF00515.vances among BPMs. The local Green’s functions are sim-ply the transfer matrix components, R12, R34, R32, R14,between any two BPMs. There are essentially unlimited ofsuch Green’s functions that help in fitting convergence andaccuracy. We could also choose eigen coupling ellipses’ tiltangles and axis ratios at all double-view BPMs location asresponse quantities. However, we usually leave these cou-pling quantities alone for after-fitting check to see if theyautomatically match between the virtual machine and thereal machine measurement to make sure the virtual modelis indeed accurate.The above variables and response quantities form a com-plete fitting system for geometric optics. Therefore, if weinclude linear dispersions at BPMs as response quantities,we should add suitable bending magnet strengths and/ororbit corrector strengths as fitting variables. However, wefound that once the geometric optics was fitted, the disper-sion was roughly matched between the virtual model andthe measurement for most cases. This encouraged us toconsider adding dispersion fitting without adding bendingor corrector magnet strengths as variables. Depending oneach case, we may turn on more normal quadrupole skewcomponents as variables to achieve dispersion fitting with-out hurting the geometric fitting.Once the virtual accelerator is obtained, instead of com-paring it to the ideal lattice model for finding and adjust-ing one or two magnets with noticeable differences, wewould go on with this virtual accelerator to search for aneasily-approachable better-optics model by pre-selectingand fitting a group of limited number of normal and skewquadrupole strengths and then create a machine operationknob for practicing the corresponding quadrupole strengthsadjustment in the real accelerator. These procedures havebeen successfully applied for PEP-II optics improvementand have made a major contribution for PEP-II luminos-ity enhancement. Examples from PEP-II results will beshown.PRECISION MEASUREMENTGEOMETRIC ORBITSLinear geometric optics is determined if one gets 4 inde-pendent linear orbits. This can be shown by the obtainablelinear mapping, Zb = RabZa, and so Rab = ZbZa−1,where the 4-by-4 matrix, Za = [~za1 , ~za2 , ~za3 , ~za4 ], represents4 independent linear orbits at location a, and Rab is thelinear map from location a to location b. Therefore, a com-plete geometric set of data must be able to provide the ex-traction of 4 independent orbits.To offset radiation damping, the most economic processfor such data acquisition would be through two orthogonalresonance excitations, one at the horizontal (eigen-plane 1)and the other at the vertical (eigen-plane 2) betatron tunes,and then take and store buffered BPM data. Since a beta-tron motion has two degrees of freedom (the phase and theamplitude), each excitation would generate a pair of conju-gate (cosine- and sine-like) betatron motion orbits. Theyare obtained from the real and imaginary parts of tune-matched FFT respectively. Therefore, a complete set of 4independent linear (X and Y) orbits can be extracted fromthe two eigen-mode excitations.DISPERSIONTo compliment the above linear geometric data acqui-sition, longitudinal oscillation at the synchrotron tune isalso resonantly excited for an additional transverse BPMdata acquisition. Dispersions at BPM locations are thenmeasured by taking a longitudinal-tune-matched (zoom-ing) FFT from such BPM turn-by-turn BPM data.RESPONSE QUANTITIES AND THEIRCORRESPONDING QUANTITIES FROMMEASUREMENTOnce the variables in the virtual lattice model, that is ~Xin Eq. 1, is given, one can update the virtual lattice trans-fer matrices. The response quantities (~Y in Eq. 1), that is,the phase advances and the Greens’ functions among BPMsand the dispersions at BPM locations, are then calculatedby projection of these updated transfer matrices or the con-catenated one-turn linear maps. Their corresponding quan-tities from measurement (~Ym in Eq. 1) are described below:Phase advancesThe orbit betatron phase at each BPM location can beobtained by taking the arctangent of the ratio of the imagi-nary part to the real part of the resonance excitation FFTmode [2]. Phase advances between adjacent BPMs canthen be calculated by subtraction. Note that the ratio of theimaginary part to the real part of the FFT will cancel the lin-ear BPM gains but not the BPM cross couplings. Thereforethe phase advances among BPMs are repeatedly calculatedduring the Least Square fitting process as the BPM crosscouplings and BPM gains are updated to correct the linearorbits.Linear Green’s functionsThe linear Green’s function are simply theRab12, Rab34, Rab14, Rab32 of the linear transfer matrix be-tween any two BPMs labeled as a and b. They aregiven in the data measurement space [3] and so to matchthese measured quantities, the variables for BPM gainsand cross couplings have to be applied to the responseGreens’ functions from the updated virtual model for theirtransformation into the data measurement space.MEASUREMENT AND IMPROVEMENTOF PEP-II STORAGE RINGSOnce the optics-matched virtual machine is obtainedthrough an SVD-enhanced Least-Square fitting [2], the up-dated transfer matrices can be concatenated into one-turnmaps at the desire locations for calculating optics parame-ters. One can also find solutions by fitting a well selectedset of normal and skew quadrupoles as well as orbit cor-rectors for improving the optics, such as reducing the betabeating and the linear coupling, optimizing beta functionsat IP, bringing the working tune to near half integer, andimproving dispersion. Furthermore, this virtual model canfeed to the lattice program LEGO and the beam-beam sim-ulation [4].Shown in Figure 1 is the PEP-II HER beta functions onNov. 22, 2005, which shows high beta beat and was sub-sequently corrected through the solution from the MIA vir-tual model. Shown in Figure 2 is the PEP-II beta functionon Mar. 16, 2006, showing that the beta beat had beenmuch improved. From the MIA accurate virtual machine,we have been able to identify a key magnet (QF5L). Thisnormal quadrupole along with the linear trombone quadsand local and global skews are used as variables in the MIAprogram for finding the solution from the virtual model.The solution is then dialed into the PEP-II HER.As mentioned above, we have been able to include dis-persion measurement in the virtual model without addingnew type of variables. Figure 3 compares dispersion fromthe virtual model and from the direct measurement for HERon Nov. 22, 2005. There is no bending magnet or orbit cor-rector involved in the fitting. The vertical dispersion beatwas subsequently improved with the MIA virtual models.MIA virtual model has also been applied to PEP-II LER.As an example, PEP-II LER major orbit steering usuallyaccompanied by a much degraded linear optics due tochange of sextupole feed-downs, which had been very dif-−500 0 500 1000 1500020406080β x (m)−500 0 500 1000 1500020406080β y (m)distance from IP(m)Figure 1: Comparing beta function between the ideal lat-tice (blue color) and the virtual machine on Nov. 22, 2005for PEP-II HER. The PEP-II HER showed high beta beat,which were subsequently corrected through solution fromthe virtual model−500 0 500 1000 1500020406080β x (m)−500 0 500 1000 1500020406080β y (m)distance from IP (m)Figure 2: Comparing beta function between the ideal lat-tice (blue color) and the virtual machine on Mar. 16, 2006for PEP-II HER. Beta beat shown in Fig. 1 has been muchimproved. From the accurate virtual machine, we havebeen able to identify a key magnet (QF5L). This normalquadrupole along with the linear trombone quads and lo-cal and global skews are used as variables for finding asolution from the virtual model.ficult to correct without an accurate optics model. With theaccurate MIA virtual model established for the LER rightafter the steering, we have been able to correct the linearoptics such that the major LER orbit steering in April, 2006is survived. Figure 5 shows the LER linear coupling char-acteristics after dialing in solutions right after the majororbit steering.CONCLUSIONWe have used a model-independent analysis (MIA) foraccurate orbit and phase advance measurement and thenuses an SVD-enhanced Least Square fitting for building ac-curate virtual models for e+, e- storage rings. MIA virtual−500 0 500 1000 1500−0.500.511.5dispx(m)−500 0 500 1000 1500−0.200.2dispy(m) Disptance from IP(m)Figure 3: Comparing dispersion between the direct mea-surement (green color) and the virtual machine on Nov. 22,2005 for PEP-II HER. No bending magnet or orbit correc-tor were added as fitting variables. The vertical dispersionbeat was subsequently improved with the MIA virtual mod-els.−1000 −500 0 500 1000−100−50050100tilt angle 1 (deg)−1000 −500 0 500 1000−100−50050100tilt angle 2 (deg)−1000 −500 0 500 100000.20.40.60.81(b/a) 1Distance from IP−1000 −500 0 500 100000.20.40.60.81(b/a) 2Distance from IP(m)Figure 4: Comparing linear coupling between the ideal lat-tice (blue color) and the virtual machine on Apr. 21 forPEP-II LER after a major orbit steering that was accom-panied by a MIA solution for linear optics correction. ThisPEP-II LER coupling is with a record low residual from theideal lattice. Top plot shows the Eigen ellipse tilt angleswhile the bottom plot shows the Eigen ellipse axis ratiosfor Eigen plane 1 and 2 respectively.model matches, very well, the real-machine linear opticsincluding dispersion. It has successfully fixed PEP-II betabeat, linear coupling, half-integer working tune. The suc-cess comes from that: (a) the SVD-enhanced Least-Squarefitting can avoid degeneracies and has a fairly fast conver-gence rate allowing for application to a fairly large system;(b) the PEP-II ring has a reasonable amount of BPMs al-lowing for extracting sufficient physical quantities for fit-ting; and (c) the linear Green’s functions among BPMs canprovide essentially unlimited fitting constraints that addsignificantly on the convergence.ACKNOWLEDGEMENTWe thank Alex Chao, John Irwin and Yuri Nosochkovfor stimulating discussions.REFERENCES[1] J. Irwin, C.X. Wang, Y. Yan, et. al., “Model-IndependentBeam Dynamics Analysis,” Phys. Rev. Lett. 82, 1684 (1999).[2] Y.T. Yan, Y. Cai, “Precision PEP-II optics measurementwith an SVD-enhanced Least-Square fitting,” Nucl. Instrum.Meth. A555:336-339, 2006.[3] J. Irwin, and Y.T. Yan, “Beamline model verification us-ing model-independent analysis” SLAC-PUB-8515 (2000),in Proc. EPAC2000, P. 151 (2000). Y.T. Yan, Y. Cai, J. Irwin,and M. Sullivan, “Linear Optics Verification and Correctionfor the PEP-II with Model-Independent Analysis,” SLAC-PUB-9368 (2002).[4] Y. Cai, et al., “Luminosity improvement at PEP-II based onoptics model and beam-beam simulation,” in these proceed-ings: MOPLS052.

Precision Measurement and Improvement of e+, e- Storage Rings

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Precision Measurement and Improvement of e+, e- Storage Rings

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