Low-emittance tuning of storage rings using normal mode beam position monitor calibration

Transcription

1 PHYSIAL REVIEW SPEIAL TOPIS - AELERATORS AND BEAMS 4, 784 () Low-emittance tning of storage rings sing normal mode beam position monitor calibration A. Wolski* Uniersity of Lierpool, Lierpool, United Kingdom D. Rbin, D. Sagan, and J. Shanks LASSE, ornell Uniersity, Ithaca, New York 4853, USA (Receied 4 April ; pblished 6 Jly ) We describe a new techniqe for low-emittance tning of electron and positron storage rings. This techniqe is based on calibration of the beam position monitors (BPMs) sing excitation of the normal modes of the beam motion, and has benefits oer conentional methods. It is relatiely fast and straightforward to apply, it can be as easily applied to a large ring as to a small ring, and the tning for low emittance becomes completely insensitie to BPM gain and alignment errors that can be difficlt to determine accrately. We discss the theory behind the techniqe, present some simlation reslts illstrating that it is highly effectie and robst for low-emittance tning, and describe the reslts of some initial experimental tests on the esrta storage ring. DOI:.3/PhysReSTAB PAS nmbers: 9.. c, 9.7. a I. INTRODUTION Lepton storage rings often reqire tning to achiee ery low ertical emittance: in light sorces, this increases the beam brightness; in colliders, it improes lminosity. A nmber of facilities in recent years hae reported ertical emittances of a few picometers [ 4]. Achieing emittances in this range reqires precise correction of machine errors that generate ertical dispersion and betatron copling. A common techniqe is to perform an analysis of the measred orbit response matrix (ORM) to determine the sorces of error [5]. This incldes diagnostic errors sch as tilts of the beam position monitors (BPMs). Howeer, data collection for ORM analysis is often rather slow, as it inoles measring the change in the closed orbit in response to each of the horizontal and ertical orbit correction (steering) magnets in the ring. In a large ring, there may be dozens of sch orbit correctors. Fitting a model to the measred data (to determine the errors present in the ring) inoles nmerically intensie comptation, and can also be time consming. For large rings, sch as those proposed, for example, for the damping rings of the International Linear ollider [6], it may become impractical to perform ORM analysis on a rotine basis to achiee and maintain the specified ertical emittance. In this paper, we describe an alternatie techniqe for low-emittance tning, based on excitation of the normal modes of the beam motion. Measrement of the trn-bytrn signals on the BPM bttons dring resonant excitation *a.wolski@lierpool.ac.k Pblished by the American Physical Society nder the terms of the reatie ommons Attribtion 3. License. Frther distribtion of this work mst maintain attribtion to the athor(s) and the pblished article s title, jornal citation, and DOI. allows the BPMs to be calibrated in terms of the local normal mode axes. Measrement and correction of the ertical mode dispersion then proides a rote to minimizing the ertical emittance. Since the BPMs are calibrated directly from the beam motion, the techniqe is intrinsically insensitie to BPM gain and alignment errors. Frthermore, gien the appropriate hardware for resonant excitation and trn-by-trn beam position measrements, data collection and analysis are ery fast and ths the techniqe can be applied as easily to a large ring as to a small ring. In Sec. II we describe the techniqe in detail, and discss the nderlying theory. We illstrate its application by presenting simlation reslts in Sec. III. Some tests of the techniqe hae already been carried ot at ornell s esrta storage ring [7]: we present some releant reslts from these tests in Sec. IV. Finally, we discss some conclsions and otline possible frther work in Sec. V. II. THEORY A. Normal mode dispersion and emittance In a lepton storage ring, the beam is preented from damping to zero emittance by qantm excitation. onsider a particle following the closed orbit. In a magnetic field (e.g. in a dipole magnet), the particle will randomly emit photons; if the dispersion is nonzero, then the change in energy of the particle reslting from a photon emission will mean that the particle is no longer on the closed orbit. Instead, it has some betatron amplitde arond the closed orbit. The aerage of the betatron amplitde (more precisely, of the betatron action) oer all particles in the beam gies the beam emittance. Particle motion in a storage ring can be coneniently described in terms of action-angle ariables. In the present 98-44==4(7)=784() 784- Pblished by the American Physical Society

2 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () work, we consider only linear motion. The relationship between action-angle and artesian ariables is pffiffiffiffiffiffiffi x J I cos I p x p ffiffiffiffiffiffiffi J I sin y ~x p ffiffiffiffiffiffiffiffi I J p ¼ N p II cos II B y ffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiii sin N ~J; () II z A p J III cos III ffiffiffiffiffiffiffiffiffi A J III sin III where N is a matrix that transforms the single-trn transfer matrix M into a block-diagonal rotation R: N MN ¼ R: () The artesian ariables are defined as follows. x and y are, respectiely, horizontal and ertical transerse coordinates in a plane perpendiclar to the reference trajectory at a gien point s; p x and p y are the normalized transerse canonical momenta: p x ¼ m _x þ qa x (3) P (and similarly for p y ), where is the relatiistic factor for a particle with rest mass m and charge q, the dot denotes the time deriatie, A x is the x component of the electromagnetic ector potential, and P is a chosen reference momentm. The longitdinal coordinate is defined by z ¼ cðt tþ; (4) where t is the time that the particle arries in the plane perpendiclar to the reference trajectory at s, is the normalized elocity of a particle with momentm P, and a particle traeling along the reference trajectory with normalized elocity arries at the point s at time t. The energy deiation is defined by ¼ E P c ; (5) where E is the total energy of the particle (for a particle with the reference momentm, ¼ ). The beam emittances are gien by aeraging the oscillation amplitdes (betatron and synchrotron actions) oer all particles in the beam: k ¼hJ k i; (6) where k ¼ I, II, or III indicates the degree of freedom. Now consider a particle that ndergoes an instantaneos change in energy, for example, reslting from photon emission. The energy deiation changes by some amont. In the approximation that the photon is emitted directly along the instantaneos direction of motion of the particle (alid for ltrarelatiistic particles), the transerse artesian coordinates and momenta, and the longitdinal coordinate, do not change: ~x ~x þ : (7) A The change in the action-angle ariables may be expressed: ~J ~J þ N : (8) A The condition for there to be no excitation of the mode II emittance (generally identified with the ertical emittance) can then be written: N36 ¼ N 46 ¼ ; (9) where Nij is the ði; jþ component of the inerse of N. Generally, it is easier to gie a direct physical interpretation to the components of the single-trn transfer matrix M than to the components of the normalizing matrix N (which can be constrcted from the eigenales of M). For example, the components of M otside the block diagonals are related to betatron copling (M 3, M 4 etc.), to horizontal dispersion (M 5, M 6, etc.), and to ertical dispersion (M 35, M 36, etc.). Althogh the condition in Eq. (9) does represent a constraint on the components of M, it is not easy to express this constraint directly in terms of the components of M. In general, it is possible for eery component of M to be nonzero, while still satisfying Eq. (9). Physically, this wold correspond to a case where the dispersion is parallel to the axis for motion in mode I. Then, nder a change in energy of a particle, only mode I motion is excited. It shold be noted that, althogh we refer to axes corresponding to motion in each of the (transerse) normal modes, a beam excited in one or other normal mode will, in general, describe an ellipse, rather than a straight line, in coordinate space when obsered oer many trns at a particlar location in a storage ring. Properly, one can only speak of a mode axis in the special case that the ellipse has infinitesimal width. That is, the ellipse is essentially a line. Howeer, if the copling in the ring is not strong, then the ellipses described by normal mode motion will hae ery narrow width, and it is reasonable to associate the major axis of the ellipse with an axis of the normal mode motion. Some jstification for this assertion is gien in Sec. IV, where we present and discss the experimental reslts. Often, low-emittance tning procedres in storage rings address the ertical dispersion and betatron copling separately. Howeer, if one can measre (and minimize) the 784-

3 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () component of the dispersion along the mode II axis, then it shold be possible to minimize the ertical emittance by correcting a single qantity (the mode II dispersion) rather than two separate qantities. Besides offering a more direct and concise correction procedre, there is the possibility of achieing a better correction. With a limited nmber of correction elements in the ring, it is conceiable that the optimm conditions for low emittance may be achieed not by trying to minimize either (or both) the ertical dispersion and betatron copling, bt by trying to align the dispersion with the mode I axis. There are nmeros formalisms for describing copled motion in accelerator beam lines, inclding storage rings; see, for example, Refs. [8 ]. The significance of the mode II dispersion for the emittance is indicated by the fact that, in the formalism of [9], the mode II emittance in a storage ring may be estimated from II q I 5;II ; () I where q 3:83 3 m is the qantm constant, is the relatiistic factor of the beam, and I and I 5;II are synchrotron radiation integrals: I ¼ I ds ; I 5;II ¼ I H II ds: () jj3 is the radis of cratre of the orbit, and the lattice fnction H II is defined by H II ¼ II II þ II II II þ II II ; () where II, II, II, II, and II are generalizations of the ncopled Twiss parameters and dispersion fnctions to the copled case. In the case of weak copling, we can expect the Twiss parameters to take ales close to those in the case of zero (or corrected) copling; howeer, the mode II dispersion may be significantly different from the ertical dispersion. This may occr, for example, if copling rotates the mode II axis so that it has a horizontal component at a location where there is large horizontal dispersion: the mode II dispersion will then acqire a significant contribtion from the horizontal dispersion, een if the ertical dispersion is zero. It is important to note that the emittance compted sing Eq. () takes into accont the effects of both ertical dispersion and betatron copling. An experimental method to measre and minimize directly the mode II dispersion, rather than the ertical dispersion and betatron copling separately, therefore offers the possibility of an elegant and effectie procedre for low-emittance tning. B. Measrement of mode II dispersion The BPMs in a storage ring are sally designed to retrn the horizontal (x) and ertical (y) position of the beam with respect to a specified laboratory coordinate system. The coordinate ales are obtained by processing the signals on the bttons indced by a beam. Generally, the signals on the BPM bttons hae a nonlinear dependence on the position of the beam. The precise form of the dependence depends on the geometry of the bttons and the acm chamber. Howeer, if the beam position aries oer a small range, then the btton signals can be assmed to ary linearly with beam position. This assmption is needed for calibrating BPMs sing normal mode beam excitation. The jstification for this can be obtained from the simlation and experimental reslts. Alignment errors and errors in the signal amplification and processing electronics lead to systematic errors in the coordinate ales retrned by a BPM. For simplicity, we consider in this section a linear model in which the errors may be represented by a gain matrix, g:! x measred ¼ g y x y actal ; (3) where g ¼ g xx g xy : (4) g yx g yy Ideally, g is the identity matrix. In practice, the components of g can ary independently of each other. The ale of g yx is particlarly significant for correction of the ertical dispersion: if it is nonzero, then a prely horizontal motion of the beam reslting from a change in beam energy (as dring a dispersion measrement) can also appear in the measred y coordinate. An attempt to correct the measred ertical dispersion wold then introdce ertical dispersion where none was, in fact, present. There are arios techniqes that can be applied to determine the gain errors (either in a linear, or in a more general nonlinear model), inclding ORM analysis [5] and BPM gain mapping [,]. Measrements of the BPM gain errors will be alid only oer a limited period of time de to sch things as alignment changes, drifts in electronics, and changes in the orbit copled with the nonlinear response of a BPM, etc. Some of the measrement techniqes (e.g. ORM analysis) inole significant time and effort. Other techniqes, particlarly those based on analysis of trn-by-trn data, may be accomplished more qickly. Howeer, it will generally be beneficial to hae a range of methods aailable that may proide reslts that are either directly comparable or complementary. Gien the need to repeat measrements periodically becase of ariation in machine conditions, techniqes in which data may be collected within a few mintes hae a strong adantage. One sch techniqe is to calibrate the BPM from normal mode motion of the beam. By obsering trn-by-trn motion with the beam excited in one or the other transerse normal mode, it is possible to calibrate the BPMs so that the ales retrned are the beam coordinates along the 784-3

4 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () BPM btton 3 lab y BPM y Normal mode BPM btton 4 the excitation in each normal mode. For Þ is the change in the signal at btton per nit change in the signal at btton, for motion in mode I ( constant). Since BPM btton normal mode axes. As discssed in Sec. II A, these are the coordinates that are most releant for low-emittance tning. The releant coordinate systems that can be defined in a BPM are illstrated schematically in Fig.. onsider a BPM with for bttons. Let the signal on btton i when a bnch passes throgh the BPM be denoted b i (i ¼,, 3, or 4). If the copling is not too large, the beam motion in normal mode I will lie along a line that we refer to as the axis, and motion in mode II will lie along a line that we refer to as the axis. Assming a linear response, which will be the case if the amplitde of excitation is not too large, the relationship between a change ð; Þ of the beam from its closed orbit position and the corresponding change in the btton signals can be B b A @ BPM btton A lab x BPM x Normal mode FIG.. oordinate systems in a BPM. The laboratory coordinate system is defined with respect to the srey positions. BPM alignment and electronics errors mean that the coordinates retrned by the BPM are with respect to the BPM axes which are not perfectly aligned with the laboratory system. opling means that resonant excitation of the transerse modes leads to motion along axes different from the laboratory or BPM axes. The for BPM bttons are labeled according to the conention sed in ESR. : (5) The matrix elements ð@b i =@Þ and ð@b i =@Þ can be obtained by obsering the beam when the beam is shaken at the mode I and mode II freqencies ia resonant excitation of the beam. When the beam is shaken at a freqency corresponding to mode I, the position ð j ; j Þ on the j th trn will hae j ¼. Similarly, when shaking at the mode II freqency, j will be zero. When shaking the beam, the absolte changes and will not be known. Howeer, we can determine the correlations between the changes in btton signals from the readings taken ¼ @ Þ ; (6) we can write Eq. (5) in the form b A ¼ B ; (7) @ @ @ ; (8) A A: (9) The components of B can be determined by correlating the changes in signals measred on different bttons while resonantly exciting the beam in one or the other of the transerse normal modes. Withot knowing the absolte amplitde of the motion, the components of cannot be determined. Howeer, they may be estimated, e.g., from modeling the BPM response to horizontal and ertical beam motion. Errors in the ales for the components of will lead to systematic errors in the measrement of the normal mode dispersion. In particlar, an error Þ will lead to a systematic error, corresponding to some scale factor, in the measrement of the mode II dispersion. Howeer, since the ltimate goal is to achiee zero mode II dispersion, this scale factor is not expected to be significant. Pt another way, if the mode II dispersion is zero, then the systematic error will not affect the measrement. The systematic error will mean that the effect of the correction will not be exactly as expected, and some iteration will be needed; howeer, if the error is not too large, the correction shold still conerge. Gien ales for the components of B (from measrements) and (from modeling), we can determine and B (defined sch that B B ¼ I, with I being the identity matrix). Then 784-4

5 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () b ¼ B b A : () Eqation () represents a BPM calibration. It allows s to determine the changes in position along the normal mode axes in a BPM, from the changes in btton signals. Applied to the btton signals taken dring a dispersion measrement, Eq. () allows s to determine the normal mode dispersion. The dispersion measrement may be made in the conentional way, by obsering the change in beam position on the (calibrated) BPMs with respect to a change in beam energy indced by ariation of the freqency of the rf caities. Note that, since the BPM is calibrated directly from normal mode motion of the beam, the normal mode dispersion measrement will be insensitie to BPM tilt and gain errors. Sch errors can, in some circmstances, be a limiting factor in emittance tning sing conentional techniqes. Nonlinear ariation of the BPM bttons response to changes in beam position (i.e. changes that are large compared with the amplitde of the resonant excitation) may mean that the calibration is a fnction of the closed orbit position. The calibration may also be a fnction of sch things as the beam crrent, the temperatre of the electronics, etc. Howeer, since the calibration is straightforward and (with the appropriate hardware for performing a resonant excitation) can be accomplished ery qickly (within a few mintes), a calibration can be performed immediately before each dispersion measrement.. orrection of mode II dispersion and minimization of mode II emittance Once the BPMs hae been calibrated to allow measrement of the mode II dispersion, correction of the dispersion may be accomplished sing conentional techniqes. For example, a set of components (ertical steering magnets, and skew qadrpoles) that affect the mode II dispersion may be identified, and a response matrix between these components and the mode II dispersion at the BPMs constrcted sing a model, or from measrements in the machine. Inerting the response matrix (by singlar ale decomposition) and applying the inerted matrix to the measred mode II dispersion gies the changes reqired to correct the mode II dispersion. It is important to note that, typically, BPMs are located in field-free regions where synchrotron radiation is not generated. Howeer, the mode II emittance will be minimized by correcting the mode II dispersion in locations (dipoles, ndlators, and wigglers) where synchrotron radiation is prodced. In principle, this oght to be taken into accont in the correction scheme, e.g., by placing a higher weight on those BPMs that are closest to the points where large amonts of synchrotron radiation are prodced. In weighting the BPMs for the correction procedre, there shold also be some accont taken of the optics. The mode II dispersion at a location with large horizontal dispersion will show a strong response to een a small amont of copling: the copling tilts the normal mode axes, and een a small tilt of the mode II axis will lead to a large component of the horizontal dispersion appearing along the mode II axis. On the other hand, where the horizontal dispersion is small, the mode II dispersion will be relatiely insensitie to changes in copling. Lowemittance storage rings generally need low horizontal dispersion in the dipoles (and in any insertion deices); bt the horizontal dispersion may reach large ales between the dipoles. To minimize the mode II emittance, therefore, a larger weight shold be gien to the mode II dispersion close to the dipoles and insertion deices, both becase this is where the mode II emittance is generated, and becase this is where the mode II dispersion will likely be least sensitie to copling. In practice, we find from simlations of esrta that, althogh weighting the BPMs has some effect on the reslts of tning based on mode II dispersion correction, there does not seem to be a ery strong impact. It is possible that with frther optimization of the way in which the weighting is applied, or in other lattices, there may be a more significant effect. The reslts from simlations and experiments presented in the following sections hae been obtained withot applying any weighting to the BPMs. It shold also be noted that, in general, emittance is generated not jst by the dispersion II, bt also by gradient of the dispersion, II ; howeer, it is sally only possible to measre the dispersion and not its gradient. The inability to measre II may limit the effectieness of tning for low emittance by correcting the mode II dispersion. It may be hoped, howeer, that if the BPMs are not too far apart, then zero (or ery low) ales for II will imply low ales also for II. The possible effectieness of low-emittance tning based on correction of only II in esrta will be demonstrated in simlations in the following section. III. SIMULATIONS A. esrta To illstrate the se of normal mode calibration of the BPMs for low-emittance tning of a storage ring, we present the reslts of some simlations based on a model of esrta. The reslts of some experiments on esrta are presented in Sec. IV. esrta is a synchrotron with 768 m circmference that can store either electrons or positrons. The beam energy can be aried between and 5 GeV. For all the stdies described here, the energy was GeV. Typical lattice fnctions (Twiss parameters and dispersion) are shown in Fig.. With these lattice fnctions, the natral emittance of esrta at GeV is approximately.7 nm

6 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () 4 β (m) s (m).5 η x (m) s (m) FIG.. Lattice fnctions in esrta. Top: horizontal (black) and ertical (red) beta fnctions. Bottom: horizontal dispersion, with BPM locations indicated by ble circles, and skew qadrpole locations indicated by ertical dashed lines. The main prpose of esrta is to inestigate electron clod effects in a parameter regime with low ertical emittance (< pm). The storage ring was deeloped from the ESRc ring by a nmber of modifications, inclding remoal of the detector (LEO) and relocation of the sperferric damping wigglers to dispersion-free regions of the lattice. Since the original ESRc ring had not been designed to reach the low emittances needed for esrta, a significant part of the esrta program consisted of deelopment of techniqes for achieing ertical emittance below pm on a reliable and rotine basis. This incldes improement of the instrmentation inclding new BPM electronics capable of trn-by-trn measrements with resoltion of a few microns, and fast beam-size monitors to allow emittance measrements with resoltion of a few picometers. Simlations for this paper were carried ot in an extended ersion of AT [3], an accelerator modeling code rnning in Matlab. The model allows a range of magnetic field strength and alignment errors to be simlated, inclding skew qadrpole fields, tilts (arond the beam axis), and ertical displacements. The x and y coordinates (in the laboratory frame) of a particle (representing the centroid of a bnch) tracked throgh the lattice can be recorded oer any nmber of trns at any specified locations. Since the calibration techniqe that we apply is based on readings from indiidal bttons, the model btton signals were generated as fnctions of the beam x and y coordinates. The BPM coordinates can then be calclated as fnctions of the btton signals. By this means, the calibration process can be simlated, and the calibration matrices B and for each BPM calclated from the reslts of tracking oer mltiple trns in the model. B. BPM model The calibration techniqe that we describe here depends on the BPM btton signals haing a linear dependence on the beam position, oer the range of beam position ariations that occr dring the calibration process. Howeer, oer the range of possible beam positions, the BPM btton signals hae a complicated nonlinear response, that depends on the geometry of the acm chamber in the region of the BPM, and on the geometry of the bttons themseles. To inclde nonlinear effects in the simlations withot needing to implement the fll complex dependence of the btton signals on the beam position, we sed a simplified model in the simlations, in which the signal b i on BPM btton i is gien by b i ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; () ðx x i Þ þðy y i Þ where ðx; yþ are the beam coordinates (with respect to the reference trajectory) and ðx i ;y i Þ are the coordinates of the g i 784-6

7 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () btton location (again with respect to the reference trajectory). g i is a gain factor, that will inclde (for example) a dependence on the bnch charge. In principle, if the bnch charge is known accrately, then the signals from only two bttons are needed to determine the beam coordinates. Howeer, in the case that the bttons are symmetrically positioned at x i ¼x ; y i ¼y () (where for x i, the pls sign holds for i ¼ and i ¼ 4, and for y i, the pls sign holds for i ¼ 3 and i ¼ 4), then aeraging the btton signals leads to the following expressions for the beam coordinates: x ¼ g g g3 g4 þ ; (3) 8x y ¼ 8y g þ g g3 g4 : (4) BPM alignment (inclding tilt) errors may be implemented in the simlation by applying some appropriate transformation to the btton coordinates ðx i ;y i Þ in Eq. (), while contining to se the nominal ales for the btton coordinates in calclating the beam coordinates from the btton signals sing Eqs. (3) and (4). Similarly, gain errors may be implemented by applying some ariation to the ales of g i in Eq. (), while contining to se the nominal ales for the gain coefficients in calclating the beam coordinates from the btton signals sing Eqs. (3) and (4). Simlated calibration data, obtained by implementing Eqs. (), (3), and (4) in a tracking code, are shown in Figs. 3 and 4. The data were generated by determining the 6D phase space coordinates of a particle with, in this case, nonzero mode I oscillation amplitde (sing the eigenectors of the single-trn matrix), and zero amplitde in the other two oscillation modes. The particle was tracked for g =.59 g =.77 g =.7 g = 9.63 µm g =.544 µm g =.6 g =.5 5 g = 8.75 µm g =.54 µm g =.96 5 g = 7.56 µm g =.46 µm g = 7.98 µm g =.465 µm g = FIG. 3. Simlation of trn-by-trn btton signals on a BPM (7W). Each plot shows a correlation between two BPM btton signals or beam coordinates (retrned by the BPM), obtained by tracking a bnch performing coherent oscillations in normal mode I. An ideal model (with no errors) of the esrta lattice is sed: tracking data are shown as ble points; fitted ellipses are shown in red; the major axes of the ellipses are shown as black lines. The ellipses are more clearly isible when copling is present (see Fig. 4). Btton signals are calclated sing Eq. (), with relatie gain errors applied to the bttons as follows: g =g ¼ :63, g 3 =g ¼ :8, g 4 =g ¼ :7. The btton locations are gien by x ¼ mm, y ¼ mm. x and y beam coordinates are calclated from Eqs. (3) and (4), bt sing nominal gain ales g ¼ g ¼ g 3 ¼ g

8 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () g =.788 g =.956 g =.7 g = 8.7 µm g =.633 µm g =.6 g =.49 g =.36 µm g =.9 µm g = g = µm g =.843 µm 5 g = µm g =.38 µm 5 5 g = FIG. 4. As Fig. 3, bt with skew qadrpole 48W trned on with strength k ¼ :3 m. seeral hndred trns in an ideal model of the lattice, with the btton signals at the BPMs, calclated sing Eq. (), recorded on each trn. Figre 3 shows a set of correlation plots (between BPM btton signals, or beam coordinates) for BPM 7W with relatie gain errors applied to the bttons. The x and y beam coordinates are calclated from the btton signals by applying Eqs. (3) and (4), bt with nominal gain ales g ¼ g ¼ g 3 ¼ g 4. The relatie btton gain errors, sed for calclating the btton signals from the actal beam position in the simlation, hae been chosen roghly to reprodce the experimental data that we show later in Fig. 7. Note that there is reasonably good agreement between the simlation and the experimental data in the gradients of the correlations between the btton signals, and between the x coordinate and the btton signals. The agreement between the simlation and the experimental data in the gradients inoling the y coordinate, howeer, is poor: this is likely the reslt of copling in the real machine, which is not inclded in the model. Withot any copling or gain errors, there will be no ariation in the y coordinate reslting from oscillation in normal mode I. Figre 4 shows the reslts of the same simlation, except with one skew qadrpole (SKQ48W) trned on, with strength k ¼ :3 m. (Figre 4 is to be compared with Fig., which shows the corresponding experimental data.) With copling generated by the skew qadrpole, the beam motion in normal mode I is now no longer along a line in coordinate space: instead, the beam describes an ellipse, which reslts in ellipses clearly isible in the correlation plots between the btton signals. Again, there is good agreement in the gradients of the major axes of the ellipses, between the simlation (Fig. 4) and the experimental data (Fig. ), except for the plots inoling the y coordinate.. Emittance calclation Exact ales for the horizontal, ertical, and longitdinal emittances in the model, taking into accont field and alignment errors leading to dispersion and copling, are obtained in AT sing the enelope method [4]. As an initial test of the theory, the emittance for seeds of skew qadrpole errors distribted arond the ring was calclated in three different ways: () sing the enelope method; () estimated sing Eq. (); (3) estimated sing Eq. () bt with a modified ersion of Eq. (), in which 784-8

9 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () Relatie nmber of seeds Relatie nmber of seeds normal mode artesian V rf = 8. MV estimated emittance / actal emittance normal mode artesian V rf =.8 MV.5.5 estimated emittance / actal emittance FIG. 5. Emittance estimated from dispersion in esrta with (top) 8. MV rf oltage, and (bottom).8 MV rf oltage. The lines show the distribtion (oer seeds of machine errors) of the ratio of the estimated mode II emittance to the actal mode II emittance; the ertical axes are scaled so that each cre has nit area. In the normal mode case, the mode II dispersion is sed, and the emittance estimated sing Eq. (). In the artesian case, a modified ersion of Eq. () is sed, in which the mode II dispersion is replaced by the ertical dispersion in Eq. (). In both cases, the exact emittance is calclated sing the enelope method. The approximations inoled in Eq. () are alid for small ales of the synchrotron tne (which is.74 at 8. MV rf, and.3 at.8 MV rf). the mode II dispersion is replaced by the ertical (y) dispersion. Figre 5 shows the distribtion of the ratio of the estimated (methods and 3) to the exact (method ) mode II emittances oer seeds of random skew qadrpole errors. Reslts are shown for two cases: with the nominal rf oltage of 8. MV in esrta, and with a redced rf oltage of.8 MV. If we consider first the case of the redced rf oltage, we see that Eq. () gies a ery good estimate of the tre mode II emittance, while the emittance calclated from modified eqations (sing the ertical rather than the mode II dispersion) shows a ery large spread. This illstrates the fact that the mode II dispersion takes accont all of the significant contribtions to the emittance generation. For larger rf oltages, the spread in the reslts from Eq. () increases: this is becase the approximations reqired to derie this eqation start to break down. In particlar, the deriation assmes that any correlations between the longitdinal and transerse coordinates of particles in the bnch can be neglected. Althogh this is tre for most storage rings, correlations between longitdinal and transerse coordinates can appear when the synchrotron tne becomes large. A large synchrotron tne reslts from strong longitdinal focsing from the rf caities, and a large momentm compaction factor. If we consider an ellipse in longitdinal phase space, the rf focsing will gie a tilt to the ellipse, while the momentm compaction factor as the particles moe arond the rest of the ring will hae the effect both of adancing particles arond the ellipse (the longitdinal phase adance), and modifying the shape of the ellipse. The stronger the rf focsing, and the larger the momentm compaction factor, the larger the tilt of the longitdinal phase space ellipse will be. For a real distribtion, this indicates that there will be a significant correlation between the energy of a particle and its longitdinal position (coordinate) within the bnch. Where dispersion is present, this will in trn lead to a correlation between the transerse and longitdinal coordinates: this is not taken into accont in the deriation of Eq. (). In esrta, the synchrotron tne is abot.3 with an rf oltage of.8 MV, increasing to.74 at an rf oltage of 8. MV. The effect of the larger synchrotron tne on the accracy of Eq. () can be clearly seen in Fig. 5. The fact that the relationship between mode II emittance and mode II dispersion expressed by Eq. () starts to break down for large rf oltages, sggests that correction of the mode II dispersion may be a more effectie techniqe for correcting the emittance in storage rings at lower ales of the synchrotron tne. Howeer, the simlation reslts presented in the following section sggest that the techniqe shold still be sefl een in esrta with the nominal rf oltage. The prpose of the simlations described in the present section is principally to illstrate the releance of the mode II dispersion for the mode II emittance, bt the reslts also gie some indication of potential limitations of the se of mode II dispersion for low-emittance tning in particlar (somewhat nsal) parameter regimes. D. Emittance tning Emittance tning simlations were performed to inestigate the likely mode II emittance that might be achieed with the magnitde of magnet alignment and BPM gain errors expected in practice in esrta. Random errors were applied, with ales haing Gassian distribtions, with rms as shown in Table I. Note that the BPM alignment and gain errors are applied as ariations in the btton coordinates ðx i ;y i Þ and gain coefficients g i, as described in Sec. III B. With errors applied in the model, the correction procedre was implemented as follows. () An initial orbit correction sing the ertical steering magnets was applied

10 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () TABLE I. The rms ales for error distribtions. Error type Distribtion rms Dipole tilt 3 rad Qadrpole tilt 3 rad Qadrpole ertical alignment 5 m Sextpole ertical alignment 5 m BPM ertical alignment m BPM tilt mrad BPM gain % Dispersion measrement resoltion mm () The normal modes were identified from the singletrn transfer matrix; particles in each of the normal modes (with initial ales of the dynamical ariables taken from the real parts of the eigenectors of the single-trn transfer matrix) were tracked for 5 trns, with the btton signals at each BPM recorded on each trn. (3) The calibration matrix B, defined in Eq. (8), for each BPM was determined from correlations between the trn-by-trn btton signals. Nominal ales, based on electromagnetic modeling of the BPM bttons and the design of the electronics, were assmed for the calibration matrix. (4) The mode II dispersion was calclated at each BPM from the change in (calibrated) BPM reading with respect to energy. That is, the change in btton readings corresponding to a change in closed orbit with respect to energy were fond, and then the change in coordinates calclated sing Eq. (). (5) A dispersion correction was applied by setting the strengths of 6 skew qadrpoles (with locations shown in Fig. ), determined from the nominal (i.e. ideal model) response matrix between the skew qadrpole strengths and the mode II dispersion at the BPMs. (6) A frther iteration of the dispersion measrement and correction was applied (by repeating steps 5). For simlations inoling a large nmber of seeds, the total time reqired for tracking becomes significant. The time reqired for tracking depends mainly on the nmber of trns sed to collect the BPM calibration data. In step, the nmber of trns was chosen to minimize the tracking time reqired in the simlation, while still achieing good accracy for the determination of the normal modes. It was fond that increasing the nmber of trns beyond 5 did not hae any significant impact on the reslts. In the experimental stdies on esrta, we sed the fll 4 trns of data retrned by the BPMs, simply becase the data were aailable, and there wold be no significant adantage in redcing the nmber of trns. We hae not inestigated how, in practice, the accracy of the reslts depends on the nmber of trns of data collected. The tning simlation was repeated sing a similar procedre to that gien aboe, bt sing the ertical dispersion instead of the mode II dispersion. The reslts from each simlation, as distribtions of the final emittance oer seeds of random errors with different magnitde of BPM gain errors, are shown in Fig. 6. learly, a dispersion correction (following an orbit correction) alone is insfficient to achiee the goal of mode II emittance less than pm if the correction is based on the ertical dispersion. This is consistent with experience of esrta operation: low-emittance tning has generally been accomplished sing correction of both (ertical) dispersion and copling, with each correction calclated and applied separately. Howeer, if the mode II dispersion is sed, then dispersion correction (following an orbit correction) seems capable, at least in simlation, of redcing the emittance well below the goal of pm. Note that the techniqe seems robst, in that for all seeds of random errors, the procedre conerged to a stable soltion. Also shown in Fig. 6 are the distribtions of the ertical dispersion and the mode II dispersion (rms of ales measred at the BPMs, inclding simlated noise), after tning based on correction of the respectie dispersion in each case. For the ertical dispersion, the reslts are again sensitie to the BPM gain errors, and in fact the gain errors limit the qality of the dispersion correction that can be achieed. The mode II dispersion measrement is insensitie to the (systematic) BPM gain errors, bt appears to be significantly affected by the (random) measrement errors: a resoltion of mm is assmed for the dispersion measrement. Becase the measred ale of the mode II dispersion following correction is limited by the resoltion of the dispersion measrement, the mode II emittance following correction is poorly correlated with the measred rms dispersion. It is therefore difficlt to set a goal for the dispersion correction: howeer, from the distribtions in Fig. 6, it appears that if a mode II dispersion of arond 5 mm (rms measred at the BPMs) can be achieed, then a mode II emittance below pm has a good probability (in the simlations, 86% of seeds gae a mode II emittance below pm after correction of the mode II dispersion). It is also significant to note that the emittance achieed after minimizing the ertical dispersion is dependent on the BPM gain errors: larger BPM gain errors tend to lead to larger final mode II emittances. This is to be expected, since gain errors in the BPMs will cople horizontal dispersion into the ertical. With large BPM copling, the ertical dispersion measrement will be sbject to significant systematic errors, limiting the qality of the correction. Howeer, the correction based on normal mode dispersion is completely insensitie to BPM gain errors: this is a conseqence of the fact that the BPMs are calibrated sing normal mode excitation of the beam. The calibration techniqe effectiely determines the appropriate relationship between changes in btton signal and changes in beam position, for the BPM to read the beam motion along the normal mode axes. 784-

11 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () Relatie nmber of seeds x 3 No BPM gain errors % BPM gain errors 5% BPM gain errors orrected mode II emittance (pm) Relatie nmber of seeds.5..5 No BPM gain errors % BPM gain errors 5% BPM gain errors orrected ertical dispersion (mm) Relatie nmber of seeds.6.4. No BPM gain errors 5% BPM gain errors % BPM gain errors orrected mode II emittance (pm) Relatie nmber of seeds.3.. No BPM gain errors 5% BPM gain errors % BPM gain errors orrected mode II dispersion (mm) FIG. 6. Distribtion (from seeds of machine errors with rms shown in Table I) of final mode II emittance (top and bottom, left) and dispersion (top and bottom, right) after correction of dispersion measred at the BPMs. The ertical axes are scaled so that each cre has nit area. Top: emittance (left) and ertical dispersion (right) after tning based on correction of ertical (y) dispersion. Bottom: emittance (left) and mode II dispersion (right) after tning based on correction of mode II dispersion. In each case, the dispersion is the rms of the dispersion measred at the BPMs (inclding measrement errors, corresponding to a resoltion of mm). Note the different horizontal scales between the top plots and the bottom plots. Note also that the emittance distribtion for the correction based on ertical dispersion is sensitie to the magnitde of the BPM gain errors, whereas the emittance distribtion for a correction based on mode II dispersion is insensitie to the BPM gain errors. Black lines correspond to no BPM gain errors; ble lines correspond to % (ertical dispersion correction) or 5% (normal mode dispersion correction) gain errors; and red lines correspond to 5% (ertical dispersion correction) or % (normal mode dispersion correction) gain errors. Similar reslts hae been obtained for simlations of other storage rings, e.g., the KEK Accelerator Test Facility [5]. IV. EXPERIMENTAL RESULTS A. BPM calibration Typical calibration data for a single BPM (BPM 7W) are shown in Fig. 7. The plots are prodced by recording the raw btton signals for 4 consectie trns, while the beam is resonantly excited in normal mode I. A similar set of plots is prodced for resonant excitation in mode II. Each plot shows the correlation between a pair of bttons. The control system also records the horizontal (x) and ertical (y) position of the beam on each trn, based on a nominal calibration of the BPM: the correlations between the coordinates and the btton signals are also shown. The amplitde of the beam oscillation is more than mm peak to peak, in the horizontal. There is some oscillation isible in the ertical, abot m peak to peak: this cold be a conseqence of some copling in the machine, or some error in the nominal BPM calibration, or both. The correlations between the btton signals are almost perfectly linear, with ery little scatter: we expect or assmption of a linear relationship between the beam position and btton signals that we sed in Sec. II B to be alid. The gradient of the correlations between each pair of bttons proides the ales for the calibration matrix B, Eq. (8). The gradients are positie or negatie according to whether the bttons are on the same horizontal side or opposite horizontal sides of the acm chamber (see Fig. ). Note that calibration data for all BPMs in the ring can be collected simltaneosly. Fitting the gradients to the correlation plots is a straightforward and ery fast procedre. To test the calibration of a BPM, we can inspect the Forier spectrm of the beam coordinates retrned by the BPM oer a nmber of consectie trns dring resonant excitation of a normal mode. If the BPM is incorrectly calibrated to read the normal mode coordinates, the spectrm of both coordinates will contain peaks at the resonant freqency corresponding to the excited mode. For a correctly calibrated BPM, the spectrm of the mode I coordinate will contain a peak only at the mode I freqency, and the spectrm of the mode II coordinate will contain a peak only at the mode II freqency. Figre 8 shows the Forier 784-

12 A. WOLSKI et al. Phys. Re. ST Accel. Beams 4, 784 () g =.63 g =.8 g =.7 g = 9.86 µm g =.7 µm g =.6 g =. g = µm g =.56 µm 3 g =.93 3 g = 7.47 µm g =.3 µm 3 g = µm g =.33 µm g =.9 FIG. 7. orrelation plots for the btton signals from BPM 7W dring resonant excitation of normal mode I. Trn-by-trn signals from the BPM bttons (and the coordinates calclated sing the nominal calibration) are shown as ble points. The points in each correlation plot are fitted with a red ellipse; the major axis of the ellipse is shown with a black line. The ellipses described by the data points are more obios if the copling is increased: compare with Fig.. ompare also with the simlation reslts shown in Fig. 3. ln(amplitde) ln(amplitde) Tne Tne x y Mode I Mode II FIG. 8. Forier spectra of beam position coordinates retrned by BPM 7W oer 4 consectie trns, with mode I resonant excitation. Note that the (fractional part of the) mode I tne is.59, and the (fractional part of the) mode II tne is.63. spectra of the coordinates retrned by BPM 7W oer 4 consectie trns, with resonant beam excitation in mode I. We see that a peak corresponding to the horizontal tne (the mode I resonant freqency) appears in the spectra of both the x and y coordinates. Howeer, sing the normal mode calibration deried from the data shown in part in Fig. 7, the mode I freqency peak is entirely absent from the mode II spectrm. Similarly, Fig. 9 shows the Forier spectra of the coordinates retrned by BPM 7W oer 4 consectie trns, with resonant beam excitation in mode II. With the nominal calibration, a peak corresponding to the mode II resonant freqency is clearly isible in the spectra of both the x and y coordinates. Howeer, with the calibration based on obseration of the trn-by-trn data, the mode II resonant freqency peak is entirely absent from the spectrm of the mode I coordinate. Note that there is a significant peak at the mode I resonant freqency, isible in the mode I coordinate: this likely reslts from real beam motion. These obserations proide confidence in the BPM calibration obtained from obseration of trnby-trn btton signals dring resonant excitation in one or the other of the normal modes. 784-

13 LOW-EMITTANE TUNING OF STORAGE RINGS USING... Phys. Re. ST Accel. Beams 4, 784 () ln(amplitde) ln(amplitde) Tne Tne B. Mode II dispersion response to changes in skew qadrpole strength To test the measrement of the mode II dispersion sing calibrated BPMs, and also to alidate the model, the change in mode II dispersion was measred, following known changes in selected skew qadrpole strengths. Figre shows the measred mode II dispersion after a standard tning procedre, inoling correction of the closed orbit, ertical dispersion, and betatron copling. The skew qadrpole strengths in a model were adjsted to fit the measred mode II dispersion. Here, an ideal model, with all magnet strengths set to their design ales, and no alignment or BPM errors, was sed for the fitting. Using a response matrix (inerted by singlar ale decomposition), strengths of all skew qadrpoles were determined that gae the best match between the mode II dispersion in the model, and the mode II dispersion measred in the machine. The strengths of the skew qadrpoles and the mode II dispersion fond in this fit in this case are also shown in Fig.. The rms (oer the ales at the BPMs) measred mode II dispersion is mm, and the fitted skew qadrpole strengths are generally less than 3 =m. Note that there is a significant residal between the fitted dispersion and the measred dispersion: this sggests that there may be only a limited ability to correct the mode II dispersion sing the skew qadrpoles. After making the initial measrement, the strength of skew qadrpole 48W was changed by k¼ :3m. The change in mode II dispersion and the changes in fitted skew qadrpole strengths are shown in Fig.. The horizontal dispersion in the lattice at the location of this skew qadrpole is close to zero. Therefore, the ertical x y Mode I Mode II FIG. 9. Forier spectra of beam position coordinates retrned by BPM 7W oer 4 consectie trns, with mode II resonant excitation. Note that the (fractional part of the) mode I tne is.59, and the (fractional part of the) mode II tne is.63. η II (m) Residal η II (m) k (m ) x 3 W 5W 6W 7W 9W 4W 8W W W Measred Fitted model 4W 6W 47W 48W RMS Measred =. m BPM nmber RMS Residal (measred fitted model) =.7 m BPM nmber FIG.. Measred mode II dispersion after correction of orbit, (ertical) dispersion, and copling. The middle plot shows a comparison between the measred dispersion, and the dispersion in a model with skew qadrpole strengths adjsted to fit the measrement. The final skew qadrpole strengths fond from the fitting are shown in the top plot (the order of the qadrpoles is the same as shown in Fig. ); the bottom plot shows the residal between the measrement and the fitted model. The rms is the root mean sqare oer the measrements at the BPMs. dispersion is not expected to change when the strength of the skew qadrpole is aried. This was erified sing conentional measrement techniqes (sing the nominal BPM calibrations). Howeer, there is a large change in the mode II dispersion (peak at.6 m). This is primarily a reslt of the copling introdced by the skew qadrpole which tilts the normal mode axes. Ths, where the horizontal dispersion is large, a significant component appears along the mode II axis. The strengths of the skew qadrpoles in a model of the machine were adjsted to fit the mode II dispersion measred after the change in skew qadrpole 48W, and the 48E 47E 6E E E 8E 4E 9E 7E 6E 5E E E 784-3