Characterization of the International Linear Collider damping ring optics
Abstract
A method is presented for characterizing the emittance dilution and dynamic aperture for an arbitrary closed lattice that includes guide field magnet errors, multipole errors and misalignments. This method, developed and tested at the Cornell Electron Storage Ring Test Accelerator (CesrTA), has been applied to the damping ring lattice for the International Linear Collider (ILC). The effectiveness of beam based emittance tuning is limited by beam position monitor (BPM) measurement errors, number of corrector magnets and their placement, and correction algorithm. The specifications for damping ring magnet alignment, multipole errors, number of BPMs, and precision in BPM measurements are shown to be consistent with the required emittances and dynamic aperture. The methodology is then used to determine the minimum number of position monitors that is required to achieve the emittance targets, and how that minimum depends on the location of the BPMs. Similarly, the maximum tolerable multipole errors are evaluated. Finally, the robustness of each BPM configuration with respect to random failures is explored.
1 Introduction
The International Linear Collider (ILC) design utilizes damping rings to cool beams delivered by the electron and positron sources before transferring to the main linacs [1]. Three of the primary requirements of the baseline damping rings are: 1) they must accept an injected bunch from the positron source with normalized phasespace amplitude and ; 2) the beams must be cooled to an equilibrium zerocurrent geometric vertical emittance ; and 3) the damping time must be short enough to provide fully damped bunch trains at a repetition rate of 5Hz.
The above requirements must be met in a real machine that includes magnet misalignments, guide field multipole errors (both systematic and random), and beam position monitor (BPM) measurement errors. Emittance tuning will be essential to achieve the target zerocurrent emittance. Quadrupoles and corrector magnets will necessarily be independently powered, allowing for localized corrections through the use of beambased measurements.
The damping rings cannot be characterized with respect to the exact set of errors they will have, as the rings are not yet built. For the purposes of this study, alignment errors in the damping rings are assumed to be randomly distributed with amplitudes dictated by survey tolerances. By simulating a large number of lattices with random distributions of errors at the appropriate levels, the damping rings can be characterized with a statistical analysis of the likelihood that the required emittances and dynamic aperture will be achieved. A configuration is deemed acceptable if 95% of the randomlymisaligned and corrected lattices meet the required vertical emittance and dynamic aperture, as per the Technical Design Report specifications.
Several methods for optics correction have been developed at storage ring light sources. By far the most widespread method is response matrix analysis (RMA), and specifically, orbit response matrix (ORM) analysis [17]. The method involves taking difference orbits where corrector strengths are varied. Both the Swiss Light Source and Australian Synchrotron have demonstrated the capability of correcting vertical emittance below 1 pm using ORM [3, 9], below the required vertical emittance for the ILC damping rings. However, the time required for ORM data acquisition scales linearly with the number of correctors in the ring. For large rings such as the proposed ILC damping rings with over 800 steering correctors, acquiring difference orbits becomes a timeconsuming process.
Another increasinglycommon class of optics correction algorithms is based on turnbyturn BPM measurements, where the beam is pinged with a shortduration kick and allowed to oscillate freely [6, 2]. Typically this data is then processed as a resonancedriving term (RDT) correction [13]. This correction technique has the benefit that correction times are roughly independent of the size of the ring. For the ILC this represents a considerable savings in time. While this method has shown promise, there are limitations. The beam decoheres due to amplitude dependent tune shift, limiting the number of useful turns of data. The decoherence may be partially mitigated by reducing the chromaticity to near zero, however measurements are still limited to a few thousand turns. As mentioned in [2], the resolution of the betatron phase measurement scales roughly inverse with the number of turns (). The limitations on the number of turns in a data set therefore directly correspond to a limitation on accuracy of betatron phase measurements.
A turnbyturnbased alternative to RDT has been developed for use at the Cornell Electron Storage Ring (CESR) for CESR Test Accelerator program (CesrTA) [10, 22]. The CESR betatron phase and coupling measurement relies on phaselocking turnbyturn kickers to resonantly excite a single bunch to an amplitude of several millimeters, thus avoiding limitations due to bunch decoherence and allowing for data sets of over 40,000 turns. The effectiveness of the betatron phase/coupling correction procedure has been evaluated at CesrTA through measurements and simulations, demonstrating good agreement and validating our understanding of the limitations of optics correction.
In this paper, the method for modeling optics correction developed at CesrTA is used to characterize the ILC damping rings. Several scenarios are evaluated, including the possibility of reducing the total BPM count and relaxing the constraints on guide field multipole errors, with respect to the baseline design. The effect of random BPM failures is analyzed to benchmark the robustness of each of the proposed BPM distributions.
The characterization is comprised of two parts: generating a randomly misaligned lattice and simulating the optics correction procedure, and analyzing the dynamic aperture of the corrected lattice as a function of guide field multipole errors. The software used in all analyses is based on the Bmad accelerator code library [18].
Only singleparticle dynamics from the beam optics are considered in the work presented here. The reported emittances therefore represent the zerocurrent limit. Other currentdependent sources of emittance dilution, such as intrabeam scattering (IBS), electron cloud, and fastion instability will increase the vertical emittance beyond this lower limit.
2 DTC04 Lattice
The baseline ILC damping ring design is the DTC04 lattice developed by Rubin et al. [21]. The lattice is a 3.2 km racetrack with a modified TMEtype arc cell. The zerodispersion straights are based on the work of Korostelev and Wolski [14]. A schematic of the ring is shown in figure 1. Optics functions are shown in figure 2. Tables 1 and 2 summarize the lattice parameters and number of magnets of each class. The quoted emittances are the horizontallike and verticallike normal modes, respectively. In the absence of coupling, these will correspond to . Decomposition into normal modes is discussed elsewhere [25, 16, 4], and as such, a derivation is omitted here.
Parameter  Value  Units 

Circumference  3239  m 
Energy  5.0  GeV 
Betatron Tunes (, )  (48.850, 26.787)  
Tune Advance per Arc  (16.418, 6.074)  
Chromaticity ()  (1.000, 0.302)  
Chromaticity per Arc  (9.074, 10.896)  
Train Repetition Rate  5  Hz 
Bunch Population  
Extracted  0.6  nm 
Extracted  pm  
Extracted Bunch Length  6  mm 
Extracted  0.11  % 
Damping Time  24  ms 
Wiggler  1.5  T 
Class  Count 

Beam Position Monitor  511 
Dipole  164 
Horizontal Steering  150 
Vertical Steering  150 
Combined H+V Steering  263 
Quadrupole  813 
Skew Quadrupole  160 
Sextupole  600 
Damping Wigglers  54 
The arc cell layout is shown in figure 3. Arc cells are comprised of one dipole, three quadrupoles, four sextupoles, one horizontal and one vertical steering corrector, one skew quadrupole, and two beam position monitors.
3 Misalignment and Correction Procedure
The effects of misalignments, BPM errors, multipoles, and choice of correction procedure are evaluated by the program ring_ma2 [22]. The ring_ma2 workflow is structured as follows:

Assign random misalignments and BPM errors with userdefined amplitudes to the ideal lattice to create a realistic machine model.

Simulate beam based measurements of optics functions including the effects of BPM measurement errors.

Compute and apply corrections for each iteration based on the simulated measurements.

After each correction iteration, record the effectiveness of the correction in terms of emittances and optics functions.
The entire procedure is repeated typically 100 times, each with a randomly chosen distribution of errors in order to generate statistics sufficient for analysis. Misalignments are allowed on any element parameter described by Bmad, including but not limited to: position offsets and angles, random field strength errors, and systematic and random multipole errors. BPM measurement errors allowed in the simulation include BPMtoquadrupole transverse offset, tilt, buttonbybutton gain error, and buttonbybutton timing errors.
The particular configuration of the guide field magnets, correctors, and BPMs is referred to as a scenario. For each scenario, 100 lattices are generated with random errors and misalignments. Each misaligned lattice is referred to as a seed. As noted above, typically 100 seeds are used for the evaluation of each scenario, and the same 100 seeds are used when evaluating multiple scenarios. That is, for a given seed, each element in a lattice for each scenario starts with the same misalignments and field errors, regardless of whether a subset of elements or detectors are vetoed or disabled during the correction.
The method used for simulating measurements with BPM errors is based on the method developed for diagnosing emittance dilution at CesrTA [22]. In the original analysis, modeling the full correction procedure includes turnbyturn tracking for several damping times in order to accurately simulate measurements of betatron phase and coupling. The turnbyturn measurements are then postprocessed identically to actual machine data into orbit, dispersion, or phase and coupling data. Though this method for simulation is as close to modeling the actual measurement process, it is very timeconsuming. In the case of the ILC damping ring lattice, this method is prohibitively slow due to the large number of elements. Two modifications have been made to the measurement simulation procedure in order to improve the efficiency of the calculations such that the characterization method developed in [22] may be applied.
First, there is a modification to tracking in the damping wigglers. When accounting for wiggler nonlinearities is required, a fullysymplectic Lie map is used for tracking through the damping wigglers [7, 8]. However, as it is presently implemented in Bmad, this tracking method is computationally intensive. When the full nonlinearities of the damping wigglers are known to not affect the resulting simulation, a simplified MADstyle “benddriftbend” wiggler model which preserves the radiation integrals can be used to reduce the time required for simulations. Radiation integrals and horizontal emittances the two wiggler models are shown in table 3. The horizontal emittances computed with the two models differ by less than 3%. The level of agreement is more than sufficient for the studies presented here.
Parameter  Lie Map  BendDriftBend  

0.553 nm  0.567 nm  2.51%  
1.08  1.08  0.00%  
0.512 m  0.515 m  0.59%  
m  m  0.44%  
m  m  1.82%  
m  m  3.13% 
Comparisons of simulations using the full wiggler model and the reduced MADstyle “benddriftbend” wiggler have shown that wiggler nonlinearities do not significantly affect the correction of optics functions or the final emittance. Therefore, the simplified “benddriftbend” wiggler model is used for all ring_ma2 studies on the ILC damping ring lattice. This reduces the time required to simulate the correction procedure for one randomlymisaligned lattice by a factor of 15.
The second modification affects the process through which optics measurements are simulated. In [22], measurements are simulated on a turnbyturn basis, applying BPM measurement errors on every turn. Simulating one betatron phase measurement requires particle tracking for over 40,000 turns, applying simulated BPM measurement errors on every turn. Even with the simplified wiggler model, this is prohibitively slow on a lattice as large as DTC04. An alternative is used in these studies where BPM errors are applied directly to the Bmadcomputed optics functions. Although not as rigorous as the full measurement simulation method, sidebyside comparisons of the two methods have shown minimal difference in the resulting optics functions. As a result, the amount of time to simulate the correction procedure for one randomlymisaligned and corrected lattice is further reduced by another factor of 25, allowing the characterization of each configuration to be completed in a reasonable amount of time.
The optics correction procedure used in this characterization was developed at CesrTA [22], and relies on the measurement and correction of the betatron phase and coupling [19] using beam position monitors (BPMs) with turnbyturn capabilities [15]. The BPM modules are capable of preprocessing phase data in parallel, therefore this method has the benefit that data acquisition time is roughly independent of the number of BPMs, and is entirely independent of the number of correctors.
The correction procedure used here is as follows:

Measure the closed orbit. Fit the data using all 826 steering correctors (150 each of dedicated horizontal and vertical steerings, and 263 combined horizontal/vertical correctors) and apply corrections to the lattice model.

Measure the betatron phase/coupling and dispersion. Correct betatron phase and horizontal dispersion to the design values using all 813 normal quadrupoles; simultaneously correct betatron coupling using all 160 skew quadrupole correctors, and apply all corrections.

Remeasure the closed orbit, coupling, and vertical dispersion; simultaneously correct all machine data using all 513 steering correctors and 160 skew quadrupole correctors and apply corrections.
The correction procedure described above is routinely used at CesrTA, therefore a direct comparison of simulation with measurement is available to validate the emittance tuning method. Details of the CesrTA characterization are available in [24, 22]. Although CesrTA has not achieved a vertical emittance below 2 pm as required for the ILC damping rings, simulated optics functions after correction are in agreement with measurements of the optics at CesrTA, and demonstrate accuracy in the modeling of the correction procedure. In practice, corrections based on betatron phase and coupling have yielded coupling and dispersion on par with that achieved using orbit response matrix analysis at CesrTA [23].
4 Error Tolerance of DTC04
The magnet misalignments, guide field errors, multipole errors, and BPM measurement error tolerances specified in the ILC Technical Design Report are applied in order to demonstrate that the baseline design satisfies the requirements for achieving vertical emittance. As will be seen, the full complement of BPMs is more than sufficient to achieve the requisite vertical emittance.
4.1 Nominal Lattice and Errors
Misalignments and BPM measurement error tolerances are summarized in table 7 in the Appendix, and are based on errors used in previous studies [21]. In addition, quadrupole and sextupole errors are now included. Multipole error coefficients are taken from measurements of PEPII guide field multipole errors by Cai [5], and are summarized in table 8 in the Appendix. These error amplitudes and multipole coefficients, along with the full complement of 511 BPMs, define the nominal ILCDR scenario.
Results of ring_ma2 studies for this scenario are shown in figure 4 and are summarized in table 4. The coupling in model lattices is characterized using the coupling matrix [20], an extension of the Edwards and Teng formalism [11]. In particular, the outofphase coupling matrix element is used, as the measurement is insensitive to BPM rotation.
Iteration  Parameter  Percentile 

Initial  1010 pm  
RMS  64.5 mm  
RMS  
12.3 pm  
RMS  6.37 mm  
RMS  
10.9 pm  
RMS  6.92 mm  
RMS  
0.224 pm  
RMS  0.502 mm  
RMS 
Including misalignments, guide field errors, multipole errors, and BPM measurement error tolerances, 95% of the resulting lattices have a vertical emittance below after corrections. This is well below the emittance budget of , though it is important to note that collective effects and other nonstatic sources of emittance dilution have not been accounted for in the calculation of the simulated emittance.
The fundamental lower limit for the vertical emittance is determined by the finite opening angle of synchrotron radiation, and for the ILC damping rings is around 0.1 pm. The zerocurrent emittances achieved after correction for the nominal scenario is close to this limit, with 95% of seeds well within a factor of three of the lower bound.
4.2 Reduced BPM Schemes
It is clear that the baseline specifications for misalignments, multipole errors, and BPM and corrector distributions, as specified in the Technical Design Report, are more than sufficient to contain the static optics contribution to the vertical emittance within the 2 pm emittance budget. It is of interest to consider whether the number of BPMs can be reduced without compromising the effectiveness of the emittance correction procedure.
The ILC damping ring lattice is wigglerdominated, with of synchrotron radiation being generated by the damping wigglers. As such, residual vertical dispersion in the wiggler straight will contribute significantly more to the vertical emittance than a comparable residual in the arcs. Preserving the 52 BPMs in the damping wiggler straight is therefore essential. The remaining 459 BPMs are either in other straight sections (159) or in the arcs (300).
The baseline arc cell design has two BPMs, indicated in figure 3. The optics functions at the two BPMs are summarized in table 5. The betatron phase advance per arc is roughly 15 ( rad), corresponding to about 10 BPMs per betatron wavelength in the arcs. The large number of BPMs and low phase advance per arc cell imply that not every BPM in the arcs may be necessary in order to maintain the ability to correct the lattice to below 2 pm vertical emittance.
Parameter  OddIndexed  EvenIndexed 

15.1 m  6.5 m  
14.9 m  23.5 m  
0.312 m  0.185 m 
Five scenarios for BPM distributions are examined. They are:

Full complement of BPMs, for a total of 511; 10 BPMs per betatron wavelength. This is the nominal scenario from Sec. 4.1.

Remove oddindexed arc BPMs, for a total of 361; 5 BPMs per betatron wavelength, at locations with

Remove evenindexed arc BPMs, for a total of 361; 5 BPMs per betatron wavelength, at locations with

Remove all oddindexed and every other evenindexed arc BPM, for a total of 287; 2.5 BPMs per betatron wavelength, at locations with

Remove all evenindexed and every other oddindexed arc BPM, for a total of 287; 2.5 BPMs per betatron wavelength, at locations with
For each BPM distribution, the misalignment and correction procedure described in Sec. 4.1 is repeated. Magnet and BPM errors are identical for the five scenarios, therefore a direct comparison of resulting corrections is possible. Correction levels achieved for lattices with these BPM distributions are shown in figure 5, and are summarized in table 6.
BPM Scheme  RMS (mm)  RMS (%)  

Scenario 1  0.224  0.502  
Scenario 2  0.224  0.547  
Scenario 3  0.225  0.670  
Scenario 4  0.263  0.892  
Scenario 5  0.269  0.838 
With half of the arc BPMs removed (scenarios 23), the correction procedure achieves vertical emittance below 0.2240.225pm for 95% of the lattices. With the removal of 3/4 of BPMs in the arcs (scenarios 45), there is a slight increase in the percentile to 0.2630.269 pm, still well below the 2 pm requirement. There is a weak preference for retaining the “evenindexed” BPMs (scenarios 2 and 4) as compared to the “oddindexed” BPMs (scenarios 3 and 5). Scenarios 3 and 5 will therefore be omitted from further discussion.
4.3 Effect of Random BPM Failures
From the proceeding discussion, it is clear that the BPM count can be reduced substantially with respect to the baseline design specification without compromising the effectiveness of the emittance tuning procedure. However, this assumes all remaining BPMs are fully functional. This is rarely the case in a real machine, and as such, the effects of random BPM failures must be explored before deeming a BPM distribution acceptable.
Again using the same random seeds as in previous simulations for generating lattice and BPM errors, a random subset of BPMs is flagged as not functional, and therefore not used in corrections. As there are multiple BPM distributions under evaluation, only the fraction of BPMs tagged as “failed” is constant between BPM distribution scenarios, rather than failing a fixed number of BPMs. The exact subset of BPMs for each test will therefore not remain the same between each BPM distribution scenario. It is not possible to have a truly random distribution of BPM failures while requiring that the same BPMs are tagged as “failed” between the three BPM distribution scenarios evaluated here, since the total number and distribution of BPMs varies from one scenario to the next. As such, BPM failure tests on each individual BPM distribution scenario are directly comparable seedbyseed, but there is a small statistical variation between the different BPM distribution scenarios for a fixed percentage of BPM failures.
It is assumed that in practice, the damping rings would not be operated if more than 10% of BPMs have failed. In this study, the fraction of BPMs tagged as “failed” is increased from 0% to 10%. Figure 6 demonstrates the evolution of the percentile , RMS , and RMS with respect to the fraction of failed BPMs for the nominal case (scenario 1) and BPM distributions with and of arc BPMs removed (scenarios 2 and 4, respectively).
The resulting emittance, dispersion, and coupling of scenarios 1 and 2 are nearly identical, within the expected statistical variation. Scenario 4, with only of arc BPMs with respect to the baseline design, shows more rapid growth in emittance, dispersion, and coupling as the fraction of failed BPMs increases. Vertical emittance growth from intrabeam scattering will increase with vertical dispersion [12], thus making scenario 4 unattractive. It is therefore concluded that one of the two BPMs in every arc cell may be removed without affecting the robustness of the emittance tuning procedure.
5 Dynamic Aperture
After a lattice model has been misaligned and corrected using the simulated emittance tuning procedure, the dynamic aperture is evaluated through tracking. The dynamic aperture is defined as the maximum stable amplitude in the transverse plane. Trajectories with initial coordinates inside the dynamic aperture boundary will remain within that boundary for at least 1000 turns. The amplitude for trajectories with initial coordinates outside the boundary will be lost within 1000 turns. The tracking is repeated for offenergy particles to compute the energy dependence of the dynamic aperture.
The full wiggler map is required for evaluating the dynamic aperture in order to account for wiggler nonlinearities. This greatly increases the required computation time for a dynamic aperture study, however individual jobs are easily parallelized. Analysis has shown minimal variation of dynamic aperture between different sets of misalignments and corrections, given the same misalignment amplitudes. Therefore only one seed for each test configuration is evaluated for dynamic aperture. Here, the random seed corresponding to the percentile lattice after misalignments and correction is used.
Figure 6(a) shows the dynamic aperture for the nominal scenario, as defined in Sec. 4.1. It is evident that the systematic and random multipole errors as specified in the ILC Technical Design Report are more than sufficient to achieve the required dynamic aperture for accepting an incoming bunch from the positron source.
The possibility that magnet manufacturing requirements may be relaxed by increasing the tolerance on magnet multipole errors is explored. Starting with the nominal DTC04 lattice characterized in Section 4.1, both systematic and random multipole coefficients are increased by a constant multiplier from 5x20x with respect to the values in table 8. All other misalignments and BPM measurement errors are identical to the nominal scenario. Results are shown in figures 6(b)–6(d).
The multipole error constraints as specified in the ILC TDR are evidently overspecified by a factor of 20.
6 Summary and Future Work
A method has been presented for determining the effectiveness of corrections for the ILC damping rings and the resulting dynamic aperture after correction, using the emittance tuning algorithm developed and used at CesrTA. Lattice models include misaligned magnets, magnet strength and multipole errors, and BPM measurement errors. The method is versatile and may be applied to any closed lattice to evaluate the sensitivity of the effectiveness of emittance corrections to changes in BPM and corrector distributions, multipole tolerances, and more.
Based on this method, the ILC damping ring design, misalignment tolerances, and BPM measurement errors as presented in the ILC Technical Design Report (TDR) have been shown to achieve the required zerocurrent vertical emittance and dynamic aperture. Note that collective effects such as intrabeam scattering, electron cloud, and fastion instability can only increase the vertical emittance above this minimum.
One of the two BPMs in every arc cell may be removed without impact on the correction capabilities, even when accounting for up to a 10% BPM failure rate. This leaves approximately 5 BPMs per betatron wavelength in the arcs. A further reduction of BPMs in the arcs to 1/4 of the original count (2.5 BPMs per betatron wavelength) causes a substantial increase in dispersion and coupling, subsequently fueling emittance growth from collective effects. Specifications for magnet multipole errors can be relaxed by a factor of twenty without compromising the dynamic aperture.
An infrastructure exists for evaluating further lattice modifications, such as sextupole optimizations, increased magnet or BPM errors, changes in corrector distributions, or a change in working point. Further lattice developments in these areas can therefore be easily evaluated for the ILC damping rings or any arbitrary closed lattice.
Acknowledgements.
This work was supported by the National Science Foundation grant PHY1002467 and Department of Energy grant DESC0006505.Appendix A Appendix: DTC04 Magnet Misalignments and Multipoles
Multipole coefficients are defined in the following way:
(1) 
where n is the multipole order, and are the normal and skew components, respectively. Multipoles are evaluated at a reference radius of cm for dipoles, 5 cm for quadrupoles, and 3.2 cm for sextupoles.
Element  Error  Amplitude  Units 
Dipole  Roll  50  rad 
Quadrupole  , Offset  25  m 
Tilt  50  rad  
k1  0.1%  %  
Sextupole  , Offset  25  m 
Tilt  25  rad  
k2  1%  %  
Wiggler  Tilt  100  rad 
Offset  100  m  
BPM  Diff. Resolution  1  m 
Abs. Resolution  50  m  
Tilt  10  mrad  
Button Gains  0.5%  %  
Button Timing  10  ps 
Element  Multipole  Systematic  Random 

Dipole  b3  
b4  
b5  
Quadrupole  a3  
a4  
a5  
a6  
a7  
a8  
a9  
a10  
a11  
a12  
a13  
a14  
b3  
b4  
b5  
b6  
b7  
b8  
b9  
b10  
b11  
b12  
b13  
b14  
Sextupole  b4  
b5  
b6  
b7  
b8  
b9  
b10  
b11  
b12  
b13  
b14 
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