Orbit Response Matrix Research Articles

Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy

The number and relative placement of BPMs and steerers with respect to the quadrupoles in a circular lattice can lead to degeneracy in the context of inverse modeling of accelerator optics. Further, the measurement uncertainties introduced by beam position monitors can propagate by the inverse modeling process in ways that prohibit the successful estimation of model errors. In this contribution, the influence of BPM and steerer placement on the conditioning of the inverse problem is studied. An analytical version of the Jacobian, linking the quadrupole gradient errors along with BPM and steerer gain errors with the orbit response matrix, is derived. It is demonstrated that this analytical version of the Jacobian can be used in place of the numerically obtained Jacobian during the fitting procedure. The approach is first tested with simulations and the findings are verified by measurement data taken on SIS18 synchrotron at GSI. The results are crosschecked with the standard numerical Jacobian approach. The quadrupole errors causing tune discrepancies observed at SIS18 are identified.

Fast orbit response matrix measurement via sine-wave excitation of correctors at Sirius

Sirius is the new 4th generation storage ring based synchrotron light source built and operated by the Brazilian Synchrotron Light Laboratory (LNLS). In this work, we report on the implementation at Sirius of a fast method for orbit response matrix (ORM) measurement which is based on sine-wave parallel excitation of orbit corrector magnets’ strength. This “AC method” has reduced the ORM measurement time from ∼ 25 minutes to 2.5–3 minutes and displayed increased precision if compared to the standard serial measurement procedure. When used as input to the Linear Optics from Closed Orbits (LOCO) correction algorithm, the AC ORM yielded similar optics corrections with less aggressive quadrupoles strength changes.

Noninvasively improving the orbit-response matrix while continuously correcting the orbit

Based on continuously recorded beam positions and corrector excitations from, for example, a closed-orbit feedback system we describe an algorithm that continuously updates an estimate of the orbit response matrix. The speed of convergence can be increased by adding very small perturbations, so-called dither, to the corrector excitations. Estimates for the rate of convergence and the asymptotically achievable accuracies are provided.

Linear optics and coupling correction with closed orbit modulation

We propose a new method to measure and correct linear optics and coupling for storage rings with beam position monitor (BPM) data. Two correctors in each transverse plane are used to modulate the closed orbit in an appropriate pattern to sample the linear optics. The orbit modulation data are fitted to the lattice model in a similar manner as the usual method of fitting the orbit response matrix. The closed orbit modulation measurements can be done much faster, and the number of sample points is not limited by the number of correctors. The method also has a number of advantages over methods that rely on turn-by-turn BPMs. A simulation is done to demonstrate the working principles of the method.

Symmetry Exploitation in Orbit Feedback Systems of Synchrotrons for Computational Efficiency

Structural symmetries in the storage ring of synchrotrons are intentionally created during the design phase of the magnetic lattices, but they are not considered in the design of control algorithms that stabilize the beam of accelerated particles. The choice of control algorithm, however, is limited by the speed requirements of the synchrotron. Standard control algorithms for synchrotrons are based on a singular value decomposition (SVD) of the orbit response matrix. SVD controllers neither exploit the structural symmetries nor exhibit any speed advantages. Based on the periodicity and the reflection properties of the betatron function, we show that these structural symmetries are inherited by the orbit response matrix. We show that the resulting block-circulant and centrosymmetric properties of the matrix can be used for different computationally efficient decompositions of the controller. We also address the case of broken symmetry due to odd placements of magnets and monitors. Our efficient decomposition could enable the use of more advanced control techniques for synchrotrons, such as control algorithms that require real-time optimization. These advanced control techniques could in turn increase the quality of research in synchrotron light sources.

Numerical simulation of NSLS-II fast orbit feedback system

We discuss recent work on the construction of a numerical simulation of the fast orbit feedback system at NSLS-II. The simulation operates in the time domain and also includes the spatial domain, i.e. all beam position monitors (BPMs) and correctors. It can accept inputs of the real beam orbit and the measured orbit response matrix. It can also add errors to every stage of the calculation. We present methods to verify the simulation results by comparing the simulation results with experimental data collected at NSLS-II. The results are in very good agreement. The effects of errors from BPMs and correctors, as well as the amplitude of excitation, on the feedback performance, are also explored. This simulation can be used to predict the behavior and performance of the system for a future upgrade.

Nonlinear optics from off-energy closed orbits

This paper describes a novel scheme for correcting the second order (chromatic sextupole) magnet lattice and its deployment on a fourth generation multibend achromat electron storage ring. The method is analogous to the well-established linear optics from closed orbits scheme, but uses an off-energy orbit response matrix to characterize the second order optics of a lattice. The matrix is constructed from the difference between two orbit response matrices measured at off-nominal energy, and is approximately linear with chromatic sextupole field strengths. This is utilized in a least squares minimization to find a model which minimizes the difference between the measured and model off-energy orbit response matrix. From this model corrections to the chromatic sextupoles of the machine can be calculated. In effect, for the MAX IV 3 GeV ring, the proposed scheme, NOECO (nonlinear optics from off-energy closed orbits), brings initial sextupole strength variations between achromats in the order of $\ifmmode\pm\else\textpm\fi{}8%$ down to the order of $\ifmmode\pm\else\textpm\fi{}1%$. The chromatic machine functions were symmetrized, the measured chromaticities approached the nominal values, and the lifetime of the beam at delivery conditions, with the same level of emittance coupling, increased to 19 h from 11 h at previously used sextupole settings.

Performance of the closed orbit feedback systems with spatial model mismatch

Closed orbit feedback (COFB) systems used for the global orbit correction rely on the pseudo-inversion of the orbit response matrix (ORM). A mismatch between the model ORM used in the controller and the actual machine ORM can affect the performance of the feedback system. In this paper, the typical sources of such model mismatch such as acceleration ramp ORM variation, intensity-dependent tune shift and beta beating are considered in simulation studies. Their effect on the performance and the stability margins are investigated for both the slow and fast regimes of a COFB system operation. The spectral radius stability condition is utilized instead of the small gain theorem to arrive at the theoretical limits of COFB stability and comparisons with simulations for SIS18 of GSI and experiments at the Cooler synchrotron (COSY) in the Forschungzentrum J\"ulich (FZJ) are also presented.

2nd Order Optics Symmetrisation through Off-Energy Orbit Response Matrix Analysis

2nd Order Optics Symmetrisation through Off-Energy Orbit Response Matrix Analysis

Improving the precision of linear optics measurements based on turn-by-turn beam position monitor data after a pulsed excitation in lepton storage rings

Beam optics control is of critical importance for machine performance and protection. Nowadays, turn-by-turn (TbT) beam position monitor (BPM) data are increasingly exploited as they allow for fast and simultaneous measurement of various optics quantities. Nevertheless, so far the best documented uncertainty of measured $\ensuremath{\beta}$-functions is of about 10\mbox{\textperthousand} rms. In this paper we compare the $\ensuremath{\beta}$-functions of the ESRF storage ring measured from two different TbT techniques---the N-BPM and the Amplitude methods---with the ones inferred from a measurement of the orbit response matrix (ORM). We show how to improve the precision of TbT techniques by refining the Fourier transform of TbT data with properly chosen excitation amplitude. The precision of the N-BPM method is further improved by refining the phase advance measurement. This represents a step forward compared to standard TbT measurements. First experimental results showing the precision of $\ensuremath{\beta}$-functions pushed down to 4\mbox{\textperthousand} both in TbT and ORM techniques are reported and commented.

Beam optics correction in the PLS-II storage ring

LOCO (Linear Optics from Closed Orbit) is a tool to analyze linear beam-optics and search for errors in the quadrupole focusing strength. Analysis with the help of LOCO reveals these errors by fitting the measured orbit-response matrix to the model orbit-response matrix. By readjusting the currents of the quadrupole-magnet power supplies to eliminate the magnet-gradient error, we can bring the machine’s optics parameter close to the design values. In this paper, we use LOCO to show how the design beam optics is restored in the PLS-II storage ring.

Measurement, analysis and correction of the closed orbit distortion in Indus-2 synchrotron radiation source

The paper presents the measurement, analysis and correction of closed orbit distortion (COD) in Indus-2 at 550 MeV injection energy and 2 GeV synchrotron radiation user run energy. The measured COD was analysed and fitted to understand major sources of errors in terms of the effective quadrupole misalignments. The rms COD was corrected down to less than 0.6 mm in both horizontal and vertical planes. A golden orbit was set for the operating synchrotron radiation beamlines. With COD correction, the injection efficiency at 550 MeV was improved by ~50% and the beam lifetime at 2 GeV was increased by ~8 h. In this paper, the method of global COD correction based on singular value decomposition (SVD) of the orbit response matrix is described. Results for the COD correction in both horizontal and vertical planes at 550 MeV injection energy and at 2 GeV synchrotron radiation user run energy are discussed.

Measurement of parameters in Indus-2 synchrotron radiation source

The paper presents the measurement of optics parameters in Indus-2 synchrotron radiation source, which include betatron tune, beta function, dispersion function, natural chromaticity, corrected chromaticity, central RF frequency, momentum compaction factor, and linear betatron coupling. Two methods were used for beta function measurement; a conventional quadrupole scan method and a method using the fitting of the orbit response matrix. A robust Levenberg-Marquardt algorithm was used for nonlinear least square fitting of the orbit response matrix. In this paper, detailed methods for the parameter measurements are described. The measured results are discussed and compared with the theoretical values obtained using accelerator simulation code Accelerator Toolbox in MATLAB.

Ultra low vertical emittance at SLS through systematic and random optimization

A new vertical emittance record of 0.9±0.4pmrad was established in the storage ring of the Swiss Light Source (SLS) at 2.4GeV by the application of a sequence of methods: Magnet girders were remotely vertically realigned in order to suppress the dipolar fields that produce the spurious dispersion in the vertical plane. A systematic optimization was then performed by measuring and correcting the betatron coupling and the vertical dispersion by means of 36 skew quadrupole correctors. The correction is performed until a limit is reached from the orbit response matrix (ORM) measurement errors and residual model deficiencies. At this point, the method of random optimization (RO), a multi-variable optimization procedure, was invoked. In the present experiment, the vertical beam size is the target function while the skew quadrupole currents act as the correction variables. The methodology of RO and the systematic correction scheme are described.

Orbit response matrix analysis and lattice periodicity restoration of the SSRF storage ring

In this paper, we present numerically and experimentally the linear beam-optics distortion in the SSRF storage ring and the correction of optics by using a number of quadrupole magnets installed in the storage ring. The measured orbit-response matrices were fitted to the model-response matrices to obtain the β and the dispersion functions in the storage ring. By readjusting the currents of quadrupole-magnet power supplies, we were able to successfully restore the optics parameters to values very close to the design ones, with rms deviations around 1%. This periodicity restoration is verified with the β function measurement.

Linear optics calibration and nonlinear optimization during the commissioning of the SSRF storage ring

Phase I commissioning of the SSRF storage ring on 3.0 GeV beam energy was started at the end of December 2007. A lot of encouraging results have been obtained so far. In this paper, calibrations of the linear optics during the commissioning are discussed, and some measured results about the nonlinearity given. Calibration procedure emphasizes correcting quadrupole magnetic coefficients with the Linear Optics from Closed Orbit (LOCO) technique. After fitting the closed orbit response matrix, the linear optics of the four test modes is substantially corrected, and the measured physical parameters agree well with the designed ones.

Calibration of the linear optics in the SSRF storage ring

Phase I commissioning of the SSRF storage ring at 3.0 GeV was ended with encouraging results. Distortions and calibrations of the linear optics during the stroage ring commissioning are discussed in this paper. The calibration procedure has reduced sextupole-additional focusing effects by minimizing closed orbit deviation and corrected quadrupole magnetic coefficients with the linear optics from closed orbit (LOCO) technique. After fitting the closed orbit response matrix, linear optics of the storage ring is substantially corrected, and the measured parameters agree well with the design. Four optics modes were commissioned, and relevant machine physics studies were carried out. Their results are summarized.