Transverse Dynamics - Measurements

Transverse Dynamics - Measurements [MCCPB, Chapter 2] • some data analysis techniques • coherent oscillations & filamentation

Data Analysis Techniques • measuring beam response to many steering corrections at many BPMs, adjusting a large number of model parameters in a global fit to best reproduce the data (LOCO code, J. Safranek) • singular-value decomposition (SVD); powerful, broad range of applications, e.g. reducing corrector strengths (V. Ziemann, 1992), orthogonal tuning knobs (P. Raimondi, 1998), BPM data analysis (J. Irwin, 1998) • principal-axes transformations (‘eigenfit’); may be superior to least-squares fit (T. Lohse, P. Emma, 1989) • model-independent analysis (MIA/PCA); extract temporal and spatial patterns from BPM data; identifies relevant physical variables (J. Irwin, C.-X. Wang, Y. Yan, 1998) • independent component analysis (ICA); more robust w.r.t. noise, coupling, and nonlinearity; extract transverse betatron phase & amplitude, dispersion, coupling, slices ( X. Huang, 2005; F. Wang, 2008; X. Pang, 2009);

orbit response measurements N BPMs reading orbit change Dx closed orbit in a storage ring M dipole correctors exciting deflection angles q trajectory Iin part of a ring or transport line

LOCO* – orbit response measurements and analysis closed orbit trajectory *J. Safranek, “Experimental determination of storage ring optics usins closed orbit response measurements”, Nucl. Inst. and Meth. A388, (1997), pg. 27.

LOCO cont’d • measured R must be adjusted to match the model; • then optical functions are obtained from matched model • - in a transfer line only the R matrix can be verified The measured R also depends on the BPM and corrector calibrations: A vector holding the weighted difference between the measured and modelled response is built from all matrix elements : 1) Data preparation: for all i and j

LOCO cont’d 2) Local gradient: • Evaluate the sensitivity w.r.t. parameters (BPM and corrector calibrations, strengths…). • Straightforward for calibrations, requires MADX runs for model parameters (quad strengths…) → linear approximation. 3) Least-square minimization: Solve the linearized equation for parameter changes Δc (based on SVD). 4) Iteration:

CERN accelerator complex

matrix sizes For a ring /line with NBPMs and Mcorrectors per plane, the minimum size of the gradient matrix Gis : (2 ×N ×M) ×(2 ×(N + M)) …with only BPM and corrector calibrations as parameters for c. SPS transfer line: N < 30, M < 30 1800 x 120 0.2 x 106 elements TT10 SPS ring: N ~ 110 , M = 108 25000 x 220 6 x 106elements LHC: N ~ 500 , M ~ 250 250000 x 1500 375 x 106 elements the complete LHC is tough to handle with all elements included: RAM + precision + CPU time

SPS example : before fit J. Wenninger

SPS example : a few fit iterations later … J. Wenninger

example : TI8 - SPS to LHC transfer line - quadrupole with wrong setting J. Wenninger

Singular Value Decomposition The singular value decomposition of a matrix is usually referred to as the SVD. This is the final and best factorization of any m × n X matrix whose entries are real numbers. The SVD factorization is of the form X = U S Vt where U is an m × m orthogonal matrix, Σ is an m × n diagonal matrix with non-negative real numbers on the diagonal, and the n × n unitary matrix Vtdenotes the transpose of the n × n orthogonal matrix V. Such a factorization is called a singular value decomposition of X. The r (nonzero) diagonal entries σi of Σ are known as the singular values of X. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by X (though the matrices U and V are not). v1, v2, ...vr is an orthonormal basis for the row space. u1, u2, ...ur is an orthonormal basis for the column space vr+1, ...vn is an orthonormal basis for the nullspace ur+1, ...umis an orthonormal basis for the left nullspace

SVD - continued eigenvector of XXt eigenvector of XtX mxn mxm nxn OR r is the rank of the matrix X = U S Vt “pseudo-inverse” X-1 = V S -1 Ut where S-1 is a diagonal matrix with elements 1/s11,…

model-independent analysis (MIA) original idea: no attempt to determine exact parameters of optics model, but use beam information and beam response to certain perturbation to monitor stability, to identify misaligned linac rf structures, etc. form huge matrix m BPMs p measurements

decompose matrix with SVD W eigenvectors form basis of temporal patterns V eigenvectors form basis of spatial patterns “signals” Singular values computed for the SLAC linac, using 5000 pulses and 130 BPMs only about 6-15 independent variables affecting the beam motion a few high-resolution BPMs BPM noise floor

independent-component analysis (ICA) ICA calculated l source signals, and determines (unknown) mixing matrix A ; source signal independence is related to “diagonality” of covariance matrices time-lagged source signal covariance matrix apply SVD: modified time-lagged co-variance matrix variance apply SVD: where W is the unitary demixing matrix and Dkis a diagonal matrix

independent-component analysis (ICA) – cont’d *J.F. Cardoso and A. Souloumiac, SIAM J. Matrix Anal. Apl. 17, 161 (1996)

Simple description of ICA Despite its wide range of applicability, ICA can be understood in terms of the classic ‘cocktail party’ problem, which ICA solves in an ingenious manner. Consider a cocktail party where many people are talking at the same time. If a microphone is present then its output is a mixture of voices. When given such a mixture, ICA identifies those individual signal components of the mixture that are unrelated. Given that the only unrelated signal components within the signal mixture are the voices of different people, this is precisely what ICA finds. ICA does not incorporate any knowledge specific to speech signals; in order to work, it requires simply that the individual voice signals are unrelated. TRENDS in Cognitive Sciences Vol.6 No.2 February 2002

coherent tune shift along a long bunch ICA MIA/PCA X. Pang et al, Phys. Rev. ST - Accel. Beams 15, 112802 (2012)

coherent oscillations & filamentation A) response to a kick: coherent damping exp. decay amplitude-dependent oscillation frequency bunch population chromaticity head-tail damping synchrotron-radiation damping

1/t LEP 45.63 GeV, damping rate 1/t vs. Ibunch for different chromaticities [A.-S. Muller] Q’=14 1/(100 turns) horizontal damping partition number Q’=2.7 1/tSR Ib

LEP: tune change during damping Q turns [A.-S. Muller]

LEP: detuning with amplitude from single kick Q 2 J [A.-S. Muller]

B) another response to a kick: filamentation R. Meller et al., SSC-N -360, 1987 ex.: Z: kick in s a =(2mw0)2 Q=Q0-ma2 both different from e-t/t amplitude in s

amplitudedecoherence factor vs. turn number oscillation amplitude 1/5 s kick 5s kick

measurement in the CERN PS (A.-S. Muller et al., EPAC 2002) simulated phase space Horizontal phase space generated with MAD at the pick-up in section 10. Two chains of stable islands (period 4 and 5) are clearly visible. The inner chain is the one studied during the measurement sessions. Upper part: Horizontal beam position vs. time measured with a pick-up for the few turns right after the kick (left) and for the last 1500 turns (right). Bottom part: Normalised phase space reconstructed from two pick-ups. a fraction of particles trapped in the resonance island

C) 3rd response to a kick: decoherence due to chromaticity (Q’>0) d>0 d<0 t=0 t=Ts/2 t=Ts

R. Meller et al., SSC-N -360, 1987 large x time small x Ts

measurement in the CERN PS (A.-S. Muller at al., EPAC 2002) Measurements of de- and recoherence for two synchrotron frequencies. Both plots are composed of several individual datasets with measurements started at different times after the kick (the current setup only allows the acquisition of about 2000 turns).

Summary data analysis techniques LOCO fits SVD MIA, ICA beam response to kick excitation coherent damping filamentation chromaticity

Computer lab, Wednesday 17 June 2015 BCM consider a simple storage ring consisting of a defocusing quadrupole and a focusing quadrupole and two bends (or drifts) use L=1 m, K=0.5 m-1 hints: 2cos(2pQ)=R11+R22 R12=bsin(2pQ) compute tune and b at focusing quadrupole compute tune shift as a function of quadrupole strength change DK over a range -0.1…+0.1 compute b as and plot vs. DK compute b as and plot vs DK

Transverse Dynamics - Measurements