Linear and Nonlinear Lattice Correction Via Betatron Phase NOCE Workshop W. Guo, E.Blum Sep. 21, 2017

Outline Introduction to the NSLS-II 2. The concept of phase correction 3. 1mr resolution and application to NSLS-II linear lattice Nonlinear correction scheme Nonlinear correction results and validation 6. Summary

NSLS-II Layout and Main Parameters 200 MeV Linac 15 nC/160-300 ns dp/p=0.5% , ε=50 µmrad Storage Ring Main Parameters C. 792 m L (straights) 6.6 & 9.3m Εx/y (w. DW) 0.9 nm/10pm αc .00037 ρ 25m I (total) 500mA U (w. DW) 0.675MeV σδ (w. DW) 0.094% δ(bucket) 4.1% Chromaticity +2, +2 frf 499.68 MHz f (landau) 1.5 GHz h 1320 Vrf 4.8 MV σl 11.5ps Q (bunch) 1.2nC t (Touschek) >3hrs f (Top-Off) 1/min Stability Δx/y<0.1 σx/y 3 GeV Booster 158m, 20 mA 1 Hz, dp/p=0.08%, ε =40 nmrad 30-cell storage ring NSLS-II schematic

Operation Status Construction started in FY08 Commissioning of the SR started in April 2014 Operation started in October 2014 Reliability has achieved >90%, user time is close to 5000 hours There are about 15 beam lines operating on a daily basis. In 2016 there were about 1000 unique users.

The Concept of Phase Correction Oscillation at BPM Define the phase advance eror: Response matrix Quadrupole strength correction Measured phase error   Benefit Fast measurement Independent of BPM gain and tilt

Relation between Phase and Amplitude * * P. Castro, PAC’93, p2103

Improved Phase Calculation Fourier Transform For signal x(n), n=1,2,3,…. N New Method 1. Naff fit for the frequency; 2. fit for the phase: Analytically:

Resolution: Vary a Quadrupole and Compare Scale factor fx=1.033, error bar 1.7mr Phase Difference Black: simulation, red: measurement error bar from 10 measurements Improved FT: Fit for the frequency and the phase

Typical Residue and Comparison with LOCO The phase error can be corrected to +/- 10 mr LOCO determines the residual beta-beat is about 1% in both planes. Possible reasons of the residue: chromatic effect and in-accurate modeling

Dynamic Aperture Optimization Frist order chromatic terms (5) Frist order geometric terms (5) D1,n Amplitude tune dependence Second order chromaticity

Sextupole Correction: the Scheme Linear lattice and coupling must be corrected. 1. Change the orbit by a horizontal corrector, or by tuning the rf frequency 2. Measure the phase change induced by the sextupoles 3. Compare with the model to obtain the phase error Repeat at many correctors and momentum offsets to break degeneracy and improve precision Measure amplitude dependent de-tuning , and nonlinear chromaticity Assemble the target function and all the response matrices to calculate the correction

Completeness of the Constraints Δψy depends on h01020, h10020, h10200, h01200, h01110, h10110, h01020, h10020 The off-momentum phase correction corrects the chromatic term: h11001, h00111, h02001, h20001, h00021, h00201,h10002 Leading order detuning terms:

Comparison with Model Upper: measurement and comparison with the model Lower: sextupole error and phase convergence

Correction Results Rms phase error before and after correction Comparison of the dynamic aperture

Algorithm Validation One of the 54 sextupole power supplies was lowered by 10 A, or ΔK2=-4.2m-3 Measured and compared with the original lattice 3. The algorithm identifies the changed power supply

Physical Meaning of the Nonlinear Correction The off-momentum lattice correction has straightforward significance The lattice correction for the orbit perturbed by a horizontal corrector can be understood as 1)The orbit wave is the same as betatron oscillation (with a disruption at the corrector location) 2)Correcting the phase is equivalent to restoring the transfer matrix for the oscillating bunch which undergoes sextupole focusing.

Summary Phase correction is complementary to LOCO for linear lattice correction; however, phase measurement is fast and independent of BPM calibration The proposed nonlinear correction approach treats the complete set of nonlinear constraints. The nonlinear correction method has been verified at NSLS-II.

Acknowledgement W.X. Cheng, J. Choi, Y. Hidaka, B. Podobedov, S. Kramer, V. Smalyuk, T. Shaftan, F. Willeke, Xi Yang, L. Yu The coordination group The operations group