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

2. 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

3. NSLS-II Layout and Main Parameters
200 MeV Linac 15 nC/ ns
dp/p=0.5% , ε=50 µmrad
Storage Ring Main Parameters
C m
L (straights) & 9.3m
Εx/y (w. DW) nm/10pm αc
ρ m
I (total) mA
U (w. DW) MeV
σδ (w. DW) %
δ(bucket) %
Chromaticity , +2
frf MHz
f (landau) GHz h
Vrf MV
σl ps
Q (bunch) nC
t (Touschek) >3hrs
f (Top-Off) /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

4. 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.

5. 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

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

7. 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:

8. 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

9. 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

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

11. 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

12. 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:

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

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

15. 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

16. 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.

17. 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.

18. 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