Linear Optics Corrections

1
Linear Optics Corrections

• RHIC Todd Satogata, Mei Bai, Jen Niedziela
• AGS Vincent Schoefer, Kevin Brown, Leif Ahrens,
  
• November 3, 2006

• Objective
• Reduce machine/model difference in optics, in
  both directions!
• Orbit Response Matrix
• AC Dipole Optics
• AGS Orbit Response Matrix

2
Response Matrix Background

• Originally used by Corbett, Lee, and Ziemann at
  SPEAR, 1993
• Popularized by Safranek at NSLS X-Ray Ring,
  1994-6
• Orbit response matrix (with coupling/nonlinear
  feeddown)
• where (?x,?y) are corrector setting changes and
  (x,y) are measured difference orbits from these
  corrector settings.
• (There are also dispersive terms in Rij when
  fRFconstant, as was the case in RHIC
  measurements.)
• Compare model and measured R response matrices,
  and iteratively make model changes to converge
  to agreement.
• Model changes include quad gradients, BPM/corr
  gains,
• Measures gradient errors, BPM/corr gain errors,
  skew errors,
• RHIC AT storage lattices include snake matrices
  from Waldo

3
Response Matrix Background

• The difference between Rmodel and Rmeasured is
  written as a single vector with length
  (NbpmsxNcorrs), and expanded in terms of variable
  changes ?vk, that include quad gradient errors,
  corrector and BPM gain errors, skew errors, etc
• This can be solved for ?vk with SVD, assuming a
  good model for . The quadrupole
  gradient error dependence is nonlinear, so the
  SVD solution is iterated through the model until
  it converges. Weight with BPM noise ?I

4
Simulation Results No BPM Noise

• Tested yellow store lattice, random 0.1 to 0.1
  quad errs
• With no BPM noise, fitting is perfect
• Lattice is therefore nondegenerate

5
Issue BPM Noise from 10 Hz

• Average orbit noise at store 30-40 um horizontal
  peak-peak
• 7800 turn averaging gives 15-20 um peak-peak
• Need 10 orbits per data point to achieve 5 um
  BPM noise, including BPM change to average orbit
  sampling period

6
Simulation Results 30 um BPM Noise

• With 30 um BPM noise, error bars are larger than
  errors
• Fits are inexact, but method still converges
  not unique
• Beta beating is reduced by two orders of magnitude

7
Simulation Results 30um BPM Noise and
BPM/Corrector Gains

• Simulation fits of BPM/corrector gains, 30 um BPM
  noise
• Fits are very good, reduce gain errors by factor
  of 20
• BPM/gain errors and optics can be fit with 30 um
  BPM noise

8
APEX Data Acquisition and ORM Data Summary (pp35)
Machine State Date BPMs Correctors
Yellow injection May 10 2006 159161 117117
Yellow store May 30 2006 8990 113116
Blue store May 30 2006 147145 112112

• Data acquisition and analysis
• Takes 1-2 hours for all correctors in either RHIC
  ring
• Both rings done in parallel for storage
  measurement
• 3 average orbits acquired for each corrector
  setting
• Need to automate analysis, tedious BPM/corrector
  alignment and orbit averaging
• Measure/monitor tunes throughout measurement
• No good blue injection ORM data acquired during
  Run-6
• Yellow injection data taken during blue
  polarimeter pumpdown

9
Beam Experiment Analysis pp35 Yellow Store Q1/8/9

• Use only arc BPMs, fit Q1/Q8/Q9 quadrupoles
• Appears to converge, chi2 reduces by factor 10
• Beta beating is 10-15 horizontal, over 40
  vertical!
• Fitted quadrupole errors are still too large by
  at least x10
• Example yi6-qf8 fit converges to a change of 5
  ! Check data

10
Beam Experiment Analysis pp35 Yellow Store Q1/4-9

• Use only arc BPMs, but now fit Q1/4-9 quadrupoles
• Appears to converge better, chi2 reduces by
  factor 20
• Beta beating is consistent with previous result,
  smoother
• Fitted quadrupole errors are even larger, up to
  8!
• Need to double-check data these errors are
  unphysical

11
Measured Yellow Store AC Dipole Beta Beat

• Measured beta beat in the Yellow ring at store
  energy fit is only yi6-qf1 has a 10 KL error!
• The gradient error is obtained by
  . Here R can be calculated from the
  model
• plan
• to finish the data analysis Yellow injection
  and Blue at store and injection
• dedicated gradient error measurement at injection

Not consistent with ORM data!
M. Bai
12
Future Plans

• Careful analysis of existing data
• Remove suspicious data, hand-match full ORM
  matrices
• Aggressive singular value cuts existing too
  permissive?
• Blind baseline APEX
• Insert and measure known quadrupole errors
• Improve BPMs, reduce data acquisition noise
• Change BPM average orbit sampling time
• Increase number of orbits acquired per corrector
  setting
• Reduce dimensionality
• Typically 20-25 of correctors are used (FNAL,
  ALS)
• Speeds up data acquisition, analysis, reduces
  memory requirements
• Coupling analysis
• Reduced dimensionality should allow this

13
APEX Requests and Requirements

• 2-3h Blue testing at injection during startup
  period
• Decouple separate/measure tunes for lattice fit
  save model
• Measure dispersion before and after ORM
  measurement
• BPMs averaging set to 7800 turns to limit 10 Hz
  (nth turn?)
• Measure BPM average orbit noise at all BPMs
• Compare difference orbit measurement/prediction,
  dispersion
• Measure ORM/correct optics/recheck if differences
  above noise
• AC dipole and ORM optics measurements
• Fit all quads to improve model, IR knobs to
  improve machine
• (Fast) blind baseline measurement for AC dipole
  analysis
• Remeasure after analysis and correction for
  validation
• 3h Storage optics measurement/correction
• Repeat above setup, including BPM noise baseline
• 6 bunches, uncogged, acquire both rings
  simultaneously
• AC dipole measurement immediately after ORM
  measurement

14
AGS ORM Measurement Data

• At extraction, near integer resonance, June 15
  2006
• 2-3mm response, need to filter out some bad data

V. Schoefer
15
AGS MAD-X and Matlab/AT Comparison
16
AGS Simulated Error ORM Fit

• Simulated/matched 5 errors in a03, c17, e03, j17
  quads
• All four simulated errors fit to high precision
• To Do Noise analysis, filter input data for LOCO
  analysis

17
Conclusions

• RHIC ORM Simulations
• RHIC storage lattice looks nondegenerate (even
  triplets)
• BPM/corrector gains, 0.1 gradient errors, and
  beta beating correction can be found with
  realistic BPM noise (30um)
• RHIC ORM Data Analysis and APEX
• Present analysis doubtful, converges to large
  (10) errors
• Hand-correlate data to triple-check for
  oversights
• Reduce BPM average orbit noise (sampling, more
  data/point)
• Parasitic setup/testing with Blue beam at
  injection
• 3h for ramping/store gradient error
  measurement/correction
• RHIC AC Dipole APEX
• Parasitic (e.g. after) above ORM APEX,
  complementary
• AGS ORM
• Studying extraction lattice, matches MAD-X
• Evaluating susceptibility to BPM noise
• Have ORM data in hand for analysis

18

19
Twiss Parameter Measurements

• Hoffstaetter, Keil, and Xiao (EPAC 2002)
• Iteratively fit cos/sin variations of (1) to
  measured Rij
• Alternate between corrector and BPM (?,?) fitting
• Least-squares is solved with SVD of

20
Response Matrix Dimensions at RHIC

• Each RHIC ring has
• 334 BPM measurements (167 per plane)
• 234 correctors (117 per plane)
• Total response matrix size is 78156 points
• APS/FNAL use only 25-40 correctors, all BPMs
  (Sajeev)
• Fitting parameters
• 334 BPM measurement gains (offsets subtract out)
• 232 corrector strength gains (assume two are
  perfect)
• 117 path length changes for horizontal correctors
• 246 main/IR quadrupole gradients (or 72 IRs)
• (144 chromatic sextupole offsets)
• Total fitting parameters 929 (more or less)
• Total size of fit matrix 78156929 70 Mpoints
• Grows larger very quickly with additional fit
  parameters
• Minimize dimensionality to improve speed