1. ILC BDS Alignment, Tuning and Feedback Studies
Glen White
SLAC/Oxford
Feb 2006
Progress report and ongoing plans for BDS alignment and tuning strategy.

2. BDS Alignment and Tuning Simulations
Using ILC-IR1 20mrad BDS deck plus extraction line.
Start with expected post-survey magnet and BPM alignment tolerances, magnet errors and BPM resolutions.
Simulate BPM-Magnet alignment using Quad-shunting technique and fits to higher-order magnet moves (Sexts, Octs).
Move to BPM readings with measured alignment and optimise orbit.
Use orthogonal knobs for correction of linear IP aberrations using Sextupole movers.
Simulation tool used: Lucretia.

3. Initial Parameter Assumptions
All magnets have an associated BPM and x, y and roll movers.
Quads have x- and y- corrector dipoles.
Magnet RMS mis-alignment: 200um.
Assume initial BPM-magnet centre alignment of 30um.
Magnet rotation: 300urad.
RMS relative magnetic strength error with respect to model: 1e-4.
Magnet mover resolution (x & y): 50nm.
All magnets on their own movers, apart from last quad/sext/oct pair which are in cryo-module and have fixed offsets.
Cryo module bpm and magnet relative offsets 10um/100urad RMS.
Assume SD0/OC0 and SF1/OC1 co-wound with above relative magnetic centre offsets.
Assume whole module has position+angle movers.
BPM resolution: 1um for all quads, assume cavities on sextupoles with 100nm resolution.
TESLA bunch parameters (gaussian beam) with 0.1 % uncorl. E spread.
Track 2000 macro-particles per bunch.

4. Alignment & Tuning Strategy
Switch off Sextupoles & Octupoles.
Apply 1-1 Steering algorithm to align beam to measured BPM centres in Quads.
Use nulling Quad-shunting technique to get BPM-Quad alignments.
Use Quad movers to put Quads in a straight line in x and y with beam going through Quad field centres.
Get BPM- Sextupole/Octupole alignment with movers and use a fit to downstream BPM responses.
Switch on Sextupoles/Octupoles.
Use global BDS pulse-pulse feedback system to keep betatron orbit centred in BPMs.
Use Sextupole knobs to tune IP waist, dispersion. Use skew quads (4 in coupling section + SQ3FF) for coupling correction.

5. BPM-Quad Alignment Nulling Quad Shunting technique:
To get BPM-Quad offsets, use downstream 10 Quad BPMs for each Quad being aligned (using ext. line BPMs for last few Quads).
Switching Quad’s power (100% - 80%), use change in downstream BPM readouts to get Quad offset.
Move Quad and repeat until detect zero-crossing (using 1-d optimiser).
For offset measurement, use weighted-fit to downstream BPM readings based on ideal model transfer functions:

6. Quadrupole Alignment Results
Left: BPM-Quad alignments (1 seed).
Right: RMS Quad position post Quad mover alignment routine.
In reality, don’t want this large drift- use lattice matrix inversion to align quads in straight line centered in vacuum chamber and to collide beams.

7. Sextupole, Octopole Alignment
Use x-, y-movers on higher-order magnets and fit 2nd, 3rd order polynomials to downstream BPM responses (for Sext, Oct respectively).
Alignment is where 2nd, 3rd derivative is 0 from fits.

8. BDS Orbit Stabilisation
A global steering algorithm throughout the BDS is used to keep the orbit centered in the magnet BPMs whilst moving Sexts/changing skew quad strengths.
Sext moves when doing IP tuning must be accompanied by set-point changes for the magnets being moved.
Feedback also counter-acts ground motion/element drift.
Correction scheme: choose set of bpm’s and correctors. Then use Matlab’s lscov routine to obtain solution to (1), where R is the matrix of coupled transfer elements , b is the vector of x and y bpm readings and c is the corrector set to solve for.
lscov does a least-squares minimisation (2); where V is proportional to the bpm covariance matrix.
(1)
(2)

9. Linear IP Tuning Knobs
Correction of linear IP aberrations (x/y waist and dispersion) performed by sextupole moves.
X-offsets in sextupoles generate additional quad-component and can be used to compensate for waist and x-dispersion aberrations.
Y-offsets general additional skew-quad components and are used to correct IP vertical dispersion.
Y-offsets also generate coupling, SQ3FF is used to compensate for this.

10. IP Tuning Knobs
Use orthogonal x-moves of SF6, SF1, SD0 to tune on x/y waist and Dx.
Use orthogonal vertical moves of SD0 and SD4 and dK for SQ3FF to tune Dy and dominant coupling term (<x’y>).
Use additional 4 skew quads in BDS coupling section to perform orthogonal tweaks of other coupling terms.
SF6
SF1
SD0 -1
0.154
0.283
-0.020
0.616 1
0.843
0.016 Dy
Waist (x)
Waist (y) Dx
SD SD4

11. Application of IP Multi-Knobs
Knobs are not completely orthogonal after adding lattice errors etc.
Iteratively apply knobs, tuning on luminosity.
Starting conditions after initial alignment:
Beamsize (x/y): ~ 620/140 nm.
Dx/y : ~ 14 / 10 um.
Waist (x/y): ~ 0.3/0.6 (s1,2/3,4)).
Coupling: ~ 0.2/0.1/0.8/0.5 (xy,x’y,xy’,x’y’).
Initially fix vertical aberrations & dominant coupling (x’y) until no further improvement possible.
Then iterate through all knobs.

12. Feedback
All Quad BPMs and correctors are used in feedback. The final doublet string is steered to with the 2 Sext BPMs.
Orbit stabilised throughout tuning process with global steering algorithm. The IP is included as a BPM with resolution 1e-9 to mimic information from the beam-beam collision which allows the feedback to keep the beam within the capture range of the fast feedback system ~100nm. Lumi stability ~0.1%.
A simple, single gain feedback is applied to the BPM readings to damp RMS errors from finite BPM resolutions. This is tuned to damp on a 50 bunch timescale- to be optimised later.

13. Results Results from single seed:
Tuning on luminosity and tuning 1 knob at a time, finding the optimal by using 1-d optimisation routine (Matlab’s fminbnd).
Convergence to beam spot size x<600nm, y<6nm- to get about 80% of geometric Luminosity is fairly quick, ~200 pulses- about 40s of beam time.
Convergence beyond this is very slow, may take many hours of optimisation to get close to design.
It may be faster to use more beam diagnostics in final tuning phase, ie. Dispersion scans and coupling measurements…
Further speed improvement also possible with fine-tuning of optimisation routines etc.
Should be possible to finally converge on very close to design Lumi. Measured normal mode emittance growth <1%.

14. Future Work Improve speed of alignment. Include GM.
Simulate 2 beams- tune on luminosity (pair signals).
Include steering routine to get initial beam collisions.
Include LINAC to get real bunch shapes.
Integrated time-evolved simulation with initial tuning + pulse-pulse FB + intra-bunch FB.
Provides information on how often re-tuning necessary and most detailed luminosity performance estimate.