1. Tolerances & Tuning of the ATF2 Final Focus Line
James Jones
ASTeC, Daresbury Laboratory

2. Use as.mad lattice (original NLC-like solution from M. Pivi)
Field Tolerances
Use as.mad lattice (original NLC-like solution from M. Pivi)
Calculate increase in beam size in both planes for the following cases:
Individual Multipole for each magnet separately
Individual Multipoles for all magnets together with the same amplitude
Individual Multipoles for all magnets with amplitude relative to maximum strength of design field
Results given in terms of K-values:
Factorials accounted for!

3. Field Tolerances – Individual Quads Multipole Errors
Normal
Tolerance for 10% beam growth due to the multipole field in an individual magnet
Multipoles from:
Order 10 (20 pole) : Red..
Order 5 (10 pole): Light Green..
Order 2 (Quad): Orange
Absolute values of Multipole strength
Small asymmetry between +ve and –ve of ~±20%
Skew

4. Field Tolerances – All Quads Multipole Errors Relative
Tolerance for 10% beam growth due to the multipole field in all Quads together, with the strength relative to the design quadrupole strength
Maximum K2 is -3.5 m-2
Normal
Skew

5. Field Tolerances - Summary
Have tolerances for all multipole components up to 20pole
Can be used to understand the requirements from the magnet designs
Data is available over a wide range of values so it is very simple to analyse the beam size increase from each multipole component separately
Analysis of the effects of combined multipole errors is more ambiguous
Requires dedicated tracking studies
Already set-up for the original Hitachi Type 5 quadrupoles (cf Cherrill Spencer!)
All data available as excel S/Sheets, data files ….

6. Position Tolerances
Calculate increase in beam size and change in spot position in both planes for the following cases:
Individual Errors on each magnet
All magnets with the same static error
All magnets with random errors, averaged over 10 seeds
The effects of the correction system were also included:
No correction at all
Correction using several kickers in the FF-line with a BPM at every quad – fast correction
Correction using linear tuning knobs ( waists, horiz. and vert. dispersion) – static correction

7. Position Tolerances – No Correction  Jitter
Analyse the beam line with random errors 3
Average change in beam size or position over 10 random seeds
Limited by time…
Estimates the random jitter levels required in a timescale less than the correction system can operate
Quadrupoles only (2% increase)
X-plane: 580nm
Y-plane: 2.7nm
Roll Angle: 1.5rad
Increasing Horizontal Error
Vertical BS
Horizontal BS

8. Position Tolerances – 3 iteration Correction  Fast Error
Same basic analysis as with no correction (with FD)
Run the SVD algorithm 3 times per random seed
No attempt to correct the dispersion
Quadrupoles only (2% increase)
X-plane: 580nm
Y-plane: 3.0nm
Roll Angle: 1.5rad
Makes only a small improvement...

9. Tuning Knobs Introduction
To achieve the nano-meter spot size, as well as nano-meter stability, it is required to ‘tune’ the ATF2 final focus line to provide a base alignment.
Tuning involves minimising the dominant error sources causing emittance blow-up, and thus beam size increase, as seen at the IP.
Dominant error sources can be analysed as 1st and 2nd (and higher) order transfer matrices
Leads to correction of these dominant matrix terms
(some) direct correlation with observables
Also analysis of the beam-matrix
Corrects directly the error in the beam size and rotation
No relation with observables

10. Traditional Tuning Analysis
First step to determine the dominant error terms is to analyse the FF line under a variety of error conditions.
From this calculate standard deviation of all matrix terms
Using MAD’s survey command
Need to convolute this information with knowledge of which terms are dominant in increasing the beam size at the IP
This is done by placing a MATRIX element within MAD, at the IP, and varying independently the 216 matrix terms
Calculate the change in beam size with each term simply as the standard deviation in the two planes.
Some terms we suspect will be important, and these are added regardless of the above analysis (T126…)

11. Tuning Knobs
Once we have knowledge of the dominant terms we can create some ‘tuning knobs’ for them.
This is generally as simple as solving several simultaneous equations. Want to produce orthogonal knobs!
Performance factor is the linearity of the ‘knob’ as well as the orthoganility, given as the ratio of primary and secondary terms, for each ‘knob’.
Most knobs optimised through the use of a Simplex algorithm which can dramatically improve the performance.
For 1st order knobs it is possible to use only three magnets to create the tuning ‘knob’, minimises the possible errors, and makes the optimisation easier.
For 2nd order knobs we generally require all 5 FF sextupoles.

12. 1st Order Tuning Knobs
Despite what I have just said, we don’t correct the first order matrix terms but actually the two Beta Function waists, and the two dispersions, as seen at the IP.
Knobs created using transverse motion of the sextupoles
These knobs work on observable parameters so correction is simply minimising directly the relevant observable.
Turns out that dispersion knob is next to useless when the machine has a large number of errors, so is not used.

13. 2nd Order Knobs
Use all 5 sextupoles to improve the orthoganility of each knob.
Use sextupole field and roll to create tuning knobs.
Required 2nd-Order knobs are:
T124 T322 T326 T344 (∆roll)
T122 T126 T166 T346 T324 (∆K2)
Since can’t directly observe these terms, use a Brent's method algorithm to optimise on the beam size.
Very computationally intensive

14. Position Tolerances – 2 iteration tuning algorithm
Use all tuning knobs for 2 iterations
Run the SVD algorithm 2 times per random seed
Quadrupoles only (2% increase)
X-plane: 79mm
Y-plane: 140nm
Roll Angle: 1.2rad
The use of the tuning knobs relaxes the tolerances to a large extent, but it must be remembered this is the effective post BBA tolerance.

15. Conclusions
Finally have a generalised method of analysing the tolerances on the ATF magnets in terms of field or position errors
Can be used with real errors to analyse effects on beam very simply
Have produced a set of specifications for the multipole components and for the position tolerances for all of the final focus line quadrupoles and sextupoles
Data is not specific to a given tolerance specification (i.e. 2% or 10% beam size increase)
Analysis using tuning knobs is ongoing, and correction works well
Analysis of other tuning scenarios, as well as model of BBA may also be useful