1. Vertical emittance measurement and modeling Correction methods
Vertical beam emittance correction with independent component analysis method
S.Y. Lee, Indiana University
XBPM workshop at NSRRC
Introduction
ICA method
Vertical emittance measurement and modeling
Correction methods
Conclusion

2. [Photons/(s-mm2-mrad2-0.1%bandwidth)]

3. Modified NSLS2
DBA Lattice
15*11.14m
15*7m
C=780.3 m
Qx = Qy =
Qx' = Qy' =
emit=2.2834nm

4. DB Non-achromat
Modified NSLS2
DB Lattice
Qx = Qy =
Qx' = Qy' =
emit= nm

5. Effective emittance
A smaller emittance in a NON_ACHROOMATIC lattice does not necessary produce a smaller effective emittance!!

6. The Quadruple-bend achromat
A simple calculation with L2=1.45L1, we find that the emittance should scale like [2/(1+31/3)]3=0.54.
It is easy to find the following facts:
Matching is relatively easy (as compared to TBA lattices)
The chromatic properties is as good as the DB lattice!
The emittance obeys the scaling law: γ2θ3, i.e. 30 cells (15 QBA cells) will give about 1.24 nm emittance without any damping wiggler!

9. How about the vertical emittance? Where it come from? How to measure?
[Photons/(s-mm2-mrad2-0.1%bandwidth)]
How about the vertical emittance?
Where it come from?
How to measure?
And more importantly, how to correct it?
But, it is normally larger than we want it to be!
The vertical emittance arises essentially from x-z coupling!
But how?

10. Guignard provided a detailed derivation of the vertical emittance in Phys. Rev. E 51, 6104 (1995).

12. QBA lattice

13. QBA lattice

16. Two main sources of the vertical emittance are the linear betatron coupling and the residual vertical dispersion function!
Linear coupling can be measured and corrected by measuring the minimum separation of the betatron tunes or by the digitized the betatron motion.

17. The vertical dispersion function arises from the horizontal dipole field errors that cause the vertical closed orbit distortion vs. the momentum deviation!

19. How to measure the vertical dispersion function?
Measure closed orbit vs. the momentum deviation
ICA method!

20. The model of BPM turn-by-turn data
The turn-by-turn beam position signal is a combination of various source signals.
For the i’th BPM
is the mixing matrix
The source signals include betatron motion, synchrotron motion and others (e.g., ground motion, power supply ripples, etc.)
Form a matrix of the BPM data (m BPMs and N turns)

21. The Principle
The source signals are assumed to be narrow-band with non-overlapping spectra, so their un-equal time covariance matrices are diagonal.
The mixing matrix A diagonalizes the un-equal time sample covariance matrices simultaneously.

22. The Algorithm* - 1
Diagonalize the equal-time covariance matrix (data whitening)
D1,D2 are diagonal
Set to remove noise
with
Construct an intermediate “whitened” data matrix
which satisfies
This is the PCA-based MIA. Matrix z is just the temporal patterns of MIA modes.
* We use the second order blind identification (SOBI) algorithm of A. Belouchrani, et al. IEEE transaction on signal processing 45, 434 (1997).

23. The Algorithm - 2
Jointly diagonalize the un-equal time covariance matrices of matrix z of selected time-lag constants.
for
Then
The columns of A (spatial vectors) and corresponding rows (temporal vectors) of s are the resulting modes.

24. Linear Lattice Functions Measurements
There are two betatron modes because each BPM sees different phase.
There is one synchrotron mode:

25. An impression of raw data; 2500 turns from turn 1; at location L01
Measurements of turn-by-turn data have traditionally being used to measure and model accelerators. Employing the independent component analysis (ICA), we were able to measure the betatron and synchrotron tunes, betatron amplitude functions, dispersion functions for the entire ramping cycle (submitted for publication).
The amplitude of oscillation is about 0.4 mm; Notice the bursts due to the pinger, which is fired about every 225 turns.
An impression of raw data; 2500 turns from turn 1; at location L01

26. 1. Data whitening with noise reduction
A is m  n mixing matrix,
n(t) is white gaussian noise.
1. Data whitening with noise reduction
with
Set to remove noise
D1,D2 are diagonal
2. Compute and jointly diagonalize covariance matrices with different time-lags
3. Compute mixing matrix A and source signals s

28. Using the tracking results as “ experimental data”, and carrying out ICA analysis, we can obtain Dz at all BPM locations.

30. Fourier analysis of the ICA results will provide information on stopbands, which is then compared with theory. (quadrupole errors)

31. Fourier analysis of the ICA results will provide information on stopbands, which is then compared with theory. (dipole errors)

32. Correction Methods
Linear coupling
Linear coupling + Dispersion stopbands

36. Conclusions
The vertical beam emittance in an electron storage ring is mainly determined by two factors: the linear betatron coupling and the spurious vertical dispersion generated by magnet errors.
We find that the contribution of spurious vertical dispersion is larger than that generated by the linear betatron coupling.
Using the independent component analysis (ICA) method, we develop stopband correction to reduce the vertical emittance. We demonstrate our method by making ICA and correction to a quadruple-bend achromatic (QBA) low emittance lattice. Six families of skew quadrupoles can effectively minimize both the vertical dispersion and the linear betatron coupling.
ICA method is a powerful tool for future studies in beam physics
Taking turn-by-turn data during the steady state and carrying out ICA analysis may be able to be resolve or calibrate the XBPM data.