DOI: 10.1201/9780429434358-5
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121
C H A P T E R 5
Linear optics
measurement and
correction - II
CONTENTS
5.1
Optics measurement with harmonic analysis . . . . . . . . . . . . . . 122
5.1.1 Phase advance measurement . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.2 Precise tune determination from turn-by-turn
BPM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.3 Determination of beta functions with phase
advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Global analysis of turn-by-turn BPM data . . . . . . . . . . . . . . . . 129
5.2.1 A model for the turn-by-turn BPM data . . . . . . . . . . 130
5.2.2 Principal component analysis (PCA) . . . . . . . . . . . . . . 131
5.2.3 Independent component analysis (ICA) . . . . . . . . . . . 137
5.3
Optics correction with turn-by-turn data . . . . . . . . . . . . . . . . . 143
5.3.1 Fitting optics functions to lattice model . . . . . . . . . . . 143
5.3.2 Fitting turn-by-turn data directly to lattice model 147
5.4
Other methods for optics correction . . . . . . . . . . . . . . . . . . . . . . . 151
5.5
Summary for linear optics correction . . . . . . . . . . . . . . . . . . . . . . 152
As discussed in the previous chapter, turn-by-turn BPM data taken with coherent beam motion sample the lattice similarly as the closed-orbit response
matrix. Compared to the latter, turn-by-turn BPM data can be collected
within seconds, instead of tens of minutes. There is no complication from
calibration errors or rolls of corrector magnets. In addition, linear optics parameters, such as beta functions and phase advances, can be directly measured
from the data, before fitting the data to the model. Advanced algorithms can
be used to process the data taken by BPMs distributed around the ring in
a consistent manner, which allows the separation of random noise, contaminating signals, and unrelated processes from the betatron motion signals.
121