144 Beam-based Correction and Optimization for Accelerators
Error propagation may also be used, for example, for beta functions and phase
advances obtained through fitting the sinusoidal model. The weights are included to adjust the relative importance of the data types. This is necessary
because there could be systematic errors in the data types. For example, beta
functions obtained with the sinusoidal fitting method and the ICA method are
subject to BPM calibration errors. In some cases, the beta functions can be
excluded by setting w βx = w βy = 0. This is acceptable because the errors in
the beta functions and the phase advances are closely related (see Eqs. (2.27)
and (2.30)) – to some extent the beta beating data could be seen as redundant
information. The weight of the dispersion function can be increased to achieve
a high precision of dispersion control.
The fitting parameters are the quadrupole gradients. These can usually
be the same quadrupole parameters as the orbit response matrix fitting. The
terms in χ
2 can be represented by the residual vector, r, such that the objective function takes the standard form, χ
2 = f (p) = r
T r. The least-square
fitting methods discussed in the previous chapter can then be applied. The
degeneracy problem due to the cross-coupling between adjacent quadrupole
magnets is common to all optics fitting, including the present case with optics functions as input data. Therefore, the constrained fitting technique in
section 4.2.7 is generally needed.
Optics correction with ICA for turn-by-turn BPM data analysis has been
experimentally demonstrated on the RHIC collider [109] and on the NSLS-II
storage ring [51, 125]. In the NSLS-II experiment 1024 turns of orbit data from
180 BPMs are decomposed into two pairs of betatron normal modes and the
synchrotron mode. The Fourier spectra of the data from one BPM (located
at a dispersive region) and the temporal patterns of the normal modes are
shown in Figure 5.12 (top plots). The amplitudes of the normal modes on the
horizontal (middle plot) and vertical (bottom plot) BPMs are also shown. The
oscillation amplitude is about 0.2 mm. Because of linear coupling, the vertical
betatron oscillation shows up on the horizontal orbit and vice versa.
From the spatial patterns, the betatron phase advances of the normal
modes are calculated. The differences of the measured phase advances with
the ideal model are shown in Figure 5.13. The values measured by ICA are
compared to the lattice model calibrated with the orbit response matrix data
taken at the same time as the turn-by-turn BPM data. The differences between
the measurements by the two methods are small. The beta functions, phase
advances, and the dispersion function are used to fit the lattice model using
Eq. (5.46). The fitted BPM gains and quadrupole gradient errors are compared
to the orbit response matrix fitting results in Figure 5.14 and 5.15, respectively.
The fitting results between the two methods are very similar. Because of the
different constraints applied, the fitted
∆K
K by ICA is slightly smaller than
LOCO, while the two fitted lattices have almost the same optics errors.
The fitted quadrupole errors by ICA were applied to the machine for optics
correction. After three iterations of corrections, each time using a new data
set of turn-by-turn BPM data taken on the updated machine, the optics errors
Error propagation may also be used, for example, for beta functions and phase
advances obtained through fitting the sinusoidal model. The weights are included to adjust the relative importance of the data types. This is necessary
because there could be systematic errors in the data types. For example, beta
functions obtained with the sinusoidal fitting method and the ICA method are
subject to BPM calibration errors. In some cases, the beta functions can be
excluded by setting w βx = w βy = 0. This is acceptable because the errors in
the beta functions and the phase advances are closely related (see Eqs. (2.27)
and (2.30)) – to some extent the beta beating data could be seen as redundant
information. The weight of the dispersion function can be increased to achieve
a high precision of dispersion control.
The fitting parameters are the quadrupole gradients. These can usually
be the same quadrupole parameters as the orbit response matrix fitting. The
terms in χ
2 can be represented by the residual vector, r, such that the objective function takes the standard form, χ
2 = f (p) = r
T r. The least-square
fitting methods discussed in the previous chapter can then be applied. The
degeneracy problem due to the cross-coupling between adjacent quadrupole
magnets is common to all optics fitting, including the present case with optics functions as input data. Therefore, the constrained fitting technique in
section 4.2.7 is generally needed.
Optics correction with ICA for turn-by-turn BPM data analysis has been
experimentally demonstrated on the RHIC collider [109] and on the NSLS-II
storage ring [51, 125]. In the NSLS-II experiment 1024 turns of orbit data from
180 BPMs are decomposed into two pairs of betatron normal modes and the
synchrotron mode. The Fourier spectra of the data from one BPM (located
at a dispersive region) and the temporal patterns of the normal modes are
shown in Figure 5.12 (top plots). The amplitudes of the normal modes on the
horizontal (middle plot) and vertical (bottom plot) BPMs are also shown. The
oscillation amplitude is about 0.2 mm. Because of linear coupling, the vertical
betatron oscillation shows up on the horizontal orbit and vice versa.
From the spatial patterns, the betatron phase advances of the normal
modes are calculated. The differences of the measured phase advances with
the ideal model are shown in Figure 5.13. The values measured by ICA are
compared to the lattice model calibrated with the orbit response matrix data
taken at the same time as the turn-by-turn BPM data. The differences between
the measurements by the two methods are small. The beta functions, phase
advances, and the dispersion function are used to fit the lattice model using
Eq. (5.46). The fitted BPM gains and quadrupole gradient errors are compared
to the orbit response matrix fitting results in Figure 5.14 and 5.15, respectively.
The fitting results between the two methods are very similar. Because of the
different constraints applied, the fitted
∆K
K by ICA is slightly smaller than
LOCO, while the two fitted lattices have almost the same optics errors.
The fitted quadrupole errors by ICA were applied to the machine for optics
correction. After three iterations of corrections, each time using a new data
set of turn-by-turn BPM data taken on the updated machine, the optics errors