DOI: 10.1201/9780429434358-6
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153
C H A P T E R 6
Coupling and nonlinear
dynamics correction
CONTENTS
6.1
Coupling measurement and correction methods . . . . . . . . . . . 154
6.1.1 Off-diagonal blocks of orbit response matrix . . . . . . 154
6.1.2 Amplitude and phase of normal modes via ICA . . . 155
6.1.3 Other methods of coupling correction . . . . . . . . . . . . . 157
6.2
Coupling correction experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3
Correction of nonlinear dynamics errors . . . . . . . . . . . . . . . . . . 161
The linear coupling between the two transverse planes is caused by the
skew quadrupole components in the lattice, which may come from rolls of
quadrupole magnets, vertical orbit offsets in sextupoles, random magnetic
field errors, or insertion devices. Linear coupling can be measured with BPMs,
through either the orbit responses of corrector magnets or the turn-by-turn
orbits of a beam undergoing coherent betatron oscillations. Quantities that
characterize the linear coupling can be derived from the BPM data and are
fitted to the lattice model to determine the linear coupling error sources. The
methods for linear coupling measurements include fitting (the off-diagonal
blocks of) the orbit response matrix, fitting the amplitudes and phase advances of the normal modes in the cross-plane, fitting the resonance driving
terms for the sum and difference resonances, and direct fitting of the turn-byturn BPM data. The coupling errors can be corrected with skew quadrupole
magnets. Skew quadrupole magnets are often coils on multi-function magnets.
In electron storage rings the linear coupling contributes to the equilibrium
vertical emittance. Another source of the vertical emittance is the spurious
vertical dispersion, which can be generated by vertical bending or coupled
from the horizontal dispersion. Sources of vertical bending include the vertical
correctors and vertical orbit offsets in quadrupoles. Skew quadrupoles and
vertical orbit offsets in sextupoles located in dispersive sections couple the
horizontal dispersion to the vertical plane. The vertical dispersion can be
corrected with skew quadrupole magnets located in dispersive regions.
153