154 Beam-based Correction and Optimization for Accelerators
Because the vertical dispersion and the linear coupling both contribute to
the vertical emittance and are both corrected with skew quadrupoles, the corrections of the two are often combined and are referred to as vertical emittance
correction or coupling correction.
Errors in a storage ring lattice also cause the nonlinear beam dynamics
behavior to deviate from the design, resulting in a reduction of the dynamic
aperture and the local momentum apertures. It would be ideal to correct the
errors and restore the nonlinear beam dynamics performance.
In this chapter we discuss the methods of coupling correction. Ideas for
nonlinear beam dynamics correction are also discussed.
6.1 COUPLING MEASUREMENT AND CORRECTION METHODS
6.1.1 Off-diagonal blocks of orbit response matrix
If there is no linear coupling, the one-turn 4 × 4 transfer matrix at a location
in a circular accelerator consists of two diagonal 2 × 2 blocks that represent
the motion of the two transverse planes, respectively. The off-diagonal blocks
are all zeros. Accordingly, the closed-orbit response of an orbit corrector is
limited in the plane of the kick. However, when linear coupling error sources
are introduced to the lattice, the one-turn transfer matrix will have non-zero
off-diagonal blocks. The values of the off-diagonal block elements are related to
the strengths and locations of the error sources. Consequently, the off-diagonal
blocks of the orbit response matrix, R xy and R yx (Eq. (4.12)), will also be
non-zero and the values of these blocks are connected to the error sources.
By fitting the skew quadrupole parameters in the lattice model to minimize the differences between the off-diagonal blocks of the measured and model
orbit response matrices, a lattice model that reproduces the linear coupling
behavior of the machine can be obtained [102]. The skew quadrupole parameters can be the gradients of actual skew quadrupoles. Corrections can be
applied to these magnets to compensate the coupling error sources, resulting
in a reduction of linear coupling in the machine. The fitting parameters can
also be the rolls of the normal quadrupole magnets. This is equivalent to fitting skew quadrupole components at the locations of the normal quadrupoles.
If large rolls on quadrupoles are found, the magnets could be realigned.
The rolls of corrector magnets and BPMs also cause cross-plane orbit responses in the measurements. These effects are not related to the linear coupling in the lattice. Because the orbit response patterns of the corrector and
BPM rolls are different from those of the skew quadrupole components, these
parameters can be determined by fitting.
In practice the off-diagonal orbit responses are almost always fitted together with the diagonal blocks such that the linear optics and coupling errors
are determined simultaneously. This is appropriate as linear optics errors have
an impact over the effects of the coupling sources, and vice versa.
Because the vertical dispersion and the linear coupling both contribute to
the vertical emittance and are both corrected with skew quadrupoles, the corrections of the two are often combined and are referred to as vertical emittance
correction or coupling correction.
Errors in a storage ring lattice also cause the nonlinear beam dynamics
behavior to deviate from the design, resulting in a reduction of the dynamic
aperture and the local momentum apertures. It would be ideal to correct the
errors and restore the nonlinear beam dynamics performance.
In this chapter we discuss the methods of coupling correction. Ideas for
nonlinear beam dynamics correction are also discussed.
6.1 COUPLING MEASUREMENT AND CORRECTION METHODS
6.1.1 Off-diagonal blocks of orbit response matrix
If there is no linear coupling, the one-turn 4 × 4 transfer matrix at a location
in a circular accelerator consists of two diagonal 2 × 2 blocks that represent
the motion of the two transverse planes, respectively. The off-diagonal blocks
are all zeros. Accordingly, the closed-orbit response of an orbit corrector is
limited in the plane of the kick. However, when linear coupling error sources
are introduced to the lattice, the one-turn transfer matrix will have non-zero
off-diagonal blocks. The values of the off-diagonal block elements are related to
the strengths and locations of the error sources. Consequently, the off-diagonal
blocks of the orbit response matrix, R xy and R yx (Eq. (4.12)), will also be
non-zero and the values of these blocks are connected to the error sources.
By fitting the skew quadrupole parameters in the lattice model to minimize the differences between the off-diagonal blocks of the measured and model
orbit response matrices, a lattice model that reproduces the linear coupling
behavior of the machine can be obtained [102]. The skew quadrupole parameters can be the gradients of actual skew quadrupoles. Corrections can be
applied to these magnets to compensate the coupling error sources, resulting
in a reduction of linear coupling in the machine. The fitting parameters can
also be the rolls of the normal quadrupole magnets. This is equivalent to fitting skew quadrupole components at the locations of the normal quadrupoles.
If large rolls on quadrupoles are found, the magnets could be realigned.
The rolls of corrector magnets and BPMs also cause cross-plane orbit responses in the measurements. These effects are not related to the linear coupling in the lattice. Because the orbit response patterns of the corrector and
BPM rolls are different from those of the skew quadrupole components, these
parameters can be determined by fitting.
In practice the off-diagonal orbit responses are almost always fitted together with the diagonal blocks such that the linear optics and coupling errors
are determined simultaneously. This is appropriate as linear optics errors have
an impact over the effects of the coupling sources, and vice versa.