222 Beam-based Correction and Optimization for Accelerators
Since a major path for lattice errors to affect the nonlinear beam dynamics
performance is by introducing betatron phase advance errors between the
sextupoles, the linear optics knobs may also be used for tuning the DA and
LMA. Linear optics can be controlled to below 1% beta beating through beambased correction. However, for the low emittance rings with a large number
of strong sextupoles, even at such a low level, optics errors can still have a
significant impact on the DA. In addition, optics correction normally only
targets the optics functions at the BPMs; there can be larger optics errors at
the sextupoles. Quadrupoles could be combined into groups to form tuning
knobs that affect the phase advances on the sextupoles with various patterns,
using the phase advance response matrix with respect to the quadrupoles.
The SPEAR3 DA optimization experiment: In the beam-based DA
optimization for the SPEAR3 storage ring [60], the injection efficiency was
used as the objective function and the injection kicker bump was reduced.
The SPEAR3 storage ring has 18 double-bend achromat (DBA) cells, 14 of
which are standard cells and the remaining 4 are matching cells. There are
two pairs of sextupoles in each cell. Originally SPEAR3 had only 4 sextupole
families: the SF/SD families for the standard cells and the SFM/SDM families
for the matching cells. For the purpose of optimizing DA/LMA for a lattice
upgrade, the SF/SD families were split into 4 families each, 3 of which consist
of SF (SD) magnets in 4 cells symmetrically distributed around the ring, while
the last one only consists of sextupoles in the two cells at the centers of the
two arcs. The sextupole family distribution in half of the ring may be labeled
M-X 2 -X 3 -X 4 -X 5 -X 4 -X 3 -X 2 -M, where M stands for the matching cells and X i
for the standard cells. The other half is in the mirror symmetry. For example,
all the SF sextupoles in X 2 cells make one family. There are a total of 10
sextupole families.
To be able to freely change the sextupole settings without changing the
chromaticities, combined knobs were created by applying singular value decomposition (SVD) to the chromaticity response matrix, R chrom = USV
T .
The chromaticities, (C x , C y )
T , depend on the sextupole families through linear relationships. The derivatives of the chromaticities with respect to the sextupole families constitute the chromaticity response matrix, which is a 2 × 10
matrix for the SPEAR3 case. It has two non-zero singular values, which corresponds to the sextupole combinations that can change chromaticities. The
other 8 singular values are zeros and the corresponding column vectors in
V form the basis vectors for the null space; any sextupole changes in the
null space do not change the chromaticities. The eight null space basis vectors are used as tuning knobs in the SPEAR3 DA optimization experiments.
Some SVD modes of the SPEAR3 chromaticity response matrix are shown in
Figure 8.12.
The injection efficiency was measured through the beam current change
in the storage ring and the intensity of the injected beam. Each data point
was collected with injection for 10 seconds to reduce the noise level. The
noise sigma for injection efficiency is about 3%. Figure 8.13 shows the evolution of the injection efficiency and 4 out of the 10 sextupole families for an
Since a major path for lattice errors to affect the nonlinear beam dynamics
performance is by introducing betatron phase advance errors between the
sextupoles, the linear optics knobs may also be used for tuning the DA and
LMA. Linear optics can be controlled to below 1% beta beating through beambased correction. However, for the low emittance rings with a large number
of strong sextupoles, even at such a low level, optics errors can still have a
significant impact on the DA. In addition, optics correction normally only
targets the optics functions at the BPMs; there can be larger optics errors at
the sextupoles. Quadrupoles could be combined into groups to form tuning
knobs that affect the phase advances on the sextupoles with various patterns,
using the phase advance response matrix with respect to the quadrupoles.
The SPEAR3 DA optimization experiment: In the beam-based DA
optimization for the SPEAR3 storage ring [60], the injection efficiency was
used as the objective function and the injection kicker bump was reduced.
The SPEAR3 storage ring has 18 double-bend achromat (DBA) cells, 14 of
which are standard cells and the remaining 4 are matching cells. There are
two pairs of sextupoles in each cell. Originally SPEAR3 had only 4 sextupole
families: the SF/SD families for the standard cells and the SFM/SDM families
for the matching cells. For the purpose of optimizing DA/LMA for a lattice
upgrade, the SF/SD families were split into 4 families each, 3 of which consist
of SF (SD) magnets in 4 cells symmetrically distributed around the ring, while
the last one only consists of sextupoles in the two cells at the centers of the
two arcs. The sextupole family distribution in half of the ring may be labeled
M-X 2 -X 3 -X 4 -X 5 -X 4 -X 3 -X 2 -M, where M stands for the matching cells and X i
for the standard cells. The other half is in the mirror symmetry. For example,
all the SF sextupoles in X 2 cells make one family. There are a total of 10
sextupole families.
To be able to freely change the sextupole settings without changing the
chromaticities, combined knobs were created by applying singular value decomposition (SVD) to the chromaticity response matrix, R chrom = USV
T .
The chromaticities, (C x , C y )
T , depend on the sextupole families through linear relationships. The derivatives of the chromaticities with respect to the sextupole families constitute the chromaticity response matrix, which is a 2 × 10
matrix for the SPEAR3 case. It has two non-zero singular values, which corresponds to the sextupole combinations that can change chromaticities. The
other 8 singular values are zeros and the corresponding column vectors in
V form the basis vectors for the null space; any sextupole changes in the
null space do not change the chromaticities. The eight null space basis vectors are used as tuning knobs in the SPEAR3 DA optimization experiments.
Some SVD modes of the SPEAR3 chromaticity response matrix are shown in
Figure 8.12.
The injection efficiency was measured through the beam current change
in the storage ring and the intensity of the injected beam. Each data point
was collected with injection for 10 seconds to reduce the noise level. The
noise sigma for injection efficiency is about 3%. Figure 8.13 shows the evolution of the injection efficiency and 4 out of the 10 sextupole families for an