82 Beam-based Correction and Optimization for Accelerators
Figure 3.4 shows the singular value spectra of the orbit response matrices
of the horizontal and vertical planes for the SPEAR3 (top plot) and NSLS-II
(bottom plot) storage rings. SPEAR3 has 58 horizontal correctors, 56 vertical
correctors, and 57 BPMS, distributed over 18 cells. NSLS-II has 180 horizontal correctors, 180 vertical correctors, and 180 BPMs over 30 cells. The
betatron tunes are [14.106, 6.177] for SPEAR3 and [33.22, 16.26] for NSLS-II.
Each SV spectrum has a pair of leading modes with comparable SV values.
The u- and v-vectors of these leading modes resemble betatron orbits of free
oscillating particles. Figure 3.5 shows the first SV mode for the SPEAR3 orbit
response matrix for both transverse planes. In the leading modes, the corrector
strengths vary according to the betatron phase to result in additive contributions to the orbit. The two leading modes have similar spatial patterns but
are out of phase by 90
◦ . These two modes contribute strongly to the real
and imaginary components of the integer stopband nearest to the betatron
tune. At the lower end of the SV spectrum, there is a floor of small singular
values. The ratio of the leading SV to the SVs on the floor is about 500 for
SPEAR3. The same ratio for NSLS-II is about 2000. The SPEAR3 horizontal
orbit response matrix has one extremely small singular value, despite having
one more corrector than the BPMs in the plane. This SV mode represents a
singularity of the orbit correction system. It corresponds to a local pattern in
the double-waist chicane area of the lattice. The singular mode needs to be
removed from the calculation of the pseudo-inverse matrix in Eq. (3.25).
An example of storage ring orbit correction with SVD is shown in Figure 3.6. In this example orbit errors were generated in the SPEAR3 lattice
by introducing random misalignment errors to all quadrupole magnets, with
the rms horizontal offset of 100 µm. The horizontal orbit errors are shown in
the top plot. Orbit correction was done with 6 SVs or 56 SVs in the calculation of the pseudo-inverse matrix. The resulting orbits after the corrections
are applied to the lattice model are shown in the middle plot. The bottom
plot shows the kick angles on all 58 correctors. Using 6 SVs, the rms kick
angle of all correctors is 0.29 mrad, which brings the rms orbit distortion from
2.35 mm to 0.71 mm. Using 56 SVs, the rms kick angle is 5 times stronger
(at 1.62 mrad), only to bring the rms orbit further down to 0.19 mm. The
example shows that the bulk of the orbit errors can be suppressed with only
a few SV modes; yet many SV modes are required in order to correct the fine
details of the orbit errors.
It is worth pointing out that the predicted residual orbit errors, x 0 +R∆θ,
have rms values of 0.72 mm for the 6-SV case and 0.004 mm for the 56-SV
case, respectively. The prediction was accurate for the few SV case, but not
as good for the 56-SV case. This is because the orbit response matrix depends
on the linear optics, which varies with the closed orbit due to the feed-down
effect of sextupoles. The orbit response matrix measured around the reference
orbit is different from the one around a closed orbit with large distortions.
In this case, typically several iterations are applied to correct the orbit. The
orbit response matrix does not need to be updated for each iteration. As long
Figure 3.4 shows the singular value spectra of the orbit response matrices
of the horizontal and vertical planes for the SPEAR3 (top plot) and NSLS-II
(bottom plot) storage rings. SPEAR3 has 58 horizontal correctors, 56 vertical
correctors, and 57 BPMS, distributed over 18 cells. NSLS-II has 180 horizontal correctors, 180 vertical correctors, and 180 BPMs over 30 cells. The
betatron tunes are [14.106, 6.177] for SPEAR3 and [33.22, 16.26] for NSLS-II.
Each SV spectrum has a pair of leading modes with comparable SV values.
The u- and v-vectors of these leading modes resemble betatron orbits of free
oscillating particles. Figure 3.5 shows the first SV mode for the SPEAR3 orbit
response matrix for both transverse planes. In the leading modes, the corrector
strengths vary according to the betatron phase to result in additive contributions to the orbit. The two leading modes have similar spatial patterns but
are out of phase by 90
◦ . These two modes contribute strongly to the real
and imaginary components of the integer stopband nearest to the betatron
tune. At the lower end of the SV spectrum, there is a floor of small singular
values. The ratio of the leading SV to the SVs on the floor is about 500 for
SPEAR3. The same ratio for NSLS-II is about 2000. The SPEAR3 horizontal
orbit response matrix has one extremely small singular value, despite having
one more corrector than the BPMs in the plane. This SV mode represents a
singularity of the orbit correction system. It corresponds to a local pattern in
the double-waist chicane area of the lattice. The singular mode needs to be
removed from the calculation of the pseudo-inverse matrix in Eq. (3.25).
An example of storage ring orbit correction with SVD is shown in Figure 3.6. In this example orbit errors were generated in the SPEAR3 lattice
by introducing random misalignment errors to all quadrupole magnets, with
the rms horizontal offset of 100 µm. The horizontal orbit errors are shown in
the top plot. Orbit correction was done with 6 SVs or 56 SVs in the calculation of the pseudo-inverse matrix. The resulting orbits after the corrections
are applied to the lattice model are shown in the middle plot. The bottom
plot shows the kick angles on all 58 correctors. Using 6 SVs, the rms kick
angle of all correctors is 0.29 mrad, which brings the rms orbit distortion from
2.35 mm to 0.71 mm. Using 56 SVs, the rms kick angle is 5 times stronger
(at 1.62 mrad), only to bring the rms orbit further down to 0.19 mm. The
example shows that the bulk of the orbit errors can be suppressed with only
a few SV modes; yet many SV modes are required in order to correct the fine
details of the orbit errors.
It is worth pointing out that the predicted residual orbit errors, x 0 +R∆θ,
have rms values of 0.72 mm for the 6-SV case and 0.004 mm for the 56-SV
case, respectively. The prediction was accurate for the few SV case, but not
as good for the 56-SV case. This is because the orbit response matrix depends
on the linear optics, which varies with the closed orbit due to the feed-down
effect of sextupoles. The orbit response matrix measured around the reference
orbit is different from the one around a closed orbit with large distortions.
In this case, typically several iterations are applied to correct the orbit. The
orbit response matrix does not need to be updated for each iteration. As long