158 Beam-based Correction and Optimization for Accelerators
of the linear coupling RDTs, the skew quadrupole errors in the lattice model
can be fitted with the least-square method, using Eq. (6.9) to calculate the
model RDT values. The measured vertical dispersion function at the BPMs
also needs to be included in the fitting input data in order to reduce the
vertical emittance contribution from the vertical dispersion.
Transfer matrices: With the full phase space coordinates determined
from two adjacent BPMs, the one-turn transfer matrices at the BPMs can be
calculated with Eq. (4.8) or by fitting parameters to construct a symplectic
transfer matrix. The two BPMs can be separated by a simple drift space, or a
short section with a few magnets. In the latter case, the model transfer matrix
between the two BPMs can be used to calculate the angle coordinates, x
and
y
(Eq. (4.5)).
The one-turn transfer matrices contain both linear optics and coupling
information. The lattice model can be fitted by minimizing the differences
between the measured and model one-turn transfer matrices at multiple BPM
locations. The skew quadrupole gradients are the fitting parameters for the
lattice model. BPM calibration errors and rolls are also included as fitting
parameters.
6.2 COUPLING CORRECTION EXPERIMENTS
Coupling correction with orbit response matrix has become a standard practice at many electron storage rings. Typically the vertical emittance can be
corrected to the level of a few picometer (pm)-rads. In some cases, the vertical
emittance has reached the sub-pm level [30].
In the NSLS-II optics correction experiment discussed in the previous chapter, BPM coupling coefficients and skew quadrupoles were also fitted for the
data sets before and after the corrections (using the ICA method) were applied to the machine. Figure 6.1 shows the ratio of the rms orbit response in
the vertical plane to that of the horizontal plane due to horizontal correctors
(H), or similarly the ratio of horizontal responses to vertical responses due to
vertical correctors (V), before and after the corrections were applied. The average value of such ratios was reduced from 0.054 to 0.025 by the corrections.
Linear coupling is also observed in the turn-by-turn BPM data. A direct
measure of the level of coupling is the amplitude ratios of the normal modes
in the cross-plane and in the primary plane, defined as
r 1 =
√
a 2 + b 2
√
A 2 + B 2
,
r 2 =
√
c 2 + d 2
√
C 2 + D 2
.
(6.10)
Figure 6.2 shows the r 1 and r 2 ratios for the turn-by-turn BPM data taken
before and after corrections in the same experiment as in Figure 6.1. The
average ratios were reduced from r 1 = 0.094 and r 2 = 0.107 to 0.046 and
0.052, respectively.
of the linear coupling RDTs, the skew quadrupole errors in the lattice model
can be fitted with the least-square method, using Eq. (6.9) to calculate the
model RDT values. The measured vertical dispersion function at the BPMs
also needs to be included in the fitting input data in order to reduce the
vertical emittance contribution from the vertical dispersion.
Transfer matrices: With the full phase space coordinates determined
from two adjacent BPMs, the one-turn transfer matrices at the BPMs can be
calculated with Eq. (4.8) or by fitting parameters to construct a symplectic
transfer matrix. The two BPMs can be separated by a simple drift space, or a
short section with a few magnets. In the latter case, the model transfer matrix
between the two BPMs can be used to calculate the angle coordinates, x
and
y
(Eq. (4.5)).
The one-turn transfer matrices contain both linear optics and coupling
information. The lattice model can be fitted by minimizing the differences
between the measured and model one-turn transfer matrices at multiple BPM
locations. The skew quadrupole gradients are the fitting parameters for the
lattice model. BPM calibration errors and rolls are also included as fitting
parameters.
6.2 COUPLING CORRECTION EXPERIMENTS
Coupling correction with orbit response matrix has become a standard practice at many electron storage rings. Typically the vertical emittance can be
corrected to the level of a few picometer (pm)-rads. In some cases, the vertical
emittance has reached the sub-pm level [30].
In the NSLS-II optics correction experiment discussed in the previous chapter, BPM coupling coefficients and skew quadrupoles were also fitted for the
data sets before and after the corrections (using the ICA method) were applied to the machine. Figure 6.1 shows the ratio of the rms orbit response in
the vertical plane to that of the horizontal plane due to horizontal correctors
(H), or similarly the ratio of horizontal responses to vertical responses due to
vertical correctors (V), before and after the corrections were applied. The average value of such ratios was reduced from 0.054 to 0.025 by the corrections.
Linear coupling is also observed in the turn-by-turn BPM data. A direct
measure of the level of coupling is the amplitude ratios of the normal modes
in the cross-plane and in the primary plane, defined as
r 1 =
√
a 2 + b 2
√
A 2 + B 2
,
r 2 =
√
c 2 + d 2
√
C 2 + D 2
.
(6.10)
Figure 6.2 shows the r 1 and r 2 ratios for the turn-by-turn BPM data taken
before and after corrections in the same experiment as in Figure 6.1. The
average ratios were reduced from r 1 = 0.094 and r 2 = 0.107 to 0.046 and
0.052, respectively.