148 Beam-based Correction and Optimization for Accelerators
over multiple turns are self consistent, which puts a constraint on the initial
phase space coordinates. Therefore, potentially, the initial phase space coordinates can also be fitted. However, since the oscillation amplitudes decrease
with time due to damping and decoherence in the experimental data, the
number of turns to be tracked with one initial coordinate should be limited.
The fitting parameters can include the quadrupole gradients and the BPM
gains when only the linear optics is concerned. Skew quadrupole gradients and
BPM rolls and crunch coefficients can also be fitted to account for the crossplane coupling.
The direct fitting scheme can also be applied to one-pass lattices. In this
case there is no summation over turns and the summation over BPMs starts
with i = 2 in Eq. (5.47). To adequately sample the phase space, a pair of
corrector magnets upstream of BPMs 0 and 1 for each transverse planes are
required to scan the beam trajectory. In each plane, it is necessary to scan
many combinations of (x, x
) or (y, y
) with phase angles that cover the full
range of [0, 2π). Different phase angles are needed because the data points with
the same phase angle, as obtained by scaling the strengths of both correctors in
proportions, are redundant information. Lattice fitting for rings can be treated
in the same manner as for the one-pass systems by considering the ring as a
transport line, using BPM data of each turn as an independent sample.
Because of the lack of temporal betatron oscillations, the methods of deriving phase advances and beta functions described earlier in this chapter do
not apply to one-pass lattices. The direct fitting scheme is particularly important for transport lines and linacs. It can also be applied to the commissioning
of storage rings before the stored beam is established. The BPM data fitting
method has been experimentally demonstrated using a section of the SPEAR3
storage ring [61] and the LCLS linac and transport lines [128].
In the LCLS linear optics fitting study [128], both horizontal and vertical trajectories are scanned on a phase space grid with two pairs of corrector
magnets upstream of the lattice of concern. For the linac section, backward
tracking was used as the pair of BPMs separated by a long drift are located
downstream of the linac. As the beam energy increases in the linac, the action
variables for (x, x
) or (y, y
) decrease. This effect is accounted for in the
tracking model. The fitted quadrupole gradients successfully recovered the
strengths of the matching quadrupoles. There were substantial differences between the fitted quadrupole gradients and the ideal model for a few magnets
as they were manually tuned during operation. For the transport lines after
the linac, both forward and backward tracking were used and the results were
consistent with the operation setting.
The direct fitting method for storage ring optics measurement can be tested
with the NSLS-II turn-by-turn BPM data discussed in the previous section.
The pair of BPMs separated by the first long straight section are used to
calculate the initial phase space coordinates. The two BPMs are separated
by a distance of 9.87 m and there are two harmonic sextupoles (SH1) in between, next to the BPMs. The nonlinear kick by the second sextupole is not
over multiple turns are self consistent, which puts a constraint on the initial
phase space coordinates. Therefore, potentially, the initial phase space coordinates can also be fitted. However, since the oscillation amplitudes decrease
with time due to damping and decoherence in the experimental data, the
number of turns to be tracked with one initial coordinate should be limited.
The fitting parameters can include the quadrupole gradients and the BPM
gains when only the linear optics is concerned. Skew quadrupole gradients and
BPM rolls and crunch coefficients can also be fitted to account for the crossplane coupling.
The direct fitting scheme can also be applied to one-pass lattices. In this
case there is no summation over turns and the summation over BPMs starts
with i = 2 in Eq. (5.47). To adequately sample the phase space, a pair of
corrector magnets upstream of BPMs 0 and 1 for each transverse planes are
required to scan the beam trajectory. In each plane, it is necessary to scan
many combinations of (x, x
) or (y, y
) with phase angles that cover the full
range of [0, 2π). Different phase angles are needed because the data points with
the same phase angle, as obtained by scaling the strengths of both correctors in
proportions, are redundant information. Lattice fitting for rings can be treated
in the same manner as for the one-pass systems by considering the ring as a
transport line, using BPM data of each turn as an independent sample.
Because of the lack of temporal betatron oscillations, the methods of deriving phase advances and beta functions described earlier in this chapter do
not apply to one-pass lattices. The direct fitting scheme is particularly important for transport lines and linacs. It can also be applied to the commissioning
of storage rings before the stored beam is established. The BPM data fitting
method has been experimentally demonstrated using a section of the SPEAR3
storage ring [61] and the LCLS linac and transport lines [128].
In the LCLS linear optics fitting study [128], both horizontal and vertical trajectories are scanned on a phase space grid with two pairs of corrector
magnets upstream of the lattice of concern. For the linac section, backward
tracking was used as the pair of BPMs separated by a long drift are located
downstream of the linac. As the beam energy increases in the linac, the action
variables for (x, x
) or (y, y
) decrease. This effect is accounted for in the
tracking model. The fitted quadrupole gradients successfully recovered the
strengths of the matching quadrupoles. There were substantial differences between the fitted quadrupole gradients and the ideal model for a few magnets
as they were manually tuned during operation. For the transport lines after
the linac, both forward and backward tracking were used and the results were
consistent with the operation setting.
The direct fitting method for storage ring optics measurement can be tested
with the NSLS-II turn-by-turn BPM data discussed in the previous section.
The pair of BPMs separated by the first long straight section are used to
calculate the initial phase space coordinates. The two BPMs are separated
by a distance of 9.87 m and there are two harmonic sextupoles (SH1) in between, next to the BPMs. The nonlinear kick by the second sextupole is not