96 Beam-based Correction and Optimization for Accelerators
can also be obtained by fitting a set of parameters from which the symplectic
matrix is constructed. The Courant-Snyder parameters can be used for the
2 × 2 transfer matrix. A 4 × 4 transfer matrix, can be constructed with 10
parameters using the procedure described in Eqs. (2.59-2.63) [105, 61].
In the case of storage rings, phase space orbit data at one location over
successive turns can be used to fit the one-turn transfer matrix with the above
method. The orbit for turn n can be seen as the data from BPM 1 and the
orbit for turn n + 1, the data from BPM 2, with n = 1, 2, · · · , N − 1. In a
storage ring light source, there is usually one pair of BPMs separated by a long
drift space for each period (where the drift space is used to house insertion
devices). Therefore, the one-turn transfer matrix at the straight section of
each period and the transfer matrix across the period can be determined. The
transfer matrices can be used to calculate the Courant-Snyder parameters and
phase advances.
The method of measuring the transfer matrix with pairs of BPMs separated by drift spaces is applicable only to a limited number of locations and
may not provide enough sampling for global optics correction. Even for storage ring light sources, the BPMs in the straight sections account for a small
fraction of BPMs. The optics information for the BPMs in the arcs is missing.
Other methods that make use of the orbit data at all BPMs are needed in
order to measure and correct the linear optics throughout the lattice.
Depending on the type of orbit data in use, these methods can be put in
two categories: methods of turn-by-turn (or pass-by-pass for one-pass systems)
BPM data and the orbit response methods.
For the turn-by-turn (TbT) methods, BPM data are taken with the beam
shifted away from the normal orbit. The beam orbit deviations propagate
through the lattice to downstream locations and are recorded by the BPMs.
Because the deviations at different locations are related through the linear
optics (Eq. (4.4)), it is possible to extract optics information from the BPM
data. In a circular accelerator, after the beam motion is excited, the subsequent free betatron oscillation moves the orbit along the phase space ellipse
according to
x(n) =
2βJ cos(2πνn + ξ),
(4.9a)
x
(n) = −
2J
β
sin(2πν + ξ) −
α
β
x(n),
(4.9b)
where ν is the betatron tune, n is the turn number, and J and ξ are the action
and phase variables given by the initial condition, respectively. Because the
fractional betatron tune is typically not equal to a low order rational number,
within a few hundreds of turns, the beam will spread out over the entire
ellipse and hence sample the phase space from all angles, as illustrated in
Figure 4.1 (a).
In a one-pass system, the sampling of the betatron phase space can be
achieved by scanning two upstream corrector magnets [33, 128]. The two
can also be obtained by fitting a set of parameters from which the symplectic
matrix is constructed. The Courant-Snyder parameters can be used for the
2 × 2 transfer matrix. A 4 × 4 transfer matrix, can be constructed with 10
parameters using the procedure described in Eqs. (2.59-2.63) [105, 61].
In the case of storage rings, phase space orbit data at one location over
successive turns can be used to fit the one-turn transfer matrix with the above
method. The orbit for turn n can be seen as the data from BPM 1 and the
orbit for turn n + 1, the data from BPM 2, with n = 1, 2, · · · , N − 1. In a
storage ring light source, there is usually one pair of BPMs separated by a long
drift space for each period (where the drift space is used to house insertion
devices). Therefore, the one-turn transfer matrix at the straight section of
each period and the transfer matrix across the period can be determined. The
transfer matrices can be used to calculate the Courant-Snyder parameters and
phase advances.
The method of measuring the transfer matrix with pairs of BPMs separated by drift spaces is applicable only to a limited number of locations and
may not provide enough sampling for global optics correction. Even for storage ring light sources, the BPMs in the straight sections account for a small
fraction of BPMs. The optics information for the BPMs in the arcs is missing.
Other methods that make use of the orbit data at all BPMs are needed in
order to measure and correct the linear optics throughout the lattice.
Depending on the type of orbit data in use, these methods can be put in
two categories: methods of turn-by-turn (or pass-by-pass for one-pass systems)
BPM data and the orbit response methods.
For the turn-by-turn (TbT) methods, BPM data are taken with the beam
shifted away from the normal orbit. The beam orbit deviations propagate
through the lattice to downstream locations and are recorded by the BPMs.
Because the deviations at different locations are related through the linear
optics (Eq. (4.4)), it is possible to extract optics information from the BPM
data. In a circular accelerator, after the beam motion is excited, the subsequent free betatron oscillation moves the orbit along the phase space ellipse
according to
x(n) =
2βJ cos(2πνn + ξ),
(4.9a)
x
(n) = −
2J
β
sin(2πν + ξ) −
α
β
x(n),
(4.9b)
where ν is the betatron tune, n is the turn number, and J and ξ are the action
and phase variables given by the initial condition, respectively. Because the
fractional betatron tune is typically not equal to a low order rational number,
within a few hundreds of turns, the beam will spread out over the entire
ellipse and hence sample the phase space from all angles, as illustrated in
Figure 4.1 (a).
In a one-pass system, the sampling of the betatron phase space can be
achieved by scanning two upstream corrector magnets [33, 128]. The two