98 Beam-based Correction and Optimization for Accelerators
In a circular accelerator, the orbit response at a BPM is given by (see
Eq. (2.8))
R ij ≡
dx
(c)
i
dθ j
= [M
(ij) (I − M
(j) )
−1 ] 12 ,
(4.11)
where M
(j) is the one-turn transfer matrix at the exit edge of the corrector magnet and [·] 12 indicates the (1, 2) element of the matrix in the square
bracket. Since the orbit response in a ring is determined by the transfer matrix, it is closely related to the linear optics. In fact, the orbit response is given
explicitly by the beta functions at the BPM and the corrector, the betatron
phase advance in between, and the betatron tune by Eq. (2.10) for the case
without linear coupling.
Each column of the orbit response matrix is a differential orbit from the
original orbit. It corresponds to one sampling point of the betatron phase space
in the same sense as the orbit shift from the origin for the turn-by-turn or passby-pass BPM data (see Figure 4.1). There is no difference between the passby-pass BPM data with corrector scans and the orbit response data (or more
accurately, trajectory response) for one-pass systems. For the case of rings, the
only difference between turn-by-turn BPM data and closed orbit responses is
that the closed orbit is abruptly changed at the location of the corrector by
the kick it applies. At all other locations, the closed orbit response represents
the free betatron motion. The closed orbit measurement is typically much
more accurate than the turn-by-turn orbit measurement as the former employs
averaging of beam signals over many turns. However, turn-by-turn BPM data
of hundreds or more turns could be collected. By properly processing the multiturn BPM data, the same precision could be achieved with turn-by-turn BPM
data as the closed orbit data in linear optics measurement.
With the measured orbit data that sample the betatron phase space, the
next step is extract the linear optics from the data. Typically this is done by
fitting a lattice model with the data. In the next section we will describe the
fitting method for the orbit response matrix.
4.2 FITTING ORBIT RESPONSE MATRIX TO LATTICE MODEL
There are two goals in analyzing the orbit data for linear optics: optics measurement and optics correction. The linear optics in a storage ring is often
described by the beta functions and the betatron phase advances. The purpose of optics measurement is to derive these functions from the data. For
optics correction, the goal is to compensate the errors in the linear optics
such that the linear optics is as close to the design optics as possible.
It would seem natural to accomplish these two goals in two steps: first
derive the optics functions with optics measurement, then use the optics functions to determine the required changes to the quadrupoles for optics correction. This is actually the case for several methods that use the turn-by-turn
In a circular accelerator, the orbit response at a BPM is given by (see
Eq. (2.8))
R ij ≡
dx
(c)
i
dθ j
= [M
(ij) (I − M
(j) )
−1 ] 12 ,
(4.11)
where M
(j) is the one-turn transfer matrix at the exit edge of the corrector magnet and [·] 12 indicates the (1, 2) element of the matrix in the square
bracket. Since the orbit response in a ring is determined by the transfer matrix, it is closely related to the linear optics. In fact, the orbit response is given
explicitly by the beta functions at the BPM and the corrector, the betatron
phase advance in between, and the betatron tune by Eq. (2.10) for the case
without linear coupling.
Each column of the orbit response matrix is a differential orbit from the
original orbit. It corresponds to one sampling point of the betatron phase space
in the same sense as the orbit shift from the origin for the turn-by-turn or passby-pass BPM data (see Figure 4.1). There is no difference between the passby-pass BPM data with corrector scans and the orbit response data (or more
accurately, trajectory response) for one-pass systems. For the case of rings, the
only difference between turn-by-turn BPM data and closed orbit responses is
that the closed orbit is abruptly changed at the location of the corrector by
the kick it applies. At all other locations, the closed orbit response represents
the free betatron motion. The closed orbit measurement is typically much
more accurate than the turn-by-turn orbit measurement as the former employs
averaging of beam signals over many turns. However, turn-by-turn BPM data
of hundreds or more turns could be collected. By properly processing the multiturn BPM data, the same precision could be achieved with turn-by-turn BPM
data as the closed orbit data in linear optics measurement.
With the measured orbit data that sample the betatron phase space, the
next step is extract the linear optics from the data. Typically this is done by
fitting a lattice model with the data. In the next section we will describe the
fitting method for the orbit response matrix.
4.2 FITTING ORBIT RESPONSE MATRIX TO LATTICE MODEL
There are two goals in analyzing the orbit data for linear optics: optics measurement and optics correction. The linear optics in a storage ring is often
described by the beta functions and the betatron phase advances. The purpose of optics measurement is to derive these functions from the data. For
optics correction, the goal is to compensate the errors in the linear optics
such that the linear optics is as close to the design optics as possible.
It would seem natural to accomplish these two goals in two steps: first
derive the optics functions with optics measurement, then use the optics functions to determine the required changes to the quadrupoles for optics correction. This is actually the case for several methods that use the turn-by-turn