Linear optics measurement and correction - I 99
BPM data (to be discussed in the next chapter). Given the connection between
the orbit responses and the optics functions (see Eq. (2.10)), it is possible to
derive the optics functions directly from the orbit response data, for example,
by fitting the data for the beta functions and the phase advances at all BPMs
and correctors. However, this is an unnecessary step if the final goal is optics
correction and it could introduce errors. The typical approach of extracting
linear optics information from the orbit response matrix data is to fit for the
quadrupole errors with a lattice model. Through fitting, the lattice model is
calibrated, which means that it is made consistent with the measurements. It
can then be used to calculate the optics functions or other lattice parameters.
When there are linear optics errors, the dispersion function will also be
distorted. Measuring the dispersion function and including it in the fitting will
help determine the optics errors and make the fitting results more effective
for restoring the design dispersion.
The method of measuring and correcting storage ring linear optics by fitting orbit response matrix data and dispersion data is commonly referred to
as Linear Optics from Closed Orbit (LOCO). It was first successfully demonstrated on the NSLS rings [102]. The method was later implemented in an
easy-to-use program [104, 95] and its fitting algorithm was updated to handle
the degeneracy issue [49, 52, 50]. It has become a widely used tool, especially
in the synchrotron light source community.
4.2.1 Measured orbit response matrix
Because quadrupole magnets affect the optics functions in both transverse
planes, the orbit response matrices in both planes are used to fit the
quadrupole errors. The orbit responses in both planes can be arranged in
one matrix
R =
R xx R xy
R yx R yy
,
(4.12)
where the first x or y in the subscript indicates the plane of the BPMs and the
second indicates the plane of the orbit correctors. For example, R xx contains
the horizontal orbit responses of the horizontal correctors, and R xy contains
the horizontal orbit responses of the vertical correctors,
R xx =
dx
dθ x
, R xy =
dx
dθ y
,
(4.13)
where x is a column vector of all horizontal BPM readings, θ x,y are the column
vectors for the horizontal and vertical kick angles, respectively, and similarly,
R yx =
dy
dθ x
, R yy =
dy
dθ y
.
(4.14)
The off-diagonal blocks R xy and R yx are non-zero if there is linear coupling
between the x and y planes or rolls of BPMs or correctors about the s-axis.
BPM data (to be discussed in the next chapter). Given the connection between
the orbit responses and the optics functions (see Eq. (2.10)), it is possible to
derive the optics functions directly from the orbit response data, for example,
by fitting the data for the beta functions and the phase advances at all BPMs
and correctors. However, this is an unnecessary step if the final goal is optics
correction and it could introduce errors. The typical approach of extracting
linear optics information from the orbit response matrix data is to fit for the
quadrupole errors with a lattice model. Through fitting, the lattice model is
calibrated, which means that it is made consistent with the measurements. It
can then be used to calculate the optics functions or other lattice parameters.
When there are linear optics errors, the dispersion function will also be
distorted. Measuring the dispersion function and including it in the fitting will
help determine the optics errors and make the fitting results more effective
for restoring the design dispersion.
The method of measuring and correcting storage ring linear optics by fitting orbit response matrix data and dispersion data is commonly referred to
as Linear Optics from Closed Orbit (LOCO). It was first successfully demonstrated on the NSLS rings [102]. The method was later implemented in an
easy-to-use program [104, 95] and its fitting algorithm was updated to handle
the degeneracy issue [49, 52, 50]. It has become a widely used tool, especially
in the synchrotron light source community.
4.2.1 Measured orbit response matrix
Because quadrupole magnets affect the optics functions in both transverse
planes, the orbit response matrices in both planes are used to fit the
quadrupole errors. The orbit responses in both planes can be arranged in
one matrix
R =
R xx R xy
R yx R yy
,
(4.12)
where the first x or y in the subscript indicates the plane of the BPMs and the
second indicates the plane of the orbit correctors. For example, R xx contains
the horizontal orbit responses of the horizontal correctors, and R xy contains
the horizontal orbit responses of the vertical correctors,
R xx =
dx
dθ x
, R xy =
dx
dθ y
,
(4.13)
where x is a column vector of all horizontal BPM readings, θ x,y are the column
vectors for the horizontal and vertical kick angles, respectively, and similarly,
R yx =
dy
dθ x
, R yy =
dy
dθ y
.
(4.14)
The off-diagonal blocks R xy and R yx are non-zero if there is linear coupling
between the x and y planes or rolls of BPMs or correctors about the s-axis.