Orbit and trajectory correction 75
the beam to go through the good field regions of the magnets and to maintain
a large acceptance by the physical apertures.
A good choice of the target orbit is to steer the beam through the magnetic
centers of the quadrupole magnets. The magnetic center of a quadrupole is
where the magnetic field crosses zero. It is typically the center of the good field
region in the magnet. When the beam goes through the quadrupole center,
it receives no angular kick from the magnet and hence the beam orbit is not
altered, regardless of the strength of the quadrupole magnet. One benefit of
such an orbit is that the strength of the quadrupole magnet, and hence the
linear optics, can be changed, without perturbing the beam orbit. In other
words, the control of linear optics is decoupled from the control of the beam
orbit.
The magnetic center of a quadrupole can be found through a procedure
called beam based alignment (BBA) [94]. BBA is based on the very fact that
the beam receives no kick at the quadrupole center. In the procedure the BPM
nearest to the quadrupole is used to register the quadrupole center. An orbit
corrector in the machine is used to change the orbit at the quadrupole. The
betatron phase advance between the corrector and the quadrupole needs to
be at an appropriate value for the corrector to be effective. At each orbit, the
strength of the quadrupole is changed to I 0 − ∆I and then I 0 + ∆I, while
the orbit shifts due to the quadrupole strength step change are recorded by
all available BPMs. For each observing BPM j, the orbit shifts induced by
the quadrupole modulation, ∆x ij , can be plotted against the orbit reading
on the nearest BPM, x i , for the i’th corrector induced orbit. The data points
will form one line for each BPM if the orbits are not far from the quadrupole
center. With a selection of observing BPMs, there will a collection of lines in
the plot, all of which cross at one particular point. The cross point indicates
the BPM reading that corresponds to the quadrupole center. Such a plot is
called a “bow tie” plot, an example of which is shown in Figure 3.3.
The BBA procedure described in the above does not need a calibration for
the quadrupole magnet or the BPMs. Nor does it utilize a lattice model. It
finds the quadrupole center directly as measured by the raw reading on the
nearest BPM. This procedure works in a circular accelerator as well as in a
one-pass lattice. In the latter case, a corrector upstream of the quadrupole is
used to change the orbit at the quadrupole, and BPMs downstream are used
to detect the orbit shifts due to the quadrupole modulation. The horizontal
and vertical offsets are determined separately.
Sometimes there is no corrector magnet conveniently located to alter the
orbit at the quadrupole of interest, or sometimes there is no BPM located
next to the quadrupole. This would more likely happen in a transport line.
In such a situation, it is still possible to find the quadrupole center by stepping its strength and monitoring the orbit changes at downstream BPMs.
Using a lattice model between the quadrupole and the BPMs, the kick angle
the beam to go through the good field regions of the magnets and to maintain
a large acceptance by the physical apertures.
A good choice of the target orbit is to steer the beam through the magnetic
centers of the quadrupole magnets. The magnetic center of a quadrupole is
where the magnetic field crosses zero. It is typically the center of the good field
region in the magnet. When the beam goes through the quadrupole center,
it receives no angular kick from the magnet and hence the beam orbit is not
altered, regardless of the strength of the quadrupole magnet. One benefit of
such an orbit is that the strength of the quadrupole magnet, and hence the
linear optics, can be changed, without perturbing the beam orbit. In other
words, the control of linear optics is decoupled from the control of the beam
orbit.
The magnetic center of a quadrupole can be found through a procedure
called beam based alignment (BBA) [94]. BBA is based on the very fact that
the beam receives no kick at the quadrupole center. In the procedure the BPM
nearest to the quadrupole is used to register the quadrupole center. An orbit
corrector in the machine is used to change the orbit at the quadrupole. The
betatron phase advance between the corrector and the quadrupole needs to
be at an appropriate value for the corrector to be effective. At each orbit, the
strength of the quadrupole is changed to I 0 − ∆I and then I 0 + ∆I, while
the orbit shifts due to the quadrupole strength step change are recorded by
all available BPMs. For each observing BPM j, the orbit shifts induced by
the quadrupole modulation, ∆x ij , can be plotted against the orbit reading
on the nearest BPM, x i , for the i’th corrector induced orbit. The data points
will form one line for each BPM if the orbits are not far from the quadrupole
center. With a selection of observing BPMs, there will a collection of lines in
the plot, all of which cross at one particular point. The cross point indicates
the BPM reading that corresponds to the quadrupole center. Such a plot is
called a “bow tie” plot, an example of which is shown in Figure 3.3.
The BBA procedure described in the above does not need a calibration for
the quadrupole magnet or the BPMs. Nor does it utilize a lattice model. It
finds the quadrupole center directly as measured by the raw reading on the
nearest BPM. This procedure works in a circular accelerator as well as in a
one-pass lattice. In the latter case, a corrector upstream of the quadrupole is
used to change the orbit at the quadrupole, and BPMs downstream are used
to detect the orbit shifts due to the quadrupole modulation. The horizontal
and vertical offsets are determined separately.
Sometimes there is no corrector magnet conveniently located to alter the
orbit at the quadrupole of interest, or sometimes there is no BPM located
next to the quadrupole. This would more likely happen in a transport line.
In such a situation, it is still possible to find the quadrupole center by stepping its strength and monitoring the orbit changes at downstream BPMs.
Using a lattice model between the quadrupole and the BPMs, the kick angle