Linear optics measurement and correction - I 93
strength of a quadrupole is varied and the corresponding betatron tune shifts
are measured. Knowing the length of the quadrupole magnet and the current
to gradient conversion rate, the beta function at the location of the quadrupole
can be calculated with Eq. (2.22). The accuracy of the method is affected by
the hysteresis of the quadrupole magnet and calibration errors.
Because of the large number of potential error sources, it is not feasible to
correct all linear optics errors at the error sources. Instead, quadrupole magnets are used as knobs to compensate the linear optics errors globally. In early
accelerators, quadrupole magnets are often powered in series. The strengths
of individual quadrupoles can be changed with shunt resistors that divert a
fraction of the current. Quadrupole magnets in newer machines are often individually powered. Their strengths can be changed by adjusting the setpoints of
the corresponding power supplies. Some machines have quadrupole correctors
beside the main quadrupole magnets.
In this chapter we will discuss the optics correction method that uses orbit
response matrix [102] as the input data. The methods that are based on turnby-turn BPM data [58, 61, 116, 125] will be discussed in the next chapter.
4.1 BEAM MEASUREMENTS FOR LINEAR OPTICS
4.1.1 Sampling linear optics with transverse beam profile
The primary goal of the linear optics of a magnet lattice is to keep the beam
focused in both transverse planes, i.e., to preserve small beam sizes through the
beam line. Linear optics concerns the transformation of the transverse beam
profile from one location to another. The transverse distribution of a particle
beam is characterized by its second order moment matrix, Σ, (see Eq. (1.80)).
The propagation of the transverse distribution through the lattice is specified
by the transfer matrix between locations. The Σ-matrices at two locations,
points 1 and 2, are related via
Σ 2 = M 21 Σ 1 M
T
21 ,
(4.1)
where M 21 is the transfer matrix from point 1 to point 2. Transfer matrices
are fundamental representation of the linear optics of a lattice. The CourantSnyder optics functions (α, β, and γ) and the betatron phase advances are
an equivalent form of the linear optics description as they can be used to
construct the transfer matrix.
In principle, the linear optics properties can be determined by beam profile
measurements, using Eq. (4.1). This is the basis of the quadrupole scan method
for emittance measurement, which is widely used in linacs or transport lines.
In this method, the transverse beam profile is measured with a screen or a
wire scanner while the strength of an upstream quadrupole magnet is varied.
The quadrupole is separated from the screen by a drift space of length L.
Letting the entrance of the quadrupole be point 1 and the screen be point 2,
the transfer matrix from point 1 to 2 can be readily calculated as a function
strength of a quadrupole is varied and the corresponding betatron tune shifts
are measured. Knowing the length of the quadrupole magnet and the current
to gradient conversion rate, the beta function at the location of the quadrupole
can be calculated with Eq. (2.22). The accuracy of the method is affected by
the hysteresis of the quadrupole magnet and calibration errors.
Because of the large number of potential error sources, it is not feasible to
correct all linear optics errors at the error sources. Instead, quadrupole magnets are used as knobs to compensate the linear optics errors globally. In early
accelerators, quadrupole magnets are often powered in series. The strengths
of individual quadrupoles can be changed with shunt resistors that divert a
fraction of the current. Quadrupole magnets in newer machines are often individually powered. Their strengths can be changed by adjusting the setpoints of
the corresponding power supplies. Some machines have quadrupole correctors
beside the main quadrupole magnets.
In this chapter we will discuss the optics correction method that uses orbit
response matrix [102] as the input data. The methods that are based on turnby-turn BPM data [58, 61, 116, 125] will be discussed in the next chapter.
4.1 BEAM MEASUREMENTS FOR LINEAR OPTICS
4.1.1 Sampling linear optics with transverse beam profile
The primary goal of the linear optics of a magnet lattice is to keep the beam
focused in both transverse planes, i.e., to preserve small beam sizes through the
beam line. Linear optics concerns the transformation of the transverse beam
profile from one location to another. The transverse distribution of a particle
beam is characterized by its second order moment matrix, Σ, (see Eq. (1.80)).
The propagation of the transverse distribution through the lattice is specified
by the transfer matrix between locations. The Σ-matrices at two locations,
points 1 and 2, are related via
Σ 2 = M 21 Σ 1 M
T
21 ,
(4.1)
where M 21 is the transfer matrix from point 1 to point 2. Transfer matrices
are fundamental representation of the linear optics of a lattice. The CourantSnyder optics functions (α, β, and γ) and the betatron phase advances are
an equivalent form of the linear optics description as they can be used to
construct the transfer matrix.
In principle, the linear optics properties can be determined by beam profile
measurements, using Eq. (4.1). This is the basis of the quadrupole scan method
for emittance measurement, which is widely used in linacs or transport lines.
In this method, the transverse beam profile is measured with a screen or a
wire scanner while the strength of an upstream quadrupole magnet is varied.
The quadrupole is separated from the screen by a drift space of length L.
Letting the entrance of the quadrupole be point 1 and the screen be point 2,
the transfer matrix from point 1 to 2 can be readily calculated as a function