DOI: 10.1201/9780429434358-4
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91
C H A P T E R 4
Linear optics
measurement and
correction - I
CONTENTS
4.1
Beam measurements for linear optics . . . . . . . . . . . . . . . . . . . . . 93
4.1.1 Sampling linear optics with transverse beam
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1.2 Sampling linear optics with beam orbit . . . . . . . . . . . 94
4.2
Fitting orbit response matrix to lattice model . . . . . . . . . . . . 98
4.2.1 Measured orbit response matrix . . . . . . . . . . . . . . . . . . . 99
4.2.2 Model orbit response matrix . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.3 Least-square fitting setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.4 Gauss-Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.6 Optics correction and an example . . . . . . . . . . . . . . . . . 106
4.2.7 Constrained least-square fitting . . . . . . . . . . . . . . . . . . . 109
4.2.8 Application of constrained fitting for optics
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
The linear optics of an accelerator beam line is of fundamental importance -
after all, the main purpose of the beam line lattice is to achieve the desired
linear optics. In a one-pass lattice, the linear optics is designed to preserve
the beam quality, avoid beam loss, and deliver certain beam characteristics at
selected locations. In a circular accelerator, the linear optics not only determines the transverse distributions of the beam, but also has significant impact
to its control and stability. A storage ring with large linear optics errors, as it
often happens during the commissioning stage of a new ring, can fail to store
the beam. In an electron storage ring, the linear optics also determines the
equilibrium distribution of the beam.
91