60 Beam-based Correction and Optimization for Accelerators
bending plane (typically horizontal) is coupled to the longitudinal motion.
The 6 × 6 transfer matrix has non-zero R 16 , R 26 elements, as given by d in
Eq. (1.33). R 51 and R 52 are non-zero and are connected with R 16 and R 26
through Eq. (2.47). The R 56 element is also non-zero. For example, the linear
terms of ∆z for a pure sector dipole are given by
∆z =
δs
γ 2
0
− x 0 sin(hs) −
x
0
h
(1 − cos(hs)) + δ(s −
1
h
sin(hs)),
(2.118)
where subscript 0 indicates coordinates at the entrance face.
Edge focusing for the dipole magnets can be included by applying the
transfer matrix in Eq. (1.35) with the proper entrance and exit angles. To
account for the finite extent of the fringe field, the vertical angle may be
corrected using the fringe field integral and the bending radius [17]. Effects of
fringe fields in quadrupoles may also be included. The dominant effect of the
soft-edge gradient variation is to scale up one transverse coordinate and scale
down its conjugate coordinate [66].
Symplectic integration:
Sextupole magnets and higher order multipoles are nonlinear elements.
There is no closed-form analytic solution to the motion in these elements.
Application of ordinary numeric integration to solve the equations of motion
through such elements can yield accurate results for one pass. However, the
solution does not preserve symplecticity of the particle motion and is thus not
ideal for long-term tracking simulation. Symplectic integration [101, 38] has
to be used for particle tracking in nonlinear elements.
The general idea of explicit symplectic integration is to split the Hamiltonian into two integrable parts, such as a drift space and a lumped kick. In
each integration step, a number of drifts and kicks are alternately applied to
the particles. Since transporting through drifts and thin-lens kicks are both
symplectic, the total transport is automatically symplectic. If the lengths of
the drifts, the strengths of the kicks, and the order of application are properly chosen (independent of the actual Hamiltonian), the integration will be
accurate as well as symplectic. A simple case is the second order symplectic
integrator illustrated in Figure 2.7. Each integration step consists of a drift,
a kick, and another drift. The lengths of the drifts are equal to one half of
the step length and the kick corresponds to the integrated magnetic field over
the step length. The symmetric configuration eliminates the first order errors
such that the leading error terms are O(L
2 ). Slicing the element into many
integration steps will increase the accuracy of the solution.
The commonly used fourth order symplectic integrator [38] is composed
of four drifts and three kicks in each step. The lengths of the four drifts
(in the order of occurrence) are α 1 L, α 2 L, α 2 L, and α 1 L, respectively, with
α 1 =
1
2(1+ζ) , α 2 =
ζ
2(1+ζ) , and ζ = 1 − 2
1/3 . The three kicks are inserted
between the drifts and their strengths corresponds to integration lengths of
β 1 L, β 2 L, and β 1 L, with β 1 = 2α 1 and β 2 = 2(α 2 − α 1 ). Note α 2 < 0 and
β 2 < 0 and hence the corresponding drift and field lengths are negative.
bending plane (typically horizontal) is coupled to the longitudinal motion.
The 6 × 6 transfer matrix has non-zero R 16 , R 26 elements, as given by d in
Eq. (1.33). R 51 and R 52 are non-zero and are connected with R 16 and R 26
through Eq. (2.47). The R 56 element is also non-zero. For example, the linear
terms of ∆z for a pure sector dipole are given by
∆z =
δs
γ 2
0
− x 0 sin(hs) −
x
0
h
(1 − cos(hs)) + δ(s −
1
h
sin(hs)),
(2.118)
where subscript 0 indicates coordinates at the entrance face.
Edge focusing for the dipole magnets can be included by applying the
transfer matrix in Eq. (1.35) with the proper entrance and exit angles. To
account for the finite extent of the fringe field, the vertical angle may be
corrected using the fringe field integral and the bending radius [17]. Effects of
fringe fields in quadrupoles may also be included. The dominant effect of the
soft-edge gradient variation is to scale up one transverse coordinate and scale
down its conjugate coordinate [66].
Symplectic integration:
Sextupole magnets and higher order multipoles are nonlinear elements.
There is no closed-form analytic solution to the motion in these elements.
Application of ordinary numeric integration to solve the equations of motion
through such elements can yield accurate results for one pass. However, the
solution does not preserve symplecticity of the particle motion and is thus not
ideal for long-term tracking simulation. Symplectic integration [101, 38] has
to be used for particle tracking in nonlinear elements.
The general idea of explicit symplectic integration is to split the Hamiltonian into two integrable parts, such as a drift space and a lumped kick. In
each integration step, a number of drifts and kicks are alternately applied to
the particles. Since transporting through drifts and thin-lens kicks are both
symplectic, the total transport is automatically symplectic. If the lengths of
the drifts, the strengths of the kicks, and the order of application are properly chosen (independent of the actual Hamiltonian), the integration will be
accurate as well as symplectic. A simple case is the second order symplectic
integrator illustrated in Figure 2.7. Each integration step consists of a drift,
a kick, and another drift. The lengths of the drifts are equal to one half of
the step length and the kick corresponds to the integrated magnetic field over
the step length. The symmetric configuration eliminates the first order errors
such that the leading error terms are O(L
2 ). Slicing the element into many
integration steps will increase the accuracy of the solution.
The commonly used fourth order symplectic integrator [38] is composed
of four drifts and three kicks in each step. The lengths of the four drifts
(in the order of occurrence) are α 1 L, α 2 L, α 2 L, and α 1 L, respectively, with
α 1 =
1
2(1+ζ) , α 2 =
ζ
2(1+ζ) , and ζ = 1 − 2
1/3 . The three kicks are inserted
between the drifts and their strengths corresponds to integration lengths of
β 1 L, β 2 L, and β 1 L, with β 1 = 2α 1 and β 2 = 2(α 2 − α 1 ). Note α 2 < 0 and
β 2 < 0 and hence the corresponding drift and field lengths are negative.