20 Beam-based Correction and Optimization for Accelerators
Eq. (1.68) also gives a formula for the phase advance
ψ 21 = tan
−1
M 12
M 11 β 1 − M 12 α 1
.
(1.71)
It is worth pointing out that the phase advance through a lattice section is not
uniquely determined by the transfer matrix of the section itself. It requires
that the C-S parameters be given at one location on the section (such as the
entrance point). In a ring lattice, the C-S parameters are naturally defined
through the periodic condition. However, in a one-pass system such as a linac
or a transport line, especially short ones that lack periodicity, the definition
of the phase advance through the line is somewhat arbitrary.
Eqs. (1.70-1.71) can be applied to the case when the section between points
1 and 2 is an infinitesimal slice of a general linear lattice element. If the length
is ds and the focusing gradient is K, the transfer matrix from point 1 to 2 is
(see Eq. (1.37))
M 21 =
1
ds
−Kds 1
+ O(ds
2 ),
(1.72)
where O(ds
2 ) stands for terms of ds
2 or higher. From Eq. (1.70) we obtain
β
= −2α, α
= Kβ − γ, γ
= 2αK,
(1.73)
where
stands for taking the derivative
d
ds . The first two equations lead to a
differential equation for the beta function
β
2
+ βK(s) −
1
β
1
4
β
2 + 1
= 0.
(1.74)
If the focusing function, K(s), is given on the cell, the beta function can be
calculated by solving for the periodic solution of Eq. (1.74).
Eqs. (1.71) and (1.72) give another useful result,
dψ
ds
=
1
β
,
(1.75)
which states that the rate of phase advance accumulation through the lattice
is inversely proportional to the beta function.
Applying Eqs. (1.70-1.71) to a lattice element, we can find out how the
C-S parameters change within or across the element. For example, in a drift
space, we have
β(s) = β 0 − 2α 0 s + γ 0 s
2 , α(s) = α 0 − γ 0 s, γ(s) = γ 0 ,
(1.76)
where the subscript 0 indicates values at the point with s = 0. If s = 0 is a
“waist”, a symmetry point where α 0 = 0 and β 0 = β
∗ , the beta function and
phase advance are given by
β(s) = β
∗ +
s
2
β ∗ ,
ψ(s) = tan
−1 s
β ∗ .
(1.77)
Eq. (1.68) also gives a formula for the phase advance
ψ 21 = tan
−1
M 12
M 11 β 1 − M 12 α 1
.
(1.71)
It is worth pointing out that the phase advance through a lattice section is not
uniquely determined by the transfer matrix of the section itself. It requires
that the C-S parameters be given at one location on the section (such as the
entrance point). In a ring lattice, the C-S parameters are naturally defined
through the periodic condition. However, in a one-pass system such as a linac
or a transport line, especially short ones that lack periodicity, the definition
of the phase advance through the line is somewhat arbitrary.
Eqs. (1.70-1.71) can be applied to the case when the section between points
1 and 2 is an infinitesimal slice of a general linear lattice element. If the length
is ds and the focusing gradient is K, the transfer matrix from point 1 to 2 is
(see Eq. (1.37))
M 21 =
1
ds
−Kds 1
+ O(ds
2 ),
(1.72)
where O(ds
2 ) stands for terms of ds
2 or higher. From Eq. (1.70) we obtain
β
= −2α, α
= Kβ − γ, γ
= 2αK,
(1.73)
where
stands for taking the derivative
d
ds . The first two equations lead to a
differential equation for the beta function
β
2
+ βK(s) −
1
β
1
4
β
2 + 1
= 0.
(1.74)
If the focusing function, K(s), is given on the cell, the beta function can be
calculated by solving for the periodic solution of Eq. (1.74).
Eqs. (1.71) and (1.72) give another useful result,
dψ
ds
=
1
β
,
(1.75)
which states that the rate of phase advance accumulation through the lattice
is inversely proportional to the beta function.
Applying Eqs. (1.70-1.71) to a lattice element, we can find out how the
C-S parameters change within or across the element. For example, in a drift
space, we have
β(s) = β 0 − 2α 0 s + γ 0 s
2 , α(s) = α 0 − γ 0 s, γ(s) = γ 0 ,
(1.76)
where the subscript 0 indicates values at the point with s = 0. If s = 0 is a
“waist”, a symmetry point where α 0 = 0 and β 0 = β
∗ , the beta function and
phase advance are given by
β(s) = β
∗ +
s
2
β ∗ ,
ψ(s) = tan
−1 s
β ∗ .
(1.77)