14 Beam-based Correction and Optimization for Accelerators
Alternatively, X n can be obtained by successively applying the transfer matrices of accelerator sections between the two points, i.e.,
X n = M n · · · M 2 M 1 X 0 ,
(1.41)
where point 0 is at the entrance of section M 1 and point n is at the exit of section M n . Therefore, the transfer matrix between two points can be calculated
by concatenating the transfer matrices of the components in between,
M(n|0) = M n · · · M 2 M 1 .
(1.42)
The transfer matrix is the Jacobian matrix of the canonical transformation
that relates the phase space coordinate vectors at the two points, i.e.,
M(n|0) ij =
∂(X n ) i
∂(X 0 ) j
.
(1.43)
It is well known that the Jacobian matrix of a canonical transformation is
symplectic, which means it satisfies the condition
M
T SM = S,
(1.44)
with the anti-symmetric matrix for the case of 2-dimensional motion given by
S =
S 2 0
0 S 2
, S 2 =
0 1
−1 0
.
(1.45)
The symplectic condition, Eq. (1.44), requires the determinant of the transfer matrix to be unity,
det M = 1.
(1.46)
This indicates that the volume enclosed by a surface in the phase space will
be preserved as the surface evolves according to the Hamiltonian.
For a full description of the transverse motion, 4 × 4 transfer matrices
are used for the transformation of coordinates (x, x
, y, y
). However, when
there is no coupling between the horizontal and vertical planes, the 2 × 2 offdiagonal blocks of the transfer matrix are zeros. In this case, the horizontal
and vertical motion are decoupled and can be described separately, with the
top and bottom 2 × 2 diagonal blocks, respectively. These transfer matrices
also satisfy the symplectic condition. It can be shown that a 2 × 2 matrix is
symplectic if and only if its determinant is unity.
The transfer matrix does not describe the effects of the nonlinear fields in
the accelerator elements. Including the nonlinear motion, the general effect of
an element or an accelerator section can be represented by a map between the
phase space coordinates at the entrance and exit faces
X f = M(X i ) ≡ MX i ,
(1.47)
Alternatively, X n can be obtained by successively applying the transfer matrices of accelerator sections between the two points, i.e.,
X n = M n · · · M 2 M 1 X 0 ,
(1.41)
where point 0 is at the entrance of section M 1 and point n is at the exit of section M n . Therefore, the transfer matrix between two points can be calculated
by concatenating the transfer matrices of the components in between,
M(n|0) = M n · · · M 2 M 1 .
(1.42)
The transfer matrix is the Jacobian matrix of the canonical transformation
that relates the phase space coordinate vectors at the two points, i.e.,
M(n|0) ij =
∂(X n ) i
∂(X 0 ) j
.
(1.43)
It is well known that the Jacobian matrix of a canonical transformation is
symplectic, which means it satisfies the condition
M
T SM = S,
(1.44)
with the anti-symmetric matrix for the case of 2-dimensional motion given by
S =
S 2 0
0 S 2
, S 2 =
0 1
−1 0
.
(1.45)
The symplectic condition, Eq. (1.44), requires the determinant of the transfer matrix to be unity,
det M = 1.
(1.46)
This indicates that the volume enclosed by a surface in the phase space will
be preserved as the surface evolves according to the Hamiltonian.
For a full description of the transverse motion, 4 × 4 transfer matrices
are used for the transformation of coordinates (x, x
, y, y
). However, when
there is no coupling between the horizontal and vertical planes, the 2 × 2 offdiagonal blocks of the transfer matrix are zeros. In this case, the horizontal
and vertical motion are decoupled and can be described separately, with the
top and bottom 2 × 2 diagonal blocks, respectively. These transfer matrices
also satisfy the symplectic condition. It can be shown that a 2 × 2 matrix is
symplectic if and only if its determinant is unity.
The transfer matrix does not describe the effects of the nonlinear fields in
the accelerator elements. Including the nonlinear motion, the general effect of
an element or an accelerator section can be represented by a map between the
phase space coordinates at the entrance and exit faces
X f = M(X i ) ≡ MX i ,
(1.47)