30 unifying physics of accelerators, lasers and plasma
FIGURE 2.11
Mechanism of the edge focusing of bending magnet in
a horizontal plane.
2.4.3 Examples of transfer matrices
The most common elements in accelerators are drifts, bending
magnets, which steer the trajectory, quadrupoles, which provide transverse focusing, and the accelerating section, which
accelerates the beam. Each of these elements can be represented by a particular transfer matrix. Let’s consider here a
couple of the simplest examples. The transfer matrix of a drift
space is simply
(
)
1 d
M =
(2.28)
0 1
Bending magnets, especially in large accelerators where
each individual bend provides only a tiny bit of bending,
have transfer matrices very close to those of drift space except
when edge focusing (illustrated in Fig. 2.11) needs to be taken
into account. In the latter, the faces of the bending magnet
are not perpendicular to the reference trajectory. As clearly
seen from the figure, this can provide additional defocusing
on the horizontal plane 1 due to edge effects.
The matrix of a focusing quadrupole is
(
√
√
)
√ 1
cos( |K|L)
sin( |K|L)
M =
√
√
|K| √
(2.29)
− |K| sin( |K|L)
cos( |K|L)
and the matrix of a defocusing quadrupole is
⎛
√
√
⎞
√ 1
⎜ cosh( KL)
sinh( KL) ⎟
⎜
⎟
⎜
K
⎟
M = ⎜ √
√
√
⎟
(2.30)
⎝
⎠
K sinh( KL)
cosh( KL)
For a thin lens, L → 0 with KL staying finite, these matrices
correspondingly become
(
)
(
)
1
0
1 0
M F =
and
M D =
(2.31)
−|K|L 1
KL 1
The higher-order elements such as sextupole and octupole
magnets are often used in accelerators for nonlinear corrections. In linear approximation their transfer matrices are
equivalent to the corresponding matrix of a drift space. They
will have, however, higher-order terms and can be described
by higher-order matrices (as will be discussed later in this
chapter).
2.4.4 Matrix formalism for transfer lines
The matrix formalism is very practical for computing propagation through transfer lines, especially since the transfer
matrix of each individual element of the beamline needs to
be calculated only once. In a purely linear transfer matrix, the
1 Vertical plane edge focusing can also occur but only in the case of a finite
vertical gap of the bending magnet.
FIGURE 2.11
Mechanism of the edge focusing of bending magnet in
a horizontal plane.
2.4.3 Examples of transfer matrices
The most common elements in accelerators are drifts, bending
magnets, which steer the trajectory, quadrupoles, which provide transverse focusing, and the accelerating section, which
accelerates the beam. Each of these elements can be represented by a particular transfer matrix. Let’s consider here a
couple of the simplest examples. The transfer matrix of a drift
space is simply
(
)
1 d
M =
(2.28)
0 1
Bending magnets, especially in large accelerators where
each individual bend provides only a tiny bit of bending,
have transfer matrices very close to those of drift space except
when edge focusing (illustrated in Fig. 2.11) needs to be taken
into account. In the latter, the faces of the bending magnet
are not perpendicular to the reference trajectory. As clearly
seen from the figure, this can provide additional defocusing
on the horizontal plane 1 due to edge effects.
The matrix of a focusing quadrupole is
(
√
√
)
√ 1
cos( |K|L)
sin( |K|L)
M =
√
√
|K| √
(2.29)
− |K| sin( |K|L)
cos( |K|L)
and the matrix of a defocusing quadrupole is
⎛
√
√
⎞
√ 1
⎜ cosh( KL)
sinh( KL) ⎟
⎜
⎟
⎜
K
⎟
M = ⎜ √
√
√
⎟
(2.30)
⎝
⎠
K sinh( KL)
cosh( KL)
For a thin lens, L → 0 with KL staying finite, these matrices
correspondingly become
(
)
(
)
1
0
1 0
M F =
and
M D =
(2.31)
−|K|L 1
KL 1
The higher-order elements such as sextupole and octupole
magnets are often used in accelerators for nonlinear corrections. In linear approximation their transfer matrices are
equivalent to the corresponding matrix of a drift space. They
will have, however, higher-order terms and can be described
by higher-order matrices (as will be discussed later in this
chapter).
2.4.4 Matrix formalism for transfer lines
The matrix formalism is very practical for computing propagation through transfer lines, especially since the transfer
matrix of each individual element of the beamline needs to
be calculated only once. In a purely linear transfer matrix, the
1 Vertical plane edge focusing can also occur but only in the case of a finite
vertical gap of the bending magnet.