Lattice Modules
229
To make the meaning of the relation clearer, let us add the drift of length L
after the cell. We obtain
1 L
0 1
(a|a) 1 (x|a) 1
(a|x) 1 (x|x) 1
=
(a|a) 1 + L(a|x) 1 (x|a) 1 + L(x|x) 1
(a|x) 1
(x|x) 1
,
and
1 L
0 1
2(x|a) 1 (a|δ) 1
2(x|x) 1 (a|δ) 1
=
2 [(x|a) 1 + L(x|x) 1 ] (a|δ) 1
2(x|x) 1 (a|δ) 1
.
When L = −(x|a) 1 /(x|x) 1 , the second half of the cell forms an image and the
dispersive ray crosses the axis. In other words, the dispersive ray behaves the
same as the axial ray from the center of the cell. The reader can check easily
that it is indeed the case for a sector bend.
9.2.1 The Double-Bend Achromat
Now let us study the simplest mirror symmetric achromat, which consists
of two bend magnets and a quadrupole in the middle. Due to the mirror
symmetry, the achromatic conditions D = D
= 0 at the end can be satisfied
requiring D
= 0 at the center.
⎛
⎜
⎝
D c
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
1
0 0
−1/2f 1 0
0
0 1
⎞
⎟
⎠
⎛
⎜
⎝
1 L 1 0
0 1 0
0 0 1
⎞
⎟
⎠
⎛
⎜
⎝
1 L Lθ/2
0 1 θ
0 0 1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ ,
⎛
⎜
⎝
D c
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
1
L + L 1
(L/2 + L 1 ) θ
−1/2f 1 − (L + L 1 ) /2f [1 − (1/2f ) (L/2 + L 1 )]θ
0
0
1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ .
=⇒ D c =
L
2
+ L 1
θ, f =
1
2
L
2
+ L 1
.
From the discussion above, the relation f = (L/2 + L 1 )/2 is simply the result
of the mirror symmetry. Even with a large bending angle where the exact
matrix of the bend has to be used, being achromatic always implies that the
center of the first bend is imaged to the center of the second bend. Since
f < (L + L 1 ) /2, it is not possible to build a FODO cell that is stable, so
a doublet or a triplet has to be used. A simple variant of the double-bend
achromat (DBA) is the triplet DBA shown in Fig. 9.5, which contains a triplet
between the bending magnets, with no quadrupoles outside.
Another type of DBA consists of two bending magnets, a focusing quadrupole
in between and a doublet outside of each bend as shown in Fig. 9.6. The cell
is symmetric about the center of QF1. Fig. 9.7 [66, 21] shows an example of
the lattice function of one example of this type of achromat.
229
To make the meaning of the relation clearer, let us add the drift of length L
after the cell. We obtain
1 L
0 1
(a|a) 1 (x|a) 1
(a|x) 1 (x|x) 1
=
(a|a) 1 + L(a|x) 1 (x|a) 1 + L(x|x) 1
(a|x) 1
(x|x) 1
,
and
1 L
0 1
2(x|a) 1 (a|δ) 1
2(x|x) 1 (a|δ) 1
=
2 [(x|a) 1 + L(x|x) 1 ] (a|δ) 1
2(x|x) 1 (a|δ) 1
.
When L = −(x|a) 1 /(x|x) 1 , the second half of the cell forms an image and the
dispersive ray crosses the axis. In other words, the dispersive ray behaves the
same as the axial ray from the center of the cell. The reader can check easily
that it is indeed the case for a sector bend.
9.2.1 The Double-Bend Achromat
Now let us study the simplest mirror symmetric achromat, which consists
of two bend magnets and a quadrupole in the middle. Due to the mirror
symmetry, the achromatic conditions D = D
= 0 at the end can be satisfied
requiring D
= 0 at the center.
⎛
⎜
⎝
D c
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
1
0 0
−1/2f 1 0
0
0 1
⎞
⎟
⎠
⎛
⎜
⎝
1 L 1 0
0 1 0
0 0 1
⎞
⎟
⎠
⎛
⎜
⎝
1 L Lθ/2
0 1 θ
0 0 1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ ,
⎛
⎜
⎝
D c
0
1
⎞
⎟
⎠ =
⎛
⎜
⎝
1
L + L 1
(L/2 + L 1 ) θ
−1/2f 1 − (L + L 1 ) /2f [1 − (1/2f ) (L/2 + L 1 )]θ
0
0
1
⎞
⎟
⎠
⎛
⎜
⎝
0
0
1
⎞
⎟
⎠ .
=⇒ D c =
L
2
+ L 1
θ, f =
1
2
L
2
+ L 1
.
From the discussion above, the relation f = (L/2 + L 1 )/2 is simply the result
of the mirror symmetry. Even with a large bending angle where the exact
matrix of the bend has to be used, being achromatic always implies that the
center of the first bend is imaged to the center of the second bend. Since
f < (L + L 1 ) /2, it is not possible to build a FODO cell that is stable, so
a doublet or a triplet has to be used. A simple variant of the double-bend
achromat (DBA) is the triplet DBA shown in Fig. 9.5, which contains a triplet
between the bending magnets, with no quadrupoles outside.
Another type of DBA consists of two bending magnets, a focusing quadrupole
in between and a doublet outside of each bend as shown in Fig. 9.6. The cell
is symmetric about the center of QF1. Fig. 9.7 [66, 21] shows an example of
the lattice function of one example of this type of achromat.