12 Beam-based Correction and Optimization for Accelerators
Z
X
δ 1
δ 2
x
Figure 1.3 Edge focusing at the entrance and exit faces of a dipole magnet. The y
and Y directions point out of the paper.
focusing. The edge focusing effect at the entrance face is represented by the
transfer matrix
X + = MX − ,
M =
1
0
0
0
h tan δ 1 1
0
0
0
0
1
0
0
0 −h tan δ 1 1
,
(1.35)
where the subscripts + and − stand for the coordinates before and after the
edge, respectively.
Edge focusing at the exit face is similar. The transfer matrices are the
same as in Eq. (1.35), except with δ 2 substituted for δ 1 , when the definition
of the entrance and exit angles follow the convention in Figure 1.3.
Quadrupole magnet: Including the effect of energy errors, the equations
of motion in a quadrupole magnet are
x
+
b 1
1 + δ
x = 0,
y
−
b 1
1 + δ
y = 0.
(1.36)
Defining K =
b1
1+δ , the solution for the case K > 0 is represented by the
transfer matrix
M FQ (s) =
cos
√
Ks
sin
√
Ks
√
K
0
0
−
√
K sin
√
Ks cos
√
Ks
0
0
0
0
cosh
√
Ks
sinh
√
Ks
√
K
0
0
√
K sinh
√
Ks cosh
√
Ks
.
(1.37)
The transfer matrix for the case with K < 0 can be obtained by swapping the
2-by-2 blocks for the horizontal and vertical planes and replacing K with |K|.
Typically for a quadrupole magnet
|K|L is much less than unity. For
example, in the SPEAR3 storage ring, the typical value of
|K|L for QF
magnets is about 0.45. For the purpose of a rough estimate, the transfer
matrix is often approximated with the thin-lens limit, by taking
|K|L → 0
Z
X
δ 1
δ 2
x
Figure 1.3 Edge focusing at the entrance and exit faces of a dipole magnet. The y
and Y directions point out of the paper.
focusing. The edge focusing effect at the entrance face is represented by the
transfer matrix
X + = MX − ,
M =
1
0
0
0
h tan δ 1 1
0
0
0
0
1
0
0
0 −h tan δ 1 1
,
(1.35)
where the subscripts + and − stand for the coordinates before and after the
edge, respectively.
Edge focusing at the exit face is similar. The transfer matrices are the
same as in Eq. (1.35), except with δ 2 substituted for δ 1 , when the definition
of the entrance and exit angles follow the convention in Figure 1.3.
Quadrupole magnet: Including the effect of energy errors, the equations
of motion in a quadrupole magnet are
x
+
b 1
1 + δ
x = 0,
y
−
b 1
1 + δ
y = 0.
(1.36)
Defining K =
b1
1+δ , the solution for the case K > 0 is represented by the
transfer matrix
M FQ (s) =
cos
√
Ks
sin
√
Ks
√
K
0
0
−
√
K sin
√
Ks cos
√
Ks
0
0
0
0
cosh
√
Ks
sinh
√
Ks
√
K
0
0
√
K sinh
√
Ks cosh
√
Ks
.
(1.37)
The transfer matrix for the case with K < 0 can be obtained by swapping the
2-by-2 blocks for the horizontal and vertical planes and replacing K with |K|.
Typically for a quadrupole magnet
|K|L is much less than unity. For
example, in the SPEAR3 storage ring, the typical value of
|K|L for QF
magnets is about 0.45. For the purpose of a rough estimate, the transfer
matrix is often approximated with the thin-lens limit, by taking
|K|L → 0