Basics of beam dynamics 11
For a pure dipole, b 1 = 0, and the solution for the transfer matrix is
M(s|0) =
cos θ ρ sin θ 0 0
−
sin θ
ρ
cos θ 0 0
0
0
1 ρθ
0
0
0 1
,
(1.30)
d(s) =
ρ(1 − cos θ) sin θ 0 0
T ,
(1.31)
where θ = hs, and the bending radius is ρ = 1/h. The pure dipole magnet
provides horizontal focusing and behaves like a drift space in the vertical plane.
For the case with K x > 0 and K y < 0, the transfer matrix and the particular solution are given by
M(s|0) =
cos k x s
sin kxs
kx
0
0
−k x sin k x s cos k x s
0
0
0
0
cosh k y s
sinh kys
ky
0
0
k y sinh k y s cosh k y s
,
(1.32)
d(s) =
1−cos kxs
k 2
x ρ
sin kxs
kxρ
0 0
T
,
(1.33)
where k x =
√
K x and k y =
−K y . In the above case, the magnet focuses in
the horizontal plane and defocuses in the vertical plane. The solution for the
case with K x < 0 and K y > 0 is similar; the magnet now defocuses in the
horizontal plane and focuses in the vertical plane.
In the case −h
2 < b 1 < 0, both K x and K y are positive; hence the dipole
magnet provides focusing for both transverse planes. It is customary to define
the focusing index n = −b 1 ρ
2 . Then the equations of motion become
x
+ (1 − n)h
2 x = 0,
y
+ nh
2 y = 0.
(1.34)
This is the case of weak focusing. Because the transverse beam size in weakfocusing accelerators tends to be very large, modern accelerators usually employ the strong focusing scheme instead.
The magnetic field in the transition region at the edges of a dipole magnet
can provide additional focusing or defocusing if the beam orbit is not perpendicular to the magnet face. If the reference orbit enters the dipole magnet with
an angle, δ 1 , with respect to the normal of the entrance face of the magnet,
as illustrated in Figure 1.3, a particle that comes to the entrance point of
the reference orbit with a horizontal offset x will see more bending (than the
reference particle) if x < 0 and less bending if x > 0. Therefore the entrance
edge gives the beam defocusing in the horizontal plane.
The magnetic field has a component, B Z ∝ y, in the direction normal to
the entrance face in the transition area in which the bending field B y goes
from zero to full strength, as is required by the condition
∂B Z
∂Y −
∂B Y
∂Z = 0. This
B Z field has a projection onto the x direction, which gives the beam vertical
For a pure dipole, b 1 = 0, and the solution for the transfer matrix is
M(s|0) =
cos θ ρ sin θ 0 0
−
sin θ
ρ
cos θ 0 0
0
0
1 ρθ
0
0
0 1
,
(1.30)
d(s) =
ρ(1 − cos θ) sin θ 0 0
T ,
(1.31)
where θ = hs, and the bending radius is ρ = 1/h. The pure dipole magnet
provides horizontal focusing and behaves like a drift space in the vertical plane.
For the case with K x > 0 and K y < 0, the transfer matrix and the particular solution are given by
M(s|0) =
cos k x s
sin kxs
kx
0
0
−k x sin k x s cos k x s
0
0
0
0
cosh k y s
sinh kys
ky
0
0
k y sinh k y s cosh k y s
,
(1.32)
d(s) =
1−cos kxs
k 2
x ρ
sin kxs
kxρ
0 0
T
,
(1.33)
where k x =
√
K x and k y =
−K y . In the above case, the magnet focuses in
the horizontal plane and defocuses in the vertical plane. The solution for the
case with K x < 0 and K y > 0 is similar; the magnet now defocuses in the
horizontal plane and focuses in the vertical plane.
In the case −h
2 < b 1 < 0, both K x and K y are positive; hence the dipole
magnet provides focusing for both transverse planes. It is customary to define
the focusing index n = −b 1 ρ
2 . Then the equations of motion become
x
+ (1 − n)h
2 x = 0,
y
+ nh
2 y = 0.
(1.34)
This is the case of weak focusing. Because the transverse beam size in weakfocusing accelerators tends to be very large, modern accelerators usually employ the strong focusing scheme instead.
The magnetic field in the transition region at the edges of a dipole magnet
can provide additional focusing or defocusing if the beam orbit is not perpendicular to the magnet face. If the reference orbit enters the dipole magnet with
an angle, δ 1 , with respect to the normal of the entrance face of the magnet,
as illustrated in Figure 1.3, a particle that comes to the entrance point of
the reference orbit with a horizontal offset x will see more bending (than the
reference particle) if x < 0 and less bending if x > 0. Therefore the entrance
edge gives the beam defocusing in the horizontal plane.
The magnetic field has a component, B Z ∝ y, in the direction normal to
the entrance face in the transition area in which the bending field B y goes
from zero to full strength, as is required by the condition
∂B Z
∂Y −
∂B Y
∂Z = 0. This
B Z field has a projection onto the x direction, which gives the beam vertical