70
An Introduction to Beam Physics
where L is the drift length, and they can be written in matrix form as
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x f
a f
y f
b f
l f
δ f
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 L 0 0 0 0
0 1 0 0 0 0
0 0 1 L 0 0
0 0 0 1 0 0
0 0 0 0 1 D
0 0 0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x i
a i
y i
b i
l i
δ i
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
where
D =
L
(2 + η 0 )
2 .
First, we observe that, as in the case of glass optics, the determinant is
unity. We also observe that the matrix can be grouped to three blocks, corresponding to the x-a (horizontal), the y-b (vertical) and the l-δ (longitudinal)
motions. So, in the linear approximation, those three submotions in the drift
are decoupled, without having any mixing. This allows us to study the entire
motion in each direction conveniently independently. Later, we often study
the motion of a system in decoupled submotions.
It is worthwhile to note that, when we take account of nonlinearity, even
the drift motion is no longer simply linear, which may sound striking. This
can be seen in the equations of motion (3.22), and the nonlinear effect comes
from the factor p 0 /p s , which contains the second order contributions of a and
b. This effect is called the kinematic correction, and it also is responsible
for nonlinear mixing of submotions in different directions even for the drift.
4.2 The Quadrupole without Fringe Fields
More interesting is the case of the quadrupole. Since the reference orbit
goes straight, we have h = 0.
4.2.1 The Electric Quadrupole
For the electric quadrupole, from Section 3.1.2, we have
V = M 2,2 cos (2φ) r
2 = M 2,2
x
2
− y
2
,
and
E x = −2M 2,2 · x, E y = 2M 2,2 · y,
while
B = 0. If M 2,2 > 0 for the positive charge beam, the field acts to
focus the beam in the horizontal (x) direction, and defocus in the vertical (y)
direction. The field description above corresponds to the case in the general
An Introduction to Beam Physics
where L is the drift length, and they can be written in matrix form as
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x f
a f
y f
b f
l f
δ f
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 L 0 0 0 0
0 1 0 0 0 0
0 0 1 L 0 0
0 0 0 1 0 0
0 0 0 0 1 D
0 0 0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x i
a i
y i
b i
l i
δ i
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
where
D =
L
(2 + η 0 )
2 .
First, we observe that, as in the case of glass optics, the determinant is
unity. We also observe that the matrix can be grouped to three blocks, corresponding to the x-a (horizontal), the y-b (vertical) and the l-δ (longitudinal)
motions. So, in the linear approximation, those three submotions in the drift
are decoupled, without having any mixing. This allows us to study the entire
motion in each direction conveniently independently. Later, we often study
the motion of a system in decoupled submotions.
It is worthwhile to note that, when we take account of nonlinearity, even
the drift motion is no longer simply linear, which may sound striking. This
can be seen in the equations of motion (3.22), and the nonlinear effect comes
from the factor p 0 /p s , which contains the second order contributions of a and
b. This effect is called the kinematic correction, and it also is responsible
for nonlinear mixing of submotions in different directions even for the drift.
4.2 The Quadrupole without Fringe Fields
More interesting is the case of the quadrupole. Since the reference orbit
goes straight, we have h = 0.
4.2.1 The Electric Quadrupole
For the electric quadrupole, from Section 3.1.2, we have
V = M 2,2 cos (2φ) r
2 = M 2,2
x
2
− y
2
,
and
E x = −2M 2,2 · x, E y = 2M 2,2 · y,
while
B = 0. If M 2,2 > 0 for the positive charge beam, the field acts to
focus the beam in the horizontal (x) direction, and defocus in the vertical (y)
direction. The field description above corresponds to the case in the general