The Linearization of the Equations of Motion
77
V
[
[
α
%\
[
V
α
(QWUDQFH
([LW
%\
[
FIGURE 4.2: Entrance and exit edge lines of a dipole magnet with edge
angle α.
When the edge angle α is sufficiently small, the effect can be approximated
as an impulsive effect that changes only the horizontal and the vertical angles
of the particle orbit but does not affect position nor the longitudinal motion.
The impulsive style of treating such an effect is called “kick,” and the kick
approximation is sometimes used when studying beam optical systems in linear approximations in a similar way as it is in glass optics in the use of the
thin lens.
As we will explain in the following, in the kick approximation, the effect of
the edge angle α for the homogeneous dipole magnet acts to change only a
and b via
a f = a i +
x i tan α
R 0
,
b f = b i −
y i tan α
R 0
,
(4.4)
where R 0 is the bending radius of the homogeneous magnet, and
1
R 0
=
B y0
χ m0
.
The same expression applies to both the entrance edge and the exit edge.
Using the abbreviation
T = tan α/R 0 ,
the matrices of the horizontal (x) and the vertical (y) kicks by the edge angle
α are described as
ˆ
M
ed
x =
1 0
T 1
,
ˆ
M
ed
y =
1 0
−T 1
.
Recalling the situation of thin glass lenses, when α > 0, the vertical kick
acts to focus the beam, and the horizontal kick acts to defocus. Combining
the horizontal and the vertical kicks, the effect of the edge is that of a thin
quadrupole of the strength −T, where always one of the directions experiences
focusing, and the other defocusing. When the sign of α is opposite, the effect
also becomes opposite.
77
V
[
[
α
%\
[
V
α
(QWUDQFH
([LW
%\
[
FIGURE 4.2: Entrance and exit edge lines of a dipole magnet with edge
angle α.
When the edge angle α is sufficiently small, the effect can be approximated
as an impulsive effect that changes only the horizontal and the vertical angles
of the particle orbit but does not affect position nor the longitudinal motion.
The impulsive style of treating such an effect is called “kick,” and the kick
approximation is sometimes used when studying beam optical systems in linear approximations in a similar way as it is in glass optics in the use of the
thin lens.
As we will explain in the following, in the kick approximation, the effect of
the edge angle α for the homogeneous dipole magnet acts to change only a
and b via
a f = a i +
x i tan α
R 0
,
b f = b i −
y i tan α
R 0
,
(4.4)
where R 0 is the bending radius of the homogeneous magnet, and
1
R 0
=
B y0
χ m0
.
The same expression applies to both the entrance edge and the exit edge.
Using the abbreviation
T = tan α/R 0 ,
the matrices of the horizontal (x) and the vertical (y) kicks by the edge angle
α are described as
ˆ
M
ed
x =
1 0
T 1
,
ˆ
M
ed
y =
1 0
−T 1
.
Recalling the situation of thin glass lenses, when α > 0, the vertical kick
acts to focus the beam, and the horizontal kick acts to defocus. Combining
the horizontal and the vertical kicks, the effect of the edge is that of a thin
quadrupole of the strength −T, where always one of the directions experiences
focusing, and the other defocusing. When the sign of α is opposite, the effect
also becomes opposite.