Resonances in Repetitive Systems
271
11.3 Linear Coupling Resonance
Linear coupling refers to mixing of linear motion between the horizontal
and vertical planes. Coupling between transversal and longitudinal motion is
present as well, but it is usually weaker. Linear coupling between transversal
planes arises from solenoids, roll of quadrupoles, vertical misalignment and
orbit offset at sextupole locations. In this section we study the effect of skew
quadrupoles, which can be present intentionally or due to the roll of normal
quadrupoles. Again, let us first assume that only one skew quadrupole is
present, which is denoted as k s . Without loss of generality, we assume that
this skew quadrupole is located at the end of a turn and that it is thin. As a
result, the linear one turn map is
⎛
⎜
⎜
⎝
x 1
a 1
y 1
b 1
⎞
⎟
⎟
⎠ =
⎛
⎜
⎜
⎝
1 0 0 0
0 1 k s l 0
0 0 1 0
k s l 0 0 1
⎞
⎟
⎟
⎠
ˆ
M
⎛
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎠ ,
where
ˆ
M =
⎛
⎜
⎜
⎝
cosμ x +α x sinμ x
β x sin μ x
0
0
−γ x sin μ x
cosμ x −α x sinμ x
0
0
0
0
c o s μ y +α y sinμ y
β y sin μ y
0
0
−γ y sin μ y
cosμ y −α y sinμ y
⎞
⎟
⎟
⎠ .
Applying the same coordinate transformation, we obtain the normalized coordinate system, which is
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
1/
√
β x
0
0
0
α x /
√
β x
√
β x
0
0
0
0
1/
β y
0
0
0 α y /
β y
β y
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
.
As a result, we have
⎛
⎜
⎜
⎜
⎝
x 1
a 1
y 1
b 1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
1 0 0 0
0 1 K 0
0 0 1 0
K 0 0 1
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
cos μ x sin μ x
0
0
− sin μ x cos μ x
0
0
0
0
cosμ y sin μ y
0
0 − sin μ y cos μ y
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
x 0
a 0
y 0
b 0
⎞
⎟
⎟
⎟
⎠
,
where K above denotes
K = k s l
β x β y .
(11.3)