46 Beam-based Correction and Optimization for Accelerators
x (mm)
-0.2
0
0.2
FFT(x)
turn
0
100
200
y (mm)
-0.2
0
0.2
0.1 0.15 0.2
FFT(y)
Figure 2.5 The coupled motion in the SPEAR3 storage ring for a particle launched
with initial offsets of x = y = 0.1 mm. The coupling is introduced by 15 skew
quadrupoles with integrated strengths randomly chosen from a Gaussian distribution
with σχ = 0.012 m
−1 . The uncoupled betatron tunes are νx = 14.106 and νy = 6.177.
As shown in Figure 2.5, the coupled motion in a ring consists of two frequency components in both transverse planes. These two frequency components correspond to two independent modes, which are called decoupled modes
or normal modes. The normal mode coordinates, ˆ
X = (u a , u
a , u b , u
b )
T are
related to the usual coordinates X = (x, x
, y, y
)
T through a linear transformation,
ˆ
X = V
−1 X,
(2.57)
where V is a 4 × 4 symplectic matrix. The one-turn transfer matrix for the
normal coordinates, ˆ
X, is block diagonal, i.e.,
ˆ
T = V
−1 TV =
M a
0
0 M b
.
(2.58)
The one-turn transfer matrices for mode a and b, M a and M b , respectively,
can be Courant-Snyder parametrized.
It has been shown that for a general 4 × 4 coupled transfer matrix [105]
T =
M m
n N
,
(2.59)
the linear transformation matrices to and from the decoupled coordinates are
given by
V =
rI
C
−C
+ rI
,
V
−1 =
rI −C
C
+
rI
,
(2.60)
x (mm)
-0.2
0
0.2
FFT(x)
turn
0
100
200
y (mm)
-0.2
0
0.2
0.1 0.15 0.2
FFT(y)
Figure 2.5 The coupled motion in the SPEAR3 storage ring for a particle launched
with initial offsets of x = y = 0.1 mm. The coupling is introduced by 15 skew
quadrupoles with integrated strengths randomly chosen from a Gaussian distribution
with σχ = 0.012 m
−1 . The uncoupled betatron tunes are νx = 14.106 and νy = 6.177.
As shown in Figure 2.5, the coupled motion in a ring consists of two frequency components in both transverse planes. These two frequency components correspond to two independent modes, which are called decoupled modes
or normal modes. The normal mode coordinates, ˆ
X = (u a , u
a , u b , u
b )
T are
related to the usual coordinates X = (x, x
, y, y
)
T through a linear transformation,
ˆ
X = V
−1 X,
(2.57)
where V is a 4 × 4 symplectic matrix. The one-turn transfer matrix for the
normal coordinates, ˆ
X, is block diagonal, i.e.,
ˆ
T = V
−1 TV =
M a
0
0 M b
.
(2.58)
The one-turn transfer matrices for mode a and b, M a and M b , respectively,
can be Courant-Snyder parametrized.
It has been shown that for a general 4 × 4 coupled transfer matrix [105]
T =
M m
n N
,
(2.59)
the linear transformation matrices to and from the decoupled coordinates are
given by
V =
rI
C
−C
+ rI
,
V
−1 =
rI −C
C
+
rI
,
(2.60)