48 Beam-based Correction and Optimization for Accelerators
The elements of the decoupling matrix C can be related to the distribution
of the coupling sources. To that end we first need to calculate matrix H. For
simplicity, we define matrix
¯
H = B
−1
x HB y ,
(2.67)
which is the equivalent of H for the normalized coordinates (¯ x, ¯
x
, ¯
y, ¯
y
) (see
Eq. (1.64)). From Eq. (2.56), it can be shown that if we define
h − = 2i sin π(ν x + ν y )e
iπ(νx−νy)
l
χ l
β x,l β y,l e
−i(Ψ x,l −Ψ y,l ) ,
(2.68)
h + = 2i sin π(ν x − ν y )e
iπ(νx+νy)
l
χ l
β x,l β y,l e
i(Ψ x,l +Ψ y,l ) ,
(2.69)
the elements of ¯
H are given by
¯
H 11 =
1
2
Im(h − + h + ), ¯
H 12 =
1
2
Re(h − − h + ),
(2.70a)
¯
H 21 = −
1
2
Re(h − + h + ), ¯
H 22 =
1
2
Im(h − − h + ).
(2.70b)
The determinant of H is thus (since ||B x,y || = 1)
||H|| = || ¯
H|| =
1
4
(|h − |
2 − |h + |
2 ).
(2.71)
Defining the coupling coefficients
G ± =
1
2π
l
χ l
β x,l β y,l e
±i(Ψ x,l ±Ψ y,l ) ,
(2.72)
we have
r
2 =
1
2
+
1
2
1 +
π
2 |G − |
2
sin
2 π(ν x − ν y )
−
π
2 |G + |
2
sin
2 π(ν x + ν y )
−
1
2
.
(2.73)
Eq. (2.73) indicates that if the betatron tunes satisfy the linear difference
resonance condition, i.e., ν x − ν y = p, where p is an integer, the G − term
dominates and r
2 = ||C|| =
1
2 . In this case the excitation of motion is equally
split between the two normal modes. If the tunes are near the linear sum
resonance, with ν x +ν y ≈ p, the G + term dominates. As |ν x +ν y −p| approaches
|G + |, r
2 and Abs(||C||) tend to infinity, where Abs(·) stands for the absolute
value. The beam motion near the linear sum resonance is unstable.
In a typical storage ring, the betatron tunes are closer to the linear difference resonance than the linear sum resonance, i.e., the fractional parts of
ν x and ν y tend to be both in the [0, 0.5] zone, or the [0.5, 1] zone. The linear
coupling is usually weak, with
|G − | | |ν x − ν y − p|,
(2.74)
The elements of the decoupling matrix C can be related to the distribution
of the coupling sources. To that end we first need to calculate matrix H. For
simplicity, we define matrix
¯
H = B
−1
x HB y ,
(2.67)
which is the equivalent of H for the normalized coordinates (¯ x, ¯
x
, ¯
y, ¯
y
) (see
Eq. (1.64)). From Eq. (2.56), it can be shown that if we define
h − = 2i sin π(ν x + ν y )e
iπ(νx−νy)
l
χ l
β x,l β y,l e
−i(Ψ x,l −Ψ y,l ) ,
(2.68)
h + = 2i sin π(ν x − ν y )e
iπ(νx+νy)
l
χ l
β x,l β y,l e
i(Ψ x,l +Ψ y,l ) ,
(2.69)
the elements of ¯
H are given by
¯
H 11 =
1
2
Im(h − + h + ), ¯
H 12 =
1
2
Re(h − − h + ),
(2.70a)
¯
H 21 = −
1
2
Re(h − + h + ), ¯
H 22 =
1
2
Im(h − − h + ).
(2.70b)
The determinant of H is thus (since ||B x,y || = 1)
||H|| = || ¯
H|| =
1
4
(|h − |
2 − |h + |
2 ).
(2.71)
Defining the coupling coefficients
G ± =
1
2π
l
χ l
β x,l β y,l e
±i(Ψ x,l ±Ψ y,l ) ,
(2.72)
we have
r
2 =
1
2
+
1
2
1 +
π
2 |G − |
2
sin
2 π(ν x − ν y )
−
π
2 |G + |
2
sin
2 π(ν x + ν y )
−
1
2
.
(2.73)
Eq. (2.73) indicates that if the betatron tunes satisfy the linear difference
resonance condition, i.e., ν x − ν y = p, where p is an integer, the G − term
dominates and r
2 = ||C|| =
1
2 . In this case the excitation of motion is equally
split between the two normal modes. If the tunes are near the linear sum
resonance, with ν x +ν y ≈ p, the G + term dominates. As |ν x +ν y −p| approaches
|G + |, r
2 and Abs(||C||) tend to infinity, where Abs(·) stands for the absolute
value. The beam motion near the linear sum resonance is unstable.
In a typical storage ring, the betatron tunes are closer to the linear difference resonance than the linear sum resonance, i.e., the fractional parts of
ν x and ν y tend to be both in the [0, 0.5] zone, or the [0.5, 1] zone. The linear
coupling is usually weak, with
|G − | | |ν x − ν y − p|,
(2.74)