Computation and Properties of Maps
127
left hand side
ˆ
X
T ˆ
J ˆ
X + ˆ
L
T
x
ˆ
J ˆ
L x =
0
( x|x)(a|a) − (x|a)(a|x)
− [(x|x)(a|a) − (x|a)(a|x)]
0
,
ˆ
Y
T ˆ
JY + ˆ
L
T
y
ˆ
J ˆ
L y =
0
( y|y)(b|b) − (b|y)(y|b)
− [(y|y)(b|b) − (b|y)(y|b)]
0
,
ˆ
D
T
x
ˆ
J ˆ
D x + ˆ
D
T
y
ˆ
J ˆ
D y + ˆ
E
T ˆ
J ˆ
E =
0 1
−1 0
,
ˆ
L
T
x
ˆ
J ˆ
L y =
0 0
0 0
,
ˆ
X
T ˆ
J ˆ
D x + ˆ
L
T
x
ˆ
J ˆ
E =
0 (l|x) + (x|x)(a|δ) − (a|x)(x|δ)
0 (l|a) + (x|a)(a|δ) − (a|a)(x|δ)
,
ˆ
Y
T ˆ
J ˆ
D y + ˆ
L
T
y
ˆ
J ˆ
E =
0 (l|y) + (y|y)(b|δ) − (b|y)(y|δ)
0
(l|b) + (y|b)(b|δ) − (b|b)(y|δ)
.
So, the conditions in eq. (5.9) are described as
(x|x)(a|a) − (x|a)(a|x) = 1,
(y|y)(b|b) − (b|y)(y|b) = 1,
(l|x) + (x|x)(a|δ) − (a|x)(x|δ) = 0,
(l|a) + (x|a)(a|δ) − (a|a)(x|δ) = 0,
(l|y) + (y|y)(b|δ) − (b|y)(y|δ) = 0,
(l|b) + (y|b)(b|δ) − (b|b)(y|δ) = 0.
(5.10)
The first two of these are familiar and describe the fact that the volume of
phase space is preserved under the linear transformations generated by particle
optical elements. The other conditions, however, represent the connection
between longitudinal and dispersive effects. For them to be satisfied requires
the use of specific scaling factors for the variables l and δ, and they are the
reason for the specific choice of the variable κ in eq. (2.1) in Section 2.1.
Next let us study the case that coupling between horizontal and vertical
planes is present, where the Jacobian matrix ˆ
M is a 4 × 4 symplectic matrix.
Similarly, we can divide ˆ
M into four blocks of 2 × 2 matrices, which is
ˆ
M =
ˆ
A ˆ
B
ˆ
C ˆ
D
.
Plugging into eq. (5.6), we obtain
ˆ
A
T ˆ
J ˆ
A + ˆ
C
T ˆ
J ˆ
C
ˆ
A
T ˆ
J ˆ
B + ˆ
C
T ˆ
J ˆ
D
ˆ
B
T ˆ
J ˆ
A + ˆ
D
T ˆ
J ˆ
C ˆ
B
T ˆ
J ˆ
B + ˆ
D
T ˆ
J ˆ
D
=
ˆ
J ˆ 0
ˆ 0 ˆ
J
,