124
An Introduction to Beam Physics
In this case, one can show that the Jacobian ˆ
M of the transfer map M, i.e.,
the matrix
ˆ
M =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂M 1 /∂z 1 ∂M 1 /∂z 2 ∂M 1 /∂z 3 ∂M 1 /∂z 4 ∂M 1 /∂z 5 ∂M 1 /∂z 6
∂M 2 /∂z 1 ∂M 2 /∂z 2 ∂M 2 /∂z 3 ∂M 2 /∂z 4 ∂M 2 /∂z 5 ∂M 2 /∂z 6
∂M 3 /∂z 1 ∂M 3 /∂z 2 ∂M 3 /∂z 3 ∂M 3 /∂z 4 ∂M 3 /∂z 5 ∂M 3 /∂z 6
∂M 4 /∂z 1 ∂M 4 /∂z 2 ∂M 4 /∂z 3 ∂M 4 /∂z 4 ∂M 4 /∂z 5 ∂M 4 /∂z 6
∂M 5 /∂z 1 ∂M 5 /∂z 2 ∂M 5 /∂z 3 ∂M 5 /∂z 4 ∂M 5 /∂z 5 ∂M 5 /∂z 6
∂M 6 /∂z 1 ∂M 6 /∂z 2 ∂M 6 /∂z 3 ∂M 6 /∂z 4 ∂M 6 /∂z 5 ∂M 6 /∂z 6
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
has to satisfy the condition
ˆ
M
T ˆ
J ˆ
M = ˆ
J,
(5.6)
where ˆ
J is the totally antisymmetric matrix
ˆ
J =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 0 0 0
−1 0 0 0 0 0
0 0 0 1 0 0
0 0 −1 0 0 0
0 0 0 0 0 1
0 0 0 0 −1 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The proof of this so-called condition of symplecticity certainly goes beyond
this volume and can be found, for example, in [5]. But we can readily see
that the symplectic condition, which mixes in a very defined way the terms
∂M i /∂z j that are themselves power series, entails a large variety of nonlinear
restrictions between the aberrations.
From eq. (5.6), we have
− ˆ
J ˆ
M
T ˆ
J ˆ
M = ˆ
I ⇒ − ˆ
M ˆ
J ˆ
M
T ˆ
J ˆ
M = ˆ
M ⇒ − ˆ
M ˆ
J ˆ
M
T ˆ
J = ˆ
I,
hence
ˆ
M ˆ
J ˆ
M
T = ˆ
J.
Furthermore, the inverse of a symplectic matrix always exists and can be
obtained easily with the help of eq. (5.6), as
ˆ
J ˆ
M
T ˆ
J ˆ
M = − ˆ
I ⇒
− ˆ
J ˆ
M
T ˆ
J
ˆ
M = ˆ
I ⇒ ˆ
M
−1 = − ˆ
J ˆ
M
T ˆ
J.
ˆ
M
−1 = − ˆ
J ˆ
M
T ˆ
J.
(5.7)
And from the following simple arithmetic
ˆ
M
−1
T ˆ
J ˆ
M
−1 =
ˆ
J ˆ
M
T ˆ
J
T ˆ
J
ˆ
J ˆ
M
T ˆ
J
= ˆ
J ˆ
M ˆ
J ˆ
J ˆ
J ˆ
M
T ˆ
J
= − ˆ
J ˆ
M ˆ
J ˆ
M
T ˆ
J = − ˆ
J ˆ
J ˆ
J = ˆ
J,