150
An Introduction to Beam Physics
Plugging eq. (6.6) into eq. (6.5), we obtain
ˆ
M (s)
T
1/
β(s) α(s)/
β(s)
0
β(s)
1/
β(s)
0
α(s)/
β(s)
β(s)
ˆ
M (s)
=
1/
√
β α/
√
β
0
√
β
1/
√
β 0
α/
√
β
√
β
.
Hence, we have the following relation
√
β −α/
√
β
0 1/
√
β
ˆ
M (s)
T
1/
β(s) α(s)/
β(s)
0
β(s)
·
1/
β(s)
0
α(s)/
β(s)
β(s)
ˆ
M (s)
√
β
0
−α/
√
β 1/
√
β
= ˆ
I.
Defining
ˆ
R(s) =
1/
β(s)
0
α(s)/
β(s)
β(s)
· ˆ
M (s) ·
√
β
0
−α/
√
β 1/
√
β
,
(6.7)
we immediately have
ˆ
R(s)
T · ˆ
R(s) = ˆ
I,
or equivalently
ˆ
R(s)
T
=
ˆ
R(s)
−1
.
The matrix ˆ
R(s) can be expressed in the extended form
ˆ
R(s) =
R 11 (s) R 12 (s)
R 21 (s) R 22 (s)
,
which entails that
ˆ
R(s)
T
=
R 11 (s) R 21 (s)
R 12 (s) R 22 (s)
,
ˆ
R(s)
−1
=
R 22 (s) −R 12 (s)
−R 21 (s) R 11 (s)
;
the latter holds due to the fact that det( ˆ
R(s)) = 1. As a result, we have
R 11 (s) R 21 (s)
R 12 (s) R 22 (s)
=
R 22 (s) −R 12 (s)
−R 21 (s) R 11 (s)
,
which leads to the relations
R 11 (s) = R 22 (s), R 12 (s) = −R 21 (s).