154
An Introduction to Beam Physics
6.3.3 The Differential Relations among the Twiss
Parameters
Now let us consider the case that s is small and to the first order β(s) = 1
β + β
s, α(s) = 1 α + α
s and φ(s) = 1 φ
s. Similarly, the transfer matrix
becomes
ˆ
M (s) = 1
⎛
⎝
1 +
β
/2β + αφ
s
β φ
s
− (1/β)
α
+
1 + α
2
φ
s 1 −
β
/2β + αφ
s
⎞
⎠ , (6.14)
where each element M ij , the i, j-th element, is computed as
M 11 (s) = 1
β + β
s
β
cos
φ
s
+ α sin
φ
s
= 1 1 +
β
2β
+ αφ
s,
M 12 (s) = 1
(β + β
s) β sin
φ
s
= 1 βφ
s,
M 21 (s) = 1 −
1
(β + β
s) β
α
s cos
φ
s
+
1 + α
α + α
s
sin
φ
s
= 1 −
1
β
α
+
1 + α
2
φ
s,
M 22 (s) = 1
β
β + β
s
cos
φ
s
−
α + α
s
sin
φ
s
= 1 1 −
β
2β
+ αφ
s.
On the other hand, the transfer matrix to the first order of s can be solved
directly from the equations of motion, which is
ˆ
M (s) = 1
1 s
−ks 1
,
(6.15)
where k is the focusing strength at s 1 = 0. Equating the corresponding terms
in eqs. (6.14) and (6.15) yields
βφ
= 1,
β
2β
+ αφ
= 0, −
1
β
α
+
1 + α
2
φ
= −k,
which can be simplified to the familiar form
φ
=
1
β
,
α
= kβ − γ,
β
= −2α,
γ
= 2kα.