transverse dynamics 35
transfer line defined by the principal trajectories
(
)
C(s) S(s)
M 1→2 =
(2.40)
C ' (s) S ' (s)
Similarly, we can write down the expression that computes
the propagation of the optics function along the transfer lines
using a matrix based on the principal trajectories
⎛
⎞ ⎛ C 2
−
S 2
β
2CS
⎞ ⎛ β 0
⎜ ⎜ ⎜
⎜ α ⎟
⎟ ⎟ ⎟ ⎜ ⎜
⎜
=
⎟ ⎜ −CC ' CS ' + SC '
SS '
α
(2.41)
⎜
⎟ ⎜
⎜
⎜
2
−
2
⎟ ⎟
⎟ ⎜ ⎜
⎟ ⎜
⎜
⎟ ⎟ ⎜
⎜ 0
⎝ γ
⎠ ⎝ C '
−2C ' S '
S '
γ 0
⎞
⎟ ⎟
⎟
The initial values of the optical functions
⎠
in
⎝
this equa
⎠ ⎟ ⎟ ⎟
tion are
either determined by the periodicity conditions as in the case
of a circular machine, or correspond to the initial values at
the entrance of the system as in the case of a transfer line.
Let’s consider two cases as examples. Drift space expresses as
(
)
⎛ β
⎞ ⎛ 1 −2s s 2 ⎞ ⎛ β 0
⎞
1 s
M = 0 1
→
⎜
⎜
⎜ ⎜ α
γ
⎟
⎟ ⎟
⎟ ⎜
⎟ ⎜
⎟
=
We can see that the function
⎜ ⎜
⎟ ⎟
has
⎜ ⎜
⎜ ⎜ 0
1
−s
0
0
1
⎟ ⎟ ⎜ ⎜ α 0
⎟ ⎟
β
⎝ ⎜
⎝
⎠
γ 0
a parabolic
⎟ ⎟ ⎜
⎜ ⎜
⎟
⎠ ⎟
beha
⎝
vior
⎟ ⎠
⎟
in
correlation to the drift length.
A thin focusing quadrupole of focal length f = 1/KL
⎛
⎞
(
)
⎛ β
⎞
1
0
0
⎜
⎟
⎛ β
1 0
⎜
0
⎞
⎟
KL 1
⎜
M =
→ ⎜
⎜ ⎜ ⎜ α
⎟ ⎟
⎟ =
⎜ ⎜ KL
1
0
⎟ ⎟ ⎜ ⎜
⎟
α
⎟ ⎟
this
⎝ γ
⎟
In
case, it is the γ function
⎜
⎟ ⎠
⎜
⎜ ⎜
)
⎟
⎟
⎜ 0
(
2
KL
2KL 1
⎟
⎝ ⎜
⎟
⎠ γ 0
tha
⎟
t has a parabolic beha
⎜
⎝ ⎜
vior
⎠
⎟ ⎟
in correlation to the inverse focal length of the quadrupole.
2.5 Phase space
2.5.1 Phase space ellipse and Courant–Snyder invariant
To summarize our discussion about betatron motion, let’s discuss the evolution of a phase space ellipse.
The solutions of the Hill’s equation
d 2 y + K y (s)y = 0
(2.42)
ds 2
discussed in Section 2.4.1 is reproduced below
.
y(s) = εβ(s) cos (φ(s) − φ)
(2.43)
y
' (s) = −
ε [sin (φ(s) − φ) + α(s) cos (φ(s) − φ)]
β(s)