Linear Phase Space Motion
153
L
β
β 1
α 1 /γ 1
FIGURE 6.11: Plot of the β function in a drift.
So as a function of L, β(L) changes quadratically. We also see readily that at
the point where
L =
α 1
γ 1
,
the beam has minimum width, and we have what is called a waist (see Fig.
6.11). On the other hand, we obtain from eq. (6.11) that
γ(L) = γ 1 ,
which reflects the fact that the divergence of the beam is not changed in a
drift. As a result, we have
β(L) =
1
γ(L)
,
which entails that
α(L) = 0
at the waist. Meanwhile the behavior of α(L), which is
α(L) = −
1
2
dβ
dL
= α 1 − Lγ 1 ,
vanishes at the waist. Finally, eq. (6.12) can be re-formulated as
β(s) = β
∗ +
s
2
β ∗ ,
(6.13)
where β
∗ is the value at the waist and s is the longitudinal distance from the
waist.