16 Beam-based Correction and Optimization for Accelerators
s
QD
QF
QF
2L
Figure 1.5 Illustration of the strong focusing principle with FODO cells.
The stability of beam motion is an important requirement for the design
of periodic lattices. For a stable lattice, a particle launched in the vicinity of
the phase space origin (representing the reference orbit) will stay around the
origin after traveling through many periods. The orbit stability of a periodic
linear lattice can be analyzed through the transfer matrix of one periodic
cell. Considering the motion in one plane, the cell transfer matrix can be
transformed to the form
M = VΛV
−1 ,
Λ = diag(λ,
1
λ
),
(1.51)
where, as a consequence of the symplecticity of M, λ and
1
λ are both eigenvalues of M, Λ is a diagonal matrix with diagonal elements λ and
1
λ , and
columns in V are the corresponding eigenvectors. For a particle with initial
coordinates X 0 , the coordinates after m cells will be
X m = M
m X 0 = VΛ
m V
−1 X 0 .
(1.52)
Because Λ
m = diag(λ
m ,
1
λ m ), the particle motion in the lattice is stable if and
only if |λ| = 1.
The eigenvalues of matrix M can be found by solving the equation det(M−
λI) = 0. For the case of one-dimensional motion, this becomes
λ
2 − Tr(M)λ + 1 = 0,
(1.53)
where Tr(M) = M 11 + M 22 is the trace of the 2 × 2 matrix, and we have used
det(M) = 1. The eigenvalues are
λ 1,2 =
1
2
Tr(M) ±
Tr(M) 2 − 4
,
(1.54)
which, combined with the |λ| = 1 requirement, leads to the stability condition
|Tr(M)| ≤ 2,
(1.55)
in which case the eigenvalues of the transfer matrix are a pair of complex
conjugates, λ 1 = e
iΦ and λ 2 = e
−iΦ , where Φ is a real angle satisfying
cos Φ =
1
2
Tr(M).
(1.56)