90 Beam-based Correction and Optimization for Accelerators
be ignored, three correctors would be enough. In this case, Eq. (3.38) leads to
θ 1
β 1 sin ψ 31 + θ 2
β 2 sin ψ 32 = 0,
(3.40a)
θ 2
β 2 sin ψ 21 + θ 3
β 3 sin ψ 31 = 0.
(3.40b)
For the simplicity of discussion, we assume the target location is at corrector
2. The position condition in Eq. (3.39) becomes
x = θ 1
β 1 β 2 sin ψ 21 .
(3.41)
Therefore, to make a large bump, it is desirable to have ψ 21 close to
π
2 +
kπ, with integer k. The kicks provided by correctors 2 and 3 are needed to
eliminate the angle and position coordinate changes after corrector 3. If both
ψ 21 and ψ 32 are equal to
π
2 modulo π, then ψ 31 is a multiple of π, and the
strength for corrector 2 is zero.