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from which the phase advance across the cell can be computed via
sin ∆ψ
2
= ± ℓ
4f
141
(6.4)
The beta functions at the focusing and defocusing quadrupoles can be obtained in
a similar manner, giving
βF =
βD =
ℓ(1 + sin ∆ψ
2 )
sin ∆ψ
ℓ(1 − sin ∆ψ
2 )
sin ∆ψ
(6.5)
(6.6)
where the subscripts F and D denote the location of the focusing and defocusing
quadrupole, respectively. For a FODO cell in the 3F region of the FEL Upgrade
Driver, ℓ = 3.11 m, 1/f = 0.91 m −1 and Eq. (6.4) gives ∆ψ = ±90 degrees. Plugging
the appropriate values into Eqs. (6.5) and (6.6) results in a beta function of 5.3 m
at the focusing quadrupole and 0.9 m at the defocusing quadrupole.
The goal in achieving point-to-point focusing is to generate a sufficient change
in the betatron phase advance while minimally affecting the beam envelopes. To
that end, consider a set of focusing perturbations at locations k each with focal
length fk. At a downstream observation point, denoted by o, the perturbations will
generate deviations in the beta function and betatron phase advance according to
δβ
βo
n
= ∓
δψn = ± 1
2π
N
k=1
N
k=1
β n k
fk
β n k
fk
sin(2∆ψ) (6.7)
sin 2 ∆ψ (6.8)
where δβ is the deviation in the beta function, δψ is the deviation in the phase
advance and where n denotes the horizontal (upper sign) or vertical (lower sign)