128 Beam-based Correction and Optimization for Accelerators
Given the connection between the beta function and the phase advance (see
Eq. (1.75)), it is possible to derive the beta function using the accurate phase
advance measurement [19].
If the transfer matrix between BPM 1 and 2 is A, from Eq. (1.68), we have
A 11
A 12
=
cos ψ 12 + α 1 sin ψ 12
β 1 sin ψ 12
,
(5.16)
where α 1 and β 1 are the Courant-Snyder parameters at BPM 1 and ψ 12 is the
phase advance between the two BPMs. Eq. (5.16) indicates that the optics
functions at a location, α 1 and β 1 , which are determined by the global lattice,
are related to the local transfer matrix and the phase advance between two
locations. While the errors in the optics functions can be large as any lattice
error around the ring contributes to them, the errors in the local transfer
matrix are expected to be small, if the two locations are next to each other
and have few magnets in between. If the transfer matrix between BPM 1 and
another location, BPM 3, B, and the phase advance between BPMs 1 and 3,
ψ 13 , are also known, the Courant-Snyder functions at BPM 1 can be solved,
using the Eq. (5.16) and its counterpart for BPMs 1 and 3. The beta function
is given by
β 1 =
cot ψ 13 − cot ψ 12
B 11 /B 12 − A 11 /A 12
.
(5.17)
The transfer matrices A and B can be calculated with the design lattice
model. Using the model lattice functions, Eq. (5.17) can also be rewritten in
the form
β
meas
1
= β
model
1
cot ψ
meas
13
− cot ψ
meas
12
cot ψ model
13
− cot ψ model
12
,
(5.18)
where subscripts “meas” and “model” are used to indicate the measured and
model values. An expression for the measured α 1 can be similarly obtained.
BPM 3 can be upstream of BPM 1, in which case, ψ 13 should be interpreted as
−ψ 31 in Eq. (5.18) and the same applies to BPM 2. The method of determining
the beta function with phase advance measurement on three BPMs is called
the 3-BPM method.
The use of model values of β and ψ in Eq. (5.18) may raise questions
about the accuracy of the measured beta function as the model values of these
global optics functions differ from the actual values before optics correction.
This should not be a concern because the ratio of β 1 and cot ψ 13 − cot ψ 12 is
locally determined. Hence, as long as the differences between the local transfer
matrices in the actual machine and the design model are small, measurement
of phase advances can be used to deduce the beta function. It is worth noting
that for the use of Eq. (5.18), the phase advances ψ 12 and ψ 13 should not be
close to 0 or
π
2 modulo π because the ratio of β 1 and cot ψ 13 − cot ψ 12 cannot
approach either zero or infinity. The optimal choice is to have phase advances
near
π
4 or
3π
4 modulo π.
Given the connection between the beta function and the phase advance (see
Eq. (1.75)), it is possible to derive the beta function using the accurate phase
advance measurement [19].
If the transfer matrix between BPM 1 and 2 is A, from Eq. (1.68), we have
A 11
A 12
=
cos ψ 12 + α 1 sin ψ 12
β 1 sin ψ 12
,
(5.16)
where α 1 and β 1 are the Courant-Snyder parameters at BPM 1 and ψ 12 is the
phase advance between the two BPMs. Eq. (5.16) indicates that the optics
functions at a location, α 1 and β 1 , which are determined by the global lattice,
are related to the local transfer matrix and the phase advance between two
locations. While the errors in the optics functions can be large as any lattice
error around the ring contributes to them, the errors in the local transfer
matrix are expected to be small, if the two locations are next to each other
and have few magnets in between. If the transfer matrix between BPM 1 and
another location, BPM 3, B, and the phase advance between BPMs 1 and 3,
ψ 13 , are also known, the Courant-Snyder functions at BPM 1 can be solved,
using the Eq. (5.16) and its counterpart for BPMs 1 and 3. The beta function
is given by
β 1 =
cot ψ 13 − cot ψ 12
B 11 /B 12 − A 11 /A 12
.
(5.17)
The transfer matrices A and B can be calculated with the design lattice
model. Using the model lattice functions, Eq. (5.17) can also be rewritten in
the form
β
meas
1
= β
model
1
cot ψ
meas
13
− cot ψ
meas
12
cot ψ model
13
− cot ψ model
12
,
(5.18)
where subscripts “meas” and “model” are used to indicate the measured and
model values. An expression for the measured α 1 can be similarly obtained.
BPM 3 can be upstream of BPM 1, in which case, ψ 13 should be interpreted as
−ψ 31 in Eq. (5.18) and the same applies to BPM 2. The method of determining
the beta function with phase advance measurement on three BPMs is called
the 3-BPM method.
The use of model values of β and ψ in Eq. (5.18) may raise questions
about the accuracy of the measured beta function as the model values of these
global optics functions differ from the actual values before optics correction.
This should not be a concern because the ratio of β 1 and cot ψ 13 − cot ψ 12 is
locally determined. Hence, as long as the differences between the local transfer
matrices in the actual machine and the design model are small, measurement
of phase advances can be used to deduce the beta function. It is worth noting
that for the use of Eq. (5.18), the phase advances ψ 12 and ψ 13 should not be
close to 0 or
π
2 modulo π because the ratio of β 1 and cot ψ 13 − cot ψ 12 cannot
approach either zero or infinity. The optimal choice is to have phase advances
near
π
4 or
3π
4 modulo π.