Coupling and nonlinear dynamics correction 157
where i is the BPM index, k is the data type, and w k is the weight factor
for the data type. There are 14 data types (4 amplitude functions, 8 phase
advance related values, 2 dispersion functions) for each BPM, covering the
linear optics, linear coupling, and dispersion errors.
The fitting parameters in the lattice model are the quadrupole gradients
and skew quadrupole gradients. BPM gains and rolls (or coupling coefficients)
are fitted as they are used to scale and rotate the spatial vector elements with
Eq. (4.17) or Eq. (4.18). For example, the (A, a) pair will be modified with
˜
A
˜
a
=
g x c x
c y g y
−1
A
a
.
(6.8)
The same transformation applies to (B, b), (c, C), and (d, D) pairs.
As the linear optics and coupling least-square fitting problem typically
suffers from the degeneracy difficulty due to the similarities between the effects
of the fitting parameters, the constrained fitting method is needed to find a
reasonable solution that can be used for correction [50].
6.1.3 Other methods of coupling correction
The linear coupling information can also be uncovered from the turn-by-turn
BPM data with other data analysis methods.
Direct fitting of turn-by-turn BPM data: The direct fitting of turnby-turn BPM data discussed in Section 5.3.2 can simultaneously determine the
linear optics and coupling errors. The fitting setup only needs to be modified
to add BPM rolls (or coupling coefficients) and skew quadrupole gradients as
fitting parameters. This approach has been demonstrated in simulation with
the SPEAR3 lattice model [61].
Resonance driving terms: The cross-plane motion can be characterized
by the resonance driving terms for the linear sum and difference resonances.
Correction of the resonance driving terms should lead to the correction of the
linear coupling [39, 47].
The RDTs that drive the linear difference and sum resonances are f 1001
and f 1010 , respectively, which are related to skew quadrupole errors through
f ∓ =
k ∆a 1,k L k
β x,k β y,k e
i(ψ x,k ∓ψ y,k )
4(1 − e i2π(νx∓νy) )
,
(6.9)
where f − ≡ f 1001 drives the linear difference resonance, f + ≡ f 1010 drives the
linear sum resonance, ∆a 1 and L are the gradients and lengths of the skew
quadrupole error sources, respectively. The corresponding spectral lines for
f 1001 are ν y on h
−
x and ν x on h
−
y , respectively. The spectral lines for f 1010 are
1 − ν y on h
−
x and 1 − ν x on h
−
y , respectively.
Using a pair of adjacent BPMs to derive the angle coordinates, the turnby-turn resonance basis coordinates, h
−
x,y , can be calculated, from which the
linear coupling RDTs can be determined. Knowing the amplitudes and phases
where i is the BPM index, k is the data type, and w k is the weight factor
for the data type. There are 14 data types (4 amplitude functions, 8 phase
advance related values, 2 dispersion functions) for each BPM, covering the
linear optics, linear coupling, and dispersion errors.
The fitting parameters in the lattice model are the quadrupole gradients
and skew quadrupole gradients. BPM gains and rolls (or coupling coefficients)
are fitted as they are used to scale and rotate the spatial vector elements with
Eq. (4.17) or Eq. (4.18). For example, the (A, a) pair will be modified with
˜
A
˜
a
=
g x c x
c y g y
−1
A
a
.
(6.8)
The same transformation applies to (B, b), (c, C), and (d, D) pairs.
As the linear optics and coupling least-square fitting problem typically
suffers from the degeneracy difficulty due to the similarities between the effects
of the fitting parameters, the constrained fitting method is needed to find a
reasonable solution that can be used for correction [50].
6.1.3 Other methods of coupling correction
The linear coupling information can also be uncovered from the turn-by-turn
BPM data with other data analysis methods.
Direct fitting of turn-by-turn BPM data: The direct fitting of turnby-turn BPM data discussed in Section 5.3.2 can simultaneously determine the
linear optics and coupling errors. The fitting setup only needs to be modified
to add BPM rolls (or coupling coefficients) and skew quadrupole gradients as
fitting parameters. This approach has been demonstrated in simulation with
the SPEAR3 lattice model [61].
Resonance driving terms: The cross-plane motion can be characterized
by the resonance driving terms for the linear sum and difference resonances.
Correction of the resonance driving terms should lead to the correction of the
linear coupling [39, 47].
The RDTs that drive the linear difference and sum resonances are f 1001
and f 1010 , respectively, which are related to skew quadrupole errors through
f ∓ =
k ∆a 1,k L k
β x,k β y,k e
i(ψ x,k ∓ψ y,k )
4(1 − e i2π(νx∓νy) )
,
(6.9)
where f − ≡ f 1001 drives the linear difference resonance, f + ≡ f 1010 drives the
linear sum resonance, ∆a 1 and L are the gradients and lengths of the skew
quadrupole error sources, respectively. The corresponding spectral lines for
f 1001 are ν y on h
−
x and ν x on h
−
y , respectively. The spectral lines for f 1010 are
1 − ν y on h
−
x and 1 − ν x on h
−
y , respectively.
Using a pair of adjacent BPMs to derive the angle coordinates, the turnby-turn resonance basis coordinates, h
−
x,y , can be calculated, from which the
linear coupling RDTs can be determined. Knowing the amplitudes and phases