Coupling and nonlinear dynamics correction 155
By including the measured vertical dispersion data in the fitting objective
χ
2 , a skew quadrupole setting that produces both the measured vertical dispersion and the cross-plane orbit responses can be determined. Both types of
errors can be corrected simultaneously after applying the corresponding corrections to the skew quadrupoles. The weight of the vertical dispersion in the
χ
2 definition can be increased in order to achieve the desired level of accuracy
of dispersion control.
6.1.2 Amplitude and phase of normal modes via ICA
With linear coupling, the coherent beam motion on the transverse planes can
be decomposed into two normal modes. In the case of weak coupling, one of
the normal modes can be identified as the horizontal betatron mode and the
other the vertical mode. The normal modes will be present on the measured
turn-by-turn orbits on both planes. The vertical betatron mode on the horizontal plane and similarly the horizontal mode on the vertical plane at any
location are related to the off-diagonal blocks of the one-turn transfer matrix, which in turn are related to the coupling error sources. Using the ICA
method to analyze the turn-by-turn BPM data, the normal modes can be
separated and their phases and amplitudes at each BPM location can be calculated. By comparing the measured phases and amplitudes to the predictions
made by the lattice model, the coupling error sources can be determined and
corrected.
At one BPM location, the coupled beam motion on the horizontal and
vertical readings can be separated into two pairs of ICA modes, which can be
written as [51, 125],
x n = A cos Ψ 1n − B sin Ψ 1n + c cos Ψ 2n − d sin Ψ 2n ,
(6.1a)
y n = a cos Ψ 1n − b sin Ψ 1n + C cos Ψ 2n − D sin Ψ 2n ,
(6.1b)
where n is the turn number, Ψ 1n = 2πν 1 n + ψ 1 , Ψ 2n = 2πν 2 n + ψ 2 , ν 1,2 are
the tunes for the two normal modes, mode 1 and 2 are the horizontal and
vertical betatron motion, respectively. The phase offsets ψ 1,2 are common to
all BPMs as they share the same temporal patterns.
The coupled motion can be predicted by the 4 × 4 one-turn transfer matrix
at the BPM, T, via X(n) = T
n X 0 , with the initial phase space coordinate
X 0 . Matrix T can be block diagonalized in the form
T = VUR 4 (ν 1 , ν 2 )U
−1 V
−1 ,
(6.2)
where V is the same as in Eq. (2.58), and U and R 4 are
U =
B a 0
0 B b
, R 4 (ν 1 , ν 2 ) =
R(2πν 1 )
0
0
R(2πν 2 )
,
(6.3)
By including the measured vertical dispersion data in the fitting objective
χ
2 , a skew quadrupole setting that produces both the measured vertical dispersion and the cross-plane orbit responses can be determined. Both types of
errors can be corrected simultaneously after applying the corresponding corrections to the skew quadrupoles. The weight of the vertical dispersion in the
χ
2 definition can be increased in order to achieve the desired level of accuracy
of dispersion control.
6.1.2 Amplitude and phase of normal modes via ICA
With linear coupling, the coherent beam motion on the transverse planes can
be decomposed into two normal modes. In the case of weak coupling, one of
the normal modes can be identified as the horizontal betatron mode and the
other the vertical mode. The normal modes will be present on the measured
turn-by-turn orbits on both planes. The vertical betatron mode on the horizontal plane and similarly the horizontal mode on the vertical plane at any
location are related to the off-diagonal blocks of the one-turn transfer matrix, which in turn are related to the coupling error sources. Using the ICA
method to analyze the turn-by-turn BPM data, the normal modes can be
separated and their phases and amplitudes at each BPM location can be calculated. By comparing the measured phases and amplitudes to the predictions
made by the lattice model, the coupling error sources can be determined and
corrected.
At one BPM location, the coupled beam motion on the horizontal and
vertical readings can be separated into two pairs of ICA modes, which can be
written as [51, 125],
x n = A cos Ψ 1n − B sin Ψ 1n + c cos Ψ 2n − d sin Ψ 2n ,
(6.1a)
y n = a cos Ψ 1n − b sin Ψ 1n + C cos Ψ 2n − D sin Ψ 2n ,
(6.1b)
where n is the turn number, Ψ 1n = 2πν 1 n + ψ 1 , Ψ 2n = 2πν 2 n + ψ 2 , ν 1,2 are
the tunes for the two normal modes, mode 1 and 2 are the horizontal and
vertical betatron motion, respectively. The phase offsets ψ 1,2 are common to
all BPMs as they share the same temporal patterns.
The coupled motion can be predicted by the 4 × 4 one-turn transfer matrix
at the BPM, T, via X(n) = T
n X 0 , with the initial phase space coordinate
X 0 . Matrix T can be block diagonalized in the form
T = VUR 4 (ν 1 , ν 2 )U
−1 V
−1 ,
(6.2)
where V is the same as in Eq. (2.58), and U and R 4 are
U =
B a 0
0 B b
, R 4 (ν 1 , ν 2 ) =
R(2πν 1 )
0
0
R(2πν 2 )
,
(6.3)