122 Beam-based Correction and Optimization for Accelerators
These methods enable accurate determination of the linear optics. Therefore,
turn-by-turn BPM data are ideal for optics measurement and correction.
In this chapter we will discuss various methods of processing turn-by-turn
BPM data for linear optics correction. These include the harmonic analysis
for betatron phase determination and the 3-BPM method for beta function
measurement [19], the model independent analysis (MIA) [65, 119] and independent component analysis (ICA) methods [58], and the method of direct
fitting of tracking data [61].
5.1 OPTICS MEASUREMENT WITH HARMONIC ANALYSIS
5.1.1 Phase advance measurement
When the beam undergoes coherent betatron oscillation, the turn-by-turn
beam orbit on either the horizontal or the vertical plane seen by a BPM is the
discrete sampling of a sinusoidal signal. For example, the turn-by-turn orbit
at BPM i is given by
x i (n) = A i sin(2πνn + ψ i + χ),
(5.1)
where n is the turn number, A i and ψ i are the oscillation amplitude and the
phase advance at the BPM, ν is the betatron tune, and χ is a phase constant
common to all BPMs for the motion. The amplitude is related to the beta
function at the BPM, β i , and the action variable, J, via A i =
√
2β i J. Other
frequency components can also show up in the data, for example, the betatron
oscillation coupled from the other transverse plane, the synchrotron oscillation through the dispersion function, and potentially signals due to nonlinear
coupling resonances. The dominant frequency component is typically the inplane betatron tune if betatron motion is excited in both planes. Figure 5.1
shows turn-by-turn orbit data on a BPM in the SPEAR3 storage ring during
an experiment.
With continuous sampling of the orbit for multiple consecutive turns, the
amplitude and phase at the BPM can be determined by calculating the projections of the raw data to the cosine and sine components of the betatron
oscillation [19],
C i =
N
n=1
x i (n) cos(2πνn), S i =
N
n=1
x i (n) sin(2πνn),
(5.2)
where N is the number of turns. With a large N , the amplitude and phase
are given by
A i =
2
N
C 2
i + S 2
i ,
(5.3)
ψ i = tan
−1 C i
S i
,
(5.4)
These methods enable accurate determination of the linear optics. Therefore,
turn-by-turn BPM data are ideal for optics measurement and correction.
In this chapter we will discuss various methods of processing turn-by-turn
BPM data for linear optics correction. These include the harmonic analysis
for betatron phase determination and the 3-BPM method for beta function
measurement [19], the model independent analysis (MIA) [65, 119] and independent component analysis (ICA) methods [58], and the method of direct
fitting of tracking data [61].
5.1 OPTICS MEASUREMENT WITH HARMONIC ANALYSIS
5.1.1 Phase advance measurement
When the beam undergoes coherent betatron oscillation, the turn-by-turn
beam orbit on either the horizontal or the vertical plane seen by a BPM is the
discrete sampling of a sinusoidal signal. For example, the turn-by-turn orbit
at BPM i is given by
x i (n) = A i sin(2πνn + ψ i + χ),
(5.1)
where n is the turn number, A i and ψ i are the oscillation amplitude and the
phase advance at the BPM, ν is the betatron tune, and χ is a phase constant
common to all BPMs for the motion. The amplitude is related to the beta
function at the BPM, β i , and the action variable, J, via A i =
√
2β i J. Other
frequency components can also show up in the data, for example, the betatron
oscillation coupled from the other transverse plane, the synchrotron oscillation through the dispersion function, and potentially signals due to nonlinear
coupling resonances. The dominant frequency component is typically the inplane betatron tune if betatron motion is excited in both planes. Figure 5.1
shows turn-by-turn orbit data on a BPM in the SPEAR3 storage ring during
an experiment.
With continuous sampling of the orbit for multiple consecutive turns, the
amplitude and phase at the BPM can be determined by calculating the projections of the raw data to the cosine and sine components of the betatron
oscillation [19],
C i =
N
n=1
x i (n) cos(2πνn), S i =
N
n=1
x i (n) sin(2πνn),
(5.2)
where N is the number of turns. With a large N , the amplitude and phase
are given by
A i =
2
N
C 2
i + S 2
i ,
(5.3)
ψ i = tan
−1 C i
S i
,
(5.4)