Linear optics measurement and correction - II 151
5.4 OTHER METHODS FOR OPTICS CORRECTION
Other methods can also be used to extract the linear optics errors from the
turn-by-turn BPM data for lattice model calibration and optics correction.
Segment-by-segment optics function fitting: With the 3-BPM or NBPM method, the Courant-Snyder parameters can be determined from the
simultaneous turn-by-turn BPM data at the BPM locations. For a large ring,
the lattice can be divided into several segments and treated separately. Using
the α and β functions at the beginning of the segment, the Courant-Snyder
parameters and phase advances at downstream BPM locations can be calculated with the lattice model. The differences of these optics functions between
the measurements and the calculation can be used to fit the quadrupole gradient errors in the lattice segment, from which the model can be calibrated and
the results can be used for optics correction. This method was used for the
LHC optics measurement and correction [3, 116] and has also been extended
to electron storage rings [73, 84].
Resonance driving terms (RDTs): In general, turn-by-turn beam motion can be decomposed into a series of harmonics of the betatron tunes, as
is shown in Eqs. (2.116)-(2.117). If the beam motion is purely linear and the
linear optics is the same as the design model, the turn-by-turn motion in the
resonance basis coordinate (using the design optics functions to obtain the
normalized betatron coordinates) is a simple rotation. However, with linear
optics errors, the motion of the resonance basis coordinates will be distorted.
The linear optics errors in the horizontal and vertical planes are characterized by the f 2000 and f 0020 terms, respectively, which are related to the
quadrupole errors distributed throughout the ring via [10, 39]
f 2000 =
k ∆b 1,k L k β x,k e
i2ψ x,k
8(1 − e i2πνx )
, f 0020 = −
k ∆b 1,k L k β y,k e
i2ψ y,k
8(1 − e i2πνy )
, (5.48)
where ∆b 1,k and L k are the normalized gradient errors and the lengths of the
quadrupole error sources, respectively, and β xy,k and ψ xy,k are the beta functions and the phase advances (between the error sources and the observation
point), respectively. The corresponding spectral lines are 1−ν x on h
−
x for f 2000
and 1 − ν y on h
−
y for f 0020 , respectively.
By using two adjacent BPMs to determine the angle coordinates, the turnby-turn resonance basis coordinates h
−
x,y can be calculated (see Eq. (2.109)).
The amplitude and phase of f 2000 are subsequently determined from the component corresponding to the 1 − ν x tune line on the Fourier spectrum of h
−
x
(see Eq. (2.116)) using NAFF or interpolated FFT. Similarly f 0020 is determined from the 1 − ν y tune line on h
−
y . If this can be done at many locations
around the ring, the RDT data can be used to fit the quadrupole errors in the
lattice with Eq. (5.48) [39, 47].
Transfer matrices: When the angle coordinates are calculated with position data from two adjacent BPMs, the phase space coordinates can be used
to determine the transfer matrices, using Eq. (4.8). From the one-turn transfer
5.4 OTHER METHODS FOR OPTICS CORRECTION
Other methods can also be used to extract the linear optics errors from the
turn-by-turn BPM data for lattice model calibration and optics correction.
Segment-by-segment optics function fitting: With the 3-BPM or NBPM method, the Courant-Snyder parameters can be determined from the
simultaneous turn-by-turn BPM data at the BPM locations. For a large ring,
the lattice can be divided into several segments and treated separately. Using
the α and β functions at the beginning of the segment, the Courant-Snyder
parameters and phase advances at downstream BPM locations can be calculated with the lattice model. The differences of these optics functions between
the measurements and the calculation can be used to fit the quadrupole gradient errors in the lattice segment, from which the model can be calibrated and
the results can be used for optics correction. This method was used for the
LHC optics measurement and correction [3, 116] and has also been extended
to electron storage rings [73, 84].
Resonance driving terms (RDTs): In general, turn-by-turn beam motion can be decomposed into a series of harmonics of the betatron tunes, as
is shown in Eqs. (2.116)-(2.117). If the beam motion is purely linear and the
linear optics is the same as the design model, the turn-by-turn motion in the
resonance basis coordinate (using the design optics functions to obtain the
normalized betatron coordinates) is a simple rotation. However, with linear
optics errors, the motion of the resonance basis coordinates will be distorted.
The linear optics errors in the horizontal and vertical planes are characterized by the f 2000 and f 0020 terms, respectively, which are related to the
quadrupole errors distributed throughout the ring via [10, 39]
f 2000 =
k ∆b 1,k L k β x,k e
i2ψ x,k
8(1 − e i2πνx )
, f 0020 = −
k ∆b 1,k L k β y,k e
i2ψ y,k
8(1 − e i2πνy )
, (5.48)
where ∆b 1,k and L k are the normalized gradient errors and the lengths of the
quadrupole error sources, respectively, and β xy,k and ψ xy,k are the beta functions and the phase advances (between the error sources and the observation
point), respectively. The corresponding spectral lines are 1−ν x on h
−
x for f 2000
and 1 − ν y on h
−
y for f 0020 , respectively.
By using two adjacent BPMs to determine the angle coordinates, the turnby-turn resonance basis coordinates h
−
x,y can be calculated (see Eq. (2.109)).
The amplitude and phase of f 2000 are subsequently determined from the component corresponding to the 1 − ν x tune line on the Fourier spectrum of h
−
x
(see Eq. (2.116)) using NAFF or interpolated FFT. Similarly f 0020 is determined from the 1 − ν y tune line on h
−
y . If this can be done at many locations
around the ring, the RDT data can be used to fit the quadrupole errors in the
lattice with Eq. (5.48) [39, 47].
Transfer matrices: When the angle coordinates are calculated with position data from two adjacent BPMs, the phase space coordinates can be used
to determine the transfer matrices, using Eq. (4.8). From the one-turn transfer