J. Laskar’s research while affiliated with CERN and other places

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Publications (5)

FIG. 1. (a) The order of accuracy ζ for each element M i;j of the quadrupole transfer matrix is plotted for various integrators where the numbers in the labels inside the parentheses (n k ,n tot ) refer to the amount of kicks n k and the total maps needed to construct the integrator n tot . The order of accuracy of μ, in a FODO cell that include dipoles, for different K Q and L Q with the use of (b) the CSABA 2 and (c) the TEAPOT 5 symplectic integrators. The area below the white dashed curve guarantees stable motion through the FODO cell.
FIG. 2. The order of accuracy of J in a FODO cell, for different K Q and L Q with the use of (a) the CSABA 2 and (b) the TEAPOT 3 symplectic integrators. The area below the white dashed curve guarantees stable motion through the FODO cell.
FIG. 3. The difference in order of accuracy of ξ x between (a) CSABA 2 & YFR 3 , (b) CSABA 2 & TEAPOT 3 , and (c) CSABA 4 & TEAPOT 5 . The difference in order of accuracy of ξ y between (d) CSABA 2 & YFR 3 , (e) CSABA 2 & TEAPOT 3 , and (f) CSABA 4 & TEAPOT 5 . The combinations of ðK Q ; L Q Þ in the white area guarantee unstable motion and the black dots are configurations where the CSABA ν is accurate up to the 16th decimal digit of ξ x or ξ y .
FIG. 5. The variation of the Oð ∂Q j ∂J j Þ for differed lattice configurations using (a) the CSABA 4 and (b) the TEAPOT 5 symplectic integrators.
Application of high order symplectic integration methods with forward integration steps in beam dynamics
  • Article
  • Full-text available

March 2022

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54 Reads

Physical Review Accelerators and Beams

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J. Laskar

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The Hamiltonian describing particle motion in an accelerator belongs to a large class of systems, the members of which can be integrated with a new set of high order symplectic integrators. One benefit of these integrators is their strong numerical stability, which results from the inclusion of only forward integration steps, independent of the order of accuracy. Using these integrators, the transfer map of any multipolar accelerator magnet is derived and presented here. From these maps, the Hamiltonian flow in different lattices is simulated and benchmarked against other well established integration schemes in the accelerator community. By comparing quantities such as the linear phase advance and action invariant, the chromaticity, as well as the working point and the tune spread with amplitude, the superiority of the novel symplectic integrators is demonstrated with respect to accuracy and integration cost.

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FIG. 1. A hypothetical ring with eight BPMs at longitudinal positions which are marked with red circles. When the mixed BPM method is employed, a sampling error δ k is introduced, due to the deviation of the BPM positions from hypothetical locations that divide the circumference of the ring in exactly eight equal parts, marked with blue circles. BPM 1 is set as the reference point.
FIG. 2. Synthesis of a one-dimensional signal of 84 000 samples (top right), from an initial two-dimensional signal of N ¼ 2000 and M ¼ 42 (top left). The first turn from five BPMs are shown (bottom left). The signals from each BPM are represented with different colors. The transformation of the five turns results in 210 samples (bottom right) which increases the sampling rate. Each period is now populated by 42 samples of the original signal, and the periods are shown between the red dashed lines.
FIG. 3. The projection of the signal used in the numerical simulations, on the complex plane. The oscillations consist of the superposition of four frequencies, with a frequency-to-frequency shift of ΔQ ¼ 0.05.
FIG. 7. The coefficients estimated from the fit of the tune error ϵ 1 to the surface of Eq. (37), along with their uncertainty. The c 1 coefficient is marked with blue circles, c 2 with blue squares, and c 3 with red diamonds. The error bars represent the 1σ standard error of the fit. The theoretical predictions for the coefficients are shown with a blue dashed and thick line for c 1 and c 2 , respectively. The theoretical estimation of C L is shown with a red line.
FIG. 8. The turn-by-turn response of the centroid of the beam, for different kick strengths. The top row depicts the horizontal oscillation and the bottom row the vertical. The left column shows the TBT data for an initial amplitude of 4σ, the middle column for 8σ, and the right column for 12σ.

+3

Refined betatron tune measurements by mixing beam position data

July 2019

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119 Reads

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17 Citations

Physical Review Accelerators and Beams

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J. Laskar

The measurement of the betatron tunes in a circular accelerator is of paramount importance due to their impact on beam dynamics. The resolution of the these measurements, when using turn-by-turn (TBT) data from beam position monitors, is greatly limited by the available number of turns in the signal. Because of decoherence from finite chromaticity and/or amplitude detuning, the transverse betatron oscillations appear to be damped in the TBT signal. On the other hand, an adequate number of samples is needed, if precise and accurate tune measurements are desired. In this paper, a method is presented that allows for very precise tune measurements within a very small number of turns. The theoretical foundation of this method is presented with results from numerical and tracking simulations and experimental TBT data which are recorded at electron and proton circular accelerators.

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FIG. 1. A hypothetical ring with eight BPMs at asymmetrical longitudinal positions, which are shown with red circles. The mixed BPM method assumes that these BPMs, are actually at the locations with the blue markers where they divide the circumference of the ring in exactly eight equal parts. This introduces an error δ k for each k BPM. BPM 1 is set as the reference point.
Figure 2: Synthesis of a one dimensional signal of 84000 samples (top right), from an initial two dimensional signal of N=2000 and M=42 (top left). The 5 first turns are shown (bottom left) with different color for each BPM. The transformation of the 5 turns results in 210 samples (bottom right), from the increase of the sampling rate. Each period is now populated by 42 samples of the original signal and the periods are shown between the red dashed lines.
Figure 3: The projection of the signal used in the numerical simulations, on the complex plane. The oscillations consist of the superposition of four frequencies, with a frequency-to-frequency shift of ∆Q = 0.05
Figure 7: The coefficients estimated from the fit of the tune error 1 to the surface of Eq. (38), along with their uncertainty. The c 1 coefficient is shown in blue circles, c 2 in blue squares and c 3 in red diamonds. The error-bars represent the 1σ standard error of the fit. The theoretical predictions for the coefficients are shown with a blue dashed and thick line for c 1 andc 2 respectively. The theoretical estimation of C L is shown with a red line.
Figure 8: The turn by turn response of the centroid of the beam, for different kick strengths. Top row depicts the horizontal oscillation and the bottom row the vertical. Left column shows the TbT data for an initial amplitude of 4σ, middle column for 8σ and right column for 12σ.

+12

Refined betatron tune measurements by mixing beam position data

March 2019

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146 Reads

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Jacques Laskar

The measurement of the betatron tunes in a circular accelerator is of paramount importance due to their impact on beam dynamics. The resolution of the these measurements, when using turn by turn (TbT) data from beam position monitors (BPMs), is greatly limited by the available number of turns in the signal. Due to decoherence from finite chromaticity and/or amplitude detuning, the transverse betatron oscillations appear to be damped in the TbT signal. On the other hand, an adequate number of samples is needed, if precise and accurate tune measurements are desired. In this paper, a method is presented that allows for very precise tune measurements within a very small number of turns. The theoretical foundation of this method is presented with results from numerical and tracking simulations but also from experimental TbT data which are recorded at electron and proton circular accelerators.

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Figure 1: Frequency map for an ideal lattice of the ESRF. The initial conditions are taken over a mesh in the horizontal (x) and the vertical (y) direction (bottom), and the corresponding frequencies are plotted in the frequency space (top). Each point is colored according to the color code associated to the values of the diffusion index D.  
Precise tune measurements from multiple beam position monitors

June 2007

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71 Reads

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1 Citation

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J. Laskar

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One of the main limitations for precise tune measurements using kicked turn-by-turn data is the beam decoherence, which can limit the available signal to a reduced number of turns. Applying Laskar's frequency analysis, on measurements from several beam position monitors, a fast and accurate determination of the real tune is possible. The efficiency of the method is demonstrated when applied in turn-by-turn data from the ESRF storage ring and CERN's super proton synchrotron. Estimates from tracking simulations and analytical considerations are further compared with the experimental results.

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Figure 4: Frequency map for the ESRF model.
Probing the non-linear dynamics of the ESRF storage ring with experimental frequency maps

June 2003

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60 Reads

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14 Citations

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L. Farvacque

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A. Ropert

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J. Laskar

A key issue for improving the performance of third generation light sources like the ESRF storage ring is the identification and correction of resonances that have a detrimental effect in the machine performance. Frequency analysis of experimental data, recorded on BPMs around the ring, is a powerful tool for studying the non-linear dynamics of an accelerator in operation. In a series of experiments, experimental frequency maps were produced using the 1000-turns measurement system. These maps revealed the resonance structure in the vicinity of the nominal working point which was limiting the dynamic aperture. Their comparison with maps produced by simulations was used as a guide for understanding the storage ring non-linear model. The possibility of using the quasi-periodic approximations of the experimental data for testing the efficiency of resonance correction schemes was finally investigated

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Citations (3)

... In the 1990s, intense efforts were devoted to improve methods for computing ω j with a precision higher than that provided by DFT, i.e., 1=N. In several cases, these developments originated in the field of celestial mechanics [1][2][3][4][5] and were then promoted to the field of accelerator physics (see, e.g., Ref. [6]). These algorithms have been implemented in a number of codes that are now of standard use for the analysis of beam dynamics [7,8] and in the analysis of massive simulation data in the form of what is called frequency map analysis [9][10][11][12][13][14][15][16][17][18][19]. ...

Reference:

Harmonic analysis of nonstationary signals with application to LHC beam measurements
Refined betatron tune measurements by mixing beam position data

Physical Review Accelerators and Beams

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J. Laskar

... The mixed BPM method has been explored in the past at proton and electron rings [15,16] for precise tune measurements with a very small number of turns. Here, the theoretical foundations of this method is presented, along with results from tracking simulations and experimental measurements. ...

Reference:

Refined betatron tune measurements by mixing beam position data
Precise tune measurements from multiple beam position monitors

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J. Laskar

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... In several cases, these developments originated in the field of celestial mechanics [1][2][3][4][5] and were then promoted to the field of accelerator physics (see, e.g., Ref. [6]). These algorithms have been implemented in a number of codes that are now of standard use for the analysis of beam dynamics [7,8] and in the analysis of massive simulation data in the form of what is called frequency map analysis [9][10][11][12][13][14][15][16][17][18][19]. ...

Reference:

Harmonic analysis of nonstationary signals with application to LHC beam measurements
Probing the non-linear dynamics of the ESRF storage ring with experimental frequency maps

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L. Farvacque

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A. Ropert

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J. Laskar