1. 1 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Application of Frequency Map Analysis for Studying Beam Transverse Dynamics Laurent S. Nadolski Accelerator Physics Group ALS frequency maps Beam data Simulation data

2. 2 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Contents Introduction to FMA and motivations Application for the SOLEIL lattice –On momentum dynamics –Off momentum dynamics Experimental frequency maps (ALS) Discussion –How to use this method for FFAG?

3. 3 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Frequency Map Analysis Motivations –Global view of the beam dynamics –Beam Lifetime –Injection Efficiency –Short and Long term stability –Particle losses –Effect of insertion devices –… Selection of a good working point

4. 4 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Frequency Map Analysis Laskar A&A1988, Icarus1990 Quasi-periodic approximation through NAFF algorithm of a complex phase space function for each degree of freedom with defined over and Numerical Analysis of Fundamental Frequency

5. 5 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 I.Very accurate representation of the “signal” (if quasi-periodic) and thus of the amplitudes II.b) Determination of frequency vector with high precision for Hanning Filter Laskar NATO-ASI 1996  Long term prediction  Accuracy gain (simulation, beam based experiments)  Diffusion coefficient related to particle diffusion Advantages of NAFF

6. 6 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Rigid pendulum Sampling effect HyperbolicElliptic

7. 7 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Accelerator 4D Dynamics Accelerator Poincaré Surface ofsection

8. 8 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 z z’ x x’ x z x0x0 z0z0 x 0 ’= 0 z 0 ’= 0 Frequency map Configuration space Phase space Tracking T NAFF F T : (x 0,z 0 ) ( x, z ) resonance Frequency map: NAFF Tracking T Computing a frequency map x z

9. 9 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Tools Tracking codes (symplectic integrators) –Simulation: Tracy II, Despot, MAD, AT, … –Nature: beam signal collected on BPM electrodes NAFF package (C, fortran, matlab) Turn number Selections –Choice dictated by Allows a good convergence near resonances Beam damping times (electrons, protons) 4D/6D –AMD Opteron 2 GHz (Soleil lattice) 0.7 s for tracking a particle over 2 x 1026 turns – 1h00 for 100x50 (enough for getting main characteristics) –s 6h45 for 400x100 Step size following a square root law (cf. Action)

10. 10 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 z x Regular areas Resonances Nonlinear or chaotic regions Fold Reading a FMA

11. 11 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 4 th order 5 th order 7 th order 9 th order Resonance network: a x + b z = c order = |a| + |b| Higher order resonance

12. 12 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Diffusion D = (1/N)*log10(||Dn||) Color code: ||Dn||< 10 -10 ||Dn||> 10 -2 Diffusion reveals as well slightly excited resonances

13. 13 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Bare lattice (no errors) WP sitting on Resonance node x + 6 z = 80 5 x = 91 x - 4 z = -23 2 x + 2 z = 57 9 x =164 x -4 z =-23 4 x =73 x +6 z =80  x +5 z =88  x +4 z =96 5 x =91  x +2 z =57  x + z =65 On-momentum Dynamics --Working point: (18.2,10.3) x z 4 x =73 x -4 z =-23 9 x =164

14. 14 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Randomly rotating 160 Quads Map fold Destroyed Coupling strongly impacts 3 x + z = 65 Resonance node excited Physical Aperture On-momentum dynamics w/ 1.9% coupling (18.2,10.3)  x + z =65 Resonance island  x + z =65 x -4 z =-23 4 x =73 x +6 z =80  x +5 z =88  x +4 z =96 5 x =91  x +2 z =57

15. 15 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Off-momentum dynamics Several approaches: –Off-momentum frequency maps –Energy/betatron-amplitude frequency maps –Touschek lifetime 4D tracking 6D tracking

16. 16 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Chromatic orbit Closed orbit ALS Example WP Particle behavior after Touschek scattering

17. 17 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Off momentum dynamics 4 x =73 excited 4 x =73 3 x + z =65 3 x - 2 z =34 3 z =31 3 x + z =65 3 x - 2 z =34  >0  <0 z 0 = 0.3mm

18. 18 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Measured versus Calculated Frequency Map Modeled Measured See resonance excitation of unallowed 5 th order resonances No strong beam loss  isolated resonances are benign D. Robin et al., PRL (85) 3

19. 19 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Frequency Maps for Different Working Points Region of strong beam loss Dangerous intersection of excited resonances D. Robin et al., PRL (85) 3

20. 20 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 FMA and FFAG Light sources: 4D tracking useful since –4D dynamics + slow longitudinal dynamics Still valid for proton FFAG? Resonant phenomena? x-y fmap at a given energy (slices during acceleration ramping up) x-  fmap –6D tracking + FMA to investigate Not very much used for 3GLS because not so important Here not synchrotron oscillation but constant acceleration –Tracking over 512 turns to get a good determination of the tunes Good tracking code with almost symplectic integrators Resonances need time to build up Definition of Dynamics aperture versus number of turns –Investigation of dynamics for large amplitude Injection efficiency FFAG are very non linear by construction Multipole errors, coupling errors

21. 21 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Conclusions FMA techniques –Gives us a global view (footprint of the dynamics) –Reveals the dynamics sensitiveness to quads, sextupoles and IDs –Reveals nicely effect of coupled resonances, specially cross term z (x) –Enables us to modify the working point to avoid resonances or regions in frequency space –Is suitable both for simulation and online data –4D tracking: on- and off- momentum dynamics Applications to FFAG ?

22. 22 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 References Tracking Codes –BETA (Loulergue – SOLEIL) –Tracy II (Nadolski – SOLEIL, Boege – SLS, Bengtsson – BNL) –AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html) Papers –H. Dumas and J. Laskar, Phys. Rev. Lett. 70, 2975-2979 –J. Laskar and D. Robin, “Application of Frequency Map Analysis to the ALS”, Particle Accelerators, 1996, Vol 54 pp. 183-192 –D. Robin and J. Laskar, “Understanding the Nonlinear Beam Dynamics of the Advanced Light Source”, Proceedings of the 1997 Computational Particle Accelerator Conference –J. Laskar, Frequency map analysis and quasiperiodic decompositions, Proceedings of Porquerolles School, sept. 01 –D. Robin et al., Global Dynamics of the Advanced Light Source Revealed through Experimental Frequency Map Analysis, PRL (85) 3 –Measuring and optimizing the momentum aperture in a particle accelerator, C. Steier et al., Phys. Rev. E (65) 056506 –L. Nadolski and J. Laskar, Review of single particle dynamics of third generation light sources through frequency map analysis, Phys. Rev. AB (6) 114801 –J. Laskar, Frequency map Analysis and Particle Accelerator, PAC03, Portland –FMA Workshop’04 proceedings, Synchrotron SOLEIL, 2004 http://www.synchrotron-soleil.fr/images/File/soleil/ToutesActualites/Archives- Workshops/2004/frequency-map/index_fma.html

23. 23 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Annexes

24. 24 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Particle Computation Frame

25. 25 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Decoherence of a particle bunch

26. 26 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007  1 = 4.38 10 -04  2 = 4.49 10 -03 Non-linear synchrotron motion Tracking 6D required +3.8%  -6%