1. FMA workshop April 1st and 2nd, 2004 L. Nadolski1 Guiding Lines for Computing and Reading a Frequency Map New FMA Calculations for SOLEIL including Insertion Device Effects Laurent S. Nadolski Pascale Brunelle and Amor Nadji

2. FMA workshop April 1st and 2nd, 2004 L. Nadolski2 z 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

3. FMA workshop April 1st and 2nd, 2004 L. Nadolski3 Tools Tracking codes –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, protrons) 4D/6D –AMD Opteron 2 GHz 0.7 s for tracking a particle over 2 x 1026 turns – 1h00 for 100x50 (enough for getting main caracteristics) – 6h45 for 400x100 (next fmap) Step size following a square root law (cf. Action)

4. FMA workshop April 1st and 2nd, 2004 L. Nadolski4 z x Regular areas Resonances Nonlinear or chaotic regions Fold Reading a FMA

5. FMA workshop April 1st and 2nd, 2004 L. Nadolski5 51015 201 5 10 15 20 25 z (mm) x (mm) Mapping

6. FMA workshop April 1st and 2nd, 2004 L. Nadolski6 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

7. FMA workshop April 1st and 2nd, 2004 L. Nadolski7 Rigid pendulum Sampling effect HyperbolicElliptic

8. FMA workshop April 1st and 2nd, 2004 L. Nadolski8 Diffusion D = (1/N)*log10(||  ||) Color code: ||  ||< 10 -10 ||  ||> 10 -2 Diffusion reveals as well slighted excited resonances

9. FMA workshop April 1st and 2nd, 2004 L. Nadolski9 Soleil Beam Dynamics investigations using FMA Working point 18.20 10.30 –Lattice: bare, errors, IDs –Optimization schemes Design of a new working point taking into account what was discovered through FMA Conclusions

10. FMA workshop April 1st and 2nd, 2004 L. Nadolski10 Optimization Method Tuneshift w/ amplitude Tuneshift w/ energy Robustness to errors multipoles coupling IDs Lattice design Fine tuning Improvement Needed Tracking NAFF NAFF suggestions Yes (x-z) fmap  injection eff. (x-  ) fmap  Lifetime Touschek computation Resonance identification 4D tracking 6D tracking Knobs 10 quadrupole families 10 sextupole families Dynamics analysis Good WP No

11. FMA workshop April 1st and 2nd, 2004 L. Nadolski11 x z  x| z=1  m z| x=1  m X or z (m) Flat vertical tune No coupling resonance crossing x - z = 8 (  = 0.1). Reference working point (18.2, 10.3) See M. Belgroune’s talk Just looking at these curves, it seems very clean …

12. FMA workshop April 1st and 2nd, 2004 L. Nadolski12 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

13. FMA workshop April 1st and 2nd, 2004 L. Nadolski13 Randomly rotating 160 Quads Map fold Destroyed Coupling strongly impacts 3 x + z = 65 Resonance node excited Physical Aperture On-momentum dynamics with 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

14. FMA workshop April 1st and 2nd, 2004 L. Nadolski14 Importance of including vaccum chamber Injection @ 14mm  x + z =65 4 x =73 x z Skew resonance excited by coupling

15. FMA workshop April 1st and 2nd, 2004 L. Nadolski15 Adding effect of 3 in-vacuum IDs  x + z =65  z = 4.5 10 - 3 ID Octupole term deeper Injection trouble if stronger

16. FMA workshop April 1st and 2nd, 2004 L. Nadolski16 Particle behavior after Touschek scattering Chromatic orbit Closed orbit ALS Example WP

17. FMA workshop April 1st and 2nd, 2004 L. Nadolski17  1 = 4.38 10 -04  2 = 4.49 10 -03 Non-linear synchrotron motion Tracking 6D required +3.8%  -6%

18. FMA workshop April 1st and 2nd, 2004 L. Nadolski18 Off momentum dynamics w/o IDs 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

19. FMA workshop April 1st and 2nd, 2004 L. Nadolski19 Off momentum dynamics w/ 3 x U20 U20 B-Roll off  x = 18 m g = 5mm 4 x =73 3 x + z =65 What’s about Effect of synchrotron radiation and damping? Very narrow resonances Synchrotron 140 turns Damping 5600 turns Loss over >400 turns Stable in 6D

20. FMA workshop April 1st and 2nd, 2004 L. Nadolski20 Coupling reduction by a factor 2 with 3 x U20

21. FMA workshop April 1st and 2nd, 2004 L. Nadolski21 Optimization of a New Point Enhanced philosophy O n momentum –3 x + z = 65 to be avoided (not shown w/o fmap) –WP to be shifted from resonance node: locus of most particles –Control of tune shift with amplitude using sextupole knobs x (J x, J z ) = a J x + b J z z (J x, J z ) = b J x + c J z Off momentum x (  ) Large energy acceptance Control of the tune shift with energy using sextupoles The 4 x = 73 resonance has to be avoided for insertion devices

22. FMA workshop April 1st and 2nd, 2004 L. Nadolski22 Energy tune shift for the new WP 18.19 / 10.29 x z dp/p x z 18.19 / 10.2918.20 / 10.30 Tune shift w/ energy optimised with sextupoles to avoid in addition the 4 x = 73 resonance for negative energy offset

23. FMA workshop April 1st and 2nd, 2004 L. Nadolski23 On momentum fmap for the WP 18.19 / 10.29 3 x - 2 z =34 WP to be slightly shifted Clean DA 1% coupling

24. FMA workshop April 1st and 2nd, 2004 L. Nadolski24 On momentum fmap for the WP 18.19 / 10.29 with 3 x U20 Map Torsion Strong diffusion related to ID roll off

25. FMA workshop April 1st and 2nd, 2004 L. Nadolski25 Off momentum fmap for the WP 18.19 / 10.29 Off-momentum map comparable to the nominal WP 2 x +2 z =57 3 x - z =26 1% coupling

26. FMA workshop April 1st and 2nd, 2004 L. Nadolski26 Off momentum fmap for the WP 18.19 / 10.29 with 3 x U20 Effect of the 2 x +2 z =57 resonance becomes « dangerous » with the 3xU20 2 x +2 z =57 2 x - z =26 2 x +2 z =57 Smoothing by synchrotron oscillations

27. FMA workshop April 1st and 2nd, 2004 L. Nadolski27 Conclusions FMA at design stage for the SOLEIL lattice –Gives us a global view (footprint of the dynamics) –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 –Importance of coupling correction to small values (below 1%) –4D/6D …

28. FMA workshop April 1st and 2nd, 2004 L. Nadolski28 Aknowledgements and references Institutes –IMCCE, ALS, SOLEIL Codes –BETA (Loulergue -- SOLEIL) –Tracy II (Nadolski -- SOLEIL, Boege -- SLS) –AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html) Papers –Frequency map analysis and quasiperiodic decompositions, J. Laskar, Proceedings of Porquerolles School, sept. 01 –Global Dynamics of the Advanced Light Source Revealed through Experimental Frequency Map Analysis, D. Robin et al., PRL (85) 3 –Measuring and optimizing the momentum aperture in a particle accelerator, C. Steier et al., Phys. Rev. E (65) 056506 –Review of single particle dynamics of third generation light sources through frequency map analysis, L. Nadolski and J. Laskar, Phys. Rev. AB (6) 114801