1. Orbits, Optics and Beam Dynamics in PEP-II Yunhai Cai Beam Physics Department SLAC March 6, 2007 ILC damping ring meeting at Frascati, Italy

2. Normalized Coordinates To normal coordinate Back to physical coordinate

3. R-Matrix in Terms of Lattice Functions M ab a a b R ab A a -1 AbAb In the “normalized ring”, the R-matrix is simply a rotation with the phase advance.

4. Closed Orbit in the “Normalized Ring” Trivial to generalized to higher dimension since (I-R) is block diagonal.

5. Closed Orbits Closed orbit at position (b) in the “normalized ring” Closed orbit at position (a) of the kick in the “normalized ring” Closed orbit at position (b) in the physical ring

6. Perturbation of a Closed Orbit Due to a Kick Closed orbit at location b is given: Here the kick is at location a. In particular, One can check directly,

7. Definition of Coupling Parameters Given one-turn matrix M, we can decouple it with a symplectic transformation: where u 1 and u 2 can be parameterized as if no coupling case and w is a symplectic matrix: There are ten independent parameters. Bar notes symplectic conjugate. g 2 =1-det(w).

8. Coupled Lattices Presentation of A is far from unique!!! There are eight independent parameters.

9. To the “Normalized Ring”

10. Horizontal Kick: 

11. Comparison to Simulation in the LER of PEP-II

12. Difference Between the Numerical and Analytical Solutions

13. Coherently Excited Betatron Motion and Turn-by-Turn data Beam excited at eigen frequency in x or y Equilibrium reached due to radiation damping or decoherence Take turn-by-turn reading at all beam position monitors up to 1024 turns The phase advances between the beam position monitors can be accurately measured J. Borer, C. Bovet, A. Burns, and G. Morpurgo, Proc. The 3 rd EPAC, p1082 (1992)

14. In addition, Four Eigen Orbits Extracted Using FFT These orthogonal orbits are the Fourier transforms of the turn-by-turn readings of beam position monitors at the driving frequency. Since the peak in the spectrum can be located accurately, they can be measured precisely as well. horizontal vertical real imaginary mode 1 real imaginary mode 2

15. R-Matrix Elements Derived from Four Orthogonal Orbits where a and b are indices for the locations of the beam position monitors, Q 12 and Q 34 are global invariance of the orbits. For general orbits, the relationship is much more complicated.

16. BPM Gains and Couplings where g x, g y are gains and  xy,  yx are cross-coupling between x and y. Measured Beam

17.  Beating correction for the High Energy Ring measured prediction implemented

18. Coupling Correction in the HER Before After

19. Dispersion Corrections in the HER Before After

20. Source that Generates the Vertical Emittance in the HER

21. Luminosity for Tilted Gaussian Beams Hour-glass effects:

22. Beam-Beam Scan at Low Beam Currents

23. Comparison to the Measurement Measured Calculated  (mrad) -10.0-17.73 a (microns)154175139 b (microns)6.434.895.62 L sp (10 30 cm -2 s - 1 mA -2 ) 5.405.335.74 Dynamic beta and emittance and hour-glass effect are not included. Dynamic beta and emittance and hour-glass effect are included.

24. Chromatic Optics for the HER Measured chromatic optics and dynamic aperture in HER –Excellent agreement between measurements and LEGO model in the chromatic optics –Improvement of understanding of nonlinear dynamics including sextupoles

25. PEP-2 LER Dynamic Aperture Simulation > 10  aperture at  p/p = 0  p/p = 5  =.00355 Single beam dynamic aperture versus tune and  p/p. Realistic MIA machine model,  * = 36 / 0.8 cm. Tune space near half-integer is limited by resonances, especially 2 x – n s, and chromatic tune spread. Best aperture at tunes.522.574. Better compensation of the 2 nd order chromatic y tune shift is needed. best aperture 2 x -2 s x + y -3 s  p/p y x 2 x -2 s 2 x - s x + y -4 s

26. Dynamic Aperture Near Half Integer There is a dynamic aperture near half integer only after a correction to the paraxial approximation is added into LEGO. seed 2544 seed: 834

27. Conclusion Turn-by-tune data from beam position monitors are very useful for constructing precision model and improving machine optics. Directly minimizing the sources (bending magnets) that generates the vertical (second-mode) emittance could be a very effective method to achieve the smallest emittance in storage rings. We find an analytical formula for the change of closed orbit in coupled lattice by a kick. It could be used to understand the coupling in the machine or to speed up the ORM fitting. We have used optics models not only to improve the linear optics but also to study the nonlinear beam dynamics in the machine. The study has shown the model has some predictive power as well.

28. Acknowledgements Thanks to my colleagues and collaborators who have contributed to this talk: –John Irwin, Yiton Yan, Yuri Nosochkov –J. Yocky, P. Raimondi