1. 6. betatron coupling sources: skew quadrupoles, solenoid fields concerns: reduction in dynamic (& effective physical) aperture; increase of intrinsic & projected y emittance in e- storage rings; degraded tuning performance; increased spot size at collision point two new eigenmodes, no longer purely x or y

2. 2 coupled linear oscillators  : coupling normal-mode coordinates: decoupled equations new eigen- frequencies frequency split: measure of strength of coupling

3. in a real storage ring, the coupling is not constant, but varies around the ring (localized sources) 2 global parameters driving terms for sum and difference resonance sources of coupling: skew quadrupole field errors, vertical orbit offset in sextupole solenoid fields (detector field, solenoids against e-cloud,…)

4. two new eigenmodes of coupled betatron oscillations; beam is tilted in x-y plane, e.g., tilt angle varies along beam line

5. two linear resonances in Hamiltonian sum resonance difference resonance uncoupled linear motion resonance driving terms: k s (s): normalized gradient of skew quadrupole L: circumference

6. minimizing the driving term improved beam lifetime increased dynamic aperture smaller emittance electron storage ring vertical emittance due to weak betatron coupling: driving term ‘including all Fourier components’ where on resonance: (Raubenheimer)

7. (A) first turn analysis difference orbits kick identify coupling source and devise correction one can fit large number of orbits & BPM data to determine skew component of each magnet measuring betatron coupling

8. (B) kick response over many turns envelopes of horizontal and vertical oscillations exhibit beating plane of kick orthogonal plane beating period define one can show that ! example ATF |  _|

9. frequency spectrum of horizontal pick up viewed on a spectrum analyzer monitoring betatron coupling at the ATF Damping Ring evolution of the peak signal in the frequency spectrum vs. time, on an oscilloscope; the slow variation reflects synchrotron motion; the fast period is due to transverse coupling; the amplitude and period of the modulation can be used to determine the driving term |  _|, in this case |  _|~0.02

10. (C) closest tune approach near the difference resonance the tunes of the two eigenmodes, in the vertical plane, are uncoupled tunes tunes can approach each other only up to distance |  _| correction strategy; use two skew quadrupoles (ideally with  x -  y )~  /2) to minimize |  _|, namely the distance of closest tune approach |  _|

11. closest tune approach in the PEP-II HER before final correction; shown are the measured fractional tunes as a function of the horizontal tune knob; the minimum tune distance is equal to the driving term |  _| of the difference resonance

12. (D) compensating the sum resonance near difference resonance, energy exchange x y near sum resonance, motion is unstable is solution for (note: these phases are not exactly the same as before but transforming into resonance basis) resonance stop band

13. in principle, |  + | could be compensated by adjusting two skew quadrupoles so as to minimize the stopband width, ideally at locations separated by minimum number of skew quadrupoles for global correction in a ring: 2 for |   | 2 for |  + | 2 for D y 6: minimum number for independent correction of 6 global effects and emittance optimization this does not yet correct the local coupling effects, which may also contribute to emittance growth, especially in lepton machines

14. (E) emittance near difference resonance for leptons near the difference resonance where measured tune difference combining the above relations yields (Guignard) recipe: infer ex from synchrotron light monitor for different values of  Q I,II ; then determine  x0 and |  _| by nonlinear fit |  _|

15. Horizontal emittance as a function of the tune separation  Q I,II at the ATF Damping Ring; the measured data and the result of a nonlinear fit are shown; fit gives  x0 ~2.44 nm, |  _|~0.037 (closest tune approach measured at the same time yielded |  _|~0.042)

16. (F) emittance near sum resonance |+||+| near the sum resonance (derived from Guignard’s expressions) alternative theoretical formula from T. Raubenheimer; simulation results from MAD (Chao formalism. probably not applicable for vicinity of sum resonance); simulation result from SAD (Ohmi-Oide-Hirata formalism); caution! 4 different answers!experiments at ATF unclear personal preference for SAD

17. (G) local coupling correction minimizing vertical closed-orbit response to horizontal steering (at KEK ATF DR); by measuring cross-plane response matrix for all dipole correctors and all BPMs, and computing skew-quad correction based on optics model (J. Urakawa, 2000)

18. (H) coupling transfer function excite beam in x detect coherent y motion used for continuous monitoring of coupling at the CERN ISR in the 1970s; is considered for LHC coupling control amplitude and phase of vertical response; complex value of  _ ISR coupling transfer function

19. mathematically exact formulation of coupling 4x4 one-turn matrix Edwards-Teng factorization new matrix U is block-diagonal; A and B are of the same form as for the uncoupled case factorization matrix V describes the coupling symplectic conjugate of C

20. block-diagonal matrices for eigenmodes are of the Courtant-Snyder type 2x2 matrices for normalization of A, B 4x4 normalization matrix normalized coupling matrix

21. if mode a is excited if mode b is excited D. Sagan & D. Rubin, PRST-AB 2, 074001 (1999)

22. the complete coupling matrix can be determined by harmonic analysis, e.g., excite beam at eigenmode frequency a, measure response in both planes over N turns and form 8 sums: the p x is obtained by combining information from two nearby BPMs exciting also the eigenmode b can serve as a test & each mode measurement gives more precise answer for half of the C ij E. Perevedentsev, 2000

23. flat versus round beams for e+e- colliders luminosity emittances could be varied by coupling: naturally flat due to synchrotron radiation beam sizes at collision point beam-beam tune shift one wants to maximize both: constraint round beams give 2x higher luminosity, but requires !

24. Summary tune measurements FFT with interpolation, Lob periodogram beam transfer functions phase locked loop multibunch spectrum  function measurements  K phase advance corrector excitation symmetry point R matrix from trajectory fit phase advance measurements multi-turn BPMs & harmonic analysis gradient errors 1 st turn, or closed-orbit distortion phase advance  bumps multiknobs beam response to kick excitation coherent damping filamentation chromaticity betatron coupling various measurement techniques