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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

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

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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,…)

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

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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

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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)

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(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

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(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 | _|

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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

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(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 | _|

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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

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(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

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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

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(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 | _|

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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)

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(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

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(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)

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(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

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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

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

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

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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

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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 !

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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

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