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For multi-turn extraction at the CERN Proton Synchrotron, the beam is transversally split into five separate beamlets. It has been shown that an AC dipole excitation effectively controls the characteristics of these beamlets. For this purpose, the AC dipole is set in resonance with the horizontal betatron tune while the horizontal tune crosses the 4th-order resonance, creating a double-resonance condition, i.e. the simultaneous resonance between the horizontal tune and the AC dipole excitation. This increases the fraction of particles that are moved from the core to the islands. Further studies have revealed that shifting the AC dipole tune slightly, thereby breaking the double-resonance condition, distributes the beam more evenly across the beamlets. This paper examines this phenomenon by establishing a Hamiltonian model for a system with such a quasi-resonant AC dipole and studying it with numerical simulations and some supporting experimental results.
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