Study of the third-order parametric resonance induced by RF phase modulation and its effectiveness in improving injection efficiency

Abstract

The on-axis swap-out injection scheme is promising for diffraction-limited storage rings, mainly because it can significantly reduce the requirement of dynamic aperture. However, this scheme requires the injection of full-charge bunches. Therefore, it may suffer from the reduction of injection efficiency caused by the so-called “injection transient instability” when high-charge bunches are needed. The idea of implementing RF modulation in a booster before extraction, driving the second-order parametric resonance to suppress the “injection transient instability” in the storage ring, was proposed and demonstrated previously. Here, we mainly present studies of third-order parametric resonance driven by RF phase modulation. Derivation of the modulated Hamiltonian around the third-order parametric resonance, the coordinates of both stable and unstable fixed points, and the ‘island tune’ are presented in detail. To double check the correctness of the theoretical results, numerical tests and tracking studies were carried out and presented. Furthermore, the results from comprehensive simulations are presented to demonstrate the improvement of injection efficiency by the proper settings of the RF phase modulation.