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Microwave instability and impedance model

Tracking simulations, with the aim of studying the microwave regime with short and intense bunches, suggest different instability mechanisms, according to the impedance model. In order to get a better insight of the source of the instability, i.e. azimuthal or radial mode coupling, we chose to follow the Vlasov-Sacherer approach to investigate the stability of the stationary solution. The generalized Sacherer's integral (1977), including mode coupling and potential well distortion, was then solved by using the "step function technique" for the expansion of the radial function, as proposed by Oide and Yokoya (1990). For illustration, the effect of the resonant frequency of a broadband resonator in the SOLEIL storage ring was studied. When the resonator frequency is much higher than the bunch spectrum width, azimuthal mode coupling can occur before radial mode coupling. When the resonator frequency is lower, radial mode coupling comes usually first, but two or more bunchlets are produced at relatively low current. The diffusion process between the bunchlets, which leads to the well-known "saw-tooth" behaviour, originates actually from a fast growing microwave instability. Lastly, the beneficial effect of an harmonic cavity on the microwave instability is estimated and discussed.