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RF voltage modulation at a finite number of discrete frequencies is described in a Hamiltonian resonance framework. The theory is applied to the problem of parasitic extraction of a fixed target beam from a high energy proton collider, using a bent crystal as a thin septum'' within an effective width of about one micron. Three modes of employment of discrete resonances are proposed.First, a single relatively strong static drive'' resonance may be used to excite a test proton so that it will penetrate deeply into the channeling crystal. Second, a moderately strong feed'' resonance with a ramped modulation tune may be used to adiabatically trap protons near the edge of the beam core, and transport them to the drive resonance. Third, several weak resonances may be overlapped to create a chaotic amplitude band, either to transport protons to the drive resonance, or to provide a pulse stretching'' buffer between a feed resonance and the drive resonance. Extraction efficiency is semi- quantitatively described in terms of characteristic penetration,'' depletion,'' and repetition'' times. simulations are used to quantitatively confirm the fundamental results of the theory, and to show that a prototypical extraction scheme using all three modes promises good extraction performance.

RF voltage modulation at discrete frequencies with applications to crystal channeling extraction