An extensive experimental study is performed to confirm fundamental resonance bands of an intense hadron beam propagating through an alternating gradient linear transport channel. The present work focuses on the most common lattice geometry called "FODO" or "doublet" that consists of two quadrupoles of opposite polarities. The tabletop ion-trap system "S-POD" (Simulator of Particle Orbit Dynamics) developed at Hiroshima University is employed to clarify the parameter-dependence of coherent beam instability. S-POD can provide a non-neutral plasma physically equivalent to a chargedparticle beam in a periodic focusing potential. In contrast with conventional experimental approaches relying on large-scale machines, it is straightforward in S-POD to control the doublet geometry characterized by the quadrupole filling factor and drift-space ratio. We verify that the resonance feature does not essentially change depending on these geometric factors. A few clear stop bands of low-order resonances always appear in the same pattern as previously found with the sinusoidal focusing model. All stop bands become widened and shift to the higher-tune side as the beam density is increased. In the spacecharge-dominated regime, the most dangerous stop band is located at the bare betatron phase advance slightly above 90 degrees. Experimental data from S-POD suggest that this severe resonance is driven mainly by the linear self-field potential rather than by nonlinear external imperfections and, therefore, unavoidable at high beam density. The instability of the third-order coherent mode generates relatively weak but noticeable stop bands near the phase advances of 60 and 120 degrees. The latter sextupole stop band is considerably enhanced by lattice imperfections. In a strongly asymmetric focusing channel, extra attention may have to be paid to some coupling resonance lines induced by the Coulomb potential. Our interpretations of experimental data are supported by theoretical predictions and systematic multiparticle simulations.

This paper addresses the generalization of the single-particle betatron resonance condition derived by Courant and Snyder more than a half century ago. A two-dimensional resonance condition including the effect of space-charge interaction was recently conjectured from one-dimensional Vlasov predictions made by Sacherer, Okamoto, and Yokoya [K. Ito et al., Phys. Rev. Accel. Beams 20, 064201 (2017)]. The condition is remarkably simple which only contains a few measurable quantities and indicates the possibility that twice as many resonance stop bands as expected from the conventional incoherent picture may exist at high beam density. Self-consistent multiparticle simulations are performed systematically to locate low-order stop bands in the tune diagram. The proposed betatron resonance formula is shown to explain the basic feature of numerical observations, which suggests that no serious incoherent resonance is activated inside the phase-space core of a dense beam matched to the external linear focusing potential. It is confirmed that the coherent tune-shift factor of any collective mode is less than unity and practically considered as a constant over the whole tune space in a typical high-intensity storage ring. The procedure for finding the optimum operating point of the ring is discussed on the basis of the coherent picture instead of the commonly used picture relying on the concept of incoherent tune spread. Despite years of theoretical efforts by many researchers, the coherent resonance concept is still not being employed for the construction of a stability tune diagram. We here provide a simple prescription to draw the diagram quickly. The present study also indicates the possibility of complete suppression of emittance exchange on particular difference resonances by choosing a proper emittance ratio.

In the Comment on our paper [Phys. Rev. Accel. Beams 22, 074201 (2019)], Hofmann claims that our conclusions are based on a questionable interpretation of Vlasov's equation and an over-interpretation of our multiparticle simulations. This assertion, however, comes largely from his misinterpretations of the essence of our work and proposed resonance condition. His criticism based on experimental data from some operating machines has also missed the point; we see no essential conflict between our arguments and experimental observations. While most of the questions raised by Hofmann have already been answered in our original paper above and previous publications, we take this opportunity to provide more information and explanation for clarity.

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A novel experimental approach has been established at Hiroshima University to explore diverse fundamental issues in accelerator physics. The compact apparatus developed in Hiroshima is called "S-POD" (Simulator of Particle Orbit Dynamics) that enables us to reproduce the complex collective behavior of an intense hadron beam in a local tabletop environment. The idea is based on the fact that the collective beam motion in a modern alternating-gradient accelerator is approximately equivalent to the motion of a non-neutral plasma confined in a linear Paul trap. A brief summary is given of some recent experimental results obtained with S-POD. Particular attention is paid to the resonant beam instability originating from the periodic nature of the electromagnetic potential in a storage ring. Several common misconceptions about collective resonances are clarified through typical S-POD data and theoretical predictions.