Uphill acceleration in a spatially modulated electrostatic field particle accelerator

Abstract

Spatially modulated electrostatic fields can be designed to efficiently accelerate particles by exploring the relationships between the amplitude, the phase velocity, the shape of the potential, and the initial velocity of the particle. The acceleration process occurs when the value of the velocity excursions of the particle surpasses the phase velocity of the carrier, as a resonant mechanism. The ponderomotive approximation based on the Lagrangian average is usually applied in this kind of system in non-accelerating regimes. The mean dynamics of the particle is well described by this approximation far from resonance. However, the approximation fails to predict some interesting features of the model near resonance, such as the uphill acceleration phenomenon. A canonical perturbation theory is more accurate in these conditions. In this work, we compare the results from the Lagrangian average and from a canonical perturbation theory, focusing in regions where the results of these two approaches differ from each other.

Ponderomotive and resonant effects in the acceleration of particles by electromagnetic modes

“…The formalism requires more of an involved procedure to obtain an accurate description of the ponderomotive regimes valid at larger field amplitudes. This will bring us to a relevant discussion on the problem of uphill acceleration of charged particles in ponderomotive effective potentials [12][13][14][15]. We shall see that uphill acceleration is always present as particles migrate from the ponderomotive to the resonant regime, at which point they jump to high-speeds due to resonant trapping.…”

“…The Hamiltonian (4) is exact for the conditions of our analysis and one sees that it resembles the purely electrostatic version discussed in Refs. [7,15]. Indeed, the effective 1D dynamics along the x-axis contains a slowly modulated high-frequency carrier term that can therefore trap and accelerate particles if proper resonant conditions are met.…”

“…Under this circumstance, where the ponderomotive approximation is no longer valid, the focus of interest turns to how much energy particles can gain from the resonant process. With the help of relativistic electrostatic models, it has been shown that the gain may be considerable [7,15,23] and the purpose of this section is to investigate the case where the inverse FEL-like configuration replaces the electrostatic model.…”

“…Recalling that v x = ∂H/∂p x we can rewrite expression (15) entirely in terms of the particle speed, and imposing the initial condition that v x = 1 at θ (mod 2π) = 0, one can obtain the maximum velocity at θ (mod 2π) = π in terms of the laser and wiggler amplitudes A l0 and A w0 for any given α. Note that we will always pick the positive root of relation (15) at θ = π. The positive root is accompanied by a negative root whose modulus is, however, always smaller due to the positive propagation direction of the trapping carrier.…”

“…Uphill acceleration has been studied in some cases with help of Lagrangian approaches [12,15]. The Lagrangian approach provides valuable information, being however entirely consistent only when the average velocity bears a monotonic relationship with the modulated potential.…”