Periodic Focusing Channel Research Articles

Mitigation of envelope instability through fast acceleration in linear accelerators

The space-charge driven envelope instability can be of great danger in high intensity accelerators. Linear accelerators were designed to avoid this instability by keeping the zero current phase advance per lattice period below 90 degrees. In this paper, we report on the acceleration effects on the instability in a periodic solenoid and radio-frequency (RF) focusing channel and a periodic quadrupole and RF focusing channel using a three-dimensional envelope model and self-consistent macroparticle simulations. The simulation results show that the envelope instability can be mitigated with a reasonable accelerating gradient in both channels. This suggests that the zero current phase advance might be above 90 degrees in linear accelerators where the accelerating gradient is sufficiently high and opens additional freedom of transverse and longitudinal focusing in the accelerator design.

Bunched Beam Envelope Instability in a Periodic Focusing Channel

The space-charge driven envelope instability presents a great danger in high intensity accelerator design. In this paper, we report on the study of bunched beam envelope instability in a periodic focusing channel using three-dimensional envelope model for a 3D uniform Waterbag distribution and a 3D Gaussian distribution. Our results show that the envelope instability stopband becomes broader with the increase of longitudinal focusing and are not sensitive to the type of distribution. Self-consistent macroparticle simulations using both distributions show similar structure in emittance growth but also extra instability stopbands. The emittance growth from the Waterbag distribution has larger stopband than that from the Gaussian distribution.

Three-dimensional envelope instability in periodic focusing channels

The space-charge driven envelope instability can be of great danger in high intensity accelerators and was studied using a two-dimensional (2D) envelope model and three-dimensional (3D) macroparticle simulations before. In this paper, we propose a three-dimensional envelope instability model to study the instability for a bunched beam in a periodic solenoid and radio-frequency (RF) focusing channel and a periodic quadrupole and RF focusing channel. This study shows that when the transverse zero current phase advance is below $90$ degrees, the beam envelope can still become unstable if the longitudinal zero current phase advance is beyond $90$ degrees. For the transverse zero current phase advance beyond $90$ degrees, the instability stopband width becomes larger with the increase of the longitudinal focusing strength and even shows different structure from the 2D case when the longitudinal zero current phase advance is beyond $90$ degrees. Breaking the symmetry of two longitudinal focusing RF cavities and the symmetry between the horizontal focusing and the vertical focusing in the transverse plane in the periodic quadrupole and RF channel makes the instability stopband broader. This suggests that a more symmetric accelerator lattice design might help reduce the range of the envelope instability in parameter space.

Structure resonances due to space charge in periodic focusing channels

The Vlasov-Poisson model is one of the most effective methods to study the space charge dominated beam evolution self-consistently in a periodic focusing channel. Since the approach to get the solution with this model is not trivial, previous studies are limited in degenerated conditions, either in smoothed channel (constant focusing) [I. Hofmann, Phys. Rev. E 57, 4713 (1998)] or in alternating gradient focusing channel with equal initial beam emittance condition in the degrees of freedom [I. Hofmann et al., Part. Accel. 13, 145 (1983); Chao Li et al., THOBA02, IPAC2016]. To establish a basis, we intentionally limit this article to the study of the pure transverse periodic focusing lattice with arbitrary initial beam condition, and the same lattice structure in both degrees of freedom, but with possibility of different focusing strengths. This will show the extension of the existing work. The full Hamiltonian is invoked for a pure transverse focusing lattice in various initial beam conditions, revealing different mode structure and additional modes beyond those of the degenerated cases. Application of the extended method to realistic lattices (including longitudinal accelerating elements) and further details will then reveal many new insights, and will be presented in later work.

Nonlinear resonance and envelope instability of intense beam in axial symmetric periodic channel

When an intense charged particle beam propagates through a given periodic focusing channel, it will experience the phenomena of nonlinear resonance, collective instability or chaotic motion with different conditions. In this paper, the collective envelope instability mechanisms are studied for symmetric beam propagation in an axially symmetric periodic channel. The beam is characterized as collectively stable if there exists a stable fixed point (SFP) located at the matched beam condition (rm,0) in (r,pr) phase space. It is found that the well-known collective envelope instability is dynamically related to the period-two orbits bifurcation of the matched SFP, meanwhile the unique stable SFP turns into an unstable saddle-node, surrounded by 1/2 resonance islands. However, higher orders of resonance (l/n, n>2) coming from period-n bifurcation will not lead to collective beam instability because a new SFP emerges immediately upon the bifurcation process. The orders of SFP bifurcation is numerically depicted by the envelope tune ν=ϕ/360, where ϕ is the eigenphase of the Poincaré tangent map T(s) in one focusing period at SFP, as functions of depressed phase advance. With strong space charge, due to these resonances from SFP bifurcation could be overlapped, mismatched beam would even show chaotic motion. For specific parameters, regular orbits, resonance islands, chaotic regions formed by resonance overlapping are clearly depicted with frequency analysis and Lyapunov spectral exponents, a method that may prove useful when extended to higher phase-space dimensions.

Space charge induced beam instability in periodic focusing channel

The transverse evolution of the envelope of an intense, unbunched ion beam in a linear periodic transport channel can be modeled for the approximation of linear self-fields by the Kapchinskij-Vladimirskij envelope equation. The envelope mismatched modes, or the second order even mode [I. Hofmann, Phys. Rew. E 57, 4713 (1998)], are the lowest order of resonance leading to collective instability that the designer should avoid, which suggests that an accelerator system should be established in the parameter region where the zero beam current phase advance σ0 less than 90°. In this paper, we systemically studied the resonance mechanisms which result in confluent resonance in quadrupole Focusing-Defocusing (FD) channel and parametric resonance in solenoid channel. We propose that the mismatch modes cannot be exactly separated in FD channel; if one mode is excited, there is always some contribution of the other. To verify the influence of the confluent resonance and parametric resonance, the 2D Poissons solver in the self-consistent particle-in-cell simulation code TOPOPIC is adopted to study the beam evolution in both channels. Our simulations results show that the emittance show significant growth both in the confluent resonance stop band and parametric resonance stop band. The influences of the higher order of resonance are also discussed.

Envelope instability and the fourth order resonance

The well-known envelope instability or the second order even collective mode [I. Hofmann, Phys. Rev. E 57, 4 (1998)] and the fourth order resonance $4\ensuremath{\sigma}=360\ifmmode^\circ\else\textdegree\fi{}$ due to the nonlinear space charge effect in high intensity beams have been studied previously. A wide stop band around 15\ifmmode^\circ\else\textdegree\fi{} is found in a pure periodic focusing channel. In addition, it is illustrated that the fourth order resonance dominates over the envelope instability and practically replaces it in the stop band [D. Jeon et al., Phys. Rev. ST Accel. Beams 12, 054204 (2009)]. In this paper, for a continuous beam with remarkable space charge, our 2D self-consistent particle-in-cell simulation work with the code topopic shows these two kinds of effects respectively in a periodic focusing defocusing (FD) channel. For a fixed tune depression $\ensuremath{\eta}=0.8$, a stop band with a width of almost 15\ifmmode^\circ\else\textdegree\fi{} is also demonstrated. Moreover, it is confirmed that analytical results of the rms envelope instability diagram are a valid tool to interpret the width of the stop band. Emittance growth rates in stop band are also well explained. It is found that, for a nearly rms matched beam, the emittance growth in the stop band is almost proportional to the saturation time of the nonlinear instability of the envelope, which happens in a quick manner and takes only a few FD cells. In contrast, the fourth order resonance is independent of rms matching and will be accompanied by beam evolution as ``a long term effect'' once the related mechanism is excited.

Design of a Power Extractor Using a Rectangular Dielectric Loaded Waveguide

We report on the design of a power extractor using a rectangular dielectric loaded (DL) waveguide. The use of a rectangular rather than a circular DL waveguide can potentially lead to higher output power, since a flat drive beam can then be used to mitigate space charge effects. This power extractor based on the LSM11 mode is predicted to be able to generate 171 MW of power with a 100 nC/bunch drive train of electron bunches separated by 0.769 ns. An output coupler is also designed which is able to extract the generated power to standard waveguides with a 94% coupling efficiency. The propagation of an intense electron beam through a long rectangular DL waveguide is investigated. Results show that, by using a periodic magnetic focusing channel surrounding the waveguide, a 20 MeV, 40 nC beam can propagate up to 129 cm without significant particle losses.

Experimental study of coherent betatron resonances with a Paul trap

Linear and nonlinear resonant instabilities of charged-particle beams traveling in periodic quadrupole focusing channels are studied experimentally with a compact non-neutral plasma trap. The present experiments are based on the idea that the collective motion of a beam in an accelerator is physically similar to that of a one-component plasma in a trap. A linear Paul trap system named ``S-POD'' (simulator for particle orbit dynamics) was developed to explore a variety of space-charge-induced phenomena. To emulate lattice-dependent effects, periodic perturbations are applied to quadrupole electrodes, which gives rise to additional resonance stop bands that shift depending on the plasma density. It is confirmed that an $m$th-order resonance takes place when the corresponding tune of an $m$th-order collective mode ${\ensuremath{\Omega}}_{m}$ is close to a half integer.

Simulation on control of beam halo-chaos by power function in the hackle periodic-focusing channel

The K-V beam through a hackle periodic-focusing magnetic field is studied using the particle-core model. The beam halo-chaos is found, and a power function controller is proposed based on mechanism of halo formation and strategy of controlling halo-chaos. Multiparticle simulation was performed to control the halo by using the power function control method. The results show that the halo-chaos and its regeneration can be eliminated effectively. We also find that the radial particle density evolvement is of uniformity at the beam's centre as long as appropriate parameters are chosen.

All-optical control of nonlinear focusing of laser beams in plasma beat wave accelerator

Nonlinear focusing of a bi-color laser in plasma can be controlled by varying the difference frequency Ω. The driven electron density perturbation forms a co-moving periodic focusing (de-focusing) channel if Ω is below (above) the electron Langmuir frequency ωp. Hence, the beam focusing is enhanced for Ω < ωp and is suppressed otherwise. In particular, a catastrophic relativistic self-focusing of a high-power laser beam can be prevented all-optically by a second, much weaker, co-propagating beam shifted in frequency by Ω > ωp. A bi-envelope equation describing the early stage of the mutual de-focusing is derived and analyzed. Later stages, characterized by a well-developed electromagnetic cascade, are investigated numerically. Stable propagation of the over-critical laser pulse over several Rayleigh lengths is predicted. The non-resonant plasma beat wave (Ω ≠ ωp) can accelerate pre-injected electrons above 100 MeV with low energy spread.

Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel

This paper is devoted to numerical simulation of a charged particle beam submitted to a strong oscillating electric field. For that, we consider a two-scale numerical approach as follows: we first recall the two-scale model which is obtained by using two-scale convergence techniques; then, we numerically solve this limit model by using a backward semi-Lagrangian method and we propose a new mesh of the phase space which allows us to simplify the solution of Poisson's equation. Finally, we present some numerical results which have been obtained by the new method, and we verify its efficiency through long time simulations.

Control of Beam Halo-Chaos by Fraction Power-Law Function in Hackle Periodic-Focusing Channel

The Kapchinsky–Vladimirsky (K-V) beam through a hackle periodic-focusing magnetic field is studied using the particle-core model. The beam halo-chaos is found, and an idea of fraction power-law function controller is proposed based on the mechanism of halo formation and the strategy of controlling halo-chaos. The method is applied to the multi-particle simulation to control the halo. The numerical results show that the halo-chaos and its regeneration can be eliminated effectively by using the fraction power-law function control method. At the same time, the radial particle density is uniform at the beam's center as long as the control method and appropriate parameter are chosen.

Control of Beam Halo-Chaos for an Intense Charged-Particle Beam Propagating Through Double Periodic Focusing Field by Soliton

We study an intense beam propagating through the double periodic focusing channel by the particle-core model, and obtain the beam envelope equation. According to the Poincare–Lyapunov theorem, we analyze the stability of beam envelope equation and find the beam halo. The soliton control method for controlling the beam halo-chaos is put forward based on mechanism of halo formation and strategy of controlling beam halo-chaos, and we also prove the validity of the control method, and furthermore, the feasible experimental project is given. We perform multiparticle simulation to control the halo by using the soliton controller. It is shown that our control method is effective. We also find the radial ion density changes when the ion beam is in the channel, not only the halo-chaos and its regeneration can be eliminated by using the nonlinear control method, but also the density uniformity can be found at beam's centre as long as an appropriate control method is chosen.

Control of beam halo-chaos by Gauss function in the triangle periodic-focusing channel

This paper studies the Kapchinsky–Vladimirsky (K–V) beam through a triangle periodic-focusing magnetic field by using the particle-core model. The beam halo-chaos is found, and an idea of Gauss function controller is proposed based on the strategy of controlling the halo-chaos. It performs multiparticle simulation to control the halo by using the Gauss function control method. The numerical results show that the halo-chaos and its regeneration can be eliminated effectively, and that the radial particle density is uniform at the centre of the beam as long as the control method and appropriate parameter are chosen.

Controlling beam halo-chaos via backstepping design

A backstepping control method is proposed for controlling beam halo-chaos in the periodic focusing channels (PFCs) of high-current ion accelerator. The analysis and numerical results show that the method, via adjusting an exterior magnetic field, is effective to control beam halo chaos with five types of initial distribution ion beams, all statistical quantities of the beam halo-chaos are largely reduced, and the uniformity of ion beam is improved. This control method has an important value of application, for the exterior magnetic field can be easily adjusted in the periodical magnetic focusing channels in experiment.

Collective instabilities and collisional effects for space charge dominated beams

This paper is organized into two parts. In the first one we analyze the equipartitioning process in the presence of collisions for a Gaussian anisotropic beam. Far from instabilities the equipartition process can be described correctly by using Landau's collisional theory, whereas when the dynamics of the system is strongly nonlinear (e.g. in the vicinity of a resonance) a hybrid dynamical–collisional equipartition mechanism occurs. In the second part of the paper we discuss shortly a systematic study of the collective instabilities in a symmetric periodic focusing channel for a KV beam by using the moments method. We emphasize differences occurring when the periodic focusing lattice is replaced by the corresponding constant focusing one.

Hermite spline interpolation on patches for parallel Vlasov beam simulations

In this paper we present a novel interpolation technique for Vlasov simulations of intense space charge dominated beams. This new technique enables to localize the cubic spline interpolation generally performed in semi-Lagrangian Vlasov codes and thus to improve the scalability of the parallel version. This new method is applied to the propagation of a potassium beam in a periodic focusing channel.

Space-charge transport limits of ion beams in periodic quadrupole focusing channels

It has been empirically observed in both experiments and particle-in-cell simulations that space-charge-dominated beams suffer strong growth in statistical phase-space area (degraded quality) and particle losses in alternating gradient quadrupole transport channels when the undepressed phase advance σ 0 increases beyond about 85 ∘ per lattice period. Although this criterion has been used extensively in practical designs of strong focusing intense beam transport lattices, the origin of the limit has not been understood. We propose a mechanism for the transport limit resulting from strongly chaotic classes of halo particle resonances near the core of the beam that allow near-edge particles to rapidly increase in oscillation amplitude when the space-charge intensity and the flutter of the matched beam envelope are both sufficiently large. When coupled with a diffuse beam edge and/or perturbations internal to the beam core that can drive particles outside the edge, this mechanism can result in large and rapid halo-driven increases in the statistical phase-space area of the beam, lost particles, and degraded transport. A core–particle model is applied to parametrically analyze this process. Extensive self-consistent particle in cell simulations is employed to better quantify properties of the space-charge limits and to verify core–particle model predictions.

Control of Halo-Chaos in Beam Transport Networkvia Neural Network Adaptation with Time-DelayedFeedback

Subject of the halo-chaos control in beam transport networks (channels) hasbecome a key concerned issue for many important applications of high-currentproton beam since 1990'. In this paper, the magnetic field adaptive controlbased on the neural network with time-delayed feedback is proposed forsuppressing beam halo-chaos in the beam transport network with periodicfocusing channels. The envelope radius of high-current proton beam iscontrolled to reach the matched beam radius by suitably selecting thecontrol structure and parameter of the neural network, adjusting thedelayed-time and control coefficient of the neural network.