Nonlinear dynamics can impact the performance of a particle accelerator in a number of different ways, depending on the type of the accelerator and the parameter regime in which it operates. Effects can range from minor changes in beam properties or behaviour, to serious limitations on beam stability and machine performance. In these notes, we provide a brief introduction to nonlinear dynamics in accelerators. After a review of some relevant results from linear dynamics, we outline some of the main ideas of nonlinear dynamics, framing the discussion in the context of two examples of different types of accelerator: a single-pass system (a bunch compressor) and a periodic system (a storage ring). We show how an understanding of the origins and nature of the nonlinear behaviour, together with the use of appropriate analysis tools, can prove useful in predicting the effects of nonlinear dynamics in different systems, and allow the design of appropriate corrections or mitigations.
Numerical simulations of beam-beam effects in the electron-positron Phi-factory DAPhNE taking into account the measured cubic nonlinearities have shown that the nonlinearities have a strong impact on the collider luminosity performance. The numerical results explain many of experimental observations made during collisions in both single and multibunch regimes.
In recent years, intense efforts have been devoted to studying how nonlinear effects can be used to shape the transverse beam distribution by means of an adiabatic crossing of nonlinear resonances. By this approach, it is possible to split the beams in the transverse plane, so that the initial single-Gaussian beam is divided into several distinct distributions. This is at the heart of the multiturn extraction process that is successfully in operation at the CERN Proton Synchrotron. Nonlinear effects can also be used to cool a beam by acting on its transverse beam distribution. In this paper, w…
An intense non-uniform particle beam exhibits strong emittance growth and halo formation in focusing channels due to nonlinear space charge forces of the beam. This phenomenon limits beam brightness and results in particle losses. The problem is connected with irreversible distortion of phase space volume of the beam in conventional focusing structures due to filamentation in phase space. Emittance growth is accompanied by halo formation in real space, which results in inevitable particle losses. A new approach for solving a self-consistent problem for a matched non-uniform beam in two-dimensi…
The fields produced by a long beam with a given transverse charge distribution in a homogeneous vacuum chamber are studied. Signals induced by a displaced finite-size beam on electrodes of a beam position monitor (BPM) are calculated and compared to those produced by a pencil beam. The non-linearities and corrections to BPM signals due to a finite transverse beam size are calculated for an arbitrary chamber cross section. Simple analytical expressions are given for a few particular transverse distributions of the beam current in a circular or rectangular chamber. Of particular interest is a ge…
Electron lenses are pulsed, magnetically confined electron beams whose current-density profile is shaped to obtain the desired effect on the circulating beam. Electron lenses were used in the Fermilab Tevatron collider for bunch-by-bunch compensation of long-range beam-beam tune shifts, for removal of uncaptured particles in the abort gap, for preliminary experiments on head-on beam-beam compensation, and for the demonstration of halo scraping with hollow electron beams. Electron lenses for beam-beam compensation are being commissioned in the Relativistic Heavy Ion Collider (RHIC) at Brookhave…
We propose to optimize the nonlinear beam dynamics of existing and future storage rings with direct online optimization techniques. This approach may have crucial importance for the implementation of diffraction limited storage rings. In this paper considerations and algorithms for the online optimization approach are discussed. We have applied this approach to experimentally improve the dynamic aperture of the SPEAR3 storage ring with the robust conjugate direction search method and the particle swarm optimization method. The dynamic aperture was improved by more than 5 mm within a short peri…
Transverse beam echoes can be excited with a single dipole kick followed by a single quadrupole kick. They have been used to measure diffusion in hadron beams and have other diagnostic capabilities. Here we develop theories of the transverse echo nonlinear in both the dipole and quadrupole kick strengths. The theories predict the maximum echo amplitudes and the optimum strength parameters. We find that the echo amplitude increases with smaller beam emittance and the asymptotic echo amplitude can exceed half the initial dipole kick amplitude. We show that multiple echoes can be observed provide…
In this paper, a new type magnet is proposed and processed to uniform the transverse beam profile. Compeared to the octupole, the new type magnet can prove a similar octupole magnet field in the middle, but the rise rate declines quickly in the edge. So that, a same uniform beam is got with less particles loss. Besides that, a mechanical structure is added to adjust the width of the middle region to satisfy different transverse dimensions, and this whould further reduce particles loss. Some numerical simulations have been done respectively with the octupole and the new type of magnet to show t…
Electron cooler is used to improve the quality of the beam in synchrotron, however it also introduces nonlinear electromagnetic field, which cause tuneshift, tunespread and may drive resonances leading to beam loss. In this paper the tuneshift and the tunespread caused by nonlinear electromagnetic field of the hollow electron beam was investigated, and the resonance driving terms of the nonlinear electromagnetic field was analysed. The differences were presented comparing with the solid electron beam. The calculations were performed for $^{238}U^{32+}$ ions of energy 1.272MeV stored in CSRm, u…
To achieve high focal spot intensities in heavy ion fusion, the ion beam must be compressed longitudinally by factors of ten to one hundred before it is focused onto the target. The longitudinal compression is achieved by imposing an initial velocity profile tilt on the drifting beam. In this paper, the problem of longitudinal drift compression of intense charged particle beams is solved analytically for the two important cases corresponding to a cold beam, and a pressure-dominated beam, using a one-dimensional warm-fluid model describing the longitudinal beam dynamics.
We have developed an efficacious algorithm for simulation of the beam-beam interaction in synchrotron colliders based on the nonlinear $δf$ method, where $δf$ is the much smaller deviation of the beam distribution from the slowly evolving main distribution $f_0$. In the presence of damping and quantum fluctuations of synchrotron radiation it has been shown that the slowly evolving part of the distribution function satisfies a Fokker-Planck equation. Its solution has been obtained in terms of a beam envelope function and an amplitude of the distribution, which satisfy a coupled system of ordina…
A numerical analysis of the self-interaction induced by a relativistic electron/positron beam in the presence of an intense external longitudinal magnetic field in plasmas is carried out. Within the context of the Plasma Wake Field theory in the overdense regime, the transverse beam-plasma dynamics is described by a quantumlike Zakharov system of equations in the long beam limit provided by the Thermal Wave Model. In the limiting case of beam spot size much larger than the plasma wavelength, the Zakharov system is reduced to a 2D Gross-Pitaevskii-type equation, where the trap potential well is…
We consider modeling for strong-strong beam-beam interactions beyond preceding linearized/perturbative methods such as soft gaussian approximation or FMM (HFMM) etc. In our approach discrete coherent modes, discovered before, and possible incoherent oscillations appear as a result of multiresolution/multiscale fast convergent decomposition in the bases of high-localized exact nonlinear modes represented by wavelets or wavelet packets functions. The constructed solutions represent the full multiscale spectrum in all internal hidden scales from slow to fast oscillating eigenmodes. Underlying var…
Due to the highly nonlinear nature of the beam-loading, it is at present not possible to analytically determine the beam parameters needed in a two-bunch plasma wakefield accelerator for maintaining a low energy spread. Therefore in this paper, by using the Broyden-Fletcher-Goldfarb-Shanno algorithm for the parameter scanning with the code QuickPIC and the polynomial regression together with k-fold cross-validation method, we obtain two fitting formulas for calculating the parameters of tri-Gaussian electron beams when minimizing the energy spread based on the beam-loading effect in a nonlinea…
We present the applications of variational--wavelet approach for the analytical/numerical treatment of the effects of insertion devices on beam dynamics. We investigate the dynamical models which have polynomial nonlinearities and variable coefficients. We construct the corresponding wavelet representation for wigglers and undulator magnets.
Plasma wakefields can enable very high accelerating gradients for frontier high energy particle accelerators, in excess of 10 GeV/m. To overcome limits on total acceleration achievable, specially shaped drive beams can be used in both linear and nonlinear plasma wakefield accelerators (PWFA), to increase the transformer ratio, implying that the drive beam deceleration is minimized relative to acceleration obtained in the wake. In this Letter, we report the results of a nonlinear PWFA, high transformer ratio experiment using high-charge, longitudinally asymmetric drive beams in a plasma cell. A…
We present the application of variational-wavelet analysis to numerical/analytical calculations of Wigner functions in (nonlinear) quasiclassical beam dynamics problems. (Naive) deformation quantization and multiresolution representations are the key points. We construct the representation via multiscale expansions in generalized coherent states or high-localized nonlinear eigenmodes in the base of compactly supported wavelets and wavelet packets.
We solve the Vlasov equation for the longitudinal distribution function and find stationary wave patterns when the distribution in the energy error is Maxwellian. In the long wavelength limit a stability criterion for linear waves has been obtained, and a Korteweg-de Vries-Burgers equation for the relevant hydrodynamic quantities has been derived.
Based on the Vlasov-Maxwell equations describing the self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. In an earlier paper [1] it has been shown that in the case of a beam with uniform phase space density the Vlasov-Maxwell equations can be replaced exactly by the macroscopic warm fluid-Maxwell equations with a triple adiabatic pressure law. In this paper we demonstrate that starting from the macroscopic fluid-Maxwell equations a nonlinear Schroedinger equation for the slowly …
The consideration of transverse dynamics of relativistic space-charge dominated beams and halo growth due to bunch oscillations is based on variational approach to rational (in dynamical variables) approximation for rms envelope equations. It allows to control contribution from each scale of underlying multiscales and represent solutions via exact nonlinear eigenmodes expansions. Our approach is based on methods provided possibility to work with well-localized bases in phase space and good convergence properties of the corresponding expansions.
We present the applications of nonlinear local harmonic analysis methods to the modelling of beam-beam interaction. Our approach is based on methods provided the possibility to work with dynamical beam localization in phase space. The consideration of Fokker-Planck or Vlasov-Maxwell models is based on a number of anzatzes, which reduce initial problems to a number of dynamical systems (with constraints) and on variational-wavelet approach to polynomial/rational approximations for reduced nonlinear dynamics. We calculate contribution to full dynamics (partition function) from all underlying sub…
In this series of eight papers we present the applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In this part, assuming a sinusoidal field variation, we consider the analytical treatment of the effects of insertion devices on beam dynamics. We investigate via wavelet approach a dynamical model which has polynomial nonlinearities and variable coefficients. We construct the corresponding wavelet representation. As examples we consider wigglers and undulator magnets. We consider the further modification of our variational appro…
