In this paper we present the applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In the general case we have the solution as a multiresolution expansion in the base of compactly supported wavelet basis. The solution is parametrized by the solutions of two reduced algebraical problems, one is nonlinear and the second is some linear problem, which is obtained from one of the next wavelet constructions: Fast Wavelet Transform, Stationary Subdivision Schemes, the method of Connection Coefficients. According to the orbit method and by using construction from the geometric quantization theory we construct the symplectic and Poisson structures associated with generalized wavelets by using metaplectic structure. We consider wavelet approach to the calculations of Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian systems and for parametrization of Arnold-Weinstein curves in Floer variational approach.
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 we consider the applications of discrete wavelet analysis technique to maps which come from discretization of continuous nonlinear polynomial problems in accelerator physics. Our main point is generalization of wavelet analysis which can be applied for both discrete and continuous cases. We give explicit multiresolution representation for solutions of discrete problems, which is correct discretization of our represen…
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 we consider variational wavelet approach for loops, invariant bases on semidirect product, KAM calculation via FWT.
In this paper we consider applications of methods from wavelet analysis to nonlinear dynamical problems related to accelerator physics. In our approach we take into account underlying algebraical, geometrical and topological structures of corresponding problems.
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 we consider application of FWT to metaplectic representation(quantum and chaotical problems) and quasiclassics.
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 paper we consider invariant formulation of nonlinear (Lagrangian or Hamiltonian) dynamics on semidirect structure (relativity or dynamical groups) and corresponding invariant calculations via CWT.
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, according to variational approach we obtain a representation for orbital particle motion in storage rings as a multiresolution (multiscales) expansion in the base of well-localized in phase space wavelet basis. By means of this "wavelet microscope" technique we can take into account contribution from each scale of resolution.
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 we consider a model for spin-orbital motion: orbital dynamics and Thomas-BMT equations for classical spin vector. We represent the solution of this dynamical system in framework of biorthogonal wavelets via variational approach. We consider a different variational approach, which is applied to each scale.
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…
In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic orbital particle motion and arbitrary variable coefficients. Then we consider more flexible variational method which is based on biorthogonal wavelet approach. Also we consider different variational approach, which is applied …
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 we consider orbital motion in transverse plane for a single particle in a circular magnetic lattice in case when we take into account multipolar expansion up to an arbitrary finite number. We reduce initial dynamical problem to the finite number (equal to the number of n-poles) of standard algebraical problem and represent all dynamical variables via an expansion in the base of periodical wavelets.
As modern accelerators demand excellent stability to magnet power supply (PS), it is necessary to decrease harmonic currents passing magnets. Aim at depressing rappel current from PS in Beijing electron-positron collider II, a wavelet-based active power filter (APF) is proposed in this paper. APF is an effective device to improve the quality of currents. As a countermeasure to these harmonic currents, the APF circuit generates a harmonic current, countervailing harmonic current from PS. An active power filter based on wavelet transform is proposed in this paper. Discrete wavelet transform is u…
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 present applications of variational-wavelet approach to nonlinear (rational) rms envelope equations. We have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic beam motion and arbitrary variable coefficients. Also we consider more flexible variational method which is based on biorthogonal wavelet approach.
We present applications of variational -- wavelet approach to nonlinear (rational) rms envelope dynamics. We have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis.
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.
We present the applications of wavelet analysis methods in constrained variational framework to calculation of dynamical aperture. We construct represention via exact nonlinear high-localized periodic eigenmodes expansions, which allows to control contribution to motion from each scale of underlying multiscale structure and consider qualitative approach to the problem.
Integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress instabilities and to mitigate the effects of space charge and magnetic field errors. To create such an accelerator lattice one has to find magnetic and/or electric field combinations leading to a stable integrable motion. This paper presents families of lattices with one invariant where bounded motion can be easily created in large volumes of the phase space. In addition, it presents two examples of integrable nonlinear accelerator lattices, realizable with longitudinal-coordinat…
We consider an application of modification of our variational-wavelet approach to some nonlinear collective model of beam/plasma physics: Vlasov/Boltzmann-like reduction from general BBGKY hierachy related to modeling of propagation of intense charged particle beams in high-intensity accelerators and transport systems. We use fast convergent multiscale variational-wavelet representations for solutions which allow to consider polynomial and rational type of nonlinearities. The solutions are represented via the multiscale decomposition in nonlinear high-localized eigenmodes (waveletons). In cont…
We consider space-charge dominated beam transport systems, where space-charge forces are the same order as external focusing forces and dynamics of the corresponding emittance growth. We consider the coherent modes of oscillations and coherent instabilities both in the different nonlinear envelope models and in initial collective dynamics picture described by Vlasov system. Our calculations are based on variation approach and multiresolution in the base of high-localized generalized coherent states/wavelets. We control contributions to dynamical processes from underlying multiscales via nonlin…
Nonlinear N-body integrable Hamiltonian systems, where N is an arbitrary number, attract the attention of mathematical physicists for the last several decades, following the discovery of some number of these systems. This paper presents a new integrable system, which can be realized in facilities such as particle accelerators. This feature makes it more attractive than many of the previous such systems with singular or unphysical forces.
Integrable systems appeared in physics long ago at the onset of classical dynamics with examples being Kepler's and other famous problems. Unfortunately, the majority of nonlinear problems turned out to be nonintegrable. In accelerator terms, any 2D nonlinear nonintegrable mapping produces chaotic motion and a complex network of stable and unstable resonances. Nevertheless, in the proximity of an integrable system the full volume of such a chaotic network is small. Thus, the integrable nonlinear motion in accelerators has the potential to introduce a large betatron tune spread to suppress inst…
We present some applications of general harmonic/wavelet analysis approach (generalized coherent states, wavelet packets) to numerical/analytical calculations in (nonlinear) quasiclassical/quantum beam dynamics problems. (Naive) deformation quantization, multiresolution representations and Wigner transform are the key points.
Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems
Independent Researcher（独立研究者）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Old Dominion University（老道明大学）；Elettra Sincrotrone Trieste（的里雅斯特电同步辐射中心）；Novosibirsk State Technical University（新西伯利亚国立技术大学）；University of Chicago（芝加哥大学）
The use of nonlinear lattices with large betatron tune spreads can increase instability and space charge thresholds due to improved Landau damping. Unfortunately, the majority of nonlinear accelerator lattices turn out to be nonintegrable, producing chaotic motion and a complex network of stable and unstable resonances. Recent advances in finding the integrable nonlinear accelerator lattices have led to a proposal to construct at Fermilab a test accelerator with strong nonlinear focusing which avoids resonances and chaotic particle motion. This presentation will outline the main challenges, th…
We present applications of variational -- wavelet approach to different forms of nonlinear (rational) rms envelope equations. We have the representation for beam bunch oscillations as a multiresolution (multiscales) expansion in the base of compactly supported wavelet bases.
The effect of nonlinearity on plasma wake wave excited by a relativistic cylindrical charged bunch is investigated. It is shown that owing to the nonlinearity the amplitude of wake wave gets modulated in the longitudinal direction. The nonlinear wavelength in the excited field changes in the transverse direction with the result that the phase front is distored and a turbulence developed. The nonlinear phase front distortion may be compensated by radial change of unperturbed plasma density.
We present applications of variational -- wavelet approach to three different models of nonlinear beam motions with underlying collective behaviour: Vlasov-Maxwell-Poisson systems, envelope dynamics, beam-beam model. We have the representation for dynamical variables as a multiresolution (multiscales) expansion via high-localized nonlinear eigenmodes in the base of compactly supported wavelet bases. Numerical modelling demonstrates formation of coherent structures and stable patterns.
We present the applications of variational--wavelet approach to nonlinear (rational) model for spin-orbital motion: orbital dynamics and Thomas-BMT equations for classical spin vector. We represent the solution of this dynamical system in framework of periodical wavelets via variational approach and multiresolution.
Experimental Demonstration of a Two-Dimensional Nonlinear Integrable System in a Particle Accelerator
Fermi National Accelerator Laboratory（费米国家加速器实验室）；Consortium for Advanced Science and Technology (CASE), University of Chicago（芝加哥大学先进科学技术联盟）；Michigan State University（密歇根州立大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Northern Illinois University（北伊利诺伊大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Laser wakefield acceleration modeling using the Lorentz-boosted frame technique in the particle-in-cell code has demonstrated orders of magnitude speedups. A convergence study was previously conducted in cases with external injection in the linear regime and without injection in the nonlinear regime, and the obtained results have shown a convergence within the percentage level. In this article, a convergence study is carried out to model electron self-injection in the 2-1/2D configuration. It is observed that the Lorentz-boosted frame technique is capable of modeling complex particle dynamics …
We generate inverse Compton scattered X-rays in both linear and nonlinear regimes with a 250 MeV laser wakefield electron accelerator and plasma mirror by retro-reflecting the unused drive laser light to scatter from the accelerated electrons. We characterize the X-rays using a CsI(Tl) voxelated scintillator that measures their total energy and divergence as a function of plasma mirror distance from the accelerator exit. At each plasma mirror position, these X-ray properties are correlated with the measured fluence and inferred intensity of the laser pulse after driving the accelerator to dete…
An alternative representation for the vector potential of the nonlinear magnetic insert for the Integrable Optics Test Accelerator (IOTA), first described in Sec. V.A. of the paper of Danilov and Nagaitsev, is determined from first principles using standard complex variable methods. In particular, it is shown that the coupled system consisting of the 2D Laplace equation and the Bertrand-Darboux equation is equivalent to a single ordinary differential equation in the complex plane, and a simple solution is constructed. The results are consistent with the paper of Danilov and Nagaitsev, and this…
The McMillan system is a novel method to increase the tune spread of a beam without decreasing its dynamic aperture due to the systems integrability. While the ideal system is based on an infinitely thin kick, the physical design requires a thick electron lens, including a solenoid. Particle transport through the lens is difficult to simulate due to the nature of the force on the circulating beam. This paper demonstrates accurate simulation of a thick McMillan lens in a solenoid using symplectic integrators derived from Yoshidas method.
We present the applications of methods from nonlinear local harmonic analysis for calculations in nonlinear collective dynamics described by different forms of Vlasov-Maxwell-Poisson equations. Our approach is based on methods provided the possibility to work with well-localized in phase space bases, which gives the most sparse representation for the general type of operators and good convergence properties. The consideration is based on a number of anzatzes, which reduce initial problems to a number of dynamical systems and on variational-wavelet approach to polynomial approximations for nonl…
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 …
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…
Laser powered dielectric structures achieve high-gradient particle acceleration by taking advantage of modern laser technology capable of producing electric fields in excess of 10GV/m. These fields can drive the bulk dielectric beyond its linear response, and break the phase synchronicity between the accelerating field and the electrons. We show how control of the pulse dispersion can be used to compensate the effect of self-phase modulation and maximize the energy gain in the laser accelerator.In our experiment, a high brightness 8MeV e-beam is used to probe accelerating fields of 1.8GV/m in …
Laser wakefield acceleration relies on the excitation of a plasma wave due to the ponderomotive force of an intense laser pulse. However, plasma wave trains in the wake of the laser have scarcely been studied directly in experiments. Here we use few-cycle shadowgraphy in conjunction with interferometry to quantify plasma waves excited by the laser within the density range of GeV-scale accelerators, i.e. a few 1e18 cm-3. While analytical models suggest a clear dependency between the non-linear plasma wavelength and the peak potential a_0, our study shows that the analytical models are only accu…
What prevents us from building super-high intensity accelerators? The answer is case-specific, but it often points to one of the following phenomena: machine resonances, various tune shifts (and spreads), and instabilities. These three phenomena are interdependent in all present machines. In this paper we propose a path toward alleviating these phenomena by making accelerators nonlinear. This idea is not new: Orlov (1963) and McMillan (1967) have proposed initial ideas on nonlinear focusing systems for accelerators. However, practical implementations of such ideas proved elusive, until recentl…
Many quantum integrable systems are obtained using an accelerator physics technique known as Ermakov (or normalized variables) transformation. This technique was used to create classical nonlinear integrable lattices for accelerators and nonlinear integrable plasma traps. Now, all classical results are carried over to a nonrelativistic quantum case.
In this paper, we get the 1D approximate analytical solution of the plasma electrostatic wake driven by the laser, and get the oscillating frequency of the wake modified by the nonlinear high order terms, and find that the frequency depends on the oscillating amplitudes of the wake and the laser which is the general nonlinear phenomenon. Finally we analyze the longitudinal beam dynamics in this electrostatic wake, and find that the high order terms don't change the topology of the longitudinal phase space.
A modified version of the Plasma Beat-Wave Accelerator scheme is introduced and analyzed, which is based on autoresonant phase-locking of the nonlinear Langmuir wave to the slowly chirped beat frequency of the driving lasers via adiabatic passage through resonance. This new scheme is designed to overcome some of the well-known limitations of previous approaches, namely relativistic detuning and nonlinear modulation or other non-uniformity or non-stationarity in the driven Langmuir wave amplitude, and sensitivity to frequency mismatch due to measurement uncertainties and density fluctuations an…
The leading Lie generators, including the chromatic effects, due to hard-edge fringe field of single multipole and solenoid are derived from the vector potentials within a Hamiltonian system. These nonlinear generators are applied to the interaction region of PEP-II to analyze the linear errors due to the feed-down from the off-centered quadrupoles and solenoid. The nonlinear effects of tune shifts at large amplitude, the synchro-betatron sidebands near half integer and their impacts on the dynamic aperture are studied in the paper.
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…
Fermilab is constructing the Integrable Optics Test Accelerator (IOTA) as the centerpiece of the Accelerator R&D Program towards high-intensity circular machines. One of the factors limiting the beam intensity in present circular accelerators is collective instabilities, which can be suppressed by a spread of betatron frequencies (tunes) through the Landau damping mechanism or by an external damper, if the instability is slow enough. The spread is usually created by octupole magnets, which introduce the tune dependence on the amplitude and, in some cases, by a chromatic spread (tune dependence…
Many quantum integrable systems are obtained using an accelerator physics technique known as Ermakov (or normalized variables) transformation. This technique was used to create classical nonlinear integrable lattices for accelerators and nonlinear integrable plasma traps. Now, all classical results are carried over to a nonrelativistic quantum case.
An electron lens is planned for the Fermilab Integrable Optics Test Accelerator as a nonlinear element for integrable dynamics, as an electron cooler, and as an electron trap to study space-charge compensation in rings. We present the main design principles and constraints for nonlinear integrable optics. A magnetic configuration of the solenoids and of the toroidal section is laid out. Single-particle tracking is used to optimize the electron path. Electron beam dynamics at high intensity is calculated with a particle-in-cell code to estimate current limits, profile distortions, and the effec…
We consider some reduction from nonlinear Vlasov-Maxwell equation to rms/rate equations for second moments related quantities. Our analysis is based on variational wavelet approach to rational (in dynamical variables) approximation. It allows to control contribution from each scale of underlying multiscales and represent solutions via multiscale exact nonlinear eigenmodes (waveletons) expansions. Our approach provides the possibility to work with well-localized bases in phase space and best convergence properties of the corresponding expansions without perturbations or/and linearization proced…
Recently, the study of integrable Hamiltonian systems has led to nonlinear accelerator lattices with one or two transverse invariants and wide stable tune spreads. These lattices may drastically improve the performance of high-intensity machines, providing Landau damping to protect the beam from instabilities, while preserving dynamic aperture. The Integrable Optics Test Accelerator (IOTA) is being built at Fermilab to study these concepts with 150-MeV pencil electron beams (single-particle dynamics) and 2.5-MeV protons (dynamics with self fields). One way to obtain a nonlinear integrable latt…
Nonlinear optics is a promising idea potentially opening the path towards achieving super high beam intensities in circular accelerators. Creation of a tune spread reaching 50% of the betatron tune would provide strong Landau damping and make the beam immune to instabilities. Recent theoretical work has identified a possible way to implement stable nonlinear optics by incorporating nonlinear focusing elements into a specially designed machine lattice. In this report we propose the design of a test accelerator for a proof-of-principle experiment. We discuss possible studies at the machine, requ…
Single particle dynamics in electron microscopes, ion or electron lithographic instruments, particle accelerators, and particle spectrographs is described by weakly nonlinear ordinary differential equations. Therefore, the linear part of the equation of motion is usually solved and the nonlinear effects are then found in successive order by iteration methods. A Hamiltonian nature of these equations can lead to simplified computations of particle transport through an optical device when a suitable computational method is used. Many ingenious microscopic and lithographic devices were found by H.…
The plasma wakefield accelerator may accelerate particles to high energy in a future linear collider with unprecedented acceleration gradients, exceeding the GeV/m range. Beams for this application would have extremely high brightness and, subject to the intense plasma ion-derived focusing, they would achieve densities high enough to induce the plasma ions to collapse into the beam volume. This non-uniform ion density gives rise to strong nonlinear focusing which may lead to deleterious beam emittance growth. The effects of ion collapse and their mitigation has been investigated recently throu…
Conventional particle accelerators use linear focusing forces for transverse confinement. As a consequence of linearity, accelerating rings are sensitive to myriad resonances and instabilities. At high beam intensity, uncontrolled resonance-driven losses can deteriorate beam quality and cause damage or radio-activation in beam line components and surrounding areas. This is currently a major limitation of achievable current densities in state-of-the-art accelerators. Incorporating nonlinear focusing forces into machine design should provide immunity to resonances through nonlinear detuning of p…
A future plasma based linear collider has the potential to reach unprecedented energies and transform our understanding of high energy physics. The extremely dense beams in such a device would cause the plasma ions to fall toward the axis. For more mild ion motion, this introduces a nonlinear perturbation to the focusing fields inside of the bubble. However, for extreme ion motion, the ion distribution collapses to a quasi-equilibrium characterized by a thin filament of extreme density on the axis which generates strong, nonlinear focusing fields. These fields can provoke unacceptable emittanc…
At Integrable Optics Test Accelerator it is possible to create a nonlinear focusing optics with one invariant of motion using just conventional magnets. 6D simulations show that this will allow to achieve a tune spread of 0.05 without significant reduction of dynamic aperture.
A high peak current, flat longitudinal phase space electron beam is desirable for efficient x-ray free electron laser (FEL) radiation in next generation light sources. To attain such a beam requires the extensive design of the linear accelerator (linac) including both linear and nonlinear effects. In this paper, we propose a lumped longitudinal beam dynamics model for fast optimization of the electron beam longitudinal phase space through the accelerator. This model is much faster than available tracking programs and also shows good agreement with the fully three-dimensional element-by-element…
