Robert Warnock

Long-term bounds on nonlinear Hamiltonian motion

Abstract In various important applications of Hamiltonian mechanics, notably in problems of accelerator design, it would be useful to set bounds on nonlinear motion for finite but very long times. Such bounds can be sought through construction of a canonical transformation to new action-angle variables (J,Ψ), such that J is nearly constant, and Ψ advances almost linearly with the time. By examining the residual change in J during a time T0 from various initial conditions in the open domain Ω of phase space, one can estimate the change in J during a much larger time T, on any orbit starting in a smaller open domain Ω0⊂Ω. A numerical realization of this idea is described. The canonical transformations, equivalent to close approximations to invariant tori, are constructed by a new method in which surfaces are fitted to computed orbits. The perturbed tunes v (winding numbers) are determined as functions of J, and the inverse function J(v) is also computed. This leads to an accurate map of resonant tune lines into J space, which serves to locate dangerous regions of phase space. Near a single strong resonance, J varies more than usual but follows the pattern expected from the isolated resonance model. All calculations proceed from a time evolution map defined by a symplectic integrator or equivalent explicit formula. As an example, an accelerator problem in 2 1 2 degrees of freedom is treated. For betatron motion in a model accelerator with strong sextupole magnets, stability for 108 turns is predicted in a region of substantial nonlinearity.

Shielded coherent synchrotron radiation and its possible effect in the Next Linear collider

Shielded coherent synchrotron radiation is discussed for two cases: (1) a beam following a curved path in a plane midway between two parallel, perfectly conducting plates, and (2) a beam circulating in a toroidal chamber with resistive walls. Wake fields and the radiated energy are computed with parameters for the high-energy bunch compressor of the Next Linear Collider (NLC).<<ETX>>

A general method for propagation of the phase space distribution, with application to the saw-tooth instability

The authors propose and illustrate a general numerical method to follow the probability distribution in phase space as a function of time. It applies to any multiparticle system governed by Liouville, Vlasov or Vlasov-Fokker-Planck dynamics. The technique, based on discretization of the local Perron-Frobenius operator, is simple in concept, easy to implement, and numerically stable in examples studied to date. The authors illustrate by treating longitudinal dynamics in electron storage rings with realistic wake field. Applied to the SLC damping rings, the method gives the observed current threshold for bunch lengthening, and several aspects of observed behavior above threshold, including the presence of a bursting or sawtooth mode. In contrast to previous particle-in-cell simulations, the authors have very low numerical noise and the ability to follow the motion over several damping times. The method has also been applied to the coherent beam-beam interaction. It appears likely that this approach will be of interest for some of the central problems of this workshop, for instance matching of space-charge dominated beams to a focusing channel, and coherent synchrotron radiation with self-consistent charge/current density.

Linear Vlasov Analysis for Stability of a Bunched Beam

We study the linearized Vlasov equation for a bunched beam subject to an arbitrary wake function. Following Oide and Yokoya, the equation is reduced to an integral equation expressed in angle-action coordinates of the distorted potential well. Numerical solution of the equation as a formal eigenvalue problem leads to difficulties, because of singular eigenmodes from the incoherent spectrum. We rephrase the equation so that it becomes non-singular in the sense of operator theory, and has only regular solutions for coherent modes. We report on a code that finds thresholds of instability by detecting zeros of the determinant of the system as they enter the upper-half frequency plane, upon increase of current. Results are compared with a time-domain integration of the nonlinear Vlasov equation with a realistic wake function for the SLC damping rings. There is close agreement between the two calculations.

Coherent Synchrotron Radiation in Storage Rings

We take a detour from the main theme of this volume and present a discussion of coherent synchrotron radiation (CSR) in the context of storage rings rather than single-pass systems. Interest in this topic has been revived by a series of measurements carried out at several light source facilities. There is strong evidence that the observed coherent signal is accompanied by a beam instability, possibly driven by CSR itself. In this paper we review a “self-consistent” model of longitudinal beam dynamics in which CSR is the only agent of collective forces. The model yields numerical solutions that appear to reproduce the main features of the observations.

Methods of stability analysis in nonlinear mechanics

We review our recent work on methods to study stability in nonlinear mechanics, especially for the problems of particle accelerators, and compare our ideas to those of other authors. We emphasize methods that (a) shows promise as practical design tools, (b) are effective when the nonlinearity is large, and (c) have a strong theoretical basis.

Progress on a Vlasov Treatment of Coherent Synchrotron Radiation from Arbitrary Planar Orbits

We report on our progress in the development of a fully self-consistent Vlasov treatment of coherent synchrotron radiation (CSR) effects on particle bunches traveling on arbitrary planar orbits. First we outline our Vlasov approach and the approximation we are currently studying. Then we discuss recent numerical results for a benchmark model studied extensively with codes by several authors.

Convergence of a Fourier-spline representation for the full-turn map generator

Single-turn data from a symplectic tracking code can be used to construct a canonical generator for a full-turn symplectic map. This construction has been carried out numerically in canonical polar coordinates, the generator being obtained as a Fourier series in angle coordinates with coefficients that are spline functions of action coordinates. Here we provide a mathematical basis for the procedure, finding sufficient conditions for the existence of the generator and convergence of the Fourier-spline expansion. The analysis gives insight concerning analytic properties of the generator, showing that in general there are branch points as a function of angle and inverse square root singularities at the origin as a function of action.

Construction of symplectic full-turn maps by application of an arbitrary tracking code

It is pointed out that a map to describe propagation of particles through any section of a nonlinear lattice can be represented as a Taylor expansion about the origin in phase space. Although the technique to compute the Taylor coefficients has been improved recently, the expansion may fail to provide adequate accuracy in regions where nonlinear effects are substantial. It is suggested that a representation of the map in angle-action coordinates, with the angle dependence given by a Fourier series and the action dependence by polynomials in I/sup 1/2/, may be more successful. Maps of this form are easily constructed by taking Fourier transforms of results from an arbitrary symplectic tracking code. Examples are given of one-turn and two-turn maps for the SLC (Stanford Linear Collider) North Damping Ring in a strongly nonlinear region. Results for accuracy and speed of evaluation of the maps are quite encouraging. It seems feasible to make accurate maps for the SSC (Superconducting Super Collider) by this method.<<ETX>>

Simulation of the microbunching instability in beam delivery systems for free electron lasers

In this paper, we examine the growth of the microbunching instability in the electron beam delivery system of a free electron laser (FEL). We present the results of two sets of simulations, one conducted using a direct Vlasov solver, the other using a particle-in-cell code Impact-Z with the number of simulation macroparticles ranging up to 100 million. Discussion is focused on the details of longitudinal dynamics and on numerical values of uncorrelated (slice) energy spread at different points in the lattice. In particular, we assess the efficacy of laser heater in suppression of the instability, and look at the interplay between physical and numerical noise in particle-based simulations.