Dynamics of McMillan mappings II. Axially symmetric map

Abstract

In this article, we investigate the transverse dynamics of a single particle in a model integrable accelerator lattice, based on a McMillan axially-symmetric electron lens. Although the McMillan e-lens has been considered as a device potentially capable of mitigating collective space charge forces, some of its fundamental properties have not been described yet. The main goal of our work is to close this gap and understand the limitations and potentials of this device. It is worth mentioning that the McMillan axially symmetric map provides the first-order approximations of dynamics for a general linear lattice plus an arbitrary thin lens with motion separable in polar coordinates. Therefore, advancements in its understanding should give us a better picture of more generic and not necessarily integrable round beams. In the first part of the article, we classify all possible regimes with stable trajectories and find the canonical action-angle variables. This provides an evaluation of the dynamical aperture, Poincaré rotation numbers as functions of amplitudes, and thus determines the spread in nonlinear tunes. Also, we provide a parameterization of invariant curves, allowing for the immediate determination of the map image forward and backward in time. The second part investigates the particle dynamics as a function of system parameters. We show that there are three fundamentally different configurations of the accelerator optics causing different regimes of nonlinear oscillations. Each regime is considered in great detail, including the limiting cases of large and small amplitudes. In addition, we analyze the dynamics in Cartesian coordinates and provide a description of observable variables and corresponding spectra.

References in corpus (6)

• IOTA (Integrable Optics Test Accelerator): Facility and Experimental Beam Physics Program
• Nonlinear Accelerator Lattices with One and Two Analytic Invariants
• Integrable four-dimensional symplectic maps of standard type
• Betatron frequency and the Poincare rotation number
• Extracting Dynamical Frequencies from Invariants of Motion in Finite-Dimensional Nonlinear Integrable Systems
• Dynamics of McMillan mappings I. McMillan multipoles

Cited by in corpus (3)

• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part II: Recovery of approximate invariant
• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part I: Perturbation Theory
• Geometry of Almost-Conserved Quantities in Symplectic Maps. Part III: Approximate Invariants in Nonlinear Accelerator Systems