Dynamics of McMillan mappings II. Axially symmetric map

TL;DR

This paper analyzes the transverse dynamics of particles in an integrable accelerator model using a McMillan axially-symmetric lens, classifying stable regimes and exploring nonlinear oscillations for different system configurations.

Contribution

It provides a comprehensive classification of stable trajectories, action-angle variables, and detailed analysis of nonlinear regimes in the McMillan axially-symmetric map, advancing understanding of this device's dynamics.

Findings

Identification of all stable trajectory regimes

Parameterization of invariant curves and dynamical aperture

Analysis of nonlinear oscillation regimes and spectra

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, Poincar\'e 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.