Complex formalism of the linear beam dynamics

TL;DR

This paper develops a comprehensive complex formalism for linear beam dynamics, representing the entire beam transport as a Möbius transformation, enabling advanced analysis of beam behavior and invariant structures.

Contribution

It introduces a general differential equation for the complex formalism and demonstrates that the beam line transformation is a Möbius transformation, extending previous partial methods.

Findings

The general transformation of a linear beam line is a complex Möbius transformation.

Invariant points are special cases of invariant circles in the transformation.

New insights into betatron functions beating in mismatched injections.

Abstract

It has long been known that the ellipse normally used to model the phase space extension of a beam in linear dynamics may be represented by a complex number which can be interpreted similarly to a complex impedance in electrical circuits, so that classical electrical methods might be used for the design of such beam transport lines. However, this method has never been fully developed, and only the transport transformation of single particular elements, like drift spaces or quadrupoles, has been presented in the past. In this paper, we complete the complex formalism of linear beam dynamics by obtaining a general differential equation and solving it, to show that the general transformation of a linear beam line is a complex Moebius transformation. This result opens the possibility of studying the effect of the beam line on complete regions of the complex plane and not only on a single point. Taking advantage of this capability of the formalism, we also obtain an important result in the theory of the transport through a periodic line, proving that the invariant points of the transformation are only a special case of a more general structure of the solution, which are the invariant circles of the one-period transformation. Among other advantages, this provides a new description of the betatron functions beating in case of a mismatched injection in a circular accelerator.