A Formalism for the Transport and Matching of Coupled Beams in Accelerators

TL;DR

This paper introduces a comprehensive formalism for transporting and matching coupled beam optics functions in accelerators, enabling better diagnosis, characterization, and control of coupled beams.

Contribution

It provides an explicit, complete formalism based on existing parametrizations, with methods for matching, extracting parameters, and analyzing resonance in coupled beam systems.

Findings

Formalism allows matching coupled and uncoupled optics.

Methods for extracting eigenmode emittances from measurements.

Expresses linear difference resonance in terms of coupled optics functions.

Abstract

Understanding transverse coupling dynamics is crucial for beam physics, accelerator design, and operations. Currently, most accelerators are designed for uncoupled beams, and coupling is treated as an error or perturbation. Many transverse ($x$-$y$) coupling parametrizations exist: Edward-Teng, Mais-Ripken, Levedev-Bogacz, and others. Here, we present an explicit and complete formalism for transporting coupled beam optics functions based on Mais-Ripken and Lebedev-Bogacz parametrizations. The formalism allows for matching generally coupled beam optics functions but applies to uncoupled optics as well. A complete transformation method for coupled optics provides easy matching routines that can be added to known beam optics codes that lack this feature. For fully coupled lattices, we present methods for extracting eigenmode emittances and other beam parameters from observables that can be measured, which is essential to diagnose and characterize the beam in a real machine. We express the linear difference resonance in terms of coupled optics functions and relate it to the coupling strength parameter with explicit emittance exchange formalism realized from the generating functions discussed here.