Explicit Thin-Lens Solution for an Arbitrary Four by Four Uncoupled Beam Transfer Matrix

TL;DR

This paper provides an explicit solution demonstrating that any four by four uncoupled beam transfer matrix can be constructed using a finite sequence of thin-lenses and drifts, advancing the understanding of beam transport line design.

Contribution

It offers a new explicit solution that proves the representation of any such transfer matrix with finite thin-lenses and drifts, addressing a longstanding open problem.

Findings

Any four by four uncoupled beam transfer matrix can be represented by finite thin-lenses and drifts.

The explicit solution advances the theoretical understanding of beam transport design.

It resolves the representation problem for thin-lens systems in beam physics.

Abstract

In the design of beam transport lines one often meets the problem of constructing a quadrupole lens system that will produce desired transfer matrices in both the horizontal and vertical planes. Nowadays this problem is typically approached with the help of computer routines, but searching for the numerical solution one has to remember that it is not proven yet that an arbitrary four by four uncoupled beam transfer matrix can be represented by using a finite number of drifts and quadrupoles (representation problem) and the answer to this questions is not known not only for more or less realistic quadrupole field models but also for the both most commonly used approximations of quadrupole focusing, namely thick and thin quadrupole lenses. In this paper we make a step forward in resolving the representation problem and, by giving an explicit solution, we prove that an arbitrary four by four uncoupled beam transfer matrix actually can be obtained as a product of a finite number of thin-lenses and drifts.