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 question 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.
Highlights
In the design of beam transfer lines, one often encounters the problem of finding a combination of quadrupole lenses and field free spaces that will produce particular transfer matrices in both the horizontal and the vertical planes
Nowadays this problem is typically approached with the help of computer routines which minimize the deviations from the desired matrices as a function of the quadrupole strengths, lengths, and distances between them
Searching for a 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 and the answer to this question is not known for more or less realistic quadrupole field models and for the both most commonly used approximations of quadrupole focusing, namely thick and thin quadrupole lenses
Summary
In the design of beam transfer lines, one often encounters the problem of finding a combination of quadrupole lenses and field free spaces (drifts) that will produce particular transfer matrices in both the horizontal and the vertical planes. The representation of the matrix of the thin-lens multiplet as a product of elementary P matrices [together with the multiplication formula (A4)] is a useful new tool for the analytical study of the properties of thin-lens systems It gives some clarification of the question why the role of the variable drift spaces and the role of the variable lens strengths are different when they are used as fitting parameters. The idea of decoupled tuning knobs by itself is not new in the field of accelerator physics (see, for example, [8,9]), our approach is new and is not based on an iterative usage of small steps in the lens strengths obtained at each iteration by linearization
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