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.

Highlights

  • C OMPLEX variable methods have been widely used in the past to conveniently represent 2-D fields and phenomena by essentially identifying the components of a planar vector with the real and imaginary parts of a complex number

  • By means of the complex formalism, we prove the existence of invariant circles under such periodic transformation, which generalizes the classical concept of invariant points of the transformation determined through the Twiss parameters methods, meaning that invariant points in a periodic transport line are invariant circles of null radius

  • Through determining and solving the general differential equation governing the beam dynamics in complex form, it has been shown that the general complex transformation of a beam line is a subgroup of the Moebius transformation

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Summary

INTRODUCTION

C OMPLEX variable methods have been widely used in the past to conveniently represent 2-D fields and phenomena by essentially identifying the components of a planar vector with the real and imaginary parts of a complex number. By means of the complex formalism, we prove the existence of invariant circles under such periodic transformation, which generalizes the classical concept of invariant points of the transformation determined through the Twiss parameters methods, meaning that invariant points in a periodic transport line are invariant circles of null radius This allows us to reinterpret and more accurately describe, predict, quantify, and control important effects measured in practice, such as betatron oscillations and their beating, among other properties. As described in the following, both in theory and illustrated through example, it becomes clear that the proposed complex formalism gives us a deeper insight on linear beam dynamics, generalizes classical concepts, and has clear implications both in theoretical beam optics and in practical computation and design of present and future beam transport lines

DEFINITION OF THE COMPLEX PARAMETERS
GENERAL DIFFERENTIAL EQUATION OF THE COMPLEX FORM
SOME USEFUL PROPERTIES OF THE MOEBIUS TRANSFORMATION
Matrix Representation of the Moebius Transformation
Circle-Preserving Properties
Fixed Points of the Transformation
EXAMPLES OF COMMON BEAM LINE ELEMENTS IN COMPLEX FORM
Drift Space of Length L
Sector Dipole
Thick Lens of Strength k and Thickness l
COMPLEX PARAMETERS FOR A CIRCULAR ACCELERATOR
Structure of the Periodic Solutions of Beam Transport in Complex Form
CONCLUSION
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