Nuclear Instruments and Methods in Physics Research B63 (1992) 319-325 North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Nuclear Instnments 4% Methods in Physics Research SectmEl Optical properties of helical undulators F. Ciocci, G. Dattoli, L. Giannessi, C. Mari and A. Torre ENEA Area Energia e Innovazione, Dip. Sviluppo Tecnologie di Punta, Centro Ricerche Energia Frascati, P.O. Box 65, 00044 Frascati, Rome, Italy Received 28 June 1991 The helical undulator is an optical element which behaves like a quadrupole (focusing an electron beam both in radial and vertical directions) and like a solenoid. In this paper we explore more deeply the optical properties of such a device and discuss problems like emittance nonconservation due to the interaction of the beam with the helical undulator. zyxwvutsrqponmlkjihgfedcbaZY 1. Introduction Undulators are currently used both in synchrotron radiation and FEL experiments. A clear understanding of the dynamical features of an electron going through the magnetic field of an undulator has already proven very useful to explain the reasons underlying the inhomogenous broadenings of the radiated spectrum [l] or to design the magnets in such a way that emittance degradation is minimized, thus preventing possible degradation of the laser oper- ation [2]. More recently a further example of the importance of such an analysis has been the proof of a much richer harmonic content, once, in the theory of undulator brightness, the effects of the betatron motion are in- cluded 131. The role of the undulators as optical elements on an electron beam line has been carefully studied since the pioneering work of Blewett and Chasman [4]. Much work has been devoted to this aspect in the case of undulators installed on storage rings, but the same effort has not been devoted to undulators operating on single passage FELs. Although an interesting body of research is in progress in connection with the effect of undulator field errors [5] and of beam emittance degra- dations [6], not much attention has been devoted to the modern Lie techniques [7], which can provide a power- ful mathematical tool to characterize both e-beam dy- namics and transport in undulator magnets. Such tech- niques have been developed to treat nonlinearities in equations of motion, hardly tractable with conventional methods. Anharmonic motion effects in undulators are due to the inclusion of the multipolar contributions, which become more and more relevant moving away from the undulator axis [8]. This paper is devoted to the case of the helical undulator, which even at the lowest order in the multi- polar expansion possesses the interesting feature of behaving like a quadrupole (focusing in both vertical and radial directions) and a solenoid. This aspect of the problem has been scarcely em- phasized and there are at least two reasons to devote a careful analysis to the problem. At first, it is a “natural exercise” to start with, to illustrate more modern the- ory of beam transport and dynamics. Secondly, it may give information on the transverse motion coupling and emittance non conservation. This paper is the first of a series dedicated to the use of algebraic techniques in the study of electron dynamics in magnetic undulators and consists of four sections. Section 2 contains some preliminary consider- ations on the analysis of transverse mode coupling, using classical evolution operators. Section 3 is devoted to the specific problem of emittance non conservation in helical undulators. Section 4 contains concluding remarks. Two appendices are finally concerned with detailed calculations. 2. Beam dynamics and transport in “quadrupole solenoid” The motion of an electron (or better of a charged particle) through a quadrupole-solenoid (QS) is char- acterized by the following Hamiltonian H=;(p;+p;) +fk,x2+3k,y2+~(ypx-~py). (2.1) The quadrupole and solenoid strenghts are character- ized by k and P’, respectively. Furthermore 9’ is 0168-583X/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved