Accurate transfer maps for realistic beamline elements: Part I, straight elements ∗ Chad E. Mitchell † and Alex J. Dragt ‡ Physics Department, University of Maryland, College Park, MD (Dated: October 23, 2018) The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole effects in the various beam-line elements. The inclu- sion of these effects requires a detailed and realistic model of the interior and fringe fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for ex- tracting this information accurately from 3-dimensional field data on a grid, as provided by various 3-dimensional finite-element field codes. Based on these realistic field models, Lie or other methods may be used to compute accurate design orbits and accurate transfer maps about these orbits. Part I of this work presents a treatment of straight-axis magnetic elements, while Part II will treat bend- ing dipoles with large sagitta. An exactly-soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks. based on these fits will be even more accurate. In this paper we will devote our attention to magnetic beam-line elements. Static electric beam-line elements, and static electric and magnetic beam-line elements such as velocity selectors (Wien filters), could be treated in a similar way. For a treatment of RF cavities, see [2]. There are two magnetic cases that it is convenient to handle separately: straight and curved. For straight magnetic elements such as solenoids, quadrupoles, sex- tupoles, octupoles, etc., and wigglers, it is convenient to employ cylindrical surfaces. For the case of curved mag- netic elements, such as dipoles with large design-orbit sagitta, it is convenient to employ the surface of a bent box with straight ends. In all cases the bounding sur- face will surround the design orbit within the beam-line element and will extend into the fringe-field regions out- side the beam-line element, thus taking into account all fringe-field effects as well as all effects within the body of the beam-line element. For the case of straight beam-line elements it is con- venient to describe the magnetic field in terms of a mag- netic scalar potential ψ . Then, if one wishes to compute transfer maps in terms of canonical coordinates, one can proceed with the aid of an associated vector potential A computed from ψ . Alternatively, if one wishes to in- tegrate noncanonical equations employing the magnetic field B , it can be obtained from the relation B = ∇ ψ . For the case of curved beam-line elements it is conve- nient to work directly with the vector potential. Its use in the case of canonical coordinates is then immediate. If instead one wishes to integrate noncanonical equations employing the magnetic field B , it can be obtained from the relation B = ∇ × A . In this paper we will treat the case of straight beam- line elements. For this purpose we will employ the sur- face of a (virtual) cylinder of uniform cross-section (cir- cular, elliptical, or rectangular) surrounding the beam, and which lies within all iron or other magnetic sources. In these cases, a Green function can be determined for the geometry of the fitted domain as a series composed $^{∗}$Work supported by U.S. Department of Energy Grant DE-FG02- 96ER40949. $^{†}$Electronic address: cemitch@umd.edu $^{‡}$Electronic address: dragt@umd.edu Realistic field data can be provided on a grid with the aid of various 3-dimensional finite element codes, some- times spot checked against measured data. But the com- putation of high-order transfer maps based on this data appears to pose an insurmountable problem: the calcula- tion of high-order transfer maps requires a knowledge of high derivatives of the field data. The direct calculation of high derivatives based only on grid data is intolera- bly sensitive to noise (due to truncation or round-off) in the grid data [1]. We will see that this problem can be solved by the use of surface methods. The effect of nu- merical noise can be overcome by fitting field data on a bounding surface far from the beam axis and continu- ing inward using the Maxwell equations. (We recall that interior fields are uniquely specified by their values on bounding surfaces.) While the process of differentiation serves to amplify the effect of numerical noise, the pro- cess of continuing inward using the Maxwell equations is smoothing . This smoothing is related to the fact that har- monic functions take their extrema on boundaries. When using surface methods, all fits are made on such bound- aries. Therefore, if these fits are accurate, interior data For the design of high-performance linear accelerators, synchrotrons, and storage and damping rings it is es- sential to have realistic electric and magnetic field infor- mation for the various beam-line elements in order to compute accurate design orbits and accurate high-order transfer maps about the design orbits. There are similar requirements for other charged-particle beam transport systems such as final focus sections, high-resolution spec- trometers, and high-resolution electron microscopes. arXiv:1001.1447v2 [physics.acc-ph] 6 May 2010 I. INTRODUCTION