Dipole fringe field thin map for compact synchrotrons

The fringe field of dipole magnets can be mportant in charged-particle beam dynamics [1,2]. For example, the edge angle effect on the vertical focusing has been parametrized by the fringe field integral introduced [1] and measured [3]. The nonlinear beam dynamics of the fringe field has also been included up to a sextupolelike potential in Ref. [2]. The fringe field effect is particularly important for compact accelerators that have a small bending radius. Applications of compact storage rings include the Inverse Compton Light Source (see, e.g., [4]), proton therapy synchrotrons, etc. The fringe field of dipole magnets typically extends to the range of the vertical magnet gap. The range of the fringe field is usually minimized to avoid magnetic field coupling due to limited available space in compact storage rings. As the range of the fringe field is reduced, parts of the fringe field effects may be minimized and others amplified. In particular, higher-order nonlinearity can become important. Often overlooked, an important feature of the dipole fringe field effect is the closed orbit deviation from the design orbit. This change of the closed orbit arises from the fact that the fringe field introduces continuously varying curvature, while the design orbit is defined by constant curvature starting from the hard edge dipole boundary. Although this fact is naive and simple, its effect can be large for compact storage rings and thus should not be disregarded. If not considered, it can cause significant misalignment errors for all other accelerator elements. As the fringe field extent decreases, the closed orbit deviation will also decrease. However, higher-order nonlinearity would increase. It is important to understand how these effects rely on the fringe field extent. Recently, the fringe field effects on nonlinear dynamics for compact rings and large emittance beams have been considered in Refs. [5,6]. The numerical method was used to extract the Taylor map or Lie map out of the 3D field data [7,8]. However, there is still a demand for an analytic expression of the fringe field map, particularly when the 3D field data are not available at the design stage. Because of its complex nonlinear effects, it is often hard to uncover the underlying physics based on simulation results alone. In addition, a good understanding of the dipole fringe field map can benefit not only storage ring design but also dipole magnet design. Theoretical studies on the influence of dipole fringe fields have been carried out by many researchers from the early 1960s through the early 1970s [9–14]. Although some earlier studies derived mapping equations up to third order of phase space variables, they often led to complicated expressions with too many integration parameters. This paper is intended to provide a simpler physics picture of fringe field effects. The Lie algebraic method is useful to study the fringe field effect [15]; e.g., the Methodical Accelerator Design ver.10 Polymorphic Tracking Code (MADX PTC) module implemented the second-order fringe field effect with a hard edge approximation [2]. However, the derivation is carried out on the pole face parallel frame, and it disregards the closed orbit deviation. This paper studies the effective thin map of the dipole soft fringe field using the Lie map method up to the next leading order of Magnus’ series and up to the 4th order of canonical variables with respect to the design orbit. We also calculate the mapping equation out of the Lie map to compare it with the simulation and an earlier study [1]. We organize this paper as follows. In Sec. II, we build a general fringe field model. Section III presents the corresponding Hamiltonian. Section IV reviews the Lie algebraic method to build an effective thin map. Section V presents the derived analytic map. Section VI compares the theory with the simulation results. Section VII discusses the fringe field induced closed orbit effect. Section VIII discusses the physics of the octupolelike potential. Section IX briefly shows the nonlinear detuning effect kilean20@gmail.com