On sector magnets or transverse electromagnetic fields in cylindrical coordinates

TL;DR

This work addresses the challenge of representing transverse electromagnetic fields in sector magnets with azimuthal translational symmetry by solving the scalar and vector Laplace equations in cylindrical coordinates. It introduces sector harmonics, built from McMillan radial harmonics and their adjoints, to form a complete, non-truncated basis that connects field coefficients to transverse power-series expansions in radial and vertical planes. The paper also develops two Hamiltonian representations for TEM fields ($t$-representation and $s$-representation) and distinguishes R- and S- elements, providing explicit expressions for normal and skew multipoles in Cartesian and cylindrical coordinates. By unifying Cartesian and cylindrical descriptions and resolving truncation ambiguities, the sector-harmonic framework offers a robust, exact basis for theoretical studies, design, and simulation of sector magnets in accelerator physics.

Abstract

The Laplace's equations for the scalar and vector potentials describing electric or magnetic fields in cylindrical coordinates with translational invariance along azimuthal coordinate are considered. The series of special functions which, when expanded in power series in radial and vertical coordinates, in lowest order replicate the harmonic homogeneous polynomials of two variables are found. These functions are based on radial harmonics found by Edwin~M.~McMillan in his more-than-40-years "forgotten" article, which will be discussed. In addition to McMillan's harmonics, second family of adjoint radial harmonics is introduced, in order to provide symmetric description between electric and magnetic fields and to describe fields and potentials in terms of same special functions. Formulas to relate any transverse fields specified by the coefficients in the power series expansion in radial or vertical planes in cylindrical coordinates with the set of new functions are provided. This result is no doubt important for potential theory while also critical for theoretical studies, design and proper modeling of sector dipoles, combined function dipoles and any general sector element for accelerator physics. All results are presented in connection with these problems.

Figures (6)

• Figure 1: Schematic plot of an equilibrium orbit for an accelerator consisting of five drift spaces and five $72^{\circ}$ bending magnets. Lab frame and local Frenet-Serret frames are shown in black and blue colors respectively. The test particle winding the equilibrium orbit shown in red.
• Figure 2: Illustration of a test particle's position vector expressed as a transverse, i.e. for fixed $q_3$, displacement from equilibrium orbit.
• Figure 3: Illustration of R- and S- elements. Elements are shown in brown. Global curvilinear coordinates with associated grid lines are shown in black. Black dashed line represent an equilibrium orbit. An example of Frenet-Serret frame attached to an equilibrium orbit drawn in blue colors. For S-element, an additional right-handed normalized cylindrical system is added and shown in cyan.
• Figure 4: Normal and skew $2n$-pole magnets in Cartesian coordinates. Each figure shows magnetic (electric) field streamlines and poles' shape in transverse cross section. North (positive electrostatic potential) and south (negative electrostatic potential) poles are shown in red and blue and are given by $(\mathcal{B,A})_n = \mp R_{\text{p}}^n$ respectively, where $R_{\text{p}}$ is the distance to the pole's tip.
• Figure 5: First five even (top row) and odd (bottom row) members of regular polynomials $\mathcal{P}_n = \rho^n$, $\mathcal{F}_n(\rho)$, $\frac{\mathcal{G}_n(\rho)}{\rho}$ and $\mathcal{G}_n(\rho)$ functions from the left to the right respectively.