Electromagnetic fields and Green’s functions in elliptical vacuum chambers

Abstract

In this paper, we discuss the electromagnetic interaction between a point charge travelling inside a waveguide of elliptical cross section, and the waveguide itself. By using a convenient expansion of the Mathieu functions, useful in particular for treating a variety of problems in applied mathematics and physics with elliptic geometry, we first obtain the longitudinal electromagnetic field of a point charge (Green's function) in free space in terms of elliptical coordinates. This expression allows, then, to calculate the scattered field due to the boundary conditions in our geometry. By summing the contribution of the direct or primary field and the indirect field scattered by the boundary, after a careful choice of some expansion expressions, we derive a novel formula of the longitudinal electric field, in any transverse position of the elliptical cross section, generated by the charge moving along the longitudinal axis of the waveguide. The obtained expression is represented in a closed form, it can be differentiated and integrated, it can be used to fully describe the radiation process of a particle beam travelling inside a waveguide of elliptical cross section, and it is valid for any elliptic geometry. The equations are used to evaluate the coupling impedance due to indirect space charge in case of elliptical geometry. In addition, they are useful as preliminary studies for the determination of the coupling impedance in different cases involving elliptic vacuum chambers, as, for example, the effect of the finite conductivity of the beam pipe wall or the geometrical variation of the vacuum chamber due to elliptic step transitions existing in some accelerators.

Space charge impedance and electromagnetic fields in elliptical vacuum chambers

“…As already discussed in Ref. [12], the Mathieu functions represent the natural way to express the electromagnetic fields inside an elliptical beam pipe. In particular, due to the symmetries with respect to the four quadrants of Fig.…”

“…From these expressions, in Ref. [12], we have obtained the longitudinal and quadrupolar impedances, in elliptic geometry, due to the indirect space charge.…”

“…In Ref. [12], starting from the longitudinal electric field of a Dirac δ-function beam distribution with total charge Q moving with velocity v ¼ βc on the axis z of the elliptical pipe, we have derived the indirect space charge longitudinal and transverse quadrupolar impedances per unit of length, which we write here as…”

“…In this paper we expand the formalism of a previous work [12], in which we derived the longitudinal and quadrupolar (detuning) indirect space charge impedance produced by a point charge traveling on the axis of a perfectly conducting elliptic vacuum chamber, presenting a model of a complete set of equations for the electromagnetic fields that allow to evaluate the longitudinal and transverse coupling impedances due to space charge at any frequency and beam energy. The equations, which show an interesting parallelism with respect to the circular case, allow to obtain the impedances for any value of ellipticity, ranging from the circular shape to the parallel plates.…”

“…IV, by applying a procedure similar to that of Ref. [12], we obtain the electric field of a dipole vertically and horizontally displaced, and, in Sec. V, we derive the transverse dipolar impedances, comparing them with the known expressions for a circular pipe and parallel plates.…”