Electromagnetic fields and Green’s functions in elliptical vacuum chambers

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.

Figure 1

Elliptic coordinates. The $\phi$ coordinate describes confocal hyperbolas that are symmetrical about the x-axis. The $\mu$ coordinate describes confocal ellipses centered on the origin of the coordinate system.

Figure 2

Comparison between the Green’s function in free space computed with Eq. (25) (blue line) and Eq. (32) (red dashed line) with 120 summation terms, as a function of the radial coordinate and $\phi =0$, for a focal distance $F=1$.

Figure 3

Comparison between the Green’s function in free space computed with Eq. (25) (blue line) and Eq. (32) (red dashed line) with 120 summation terms, as a function of the radial coordinate and $\phi =\pi /2$, for a focal distance $F=1$.

Figure 4

Comparison between the longitudinal electric field per unit of charge computed with Eq. (38) (blue line) and by CST (red dashed line) with 40 summation terms, calculated in $\mu =0.1$ at 100 MHz as a function of $\phi$.

Figure 5

Longitudinal electric field per unit of charge in the elliptic beam pipe of the SPS accelerator at the injection energy, calculated in ${\mu }_0=0.1$ at 100 MHz as a function of $\phi$.

Figure 6

Longitudinal indirect space charge per unit of length for $b=3.5\mathrm{cm}$ and $\beta =0.9$ as a function of frequency. Comparison between IW2D and Eq. (38) for circular pipe (left) and parallel plates (right).

Figure 7

Longitudinal form factor as a function of $q_r=(a-b)/(a+b)$ for the PS (left) and the SPS (right) injection energies at different frequencies. The vertical black lines correspond to the machines beam pipe geometries.

Figure 8

Longitudinal indirect space charge impedance as a function of $q_r=(a-b)/(a+b)$ for the PS accelerator for a frequency of 100 kHz. Comparison between Eq. (40) and Eq. (41) by using the equivalent radius defined in Eq. (42).

Figure 9

Transverse indirect space charge per unit of length for $b=0.35\mathrm{cm}$ and $\beta =0.9$ as a function of frequency for different $q_r=(a-b)/(a+b)$ compared with the circular and flat beam pipe given by IW2D, and with theory which uses the Laslett coefficients at low frequency.