New method of calculating the wakefields of a point charge in a waveguide of arbitrary cross section

A new method for calculating the Cherenkov wakefield acting on a point charged particle passing through a longitudinally homogeneous structure lined with layer(s) of an arbitrary retarding (dielectric, resistive, or corrugated) material has been developed. In this paper we present a rigorous derivation of the expressions for the fields that are valid at the cross section of the particle on the basis of a conformal mapping method. This new formalism allows reduction of the loss factor calculation to a simple derivation of a conformal mapping function from the arbitrary cross section onto a circular disc. We generalize these results to the case of a bunch with an arbitrary transverse distribution by deriving a two-dimensional Green function at the cross section of the particle. Consequently, for the first time analytical expressions for the transverse distributions of the electric field $E_z$ for the most commonly used cylindrical, planar and elliptical cross section geometries are found. The proposed approach significantly decreases simulation time and opens new possibilities in optimizing wakefield effects resulting from short charged particle bunches for FEL and Linear Collider applications.

Figure 1

(a) Definition of the conformal map from the $\omega$ plane to the $\chi$ plane. (b) Schematic of the shock wave formation process.

Figure 2

Dependence of $E_z$ on azimuthal angle $\varphi -{\varphi }_0$ normalized by a factor of $-2Q/a^2$ for the case of an off axis displaced particle. Orange line: $r=r_0$; Green line: $r=a/10$; Blue line: $r=a$. Points are results for the direct simulation using the mode decomposition method [23, 24] (1000 radial and 15 azimuthal modes were taken in to account) and the solid lines are calculated using the formula (51).

Figure 3

Y-dependence (left) and X-dependence (right) of $E_z$ normalized by $-2Q/a^2\times {\pi }^2/8$ in the case where the test charge is located at $x=0.4\mathrm{cm}$ from the center for the y-plot (left) and $y=3.8\mathrm{cm}$ for the x-plot (right). Points are computed using the mode decomposition method [14] (array of 5 by 500 modes were taken into account) and solid lines are obtained from formula (56).

Figure 4

Dependence on the waveguide relative width $a/w$ of the normalized longitudinal electric field $E_z$ at the particle in a rectangular waveguide. Solid curve: dependence of formula (59) on $a/w$ normalized by $-\frac{2Q}{a^2}\frac{{\pi }^2}{16}$; points: results using the mode decomposition method [14] and also normalized by $-\frac{2Q}{a^2}\frac{{\pi }^2}{16}$.

Figure 5

Longitudinal electric field $E_z$ at the particle in the elliptical structure normalized by $-\frac{2Q}{a^2}$, showing its dependence on the ellipse flattering factor $g=1-a/b$.