Calculation of the Cherenkov fields in the cross-section of a short relativistic bunch

TL;DR

This paper derives formulas for Cherenkov fields acting on relativistic electron bunches in waveguides, showing the fields are independent of material properties and extending the analysis to transversely distributed bunches using conformal mapping.

Contribution

It provides a rigorous derivation of Cherenkov field formulas at the bunch cross-section and generalizes previous results to transversely distributed bunches with a new Green function.

Findings

Cherenkov field at the bunch point is material-independent.

Formulas valid for arbitrary waveguide cross-sections.

Extension to transversely distributed bunches using Green function.

Abstract

Recently, a new method for calculating the Cherenkov field acting on a point-like electron bunch passing through longitudinally homogeneous structures lined with arbitrary slowdown layers was proposed, where the formalism was obtained though consideration of a general integral relation that allows calculation of the fields at the vicinity of a point-like bunch. It demonstrates that the Cherenkov field at the point of the short relativistic bunch does not depend on the waveguide system material and is a constant for any given transverse dimensions and cross-section shapes of waveguides. With this paper we present a strict derivation of the fields formulas valid at the cross-section of a bunch on the basis of a conformal mapping method. We generalize the results of the previous paper to the case of transversely distributed bunch by deriving a two-dimensional Green function at the cross-section of a bunch. Comparison of the results proving validity of the method is given in the appendix.