Structure of electromagnetic field excited by an electron bunch in a semi-infinite dielectric-filled waveguide.

The exact solution of a problem on electromagnetic field excitation by a thin annular charged bunch in a semi-infinite round cylindrical waveguide with metal sidewalls and solid homogeneous dielectric filling is obtained. Expressions for all components of electromagnetic field are derived. These formulas describe the excited field at any point and any moment of time. In contrast to previous works, where asymptotic methods (saddle-point technique) were used, we applied a number of successive conformal transformations of integration area in order to carry out the inverse Fourier transformation. Integration along the initial infinite straight-line contour was substituted by integration along the closed circular contour. This allowed us to separate out the integral presentation of the cylindrical Bessel function of first kind and obtain the final solution in the form of infinite converging series. The process of integration is presented in detail. Both cases, when the Cherenkov resonance condition is satisfied and when this condition is not satisfied, are considered. Spatial pictures of field excited by a finite-size electron bunch are calculated numerically and discussed. In the case of the Cherenkov resonance the drift of excited wake field after the bunch with group velocity is demonstrated, and in the nonresonance case the appearance of impulse of transition radiation and the presence of precursor of the signal are shown.