Four-Vector Potential for Point Charge Moving Arbitrarily in Cylindrical Waveguide

Using the method of Green functions we found four-vector potential for a point charge moving along an arbitrary path in a perfectly conducting cylindrical waveguide. Solutions are expressed analytically through the Green functions of the d´Alembert operator with Dirichlet and Neumann boundary conditions. It is shown that if the waveguide is excited only by a longitudinal current density component and/or non-zero charge density (transverse current components) the obtained solution reduces to the well-known expressions for TM (TE) cylindrical waveguide modes. If transverse and longitudinal current density components and non-zero charge density are simultaneously present there, then the radial structure of the excited electromagnetic field coincides with that of the superposition of TM and TE cylindrical waveguide modes. The results thus obtained allow one an ab initio calculation of the forces acting on an arbitrarily moving relativistic charge from the induced-by-itself charges and currents at the waveguide walls. They also provide a basis for solution of a rigorous self-consistent problem on the propagation of relativistic electron beams in an external electromagnetic field by the particle-in-cell method