Bunch Imaging at the Open End of an Embedded Circular Waveguide

We consider a charged particle bunch passing along the axis of a semi-infinite open-ended circular vacuum waveguide placed into a collinear infinite waveguide with larger radius. The rigorous solution of this problem is constructed using residue-calculus technique. Analytical expressions for the Fourier harmonics of the field components are obtained in each domain of the structure. Numerical integration over frequency is also performed and time-domain field dependencies are obtained. The analytical results are compared with the results of the numerical simulation and an excellent agreement is observed. In particular, we report on the effect of a “bunch image,” a radiation pulse is generated in the coaxial area and has the form of the transverse electric field correlating with the longitudinal charge distribution within the bunch.