Interaction of the Transition Radiation of a Charged Particle with Bi-Periodically Modulated Medium in a Waveguide

This paper is devoted to a theoretical study of the interaction of transition radiation of a moving charged particle with bi-periodically modulated in space dielectric filling of a regular ideal waveguide. We assume that the particle moves at a constant velocity, direction of which is perpendicular to the axis of waveguide. From the system of Maxwell's equations, a wave equations are obtained whose solutions leads to an analytical expressions for the transverse magnetic (TM) and electric (TE) fields of radiation with an accuracy of terms, which is proportional to small modulation depths. It is shown that in this approximation the TM and TE fields in the waveguide are a superposition of spatial harmonics (zero and plus - minus the first harmonics). The energies of the transition radiation of a charged particle in the domain of non-resonant interaction between the radiation and modulation waves of the waveguide filling is calculated (the first-order Wulf-Bragg condition for waves reflected from inhomogeneities of filling is not satisfied). It is shown that the energies of transition radiation at the fundamental (zero) harmonic does not depend on the depths of modulation of the filling, while at the side (plus - minus the first) harmonics they are proportional to the depths of modulation. The Cherenkov radiation frequency is found for the case of a rectangular waveguide.