Propagation of Electromagnetic Waves in a Waveguide with Space-Time Multiperiodically Modulated Magnetodielectric Filling

The properties of electromagnetic waves in an ideal regular waveguide that is characterized by arbitrary cross-section are studied. It is assumed, that the filling of waveguide is multiperiodically modulated in space and time, and the modulation depths are small. By this the modulation of the waveguide filling does not lead to the interaction between different waveguide modes. The wave equations for the longitudinal components of the electric and magnetic vectors are obtained based on the system of Maxwell's equations. They describe the waveguide's transverse-electric (TE) and transverse-magnetic (TM) fields. These wave equations are partial differential equations of second order with periodic coefficients. Using change of variables, they are reduced to ordinary differential equations with periodic coefficients of the Mathieu-Hill type. Under the assumption of small modulation depths of the waveguide filling solutions of these equations are found in the first approximation with respect to the modulation depths in the frequency domain of the “weak” interaction between the signal wave and the modulation wave, when the first-order Wulff-Bragg condition between the waves, reflected from the filling inhomogeneities, is not satisfied. The analytical expressions, obtained in this article for the TE and TM fields, show that in the first approximation in terms of small modulation depths the TE and TM fields are given as a set of three space-time harmonics with specified complex amplitudes and frequencies.