Electromagnetic waves in a waveguide with periodically modulated magnetodielectric filling

The propagation of electromagnetic waves in an ideal regular waveguide, filling of which is periodically modulated in space and time, is considered. It is assumed that the modulation depths are small and the modulation of the waveguide filling does not lead to the interaction between different waveguide modes. Wave equations are obtained for transverse-electric (TE) and transverse-magnetic (TM) fields in the waveguide with respect to the longitudinal components of the magnetic and electric vectors, respectively, are obtained. They represent second order partial differential equations with periodic coefficients. By changing the variables these equations are reduced to ordinary differential equations with periodic coefficients of the Mathieu-Hill type. Solutions of these equations are found in the first approximation with respect to small modulation depths in the region of “weak” interaction between the signal wave and the modulation wave (the Wulff-Bragg condition is not satisfied). The obtained results show that TE and TM fields in the waveguide in the above approximation are represented as the sum of three space-time harmonics (zero and plus and minus first) with complicated amplitudes and frequencies.