Interaction of Transverse-magnetic Waves with Space-time Multiperiodic Filling of the Waveguide

In this article the propagation of transverse-magnetic (TM) waves in the waveguide of arbitrary cross-section filled with space-time multiperiodic medium is considered. It is assumed that the modulation indexes m₁, m₂ and m₃ of the non-magnetic waveguide filling are small quantities. From Maxwell's equations the partial differential equation of second order for the longitudinal component of electric vector E_z, which describes the TM field in the waveguide, is received. By changing variables this equation reduces to an ordinary second-order differential equation with periodic coefficients of Mathieu-Hill type. The analytical solution of this equation is obtained with an accuracy of small modulation indexes in the first degree inclusive in the frequency region of “weak” interaction between the signal wave and the waveguide modulation wave (Wulff-Bragg condition is not satisfied during the interference of waves reflected from the seals of waveguide filling). The result obtained in this article shows that the TM field in the waveguide is a superposition of space-time harmonics with different amplitudes (the basic and lateral harmonics). At the same time the amplitude on the zero harmonic does not depend on the modulation indexes, while on the plus and minus first harmonics the amplitudes are proportional to the modulation indexes in the first degree. An expression is also obtained for the frequency around which there is a “strong” (resonant) interaction between the signal wave and the modulation wave, when the waveguide filling is multi-periodically modulated in space, an expression is obtained using the limit transition.