Propagation of electromagnetic waves in a regular waveguide with multiperiodically modulated media

The propagation of signal electromagnetic wave in a regular waveguide of arbitrary cross-section is discussed. It is assumed that the axis of the waveguide coincides with the OZ axis of some Cartesian coordinate system and the dielectric filling of the waveguide is modulated in the space (on Z coordinate) according to the multiperiodic law. The wave equations for the longitudinal components of the electric vector (E
_z
(x,y,
_Z
,t)) and of the magnetic vector (H
_z
(x,y,
_Z
,t)), which describe the transverse-magnetic(TM) and transverse-electric (TE) fields in the waveguide, is received. They represent differential equations of the second order in partial derivatives with periodic coefficients. The solution of the wave equations in an assumption of small indexes of modulation of filling (m
_1ε
≪1, m
_2ε
≪1, m
_3ε
≪1) leads to the analytical expressions for TM and TE fields in the waveguide in the region of "weak" (not resonance) interaction between the signal wave and the modulation wave. From the obtained expressions follows that TE and TM fields are a superposition of waves that have different amplitudes.