Interaction of a Transverse Electric Signal Wave with a Spatially Periodically Modulated Magnetodielectric Plate in a Waveguide

The propagation of signal transverse electric (TE) wave in an ideal regular waveguide of arbitrary cross section, where a magnetodielectric plate with a length of $2 d(d\gt0)$ is placed, is considered. Supposed that the permittivity and permeability of the plate are periodically modulated by the z coordinate according to the periodic law. It is assumed that modulation depths $\boldsymbol{m}_{\varepsilon}$ and $\boldsymbol{m}_\mu$ are small. Based on the boundary conditions at the plate boundaries, a system of algebraic equations is obtained for unknown amplitudes of waves reflected and transmitted from the boundaries of the plate. The solution of this system leads to cumbersome analytical expressions for unknown amplitudes (generalized Fresnel formulas). A particular case of thin plate in a waveguide (the wavelength in the plate is much greater than 2d) is considered and the Fresnel formulas are obtained up to terms proportional to d in this case. Note that the results obtained in this work are valid in the frequency region of “weak” interaction of the signal wave with the modulation wave of the plate, when the first-order Wulff-Bragg condition is not satisfied.