Wave Propagation in Sinusoidally Stratified Dielectric Media

The dispersion properties and the fields of electromagnetic waves are investigated for propagation in a stratified infinite medium. The stratification is characterized by a dielectric constant which, along one coordinate, is modulated sinusoidally about an average value. A systematic and comprehensive study is presented for the case of H modes for which the pertinent wave equation is in the form of a Mathieu differential equation. The modes and dispersion characteristics are analyzed in terms of a "stability" chart, which is customary in the study of the Mathieu equation. Results are obtained for an unbounded medium and for a waveguide filled with the modulated medium. Also, the reflection occurring at an interface between free space and a semi-infinite medium of this type is examined. In addition to these rigorous results for arbitrary values of modulation, simple analytical expressions are given for all of these cases where the modulation in the dielectric is small. It is shown that the fields are then expressible in terms of the fundamental and the two nearest space harmonics. The fields within a unit cell in the stratified medium are calculated for both small and large modulation and for frequencies up through the second pass band. It is of interest that the variation of the fields is not, in general, simply related to the variation of the dielectric constant within a cell.