Scattering of Electromagnetic Waves by a Sinusoidally Stratified Half-Space: I. Formal Solution and Analytic Approximations

A detailed investigation of the scattering of a plane wave incident on a periodically stratified medium is presented. The periodic medium occupies the x > 0 region, and the stratification occurs along the z direction and is produced by a sinusoidal modulation of the dielectric constant. By choosing the polarization parallel to the y coordinate, the fields in the stratified medium appear as solutions of a wave equation of the Mathieu type. The geometry employed herein is shown to serve as a basic configuration for a large number of applications dealing with optical–acoustic interactions, reflection gratings, periodic antennas, and others. The modulated half-space therefore represents a canonical problem which leads to rigorous solutions for the present polarization and to good approximations for the alternative polarization.By viewing the solution in terms of waves of the Floquet type, it is shown that the field consists of modes that travel independently and are coupled to each other only at the air–dielectric interface. The coupling mechanism is represented in terms of an equivalent network which lends insight into both qualitative and quantitative aspects of the diffracted field. Graphical constructions using dispersion diagrams, which greatly facilitate the understanding and interpretation of various field properties, are presented. Explicit analytic results are obtained for small modulation and it is shown that the field is then given primarily in terms of only three modes. When only one mode propagates in each half-space region, the interface network reduces to a simple circuit with interesting properties. While possessing considerable interest by themselves, the present results also form the basis for an extensive investigation of the field behavior at the Rayleigh and Bragg regimes. The latter aspects will be discussed in Part II of this study.