Modal expansion theory by almost periodic functions for dielectric gratings

A mode theory is presented for analyzing diffraction of a TE-polarized plane wave by a dielectric grating. The dielectric constant of the grating varies sinusoidally in the transverse plane, thereby allowing the field of the incident wave to be expressed as an infinite series as functions of the grating modes. The modes are formulated as Mathieu functions to account for their near-periodicity. A system of equations is obtained which considers only reflected and transmitted waves and applied in sample calculations of the diffraction efficiency relative to the grating thickness or incident angle. A change in the grating thickness is found to induce a periodic undulation, while diffraction causes a slow undulation. Certain grating thicknesses are shown to generate surface wave excitation, i.e., anomalous diffraction.