Effective-medium theory for one-dimensional gratings

Based on a mode-expansion theory under single-mode approximation, we derived the scatterings parameters for a general one-dimensional photonic grating composed of two different materials, and then established an effective-medium theory for such a composite by equating the obtained scattering parameters to those of a homogeneous medium. Our effective-medium theory well describes the grating structures with general material and geometrical parameters, and recovers two previous formulas, which are valid only at certain limiting conditions. The theory is justified by full-wave simulations and microwave experiments.