Depth dependence of the effective properties of subwavelength gratings

Effective-medium theory is often helpful for understanding the behavior of subwavelength gratings with small period-to-wavelength ratios. By analytically solving Maxwell's equations in the small-depth limit, we show effective properties are shown to depend not only on the grating structure but also on the optical indices of the surrounding media. A simple expression for the effective indices of one-dimensional (1-D) and two-dimensional (2-D) gratings with arbitrary depths is proposed. that for TE polarization of 1-D gratings the depth dependence of the effective index prediction is accurate. For TM polarization of 1-D volume gratings and for 2-D volume gratings a slight deviation between rigorous computation and the prediction is observed for approximately quarter-wave depths. surface-relief gratings and for 2-D surface-relief gratings, no closed-form expression for the depth dependence of the effective indices is found, but a very sharp variation of the effective index at small thicknesses is predicted. This prediction is confirmed by rigorous computation. (C) 1997 Optical Society of America [S0740-3232(97)01502-0]