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 that the effective properties of subwavelength gratings strongly depend on the grating depth. Moreover, the 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. Comparison with rigorous computations shows 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. For TM polarization of 1-D 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.
© 1997 Optical Society of America
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