The effect of the refractive index of the substrate together with the incident polarization on the optimization of sawtooth surface-relief gratings (SRGs) is investigated. The global optimum diffraction efficiencies of the Ϫ1st forward-diffracted order of sawtooth SRGs are 63.3% occurring at n 2 ϭ 1.47 for TE polarization and 73.8% occurring at n 2 ϭ 2.88 for TM polarization. Incident TE polarization has higher optimum diffraction efficiency than TM polarization for all n 2 Ͻ 1.85. In contrast,

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... TM polarization has higher optimum diffraction efficiency than TE polarization for all n 2 Ͼ 1.85. A polymer ͑n 2 ϭ 1.5͒ optimum sawtooth SRG exhibits 62.6% efficiency for TE polarization. A silicon ͑n 2 ϭ 3.475͒ optimum sawtooth SRG exhibits 68.6% efficiency for TM polarization. These sawtooth SRGs are compared to right-angle-face trapezoidal SRGs. It is found that the optimum profiles of right-angle-face trapezoidal SRGs have only very slightly increased efficiencies over sawtooth SRGs ͑0.04% for TE and 0.55% for TM).