Bragg condition for asymmetric dielectric surface-relief gratings

Abstract

Holographic planar gratings are fabricated by inducing perturbation in the refractive index (permittivity) in the medium. The Bragg condition for slanted or unslanted holographic gratings is given by 2nL cos(ϕ ‒ θ) = λ0, where L is the grating period, ϕ is the slant angle, θ is the angle of incidence measured in the grating medium, λ0 is the free-space wavelength, n is the refractive index of the grating medium neglecting the grating modulation. This approximation is generally valid since the index does not vary through the grating thickness and the index modulation is very small. Surface-relief gratings are fabricated by producing periodic perturbation in the surface of the grating medium. The above Bragg condition may be used for symmetric surface-relief gratings (ϕ = 90) in the form 2L sinθ0 = λ0, where θ0 is the angle of incidence in air. For asymmetric gratings, the average refractive index or permittivity of the grating varies though the grating thickness and is significantly different from the refractive indices of two regions forming the grating. The slant angle of the grating and the grating period needed in the Bragg condition are not defined. In this work, a condition for maximum diffraction efficiency for asymmetric surface-relief gratings is proposed. The predicted angles of incidence for maximum diffraction efficiency for several grating profiles are compared with the exact angles obtained using the rigorous coupled-wave analysis. Satisfactory agreement is shown for both TE and TM polarization for the various grating profiles considered. The proposed condition is shown to be a reliable indicator of the maximum efficiency.