Validity of the scalar theory in the design of diffractive optical elements

Abstract

Scalar diffraction theory based on Fourier optics is widely used for the analysis and the design of diffractive optical elements. The approach, which involves Fourier expansion of the transmittance function of the grating structure, is believed to be valid for the analysis of gratings with large grating spacing and small depth at near normal incidence. Recent uses for diffractive optical elements such as fast lenses require grating periods comparable to the light wavelength. In this work, the range of validity of the approximate scalar Fourier optics approach in the analysis of these structures is investigated. Exact diffraction characteristics of several multilevel binary dielectric gratings are calculated using the rigorous vector electromagnetic-based coupled wave approach as a function of the grating period to light wavelength ratio and of the grating refractive index for both TE and TM polarizations. Comparisons with those characteristics obtained by the approximate scalar diffraction theory (corrected to account for specular reflection) are made. It is shown that the error in the diffraction efficiency calculated by the approximate scalar analysis relative to that obtained by the rigorous vector analysis is >15% for a grating period light wavelength ratio of more than 10. It is also shown that the error increases as the number of binary levels or the grating substrate index increases. As the ratio between the grating period and the light wavelength decreases (<5) strong polarization dependence is observed.