A Study of Uniform Approximation of Equivalent Permittivity for Index-Modulated Gratings

摘要

It is well known that surface-relief dielectric gratings with rectangular profile can be treated by uniform approximation of the equivalent permittivity when the periodicity is very small compared with the wavelength. In optics, this phenomenon is the equivalent anisotropic effects or the form birefringence. When the periodicity is very small, the equivalent anisotropic effects will be shown in index-modulated gratings. In this paper, the uniform approximation is described for the electromagnetic scattering problem of index-modulated gratings. The scattering properties of dielectric slabs are calculated by transmission-line theory and the equivalent permittivity obtained from our proposed formulation of the uniform approximation. Scattering by index-modulated gratings is analyzed rigorously by matrix eigenvalue calculations using the Fourier expansion method and spatial harmonics expansions. When the periodicity is very small, the results are in good agreement. By investigating the difference between the equivalent permittivity and the numerical values corresponding to the permittivity of the index-modulated gratings, the conditions of applicability of the uniform approximation are shown. The equivalent anisotropic effects of various profiles are compared.