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

Abstract

It is well known that surface-relief dielectric gratings with rectangular profile can be treated by uniform approximation of equivalent permittivity when the periodicity is very small compared with a 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 electromagnetic scattering problem by index modulated gratings. Scattering properties by dielectric slabs are calculated by transmission-line theory and the equivalent permittivity obtained from our proposed formulation of uniform approximation. Scattering by index modulated gratings are analyzed rigorously by the matrix eigenvalue calculations by using Fourier expansion method and the spatial harmonics expansions. When the periodicity is small, both results are in good agreement. By investigating the difference between the equivalent permittivity and the numerical values corresponding to the permittivity of index modulated gratings, the applicable condition of the uniform approximation is shown. And the equivalent anisotropic effects of various permittivity profiles are compared.