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In this paper, uniform approximation of equivalent permittivity is described for electromagnetic scattering problem by index modulated gratings. An index modulated grating is given by periodicity Λ in the z direction and thickness d that is uniform in the y axis. The incident wave is given by wavelength λ and an incident angle θi from x axis. The incident region I and substrate region III are lossless materials, and the relative permittivities are set to be ε1(= 1) and ε3, respectively. The region II is the grating layer with permittivity profiles of sinusoidal, triangular, asymmetric triangular and asymmetric trapezoidal functions ε(z) expressed by using degree of index modulation δ and medium complex permittivity ε. When the periodicity Λ is very small compared with the incident wavelength λ, we present versatile approximation formulations concerning equivalent permittivity εTE for TE wave and εTM for TM wave as follows:

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