Effective-medium permittivity of one-dimensional gratings of vertically invariant and laterally arbitrary permittivity distribution

Derivation of effective-medium permittivities (in the traditional sense) of one-dimensional gratings is revisited. In the present derivation, carried out in Fourier-space and for conical mounting, systematic treatment and mathematical rigour are emphasized. An error of a previous paper published in this journal concerning the 2nd-order effective permittivity in TM polarization is corrected. The derived formulas of the 4^th-order $E_{\mathrm{\perp }}$ effective permittivity and the 2nd-order $H_{\mathrm{\perp }}$ effective permittivity, when restricted to TE and TM polarizations respectively, agree well with the previous results. The symmetry of the effective permittivities with respect to the sign of the Floquet exponential factor and the sign of grating period is noted and proven.