Fourier factorization in the constitutive relations for modeling diffraction gratings with continuous permittivity profiles

The validity of the Fourier factorization (FF) rules used in the constitutive relations in the Fourier modal method is analyzed for one-dimensional diffraction gratings made as a thin film with uniform thickness and several examples of continuous lateral profiles of permittivity. The comparison of the method using the correct FF rules with other FF choices demonstrates the validity of the FF rules for any inhomogeneous, not only discontinuous profile.