A variant on eigenmode method in periodic crossed gratings

The eigenvalue problem discussed in Noponen and Turunen [Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494-2502] for crossed gratings is analyzed in a different way using the three components of the electric field instead of the components Ex,y, Hx,y. As a result, half as many eigenvectors are needed.