Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings

The Fourier modal method for modelling crossed gratings is revisited in the case of symmetrical grating profiles. It is shown that when a plane wave is symmetrically incident on a grating, whose surface profile is mirror-reflection symmetric with respect to the two planes that are perpendicular to the grating plane and contain the two orthogonal periodic directions of the grating, the computational effort can be substantially reduced. For example, at normal incidence with polarization parallel to one of the periodic directions, the special formulation takes only about 1/64 of the computation time that the general formulation takes. Because the use of the symmetry conserves memory and speeds up computation, more accurate results can be obtained and more difficult problems can be treated.