Improving the Fourier modal method for crossed gratings with C3 symmetry by using a group-theoretic approach

Following a group-theoretic approach that we have developed recently, we reformulate the Fourier modal method for crossed gratings with C3 symmetry, i.e., grating structures that are invariant after rotations about the normal of the mean grating plane through angles n(2π/3). By exploiting the structural symmetry of the grating, a general diffraction problem can be decomposed into a linear combination of three symmetrical basis problems. Hence the total diffracted field can be obtained as a superposition of the solutions of three symmetry modes. It is shown theoretically and numerically that when the incident mounting and the truncation scheme make the truncated reciprocal lattice of the diffracted field also have the C3 symmetry, the number of unknowns in each symmetrical basis problem is cut by 2/3. Therefore, the maximum effective truncation number of the new algorithm is tripled and the total computation time is reduced by a factor of 9. For the case of normal incidence with arbitrary polarization, the reduction factor can be further increased to 27/2. A numerical example is provided to illustrate the effectiveness of the new formulation.