Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with equilateral triangular symmetry

The Fourier modal method for crossed gratings is reformulated by use of a group-theoretic approach when the grating structures have the equilateral triangular symmetry. In the new formulation, a grating problem is first decomposed into four symmetrical basis problems whose field distributions are the symmetry modes (two are nondegenerate and the other two are doubly degenerate) of the grating. Then the symmetry relations of fields in the symmetry modes are used to reduce the number of unknowns in numerical computation. After the symmetrical basis problems are solved, their solutions are superposed to get the solution of the original problem. It is shown that when the grating is at some incident mountings, the memory occupation can be saved by 2/3 and the computation time can be reduced to 1/12 to 1/13.5 of the original one for different incident cases. Numerical examples are given to illustrate the effectiveness of the new formulation and verify the improved computation efficiency.