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

The Fourier modal method for crossed gratings with square symmetry is reformulated by use of a group-theoretic approach that we developed recently. In the new formulation, a crossed-grating problem is decomposed into six symmetrical basis problems whose field distributions are the symmetry modes of the grating. Then the symmetrical basis problems are solved with symmetry simplifications, whose solutions are superposed to get the solution of the original problem. Theoretical and numerical results show that when the grating is at some Littrow mountings, the computation efficiency can be improved effectively: The memory occupation is reduced by 3/4 and the computation time is reduced by a factor from 25.6 to 64 in different incident cases. Numerical examples are given to show the effectiveness of the new formulation.