Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry

Using a group-theoretic approach that we developed recently, we reformulate the Fourier modal method for crossed gratings with C4 symmetry, i.e., the two-dimensionally periodic structures that are invariant after rotations about the normal of the mean grating plane through angles nπ/2 where n is any integer. With this approach a crossed-grating problem can be decomposed into four 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 of the algorithm can be improved effectively: the memory occupation and time consumption are reduced to one-quarter and one-16th of the original ones, respectively; for a special case of normal incidence, the time-saving ratio is further reduced to 1/32.