Using symmetries of grating groove profiles to reduce computation cost of the C method

This work reduces the computation cost of the C method by taking into account the symmetries of grating grooves. All one-dimensionally periodic, single-interface, surface-relief gratings are classified into five categories according to the symmetries of the planar periodic curves describing the interface. The five categories are reflection symmetry, inversion symmetry, reflection-translation symmetry, complete symmetry (i.e., simultaneous existence of all three aforementioned symmetries), and no symmetry. Reductions of the eigenvalue problem in the C method are first carried out in real space and then in Fourier space by taking advantage of the four types of symmetries. The reflection-translation symmetry can be used without any restriction on the incident angle, but the other symmetries require a Littrow mounting; simultaneous use of the reflection-translation symmetry with any other symmetry further requires an even-order Littrow mounting. The types of eigenfunctions to be solved and boundary conditions to be matched, as well as the time reduction ratios in solving the eigenvalue problem, are given for all possible combinations of groove symmetries and incident configurations. The time reduction ratios range from 1/4 to 1/64.