Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings

We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain.