Analytical treatment of the field discontinuities within the Fourier space methods in the grating diffraction theory

In our work we develop an idea for the analytical treatment of discontinuities within the Fourier space methods for solving the electromagnetic diffraction problem, and demonstrate the reformulated Fourier Modal Method as applied to 1D and 2D periodic structures. The obtained formulations replace the need in the Li's factorization rules to operate only on unknown Fourier vectors of continuous functions, and allow for the natural computations of the inner fields free from the Gibbs phenomenon for any number of harmonics. In addition, the new formulations appear to be free of intermediate matrix inversions paving the way for the application of modern powerful methods of numerical linear algebra.