Fast semi-analytical solution of Maxwell's equations in Born approximation for periodic structures

We propose a fast semi-analytical approach for solving Maxwell's equations in Born approximation based on the Fourier modal method (FMM). We show that, as a result of Born approximation, most matrices in the FMM algorithm become diagonal, thus allowing a reduction of computational complexity from cubic to linear. Moreover, due to the analytical representation of the solution in the vertical direction, the number of degrees of freedom in this direction is independent of the wavelength. The method is derived for planar illumination with two basic polarizations (TE/TM) and an arbitrary 2D geometry infinitely periodic in one horizontal direction.