Arbitrary shaped periodic anisotropic media: new presentation of Maxwell equations in the truncated Fourier space

Abstract

We establish the most general differential equations satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived using the recently published Fast Fourier Factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classical isotropic gratings arises as a particular case of the derived equations, while the case of anisotropic classical gratings has been published in a separate paper. The equations can be resolved either through the classical differential theory or through the modal method after staircase approximation of the groove profile. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two- dimensional (2-D) arbitrary-shaped periodic surfaces, appear as particular cases of the theory, as well as three- dimensional photonic crystals. The method can be extended to non-periodic media through the use of Fourier transform.