Fast modal method for crossed grating computation, combining finite formulation of Maxwell equations with polynomial approximated constitutive relations

We present a modal method for the fast analysis of 2D-layered gratings. It combines exact discrete formulations of Maxwell equations in 2D space with polynomial approximations of the constitutive equations, and provides a sparse formulation of the eigenvalue equations. In specific cases, the use of sparse matrices allows us to calculate the electromagnetic response while solving only a small fraction of the eigenmodes. This significantly increases computational speed up to 100×, as shown on numerical examples of both dielectric and metallic subwavelength gratings.