Chebyshev collocation Dirichlet-to-Neumann map method for diffraction gratings

For diffraction gratings with layered refractive index profiles, the Fourier modal method is widely used. However, it is quite expensive to calculate the eigenmodes for each layer, especially when the structure involves absorptive media. We develop an efficient method that avoids the eigenvalue problems based on the so-called Dirichlet-to-Neumann (DtN) map. For each layer, the DtN map is an operator that maps the wave field to its normal derivative on one period of the boundaries of the layer, and it is approximated by a matrix. An efficient procedure for computing the DtN map is developed based on a Chebyshev collocation method and a fourth- order finite difference method for discretizing the uniform and the periodic directions, respectively. The efficiency and accuracy of our method are illustrated by numerical examples.