Efficient method for the solution of Maxwell’s equations for nanostructured materials

Abstract

The calculation of the electromagnetic field in nanostructured materials and nano-optoelectronic devices, when the wavelength of the incident radiation is comparable with the size of the structural elements, requires the exact solution of Maxwell's equations. In this case, a very promising numerical approach is the spectral element method, which combines the geometric flexibility of finite elements with high precision of spectral methods. In this paper the implementation of the spectral element method based on the Dirichlet-to-Neumann map for solving Maxwell’s equations is discussed. The application of the method for two-dimensional periodic structures, such as diffraction gratings and a metal nanowire array in a dielectric matrix, is demonstrated.

Improving accuracy of the numerical solution of Maxwell's equations by processing edge singularities of the electromagnetic field

“…Speaking about the spectral element method, we adhere to the terminology of the book [13], referring to methods that use the spectral approach inside the subdomains, not only limiting by the Galerkin method in a weak form, as in the original proposal [25]. A formalism based on the application of the Dirichlet-to-Neumann (DtN) map, which we shall use, was previously discussed by us in [29], here we only briefly outline the main points.…”

“…, and 2 0 − ≡ c are known and specify the zero order incident wave in the superstrate and the upward going wave in the substrate, respectively. Instead of equating the field at grid points, the Galerkin method can be implemented by equating field integrals with test functions on both sides of the boundary, for more details see [29]. After the magnetic field at the boundary of the entire domain is obtained, we can, using Eq.…”