Generalization and modularization of two-dimensional adaptive coordinate transformations for the Fourier modal method.

Jens Küchenmeister

doi:10.1364/oe.22.009404

Abstract

The Fourier modal method (FMM) has advanced greatly by using adaptive coordinates and adaptive spatial resolution. The convergence characteristics were shown to be improved significantly, a construction principle for suitable meshes was demonstrated and a guideline for the optimal choice of the coordinate transformation parameters was found. However, the construction guidelines published so far rely on a certain restriction that is overcome with the formulation presented in this paper. Moreover, a modularization principle is formulated that significantly eases the construction of coordinate transformations in unit cells with reappearing shapes and complex sub-structures.

Highlights

Periodic arrangements of nanostructures haven proven to be a rich field of research [1]
One of the solvers that has proven to be of great applicability is the Fourier modal method (FMM)
The system is sliced into layers with constant permittivity in z-direction, each of which lead to an eigenvalue problem representing Maxwell’s curl equations

Summary

Introduction

Periodic arrangements of nanostructures haven proven to be a rich field of research [1]. The system is sliced into layers with constant permittivity in z-direction, each of which lead to an eigenvalue problem representing Maxwell’s curl equations This allows expanding the fields into eigenmodes. It constitutes an option for unit cells with repeating shapes within the unit cell or shapes with complex substructures. This section discusses the impact of coordinate transformations on Maxwell’s equations and the consequences for the Fourier modal method. Since this is not the first publication on this topic, I keep the section short and use the notation already used in [9,10,11,12,13]. This means that the surfaces are grid-aligned in the effective permittivity and the coordinate line density at the surface of the structure is increased (see [11] for details and illustration)

Generalization of two-dimensional mesh construction

Solution for piecewise linear mappings

Modularized construction of two-dimensional meshes

Conclusion