Three-dimensional adaptive coordinate transformations for the Fourier modal method

Jens Küchenmeister

doi:10.1364/oe.22.001342

Abstract

The concepts of adaptive coordinates and adaptive spatial resolution have proved to be a valuable tool to improve the convergence characteristics of the Fourier Modal Method (FMM), especially for metallo-dielectric systems. Yet, only two-dimensional adaptive coordinates were used so far. This paper presents the first systematic construction of three-dimensional adaptive coordinate and adaptive spatial resolution transformations in the context of the FMM. For that, the construction of a three-dimensional mesh for a periodic system consisting of two layers of mutually rotated, metallic crosses is discussed. The main impact of this method is that it can be used with any classic FMM code that is able to solve the large FMM eigenproblem. Since the transformation starts and ends in a Cartesian mesh, only the transformed material tensors need to be computed and entered into an existing FMM code.

Highlights

Periodic nanostructures gathered a tremendous amount of interest in the past decade [1]
This paper presents the first systematic construction of three-dimensional adaptive coordinate and adaptive spatial resolution transformations in the context of the Fourier Modal Method (FMM)
In the three layers, we gradually introduce the adaptive spatial resolution (ASR) as discussed in great detail in [11]. “1/3 ASR” figuratively means that the ASR has reached a third of its desired strength, see Figs. 4(c)–4(e)

Summary

Introduction

Periodic nanostructures gathered a tremendous amount of interest in the past decade [1]. These problems can be tackled by designing a three-dimensional adaptive coordinate transformation This method trades an increased amount of slices in the method for an accurate representation of the structure’s surface in all three dimensions. In this paper, such a three-dimensional transformation is designed for a system that has gathered an extensive amount of interest in recent years, two periodic layers of mutually rotated, metallic crosses [12]. These structures are known for their strong optical activity. The full anisotropic FMM eigenvalue problem has to be solved

Three-dimensional adaptive coordinates

Two-dimensional mesh for a rotated cross

Constructing the three-dimensional transformation

Three-dimensional adaptive spatial resolution

Conclusion