# Modeling open nanophotonic systems using the Fourier modal method: Generalization to 3D Cartesian coordinates

## TL;DR

This paper extends the Fourier modal method to 3D Cartesian coordinates for open nanophotonic systems, introducing a non-uniform Fourier space sampling technique that enhances accuracy and convergence in modeling open 3D structures.

## Contribution

The authors generalize the open boundary Fourier modal method to 3D Cartesian coordinates with a novel non-uniform Fourier space sampling, improving modeling accuracy for open nanophotonic systems.

## Key findings

- Enhanced accuracy in modeling radiation modes
- Improved convergence compared to conventional methods
- Effective application to various 3D optical structures

## Abstract

Recently, an open geometry Fourier modal method based on a new combination of an open boundary condition and a non-uniform $k$-space discretization was introduced for rotationally symmetric structures providing a more efficient approach for modeling nanowires and micropillar cavities [J. Opt. Soc. Am. A 33, 1298 (2016)]. Here, we generalize the approach to three-dimensional (3D) Cartesian coordinates allowing for the modeling of rectangular geometries in open space. The open boundary condition is a consequence of having an infinite computational domain described using basis functions that expand the whole space. The strength of the method lies in discretizing the Fourier integrals using a non-uniform circular "dartboard" sampling of the Fourier $k$ space. We show that our sampling technique leads to a more accurate description of the continuum of the radiation modes that leak out from the structure. We also compare our approach to conventional discretization with direct and inverse factorization rules commonly used in established Fourier modal methods. We apply our method to a variety of optical waveguide structures and demonstrate that the method leads to a significantly improved convergence enabling more accurate and efficient modeling of open 3D nanophotonic structures.