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Recently, an open geometry Fourier modal method based on a new combination ofan open boundary condition and a non-uniform $k$-space discretization wasintroduced for rotationally symmetric structures providing a more efficientapproach for modeling nanowires and micropillar cavities [J. Opt. Soc. Am. A33, 1298 (2016)]. Here, we generalize the approach to three-dimensional (3D)Cartesian coordinates allowing for the modeling of rectangular geometries inopen space. The open boundary condition is a consequence of having an infinitecomputational domain described using basis functions that expand the wholespace. The strength of the method lies in discretizing the Fourier integralsusing a non-uniform circular "dartboard" sampling of the Fourier $k$ space. Weshow that our sampling technique leads to a more accurate description of thecontinuum of the radiation modes that leak out from the structure. We alsocompare our approach to conventional discretization with direct and inversefactorization rules commonly used in established Fourier modal methods. Weapply our method to a variety of optical waveguide structures and demonstratethat the method leads to a significantly improved convergence enabling moreaccurate and efficient modeling of open 3D nanophotonic structures.

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