Abstract
A detailed analysis of the B-spline Modal Method (BMM) for one- and two-dimensional diffraction gratings and a comparison to the Fourier Modal Method (FMM) is presented. Owing to its intrinsic capability to accurately resolve discontinuities, BMM avoids the notorious problems of FMM that are associated with the Gibbs phenomenon. As a result, BMM facilitates significantly more efficient eigenmode computations. With regard to BMM-based transmission and reflection computations, it is demonstrated that a novel Galerkin approach (in conjunction with a scattering-matrix algorithm) allows for an improved field matching between different layers. This approach is superior relative to the traditional point-wise field matching. Moreover, only this novel Galerkin approach allows for an competitive extension of BMM to the case of two-dimensional diffraction gratings. These improvements will be very useful for high-accuracy grating computations in general and for the analysis of associated electromagnetic field profiles in particular.
Highlights
Transmittance and reflectance computations from periodic photonic structures such as diffraction gratings and photonic crystals are of considerable interest as appropriately designed structures facilitate a far-reaching control over light propagation and light-matter interaction [1]
With regard to B-spline Modal Method (BMM)-based transmission and reflection computations, it is demonstrated that a novel Galerkin approach allows for an improved field matching between different layers
We have shown that BMM computations offer significant advantages over Fourier Modal Method (FMM) computations
Summary
Transmittance and reflectance computations from periodic photonic structures such as diffraction gratings and photonic crystals are of considerable interest as appropriately designed structures facilitate a far-reaching control over light propagation and light-matter interaction [1]. As a matter of fact, the basic approach of modal methods such as FMM or BMM is that, in propagation direction, the structure of interest is approximated by several layers, where each layer is homogeneous along the stacking direction, e.g., the z-direction. This allows, in each layer, a plane wave ansatz eiλz for the z-dependence that transforms Maxwell equations in frequency domain into an eigenvalue problem for the corresponding propagation constants λ and associated eigenmodes.
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