B-spline modal method: A polynomial approach compared to the Fourier modal method

Abstract

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.

Electromagnetic modelization of spherical focusing on a one-dimensional grating thanks to a conical B-spline modal method

“…The matrices of tangential electric and magnetic fields are where each submatrix is obtained by the eigenvector matrices computed when solving Eqs. (9) and (10). Then, the S matrices of the reflected and transmitted coefficients at the inter- , where the superscript of the submatrix indicates the incoming and the outgoing polarization states.…”

“…More recently, the B-spline modal method (BMM) was introduced [8][9][10][11]. It is based on the resolution of the Maxwell and constitutive equations using a B-spline approximation of the field in each layer.…”

“…Moreover, the eigenvalue equation obtained is sparse, which allows us to increase the calculation speed by solving the problem for only a few eigenmodes. While the generalization of BMM to crossed gratings has been recently reported [10], it has not been adapted yet to conical mountings.…”

Simulating the focusing of light onto 1D nanostructures with a B-spline modal method

“…[1][2][3][4] The B-Spline modal method (BMM), which was recently developped, [5][6][7][8][9] is an interesting alternative to the FMM. Indeed, it enables the use of algorithms for solving the eigenvalue equations which are specific to sparse matrices, thus reducing significantly the calculation time.…”

Efficient implementation of B-spline modal method for lamellar gratings

“…Edee and co-workers [14,15] were the first to implement the modal method with such a basis. Since then, this way has also been explored by a few researchers [16,17]. In this paper, we propose an efficient implementation of the B-spline modal method (BMM) that gives dramatic improvements in terms of numerical convergence compared to previous formulations.…”