Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation

Abstract

In this paper we present an extension of the modal method by Gegenbauer expansion (MMGE) [J. Opt. Soc. Am. A28, 2006 (2011)], [Progress Electromagn. Res.133, 17 (2013)] to the study of nonperiodic problems. The nonperiodicity is introduced through the perfectly matched layers (PMLs) concept, which can be introduced in an equivalent way either by a change of coordinates or by the use of a uniaxial anisotropic medium. These PMLs can generate strong irregularities of the electromagnetic fields that can significantly alter the convergence and stability of the numerical scheme. This is the case, e.g., for the famous Fourier modal method, especially when using complex stretching coordinates. In this work, it will be shown that the MMGE equipped with PMLs is a robust approach because of its natural immunity against spurious modes.

Mode solver based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections

“…In the current study, the computational domain is a circle of radius u 2 4λ, and the scaling factor χ allows this domain to be enlarged from u 2 to 2χu 2 . This parameter works as a real PMLs [3]. In regard to these results, we first remark that the convergence with respect to N defines an N max value: the number of polynomial needed to describe the hole structure for any value of χ.…”

“…A reader who desires further explanation on these PMLs should refer to [3]. In addition, all the calculations related to the complex variable are presented in Appendix A.…”

“…In each region, functions representing each electromagneticfield component can be expanded as a linear combination of Gegenbauer polynomials, and thereby the continuity conditions of certain electromagnetic-field components are explicitly enforced. This method has been successfully extended to nonperiodic structures by the introduction of complex coordinates, which allow the simulation of perfectly matched layers (PMLs) [3]. Until now, the MMGE has only been applied to Maxwell's equations in Cartesian-coordinate form.…”

“…We then present the coordinate transform and the metric tensor generated by this transformation. In doing this, we also briefly introduce a useful transformation (a complex coordinate transform) that allows the simulation of PMLs in MMGE [3]. While keeping in mind the basic principles of the MMGE, we show in Section 3 how this numerical method is suitable for solving the problem of a waveguide with an arbitrary cross section.…”

Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating

“…In earlier papers [1][2][3][4], we present a modal method based on the use of Gegenbauer polynomial [5] expansion for the analysis of a 1D structure with piecewise homogeneous media. As with other modal methods, the computed eigenmodes and propagation constants are obtained by searching the eigenvalues and eigenvectors of a matrix, which are derived from Maxwell's equations by using the method of moments.…”