Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization

Abstract

A modified formulation of Maxwell's equations is presented that includes a complex and nonlinear coordinate transform along one or two Cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow one to map an infinite space to a finite space and to specify graded perfectly matched absorbing boundaries that allow the outgoing wave condition to be satisfied. The approach is validated by numerical results obtained by using Fourier-modal methods and shows enhanced convergence rate and accuracy.

Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method

“…We mention that although PMLs are originally designed for hyperbolic equations, they have also been recently successfully applied to the heat equation (Lantos and Nataf, 2010). Lalanne and co-workers (Lalanne and Silberstein, 2000;Silberstein et al, 2001;Hugonin and Lalanne, 2005) were the first to use absorbing layers and PMLs with Fourier modal methods. In (Hugonin and Lalanne, 2005) they chose trigonometric stretching functions whose Fourier coefficients can be computed analytically.…”

“…Lalanne and co-workers (Lalanne and Silberstein, 2000;Silberstein et al, 2001;Hugonin and Lalanne, 2005) were the first to use absorbing layers and PMLs with Fourier modal methods. In (Hugonin and Lalanne, 2005) they chose trigonometric stretching functions whose Fourier coefficients can be computed analytically. Many other forms for f (and implicitly for β) have been suggested (Berenger, 1996;Collino and Monk, 1998a;Petropoulos, 2003).…”

“…It builds upon the aperiodic Fourier modal method developed and refined by Lalanne and co-workers (Lalanne and Silberstein, 2000;Silberstein et al, 2001;Cao et al, 2002;Hugonin and Lalanne, 2005;Lecamp et al, 2007) with the key difference that the AFMM-CFF solves Maxwell's equations formulated in terms of a contrast (scattered) field instead of a total field. This reformulation allows prescription of arbitrary incoming fields onto the structure of interest (Pisarenco et al, 2010).…”

Resonant-state expansion of light propagation in nonuniform waveguides

“…One popular approach is the aperiodic Fourier modal method (a-FMM) [5][6][7][8], a generalization of the standard FMM [9][10][11], which allows to treat an open WG by introducing an artificial periodicity and a perfectly matched layer (PML) [6,12].…”

Vertical-cavity in-plane heterostructures: Physics and applications

“…24 For simulations, an in-house developed three-dimensional (3D) simulator was used, 25,26 which is based on a rigorous coupled wave analysis method 27,28 and employs absorbing boundary conditions. 29 The band edge profiles in Figs. 2-4 are found for each section of the VCI heterostructure, by assuming a grating that is periodic in the x-direction and infinitely long in the y-direction.…”