Numerically stable eigenmode extraction in 3-D periodic metamaterials

A numerical method is presented to compute the eigenmodes supported by 3-D metamaterials using the method of moments. The method relies on interstitial equivalent currents between layers. First, a parabolic formulation is presented. Then, we present an iterative technique that can be used to linearize the problem. In this way, all the eigenmodes characterized by their transmission coefficients and equivalent interstitial currents can be found using a simple eigenvalue decomposition of a matrix. The accuracy that can be achieved is limited only by the quality of simulation, and we demonstrate that the error introduced when linearizing the problem decreases doubly exponentially with respect to the time devoted to the iterative process. We also draw a mathematical link and distinguish the proposed method from other transfer-matrix-based methods available in the literature.