E-B Eigenmode Formulation for the Analysis of Lossy and Evanescent Modes in Periodic Structures and Metamaterials

We present here a 3-D E-B linear eigenmode formulation for the analysis of periodic structures, including lossy periodic waveguides, photonic crystals, and metamaterials. The proposed finite-element formulation utilizes proper H (curl) vector basis functions for the electric field and H (div) vector basis functions for magnetic flux density. The resulting eigenmode problem is linear, and it enforces periodic boundary conditions in a simple manner via appropriate field transformations and does not exhibit spurious modes. Moreover, it can provide accurate complex-k dispersion diagrams, in terms of both propagation and attenuation constant, while it is also able to deal with either propagating or evanescent modes, including those with oblique propagation directions.