Efficient determination of the left-eigenvectors for the Method of Lines

The efficient determination of left eigenvectors in the method of lines (MoL) is described in this paper. The electromagnetic fields are expanded into eigenmodes and the eigenmodes are determined from an explicit matrix eigenvector problem. To study complicated structures with a moderate numerical effort, the analysis is done with a reduced set of these eigenmodes. The enforcements of the continuity of the transverse electric and magnetic fields at interfaces leads to expressions with rectangular matrices. Now left eigenvectors can be considered as inverse of these rectangular matrices. Until now, the left eigenvectors were determined from a second explicit eigenvalue problem. Here, it is shown how they can be determined with simple matrix products from previously determined right eigenvectors. This is done by utilizing the relation between the transverse electric and magnetic fields. The derived formulas hold for structures with Dirichlet, Neumann or periodic boundary conditions and the materials may be lossy. Open structures are modeled with perfectly matched layers (PML). To verify the expressions, various devices that contain such PMLs and lossy metals were studied. In all cases, error measures show that the algorithm derived in this paper works very well.