Analysis of dispersion splitting on light line by characteristic roots of retarded circuit equation

To design electromagnetic phenomena in complex metallic structures such as metamaterials, it is useful to analyze them by equivalent circuit models. Although general equivalent circuit models do not have retardations of electromagnetic couplings, the model derived theoretically from Maxwell's equation includes the retardation. The retardation causes the singularity on the light line of the dispersion curve. In order to find the dispersion curve, we usually pick up the most dominant root in the infinite characteristic roots. However, considering the other characteristic roots, we find another dispersion curve and clarify the coupling between the light line and plasmonic modes.