An automated phase correction algorithm for retrieving permittivity and permeability of electromagnetic metamaterials

To retrieve complex-valued effective permittivity and permeability of electromagnetic metamaterials (EMMs) based on resonant effect from scattering parameters using a complex logarithmic function is not inevitable. When complex values are expressed in terms of magnitude and phase, an infinite number of permissible phase angles is permissible due to the multi-valued property of complex logarithmic functions. Special attention needs to be paid to ensure continuity of the effective permittivity and permeability of lossy metamaterials as frequency sweeps. In this paper, an automated phase correction (APC) algorithm is proposed to properly trace and compensate phase angles of the complex logarithmic function which may experience abrupt phase jumps near the resonant frequency region of the concerned EMMs, and hence the continuity of the effective optical properties of lossy metamaterials is ensured. The algorithm is then verified to extract effective optical properties from the simulated scattering parameters of the four different types of metamaterial media: a cut-wire cell array, a split ring resonator (SRR) cell array, an electric-LC (E-LC) resonator cell array, and a combined SRR and wire cell array respectively. The results demonstrate that the proposed algorithm is highly accurate and effective.