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The constitutive parameters of single negative (only electric permittivity or only permeability less than zero) metamaterials and wires-split ring resonator metamaterials can usually be retrieved by the S parameter method. Due to the cross polarization phenomenon in magnetoelectric coupling metamaterial, only two constitutive parameters of permittivity and permeability cannot accurately describe the electromagnetic characteristics of the magnetoelectric coupling metamaterial. The traditional S parameter retrieval method starts with the assumption that the metamaterial has only two constitutive parameters of permittivity and permeability, so the method to retrieve the constitutive parameters of magnetoelectric coupling metamaterials is obviously restricted. In this paper, the electric component and magnetic component in a magnetoelectric coupling metamaterial cell are treated as being equivalent to the surface current and surface magnetic flow, respectively. By deriving the relationship of the average electric flux density and the average magnetic flux density to the external electromagnetic field, we obtain a constitutive parameters matrix (2×2) of the magnetoelectric coupling metamaterial, and find analytical formulas for the relationship between these four constitutive parameters of the magnetoelectric coupling metamaterial and the parameters such as the permittivity of the electric component, permeability of magnetic element, spatial dispersion, and coupling coefficient, and then deduce the refractive index formula. We use the refractive index formula to nonlinearly fit retrieval refractive index curves, and find very good agreement between the refractive index values theoretically predicted by analytical formulas and those obtained from numerical retrievals based on full-wave simulations, thereby verifying the proposed constitutive matrix analytic formula and the refractive index expression. According to the fitting results, we obtain the frequency response curves of the four electromagnetic parameters in constitutive matrix. The proposed method of retrieving the constitutive matrix parameters will provide an important theoretical reference for the researchers engaged in analyzing and studying the coupling phenomenon between electric component and magnetic component in a magnetoelectric metamaterial cell.

Based on Maxwell's curl equations, the electric component and the magnetic component in magnetoelectric metamaterial plate are equivalent to the surface electric current and the surface magnetic current respectively. By calculating the total electric field and the total magnetic field in a magnetoelectric metamaterial plate generated by these periodic surface electric currents and magnetic currents, we obtain two equations about the surface current density and the surface magnetic current density, and thus deduce the analytical formulas for relationship between the refractive index of periodic magnetoelectric metamaterial and permeability of the magnetic component, and that between permittivity of the electric component and spatial dispersion. Unlike traditional index formula, the analytical formulas fully consider the spatial dispersion and the interaction of the electrical component and the magnetic component. The theoretical curves for refractive index are found to be in good agreement with the retrieval curves from simulation data, which shows that the analytical formulas for the refractive index can correctly describe the negative refraction characteristics of the magnetoelectric metamaterials. Our work will provide important theoretical reference for researchers to analyze interaction between electromagnetic components and to design magnetoelectric metamaterials with negative refractive indexes which meet certain requirements.

The thin magnetic resonance dielectric plate is equivalent to a surface magnetic current. Using periodic boundary conditions, the exponential form of the surface magnetic current density is given. The dispersion relation and Bloch impedance of a periodic magnetic resonator metamaterial are derived by calculating the total electric and magnetic fields at the different positions excited by the infinite number of surface magnetic currents, and thus the theoretical formulas for Bloch constitutive parameters are obtained. Since the electric anti-resonance influence on the Bloch permittivity and permeability of magnetic resonator metamaterial is considered, thus the Bloch constitutive parameter difference between theoretical values and retrieval results based on simulations is very small, which shows that the Bloch constitutive parameter formula derived in the paper is very effective to describe the electromagnetic properties of the periodic magnetic resonant material. These theoretical formulas will provide important theoretical basis for the interpretation of the magnetic resonance phenomenon, the design and optimization of the periodic magnetic resonant material.

The relations between the scattering matrix and relative Bloch permeability of periodic symmetric magnetic resonance metamaterial are investigated in detail by means of network theory and Floquet's theorem, and we retrieve the Bloch constitutive parameters for four split ring resonators using the relations. Full wave simulations in conjunction with numerical retrieval algorithms show that when we load the interdigital capacitance at the gap of split ring resonator, the magnetic resonance frequency drastically lowers. Our work provides not only theoretical foundation for the Bloch permeability parameter retrieval from periodic metamaterials, but also a novel design way of miniaturizing magnetic resonance metamaterial configurations.