Analytic expressions for the constitutive parameters of magnetoelectric metamaterials

Electromagnetic metamaterials are artificially structured media typically composed of arrays of resonant electromagnetic circuits, the dimension and spacing of which are considerably smaller than the free-space wavelengths of operation. The constitutive parameters for metamaterials, which can be obtained using full-wave simulations in conjunction with numerical retrieval algorithms, exhibit artifacts related to the finite size of the metamaterial cell relative to the wavelength. Liu et al. [R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, Phys. Rev. E 76, 026606 (2007)] showed that the complicated, frequency-dependent forms of the constitutive parameters can be described by a set of relatively simple analytical expressions. These expressions provide useful insight and can serve as the basis for more intelligent interpolation or optimization schemes. Here, we show that the same analytical expressions can be obtained using a transfer-matrix formalism applied to a one-dimensional periodic array of thin, resonant, dielectric, or magnetic sheets. The transfer-matrix formalism breaks down, however, when both electric and magnetic responses are present in the same unit cell, as it neglects the magnetoelectric coupling between unit cells [C. R. Simovski, Metamaterials 1, 62 (2007)]. We show that an alternative analytical approach based on the same physical model must be applied for such structures. Furthermore, in addition to the intercell coupling, electric and magnetic resonators within a unit cell may also exhibit magnetoelectric coupling. For such cells, we find an analytical expression for the effective index, which displays markedly characteristic dispersion features that depend on the strength of the coupling coefficient. We illustrate the applicability of the derived expressions by comparing to full-wave simulations on magnetoelectric unit cells. We conclude that the design of metamaterials with tailored simultaneous electric and magnetic response—such as negative index materials—will generally be complicated by potentially unwanted magnetoelectric coupling.

Figure 1

Periodic system of electrically or magnetically polarizable sheets. The sheets have thickness $l$ and are spaced a distance $d$ apart. An electromagnetic wave is assumed to propagate along the direction normal to the sheets.

Figure 2

(Color online) Phase advance $\left(\theta =\alpha d\right)$ as a function of frequency for a medium comprising electrically resonant slabs separated by regions of free space. Solid line corresponds to the exact dispersion relation of the slab model, Eq. (13), while dashed line corresponds to the approximate expression found in Eq. (16).

Figure 3

(Color online) Plot of phase advance $\theta =\alpha d$ as a function of frequency near the resonance of a magnetoelectric structure, consisting of an array of thin slabs having simultaneous resonances in the permittivity and permeability. The resonance frequencies are both at 10 GHz, with all other parameters identical. Solid line corresponds to the exact dispersion relation of Eq. (50), while red dashed line corresponds to the approximate dispersion relation in Eq. (52). For the region where $kd<1$, the curves for the two cases are nearly identical. The dispersion relation developed assuming no magnetic electric coupling [Eq. (33)] leads to a curve with very different behavior (short dashed orange curve).

Figure 4

Unit cell composed of interacting electric and magnetic sheets. Magnetic currents in the magnetic layer produce an electric field that drives currents in the electric layer, while electric currents in the electric layer produce a magnetic field that drives currents in the magnetic layer.

Figure 5

(Color online) Plots of the real (blue curves) and imaginary (red curves) parts of the effective refractive index corresponding to a unit cell with matched resonant permittivity and permeability. Each plot corresponds to Eq. (63) with a different value of $\kappa$ assumed. In all cases, $\kappa$ is real, having values 0.0, 0.1, 0.5, 0.8, 1.0, and 1.4, from top to bottom.

Figure 6

(Color online) Plots of the real (blue curves) and imaginary (red curves) parts of the refractive index corresponding to a unit cell with matched resonant permittivity and permeability. Each plot corresponds to Eq. (63) with a different value of $\kappa$ assumed. In all cases, $\kappa$ is imaginary, having values 0, $0.1i$, $0.5i$, $0.8i$, $1.0i$, and $1.4i$ from top to bottom.

Figure 7

(Color online) (Left) ELC and MLC structures. The wave is assumed to propagate between the two patterned surfaces, polarized such that the electric field excites the ELC while the magnetic field excites the MLC. (Right) ELC consists of a conducting ring, as shown, with dimensions $d=3\text{mm}$, $p=2.6\text{mm}$, $l=0.2\text{mm}$, $b=0.25\text{mm}$, and $g=0.02\text{mm}$. The conductor thickness is assumed as $17\mu \text{m}$. The MLC has precisely the same geometry, but with a magnetic conductivity rather than an electrical conductivity. To lower the resonance frequency, a piece of either dielectric or permeable material is placed within the gaps, with $\epsilon =6$ for the ELC and $\mu =6$ for the MLC. Periodic boundary conditions are applied on the sides of the unit cell (perpendicular to the propagation direction) to simulate a metamaterial array infinite in the lateral dimensions.

Figure 8

(Color online) Retrieved Bloch (a) permittivity and (b) permeability for the ELC alone; retrieved Bloch (c) permittivity and (d) permeability for the MLC alone; (e) retrieved index of refraction for the ELC and MLC together, but with resonances slightly detuned; (f) retrieved index of refraction for the ELC and MLC with resonances matched. Solid blue and red lines correspond to the real and imaginary parts of the given parameter retrieved from full-wave simulations, while blue and red dashed lines are fits to the real and imaginary parts of the same parameter using Eq. (24) or (31). The real and imaginary parts of the index (solid blue and red lines) are compared to Eq. (63) (dashed blue and red lines). In (g), the lighter blue and red dashed curves correspond to Eq. (63) plotted with $\kappa =0$, illustrating the very different behavior expected from coupled vs uncoupled oscillators.

Figure 9

(Color online) (Left) Dual-gap ELC and MLC structures. The wave is assumed to propagate between the two patterned surfaces, polarized such that the electric field excites the ELC while the magnetic field excites the MLC. (Right) ELC consists of a conducting ring, as shown, with dimensions $d=3\text{mm}$, $p=2.6\text{mm}$, $l=0.2\text{mm}$, $b=0.4\text{mm}$, and $g=0.03\text{mm}$. The conductor thickness is assumed as $17\mu \text{m}$. The MLC has precisely the same geometry, but with a magnetic conductivity rather than an electrical conductivity. Also, $b=0.45$ for the MLC, in order to tune the resonances to the same frequency. Periodic boundary conditions are applied on the sides of the unit cell (perpendicular to the propagation direction) to simulate a metamaterial array infinite in the lateral dimensions.

Figure 10

(Color online) Retrieved Bloch (a) permittivity and (b) permeability for the ELC alone; retrieved Bloch (c) permittivity and (d) permeability for the MLC alone; (e) retrieved index of refraction for the ELC and MLC together with resonances matched, arranged as in Fig. 9. Solid blue and red lines correspond to the real and imaginary parts of the given parameter retrieved from full-wave simulations, while the blue and red dashed lines are fits to the real and imaginary parts of the same parameter using Eq. (24) or (31). The real and imaginary parts of the index (solid blue and red lines) are compared to Eq. (63) (dashed blue and red lines). In (e), the lighter blue and red dashed curves correspond to Eq. (63) plotted with $\kappa =0$, illustrating the very different behavior expected from uncoupled vs coupled oscillators.

Figure 11

(Color online) (Left) ELC and ESRR unit cells. (Right) Geometrical parameters for the ELC are identical to those in Fig. 9, while the ESRR has the dimensions $d=3\text{mm}$, $p=2.6\text{mm}$, $l=0.2\text{mm}$, $g=0.02\text{mm}$, and $b=0.65\text{mm}$. Both the ELC and the ESRR are modeled as metals, with an electrical conductivity corresponding to that of copper. The SRR and ELC are placed symmetrically across the midplane of the cell, with a variable distance between them in the direction perpendicular to propagation.

Figure 12

(Color online) Retrieved Bloch (a) permittivity and (b) permeability for the ELC alone; retrieved Bloch (c) permittivity and (d) permeability for the ESRR alone; (e) retrieved index of refraction for the ELC and ESRR combined, arranged as in Fig. 11, separated by 0.332 mm; (f) retrieved index of refraction for the ELC and ESRR combined, separated by 1.2 mm; (g) retrieved refractive index for the ELC and ESRR combined, separated by 0.3 mm. Solid blue and red lines correspond to the real and imaginary parts of the given parameter retrieved from full-wave simulations, while the blue and red dashed lines are fits to the real and imaginary parts of the same parameter using Eq. (24) or (31). The real and imaginary parts of the index (solid blue and red lines) are compared to Eq. (63) (dashed blue and red lines). The curve fits make use of the parameters shown in Table . The qualitative features observed in these plots can be compared to those in Figs. 5, 6.