Retrieval of the effective constitutive parameters of bianisotropic metamaterials

We propose a method to retrieve the effective constitutive parameters of a slab of bianisotropic metamaterial composed of split-ring resonators from the measurement of the $S$ parameters. Analytical inversion equations are derived for homogeneous lossless bianisotropic media, and a numerical retrieval approach is presented for the case of lossy bianisotropic media. The method is verified both analytically and numerically, and it is shown that the results for various split-ring resonator metamaterials qualitatively corroborate the conclusions found in published papers. The proposed retrieval method can be used as a valuable tool for the study of anisotropic and bianisotropic properties of metamaterials.

Figure 1

Unit cell of a metamaterial composed of the edge-coupled SRR upon which six incidences are used to obtain the $S$ parameters from finite-difference time-domain simulations. (a) Structure of the edge-coupled SRR: The unit cell is a cube of edge $a=5.25\mathrm{mm}$. The parameters of the SRR are $g=d=0.125\mathrm{mm}$, $c=0.25\mathrm{mm}$, and $w=3.75\mathrm{mm}$. The background medium is free space. (b) Modes of multiple incidences: When the wave is incident normally on the unit cell, the unit cell is periodically repeated in the plane perpendicular to the incidence direction, thus forming a slab.

Figure 2

Comparison of the analytical and the retrieved results for a lossless homogeneous medium. The curves with $◻$ and $엯$ are the retrieval results using method 1, and the curves with $\times$ and $+$ are from method 2. Note that the markers in the figure are hard to distinguish because the results are almost identical for the two methods.

Figure 3

Comparison of the analytical and the retrieved results for a lossy homogeneous medium. The curves with $◻$ and $엯$ are the retrieval results using method 1, and the curves with $\times$ and $+$ are from method 2. Note that the markers in the figure are hard to distinguish because the results are almost identical for the two methods.

Figure 4

Retrieval results for a lossless edge-coupled SRR metamaterial whose unit cell is shown in Fig. 1a, using a retrieval method not considering the bianisotropy. The retrieved ${\mu }_x$ (a) and ${ϵ}_x$ (c) show negative imaginary parts around the resonance, which violates physical laws [15] and therefore indicates that the results are not reliable in the corresponding region. Those results difficult to read within the resonance band are not shown in (e) and (f). The shaded region indicates the frequency range where the mismatch of either ${\mu }_y$ or ${ϵ}_z$ exceeds the threshold $(M_R>0.25)$.

Figure 5

Retrieval results for a lossless edge-coupled SRR metamaterial whose unit cell is shown in Fig. 1a, using lossless retrieval for bianisotropic media. The subscripts 1 and 2 denote the results obtained from the proposed methods 1 and 2, respectively. Those results difficult to read within the resonance band are not shown here. The shaded region indicates the frequency range where the mismatch of either ${\mu }_y$ or ${\xi }_0$ exceeds the threshold $(M_R>0.25)$.

Figure 6

Retrieval results for a lossy edge-coupled SRR metamaterial, whose unit cell is the same as that in Fig. 1a except that the background material is lossy ($\sigma =0.042\mathrm{S}∕\mathrm{m}$, ${ϵ}_r=3.4$). The shaded region indicates the frequency range where the mismatch of either ${\mu }_y$ or ${\xi }_0$ exceeds the threshold $(M_R>0.25)$.

Figure 7

Retrieved results for a broadside-coupled SRR metamaterial, where a negligible ${\xi }_0$ is observed (except around the resonance 5.9 GHz–6.7 GHz; see Table .) Those results difficult to read within the resonance band are not shown here. The shaded region indicates the frequency range where the mismatch of either ${\mu }_y$ or ${\xi }_0$ exceeds the threshold $(M_R>0.25)$.