CST – Computer Simulation Technology

Investigation of Backward Wave Propagation on LHM Split Ring Resonators (SRRs)

Recent works [1,2,3] have predicted that a dense array of Split Ring Resonators (SRRs) can be considered as a gyrotropic bianisotropic media with magneto-electric coupling effects. This application article describes work carried out at the Laboratoire d'Electronique et Electromagnetisme (L2E), where simulations of such SRRs with CST MICROWAVE STUDIO® (CST MWS) have demonstrated the propagation of a backward wave component in a certain frequency range where the structure behaves like a medium with a negative refractive index.

The periodic structure under consideration is shown in Figure 1. A plane wave is introduced along the z-axis whereas the electric and magnetic fields have been polarized along the x- and y-axes respectively.

Figure 1: Dense array of copper SRRs backed by an epoxy substrate...

The frequency dependent effective refractive index of the equivalent metamaterial structure can be extracted by considering a single element rather than performing a time-consuming simulation of the entire array.

The relations for the refractive index n and wave impedance Z are given by the expressions in Figure 2, where S21 and S11 represent the transmission and reflection coefficients, and j and k are the complex number and the wavenumber respectively of the incident wave in free space.

Figure 2: Calculation of refractive index n and wave impedance Z

The S-parameters are calculated by simulating a single element of the array using the frequency domain solver in CST MWS. Unit cell boundary conditions with a full Floquet port mode implementation allow the consideration of the infinite SRR structure as shown in Figure 3. The resulting S-parameters are shown in Figure 4, and these then lead to the effective refractive index in Figure 5.

Figure 3: Single element of the periodic structure with the Unit Cell boundary conditions taking into account the EM-field periodicity given by the Floquet modes. The structure is described as an infinite periodic SRR array

Figure 4: S-parameters of the structure depicted in Figure 3. Red: S11; Green: S21

Figure 5: Effective refractive index versus frequency, as calculated from the S-parameters given in Figure 4

The effective permittivity and permeability are generally calculated from the relations εeff = n/Z and µeff = n.Z, but for bianisotropic media the extraction is more complicated since a magneto-electric parameter must be integrated [3].

From Figures 4 and 5 one can observe that the first and second resonances reveal all frequency bands where the refractive index is negative. Nevertheless, the backward wave propagation occurs only in the second resonance since the phase velocity behaves as a negative eigenmode [4]. Figure 6 confirms the evidence of this backward wave propagation: the phase of the right-to-left propagating wave appears to move from left to right within the structure.

Figure 6: Evidence of backward wave propagation inside the SRR array showing the negative refractive index of the equivalent medium

The backward wave propagation is due to the so-called Faraday Effect, since an induced quasistatic magnetic Hz field oscillates around a mean value of about 1.1 mA/m, as shown in Figure 7. This result has been extracted automatically from CST MWS simulation results using magnetic probes placed along the structure. This effect is reciprocal since the same Hz field is spontaneously induced along the z-axis.

Figure 7: Faraday effect induced along the structure

Looking carefully into the structure, one can see that the distribution of E-fields is the sum of two elliptically polarized electric fields (right and left handed) along the z-axis as shown in Figure 8. In fact each SRR cell behaves as a pseudo-chiral ferrite.

Figure 8: The linearly polarized E-field introduced along the x-axis leads to elliptically polarized electric fields inside the SRR structure indicating a gyrotropic effect

As an example, the lens of negative refractive index in Figure 9 is illuminated by a plane wave at 14.85 GHz and shows a focal point. A positive index would lead to the well known divergent lens.

Figure 9: Negative lens illuminated by a plane wave showing a focalization similar to a convergent lens

A gyrotropic bianisotropic medium has composed of a SRR array been proposed. An analysis using an extraction method calculating the phase velocity parameter has predicted the frequency range where the refractive index can be negative, thus leading to a backward wave propagation phenomenon. The results given by CST MWS have demonstrated the strong contribution of the Faraday Effect in the equivalent bianisotropic medium.


[1] R. Marques, F. Medina, R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials”, Phys. Rev. B 65 (2002) 144440-1-6

[2] C.W. Qiu, H.Y Yao, L.W Li; S. Zouhdi, T.S. Yeo, “Routes to left-handed materials by magnetoelectric couplings”, Phys. Review B75, 245214, 2007.

[3] S. A. Tretyakov, C. R. Simovski and M. Hudlicka, “Bianisotropic route to the realization and matching of backward-wave metamaterial slabs”, Phys. Review B75, 14504, 2007.

[4] H. Talleb, Z. E. Djeffal, D. Lautru, V. F. Hanna , “ Investigation of backward-wave propagation on LHM Split Ring Resonators”, Conference Meta’10, February 2010, Cairo, Egypt.

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