Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials

We study the frequency dependence of the effective electromagnetic parameters of left-handed and related metamaterials of the split ring resonator and wire type. We show that the reduced translational symmetry (periodic structure) inherent to these metamaterials influences their effective electromagnetic response. To anticipate this periodicity, we formulate a periodic effective medium model which enables us to distinguish the resonant behavior of electromagnetic parameters from effects of the periodicity of the structure. We use this model for the analysis of numerical data for the transmission and reflection of periodic arrays of split ring resonators, thin metallic wires, cut wires, as well as the left-handed structures. The present method enables us to identify the origin of the previously observed resonance-antiresonance coupling as well as the occurrence of negative imaginary parts in the effective permittivities and permeabilities of those materials. Our analysis shows that the periodicity of the structure can be neglected only for the wavelength of the electromagnetic wave larger than 30 space periods of the investigated structure.

Figure 1

The layout of the single unit cell $\left(a\right)$ and of a finite slab of the model periodic medium are shown. The shaded regions indicate the homogeneous core of the width $d$ which is characterized by the chosen appropriately model functions $\mu \left(\omega \right)$ and $\epsilon \left(\omega \right)$, sandwiched by two vacuum slabs. $L$ is the length of a single unit cell and $N$ the number of unit cells in the slab in propagations direction. Periodic boundary conditions apply in the directions perpendicular to the propagation direction $z$.

Figure 2

(Color online.) The HEM inversion [Eqs. (24, 25)] of the analytic continuum PEM scattering amplitudes [Eqs. (17, 18)] for model SRR-type material parameters ${\omega }_m=0.13$, ${\omega }_{\mathit{mp}}=0.16$ for the magnetic and ${\omega }_e=0.4$, ${\omega }_{\mathit{ep}}=0.5$ for the electric response and $\gamma ={10}^{-4}$ [see Eqs. (29, 30)]. The homogeneous core located in the middle of the unit cell was $d=L∕10$ thick. The retrieved real (red, purple) and imaginary (green, turquoise) parts of effective parameters are shown as a function of frequency $\omega$. The dashed lines show the real (purple) and imaginary (turquoise) parts of the anticipated homogeneous parameters [Eqs. (29, 30)] and corresponding index of refraction and impedance. The dash-dotted black lines in $\mathrm{Re}{n}_{\mathrm{eff}}$ indicate the upper edge of the Brillouin zone, $n_{\text{edge}}=k_{\text{edge}}∕k=m\pi ∕\left(kL\right)$.

Figure 3

(Color online.) The HEM inversion [Eqs. (10, 11)] of the analytic lattice PEM scattering amplitudes [Eq. (46)] for model SRR-type material parameters ${\omega }_m=0.13$, ${\omega }_{mp}=0.16$ for the magnetic, and ${\omega }_e=0.4$, ${\omega }_{ep}=0.5$ for the electric response and $\gamma ={10}^{-4}$ [see Eqs. (29, 30)]. The homogeneous core located in the middle of the unit cell was $d=L∕10$ thick. The retrieved real (red, purple) and imaginary (green, turquoise) parts of effective parameters are shown as a function of frequency $\omega$. Note the reduction of the multiple band gaps seen in Fig. 2 around the resonances to a single gap before each resonance. The dashed lines show the real (purple) and imaginary (turquoise) parts of the anticipated homogeneous parameters [Eqs. (29, 30)] and corresponding index of refraction and impedance. The dash-dotted black lines in $\mathrm{Re}{n}_{\mathrm{eff}}$ indicates the upper edge of the Brillouin zone $n_{\text{edge}}=k_{\text{edge}}∕k=m\pi ∕\left(kL\right)$.

Figure 4

(Color online.) The HEM inversion [Eqs. (10, 11)] of the analytic lattice PEM scattering amplitudes [Eq. (46)] for model LHM-type material parameters ${\omega }_m=0.13$, ${\omega }_{mp}=0.16$ for the magnetic, and ${\omega }_e=0.4$, ${\omega }_{ep}=0.5$, ${\omega }_p=0.27$ for the electric response, and $\gamma ={10}^{-4}$ [see Eqs. (29, 31)]. The homogeneous core located in the middle of the unit cell was $d=L∕10$ thick. The retrieved real (red, purple) and imaginary (green, turquoise) parts of effective parameters are shown as a function of frequency $\omega$. The dashed lines show the real (purple) and imaginary (turquoise) parts of the anticipated homogeneous parameters [Eqs. (29, 31)] and corresponding index of refraction and impedance. The dash-dotted black lines in $\mathrm{Re}{n}_{\mathrm{eff}}$ indicate the edges of the Brillouin zone, $n_{\text{edge}}=k_{\text{edge}}∕k=m\pi ∕\left(kL\right)$.

Figure 5

(Color online.) For the simulated SRR metamaterial the effective index of refraction $n_{\mathrm{eff}}\left(\omega \right)$ and impedance $z_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [see Eqs. (51, 52)] obtained from the first unit cell data. Note the different positions of the resonance for $n_{\mathrm{eff}}\left(\omega \right)$ and $z_{\mathrm{eff}}\left(\omega \right)$.

Figure 6

(Color online.) For the simulated SRR metamaterial the effective permittivity ${\epsilon }_{\mathrm{eff}}\left(\omega \right)$ and permeability ${\mu }_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [see Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [Eqs. (51, 52)] obtained from the first unit cell data. Note the antiresonant behavior of the permittivity and the misshapen magnetic resonance in the frequency interval where $n_{\mathrm{eff}}\left(\omega \right)$ reaches the edge of the Brillouin zone.

Figure 7

(Color online.) For the simulated low-frequency SRR metamaterial the effective index of refraction $n_{\mathrm{eff}}\left(\omega \right)$ and impedance $z_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [see. Eqs. (51, 52)] obtained from the first unit cell data. Note that far away from the edge of the Brillouin zone HEM and PEM approximation, and the expected homogeneous medium behavior coincide.

Figure 8

(Color online.) For the simulated low-frequency SRR metamaterial the effective permittivity ${\epsilon }_{\mathrm{eff}}\left(\omega \right)$ and permeability ${\mu }_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [see. Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [Eqs. (51, 52)] obtained from the first unit cell data. In the low-frequency limit the resonance/antiresonance coupling as well as the negative imaginary parts disappear.

Figure 9

(Color online.) For the simulated off-plane LHM metamaterial the effective index of refraction $n_{\mathrm{eff}}\left(\omega \right)$ and impedance $z_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [see. Eqs. (51, 52)] obtained from the first unit cell data.

Figure 10

(Color online.) For the simulated off-plane LHM metamaterial the effective permittivity ${\epsilon }_{\mathrm{eff}}\left(\omega \right)$ and permeability ${\mu }_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [see. Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [Eqs. (51, 52)] obtained from the first unit cell data.

Figure 11

(Color online.) For the simulated continuous wire metamaterial the effective index of refraction $n_{\mathrm{eff}}\left(\omega \right)$ and impedance $z_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [see Eqs. (51, 52)] obtained from the first unit cell data.

Figure 12

(Color online.) For the simulated continuous wire metamaterial the effective permittivity ${\epsilon }_{\mathrm{eff}}\left(\omega \right)$ and permeability ${\mu }_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [see Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [Eqs. (51, 52)] obtained from the first unit cell data.

Figure 13

(Color online.) For the simulated cut-wire metamaterial the effective index of refraction $n_{\mathrm{eff}}\left(\omega \right)$ and impedance $z_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [see Eqs. (51, 52)] obtained from the first unit cell data. The weak additional structure in the PEM approximation close to the resonance is indicating the beginning breakdown of the approximation of the metamaterial within our most simple periodic medium model.

Figure 14

(Color online.) For the simulated cut-wire metamaterial the effective permittivity ${\epsilon }_{\mathrm{eff}}\left(\omega \right)$ and permeability ${\mu }_{\mathrm{eff}}\left(\omega \right)$ are shown. The colored curves represent the HEM approximation [see Eqs. (10, 11)] of the simulation data for the first three unit cells, the solid black line the HEM(PEM) approximation, and the dashed line the PEM approximation [Eqs. (51, 52)] obtained from the first unit cell data.