Figure 1

(Color online) Geometry of the grid mesh for the FDFD method. The nodes are spaced by $\Delta x$ and $\Delta y$ along the $x$ and $y$ directions, respectively. The shaded region represents a portion of the dielectric inclusion with dielectric permittivity ${\epsilon }_r$.Reuse & Permissions

Figure 2

(Color online) (a) Quasistatic effective permittivity as a function of the volume fraction $f_V$ (in percentage) of the inclusions for different values of the permittivity ${\epsilon }_r$. (b) Effective permittivity as a function of ${\epsilon }_r$ for different values of $f_V$. The discrete symbols in (a) and (b) correspond to the values computed with the homogenization method and the solid lines were obtained from the slope of the band structure of the material at the origin of the Brillouin zone. The geometry of the unit cell of the two-dimensional metamaterial is shown in the inset: the unit cell consists of a cylindrical inclusion with circular cross section, normalized radius $R/a$ and permittivity ${\epsilon }_r$.Reuse & Permissions

Figure 3

(Color online) Real parts of the effective permittivity ${\epsilon }_{eff}$ (green lighter curves) and permeability ${\mu }_{eff}$ (blue darker curves) as a function of the normalized frequency $\omega a/c$. The discrete symbols correspond to the values extracted with the FDFD method and the solid lines are the values obtained using the Clausius-Mossotti formula. The high-index cylindrical-shaped inclusion has a normalized radius $R=0.4a$ and permittivity ${\epsilon }_r=56$. The host material is air.Reuse & Permissions

Figure 4

(Color online) Real and imaginary parts of the effective permittivity ${\epsilon }_{eff}/{\epsilon }_0={\epsilon }^{\prime }-j{\epsilon }^{″}$ as a function of the normalized frequency $\omega /{\omega }_p$, for a mixture with plasmonic cylinders arranged in a regular lattice, for different values of the damping frequency: $\Gamma /{\omega }_p=0.001$ (black curves), $\Gamma /{\omega }_p=0.01$ (green curves), and $\Gamma /{\omega }_p=0.1$ (blue curves). The solid lines correspond to the real part of ${\epsilon }_{eff}$ while the dashed lines represent the imaginary part. The host material is air. The inset shows the real part of the effective magnetic permeability $\left(\Gamma /{\omega }_p=0.001\right)$ of the system in the vicinity of $\omega /{\omega }_p=0.64$.Reuse & Permissions

Figure 5

(Color online) Real part of the effective quasistatic permittivity as a function of the real part of the permittivity of the inclusion ${\epsilon }_r^{\prime }$, with ${\epsilon }_r^{″}=0$ (solid green curve and discrete diamond-shaped green symbols), $-{\epsilon }_r^{″}/{\epsilon }_r^{\prime }=0.1$ (solid blue curve and discrete circle-shaped blue symbols). The geometry of the unit cell of the two-dimensional metamaterial is shown in the inset: the unit cell consists of a square-shaped inclusion with negative permittivity ${\epsilon }_r={\epsilon }_r^{\prime }-j{\epsilon }_r^{″}$ and sharp corners. The solid curves were extracted with the FDFD method while the discrete symbols are from Ref. 48. The host material is air.Reuse & Permissions

Figure 6

(Color online) Real part of the effective permeability ${\mu }_{eff}$ as a function of the permittivity ${\epsilon }_r$ of a square lattice of dielectric cylinders embedded in a host medium with permittivity near to zero $\left({\epsilon }_h\approx 0\right)$. The cylindrical-shaped inclusions have normalized radius $R=0.4a$. The solid curve was obtained using Eq. (15) and the discrete symbols where obtained with the FDFD method.Reuse & Permissions

Figure 7

(Color online) (a) Real parts of the effective permittivity ${\epsilon }_{eff}$ (green curves) and permeability ${\mu }_{eff}$ (blue curves) as a function of the normalized frequency $\omega /{\omega }_p$. The discrete symbols correspond to the values extracted with the FDFD method and the solid lines were obtained using the Clausius-Mossotti formula. The inclusion has a normalized radius $R=0.4a$ and permittivity ${\epsilon }_r=56$ and the host medium is characterized by a Drude-type dispersion model. (b) normalized amplitude of the electric field component $E_y$ in the unit cell (when the current source is directed along $y$) at the frequency where the effective permittivity hits a resonance (ii) and when ${\epsilon }_{eff}\approx 0$ (i).Reuse & Permissions

Figure 8

(Color online) (a) Real part of the effective permittivity ${\mu }_{eff}$ as a function of the normalized frequency $\omega a/c$. Solid line: homogenization method used in this work; discrete symbols: data extracted using the inversion of the reflection and transmission coefficients (Refs. 16, 17). The geometry of the unit cell is shown in the inset and consists of a U-shaped plasmonic inclusion with complex permittivity ${\epsilon }_r$. The arms of the horseshoe have a normalized thickness $b=0.18a$ and normalized length $L=0.79a$. The distance between the two arms is $d=0.26a$. (b) Real parts of ${\epsilon }_{eff,xx}\left(\omega ,\mathbf{k}=0\right)$ (blue solid curve) and ${\epsilon }_{eff,yy}\left(\omega ,\mathbf{k}=0\right)$ (green dashed curve) as a function of the frequency. The host medium is air. (c) and (d) normalized amplitude of the magnetic field $H_z$ in the unit cell, when the current source is directed along $x$ and $y$, respectively, at $\omega a/c\approx 1.47$. (e) and (f) represent the real part of the electric field vector $\mathbf{E}$ in the unit cell, when the current source is directed along $x$ and $y$, respectively.Reuse & Permissions

Figure 9

(Color online) Nonlocal dielectric function along the $x$ direction (green dashed line), effective permeability (black dotted line), and magnetoelectric parameter (blue solid line) for the horseshoe geometry of Fig. 8a.Reuse & Permissions

Figure 10

(Color online) Real part of the effective permittivity ${\mu }_{eff}$ as a function of the normalized frequency $\omega a/c$ considering different formulas for the extraction of ${\mu }_{eff}$.Reuse & Permissions

Figure 11

(Color online) Required permittivity ${\epsilon }_r$ for the horseshoe inclusion as a function of the normalized resonant frequency ${\omega }_{r}a/c$. The inset shows the normalized resonant wavelength ${\lambda }_r/a$ as a function of the normalized skin depth $\delta /b$ of the metal. The circle-shaped symbols correspond to the values extracted with the method of inversion of the reflection and transmission coefficients (Refs. 16, 17) and the star-shaped symbols were extracted with the FDFD method.Reuse & Permissions