Generalized effective-medium theory for metamaterials

We present an effective-medium model for calculating the frequency-dependent effective permittivity $\epsilon \left(\omega \right)$ and permeability $\mu \left(\omega \right)$ of metamaterial composites containing spherical particles with arbitrary permittivity and permeability. The model is derived from the zero-scattering condition within the dipole approximation, but does not invoke any additional long-wavelength approximations. As a result, it captures the effects of spatial dispersion and predicts a finite effective refractive index and antiresonances in $\epsilon \left(\omega \right)$ and $\mu \left(\omega \right)$, in agreement with numerical finite-element calculations.

Figure 1

Definition of terms for the GEM model. The parameters of the effective medium ${\epsilon }_1$ and ${\mu }_1$ are determined by the condition that electromagnetic plane waves incident from the effective medium on the core-shell structure do not scatter.

Figure 2

Comparison of calculated real values of the effective (a) impedance, (b) index, (c) permeability, and (d) permittivity of the metamaterial composite of dielectric spheres with ${\epsilon }_3=50$ and ${\mu }_3=1$, embedded in vacuum with a volume fraction of $f=0.25$.

Figure 3

Calculated magnetic and electric field around the core-shell described in Fig. 2, embedded in a medium with effective parameters ${\epsilon }_1$ and ${\mu }_1$ calculated in HFSS (a) and (b) (equivalent to GEM in this case) and the Lewin model (c) and (d), at a frequency just below the first resonance at $a/\lambda \approx 0.17$, for a plane wave incident from the left.

Figure 4

Comparison of calculated real values of the effective (a) impedance, (b) index, (c) permeability, and (d) permittivity of the metamaterial composite of dielectric spheres with ${\epsilon }_3=12$ and ${\mu }_3=1$, embedded in vacuum with a volume fraction of $f=0.25$.

Figure 5

Comparison of the normalized scattering cross sections, calculated from Eq. (1), for core-shells with ${\epsilon }_2={\mu }_2=1$ and $f=0.25$ and (a) ${\epsilon }_3=50$ and ${\mu }_3=1$, and (b) ${\epsilon }_3=12$ and ${\mu }_3=1$, embedded in a medium with effective ${\epsilon }_1$ and ${\mu }_1$ values obtained from the three EM models.