Retrieving effective parameters for metamaterials at oblique incidence

We introduce a technique to retrieve effective metamaterial parameters for arbitrary angles of incidence. It employs the complex reflection and/or transmission coefficients of a finite slab. Explicit expressions for both polarizations are derived and the constraints to be met for obtaining unique solutions are discussed. The method is applied to the fishnet structure. It turns out that all retrieved parameters strongly depend on the lateral wave vector component due to the complexity of the metamaterial structure. Thus, these parameters are mere wave parameters rather than global material parameters. The physical effects behind this behavior are very likely anisotropy and spatial dispersion.

Figure 1

(Color online) Sketch of the geometry under consideration. Top: the fishnet structure and the definition of the polarization. Bottom: cross section through the MM slab and definition of the wave vector components.

Figure 2

(Color online) Real and imaginary parts of the effective permittivity $\epsilon$, the effective permeability $\mu$, the effective normal component $k$ of the wave vector, the effective refractive index $n$, and the effective impedance $Z$ as a function of the wave number and the number of MM layers. The results are shown at a fixed transverse wave number of the incident field of $k_x=0.8727\mu {\text{m}}^{-1}$ and TE polarization. The lines are labeled as follows: 1 MM layer—blue solid line, 2 MM layers—green dashed line, 3 MM layers—red dotted line, and 4 MM layers—black dashed-dotted line.

Figure 3

(Color online) Real and imaginary parts of the effective permittivity $\epsilon$, the effective permeability $\mu$, the effective refractive index $n$, and the effective impedance $Z$ as a function of the angle of incidence for different numbers of the MM layers. Results are shown for a fixed wave number of $\overline{\nu }=6957{\text{cm}}^{-1}$. The lines are labeled as follows: 1 MM layer—blue solid line, 2 MM layers—green dashed line, 3 MM layers—red dotted line, and 4 MM layers—black dashed-dotted line.

Figure 4

(Color online) Amplitude and phase of the transmitted and reflected waves at a MM slab made of four layers as a function of the wave number for various transverse wave vector components $k_x$. The lines are labeled as follows: $k_{x1}=0.8727\mu {\text{m}}^{-1}$—blue solid line, $k_{x2}=2k_{x1}$—green dashed line, $k_{x3}=3k_{x1}$—red dotted line, and $k_{x4}=4k_{x1}$—black dashed-dotted line.

Figure 5

(Color online) Real and imaginary parts of the permittivity $\epsilon$, the permeability $\mu$, the effective normal component $k$ of the wave vector, the refractive index $n$, and the impedance $Z$ versus the wave number $\overline{\nu }=1/\lambda$ for different values of $k_x$. The lines are labeled as follows: $k_{x1}=0.8727\mu {\text{m}}^{-1}$—blue solid line, $k_{x2}=2k_{x1}$—green dashed line, $k_{x3}=3k_{x1}$—red dotted line, $k_{x4}=4k_{x1}$—black dashed-dotted line.