Negative refraction, negative phase velocity, and counterposition in bianisotropic materials and metamaterials

The planewave response of a linear passive material generally cannot be characterized by a single scalar refractive index, as directionality of energy flow and multiple wave vectors may need to be considered. This is especially significant for materials which support negative refraction, negative phase velocity, and counterposition. By means of a numerical example based on a commonly studied bianisotropic material, our theoretical investigation revealed that (i) negative (positive) refraction can arise even though the phase velocity is positive (negative), (ii) counterposition can arise in instances of positive and negative refraction, (iii) the phase velocity and time-averaged Poynting vectors can be mutually orthogonal, and (iv) whether or not negative refraction occurs can depend on the state of polarization and angle of incidence. A further numerical example revealed that negative phase velocity and positive refraction can coexist even in a simple isotropic dielectric material.

Figure 1

(a) Real part of $k_{zj}/k_0$ and (b) the angle (in degree) between the real part of ${\underset{̱}{k}}_j$ and the positive $z$ axis both plotted against $\kappa /k_0$. The solid curves correspond to $j=1$ and the dashed curves to $j=2$. As $\text{Im}{k}_{zj}>0$ for all $\kappa /k_0∊\left[0,1\right)$ and both values of $j$, the imaginary parts of $k_{z1}/k_0$ and $k_{z2}/k_0$ have not been plotted here.

Figure 2

(a) The $x$ component of the normalized time-averaged Poynting vector and (b) the angle (in degree) between the time-averaged Poynting vector and the positive $z$ axis, both plotted against $\kappa /k_0$. The solid curves correspond to $j=1$ and the dashed curves to $j=2$. As $\underset{̱}{z}•{\underset{̱}{P}}_j>0$ for all $\kappa /k_0∊\left[0,1\right)$ and both values of $j$, those quantities have not been plotted here.

Figure 3

The quantity $\text{Re}\left({\underset{̱}{k}}_j\right)•{\underset{̱}{P}}_j/\left(k_0\left|{\underset{̱}{P}}_j\right|\right)$ plotted against $\left(\kappa /k_0\right)∊\left[0,1\right)$. The solid curve corresponds to $j=1$ and the dashed curve to $j=2$.