Electromagnetic modes in cavities made of negative-index metamaterials

We discuss electromagnetic modes in cavities formed by metamaterials with negative refraction and demonstrate that the straightforward approach to substitute negative values of the electric permittivity and the magnetic permeability leads to quasibound states with a negative quality factor. To ensure positive quality factors and a consistent physical interpretation of the quasibound states it is essential to include the frequency dispersion of the permittivity and the permeability, as required by positive field energy and causality. The basic mode equation and the boundary conditions including linear frequency dispersion are derived. As an example we consider a disklike cavity with deformed cross-sectional shape. The transition from the unphysical nondispersive case with negative quality factors to the dispersive case with positive quality factors is demonstrated numerically and in an analytical perturbative treatment.

Figure 1

Near-field intensity pattern of an electromagnetic mode with normalized frequency $\Omega R/c=44.9376-i0.0622$ in a conventional cavity with $\varepsilon =9/4$ and $\mu =1$ ($n=1.5$) surrounded by vacuum with $\varepsilon =\mu =1$. Arrows illustrate the direction of the energy flow.

Figure 2

(a) Near-field intensity pattern of an electromagnetic mode with $\Omega R/c=45.0966-i0.0514$ in a NIM cavity with $\varepsilon ({\omega }_{\mathrm{r}})=-9/4$, $\mu ({\omega }_{\mathrm{r}})=-1$ ($|n({\omega }_{\mathrm{r}})|=1.5$), ${\omega }_{\mathrm{r}}R/c=45.1$, and linear frequency dispersion ${\alpha }_{\varepsilon }={\alpha }_{\mu }=4.1$. Arrows illustrate the direction of the energy flow. (b) Unphysical mode with ${\Omega }_{0}R/c=45.1060+i0.1647$ in a NIM cavity with $\varepsilon =-9/4$ and $\mu =-1$ ($\left|n\right|=1.5$) calculated without frequency dispersion.

Figure 3

Goos-Hänchen shift in (a) a conventional cavity (cf. Fig. 1), and (b) a NIM cavity [cf. Fig. 2a]. The dotted line is the center of a Gaussian beam being reflected at the dielectric interface. The appearance of the Goos-Hänchen shift along the boundary is evident. The mode pattern is the full numerical solution of the respective mode equation.

Figure 4

$Q/(-Q_0)$ (upper panel) and overlap difference $1-S$ [lower panel, cf. Eq. (38)] vs. linear frequency dispersion ${\alpha }_n={\alpha }_{\varepsilon }={\alpha }_{\mu }$. All quantities are dimensionless. Here, $Q_0=-137$ is the quality factor in the nondispersive case ${\alpha }_n=0$. In the regime ${\alpha }_n>1$ the electromagnetic field energy is positive and for ${\alpha }_n>4$ also causality holds. The empty dots mark the results of the full solution of the mode equation (32). The solid line is the theoretical prediction in Eq. (39).