Guided modes of elliptical metamaterial waveguides

The propagation of guided electromagnetic waves in open elliptical metamaterial waveguide structures is investigated. The waveguide contains a negative-index media core, where the permittivity $\epsilon$ and permeability $\mu$ are negative over a given bandwidth. The allowed mode spectrum for these structures is numerically calculated by solving a dispersion relation that is expressed in terms of Mathieu functions. By probing certain regions of parameter space, we find the possibility exists to have extremely localized waves that transmit along the surface of the waveguide.

Figure 1

(Color online) Cross section of the open elliptical waveguide structure. Within the metamaterial core, ${\epsilon }_1$ and ${\mu }_1$ are in general both negative and frequency dispersive. The outer region is air; ${\epsilon }_2=1$ and ${\mu }_2=1$.

Figure 2

(Color online) Surface-wave spectrum for the even and odd hybrid modes of an elliptical waveguide filled with NIM and embedded in air (top panel). The bottom panel is the same configuration except with positive ${\epsilon }_1\left(\omega \right)$ and ${\mu }_1\left(\omega \right)$. The propagation constant $k_z$ is normalized by the free-space wave number $k_0$ and is plotted as a function of the operating frequency for a cross section corresponding to an ellipticity of $e=0.44$. We take the characteristic frequencies in the material dispersion to be ${\omega }_p=8\left(2\pi \right)\mathrm{GHz}$ and ${\omega }_m=7\left(2\pi \right)\mathrm{GHz}$, which corresponds to ${\mu }_1,{\epsilon }_1\le -1$ within the frequency range $f\le {\omega }_m∕\left(2\pi \sqrt{2}\right)$. The mode spectra illustrate the slow and fast modes, separated by the $q_1=0$ curve. If the NIM core is replaced with media corresponding to positive ${\epsilon }_1$ and ${\mu }_1$ (bottom panel), the slow-wave solution is absent and the curves vary in an opposite sense.

Figure 3

(Color online) (a) The total power, $P=P_1+P_2$ (normalized by $\mid P_1\mid +\mid P_2\mid$) versus the frequency for (odd) slow wave modes. The eccentricity, $e$, is set at $e=0.59$. The propagation constant, $k_z$, increases in going from $\mathrm{A}\to \mathrm{B}$, in accordance with the mode dispersion diagram in the inset. In (b) energy-density reversal is shown by means of the Poynting vector, $S_z$, normalized by its maximum, and as a function of the dimensionless coordinate, ${\xi }^{\prime }\equiv \xi ∕{\xi }_0$. The dashed vertical line identifies the NIM-air boundary. The frequency is fixed at $f=5.28\mathrm{GHz}$, and $k_z=1.1$. The spatial range is for the angle $\eta =\pi ∕6$.

Figure 4

(Color online) The slow-wave effective mode area $A_{\mathrm{eff}}$ (normalized by the wavelength ${\lambda }^2$) as a function of the eccentricity $e$.

Figure 5

(Color online) The normalized electric field distributions for even wave modes near the cutoff in an elliptical waveguide as a function of (a) angular coordinate $\eta$ and (b) normalized radial coordinate ${\xi }^{\prime }\equiv \xi ∕{\xi }_0$. The frequency is set at $3.25\mathrm{GHz}$, and the ellipticity $e$ has the value $e=0.14$, corresponding to ${\xi }_0=2.65$. The vertical dashed line at ${\xi }^{\prime }=1$ identifies the waveguide boundary.

Figure 6

(Color online) The spatial and angular variation of the electric field, with the same parameters as in Fig. 5, except at the higher frequency, $f=5.28\mathrm{GHz}$. These localized slow waves decay away from the outer side of the interface rapidly, consistent with dispersion curves of Fig. 2.

Figure 7

(Color online) Contour plots of the even transverse electric field amplitudes for $e=0.59$ and $f=5.3\mathrm{GHz}$. The top and bottom panels correspond to $E_{\eta }$ and $E_{\xi }$, respectively. Bright areas indicate larger field intensities. Each component is normalized to its respective maximum for clarity.