Left-Handed Interfaces for Electromagnetic Surface Waves

We show that surface electromagnetic waves (SEMWs) propagating along two-dimensional (2D) interfaces separating different metamaterials can behave analogously to 3D electromagnetic waves in either usual or left-handed media, depending on the permeabilities and/or permittivities of the two materials forming the interface. We derive the conditions when SEMWs carry energy opposite to the phase velocity. In analogy to three-dimensional (3D) left-handed media, we derive both an anomalous Cherenkov emission and a reversed Doppler effect. We also predict a negative refraction at the boundary between two different interfaces, which can be useful for perfect 2D lensing.

Figure 1

Schematic diagram of the relative orientations of the electric ${\mathbf{E}}_{\pm }^{(0)}$ and magnetic ${\mathbf{H}}_{\pm }^{(0)}$ field amplitudes respective to the wave vector $\mathbf{q}$ and energy flux densities ${\mathbf{S}}_{\pm }$ in the neighboring media for TE (a) and TM (b) surface waves. The color gradient illustrates the decay of the SEMWs away from the interface. Note that for interfaces separating any kind of media (for instance, usual dielectric and metal, but not only LHM and RHM), the vectors $\mathrm{Re}\mathbf{E}({\mathbf{r}}_{\perp },z,t)$, $\mathrm{Re}\mathbf{H}({\mathbf{r}}_{\perp },z,t)$, and $\mathbf{q}$ form the right triplet in one and the left triplet in the other media. Moreover, the Pointing vector is parallel to $\mathbf{q}$ in one media and antiparallel in the other.

Figure 2

Regions in the ($|{\epsilon }_{-}/{\epsilon }_{+}|$, $|{\mu }_{-}/{\mu }_{+}|$) plane where TE and TM SEMWs exist are marked using different patterns. No surface waves occur in the white regions. Red (blue) regions correspond to left-handed (right-handed) interfaces LHI (RHI): (a) RHM-to-metal, (b) RHM-to-meta-dielectric, (c) meta-dielectric-to-LHM, (d) metal-to-LHM, (e) RHM-to-LHM, (f) metal-to-meta-dielectric. For cases (c)–(f), there exist regions where LHI can be realized. The hyperbola in panel (e) is $|{\epsilon }_{+}/{\epsilon }_{-}|=|{\mu }_{-}/{\mu }_{+}|$.

Figure 3

Reflection and refraction of a SEMW at the boundary $x=0$ between a RHI at $x<0$ and a LHI at $x>0$. The wave vectors and energy fluxes are schematically shown as a side view (a), top view (b), and 3D diagram (c).

Figure 4

Frequency dependence of the refraction index $n_{\mathrm{eff}}$ [dashed line in (a)], phase velocity $v_{\mathrm{phase}}$ [solid line in (a)], and group velocity $v_{\mathrm{group}}$ [in (b)] for surface waves propagating along an interface separating media with permittivities ${\epsilon }_{+}$, ${\epsilon }_{-}$ and permeabilities ${\mu }_{+}$, ${\mu }_{-}$. Also, ${\epsilon }_{+}$ and ${\mu }_{-}$ are shown in the inset; and ${\epsilon }_{-}={\mu }_{+}=1$. Both TE and TM modes exist in different frequency regions. The frequency interval where the group velocity is negative (multiplied by five, to see this better) corresponds to the LHI where the total energy flux is directed opposite to the phase velocity. The parameters used are $F=0.56$, and ${\omega }_p/{\omega }_0=2.5$ (cf. 10 ).