Negative refraction in anisotropic waveguides made from quantum metamaterials

We describe two metamaterial waveguide designs which display negative refraction. Both use quantum well nanostructures to engineer tunable resonances in the dielectric response, which can be designed using the rules of quantum mechanics. One uses coupling to a “plasmonic” resonance in adjacent metallic layers to enhance the size of the negative refraction effect. The negative refraction appears close to the quantum well intersubband transition energy, and is sensitive to the two-dimensional concentration of electrons within the wells and in adjacent conducting layers, thus offering the potential for switchable devices. These “quantum metamaterials” are highly distributed multilayered optically anisotropic systems with comparatively low absorption in the negatively refracting spectral region, and they can readily be grown by conventional semiconductor epitaxial growth technologies.

Figure 1

Schematic of the overall waveguide comprising a central slab, of thickness $d$, whose anisotropic dielectric response tensor, ${\overline{\epsilon }}_2$, is determined by its nanoscale layered structure. It is surrounded by two semi-infinite dielectric layers with the same real, isotropic, and positive dielectric constant, ${\epsilon }_1$.

Figure 2

(Color online) Detail of the layers making a single period of a “plasmonically coupled” quantum metamaterial multilayer which, in turn, makes up the central slab in Fig. 1. Layer 1 acts as a bulk metallic conducting layer whose collective plasma resonance couples electrostatically to the single electron intersubband transitions in the QW, layer 3. The resulting quantum metamaterial behaves as an anisotropic effective medium with the dielectric response tensor ${\overline{\epsilon }}_2$ as described in the text.

Figure 3

Dispersion curves for modes propagating in the plane of the plasmonically coupled quantum metamaterial waveguide. Dotted line, case (i) with the QWs doped to a sheet electron density, $n_s=5\times {10}^{12}{\mathrm{cm}}^{-2}$. Dashed line, case (ii), with the QWs undoped, $n_s=0$. The intersubband transition in the QW induces a region of negative group velocity at energies around $121\mathrm{meV}∕1.84\times {10}^{14}{\mathrm{s}}^{-1}$. Solid line, light line in vacuum.

Figure 4

Absorption coefficient, $\omega$, for radiation traveling in the plane of the plasmonically coupled MQW quantum metamaterial waveguide $(\alpha =2k_x^{\left(i\right)})$. Sheet electron density in quantum wells, $n_s=5\times {10}^{12}{\mathrm{cm}}^{-2}$ (dashed line) and $n_s=0$ (solid line).

Figure 5

Dispersion curves for modes propagating in the plane of the MQW waveguide. Dashed line, QWs doped at a sheet electron density, $n_s=4\times {10}^{11}{\mathrm{cm}}^{-2}$. Solid line, undoped QWs, $n_s=0$. The ISBT induces a region of negative group velocity around $\omega =1.84\times {10}^{14}{\mathrm{s}}^{-1}∕121\mathrm{meV}$. Note that, because the “kink” (at $\omega \sim 125\mathrm{meV}$) in the dispersion curve of the doped sample lies to the left of the (dotted) light line of the cladding material, these modes can be coupled into directly with incident plane-wave beams.

Figure 6

Absorption coefficient of the MQW waveguide $(\alpha =2k_x^{\left(i\right)})$. Sheet electron density, $n_s=4\times {10}^{11}{\mathrm{cm}}^{-2}$.

Figure 7

Mode profiles of the ${\mathrm{H}}^2$ field for the LRP-like mode of the plasmonially coupled quantum metamaterial waveguide (dashed line) and ${\mathrm{TM}}_0$–like mode of the MQW dielectric slab waveguide (solid line). The QWs are undoped in $(n_s=0)$ in both cases, and $\omega =121\mathrm{meV}$. In the plasmonically coupled sample the fields in the LRP mode are laterally compressed by a factor of $\sim 2.6$ over the ${\mathrm{TM}}_0$ type modes supported by the undoped MQW slab waveguide.