Ultralong-range plasmonic waveguides using quasi-two-dimensional metallic layers

We calculate the bound plasmonic modes of a “quantum metamaterial” slab, composed of multiple quasi-two-dimensional electron gas (Q2DEG) layers, whose thickness is much smaller than the optical wavelength. For the first order transverse magnetic optical and surface plasmonic modes, we find propagation constants which are independent of both the electron density and scattering rates in the Q2DEGs. This leads to extremely long propagation distances. In a detailed case study of a structure comprising a slab of $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ multiple-quantum-well (MQW) material, we find propagation lengths of hundreds of millimeters. In addition, the electric field enhancement associated with the plasmonic resonance is found to be sufficient to induce the condition of “strong coupling” between the slab modes and the intersubband transitions in the MQWs.

Figure 1

A schematic representation of a Q2DEG slab (dielectric constant ${\epsilon }_2$), of thickness $d$, surrounded by two semi-infinite dielectric layers with real and positive dielectric constant ${\epsilon }_1$.

Figure 2

The (a) magnetic and (b) electric fields of the symmetric LRP mode supported by a slab of Q2DEG material with a dielectric response ${\mathbf{\epsilon }}_2$ for comparison with the (c) magnetic and (d) electric fields of the asymmetric SRP modes. $E_x$ is the component of the electric field which is parallel to the two-dimensional electron gas. The asymmetric SRP mode has a large fraction of its $E_x$ field component in the slab, but the symmetric LRP has its $E_x$ screened out of the slab by the charges in the Q2DEG, dramatically cutting absorption losses.

Figure 3

Schematic of the prism coupling scheme used in the TfM calculation. The region above the slab, of dielectric tensor ${\mathbf{\epsilon }}_2$, is now bounded by a coupling layer, of thickness $d_1$ and isotropic dielectric constant ${\epsilon }_1$. The layer above that is the semi-infinite prism coupling layer, typically a semiconductor substrate, many wavelengths thick and with an isotropic dielectric constant ${\epsilon }_3>{\epsilon }_1$. ${\theta }_0$ is the angle of incidence from the prism coupler. In the text, we assume that $d_2$ is equal to $d_1$ and we have added a thin capping layer of GaAs to the design to ensure long-term chemical stability.

Figure 4

Mode profiles of the $H^2$ field as a function of doping densities for the MQW described in the text: (a) $n_s={10}^{13}{\mathrm{cm}}^{-2}$ and (b) $n_s=2.5\times {10}^{12}{\mathrm{cm}}^{-2}$. Dashed lines show the position and width of the MQW slab. In (a), the fields inside the MQW are decaying fields corresponding to a LRP mode, while in (b), the field distribution peaks inside the MQW slab correspond to those in a ${\mathrm{TM}}_0$-like mode.

Figure 5

Calculated propagation distances as a function of free-space wavelength for the $600\mathrm{nm}$ thick MQW slab described in the text. 2D doping densities: (bold) $n_s={10}^{13}{\mathrm{cm}}^{-2}$, (dashed) $n_s=7.5\times {10}^{12}{\mathrm{cm}}^{-2}$, (dotted) $n_s=5\times {10}^{12}{\mathrm{cm}}^{-2}$, and (dot-dash) $n_s=2.5\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 6

(a) Dispersion curves for the ${\mathrm{TM}}_0$-ISBT polaritons as a function of incidence angle of incoming light in the prism-coupling scheme of Fig. 3; ${\mathrm{TM}}_0$ mode light is the bold line, and ISBT light is the dashed line. (b) Reflection spectrum from the prism/coupling-layer interface at the incidence angle $({\theta }_0=64.75°)$. When the ${\mathrm{TM}}_0$ slab mode and the ISBT have the same energy, the two excitations hybridize into two new polariton admixtures whose splitting $(\sim 70\mathrm{meV})$ is much larger than either of their linewidths. This is an example of strong coupling behavior.