Lossless propagation of PT graphene plasmons

Abstract

Graphene supports surface plasmon polaritons (SPPs) with extreme field confinement and electrical tunability, but these waves are typically short-lived due to ohmic loss in the sheet. We show that embedding graphene in an active dielectric can counteract this loss and we derive closed-form design rules to do so, based on gain-assisted plasmonics and plasmonic amplification concepts. Specifically, from the full Maxwell model of a conductive sheet we obtain (i) the exact gain required for lossless plasmon propagation, and (ii) a second critical gain that marks the $\mathcal{PT}$-symmetric threshold, the exceptional point separating propagating and forbidden SPP regimes. The formulas are expressed directly in terms of the complex conductivity of graphene and the surrounding media, making them easy to evaluate and implement. We verify the theory with full-wave eigenmode calculations (COMSOL), showing dispersion and attenuation/amplification trends with and without gain for our plasmonic structures, finding a practical route to engineer long-range, tunable, lossless graphene plasmonics and to map/target non-Hermitian operating phases for device design in single- and double- layer graphene surfaces.

Figures (6)

• Figure 1: (a) Frequency dependence of the parity--time symmetric permittivity parameter $\varepsilon_\mathcal{PT}$. (b) Dependence of $\varepsilon_\mathcal{PT}=-\Im(\varepsilon_2)$ on $\varepsilon_d=\Re(\varepsilon_2)$ for three representative frequencies.
• Figure 2: Dispersion relation $q(f)$ of a surface plasmon in a single graphene layer embedded in an active dielectric enviroment.
• Figure 3: Propagation length $L(q)$ of our graphene plasmonic systems. We can see that ohmic losses our counterbalanced in the presence of an active dielectric environment leading to amplifying the propagation distance of the system. Here the examined cases are for SLG and DLG surfaces.
• Figure 4: Spatial distribution of the electric field $E_y$ component of graphene plasmons at 42 THz. The color map shows $E_y$ (V/m). The overlaid arrows represent the total electric field vectors, highlighting the oscillatory near-field pattern localized around the graphene sheet. On the left panel we see attenuation, while on the right panel the active dielectric enhances the mode leading to lossless propagation.
• Figure 5: Spatial distribution of the electric field $E_i,\space i = x,y$ component of graphene plasmons. The color map shows $E_i$ (V/m). The overlaid arrows represent the total electric field vectors, highlighting the oscillatory near-field pattern localized around the graphene sheet. In 5a(5c) we see attenuation at the acoustic(optical) mode of DLG in the absence of a active dielectric environment , while in 5b(5d) the active dielectric enhances the modes leading to lossless propagation.