Theory of Light Amplification in Active Fishnet Metamaterials

We establish a theory that traces light amplification in an active double-fishnet metamaterial back to its microscopic origins. Based on ab initio calculations of the light and plasmon fields we extract energy rates and conversion efficiencies associated with gain and loss channels directly from Poynting’s theorem. We find that for the negative refractive index mode both radiative loss and gain outweigh resistive loss by more than a factor of 2, opening a broad window of steady-state amplification (free of instabilities) accessible even when a gain reduction close to the metal is taken into account.

Figure 1

Schematic of an active plasmonic metamaterial illuminated by an incident plane wave of intensity $I$ that creates a power flux into the volume $V$ through the surface $A_{\mathrm{z}}$. According to Poynting’s theorem [see Eq. (1)] the electromagnetic energy $U$ inside the volume changes due to energy transfer into and out of various loss and gain channels. Note that for normal incidence onto an infinite film, the flux between neighboring volumes cancels.

Figure 2

Conversion efficiencies over pump duration ${\tau }_p$ for gap sizes of $0–10\mathrm{nm}$ (thick lines: 5 nm). (a) Left axis: total absorption efficiency $\alpha$ (dash-dotted line), external pump efficiency $\alpha {\eta }_g$ (dashed line), and dissipative efficiency $\alpha {\eta }_f$ (solid line); absorption coefficient $A$ (squares) obtained from ATR calculations; right axis: total inversion $⟨N_2-N_1⟩$ (dotted line) of the gain material. (b) Pump intensity $I_p$ over ${\tau }_p$ indicating the switch-off times (dashed lines). (c) Spatial inversion profiles, $[N_2(\mathbf{r})-N_1(\mathbf{r})]/N$, for four different pump times (i)–(iv): (top row) $x\mathrm{\text{−}}y$ cross section in the middle between the fishnet metal films; (bottom row) $x\mathrm{\text{−}}z$ cross section through the centers of the holes.

Figure 3

(a) Rate dynamics during probe process for 5 nm gap in amplifying regime: Gain rate ${\Gamma }_g$ (dashed line), dissipative-loss rate ${\Gamma }_f$ (solid line), outflux rate $\Lambda$ (dash-dotted line), and energy decay rate ${\Gamma }_t$ (dotted line). (b) Probe pulse intensity $I_s$ (dashed line) and energy $U$ inside the volume (solid line) over time, with highlighted regimes of continuous excitation (CE) and free decay (FD). (c),(d) Extracted rates over total inversion in the gain medium for gap size $0–10\mathrm{nm}$ (thick lines: 5 nm) under CE (c) and in FD (d) [colors as in (a)].

Figure 4

Absorption ($A$) and transmission spectra ($T$) for pump times ${\tau }_p=2.7–180\mathrm{ps}$ (lighter to darker shades, black arrows) for 5 nm gap size. The $Q$ factor calculated from the width of the transmission resonance is given for ${\tau }_p=180\mathrm{ps}$. Inset: real ($n_{\mathrm{eff}}^{\prime }$) and imaginary parts ($n_{\mathrm{eff}}^{\prime \prime }$) of the effective refractive index retrieved from the complex ATR spectra.