Model for describing plasmonic nanolasers using Maxwell-Liouville equations with finite-difference time-domain calculations

We present a theoretical study of lasing action when plasmonic metallic structures that show lattice plasmon resonances are embedded in a gain medium. Our model combines classical electrodynamics for arrays of gold nanoparticles with a four-level quantum Liouville model of the laser dye photophysics. A numerical solution was implemented using finite-difference time-domain calculations coupled with a finite-difference solution to the Liouville equation. A particular focus of this work is the influence of dephasing in the quantum dynamics on the emission intensity at the threshold for lasing. We find that dephasing in the quantum system leads to reduced lasing emission, but with little effect on the long-term population inversion. Both electronic and vibrational dephasing is considered, but only electronic dephasing is significant, with the fully dephased result appearing for dephasing times comparable to plasmon dephasing ($\sim 10$ fs) while fully coherent results involve $>100$ ps dephasing times as determined by the rate of stimulated emission. There are factor-of-2 differences between the Maxwell-Liouville results (greater emission intensities and narrower widths) compared to the corresponding results of rate-equation models of the dye states, which indicates the importance of using the Maxwell-Liouville approach in modeling these systems. We also examine rate-equation models with and without constraints arising from the Pauli exclusion principle, and we find relatively small effects.

Figure 1

Schematic representation of the nanoparticle array laser.

Figure 2

Scheme showing the energy-transfer process for the optically excited four-level gain medium coupled to lattice plasmons in nanoparticle arrays: The dashed lines are for spontaneous transitions and continuous lines for stimulated transitions.

Figure 3

(a) Extinction spectra of a passive structure. (b) Emission spectra of an active structure. (c) Time dependence of the occupation densities of the gain medium. (d) Time dependence of the normalized population inversion. The input pulse energy for these calculations is equal to 60 $\mathrm{mJ}{\mathrm{cm}}^{-2}$, corresponding to the threshold for lasing in this model.

Figure 4

Comparison between the impact of various types of dephasing on emission. Plots of (a) full emission profile and (b) zoomed version of stimulated emission peak. Red solid line, without any kind of dephasing; blue solid line, complete dephasing; green dotted line, only vibrational dephasing; orange dotted line, only electronic dephasing. Threshold for an input energy is equal to 60 $\mathrm{mJ}{\mathrm{cm}}^{-2}$.

Figure 5

Comparison between the impact of various types of dephasing on steady-state population inversion as a function of time. (b) Plot of the first 30 fs of the dynamics. Red solid line, without any kind of dephasing; blue solid line, complete dephasing; green dotted line, only vibrational dephasing; orange dotted line, only electronic dephasing. The input pulse energy is 60 $\mathrm{mJ}{\mathrm{cm}}^{-2}$, which is just above the laser threshold.

Figure 6

Plot of stimulated emission peak (a) for different vibrational dephasing time scales, with constant 2 fs electronic dephasing, and (b) for different electronic dephasing time scales, with constant 200-fs vibrational dephasing. In the inset, the curves are ordered from top to bottom, going from 200 ps (top) to 2 fs (bottom).

Figure 7

(a) Stimulated emission peak and (b) steady-state condition of population inversion. The emission profile is normalized to the same input energy.