Emission dynamics and spectrum of a nanoshell-based plasmonic nanolaser spaser

TL;DR

This work presents a time-domain, self-consistent model of a nanoshell plasmonic nanolaser with gain confined to the core and a thin metal shell. By coupling Drude metal dynamics with a two-level gain medium under a rotating-wave approximation, the authors identify a lasing threshold defined by det[\mathbf{A}(\widetilde{N})]=0 and show that at threshold the emission aligns with a plasmon resonance, while above threshold the system exhibits a frequency pull-out $\omega_{em}=\omega-\Omega(\omega)$ and settles into a steady-state lasing with a shifted frequency. The maximal emission spectrum is computed by sweeping the carrier frequency, revealing a strongly asymmetric, one-sided spectrum that broadens linearly with gain and can be tuned across the visible range via the nanoshell aspect ratio $\rho$. The study also discusses the limitations of assuming independent modes and the potential implications for linewidth, suggesting that the actual spectrum may be narrower due to mode competition. Overall, the results provide a comprehensive, physics-based map of the emission dynamics and spectral properties of nanoshell-based spasers under realistic, time-dependent conditions, with clear pathways to tunable nanoscale light sources.

Abstract

We study theoretically the emission and lasing properties of a single nanoshell spaser nanoparticle, or plasmonic nanolaser, made of an active core (gain material) and a plasmonic metal shell. Based on an analytical framework coupling together time-dependent equations for the gain and the metal, we calculate the lasing threshold with the help of an instability analysis. We characterize the regime under the threshold, where the nanoshell behaves as an optical amplifier when excited by an incident probe field. We then investigate in depth the non-linear lasing regime above the threshold, under autonomous conditions (free lasing without external drive), by computing the system's dynamics both in the transient state and in the final steady state. We show that at threshold, the lasing starts at one frequency only, usually one of the plasmon resonances of the nanoshell; then as the gain is further raised, the emission widens to other frequencies. This differs significantly from previous findings in the literature, which found only one emission wavelength above threshold. We proceed to calculate the complete (maximal) emission spectrum of the nanolaser as well as its emission linewidth, both of which are evidenced to be affected by unusually strong frequency shifts (pull-out) effects. We find that the nanolaser emission is highly asymmetrical spectrally and only occurs on one side (high-frequency) of the plasmon resonance. Finally, we show that the spectral position of the emission line can be tuned across the whole visible range, by changing the geometrical aspect ratio of the nanoshell.

Figures (12)

• Figure 1: A spherical nanoshell particle, with a gain medium filling the core and a metallic shell, placed into an external medium.
• Figure 2: (a-c): Plots of the real part of the eigenvalue $\kappa_3$ in the spectrum of the geometry matrix ${\bf A}$, respectively below, at, and above threshold. The real part of the eigenvalue is always strictly negative below threshold ($G<G_\text{th}$), exactly null at threshold for $\hbar \omega=\hbar\omega_\text{th}\simeq 2.813$ eV and $G=G_\text{th}\simeq 0.135$, and then strictly positive over a range of frequencies for $G>G_\text{th}$. Other eigenvalues (not shown) always keep a negative real part. (d-e): Real and imaginary parts of the determinant $\det({\bf A})$, respectively below, at and above threshold. The situation shown in (e), where both parts cancel simultaneously defines the value for the frequency of lasing at threshold $\omega_\text{th}=\omega_\text{res}$ and the gain value at threshold $G_\text{th}$, according to eq \ref{['eq:lasingcond']}. Vertical lines in all plots are guides for the eye showing the position of $\omega_\text{th}$. Parameters values are (see main text for explanations): $a=10$ nm, $\rho=0.6$, $\hbar\omega_\text{p}=9.6$ eV, $\hbar\gamma = 0.0114$ eV, $\epsilon_\infty/\epsilon_0=5.3$, $\epsilon_\text{b}/\epsilon_0=2.1316$, $\epsilon_\text{e}/\epsilon_0=1.7689$, $\mu=10$ D, $\hbar\Delta= 2\hbar/\tau_2 =0.15$ eV, $\tau_1=5\,\tau_2$, $\widetilde{N} = 1$.
• Figure 3: Lasing threshold conditions for a gain-doped silver nanoshell, as a function of the aspect ratio $\rho$. (a) Threshold gain value $G_\text{th}$; (b) Lasing frequency at threshold $\omega_\text{th}$. Colors on the nanoparticle drawings correspond to the lasing frequency at threshold for $\rho = 0.4, 0.6, 0.8$. Parameters other than $\rho$ are the same as in Fig. \ref{['fg:eigen']}.
• Figure 4: Numerical solutions for the time evolution of the nanoshell's response below threshold, calculated at a fixed frequency $\hbar\omega=2.811$ eV, first with $G=0$ (no gain) for $t \leq 2$ ps, and then $G=0.25\,G_\text{th}$ for $t > 10$ ps. (a) Population inversion $N(t)$ versus time. Real and imaginary part of the dielectric polarization modes (b) $q_0$; (c) $q_1$; (d) $q_2$ versus time. Parameters are the same as in Fig. \ref{['fg:eigen']}.
• Figure 5: Enhancement of the nanoshell polarizability $\alpha= \alpha' + i \alpha"$ for increasing values of the gain in the core. Parameters are the same as in Fig. \ref{['fg:eigen']}. (a) $G=0$ (no gain); (b) $G=0.35 \,G_\text{th}$; (c) $G=0.75 \, G_\text{th}$.