Limitations of Particle-Based Spasers

We present a semiclassical analytic model for spherical core-shell surface plasmon lasers. Within this model, we drop the widely used one-mode approximations in favor of fully electromagnetic Mie theory. This allows for incorporation of realistic gain relaxation rates that so far are massively underestimated. Especially, higher order modes can undermine and even reverse the beneficial effects of the strong Purcell effect in such systems. Our model gives a clear view on gain and resonator requirements, as well as on the output characteristics that will help experimenters to design more efficient particle-based spasers.

Figure 1

Physical model of the gain medium and loss processes for a spaser that operates with emitters and the dipolar resonances of a metal sphere. (a) The gain-medium model considers dipolar emitters ($\overrightarrow{d}$) in proximity to a lossy plasmonic resonator, specifically the decay rates into radiative (${\gamma }_{\mathrm{rad}}$) and nonradiative (${\gamma }_{\mathrm{nrad}}$) channels or into different modes like off-resonant higher order modes and the resonator mode (${\gamma }_{\mathrm{res}}$). (b) The emitters are randomly but homogeneously incorporated in a shell around the sphere of thickness $d_{\text{shell}}$ (between the radii $R$ and $R_2$) except for the emitter-free spacing layer (gray) of thickness $d_{\mathrm{free}}$ ($R$ to $R_1$). The part of the shell that hosts emitters is treated as a series of layers (with index $l$ and accordingly with individual decay rates ${\gamma }_{\mathrm{res},l}$ and ${\gamma }_{\mathrm{tot},l}$) to account for radial changes in the inversion.

Figure 2

Angular- and orientation-averaged decay rates of a dipolar emitter as a function of layer $l$ inside the shell (equivalent to the distance to the sphere) and quality factors $Q_{\mathrm{res}}$ of higher order modes in the gold nanosphere. The emission frequency of the emitter is resonant with the sphere’s dipole modes. All rates are normalized to the corresponding vacuum decay rate ${\gamma }_0$. (a) The red curve shows the (relative) total decay rate ${\overline{\gamma }}_{\mathrm{tot}}/{\gamma }_0$. The sparsely dashed black curve shows the nonradiative decay rate ${\overline{\gamma }}_{\mathrm{nrad}}/{\gamma }_0$ that almost perfectly overlaps with ${\overline{\gamma }}_{\mathrm{tot}}/{\gamma }_0$. The green curve shows the decay rate into a single spaser mode $\mathrm{\Gamma }={\overline{\gamma }}_{\mathrm{res}}/{\gamma }_0$, i.e., the Purcell factor. The tighter dashed black line shows the nonradiative part of $\mathrm{\Gamma }$, showing also that $\mathrm{\Gamma }$ is almost exclusively nonradiative. (b) Ratio of the decay rate into the spaser mode to the total decay rate ${\overline{\gamma }}_{\mathrm{res}}/{\overline{\gamma }}_{\mathrm{tot}}$, i.e., the $\beta$ factor. (c) Quality factors of the first 40 multipolar Mie modes. Higher order modes have $Q_{\mathrm{res}}$ comparable with that of the dipolar resonance. (d) $Q_{\mathrm{res}}$ as a function of frequency. The resonances lie very close to each other; i.e., they condensate [27, 36].

Figure 3

Stationary solution of the rate equations: (a),(b) without ($d_{\mathrm{free}}=0.1\mathrm{nm}$) and (c),(d) with a 2 nm spacing layer between the emitters and the sphere surface. (a),(c) Plasmon number $n$ in the resonator mode of a core-shell particle as a function of the pump rate per emitter (double-logarithmic plot) for three different emitter densities in the shell corresponding to $1{\rho }_{\min }$, $3{\rho }_{\min }$, and $25{\rho }_{\min }$ (orange, red, and purple lines, respectively) with ${\rho }_{\min }\approx 1.05\times 10^{20}{\mathrm{cm}}^{-3}$ (see text). The horizontal lines mark the threshold population numbers [solid, $n=1$; dashed, $n={\overline{\beta }}^{-1}$ with ${\overline{\beta }}^{-1}\approx 500$ in (a) and ${\overline{\beta }}^{-1}\approx 7$ in (c)]. (b),(d) Rate of absorbed or gained plasmons ($D_l{\overline{\gamma }}_{\mathrm{res},l}$) per layer $l$ as a function of the excitation rate corresponding to the red curves with $3{\rho }_{\min }$ in (a) and (b).