Yok S. It is commonly believed, however, that the high losses in metals are prohibitive for laser operation in small metallic cavities. Here we report for the first time laser operation in an electrically pumped metallic-coated nanocavity formed by a semiconductor heterostructure encapsulated in a thin gold film. The demonstrated lasers show a low threshold current and their dimensions are smaller than the smallest electrically pumped lasers reported so far. With dimensions comparable to state-of-the-art electronic transistors and operating at low power and high speed, they are a strong contender as basic elements in digital photonic very large-scale integration.

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Both panels include schematic views of the corresponding core—shell nanoparticles in the top-left part. The observed linear dependence above threshold confirms that the two considered configurations are indeed lasing. At the same time the lasing threshold is reduced by a factor of about 24 in the elongated case with respect to the spherical one values of 0. A detailed analysis of the variation of these enhancement factors with Lrod is provided below.

Of special interest is the comparison of the above numerical results for the spherical case with those reported experimentally by Noginov et al. Equivalently, we did not observe any signature of laser action at those pump intensity levels for the value of the concentration of dye molecules reported by Noginov and coauthors.

This discrepancy can be ascribed to the fact that an additional optical feedback mechanism, beyond the one associated with the LSP supported by a single nanoparticle such as the reillumination among neighboring nanoparticles present in the experimental realization , could be playing an important role in the experimental laser observations reported in ref Alternatively, the active molecules in the experimental configuration could be not uniformly distributed over the whole silica shell volume, but concentrated in a layer of smaller volume close to the metallic core.

Finally, we also note that the difference between our simulations and the experimental data of ref 13 cannot be accounted for in terms of the so-called Purcell effect. Instead, below we propose a novel route, based on tailoring the shape of the nanoparticles, which allows reaching lasing action at or even below the above-mentioned pump intensity levels using dye molecule concentrations as low as those used in ref To further investigate how lasing action is realized in the considered class of systems, we study how the above-described lasing dynamics is linked to the population inversion of the corresponding lasing transition.

As expected, before the first lasing spike occurs, the averaged population inversion grows almost linearly with time. This corresponds to the regime in which the population of the upper level of the relevant lasing transition is increasing the system is accumulating population inversion , and the whole system effectively behaves as an optical amplifier.

At that time the population inversion becomes large enough so its associated optical gain can overcome all the losses present in the system both radiative and ohmic. This burst, in turn, leads to a significant depletion of the population inversion a significant amount of the upper-level population of the laser transition decays via stimulated emission , leading to a dramatic drop of the laser signal. After that, it starts a subsequent recovery of the population inversion, until, again when enough population inversion is accumulated , a second spike of the laser signal occurs, accompanied by the corresponding drop in the population inversion.

This series of bursts and subsequent drops of the population inversion takes place sequentially for larger times smoother spikes and drops of the lasing signal and population inversion are obtained until the steady state of the laser is finally reached. This distribution starts changing quickly once we enter into the lasing regime. Specifically, after the lasing onset, the regions of high electric-field intensity are the ones that experience faster depopulation of the lasing transition these regions feature enhanced stimulated emission rates , while regions with low intensities retain most of the population inversion they accumulated before lasing action starts.

As shown, as time grows, the population inversion distribution starts increasingly resembling the complementary profile of the above-mentioned LSP field profile we find maxima of the population inversion at the minima of the field and vice versa. This particular population inversion distribution settles down in the steady state, giving rise to a subwavelength-scale highly nonuniform gain distribution whose spatial average leads to the effective lasing response of the system for long times.

This behavior can be seen as a novel instance at the subwavelength scale of the well-known lasing processes observed in traditional active systems. Both magnitudes have been normalized to the corresponding values for the spherical configuration.

However, for values of Lrod greater than 20 nm, the lasing threshold becomes much less sensitive to elongation. We believe this result could have a significant influence for further engineering and optimization of spasers based on metallic nanoparticles.

Figure 5 a Simulation results for the steady-state values of the lasing threshold blue squares, left axis and the slope efficiency red squares, right axis as a function of the nanoparticle elongation Lrod.

The same volume of the active shell is assumed for all Lrod values. The panel also includes the results for two alternative conditions to the same-volume condition applied to the shell: green squares correspond to assuming the same shell thickness for all Lrod values, whereas black squares correspond to imposing the same molecule number for all elongations.

Much in the same way as occurs in conventional macroscopic laser sources, 59 we expect that the above-described numerical results can be qualitatively understood in terms of the temporal and spatial light confinement properties of the studied structure. In particular, we expect the quality factor of the longitudinal LSP supported by the nanoparticle describing the temporal confinement properties of the system and the fraction of field energy residing in the gain medium the so-called confinement factor in conventional laser rate equation approaches 59 are the key parameters to account for the evolution of the system lasing characteristics with elongation.

To obtain specific analytical expressions of the dependence of both the laser threshold and the slope efficiency on these two magnitudes, we apply a coupled-mode theory CMT analysis to the problem. As seen, although CMT cannot accurately reproduce the actual values obtained in the full simulations, it does capture the overall dependence of the lasing characteristics on Lrod including the location of the optimal elongation value in each case.

In view of the expression for S provided above, this result implies that the observed nonmonotonic dependence of the slope efficiency on the nanoparticle elongation relies on the evolution with Lrod of the quality factor Q. This analysis also provides insight into the dependence on Lrod of the lasing threshold for large elongation values.

Finally, we note that the above CMT generalizes previous analytical approaches based on quasi-static analytical descriptions. Black squares in the same panels correspond to similar CMT calculations but now assuming the same number of dye molecules for all elongation values the concentration of molecules is the same in all cases. Despite the relative variation in the magnitude of the slope efficiency and the lasing threshold which can be ascribed to the difference in gain medium volume among configurations , the overall trend of the lasing characteristics is maintained, which supports the general character of our findings.

Conclusions In conclusion, by using detailed simulations based on a time-domain generalization of the FEM method, we have analyzed the spatio-temporal dynamics of lasing action in spasers based on core—shell metallic nanoparticles. We have particularly focused on studying how the lasing characteristics of this class of structures are influenced by the nanoparticle shape.

We have found that both the lasing threshold and the slope efficiency of conventional spherical spasers can be significantly improved simply by elongating the nanoparticle. In this context, we have also found that the enhancement of the laser characteristics is maintained across a broad range of elongation values.

Moreover, we have used an analytical coupled-mode theory to explain these findings in terms of the spatial and temporal light confinement properties of the LSP modes supported by the nanoparticles. We expect this work to stimulate further theoretical and numerical investigations on laser light generation assisted by localized surface plasmons and particularly on how nanoparticle shape optimization can be used to tailor lasing emission at the subwavelength scale.

In particular, future work includes the extension of the reported time-domain semiclassical approach to a fully quantum-mechanical model able to account for quantum confinement and electron mean free path effects. Due to the versatility of metallic nanoparticles in a number of contexts beyond lasing, we believe this work could be of relevance across a broad spectrum of different areas, including molecular sensing, photovoltaics, nanoscale microscopy, and emerging quantum technologies.

Within the metallic regions of the system, P r, t is computed through the conventional Drude—Lorentz form. These contributions arise from the stimulated absorption Pa r, t and emission Pe r, t of photons in the gain medium. Thus, by solving the coupled set of nonlinear equations given by eqs 1 — 6, we obtain the whole spatio-temporal dynamics of the studied systems, including all the characteristics of their eventual laser emission.

In order to reduce the computational requirements of this problem, we introduce the two additional steps summarized in the main text. We also assume that a similar expansion holds for P r, t. These expansions enable tracing the fast optical oscillations out of the problem, and, consequently, reduce the total simulation time. We have checked numerically that our results do not depend on the amplitude or duration of that seed excitation.

Second, we rewrite eq 1 in the so-called weak form. This yields

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## Lasing in metallic-coated nanocavities

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## Spatio-temporal Modeling of Lasing Action in Core–Shell Metallic Nanoparticles

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