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

1 Introduction

With the advent of nanotechnology, one key aspect of research efforts in the past 30 years has been to generate, shape and manipulate light at subwavelength scales, much below the diffraction limit. An instrumental physical phenomenon towards this aim is surface plasmon polaritons, which are localized excitons where the electromagnetic field is coupled to electronic oscillations in a metal [1]. If the metallic structure is at the nanoscale, then the associated fields are also generated at that same scale, and depending on cases, they may or may not be able to produce far-field radiation.

However, the use of metals at optical frequencies inevitably comes at the cost of significant Ohmic losses hindering the performance of plasmonic devices. One way to partially circumvent this fundamental issue is to use a gain material (active medium), placed at a distance close enough to the metallic structure: the gain medium can then transfer energy to the metal radiatively and/or non-radiatively. This strategy has proved to significantly improve properties and amplify the responses of device in various applications [2], [3], [4], [5], [6], [7], [8], [9].

When the quantity of gain provided by the active medium is in excess to losses in the plasmonic system, one may enter a regime of nanolasing, i.e. the generation of coherent light at the nanoscale via the stimulated amplification of plasmons. Starting with the seminal concept of the spaser introduced in 2003 [10], quickly followed by the first experimental realizations of lasing spasers in 2009 [11], [12], the field of nanolasers has since been highly active, as is testified by the flurry of reviews published over recent years [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].

Amongst the variety of geometries and schemes described in the literature, plasmonic nanolasers based on metallic nanoparticles (combining localized plasmons with gain within a nanoparticle) are especially attractive [23], due to the ease of mass fabrication of such structures with the help of bottom-up colloidal chemistry and self-assembly techniques [24], [25]. One disadvantage of particle-based designs, however, is the difficulty to geometrically pack the necessary amount of gain to obtain the lasing [23], [26]. In particular, there has been an ongoing debate about the true nature of the pioneering experiments by Noginov et al. [11] (lasing vs. random lasing, or some hybrid situation), and experimental realizations including metallic nanoparticles actually remain scarce [23].

In the wait for an improvement in the experimental situation, theoretical efforts have nonetheless been exploring lasing when nanoparticles are coupled to gain in various geometries: spheres, core-shells, multiple core-shells, ellipsoids …, using more or less refined analytical descriptions, or numerical simulation tools like the finite-element method [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49]. Most of these works were carried out assuming that the electromagnetic response for the materials involved could be faithfully accounted for using standard electrical permittivities; in particular for the gain medium, either a linear, Lorentzian permittivity or a non-linear saturated version were employed. When such an assumption is made from the start, however, it results in leaving out situations where more complex temporal dynamics may deploy. Since the latter is not uncommon in lasers, it is therefore necessary to rather make use of a fully time-dependent description to obtain a full understanding of the lasing regime. A few numerical studies have integrated all time and space-dependent effects using four-level population dynamics for gain carriers locally coupled to Maxwell equations [37], [38], [39], [40]. Powerful as these are, full-wave simulations are not always transparent in terms of understanding the physical mechanisms at work, and a complementary model-based approach is undoubtedly useful.

In a past work [31], we studied the situation of a metallic sphere immersed in an unbounded gain medium within the help of a time and space-dependent model. We proved the existence of a lasing threshold as the onset of an instability arising under zero driving field, starting first with the dipolar mode, and discussed how a mode cascade would subsequently occur due to a spatial hole-burning effect similar to laser physics, triggering many higher multipolar modes into emission. This multi-modal complication linked to the chosen geometry prevented us from studying the complete nanolaser’s dynamics above threshold.

In the present article, we consider another geometry, closer to experiments, namely a single nanoshell where the gain medium is placed inside the core of the nanoparticle and is surrounded by a thin shell of metal (as shown in Figure 1). In this case, as shall be explained, the dipolar is the only one that can emit in the lasing regime, making the analysis easier; based on the same model as earlier, we are then able to provide a full model-based characterization of the nonlinear lasing state for a nanoparticle, unveiling specific novel effects which, to the best of our knowledge, have remained unnoticed in the literature.

Figure 1:

A spherical nanoshell particle, with a gain medium filling the core and a metallic shell, placed into an external medium.

Finally, and before we start presenting the model, a word on vocabulary: in the following, we shall make no distinction between the intrinsically entangled notions of “spasing” and “lasing”. In the specific context of nanoparticle-based plasmonic lasers, these notions are two sides of the same coin: the phenomenon of spasing focuses on the coherent excitation of plasmons in the nanoparticle, while lasing focuses on the emission of photons in the far-field associated with these plasmon oscillations [17], [50]. In this article, we shall mostly use the latter wording, as we will be primarily concerned with the emission of light by the nanoshell particle.

2 Dynamical equations for materials

To ensure that our model is able to capture spatial and temporal variations of the fields, our first step is to formulate dynamical constitutive equations for the materials comprised in the nanoshell, i.e., for the metal and the gain medium. This formalism has been previously described in ref [31], and the interested reader is referred to the Supplementary Material of the present article for detailed derivations (Supplementary Material). We give below a summary of the important steps.

3 Nanoshell model

4 Lasing threshold

Classically, we define the lasing threshold as the minimal quantity of gain G = G _th to be provided to the system, in order to observe the rise of a self-oscillation of the nanoshell [31], [41], i.e., the rise of non-zero fields inside and outside the particle in the absence of an exciting probe field. We thus take E ₀ = 0 and b = 0.

Right at the onset of the self-oscillation, fields are small and therefore the “small-signal” approximation is valid: one can neglect the r.h.s. of equation (23). After a transient of duration ∼τ ₁, N will reach the stationary value N = N ̃ , which is independent of all variables q _i . Then, the geometry matrix becomes a constant matrix A ( N ̃ ) , meaning that equation (22) is now a linear differential equation with generic solutions

where κ _n are the eigenvalues of A ( N ̃ ) and q ̂ n are associated eigenvectors.

Above the threshold, the solution should be exponentially growing (i.e., at least one eigenvalue has a positive real part) as a result of the self-oscillation instability. Below the threshold, the solution should be on the contrary exponentially decaying (i.e., all eigenvalues have a strictly negative real part), as no field should emerge in the absence of external excitation (no self-oscillation). We refer the reader to ref [31] for a similar analysis carried out in the case of a metal sphere immersed in a gain medium.

The lasing threshold is thus characterized by the tipping point where one null eigenvalue appears in the spectrum of A. The lasing condition therefore simply writes:

As this condition implies that both the real and imaginary part of det [ A ( N ̃ ) ] be cancelled simultaneously, it determines both the threshold gain value, G = G _th, and the lasing frequency of the nanoshell at the threshold, ω = ω _th. After some cumbersome calculations (see Supplementary Material for details), condition (26) can be rewritten under the simple form:

which needs to be solved to find the couple (ω _th, G _th).

We note that this condition is the same that we proposed in ref [26], where a more intuitive argument was followed. It is useful to briefly remind of this argument, which proceeded from considering the classical formula for the quasi-static polarizability α for a nanoshell particle [58]:

This polarizability allows to calculate the value of the nanoshell’s total dipolar moment P when it is excited by a probe field E ₀, as

The self-oscillation of the nanolaser is then defined as the situation when this dipolar moment remains finite ( P ≠ 0 ) even if the probe field is made to vanish (E ₀ → 0). This is possible only if the polarizability α becomes singular at the lasing threshold, or in other words, if its denominator cancels out, which is exactly the same as condition (27).

On a rigorous standpoint, the use of the classical expression (28) for polarizability remains unsubstantiated at this stage, but it will be justified fully in the next section “Below threshold”.

It furthermore appears from the above argument on polarizability that the lasing frequency at threshold ω _th is in fact no other than one of the natural plasmon resonance frequencies ω _res of the nanoshell, which are usually also found by cancelling the denominator of α, i.e.,

Two important remarks are in order at this point. Firstly, these natural plasmon resonance frequencies are those of the nanoparticle in the presence of gain inside the core, with the gain level set at G = G _th. Therefore, the actual values found for ω _res, and hence ω _th, will depend on the level of gain and on the positioning of the centerline frequency ω _g of the gain emission spectrum, through the presence of ϵ _g(ω) in the denominator of (28). The optimal situation allowing the lowest lasing threshold is obtained when the gain provides its peak value to one of the resonances, i.e. it has been chosen to be centered exactly on the same frequency. In this case, comparing to equation (30), one has an additional equality:

In the rest of this paper, we will always assume such optimal gain positioning. In this case, it can be shown that ω _res and ω _th are independent of the gain level, and are the same as the resonant frequencies of the nanoshell with zero gain (found by solving equation (27) with ϵ _g = ϵ _b). Note, however, that the resonant frequencies still depend on the nanoshell’s aspect ratio ρ, so that the gain centerline should be adjusted each time ρ is modified, to remain optimal. Effects linked to gain detuning (non-optimal positioning) with respect to the nanoshell resonances are left for future work.

Secondly, nanoshells exhibit two plasmonic resonances, one symmetric (lower frequency) and one anti-symmetric (higher frequency) [1]. In principle, both resonances can be brought to lasing. In this work, we arbitrarily choose to focus on provoking the lasing of the symmetric resonance only, by centering the gain spectrum on the lowest frequency solution of equation (27). For nanoshells with aspect ratios ρ around 0.6, which we will be mostly concerned with, the symmetric resonance is the one necessitating the lowest gain level G _th to cross the lasing threshold. Again, a more general approach of the lasing properties of both resonances, with proper comparisons of their lasing thresholds and intensities etc., is left for future work.

Let us now illustrate our findings about the lasing threshold on a realistic example: we consider a nanoshell of external radius a = 10 nm, internal radius 6 nm, and aspect ratio ρ = 0.6. The shell is assumed to be made of silver, with the following parameters: ℏω _p = 9.6 eV, ℏγ = 0.0228 eV and ϵ _∞/ϵ ₀ = 5.3. The core is made of silica (ϵ _b/ϵ ₀ = 2.1316), doped with gain elements for which we set μ = 10 D, ℏΔ = 2ℏ/τ ₂ = 0.15 eV (close to experimental linewidths observed for dyes), corresponding to τ ₂ ≃ 0.06 ps, and τ ₁ = 5 τ ₂. We assume that the pump rate is much faster than spontaneous emission in the emitters (strong pumping), so that we take N ̃ = 1 . Finally, the external medium is taken to be water (ϵ _e/ϵ ₀ = 1.7689).

Solving equation (27) for the threshold conditions with the above numerical values, we find ℏω _th = ℏω _res ≃ 2.813 eV and G _th ≃ 0.135. (And we take ω _g = ω _th as per the optimal gain condition.) Figure 2 shows the evolution of the eigenvalues of A when G is increased for values below, at and above the threshold: it can be seen that the crossing of the lasing threshold is manifested by the appearance of a positive real part in one eigenvalue.

Figure 2:

Eigenvalue behavior and determinant analysis of matrix A near lasing threshold. (a–c) Plots of the real part of the eigenvalue κ ₃ in the spectrum of the geometry matrix A, respectively, below, at, and above threshold. The real part of the eigenvalue is always strictly negative below threshold (G < G _th), exactly null at threshold for ℏω = ℏω _th ≃ 2.813 eV and G = G _th ≃ 0.135, and then strictly positive over a range of frequencies for G > G _th. Other eigenvalues (not shown) always keep a negative real part. (d–e) Real and imaginary parts of the determinant det(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 ω _th = ω _res and the gain value at threshold G _th, according to equation (26). Vertical lines in all plots are guides for the eye showing the position of ω _th. Parameters values are (see main text for explanations): a = 10 nm, ρ = 0.6, ℏω _p = 9.6 eV, ℏγ = 0.0228 eV, ϵ _∞/ϵ ₀ = 5.3, ϵ _b/ϵ ₀ = 2.1316, ϵ _e/ϵ ₀ = 1.7689, μ = 10 D, ℏΔ = 2ℏ/τ ₂ = 0.15 eV, τ ₁ = 5 τ ₂, N ̃ = 1 .

Figure 3 shows changes in the gain and frequency at threshold when the aspect ratio ρ of the nanoshell is varied from 0.4 to 0.8. Under optimal gain positioning, both ω _th and G _th are functions of ρ only [26]. As is known about the symmetric resonance of plasmonic nanoshells, when ρ is increased (thinner shells), the resonance frequency ω _res redshifts, and so does the lasing frequency at threshold since they are equal according to equation (30). Simultaneously, the gain G _th required to cross the lasing threshold decreases, since the quantity of metal becomes less and less; lasing becomes easier as the Ohmic losses that need to be overcome are progressively reduced.

Figure 3:

Lasing threshold conditions for a gain-doped silver nanoshell, as a function of the aspect ratio ρ. (a) Threshold gain value G _th; (b) lasing frequency at threshold ω _th. Colors on the nanoparticle drawings correspond to the lasing frequency at threshold for ρ = 0.4, 0.6, 0.8. Parameters other than ρ are the same as in Figure 2.

5 Below the lasing threshold

We now consider the behaviour with time of the gain-doped nanoshell below the lasing threshold, i.e., G < G _th.

We place ourselves in the small-signal regime, where gain saturation is negligible and fields remain small with respect to E _sat, which guarantees that the r.h.s. of equation (23) can be approximated to zero. We assume that the nanoshell is excited with an incoming probe field E ₀, taken as a harmonic plane wave with a definite frequency ω and constant magnitude.

5.1 Time-dependent analysis

For simplicity, let us start with considering a situation where the population inversion at the initial time t ₀ is N ( t 0 ) = N ̃ : according to equation (23), N does not evolve through time, i.e., N ( t ) = N ̃ = const . Then, A(N) = A is a constant matrix in time and the differential system (22) is linear. The complete evolution of the system over time is given by the sum of exponentials of equation (25) (usually called “homogeneous” solution), plus a constant term stemming from b (usually called “particular” solution):

(32) q ( t ) = ∑ n = 1 n = 3 q ̂ n e κ n t + q part .

The obvious particular solution is the constant vector:

(33) q part = − A − 1 b .

Since below threshold, all eigenvalues κ _i of the matrix A have a strictly negative real part, the homogeneous part of the solution (32) represents an exponentially-decaying transient response. After it has vanished out, only the constant part q _part remains, which must represent the steady-state response of the nanoshell. This steady state is linearly related to b and therefore proportional to E ₀. Equation (33) can be easily solved numerically, from which all steady-state values for the p _i are found using equations (44)–(47), completing the solution by providing the value of all fields and polarizations in all domains of space.

Now, let us assume that the initial value of the population inversion N ( t 0 ) ≠ N ̃ , which is the general case: then N(t) will have a dynamics of its own, evolving in accordance to (23) over a typical duration ∼τ ₁, until it reaches its final value N = N ̃ . The transient evolution of the nanoshell is then modified slightly with respect to the previous case where we had N ( t ) = N ̃ . But it can be proved that, because now N ( t ) ≤ N ̃ at all times, the transient will still be quickly vanishing. Therefore, the final steady state remains unaffected and still given by equation (33).

Figure 4 illustrates this latter case of the time evolution of the nanoshell below threshold, with N = N(t). We consider again the earlier example of a silver nanoshell with a gain-doped silica core (parameters are the same as in Figure 2). In this case, G _th ≃ 0.1349 and ℏω _th = ℏω _res ≃ 2.8122 eV, as obtained from the previous Section. We show the time evolution of the system computed from a numerical integration of equations (22) and (23), for a fixed frequency ℏω = 2.811 eV. At t = 0, the probe field E ₀ is shone on the nanoparticle, which is assumed to initially have zero fields (q _i = 0) and no population inversion (N = 0). To start with, we assume N ̃ = 0 so that the gain value is set at G = 0. The probe field magnitude is chosen very small to ensure that the systems stays well into the small-signal regime: E ₀ = 10⁻⁸ E _sat. We see that after a short decaying transient due to equation (25), all values for the field mode amplitudes converge to their final values. This gives the steady-state response of the bare nanoparticle in the absence of gain, without any effect of the gain elements in the core. Then at t = 2 ps, conditions are changed: the gain value is set to G = 0.25G _th (sub-threshold level), and N ̃ is set to 1. It can be observed that N(t) increases quickly from 0 to reach its final value N = N ̃ = 1 , while all other variables undergo a decaying transient evolution as explained above, converging to a new final value: this is now the steady state of the nanoparticle in the presence of gain. We note that all final values obtained in the presence of gain (for t ≳ 3 ps) are larger in absolute value than the ones without gain (t ≲ 2 ps). This means that the response of the nanoparticle is amplified with the help of gain, as compared to the situation without gain.

Figure 4:

Numerical solutions for the time evolution of the nanoshell’s response below threshold, calculated at a fixed frequency ℏω = 2.811 eV, first with G = 0 (no gain) for t ≤ 2 ps, and then G = 0.25 G _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 Figure 2.

Summarizing, our conclusions on the response of the gain-doped nanoshell below the lasing threshold are the following: (a) In the presence of an excitation of amplitude E ₀, and in the small-regime signal, the response of the nanoshell is linear, proportional to E ₀ and synchronized with it, i.e., oscillating with the same frequency ω; (b) If there is no external excitation (E ₀ = 0), the steady state of the nanoshell is null, i.e., there is no self-oscillation (as expected). These facts are often taken as granted in the literature on active nanoparticles below threshold. Our point here, however, is to lay out the proper mathematical justification supporting them, as they will soon be challenged when we switch to studying the situation above the lasing threshold.

5.2 Steady-state polarizability

The steady-state response of the nanoshell under external excitation can in fact be expressed in a much more familiar way, if one expresses the particle’s external scattered field p ₃ by solving equations (44)–(47). After some calculations (see Supplementary Material for details), one finds that p ₃ is proportional to E ₀:

(34) p 3 = 1 4 π a 3 α E 0 ,

where the polarizability α has the same classical expression as shown in equation (28). In other words, this proves that under the threshold, the steady-state response of the nanoshell is simply given by the usual formula for polarizability, with the permittivites ϵ _m and ϵ _g given by their steady-state values from equations (10) and (13). By legitimating the use of the classical quasi-static polarizability, this also legitimates in retrospect the intuitive argument that the lasing threshold can be understood as the vanishing of the denominator of α in equation (28), as was done in the previous section.

It is interesting to plot the evolution of the polarizability α(ω) of the nanoshell for increasing values of the gain levels G under the threshold, see Figure 5. It can be seen that below threshold, the effect of gain when it is increased, is simply to enhance the natural plasmon response of the nanoshell and improve its resonance quality.

Figure 5:

Enhancement of the nanoshell polarizability α = α′ + iα″ for increasing values of the gain in the core. Parameters are the same as in Figure 2. (a) G = 0 (no gain); (b) G = 0.35 G _th; (c) G = 0.75 G _th.

This intuitive approach of describing gain-doped nanoparticles below threshold based on their quasi-static polarizability, is actually valid across other geometries beyond nanoshells (as long as a steady state with well-defined permittivities ϵ _m and ϵ _g exists in the final state, which is the generically true). Some of us had already demonstrated this fact in the case of a homogeneous, plasmonic sphere immersed in a gain medium [31]; calculations by us (not shown) also prove that this is true for core–shell nanoparticles with a metal core and gain shell. This justifies in hindsight all calculations that were made in ref [26] about gain-enhanced nanoparticles, based on this assumption. Note also that situations where nanoparticles are too large to be describable only with the quasi-static polarizability (i.e., multipolar modes are required) are studied in detail in ref [54].

Finally, we note that Equation (34) is valid both in the small-signal regime that we studied, but also in the large-signal regime, when |E _g| is comparable to E _sat. In the latter case, one needs to call upon the saturated permittivity (11) for the gain medium, making the polarizability α = α(E _g) nonlinear; Reference [55] has a complete analysis of the behavior of a metallic nanoparticle with gain in this situation.

Therefore, below the lasing threshold, we conclude that, after a short-lived transient, the nanoshell simply acts as an “optical amplifier”, whereby the natural plasmon of the nanoshell can be significantly amplified and improved thanks to the assistance of optical gain. As this amplification regime was already studied in depth in a previous work [26], based on the simpler quasi-static polarizability approach, for various metals, geometries and aspect ratios, we need not delve into it further, and readily move to studying the lasing regime of the nanoshell when gain is raised above the threshold.

6 Above the lasing threshold

We now turn to describing the situation of the nanoshell when it is pumped above its lasing threshold, i.e., when G > G _th.

6.1 Time dynamics of the laser emission

As explained in the “Lasing threshold” section, at threshold, an autonomous self-oscillation instability sets in, whereby fields become can become non-zero and start growing even in the absence of any exciting external probe field. The existence of such a sustained, autonomous self-oscillation state is indeed a central concept for lasers in general, and it is therefore crucial that its properties are fully studied and understood in the present case of lasing nanoshells. This is why, contrary to the previous section, we will here be concerned only with the free lasing state of the nanoshell, i.e., when the probe field is zero (E ₀ = 0), discarding any situations of external forcing. (Such situations will be briefly discussed in the conclusion of this article.)

Looking back at Figure 2, we observe that when the gain level G is that at threshold exactly, only the frequency ω _th = ω _res is unstable, but when G is increased above threshold, a wider and wider range of frequencies becomes unstable (i.e., there is a wider range where one eigenvalue of the matrix A has a positive real part). For example, when G = 1.01 G _th, the unstable range lies approximately between 2.6 and 3 eV. Therefore, we expect the lasing instability to grow over a finite range of frequencies.

To check this, and compare to the situation below threshold, we choose the same frequency as in Figure 4 (ℏω = 2.811 eV < ℏω _th), and numerically solve the governing equations (22) and (23) for G = 1.01 G _th. Results are shown in Figure 6.

Figure 6:

Time dynamics of field and polarization amplitudes above the lasing threshold, for a gain value G = 1.01 G _th and a frequency ℏω = 2.811 eV < ℏω _th. A t = 0, the gain is G = 0, then it is set to G = 1.01 G _th at t = 10 ps. Displayed are the real and imaginary part of: (a) q ₀(t); (b) q ₁(t); (c) q ₂(t); (d) p ₃(t), normalized to ϵ ₀ E _sat for polarizations and E _sat for fields. Diverging envelopes correspond to the initial exponential growth computed from equation (25) when N = N ̃ . Once gain saturation effects take place, the signal growth saturates and separates from the exponential envelopes, finally reaching a stable oscillatory state. Parameters are the same as in Figure 2.

(Computational details on how these results were obtained numerically can be found in the Supplementary Material.) We see that the variables q ₁, q ₂, q ₃ first follow an exponential growth (led by the positive eigenvalue in A); then this growth saturates and a stable long-term state is finally reached. We observe that this final state is oscillatory, with all magnitudes |q _i (t)| reaching a constant value; oscillations are purely sinusoidal with the same constant frequency Ω for all variables q _i (t). The value of the frequency is obtained from a Fourier transform of the signals, see Figure 7: ℏΩ ≃ − 1.2 × 10⁻³ eV. As before, from the q _i (t), the field components p _i (t) in all regions of space can be computed through the boundary conditions (44)–(47). Because these conditions are algebraic and linear, we find that all fields p _i (t) follow the same temporal evolution as the q _i (t), namely, an initial exponential growth and a saturation into a final oscillatory state with the same frequency Ω: Figure 6-(d) displays the evolution of p ₃(t), which corresponds to the external field emitted by the nanoshell.

Figure 7:

Fourier analysis of q ₀(t) in the final oscillatory state shown in Figure 6, yielding a single oscillation frequency ℏΩ ≈ −0.0012 eV. Similar analyses conducted on q ₁(t), q ₂(t), or p ₃(t), result in the same value for Ω.

Thus, the nanoshell is able to maintain a stable emitted field |p ₃| ≠ 0, proving that it is indeed acting a nanolaser above the lasing threshold; and it is capable of doing so in an autonomous way, i.e., it reaches a free lasing state without any external excitation.

Figure 8 shows the intensity of emission radiated by the nanoshell in the lasing state I _em(t) = |p ₃(t)|² at the same gain level and frequency (G = 1.01 G _th and ℏω = 2.811 eV), normalized to the saturation intensity I _sat = |E _sat|². The emitted intensity can be seen to pick up gradually and then reach a constant steady-state value, at the same time as the q _i and p _i reach their final oscillatory state. We also plot the evolution of the population inversion N(t): initially, all fields are small and therefore, N quickly comes close to N ̃ . But as the emitted intensity I _em increases, alongside with all fields inside the nanoshell, the saturation term Im q 0 p 0 * / ( n ℏ ) in equation (23) becomes significant, leading to a decrease in the value of N(t) (corresponding to a depletion in the higher level of the two-level emitters due to stimulated emission). This decrease, in turn, limits the increase in the nanoshell’s emitted intensity. Finally, an equilibrium is found and N(t) stabilizes to a steady-state value where losses are exactly compensated by the pumping, as happens in a conventional laser. As expected (also from conventional lasers), this equilibrium is non-linear in nature due to the form of the saturation term Im q 0 p 0 * / ( n ℏ ) .

Figure 8:

Time dynamics of the population inversion and emitted intensity of the nanoshell above the lasing threshold, for a gain value G = 1.01 G _th and a frequency ℏω = 2.811 eV < ℏω _th (same conditions as in Figure 6). A t = 0, the gain is G = 0, then it is set to G = 1.01 G _th at t = 2 ps. Immediately, N rises due to pumping, up to its maximal allowed value N = N ̃ = 1 . Then N remains constant while the lasing instability slowly grows. Once the emitted intensity I _em picks up, saturation terms come into play and decrease the value of N, until a steady state is finally reached for both quantities.

Let us now explore other frequencies in the unstable range. Figure 9 shows results for two other situations, for a frequency ω = ω _th = ω _res and for a frequency ℏω = 2.813 eV > ℏω _th (always keeping G = 1.01 G _th as before). For both frequencies, all polarization and field modes q _i (t) and p _i (t) are found to follow the same generic trends as just before, i.e., an exponential growth saturating into a final state. (For conciseness, only the external field p ₃(t) has been plotted in the Figure.) For ℏω = ℏω _th = ℏω _res = 2.8122 eV, no oscillations are observed, i.e., the final state has steady values with Ω(ω _th) = 0, see Figure 9-(a) and (b). For ℏω = 2.813 eV > ℏω _th, as shown in Figure 9-(c), the final state is found to be oscillatory again with a frequency ℏΩ ≃ + 8 × 10⁻⁴ eV (obtained by Fourier transform, not shown).

Figure 9:

Time dynamics of the emitted field p ₃(t), emitted intensity I _em(t) = |p ₃(t)|² and population inversion N(t) above the lasing threshold (G = 1.01 G _th) for two different frequencies: (a) and (b) ℏω = ℏω _th = 2.812 eV; (c) and (d) ℏω = 2.813 eV > ℏω _th. Parameters are the same as in Figure 2.

We thus observe that the value of Ω depends on the frequency, i.e. Ω ≡ Ω(ω), with Ω → 0 when ω = ω _th, and a sign change when ω > ω _th or ω < ω _th (resp. Ω > 0 or Ω < 0).

In light of these findings, we can conclude that in the final lasing state above threshold, the variables in the system take on the following generic form:

(35) q i ( t ) = q i ss e i Ω t ,

(36) p i ( t ) = p i ss e i Ω t ,

(37) N = N ss ,

(38) I em = I em ss = | p 3 ss | 2 ,

where all quantities with the superscript ‘ss’ for “steady state” are constants and the frequency Ω depends on ω. From (35), we deduce that the vector q = [ q 0 , q 1 , q 2 ] T defining the electromagnetic state of the system, as introduced in equation (62), has the following form in the final state:

(39) q ( t ) = Q e i Ω t ,

where Q is the constant vector q 0 ss , q 1 ss , q 2 ss T .

7 Maximal emission spectrum of the nanolaser

With the help of the results obtained in the previous Section, we are now in a position to build the spectrum of emission of the nanoshell laser above its lasing threshold.

The procedure to calculate the spectrum of emission, is to compute the whole set of emission frequencies ω _em = ω − Ω of the nanoshell, by scanning the value of the parameter ω over the unstable range and finding the value of Ω(ω) as explained above; then, for each value found for ω _em, one calculates the value of the steady-state intensity of emission I em ( ω em ) = | p 3 ss | 2 according to equation (38), by picking the final value p 3 ss as obtained from the corresponding numerical solution.

We emphasize that the intensity I _em(ω _em) obtained for any ω _em frequency is here calculated by taking each of them separately from all others, i.e., without considering the effect of simultaneous emission in other ω _em frequencies. In other words, we are assuming that each frequency is able to lase independently from others, and is free to use all of the available gain brought by external pumping at will, without competition from other frequencies. To which extent this assumption may hold true will be considered in the “Discussion” section; what can be said for certain at this stage is that under this independency assumption, all ω _em frequencies in the spectrum are able to lase up to their full capacity-while all frequencies outside this spectrum cannot lase at all (since they are not unstable by self-oscillation). Therefore, the spectrum that we calculate here really represents the maximal spectrum of emission of the nanolaser, in the sense of the widest possible that can be expected, with maximal possible intensities for all unstable frequencies.

In Figure 10-(a)–(c), we plot this maximal spectrum of emission I _em(ω _em) as obtained from the procedure exposed above, for a nanoshell with aspect ratio ρ = 0.6 and for three increasing gain levels: G = 1.25 G _th, G = 1.50 G _th, and G = 1.75 G _th. One observes that the emission lines are relatively thin, and asymmetric in shape, with the intensity peak located at the lower-frequency edge [note that this holds under the condition of optimal gain positioning, see equation (31)]. We also find that this sharp, lower-frequency edge of the emission line (with the associated intensity peak) corresponds precisely to the lasing frequency at threshold ω _th = ω _res.

We therefore come to the following striking conclusion: due to the frequency displacement Ω, the resulting spectrum of emission is one-sided with respect to the plasmon resonance, i.e., the emission is strictly restricted to frequencies such that

The numerical reason for this is because the values we compute for Ω(ω) are always such that Ω ≤ ω − ω _res; which means that, in accordance with equation (41), we always have ω _em − ω _res ≥ 0 for emission frequencies.

Figure 10:

Emission spectrum of the nanoshell I _em as a function of the emission frequency ℏω _em in the lasing steady state, for increasing gain levels: (a) G = 1.25 G _th; (b) G = 1.5 G _th; (c) G = 1.75 G _th. (d) Emission linewidth Δℏω _em and peak intensity I _max as a function of gain. Vertical dashed lines mark the points corresponding to the spectra shown in (a), (b), and (c). The linewidth is taken as the full width measured at the base of the emission spectrum. Parameters are the same as in Figure 2.

Finally, without much surprise, as the gain level is increased from Figure 10-(a)–(c), we see that the intensity of the nanolaser emission increases significantly, and that the range of emission widens as well. However, all in all, the aspect ratio of the emission band remains globally similar, meaning that the quality of the nanolaser emission does not depend much on the level of gain provided to the system.

Figure 10-(d) shows the evolution of the maximum (peak) lasing intensity in the spectrum, I _max, and the linewidth of the emission, Δℏω _em, as functions of the gain level G normalized to the gain threshold G _th. As expected, the emission appears when G/G _th = 1. From that point onward, both the maximum emission intensity and the width monotonously increase. We observe in particular that the increase of I _max versus G/G _th is strictly linear. It is interesting to consider the typical values obtained for the nanolaser linewidth: the observed values for Δℏω _em are of the order of a few 10⁻² eV, which correspond to a linewidth of a few nanometers; for example, for G = 1.5 G _th, we have Δℏω _em ≃ 0.05 eV, which gives a linewidth of around 8 nm. (Let us recall, as explained earlier, that this linewidth should be considered as the widest possible to be expected from the nanolaser, see the “Discussion” section.)

One well-known feature of nanoshells is that the position of their plasmonic resonances can be easily tuned by changing the thickness of the metallic shell (i.e., by changing the nanoparticle’s aspect ratio ρ): how does this reflect in the emission spectrum of the nanolaser? Figure 11 shows the evolution of the maximal emission spectrum as a function of ρ, when the shell size is modified, keeping the external radius of the nanoparticle constant (a = 10 nm). One can see that the spectra are indeed strongly dependent on ρ and cover most of the visible region, from green (ρ = 0.8) to violet (ρ = 0.4). Hence, the specific nanoshell geometry indeed makes a versatile choice for applications.

Figure 11:

Emission spectra I _em(ω _em) of lasing nanoshells for varying aspect ratios ρ, with a gain level set at G = 1.2 G _th, and normalized to I _sat. Nanoshell drawings bear the actual colors corresponding to the emission spectrum. All parameters beside ρ are the same as in Figure 2.

We also note that the peak intensity of the emission increases strongly as ρ is increased (from violet to green). This is because larger ρ values correspond to thinner metallic shells, and therefore to smaller associated Ohmic losses responsible for dampening the emission. If we compare absolute gain levels, taking into account the values found earlier for G _th (see Figure 3), for the violet emission (ρ = 0.4), we have an absolute gain value G = 1.2 G _th ≃ 0.35, while for the green emission (ρ = 0.8), we have G = 1.2 G _th ≃ 0.11. In other words, a green-lasing nanoshell is much more efficient than a violet one, as it delivers an emission seven times more intense with a quantity of gain more than three times smaller.

8 Summary

We studied the emission and lasing properties of a nanoshell particle made of an externally pumped, active core and a plasmonic shell, with the help of a set of space and time-dependent governing equations. These coupled equations are, on one hand, the Drude equation of motion for the free electrons within the metallic part, and the other hand, the optical Bloch equations, accounting for population changes occurring between the two electronic levels in the gain material part. In the nanoshell geometry specifically, the dipolar mode is the only one to be excited and emitting, including in the lasing regime. Therefore, we used a quasi-static description for fields based on an expansion in spherical harmonics, keeping only dipolar terms. The set of governing equations was then projected onto these dipolar terms and put under matrix form to allow for numerical solving.

With the help of a linear instability analysis of the governing equations, we first demonstrated the existence, and then calculated the value, of the gain threshold value G _th above which the nanoshell hosts a self-oscillation instability (i.e., the emission of a lasing field in the absence of any exciting probe field). Below this threshold, no instability exists and the nanoparticle can only react to an external excitation.

We then studied the situation prevailing when the gain level is lower than the threshold (G < G _th). When submitted to a probe field, the nanoparticle produces transient fields which rapidly decay to a steady-state response, which is linearly proportional to the amplitude of the exciting probe field E ₀. One may then use the standard quasi-static formula for the polarizability of a nanoshell, equation (28), to calculate the dipolar moment of the particle relative to E ₀, with the help of the gain-dependent permittivity of equations (13) or (11). In this regime, the nanoshell acts as a plasmonic amplifier, i.e., it synchronizes and responds linearly to the external field, and its response becomes more and more intense and sharp as more gain G is provided.

Next, we studied in depth the lasing regime above the threshold (G > G _th). We exclusively considered autonomous situations, where the nanoshell oscillates freely in the absence of any externally-imposed field. The self-oscillation of the nanoshell initially grows exponentially, according to the results of the linear instability analysis made previously. After a while, stimulated emission due to the lasing process starts exceeding the capacity of the pump, effectively reducing the population inversion (saturation effect) and limiting the intensity growth. This brings the particle to a final state of steady-state emission.

This final lasing state is characterized by the following salient features:

1. The determination of the actual values of the emission wavelengths ω _em within the spectrum requires extra care, since the nanolaser is free to choose its lasing frequency. The actual wavelength of emission ω _em is given as ω _em = ω − Ω(ω), where Ω(ω) is a frequency shift (frequency-pulling effect) from the rotating-wave frequency ω. The shift Ω is computed by Fourier analysis from the numerical steady-state solution.

2. For a given nanoshell, the range (spectrum) of emission wavelengths ω _em depends on the applied gain G. When G is increased, and the threshold is crossed, the lasing start at a single frequency ω _th = ω _res, where ω _res is one of the nanoparticle’s plasmon resonance frequencies. Then, as G is further increased above threshold, the emission range Δℏω _em widens. We calculated the corresponding maximal (widest possible) spectrum of emission of the nanolaser and find typical linewidths in the range 5–10 nm. Simultaneously, we find that the peak lasing intensity I _max increases linearly as the level of gain G is increased.

3. Due to the action of the frequency shift Ω moving unstable frequencies around, we observe that the final (maximal) spectrum of emission of the nanolaser is one-sided with respect to the plasmon resonance frequency ω _res, that is, the lasing occurs only for frequencies ω _em ≥ ω _res.

4. The color of the nanolaser emission can be tuned in a versatile way across the visible range by choosing nanoshells of various aspect ratios ρ. Nanoshells with thinner metallic shells (emitting on the low-energy end of the visible) are more efficient, i.e., much more intense with less required gain, than those with thicker shells (emitting on the high-energy end of the visible).

9 Discussion

The results summarized just above now require some thorough discussion to assess their physical significance and validity.

Regarding our results on the determination of the lasing threshold and the amplification regime below this threshold: they globally confirm earlier similar findings in the literature, justifying in particular the common use made of the standard polarizability formula and of steady-state permittivities for the gain material (whether linear or saturated).

Let us now turn to the lasing regime, above threshold. It is the first time, to our knowledge, that the nonlinear lasing steady state of a nanoshell, alongside with the dynamics leading to it, has been fully characterized. It is also the first time that the maximal spectrum of emission of a lasing nanoshell has been calculated.

Several earlier works have studied the nanoshell geometry in the lasing regime, mostly focusing on driven situations where the nanoshell is under the action of a probe field. Authors of Refs. [41], [42] in particular have briefly considered the autonomous (free) situation, writing equations closely similar to our set of equations (6)–(8). They looked for the steady-state regime of lasing using the steady-state expression of the permittivity ϵ _g(ω) in the gain region, including saturation effects, equation (11), and concluded that autonomous lasing can only occur at one single frequency ω = ω _res, equal to one of the plasmonic resonance frequencies of the nanoshell, for all gain levels above the threshold. Our findings show that this line of reasoning is incomplete. We also find that lasing at one single frequency occurs only when the gain level is set right at the threshold value, but then the spectrum of emission widens as G is increased above the threshold, with a finite linewidth Δℏω _em. The reason why this fact was overlooked is because the use of a steady-state permittivity ϵ _g(ω) in the lasing regime is incorrect; steady-state permittivies follow from cancelling all time derivatives in equations (6)–(8), or equivalently, in equations (22) and (23). However, we have found that the final state of lasing is a steady state with a shifted frequency with respect to the frequency ω used in the rotating-wave approximation. This means that, when expressed in terms of a carrier wave at frequency ω, field amplitudes show an extra oscillation at frequency Ω, and therefore the aforementioned time derivatives are non-zero (except for the equation on N). The only frequency where there is no extra oscillation (i.e., where Ω = 0 and time derivatives do cancel) is ω _em = ω _res = ω _th, as seen in Figure 9-(a). Therefore, the conclusions about the free lasing regime of nanoshells as written in refs [41], [42] do only apply to that specific frequency but miss out on the rest of the spectrum. We emphazise, however, that the authors of ref [41] have correctly predicted the linear dependence of the peak intensity I _max versus the gain level G/G _th seen in Figure 10-(d), because that peak intensity indeed occurs at the specific frequency ω _em = ω _res (under conditions of optimal gain positioning). Beyond these considerations on the free lasing regime, to what extent the existence of other emission frequencies ω _em ≠ ω _res should also modify conclusions drawn in refs [41], [42] about the driven (forced) regime of oscillation, is an open question at this stage.

One striking and novel result of our study is that the emission of the nanolaser is strongly asymmetrical, since it only occurs on the high-frequency side of the nanoshell’s natural plasmon resonance (ω _em ≥ ω _res). This is due to the effect of the frequency shift Ω (also known as a frequency pull-out) which translates and “folds” the initially symmetric unstable range of Figure 2 to that one side only of the plasmon resonance. (Note that we did not find numerically any situation where this pull-out effect would result in a spectrum located on the opposite, lower frequency-side of ω _res.) We are not aware, to the best of our knowledge, of any similar claim made explicitly in the existing literature on nanolasers, either theoretically or experimentally. Let us underline that it is uncertain whether this phenomenon would extend to other geometries than nanoshells, and that in any case, exhibiting this effect experimentally would be challenging, since most experiments are made on collections of individual nanolasers: any statistical dispersion in the structural properties of the nanoresonators will certainly smear out the asymmetry of the emission line. We note, however, the existence of sharply asymmetric emission spectra on the high- or low-frequency side for example in refs [39], [59].

Finally, a discussion is due on the actual spectrum of emission to be expected from the nanoshell laser. In our study, we have calculated this spectrum under the assumption of independent emission of the various frequencies ω _em composing it, meaning that all frequencies are allowed to consume energy from the two-level system as if they were alone. They would therefore all grow to their maximal capacity, which is why we called our calculated spectrum “maximal”, i.e. the widest one with all frequencies emitting to the highest possible intensity. However, this independency hypothesis is clearly incorrect because all ω _em-frequencies in fact draw energy from the same reservoir of excited electrons represented by the population inversion N. Whatever is consumed by one frequency (making N decrease), is not available to another one, and thus these “modes” are truly competing for the same energy resource. This is because the dispersion in the ω _em-frequencies originates in the finite width of the gain curve feeding the nanolaser, as illustrated by the Lorentzian curve of equation (13), which is well-known in the literature on classical (macroscopic) lasers as a situation of homogeneous broadening. In such a situation, it is established that the numerous laser modes inside the initially unstable range of the spectrum will indeed compete, and only the mode with the fastest growing rate will ultimately survive – usually the first mode to reach the threshold (see for example Chap. 8 in ref [60] or Chap. 11 in ref [61]). This winning mode will then sharpen by several orders of magnitude, as it remains alone in the cavity and may consume all the available gain inside it at will, until reaching some final limit of acuteness which will be discussed below. Therefore, the actual width of emission of a homogeneously broadened laser is set by the ultimate width of this surviving mode only, not by the initial width of the gain curve or anything of that order.

If we were to apply this line of thought in our case, this would mean that only the frequency at ω _em = ω _res (where the intensity is maximal) will eventually survive in the final lasing state of the nanolaser, eliminating all other ω _em frequencies, resulting in a width of emission possibly much thinner than the spectra presented in Figures 10 and 11. In the present stage, it is unclear to what extent, if any, this scenario should apply here. Several qualitative facts should indeed be taken into consideration. Firstly, the classical scenario applies well to situations with spectrally well-defined cavity modes, much thinner than the unstable gain curve, whereas we are here in presence of several frequencies acting within a rather wide mode (the dipolar plasmon resonance) inside an unstable range. Secondly, the curve of the growth rates for unstable frequencies, given by Re(κ ₃) in Figure 2, is very flat around its maximum: therefore, it may be that the contrast between the central frequency ω _th = ω _res and neighbouring ones is not significant enough for the former to dominate the competition. Thirdly, in the classical homogeneously broadened laser scenario, the final surviving mode controls the final width of emission because it is able to become much thinner than the homogeneously broadened gain transition. The well-known Schawlow–Townes formula [60], [61] actually expresses the theoretical value Δω _em of the limiting linewidth of the surviving mode, in the most ideal case, as

where ω _em is the central emission frequency of the mode, Δω _cav is the width of the passive resonant cavity mode at the origin of the surviving mode, and P _out the power output of the laser. In the case of a classical laser, we may typically have Δω _cav ≃ 1 MHz ≃ 10⁻⁸ eV and P _out ≃ 1 mW, which results in Δω _em ≃ 10⁻⁴ Hz ≃ 10⁻¹⁸ eV. This theoretical value is far from reached in practice due to all types of imperfections. Nonetheless, let us evaluate what could be expected in the case of a nanolaser, assuming this formula retains some physical relevance (at least in spirit). For a lasing nanoshell, as seen on Figure 5-(a), we now have Δω _cav ≃ 10⁻² eV ≃ 10⁶ MHz, while we may take P _out ≃ 10⁻⁴ mW [41], yielding Δω _em ≃ 10¹² Hz ≃ 10⁻² eV. This last value not only is immensely larger than the equivalent for a classical laser, but in fact, also lies in the same range as the spectral widths already exhibited in Figures 10 and 11 (typically a few 10⁻² eV). This would suggest that, should the classical scenario for homogeneously broadened transitions apply, the single surviving mode would barely sharpen; and therefore, the final emission width of the nanoshell could possibly not change much, if at all, in comparison to the maximal spectral width as we calculated it.

To definitely conclude on this point, the only way to calculate the actual final width of emission of the lasing nanoshell would be to compute the growth dynamics of the whole set of unstable frequencies in the spectrum, taken all together (not independently), fully accounting for their competitive effect on the population inversion N. This is a challenging task that we leave for future work.

Nonetheless, we shall close this discussion by emphasizing that the (maximal) emission widths found in this work, which are in the range 5–10 nm (see Figure 10), are as such already comparable to the typical values measured experimentally on actual nanolasers [11], [13], [14], [15], [18], [20], [21], [22], [23], [59].

10 Conclusions

To conclude, in this paper, we have unveiled a study of the properties of emission of a nanolaser in the nanoshell geometry, with gain in the core, and metal in the shell, both under and above the lasing threshold. For the first time, the free lasing regime was carefully studied, both in the dynamical transient regime and in the non-linear steady state, showing that strong frequency shifts effects (pull-out) shape up the spectrum of emission of the particle. These novel theoretical results add to the knowledge on one of the most promising geometries for nanolasers, in the hope to bring real-world applications within this thriving field one step closer.

11 Methods

We here present some of the intermediate technical steps required to obtain the final set of governing equations in matrix form, as shown in equations (22) and (23).

From the dipolar description of the nanoshell displayed in equations (16)–(20), one first needs to compute the amplitudes p _0,1,2,3 by enforcing continuity of the tangential electrical field and normal displacement at the boundaries r = a and r = ρa. This procedure yields the following expressions relating p _0,1,2,3 to q _0,1,2:

Detailed calculations to obtain these relations can be found in the Supplementary Material. Equations (44)–(47) allow to calculate the time-dependent values of all electrical field components p _i (t) from the knowledge of the polarization mode components q _i (t). The latter are known from the resolution of the governing set of equations of the system, equations (22) and (23).

We can now substitute equations (16)–(20) and (44)–(47) into the dynamical equations (6)–(8). Through this procedure, we produce a system of equations determining the time evolution of the mode amplitudes q _0,1,2 pertaining to the polarizations in the nanoparticle:

and the equation for the evolution of the population inversion N:

In the set of equations (49) and (50), we have defined the shorthand notations:

Since relations (44)–(47) make a linear system, we can write p ₀, p ₁, p ₂ and p ₃ as linear combinations of q ₀, q ₁, q ₂ and E ₀, namely:

The above coefficients p _ij are real constants, whose analytical expressions involve combinations of the four parameters ρ, ϵ _b, ϵ _∞ and ϵ _e only. (Full expressions are given in the Supplementary Material.)

We can then use these coefficients p _ij to define the following matrix A(N):

and the vector b(N, E ₀):

Collecting the mode amplitudes q _i into a vector q

the system of equations (49)–(51) can be rewritten in the following matrix form:

This gives the governing set of equations for the nanoshell’s dynamics, as shown in equations (22) and (23).

We can see that the matrix A(N) encodes all the information about the nanoshell geometry of the system, via the coefficients p _ij . For other geometries like a homogeneous nanolaser sphere, or a core–shell one, the matrix A would admit different components from those of equation (60), but the global formalism of equations (22) and (23) will remain unchanged (as long as fields keep irrotational). Importantly, A also explicitly depends on the population inversion N = N(q, t), which in general is time-dependent, and, most importantly, depends non-linearly on the q _i via equation (23) and relations (44)–(47). The A matrix also depends on the frequency ω and on the level of gain through the factor Γ_g ∝ G.

The vector b(N, E ₀) depends on N as well and on the excitation by the probe field E ₀.