Gain clamping in random lasers

Random lasers (RLs) are based on the propagation of light in scattering materials with gain [1–4]. Due to the absence of a well-defined optical cavity, which usually is composed of a couple of mirrors, the RL emission was considered an amplification of the spontaneous emission rather than a true laser in the sense of oscillations [5]. One definitely demonstration of oscillations was provided by the photon statistics, in which the photon number distribution changes from Bose–Einstein to Poisson [6] as the gain is increased. Actually, the RLs are characterized by the increasing of the input–output slope efficiency, bandwidth narrowing, and shortening of the decay time of the upper level of the RL transition [7, 8]. Furthermore, the RL threshold can be found by statistical mechanics analyses [9]. Upon identical experimental realizations, the output spectra of the RLs can change dramatically from one experimental to another [10–14]. For optical gain below the threshold, the probability distribution for the output intensity is Gaussian. At the onset of laser emission the distribution can present Lévy statistics, which can be used to characterize the RL threshold [15]. However, it seems that it is not a universal feature, since some RLs (those for larger scattering strength) do not present this feature [14]. In another interdisciplinary study, there is an analogy between the optical modes of the RLs and the spins of magnetic systems [16, 17]. In the experimental realization, instead of looking only to the output intensity, the full spectra of the RL transition are analyzed [18]. By defining a correlation function (q) among the spectra obtained under the same starting conditions (replicas), the shape of the probability distribution of q is the analogue of the Edward–Anderson parameter of the spin-glass theory, and it has been experimentally observed a phase transition from a photonic-paramagnetic to a photonic spin-glass phase at the onset of the RL emission [18–20]. This parameter has been demonstrated to be robust over the different RL architectures [21–23]. Although there are many ways to characterize the RL action, as mentioned recently by Sapienza [24], the phenomenon called 'gain clamping', which is known for conventional lasers, remains unexplored for RLs. The gain clamping is the reduction of the population inversion (and the gain) to the threshold value due to the increase of the lasing field.

Here we investigate the gain clamping in different kinds of RLs. The systems analyzed here present distinct features such as lifetime of the upper level of the RL transition and quantum efficiency. Despite of that, the gain clamping was observed in both systems, and the aspects are discussed based on a rate equation model for the atomic ⁵ population, the energy density at the RL transition and a fluorescence channel.

In order to make clear what the gain clamping is, consider a four-level laser system in which the optical excitation is resonant with the ground-state transition |0> → |3> (see inset of figure 1). Consider also that the state |2> is so close to the state |3> that very fast nonradiative relaxations from |3> populate the state |2> (relaxation time τ₃ ≈ 10 ns). Taking |2> as a metastable state (total lifetime τ₂ ≈ 20 µs—including both radiative and nonradiative transitions), and considering fast nonradiative relaxation from |1> to |0> (τ₁ ≈ 10 ns), it is known that this system can work very well for laser emission. The following rate equation system can cope with the system dynamics, where n₀, n₁, n₂, and n₃ are the population densities of the nondegenerate states |0>, |1>, |2>, and |3>, respectively, and the dot above n_j represents time derivative

$$\begin{align} {{\dot n}_3} &= {\text{ }}K\left( t \right){\sigma _{03}}\left( {{n_0}-{\text{ }}{n_3}} \right)/h{\nu _{03}}-{\text{ }}{n_3}/{\tau _3}\nonumber \\ {{\dot n}_2} &= {\text{ }}{n_3}/{\tau _3}-{\text{ }}{n_2}/{\tau _2}-c{\sigma _{21}}\left( {{n_2}-{\text{ }}{n_1}} \right){E_{21}}/h{\nu _{21}}\nonumber \\ {{\dot n}_1} &= {\text{ }}{f_{21}}{n_2}/{\tau _2} + c{\sigma _{21}}\left( {{n_2}-{\text{ }}{n_1}} \right){E_{21}}/h{\nu _{21}}-{\text{ }}{n_1}/{\tau _1}\nonumber \\ {{\dot n}_0} &= {\text{ }}-{\text{ }}K\left( t \right){\sigma _{03}}\left( {{n_0}-{\text{ }}{n_3}} \right)/h{\nu _{03}} + {\text{ }}{f_{20}}{n_2}/{\tau _2} + {\text{ }}{n_1}/{\tau _1} .\end{align} \tag{ 1 }$$

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In these equations, K(t) is the excitation power density (∫K(t)dt gives the excitation pulse energy (EPE) density), σ₀₃ the resonant absorption cross-section at the excitation wavelength, σ₂₁ the stimulated emission cross-section at the RL transition, c the speed of light, and hν₀₃ and hν₂₁ are the photon energy at the absorption and lasing wavelengths, respectively. We considered relaxations (radiative and nonradiative) from the state |2> with fractions f₂₁ and f₂₀ ending at levels |1> and |0>, respectively. Then, we introduced equations for the energy density (E₂₁) at the RL transition, and for the fluorescence (E₂₀) at the transition |2> → |0>:

$$\begin{align} {{\dot E}_{21}} & = {\text{ }}\beta {f_{21}}{n_2}/{\tau _2} + c{\sigma _{21}}\left( {{n_2}-{\text{ }}{n_1}} \right){E_{21}}-{E_{21}}/{\tau _{21}} \nonumber \\ {{\dot E}_{20}} & = {\text{ }}{f_{20}}{n_2}/{\tau _2}-{\text{ }}{E_{20}}/{\tau _{20}}, \end{align} \tag{ 2 }$$

where β (0 < β ⩽ 1) is a dimensionless parameter introduced to consider that an ion relaxing at the transition |2> → |1> does not necessarily create a photon at the random medium (for example due to nonradiative relaxations), and τ₂₁ is the relaxation time of the photons out of the random medium, which in a conventional laser is linked to the mirrors reflectivity [25]. This approach is based on that introduced by Noginov et al for RLs [26], which has been describing quite well the experimental observations [27] including parametric effects [28, 29]. In their approach, τ₂₁ is related to the dwell time (≈10 ps) of photons in the disordered materials. The main difference from here to [26] is that we did not simplify the equations for the ionic densities to only one equation for the upper level of the RL transition (|2>), and we introduced an equation for the spontaneous emission (E₂₀) due to other transition (|2> → |0>) starting from same level (|2>) of the RL transition. For simplicity, we considered the same relaxation time for E₂₁ (τ₂₁ = τ₂₀), but neglected any possibility of stimulated emission at this transition, which assure only fluorescence at the transition |2> → |0>. We considered typical parameters for the Nd³⁺ ions [26]: σ₀₃ = 3 × 10⁻²¹ cm²; σ₂₁ = 1 × 10⁻¹⁸ cm²; n₀ + n₁ + n₂+ n₃ = 5 × 10²¹ cm⁻³; hν₀₃ = 4 × 10⁻¹⁹ J; hν₂₁ = 2 × 10⁻¹⁹ J; β = 0.01. The parameters f₂₁ and f₂₀ were fixed at 0.6 and 0.4, respectively. In most of the investigated RLs, the excitation source was due to Q-switched lasers with pulse duration of the order of ≈10 ns. In order to mimic their temporal behavior, the photon flux K(t) in equation (1) is considered a Gaussian pulse with duration of 4.1 ns. The integration of K(t) over time provides the energy density of the excitation pulse.

Most of the observed RL features are described by the model above as follows. Monitoring the output energy at the RL transition (power integrated over time) one has an increasing in the slope efficiency, which transition at a given EPE threshold (EPE_th) is abrupt (figure 1). Also, by monitoring the time decay of the upper level of the RL transition (|2>) one has for an EPE lower than the threshold a single exponential decay, which is typical of fluorescence; on the other hand, crossing the EPE threshold one has the shortening of the decay time as follows. The temporal behavior of n₂ is shown in figure 2(a) for three EPE. For an EPE lower than the threshold, one can see the build-up of n₂ in the showed time scale. Due to the large lifetime (τ₂ = 20 µs) of the level |2>, n₂ decays exponentially on this time scale. Similarly, the intensities at the transitions from the |2> state to the |1> and |0> ones decay exponentially because for EPE below the threshold one has only spontaneous emission and the trapping of light is weak (τ₂₁ = τ₂₀ = 10 ps). On the other hand, when n₂ is larger than the threshold condition (gain = losses for conventional lasers; and number of generated photons per unit of time equals the number of escaping photons of a given excited volume [1, 2]) the stimulated emission takes place (figure 2(b)) pushing the gain down to the threshold (indicated in figure 2(a) by a horizontal dashed line)—that is the gain clamping. At the onset of RL emission, one has the emergence of a short pulse with duration of ≈400 ps, which coincides with the decrease of the gain due to the stimulated emission (figure 2(b)). Under gain above threshold, n₂ relaxes mainly by stimulated emission at the RL transition. Thus, the stimulated emission inhibits the other transition from the level |2> to level |0> as well as the nonradiative relaxations which do not create photons at the random medium (accounted by the parameter β). As a consequence, the intensity of the emission due to the transition |2> → |1> increases abruptly (figure 1). Increasing the EPE further the threshold, one has a competition between the stimulated emission to decrease the gain, and the excitation pulse to increase the gain. As a consequence, the time behavior of n₂ presents some oscillations (figure 2(a)), and the RL intensity possess several short peaks, which are known as relaxation oscillations (figure 2(b)). Since the fluorescence at the transition |2> → |0> depends on n₂, it would also show the same oscillations (figure 2(c)). But, usually, the intensity of the fluorescence is so weak that is not measured experimentally (in order to resolve it experimentally, fast detection systems are required which work well for large intensities). At the end of the excitation pulse the n₂ reaches back the threshold (figure 2(a)) and the system relaxes as if there were no stimulated emission. Then, the fluorescence (energy) at the transition |2> → |0> presents a saturation behavior (figure 1), which is an evidence of the gain clamping.

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Another evidence of RL emission, not coped with the model above, is the bandwidth narrowing, i.e. increasing the gain, the RL spectra narrow due to the exponential amplification of the gain curve. For homogeneous broadening lasers, the gain presents a Lorentzian dependence with the light frequency (due to the finite lifetime), and it collapses to a single frequency at an EPE larger than the threshold. For nonhomogeneous broadening, one can have more than one lasing wavelength [30, 31]. For example, due to aggregates formation in dyes [30] or non-equivalent site symmetries for Nd³⁺ ions in crystalline structure [31].

At this point, it worth emphasizing the gain clamping is one mechanism of gain saturation. The clamping is a tendency of the stimulated emission to push down the gain towards the threshold because the growing of the optical field increases the stimulated emission rate [32]. On the other hand, one can have gain saturation by population redistribution among the energy levels [33]. For example, in the rate equation system (equation (1)), if one consider a large time (τ₁) of the lower level of the laser transition (|1>), as the ions at the level |2> relax, one has accumulation at the lower level, which decreases the population inversion (n₂ − n₁), a requirement for amplification. In the following, we present the experimental results concerning the gain clamping in RLs.

3.1. Experimental setup

3.2. Experimental results for the Nd_0.60Y_0.40Al₃(BO₃)₄ nanocrystals

Solid-state RLs based on Nd³⁺ ions have been investigated [7] since the first experimental evidences in 1986 [3], and one promising application is in near-infrared biomedical imaging systems [36]. Here, the excitation at 532 nm (ground-state absorption ⁴I_9/2 → ⁴G_7/2) of Nd_0.6Y_0.40Al₃(BO₃)₄ provided down-conversion fluorescence around 885 nm, corresponding to the Nd³⁺ electronic transition ⁴F_3/2 → ⁴I_9/2, and RL emission at around 1064 nm due to the ⁴F_3/2 → ⁴I_11/2 transition—see [37, 38] for detailed description of the excitation scheme. This excitation route can be simplified to a four-level system, in which the states ⁴I_9/2, ⁴I_11/2, ⁴F_3/2 and ⁴G_7/2 are represented by the |0>, |1>, |2> and |3>, respectively, in the inset of figure 1. Other down-conversion emissions are expected at wavelengths lower than 885 nm and large than 1100 nm. For example, at 1320 nm due to the Nd³⁺ transition ⁴F_3/2 → ⁴I_13/2. It is also possible to observe up-conversion emissions, in which the energy of the generated photons are larger than the excitation ones [39]. However, those emissions are out of the scope of the present letter. Figure 3 shows single-shot spectra of the excitation laser at 532 nm, the fluorescence (around 885 nm), and the RL transition (around 1064 nm) for three different EPEs. For each EPE, four independent experimental realizations are displayed.

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The output fluctuation analyses constitute a current trend in the research of lasers/RLs [12, 15, 18, 25, 40, 41]. One can see in figure 3 that the excitation laser is unstable from shot-to-shot. That characteristic is intrinsic for some lasers, and is attributed to gain competition and nonlinear interaction among the optical modes [25, 41]. For an EPE below the onset of RL emission (2.52 mJ), the fluctuation of the fluorescence and the emission at around 1064 nm is small and correlated with the excitation pulse (first row of figure 3). On the other hand, for an EPE (5.75 mJ) larger than a given threshold (≈4.0 mJ—discussed below), one can notice the narrowing of the spectra around 1064 nm (second row of figure 3), which is an evidence of RL emission. Notice also that the fluorescence around 885 nm is quite stable, while the RL presents strong and uncorrelated fluctuations with the excitation laser. As in a conventional laser, the fluctuation of the RL spectra is associated with the gain competition and nonlinear interaction among the optical modes [14, 16]. On the other hand, the fluctuation of the fluorescence is small for gain above threshold because of the gain clamping [32, 42]. According to figure 1 the slope efficiency of the fluorescence for gain above the threshold is close to zero, which implies that the fluctuation of the excitation energy would provide almost zero fluctuation for the fluorescence at the transition |2> → |0> (⁴F_3/2 → ⁴I_9/2 for the Nd³⁺). For large EPE the fluctuation of the RL shows some correlation with the excitation laser (third row), which we associate to a self-averaging mechanism in the competition of the optical modes from the available gain [14]. Our observations about the correlations between the RL and the excitation laser are in agreement with the work of Zhu et al [13], where the authors performed a detailed study about those correlations.

The input–output energy characterization of the Nd³⁺ based RL is presented in figure 4. Increasing the EPE, one has the abrupt increase of the slope efficiency, which defines the threshold (≈4.0 mJ). On the other hand, the fluorescence around 885 nm presents a decrease in the slope efficiency, which evidences the gain clamping. Notice the good agreement between the experimental results of figure 4 and the theoretical ones in figure 1. It is worth emphasizing that the experimental results display a small growth of the fluorescence for gain above threshold, which is attributed to the fact that only a fraction of the excited volume participates from the RL. It is assigned to the spatial profile of the excitation laser, which is close to a Gaussian (TEM₀₀), as well as the diffusion of excitation photons, which increases the excited volume. As a consequence, the RL takes place near the center of the excitation beam, which implies that the excited volume out of the center is not subject to the gain clamping, and thus presents a linear contribution to the fluorescence at 885 nm as the EPE is increased further the RL threshold. Similar evidences were found by Noginov et al [26] by monitoring the fluorescence at 1054.4 nm (due to Stark levels transitions between the ⁴F_3/2 and ⁴I_11/2 manifolds) close to the RL emission at 1063.1 nm in NdAl₃(BO₃)₄ powders. The authors observed the saturation behavior without mentioning the clamping as the responsible mechanism.

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3.3. Experimental results for rhodamine 6G and TiO₂ nanoparticles in PMMA

Due to the large quantum yields of laser dyes, RLs based on them are among the most investigated. Several architectures have been demonstrated to support dye based RLs. For example, colloidal solutions of laser dyes and TiO₂ nanoparticles [4, 20], porous systems infiltrated with laser dyes [43, 44], hollow-core photonic optical fibers infiltrated with dye and TiO₂ nanoparticles [45], biological tissues [46–49], and exotic materials as butterfly [50] and cicada wings [51]. In this section, we investigate the gain clamping in a RL based on a PMMA film with rhodamine 6G molecules, and TiO₂ nanoparticles to provide the scattering. The excitation of rhodamine 6G molecules at 532 nm provides singlet-singlet (S₀ → S₁) absorption to some energy level (|3>) within the S₁ band, which, followed by nonradiative relaxations, reach the bottom of the excited S₁ band (level |2>). From that, one has relaxations to the S₀ manifold, and the RL transition ends in a level (|1>) above the ground state (|0>). Because of that, the four-level model presented above can be used for the rhodamine 6G based RL.

The excitation of the PMMA sample with rhodamine 6G and TiO₂ nanoparticles was accompanied with the bandwidth narrowing and increasing in the slope efficiency for the RL peak. Three emission spectra are shown in figure 5(a) for different EPEs. For low EPE the spectrum is broad (from ≈560 nm to ≈700 nm). Increasing the EPE one can notice the bandwidth narrowing and the emergence of a short peak (≈7 nm), which is characteristic of dye based RLs [4, 20, 45]. In contrast with the Nd³⁺ case, here there is just one emission band, i.e. other fluorescence band due to other relaxation transition from the same starting level is absent. Because of that, to compare the RL (600 nm) with the fluorescence in order to analyze the gain clamping, we select the intensity at 585 nm, which is close to the RL peak (600 nm), and thus expected to be clamped by the RL transition. To avoid any hidden contribution to the fluorescence analyses from molecular aggregates and reabsorption/reemission effects, which are known to redshift the RL wavelength [30, 52, 53], we selected the fluorescent emission at 585 nm whose wavelength is shorter than the RL peak.

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The input–output energies of the RL (600 nm) and the fluorescence (585 nm) as a function of the EPE present distinct features (figure 5(b)). The slope efficiency for the RL increased at a given threshold (≈0.16 mJ), while the fluorescence exhibited a saturation behavior, which we associate with the gain clamping. In contrast with the Nd³⁺ behavior, the increasing in the slope efficiency and the gain clamping are not pronounced. We attributed those differences to the large quantum yield and short lifetime of the rhodamine 6G molecules. In fact the simulation of the rate equation model by changing only the parameters β and τ₂ to 0.3 and 4 ns, respectively, provides the results in the inset of figure 5(b), which are similar to the experimental ones in figure 5(b). β influences the increasing in the slope efficiency, since the stimulated emission inhibits nonradiative relaxations from the upper level of the RL transition [27]. The nonradiative relaxations are considerable for the Nd_0.60Y_0.40Al₃(BO₃)₄ nanocrystals due to the low energy gap between the ⁴F_3/5 and ⁴I_15/2 states [54]. On the other hand, the rhodamine 6G presents only one main electronic transition (S₁ → S₀) whose energy gap is high. Because of that, β is considerably larger for the rhodamine 6G. Consequently, the increase of the slope efficiency is smooth for the rhodamine 6G and abrupt for the Nd³⁺ based RL. Regarding the differences in the gain clamping behavior, for the rhodamine 6G the clamping is less evident because the spontaneous emission decay rate (n₂/τ₂) is on the order of the stimulated emission one [cσ₂₁(n₂− n₁)E₂₁/hν₂₁]. Increasing τ₂ in the simulations, we observed a bending of the fluorescence intensity curve towards a horizontal line, as in figure 1. In spite of subtle change in the fluorescence emission for the rhodamine 6G based RL as the gain increases, the gain clamping, which concerns the present work, is observed and can be used as a tool to characterize the RL.

We investigated the gain clamping in two different RLs: the first, based on Nd³⁺ ions in crystalline nanoparticles [YAl₃(BO₃)₄]; and the second based on rhodamine 6G molecules and titanium dioxide (TiO₂) nanoparticles as scatterers. Despite these systems present distinct features such as quantum yields and fluorescence lifetimes, the gain clamping was observed in both, and corroborated with numerical simulations based on a rate equation system for the atomic populations, and the energies at the RL and a fluorescence emission, whose transitions departure from the same level. In both systems, at the onset of the RL emission, the fluorescence demonstrated a saturation behavior, evidencing the gain clamping. Because of that saturation behavior, the fluorescence presents low fluctuation for EPE larger than the threshold, on the other hand of the RL emission, which shows large fluctuations. The fluctuation of the RL emission, especially at the vicinity of the EPE threshold, is associated with mode competition for the available gain and nonlinear interaction among them, which has been investigated within the framework of the statistical mechanics of complex systems [11, 16, 18, 19, 25, 40]. Then, the gain clamping can be used not only to determine the RL threshold, but to support the discussion about the output fluctuations in RLs.

We acknowledge the financial support from the Brazilian agencies: Fundação de Amparo à Pesquisa do Estado de Alagoas (FAPEAL); Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior (CAPES)—Finance Code 001, Fundação de Amparo à Pesquisa do Estado de Goiás (FAPEG), Financiadora de Estudos e Projetos (FINEP), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) through the Grant No. 427606/2016-0; scholarships in Research Productivity 2 under the Nos. 303990/2019-8 and 305277/2017-0; National Institute of Photonics (INCT de Fotônica). C V T M and R F S thank CNPq for their Scientific Initiation scholarship. We thank to J G B Cavalcante, and M A Nascimento for technical support.