## Abstract

The features of fluorescence emission in a dye-doped dense multiple scattered medium under pulsed laser pumping are considered in terms of confined excitation in small zones associated with laser speckles occurring in a pumped medium. The results of numerical modeling of the fluorescence emission kinetics are compared to the experimental data obtained using the rhodamine 6G-doped layers of the densely packed TiO_{2} (anatase) particles pumped at 532 nm by 10 ns laser pulses. The intensity of pump radiation during the action of laser pulses was varied from 1·10^{5} W/cm^{2} to 5·10^{7} W/cm^{2}. In the recovery of the ratios of stimulated to a spontaneous emission, the spectra of the stimulated component were fitted using the spectral function derived by R. Dicke. In the framework of the considered concept, saturation of the ratio of the stimulated to a spontaneous emission and linear growth of an integrated fluorescence output with a practically unchangeable half-width of the emission spectra at high pump intensities are interpreted.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The phenomenon of random lasing, first theoretically predicted by V.S. Letokhov in 1967 [1], was an object of intense experimental and theoretical research in the 90s of the past century and the first decade of the present century. During this period, significant contributions to a better understanding of fundamental features of this effect at microscopic and macroscopic levels were made by a number of research groups (see, e.g., [2–28]). At present, the initial euphoria over a possible usage of cavityless random lasers in various fields of modern photonic technologies has significantly subsided (mainly due to low temporal and spatial coherence of radiation emitted by these systems). Nevertheless, an interest in the further studies of random lasing in fluorescing multiply scattering media continues to remain at a fairly high level. This interest is particularly driven by an appropriately high potential of the phenomenon from the point of view of morphological and functional diagnostics of the media with a complex structure (even in biomedical applications as a tool for differentiation of normal and pathological tissues, and biochemical characterization [29–32]). In addition, further studies of random lasing may contribute to a further refinement of the fundamental concepts of light transport in complex media under conditions of a strong influence of nonlinear, resonant and cooperative phenomena on interaction of radiation with the matter.

A classical random lasing medium is a multiple scattering randomly inhomogeneous medium with a high quantum fluorescence yield, which is pumped into the absorption band of a fluorescent component by an external source. When intensity of pump radiation reaches a certain threshold level, the pumped medium switches from a spontaneous fluorescence to an expressed stimulated emission. This transition is accompanied by a significant narrowing in the emission spectrum of the medium, and the random lasing threshold can be defined as the pump energy ${E_{p,\exp }}$ or intensity ${I_{p,\exp }}$ corresponding to the cusp of $\Delta {\lambda _{em}}({E_{p,\exp }})$ or $\Delta {\lambda _{em}}({I_{p,\exp }})$ dependence; here $\Delta {\lambda _{em}}$ is the full width at half maximum (FWHM) of the emission spectrum. Also, the threshold is defined as ${E_{p,\exp }}$ or ${I_{p,\exp }}$ at which the linewidth of the luminescence spectrum reduces to half of the initial width observed in the weak excitation limit (see, e.g. [5]). Numerous experimental data (see, e.g. [2, 5, 20, 21, etc.]) allow us to identify the main features in the behavior of $\Delta {\lambda _{em}}$ depending on ${I_{p,\exp }}$ for the various random fluorescent media: a slow decrease with an increase in the pump fluence rate below the threshold, an abrupt decay of the FWHM value in the threshold domain, and practically constant pump-independent values of $\Delta {\lambda _{em}}$ above the threshold. Typical values of the spectral quality factor ${Q_{sp}} = {\overline \lambda _{em}}/\Delta {\lambda _{em}}$ for these media above the threshold, which were estimated using numerously presented emission spectra far above the threshold, are in the range from 50 to 100. Note that these emission spectra are typically characterized by a significant contribution of a broadband component associated with spontaneous fluorescence even when the pump intensities significantly exceed the estimated threshold value.

Additionally, the reported values of the Purcell factor (${F_p}$) [33] for such emitting systems range from 5 to 20 [34,35], except in the case of localization of fluorescence emitters in the surroundings with expressed resonant properties or strong optical coupling. Following from the typical values of ${Q_{sp}}$ and ${F_p}$ for the considered random lasing systems above the threshold, we can conclude that the volumes of effective cavities associated with generation of a narrow-band radiation in the random lasing mode must satisfy the following condition: ${V_{em}} < \bar{\lambda }_{em}^3,\lambda _p^3$, where ${\lambda _p}$ is the wavelength of pump radiation. For a more detailed consideration of this point see the Section “Discussion of the results”.

On the other hand, pumping of the random media by a pulse-modulated laser light must cause the expressed speckle patterning of the pumping light field inside the medium. Typically, laser radiation with the pulse durations in the nanosecond range is characterized by the values of the coherence length ${l_c}$ significantly exceeding the average propagation paths $\left\langle s \right\rangle$ of pump radiation in the dense random media. A characteristic size of the coherence volume, or the speckle size in the case of a diffusion mode of the coherent light propagation in such multiply scattering system occurs less than the wavelength of light due to a near-isotropic angular spectrum of a multiply scattered light. Even for an almost totally depolarized pumping light field in the medium due to a multiple scattering, we must expect occurrence of a certain number of randomly distributed “hot zones”, or high-intensity speckles beginning from the relatively low pump intensities. In contrast with excitation in the bulk homogeneous fluorophors, fluorescence emission in the confined high-intensity zones of the medium must be strongly influenced by a number of specific factors, such as depletion of the ground state of fluorescence emitters in the course of a pumping pulse action and radiation losses from the emitting zone.

Interactions between the pump and fluorescence light field in the random lasing media above the threshold were previously considered in [6, 9, 22, etc.] in the framework of diffusion approximation of the radiative transfer theory [36]. The light diffusion equations were used to describe spatial-temporal variations of both light fields during the pulsed pumping, and the kinetic equation for a two-level system (dye molecules in the random multiple scattering medium) was applied to establish interconnections between the absorption and emission rates of the pump and fluorescence radiation. Note, however, that the diffusion approximation does not allow for taking into account the effect of speckle-associated graining of the pumping light field due to stochastic interference of the multiple scattered light waves.

In this work, we analyze a possible role of this graining as a limiting factor for a stimulated emission channel of the pump-induced fluorescence in the dense random media. The results of the analysis are compared to the experimental data on fluorescence excitation in the dye-saturated layers of the densely packed high-efficient scatterers. Morphology of the used samples allowed us to expect the expressed diffusion mode of the laser light transport in the pumped layer. Consequently, we must expect a fine granular structure of the pumping light field in these systems.

## 2. Experimental technique and results

#### 2.1 Experimental technique

The experimental studies were focused on the analysis of the ratio of the stimulated to spontaneous fluorescence in the fluorescent layers of densely packed anatase nano- and micro-particles depending on the intensity ${I_p}$ of the pumping laser radiation. The particular interest was paid to the behavior of the stimulated-to-spontaneous emission ratio within the interval of pump intensities corresponding to abrupt decrease in the FWHM value $\Delta {\lambda _{em}}$ of the fluorescence emission spectra. As mentioned above, the cusp of $\Delta {\lambda _{em}}({I_p})$ dependence is traditionally associated with a random lasing threshold. The arrangement of our experimental study was in general similar to the numerous experiments previously carried out by various research groups. However, optical properties of the examined samples in combination with the approaches to interpretation of the obtained empirical data allowed us to validate the discussed concept of a strong influence of the pump light speckling on the fluorescence response of the dense random media.

The examined samples were prepared as the layers of densely packed titania (anatase) particles doped by the solution of Rhodamine 6G (R6G) in the water. The first group of samples was prepared using the polydisperse anatase nanopowder (the product # 637254 from Sigma Aldrich Inc., the average size of particles is less than 25 nm), and a second group was made on the basis of a more coarse-grained polydisperse anatase powder with a non-specified average size of particles (the product # 10122392 from Fisher Scientific UK Ltd). Note that, by many reasons, the knowledge of particle size distributions in the raw powders cannot be used for evaluation of optical transport parameters of prepared samples (in particular, the transport mean free path ${l^\ast }$[36] controlling light propagation in the examined layer). Therefore, the examined samples were specified in the wet state without the dye adding using diffuse transmittance measurements in the spectral range from 500 nm to 750 nm. The powders were put into open cylindrical containers (1 mm in the height and 20 mm in the inner diameter) with glass bottom walls to provide the layers with small variations in the thickness across the area. The layers were slightly pressed and carefully saturated by the necessary amounts of distilled water. After this, the surfaces of the prepared water-saturated layers were inspected the optical coherence tomograph (OCT) OSC1300SS. The inspection was provided to evaluate the average sample thickness and thickness variations across the sample areas. It was found that the average thickness of water-saturated layers prepared using the Sigma Aldrich product (sample #1) is $\bar{L} \approx$ 940 ± 38 µm. Similarly, $\bar{L} \approx$ 830 ± 33 µm for sample #2 were prepared using the Fisher Scientific product. For both groups of samples, the thickness variations $\delta L$ across the areas did not exceed 5% of the average thickness. In combination with mass-volumetric measurements during preparation of the samples, the OCT inspection allowed us to evaluate the volume fractions of the solid phase (particles) in the prepared samples. The reason for application of the distilled water instead of the dye solution for a sample preparation in the case of diffuse transmittance measurements is to exclude the influence of the sample fluorescence on the acquired diffuse transmittance spectra. Note also that the refractive index of the dye solution for the used R6G concentration (see below) does not significantly differ from that for the distilled water. In addition, as shown below, the ${l^\ast }$ values for the examined samples occur significantly lesser than the expected absorption length ${l_a}$ of laser radiation in the R6G solution. These circumstances give us a possibility to expand the obtained results for the water-saturated samples to the samples with a similar morphology saturated by the dye solution. Rough estimates of the volume fraction of anatase nanoparticles in the prepared samples give values of the order of 0.20–0.25. At such high values of the volume fraction, the average distance between neighboring nanoparticles in the sample is comparable to the average particle size. Accordingly, the surrounding neighbors block any significant displacements of particles relative to their initial positions and the samples can be considered as relatively stable highly porous structures filled with a dye solution.

The measurements of the diffuse transmittance ${T_d}(\lambda )$ of water-saturated powder layers were carried out through transparent bottom walls of the containers using the broadband light source (Ocean Optics HL-2000) and the integrating sphere (ThorLabs IS 236A-4) coupled with the portable spectrometer (Ocean Optics QE65000) using the fiber-optic patch cord (Ocean Optics P200-5-UV-VIS). After a careful calibration, the ${T_d}(\lambda )$ values measured at 532 nm (the wavelength of the laser light used for pumping), and 570 nm (the wavelength in the vicinity of the fluorescence emission maximum) were used for recovery of the mean transport path length ${l^\ast }.$The obtained ${T_d}(\lambda )$ values were equal to ≈ (6.2 ± 0.4)·10^{−3} at *λ=*532 nm and ≈ (7.0 ± 0.5)·10^{−3} at *λ=*570 nm for the sample #1, and ≈ (2.4 ± 0.3)·10^{−3} at *λ=*532 nm and ≈ (2.7 ± 0.3)·10^{−3} at *λ=*570 nm for the sample #2.

The diffuse transmittance of a randomly-structured layer can be written as (see, e.g., [37,38])

- both components of the systems (water and anatase nanoparticles) are weakly absorbing in the examined spectral interval;

- characteristic sizes of the structure heterogeneities in the examined layers are expected to be significantly smaller than the wavelengths of the interest; this corresponds to the so-called low-frequency limit, for which discrepancies between the MG model and more sophisticated models applicable for large-sized particles are not very significant. The results of the samples characterization are presented in Table 1.

We used purified R6G to prepare the dye solution. At the first stage of purification, the 2.1·10^{−3} M water solution of the raw dye (the product #160027 from JSC LenReactiv company, Russian Federation; an analytical reagent grade) was mixed with the 1 M sodium hydrate solution to separate heavy metallic impurities. After filtration of the mixture, it was treated using a strong hydrochloric acid for dye crystallization during 7–10 days. The purification process was repeated four times; as a result, the fluorescence yield significantly increased compared to the raw dye. Finally, the purified dye was used to prepare the water solution with the molar concentration of 3.4·10^{−3} M. The layers in containers were carefully saturated by the dye solution similar to the procedure used for diffuse transmittance measurements.

The fluorescence spectra of the prepared samples under the pulse-periodic laser pumping at 532 nm were acquired using the experimental scheme similar to those traditionally applied in the numerous studies of the random lasing beginning from the work of Lawandy et al. [2]. The experimental setup is shown in Fig. 1.

Upper surfaces of the prepared samples in the containers (6) were irradiated by a converging laser beam passed through a convex lens (5) with the focal length of 150 mm. The diameter ${d_{sp}}$ of the light spot on the sample surfaces was equal to ≈ 1.14 mm. The spot diameter was preliminarily determined when assembling the setup for the used configuration “the laser (1) output window - the beam splitter (3) – the right-angle totally reflecting prism (4) – the lens (5) – the sample (6) surface”. The laterally translating Foucault knife was placed in the irradiated plane at the sample position instead of a sample, and the fraction of the transmitted energy of laser pulses was measured depending on the knife-edge lateral position. An analysis of the obtained dependences for the applied range of pump energies showed that the energy density distribution over the irradiating spot is close to the Gaussian profile and the effective spot size corresponding to 95% of the pump energy is (1.14 ± 0.03) mm. The second harmonics of the Q-switched YAG-Nd laser [(1), the product LS-2134 of Lotis TII company, Republic of Belarus, the pulse energy ${E_i}$ varied in the range from 0.08 mJ to 7.5 mJ, the pulse duration ${\tau _i}$ 10 ns, the repetition rate equals to 10 Hz] is applied for irradiation of the samples. The energy of the laser pulses during irradiation was measured using the laser energy/power meter Gentec Maestro with the QE12HR-MB head (2) in combination with a beam splitter (3). In order to prevent the sample degradation (photo-bleaching of the dye, liquid phase evaporation, changes in the positions of anatase particles due to thermal effects, etc.) during a long-term irradiation, each single experiment was carried out using five-fold pulse irradiation of the samples. After this, the examined sample was translated in the lateral direction to the new position. The translation step was three times larger than the diameter of the illuminating spot. No visible changes in the structure of the samples occurred as a result of such procedure. The average intensity ${I_{p,\exp }}$ of pumping radiation falling onto the sample surfaces during the action of laser pulses was evaluated to characterize the pump conditions in each experiment: ${I_{p,\exp }} \approx {{4{E_i}} / {\pi {\tau _i}d_{sp}^2}}$. Fluorescence emission spectra were recorded using the portable spectrometer Ocean Optics QE65000 with the fiber-optic patch cord Ocean Optics P200-5-UV-VIS (8). An additional cut-off filter [(7), the orange glass OS12 filter, the thickness is 2 mm, the cut-off wavelength is 540 nm, the transmittance is no less than 0.94 above the cut-off wavelength] was used to minimize the contribution of pump radiation to the recorded signal.

Figure 2, a displays the examples of the typical fluorescence spectra acquired in the cases of a small (1) and large (2–4) pump intensity. The dependencies of the WFHM values of the emission spectra on the pump intensity obtained for the samples #1 and #2 are shown in Fig. 2(b).

These shapes are rather conventional and similar to the $\Delta {\lambda _{em}}({I_{p,\exp }})$ shapes for a wide variety of the random lasing systems studied in the numerous works during the past three decades. However, a remarkable feature of fluorescence emission for the samples #1 and #2 is related to the closeness of the $\Delta {\lambda _{em}}({I_{p,\exp }})$ dependencies; despite a significant difference in the ${l^\ast }$ values (see Table 1), the threshold intensities differ insignificantly (${I_{th}} \approx$ 9.2·10^{6} W/cm^{2} for the sample #1 against ${I_{th}} \approx$ 7.1·10^{6} W/cm^{2} for the sample #2). Note also very close and practically unchangeable FWHM values for both examined systems in the case of the pump intensities significantly exceeding the threshold value.

A decrease in the dye concentration causes an abruptly increasing threshold value of the pump intensity for the examined samples; in particular, the curve (3) in Fig. 2(b), obtained for the sample #1 saturated by 1.7·10^{−3} M dye solution, exhibits a more than tenfold increase in ${I_{th}}$. With lesser concentrations of R6G, the threshold cannot be reached even for the pump intensities approaching the radiation resistance of the samples.

#### 2.2 Decomposition of fluorescence spectra

In the further analysis, we considered the normalized fluorescence spectra ${\tilde{I}_f}(\lambda ,{I_{p,\exp }})$ of the examined systems as the sums of two specified spectral functions, ${\tilde{I}_{f,sp}}(\lambda )$ and ${\tilde{I}_{\lambda ,co}}(\lambda ,{I_{p,\exp }})$ with the pump-dependent weighting factor $\phi ({I_{p,\exp }})$:

Compared to the Lorentzian spectral approximation, ${\tilde{I}_D}(\lambda )$ provides a much better fit for the stimulated-emission-induced spectral feature at ≈ 573 nm with the increasing pump intensity ${I_p}$. In particular, the attempts to fit the shape of this feature by the Lorentzian distribution under condition of equality of $\Delta {\lambda _{em}}$ for the empirical and fitting spectral data lead to anomalously large values of ${\tilde{I}_{f,co}}({\lambda ,{I_{p,\exp }}} )$ in the short-wavelength region (below 570 nm) compared to the experimental data. The Gaussian fitting provides the accuracy of approximation comparable with that for the Dicke spectral function thoughbut its application seems rather physically unfounded.

Figure 3 displays the examples of fitting of the normalized fluorescence spectra for the examined systems under various pump intensities using the normalized Dicke function as ${\tilde{I}_{\lambda ,co}}({\lambda ,{I_{p,\exp }}} )$.

Best values of the adjustable parameters ${\lambda _0}({{I_{p,\exp }}} ),{\varsigma _\lambda }({{I_{p,\exp }}} ),$ and $\varphi ({{I_{p,\exp }}} )$ were determined using the LS (least squares) technique. Note that some discrepancies between the experimental data and fitting curves occur at high pump intensities in the spectral range between 580 nm and 600 nm (see Fig. 3, b). The presumable reason for occurrence of these discrepancies above the threshold pump intensity ${I_{th}}$ is a “two-state” character of the applied approximation. In terms of the “two-state” approach, the ensemble of fluorescence emitters is divided into the “cooperative” and “spontaneous” sub-ensembles without taking into account a possible occurrence of the intermediate states. However, this feature is not crucial for the further analysis.

The obtained values of the weighting factor $\varphi ({{I_{p,\exp }}} )$ in the fitting model (2) can be used to recover the pump-dependent ratio ${\bar{\Psi }_{exp }}({{I_{p,\exp }}} )$ of the stimulated to spontaneous emission from the fluorescence spectra. The introduced notation (the sign of averaging) means that the recovered values characterize the ratios of a stimulated to a spontaneous emission averaged over the durations of the pumping laser pulses.

Considering the absolute values of the model spectral densities ${I_{f,sp}}(\lambda )$ and ${I_{f,co}}(\lambda )$ measured in [J·cm^{-2}·s^{-1}·nm^{-1}], we can express a corresponding flux of the fluorescence photons ${N_f}$ ([1·cm^{-2}·s^{-1}]) as

^{−3}M dye solution; the stimulated-to-spontaneous emission ratios rapidly rise up with an increase in the pump intensity below the threshold values and tend to saturate above the threshold. The ratio values are relatively small even for the pump intensities sufficiently above the threshold.

Following [9], we can refer the spectral ratio ${{{S_{co}}({{I_{p,\exp }}} )} / {{S_{sp}}({{I_{p,\exp }}} )}}$ to universally adopted quality parameter of the laser radiation, such as the β factor. In contrast with high-finesse-valued cavity-based systems (the typical β values are in the range from 10^{−5} to 10^{−3}), the fluorescent random media above the pump threshold ${I_{th}}$ are characterized by large values of the β factor typically occurring between 0.1 and 0.2 (see, e.g., [9]). Note that the extreme β value in the absence of the stimulated emission is equal to 1. In our case, analysis of the obtained fluorescence spectra shows that the ratios ${{{S_{co}}({{I_{p,\exp }}} )} / {{S_{sp}}({{I_{p,\exp }}} )}}$ are practically independent on the pump intensity in the vicinity and above the threshold. The estimated values $\beta \approx { {{{{S_{co}}({{I_{p,\exp }}} )} / {{S_{sp}}({{I_{p,\exp }}} )}}} |_{{I_{p,\exp }} \ge {I_{th}}}}$ are approximately equal to 0.19 ± 0.03 for the samples #1 and 0.22 ± 0.04 for the sample #2.

## 3. Discussion of the results

As mentioned above, the general property of the dye-doped fluorescent random media pumped by the laser light is a small and practically unchangeable value of the spectral quality factor ${Q_{sp}} = {{{{\bar{\lambda }}_{em}}} / {\Delta {\lambda _{em}}}}$ above a certain threshold value of the pump intensity. Typically, for a wide variety of these media based on various scattering systems and various dyes with a high fluorescence yield, the values of ${Q_{sp}}$ corresponding to the experimentally measured parameters ${\bar{\lambda }_{em}}$ and $\Delta {\lambda _{em}}$ of fluorescence emission above the threshold are ranging from 50 to 100. In our case, the spectral quality values for the samples #1 and #2 are also small and close to each other (≈ 63.6 versus ≈ 70.8). In addition, the experimentally obtained values of the Purcell factor [33] ${F_P} = 3{Q_{sp}} \cdot \{{{{{{({{{{{\bar{\lambda }}_{em}}} / {{n_r}}}} )}^3}} / {4{\pi^2}V}}} \}$ are in the range from 5 to 10 for the random multiple scattering fluorescent media (see, e.g., [34,35]). Here ${n_r}$ is a refractive index of the medium and *V* is the light-emitting volume considered as a low-finesse-valued optical cavity.

Thus, we can estimate a characteristic size of such fluorescence-emitting cavity of a near-spherical shape as

On the other hand, the pumping light is characterized by the coherence length ${l_c}$ sufficient for occurrence of a speckle modulation of a multiple scattered laser radiation in the pumped media. Indeed, typical values of ${l_c}$ in the case of pulsed pumping in the nanosecond range of pulse durations are not less than several tens of centimeters, whereas the propagation paths of laser radiation in the pumped layers are sufficiently lesser (see Table 1). As a result, this leads to formation of 3D speckle patterns with significant fluctuations of the local ${I_p}$ values in the pumped layers. The average size of these pump-induced speckles is comparable with the wavelength of pump radiation due to a multiple scattering, and such stochastic confinement of fluorescence excitation must cause a crucial influence on the basic properties of the overall fluorescence response of the pumped system. Further consideration of this feature will be provided via the following steps:

- analysis of fluorescence excitation in the confined zone associated with a separate pump-induced speckle;

- consideration of the first-order statistics of local intensity values in the ensemble of pump-induced speckles and ensemble averaging of single-speckle parameters of the fluorescence emission;

- collation of the empirical and modeled data.

#### 3.1 Kinetic model of confined fluorescence excitation

We consider a small (${\propto} \lambda _p^3$) volume ${V_{em}}$ in the pumped layer of the dye-doped random media which is associated with a low-finesse micro-cavity and characterized in every time moment by the current energy densities of the pump (${W_p}(t )$) and fluorescence (${\left\langle {{W_f}(t )} \right\rangle _\lambda }$) radiation averaged over the pumped local volume. In addition, the energy density ${\left\langle {{W_f}(t )} \right\rangle _\lambda }$ of fluorescence radiation is averaged over the fluorescence emission spectrum. The energy densities ${W_p}(t )$ and ${\left\langle {{W_f}(t )} \right\rangle _\lambda }$ are related to the values of pump and fluorescence intensity as ${I_p}(t )= {W_p}(t )v,{\left\langle {{I_f}(t )} \right\rangle _\lambda } = {\left\langle {{W_f}(t )} \right\rangle _\lambda }v$, where *v* is the light velocity in the medium. We describe the evolution of the excited state population for the ensemble of dye molecules in the pumped volume ${V_p}$ using a kinetic equation for a two-level system with the depleted ground state

The considered model can be compared to the other approaches previously used for modeling of the kinetics of random lasing in the fluorescent random media (see, e.g., [6,9,22]). These approaches are primarily based on application of the diffusion approximation to describe the pump light and fluorescence transport in the pumped systems without taking into account the granular structure of the pumping light field, which occurs due to the laser light coherence. However, strong fluctuations of the local values of the pump energy density due to stochastic interference of a multiple scattered laser radiation, must cause an expressed granular structure of the excited fluorescence field. This conclusion is supported by some previously reported results (see, e.g., [7]). Additionally, strong confinement of the pumping zones due to granulation of the laser light necessarily requires for accounting the depletion of the ground state [terms 1–*f* in Eqs. (11) and (12)] due to a finite number of the dye molecules in the pumping “hot” zones. Usually this factor was not taken into account in the previous works on random lasing (see, e.g., [6,9,22]).

#### 3.2 Evaluation of the model parameters

In the considered model, the rate of down-up transitions in a two-level system for the given pump intensity ${I_p}(t )$ is determined by the ratio of the absorption cross-section ${\sigma _a}({{\lambda_p}} )$ of the used fluorophore (in our case, Rhodamine 6G) at the pump wavelength (${\lambda _p} = $ 532 nm) to the pump photon energy (≈ 2.33 eV ≈ 3.73·10^{−19} J). The absorption cross-section ${\sigma _a}$ of R6G molecules can be obtained using the dye molar extinction coefficient defined for the given solvent (water). In our opinion, most reliable data were published in the monograph dedicated to dye lasers [42]. It is important that the applied wavelength of excitation radiation (532 nm) is in the vicinity of the isobestic point of the R6G-water solution in the moderate range of dye concentrations (from 3·10^{−6} M/l to 7.6·10^{−3} M/l). Under these conditions, the molar extinction of R6G water solutions is approximately equal to 5.5·10^{4} l/M·cm. This gives an absorption cross-section for our conditions of fluorescence excitation approximately equal to 9.13·10^{−17} cm^{2}.

Relating the decay rate of the spontaneous fluorescence $\delta$, we can refer to the numerously published data on fluorescence kinetics in the various pulse-pumped R6G solutions (see, e.g., [41,43,44]). Typically, these publications present characteristic times of the R6G fluorescence decay closely scattered between 3.6 and 4.0 ns (except the cases of very large-scale dye concentrations exceeding 10^{−2} M, [41]). Therefore, after averaging over this large collection of data, we estimated the parameter $\delta$ for the considered model as $\delta = \tau _s^{ - 1} \approx $ 2.56·10^{8} s^{-1}, where ${\tau _s}$ is the data-averaged decay time for the spontaneous fluorescence (${\tau _s} \approx $ 3.9 ns).

The wavelength-dependent stimulated emission cross-section ${\sigma _{st}}({{\lambda_{em}}} )$ can be evaluated from the experimentally obtained fluorescence spectra ${I_f}({{\lambda_{em}}} )$ using the following known relationship [45,46]:

^{−17}cm

^{-2}, which was used in the further modeling of fluorescence kinetics for the examined samples. Note that the estimated values of the absorption cross-section ${\sigma _a}$ and the wavelength-averaged cross-section ${\left\langle {{\sigma_{st}}} \right\rangle _\lambda }$ are expectedly close to each other.

The wavelength-averaged self-absorption cross-section ${\left\langle {{\sigma_{sa}}} \right\rangle _\lambda }$ was evaluated for the examined samples basing on the numerous empirical data relating the long-wavelength R6G absorption above ${\lambda _{\min }}$ = 540 nm. It should be noted that ${\left\langle {{\sigma_{sa}}} \right\rangle _\lambda }$ occurs significantly smaller (at least 15-20 times) than the absorption cross-section at the pump wavelength. Therefore, it was assumed that this channel of population exchange associated with the term $\left( {{{{{\left\langle {{\sigma_{sa}}} \right\rangle }_\lambda }} / {{{\left\langle {h{\nu_f}} \right\rangle }_\lambda }}}} \right){I_f}(t )\{{1 - f} \}$ in Eqs. (11) and (12) is much less efficient compared to other channels and, therefore we, dropped out this contribution from the modeling.

Regarding the cross-section of the radiation losses ${\sigma _{rad}}$, we can roughly estimate the upper value of these losses using the following simplifying assumptions:

- at the moment, the ensemble of fluorescence photons is uniformly distributed in the emitting volume with the density of ; the volume has the spherical shape with the diameter of ${d_{em}}$;

- the momentum vectors of fluorescence photons in the emitting volume are characterized by a uniform angular distribution;

- the density of fluorescence photons outside the volume is equal to 0;

- transport of fluorescence photons is ballistic at the scales comparable with ${d_{em}}$ due to the condition ${d_{em}} < < {l^\ast }$.

Under these assumptions, the flux density of photons escaping from the volume is approximately equal to $0.5{\tilde{N}_{ph}}v$, where *v* is the light velocity in the medium and the factor 0.5 occurs due to an uniform distribution of the photon momentum vectors. Thus, the total flux of escaping photons is approximately equal to $2\pi d_e^2{\tilde{N}_{ph}}v$. The negative contribution of radiation losses to the photon density in the emitting volume ca${\tilde{N}_{ph}}$n be obtained as

A detailed quantitative analysis of reduction of radiation losses $\left\langle {{\sigma_{rad}}} \right\rangle $ with an increasing pump intensity is outside the scope of this work. In the modeling, we used a cross-section of radiation losses $\left\langle {{\sigma_{rad}}} \right\rangle $ as a varying free parameter. Note that for the examined samples, the estimated extreme value $\sigma _{rad}^{\max }$ is of the order of (4·10^{−15} ÷8·10^{−15}) cm^{2}.

#### 3.3 First-order statistics of a speckle intensity in the pumping light field

In the framework of the considered concept, spontaneous and stimulated fluorescence emission in the laser-pumped random media mainly occurs in the speckle-associated “hot zones” randomly distributed in the pumped volume and considered as low-finesse micro-cavities. These speckle-associated randomly distributed micro-cavities result from stochastic interference of the multiple scattered coherent light waves propagating along the non-correlated random paths in the pumped volume. Rough estimates showed that under the used experimental conditions, the coherence length of the pumping light significantly exceeds a characteristic propagation path of laser light in pumped layers. Consequently, it can be concluded that effect of a speckle blur due to a finite coherence length of the propagating radiation is subtle. Therefore, considerable differences between the local values of electromagnetic energy density for a pair of arbitrarily chosen observation points can be expected.

Consideration of the probability density distribution $\rho ({{W_p}} )$ of local values relating the pumping energy density was carried out under the following assumptions:

- the angular distributions of the pumping flux in the pumped layers are almost isotropic;

- the pumping light is almost totally depolarized due to a multiple scattering.

Under these assumptions, we can present a random value of the electric field amplitude for the pumping light in an arbitrarily chosen point as a superposition of three statistically independent orthogonally directed components:

The real and imaginary parts of ${\bar{E}_1}({\bar{r}} ),{\bar{E}_2}({\bar{r}} ),{\bar{E}_3}({\bar{r}} )$ are normally distributed random variables with zero mean values and equal variances. Correspondingly, we can express ${W_p}({\bar{r}} )$ as where ${W_{p1,2,3}}({\bar{r}} )\propto {|{{{\bar{E}}_{1,2,3}}({\bar{r}} )} |^2}$. In the framework of statistical optics, each orthogonal component ${\bar{E}_1}({\bar{r}} ),{\bar{E}_2}({\bar{r}} ),{\bar{E}_3}({\bar{r}} )$ can be considered as corresponding to realization of the fully developed speckle pattern with a negative exponential probability distribution of local intensity (or energy density) values:^{8}and 1·10

^{9}. The relative number of high intensity micro-cavities with ${I_p} \ge {I_{th}}$ is obtained as $\int\limits_{{I_{th}}}^\infty {\rho ({{I_p}} )d{I_p} \approx }$ 4·10

^{−5}. This means that under these pumping conditions, several thousand speckle-associated micro-cavities in the layer emit fluorescence radiation with a significant contribution from the stimulated component. On the other hand, a dominating exponential decay of $\rho ({{I_p}} )$ for the large ${I_p}$ values causes occurrence of the critical pump intensity ${I_{p,cr}}$, for which contribution of the “hot” cavities becomes negligible. We estimated this critical value using the analysis of the probability density distribution (21) as ${I_{p,cr}} \approx 0.14{I_{th}}$. It should be noted that this estimation fairly agrees with the behavior of the recovered ratios ${\bar{\Psi }_{\exp }}({{I_p}} )$ of a stimulated to a spontaneous emission in the region of small pump intensities (see the inset in Fig. 4, the corresponding critical values for the samples #1 and #2 are marked by the colored dashed arrows).

An important question is related to the temporal stability of the ensembles of stochastically distributed micro-cavities, which is determined by the correlation time of speckle intensity fluctuations in the pumped layer. We assume these ensembles stable during the action of each laser pulse; consequently, temporal dependencies of ${I_p}$ for each micro-cavity (speckle) are assumed corresponding to the temporal dependence of pump intensity in the laser pulses. As discussed below (see the subsection 3.5), this assumption should be valid in the case of small and moderate (around the threshold value ${I_{th}}$) pump intensities, when the effect of pumped volume expansion is insignificant.

#### 3.4 Collation of experimental and modelled data

The considered kinetic model [Eqs. (11)–(13)] was applied for the numerical simulation of the time-integrated values ${\bar{\Psi }_{\bmod }}$ for a single speckle-associated micro-cavity depending on the pump intensity ${I_p}$ and the ensemble-averaged cross-section of radiation losses $\left\langle {{\sigma_{rad}}} \right\rangle $. In the simulation, the ${I_p}$ values were varied between 1·10^{4} W/cm^{2} and 5·10^{8} W/cm^{2} with the step of 1·10^{4} W/cm^{2}, and cross-section values were changed from 2.0·10^{−15} cm^{-2} to 1.0·10^{−16} cm^{-2} with the step of 5.0·10^{−17} cm^{-2}. After collection of the dataset ${\bar{\Psi }_{\bmod }}\left( {{I_p},\left\langle {{\sigma_{rad}}} \right\rangle } \right)$, the averaging over intensity probability distributions (21) was carried out

^{5}W/cm

^{2}to 5·10

^{7}W/cm

^{2}with the step of 1·10

^{5}W/cm

^{2}. The modeling results in the form of 2D color map are presented in Fig. 7 compared to the dependencies ${\bar{\Psi }_{\exp }}({{I_{p,\exp }}} )$ recovered from the experimental data (Fig. 4). In the course of plotting the recovered values ${\bar{\Psi }_{\exp }}({{I_{p,\exp }}} ),$ it was taken into account that the average intensity $\left\langle {{I_p}} \right\rangle$ across the pumped layer is reduced compared to ${I_{p,\exp }}$: $\left\langle {{I_p}} \right\rangle = {K_r}{I_{p,\exp }}$ by a certain reduction factor ${K_r} < 1$ dependent on the surface reflectivity and depth distribution of the total intensity in the pumped layer. Assuming an exponentially decaying depth distribution of total intensity in the pumped layer, and using the estimated value ${n_{ef}}$ for the examined samples (Table 1), we obtained an approximate estimate of the reduction factor as ${K_r} \approx$ 0.58. Positions of the plotted values ${\bar{\Psi }_{\exp }}({{I_{p,\exp }}} )$ on the 2D map were determined by the criterion of matching the ${\bar{\Psi }_{\exp }}({{K_r}{I_{p,\exp }}} )$ and ${\bar{\Psi }_{\bmod }}\left( {\left\langle {{I_p}} \right\rangle ,\left\langle {{\sigma_{rad}}} \right\rangle } \right)$ values under the condition ${K_r}{I_{p,\exp }} = \left\langle {{I_p}} \right\rangle $.

Comparison of the recovered and modeled $\bar{\Psi }$ values clearly indicates saturation of a stimulated to a spontaneous emission ratio at the large pump intensities. The average cross-section of radiation losses as a model phenomenological parameter also saturates approaching to the value slightly exceeding the wavelength-averaged cross-section ${\left\langle {{\sigma_{st}}} \right\rangle _\lambda }$ of the stimulated emission ($\left\langle {{\sigma_{rad}}} \right\rangle \approx$ 1.27·10^{−16} cm^{-2} at $\left\langle {{I_p}} \right\rangle \approx $ 4.2·10^{7} W/cm^{2} against ${\left\langle {{\sigma_{st}}} \right\rangle _\lambda } \approx $8.03·10^{−17} cm^{-2}). A decrease in the pump intensity below the threshold value ${I_{th}}$ is accompanied by an expected rapid increase in the average cross-section $\left\langle {{\sigma_{rad}}} \right\rangle$. Note that the $\left\langle {{\sigma_{rad}}} \right\rangle$ values for the samples #1 and #2 recovered from the 2D map for the lowest pump intensity $\left\langle {{I_p}} \right\rangle \approx $ 1.2·10^{6} W/cm^{2} (${\left\langle {{\sigma_{rad}}} \right\rangle _{\# 1}} \approx$2.0·10^{−15} cm^{-2} and ${\left\langle {{\sigma_{rad}}} \right\rangle _{\# 2}} \approx$1.15·10^{−15} cm^{-2}) are comparable in the order of magnitude with the extreme cross-section $\sigma _{rad}^{\max }$ of radiation losses for the isolated spherical micro-cavity.

In the framework of the considered phenomenological model, the extreme value $\sigma _{rad}^{\max }$ is inversely proportional to the dye concentration, $\sigma _{rad}^{\max } \propto {1 / {{n_0}}}$. Therefore, we should expect an increase in the average value $\left\langle {{\sigma_{rad}}} \right\rangle$ with a decreasing dye concentration at the fixed pump intensity. This effect is illustrated (Fig. 8) by the same 2D color map as in Fig. 7 with the plotted dependence ${\bar{\Psi }_{\exp }}({{K_r}{I_{p,\exp }}} )$ for the sample #1 with a reduced dye concentration [1.7·10^{−3} M, see the curve (3) in Fig. 2(b)]. Comparing Figs. 7 and 8, we can see a stronger influence of ${n_0}$ on the phenomenological parameter $\left\langle {{\sigma_{rad}}} \right\rangle$ than the predicted by the simple model considered above.

It should be noted that effect of a dye concentration on the kinetics of pulsed excitation of fluorescence in the random media is not unambiguously defined, and an increase in ${n_0}$ above a certain value can provoke significant suppression of excitation of the stimulated emission. As an example, we can refer to the results on the random lasing excitation in colloidal suspensions of submicrometer-sized rutile particles in methanol solutions of R6G [47]. A major difference between our samples and these suspensions is a significantly larger dye concentration in the latter case (approximately 4.18·10^{−2} M).

The presented experimental data show similar saturation of the FWHM values with an increase in the pump energy above the threshold, and a similar low enhancement of the spectral quality factor ${Q_{sp}}$ in the random lasing mode (from ≈ 9 far below the threshold to ≈ 95 at the high pump energies). But the reported threshold energy densities at 532 nm for 10 ns pumping pulses significantly exceed (at least in 10 times) the threshold energy densities referred to our case. The reason for this discrepancy is related to the effect of fluorescence quenching at high concentrations of dye molecules. It is well established (see, e.g., [41]) that an increase in the molar concentration of R6G dye in various solutions above 10^{−2} M leads to a dramatic decrease in fluorescence lifetimes, and the decaying quantum yield due to increasing influence of a non-radiative deactivation of the excited molecules. This should lead to a switching on an effective channel of a non-radiation depletion of the excited state. In particular, the fluorescence lifetime of R6G molecules in methanol for 0.04 M dye concentration measured by A. Penzhofer and Y. Lu is approximately equal to 357 ps. The corresponding contribution to the rate of spontaneous decay of the excited state population for the suspensions [47] should be at least ten times larger than presented in our case. This increasing spontaneous decay presumably leads to an increase in the threshold energy density.

#### 3.5 Expansion of the pumped volume and decrease in the lifetime of speckle-associated micro-cavities at high pump intensities

We assumed that in the course of modeling the temporal dynamics of pump radiation ${I_p}(t )$ in each separate fluorescence-emitting volume is governed only by the shape of the incoming laser pulse ${I_{p,\exp }}(t )$. In other words, the lifetime ${\tau _{lt}}$ of speckle-associated micro-cavities is strictly determined by the duration of the pumping pulse: ${\tau _{lt}} = {\tau _i}$. The main factor which can reduce ${\tau _{lt}}$ compared to ${\tau _i}$ is an expressed time-dependent dynamics of laser-induced speckles characterized by the speckle decorrelation time ${\tau _c} < {\tau _i}$. Temporal decorrelations of speckles during the action of laser pulses can be caused by the following reasons: the finite bandwidth of pump radiation, local translational or rotational motions of the scattering sites, and changes in configuration of the scatter ensemble due to appearance or disappearance of part of the scattering sites. Brief considerations show that the reduction effect related to a partial coherence of the pump light, and microscopic motions of the scattering sites is rather subtle. Indeed, it was mentioned above that usually the coherence length of pump radiation significantly exceeds the average propagation path of light in the medium: ${l_c} > > \left\langle s \right\rangle$. Additionally, the reduction of ${\tau _{lt}}$ due to the local motions of scattering sites (e.g., the Brownian dynamics) is also negligible. In our case of “steady-structured” samples consisting of densely packed particles, each scattering site is “arrested” by the neighboring sites. Even in the case of multiple scattering fluorescent suspensions of nanoparticles frequently used in the experimental studies of random lasing, the estimated values of the speckle decorrelation time ${\tau _c}$ occur significantly larger than ${\tau _i}$ (tens of microseconds against nanoseconds).

On the contrary, the effect of scatter ensembles reconfiguration due to absorption reduction in the pumped layer and its expansion should lead to a significant reduction of the speckle lifetime at a large pump intensity. In our opinion, this expansion manifests itself in the behavior of the normalized values of the wavelength-integrated fluorescence output $\tilde{S}({{I_{p,\exp }}} )= {{\int\limits_{{\lambda _{\min }}}^{{\lambda _{\max }}} {{I_f}({\lambda ,{I_{p,\exp }}} )d\lambda } } / {{S_{norm}}}}$ depending on the pump intensity ${I_{p,\exp }}$ (Fig. 9). These dependencies were recovered from the experimentally obtained emission spectra for the samples #1, #2. Two characteristic emission modes are clearly identified for both data sets: the gradual saturation of emission with an increasing pump intensity below and in the vicinity of the threshold ${I_{th}}$ (I), and an approximately linear increase in $\tilde{S}({{I_{p,\exp }}} )$ at high pump intensities (II). The values ${S_{norm}} = \int\limits_{{\lambda _{\min }}}^{{\lambda _{\max }}} {{I_{em}}({\lambda ,{I_{p,\exp }}} )d\lambda }$ corresponding to the crossover between these modes were used as normalization parameters for the both datasets. Despite a significant difference in the values of transport mean free path (Table 1), the behavior of $\tilde{S}({{I_{p,\exp }}} )$ is similar for the samples #1 and #2. A linear growth of the integrated output with an increasing pump energy was usually interpreted as manifestation of expressed random lasing (see, e.g., [9]). However, we can consider another interpretation of this linear growth in terms of the discussed here concept. At low pump intensities, the pumped layer gradually approaches to the certain saturation level with the increasing pump intensity ${I_{p,\exp }}$. The dye absorbance at this level is reduced due to depletion of the ground state of dye molecules. A further increase in ${I_{p,\exp }}$ must cause expansion of this extreme saturated state on a larger volume in the pumped layer (primarily due to an increasing penetration depth for the pump radiation). Based on the general physical considerations, we can claim an approximately linear relationship between the volume in the pumped layer, which is characterized by an extreme fluorescence output, and the incoming pump energy above the crossover between the emission modes I and II. In particular, this assumption is supported by experimentally observed features in the behavior of the fluorescence output at the high pump intensities ($\tilde{S}({{I_{p,\exp }}} )\propto {I_{p,\exp }}$,$\Delta {\lambda _{em}}({{I_{p,\exp }}} )\approx const$, etc.). Additionally, the speckle decorrelation time, or the lifetime of speckle-associated micro-cavities falls down due to incorporation of new scattering sites to the scattering ensemble.

In addition to the considered limitation of stimulated emission due to speckle modulation of pump radiation, the influence of other limiting factors can also be discussed. One of these factors could be the photo-bleaching of the dye by pump radiation. However, due to the high photochemical stability of Rhodamine 6G and the short duration of the repetitively pulsed laser action in our experiments, this effect is practically absent. The interconversion effect associated with non-radiative singlet-triplet transitions of excited dye molecules and interconversion-induced triplet-triplet absorption can also reduce the fluorescence yield (and, accordingly, the stimulated-to-spontaneous emission ratio). This effect is negligible if the following condition is met [48]:

where ${N_s},{N_t}$ are the populations of the excited singlet and ground triplet states of the dye molecules and ${\sigma _{ss}},{\sigma _{tt}}$ are the absorption cross-sections for the singlet-singlet and triplet-triplet transitions. When short laser pulses with steep edges pump the dye, this condition is fulfilled as in our case.## 4. Conclusions

Based on the presented analysis of experimental and theoretical data, we can conclude that the concept of speckle-induced confined excitation of fluorescence in the laser-pumped dense random media provides a fair interpretation of the basic features in the behavior of a stimulated and spontaneous emission depending on the pump conditions. These features are relatively small values of the ratio of a stimulated to a spontaneous emission, and a large bandwidth of the stimulated emission even in the case of large pump intensities. This results in large values of the β factor and small values of spectral quality of the emitted fluorescence for these systems. Transition from saturation of the integrated fluorescence output at low pump intensities to its linear growth for the large values of intensity, which is commonly identified as occurrence of the random lasing, can be interpreted in terms of the limiting state of fluorescence output for the given pumped volume, which corresponds to a certain threshold value of the pump energy. With an increase in the pump energy, this limiting state expands to a larger volume in the pumped layer proportionally to the difference between the applied pump energy and the above mentioned threshold value. In contrast with the integrated fluorescence output, spectral parameters of the emitted fluorescence (in particular, the FWHM value of the emission spectrum and spectral quality factor) remain practically unchangeable in the course of this expansion.

In our opinion, the remarkable result obtained in this work is related to the established critical value of the pump intensity, which is approximately equal to 0.14 of the threshold value determined by the cusp position of $\Delta {\lambda _{em}}({{I_{p,\exp }}} )$. Below this critical value, the probability to find a speckle-associated micro-cavity with a remarkable stimulated emission falls down to zero. This feature clearly manifests itself for the obtained dependencies of the ratio of a stimulated to a spontaneous emission on the pump intensity.

The obtained results can be useful for further development of fluorescence diagnostic techniques for applications in biomedicine and material science.

## Funding

Council on grants of the President of the Russian Federation (MK-2181.2020.2); Russian Foundation for Basic Research (19-32-90221).

## Acknowledgments

The authors are grateful to Prof. A.Kh. Askarova for literary editing of a manuscript text. D.A. Zimnyakov acknowledges that the results of modeling the kinetics of fluorescence responses were obtained within the framework of the State Assignment of the Institute for Problems of Precision Mechanics and Control, Russian Academy of Sciences.

## Disclosures

The authors declare that there are no conflict of interest relating this article.

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