1. Introduction

Random lasers (RLs) have been reported extensively in various complex media ranging from weakly to strongly scattering materials [1]. Without a requirement of a reflective cavity as in conventional lasers, the optical feedback in RLs solely depends on the multiple light scattering, stemming from random fluctuations of the dielectric constant [2]. The RL applications have been extended from illumination sources to biosensors in biological field where the biological tissues are nature scattering media [3]. The cytometry of apoptosis [4], cancer diagnosis [5–7] and cancer therapy monitoring [8,9] are achievable with the RLs.

Despite the complex media of RLs differ considerably, the investigated RL spectral parameters are restricted: the line width, the emission intensity and the wavelength, as well as the threshold pump energy for lasing [1]. As known from conventional lasers, the lasing threshold corresponds to the condition where the total amount of the amplification compensates losses at a given lasing mode. It usually can be characterized with an abruptly collapsed line width or enhanced emission intensity in terms of pump energy. However, a precise extraction of the lasing threshold in a diffusive RL (DRL) is not always desirable [2,10–12]. DRL is a special open system where light undergoes a diffusive motion within the scattering gain medium and might experience an extreme leakage at boundaries [2,10–12]. In a DRL, the optical feedback for the light amplification relies on the light dwelling path length, rather than a closed trajectory like in a laser cavity or in the Anderson localization regime [13]. To compensate the significantly large losses caused by the incomplete feedback, the abruptly changed trend of intensity and line width easily becomes progressive when crossing the DRL threshold, making the identification of lasing threshold challenging [14–16].

The incomplete and stochastic feedback of light diffusion in DRLs induces other particular features. It has been found that the modal nature of DRLs is extended in general [2,10]. The multiple extended modes compete for the available gain for lasing, randomly couple with each other and might be modified by the leaky boundaries, constructing an unpredictable complex system [17–19]. In the case of numerous overlapped modes which tend to be averaged out, no lasing modes but an overall line width narrowing around the maximum gain are observed when reaching the lasing threshold [2]. On the contrary, the modes which are spectrally visualized survive the mode competition [2].

In fact, the random nature of the mode interactions in DRLs gives rise to the strong intensity fluctuation around the lasing threshold [17,18,20–22]. Intriguingly, when fitting the fluctuated intensity distribution with an $\alpha$-stable distribution function, statistics turn out a Lévy type ($0 \,<\,\alpha \,<\, 2$) near the lasing threshold and a Gaussian type ($\alpha = 2$) far below or above the threshold [21]. A Lévy threshold regime is therefore revealed [20]. Furthermore, the first transition point from Gaussian to Lévy regime is denoted as the onset of the random lasing, i.e., the DRL threshold point [21]. This threshold point is in a good agreement with the one disclosed from the standard input-output intensity curves, thus proved a robust RL threshold indicator [21].

The replica symmetry breaking (RSB) effect from the spin glass theory is another efficient RL threshold indicator [22–24]. RSB describes the effect that identical replicas of a complex system under the same conditions manifest completely different dynamics, revealing the hidden interplay between the system disorder and fluctuations [25]. The importance of RSB was recognized by 2021 Nobel Prize for physics as it is an elegant and powerful approach on revealing intrinsic phase transition of a complex system such as condensed matter, biophotonics and social dynamics [25]. Long after the prediction of the RSB effect, the first experimental observation was demonstrated only until 2015 in RLs [23]. The RSB effect is characterized by the distribution shape splitting of an order parameter named the Parisi overlap $q$ ($q$ in the case of RLs is defined as the fluctuation correlation between spectrum $\alpha$ and $\beta$) [23]:

In a short summary, the Lévy statistics and RSB effect are more advantageous and reliable in the identification of DRL threshold point compared to the fluctuated spectral intensity and line width. In particular, the Lévy statistics reveal a special threshold regime of DRLs. But their limitations are also obvious: firstly $\alpha$ and $P(q)$ are not readily available and the statistical calculation requires repeated measurements; secondly, the fluctuation statistics might fail when fluctuations are weak, for instance, when the temporally or spatially large pumping causes the self-averaging effect and reduces the fluctuations [21]. In addition, the quenched disorder system such as a solid-state RL was considered necessary to bring about the RSB effect, while a liquid-state RL is rather not satisfying the requirement of identical replicas because the realization of disorders is time-changed [24]. Taking all of the above into consideration, a straightforward approach is preferred and proposed in this study for the aim of DRL threshold characterization. This approach depends on the spectral feature of the peak wavelength shift, i.e., tunability.

The tunability is often observed in a dye-based RL when the used dye shows an overlap between the absorption and fluorescence spectra [28–30]. As a result, the blue part of the fluorescence has the chance to be reabsorbed and then reemitted at the red part of the spectrum with longer wavelengths, yielding the peak wavelength redshift [31]. In a DRL, a well-defined cavity for wavelength selection does not exist. Although in some cases, the mode interaction might modify the lasing wavelengths [32], the DRL wavelength is still mainly determined by the gain, e.g., the dye fluorescence. Hence, the reabsorption and reemission of fluorescence also account for the peak wavelength redshift in DRLs [33].

For the generation of peak shift, the local pumping is critical too. The fluorescence reabsorption only happens with the unpumped dye molecules within a light diffusion volume [34]. A global pumping otherwise excites all the fluorescent dye leaving no space for the reabsorption and reemission effect as well as the peak redshift. Besides the experimental requirement, it has been explored that changes of dye concentrations [35] or the concentration ratio of non-fluorescent absorber and fluorescent dye [36] have an influence on the peak shift too. The changes of scattering strength give rise to the peak shift as well [30,35]. The RL peak shift is therefore proved a fruitful parameter for optical sensing, for instance, to detect IgG concentration changes [37] and to differentiate cancerous thyroid tissues [7].

Therefore, in addition to the aim of threshold characterization using the peak shift, the investigation of the influence factors on the peak shift and threshold behaviors is another goal of this study. In fact, these factors also influence the RL fluctuation statistics. For instance, it is found that the physical quantities behind the deterministic parameter $\alpha$ of Lévy statistics are indeed the mean free path of light propagation in the gain medium $\,<\,l\,>$ and the gain length $l_g$ [20]. The experimental and material conditions such as scattering strength and pump energy density (pump energy per pump spot area) are considered the influence factors of these two quantities, and thus affecting the occurrence and crossovers of the threshold regimes in DRLs. This has already been evidenced experimentally [38,39]. To give an example, the Gaussian-Lévy-Gaussian regimes and their crossovers are clearly identified at small pump spot size, while the well-above-threshold Gaussian regime is hard to achieve with the increasing pump spot size [38].

Two of the influence factors, the non-fluorescent absorption and the pump energy, were varied to achieve the second goal of current study. In detail, Indian black ink was added to the DRL phantoms to change the non-fluorescent absorption. The ink as a source of absorption losses does have an effect on the DRL lasing behaviors. The Lévy statistics and RSB effect were additionally explored in this study to validate the ability of peak shift on the DRL threshold identification. Eventually, with the experimental verification and the statistical validation mentioned above, it can be shown that the peak shift is capable to reveal the DRL threshold regime, as well as to illustrate the laser phenomena of mode competition and intensity saturation.

2. Materials and methods

2.1 RL experimental setup and measurements

The experimental setup is shown in Fig. 1(a). The setup mainly consists of a light source for excitation, optics to collect the light emission and a spectrograph to acquire RL spectra.

 

Fig. 1. Experimental setup (a) and pump laser intensity drift (b). (a) The pump light from the pulsed Nd:YAG laser is focused on the phantom, and the back-scattered light is collected and guided to the spectrograph. M: mirror; L: lens; NF: notch filter; NDF: neutral density filter. (b) The coefficient of variation of intensity sum indicates that the fluctuation is quite low. The pump laser is therefore considered to fulfill the replica nature of the experiment.

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A frequency-doubled Nd:YAG pulsed laser with a working wavelength of 532 nm and pulse duration of 5 ns (Q-smart 450, Quantel) was used for excitation. By adjusting the Q-switch delay, the pump energy was increased from 1.82 mJ to 40.86 mJ in a trend of half Gaussian distribution. The wavelength spectra of the pump laser from 100 excitation shots were recorded at pump energy of 21.65 mJ and shown in Fig. 1(b). The coefficient of variation (CV) of intensity sum indicates that the fluctuation is quite low. The pump laser is therefore considered to fulfill the replica nature of the experiment for the statistical study of the random laser intensity fluctuation.

The sample of 75 ml solution was filled in a beaker with a diameter of 50 mm. A beaker rather than a cuvette was chosen to demonstrate a semi-infinite turbid sample. The large size of the sample can ensure that the light dwelling path is only influenced by the optical properties rather than the boundaries of the sample. To avoid the wall effect of the beaker, a laser top-pumping configuration was applied. The laser focal spot on the phantom surface was around 0.25 mm in diameter. The usage of bulky phantom and local pumping ensures the diffusion regime for the generation of peak wavelength shift.

The back-scattered light from liquid phantoms was collected and coupled into a fiber and sent to a spectrograph (Mechelle Me5000 Echelle, Andor). The spectrograph has a spectral range from 200 to 975 nm with a spectral resolving power ($\lambda$/$\Delta \lambda$) of 6000 which enables a high spectral resolution (resolution of 0.1 nm at 600 nm). The spectral acquisition was triggered by the flash lamp of the pump laser at 1/3 Hz repetition rate which is low enough to allow the dye relaxation.

2.2 DRL phantoms and optical properties

Rhodamine 6G (R6G; Sigma Aldrich, Germany), Intralipid (IL; Fresenius Kabi, Germany) and distilled water served as fluorescent dye, scatterer and solvent, respectively. After mixing, the IL scatterers distributed uniformly in the phantoms. Additional Indian black ink (Royal Talens, Netherland) was added to the liquid phantoms to provide the non-fluorescent absorption.

The R6G concentration was kept at 2$\times$10$^{-4}$ g/ml. The fluorescence peak is located at around 560 nm for such a concentration of R6G in water [40]. Due to the reabsorption and reemission effect, the RL peak wavelength is always larger than 560 nm. The volume concentration of IL was fixed at 5% where the first local maximum of RL intensity with increasing IL concentration was found in the previous study [30]. Phantoms with too high or too low IL concentrations require too high or too fine ink concentrations to visualize the transition dynamics between non-lasing and lasing threshold regimes. In this context, the volume concentration of ink was selected from 0% to 0.13%. A RL cannot be generated with further increased ink concentration under the current experimental conditions of pump energy and IL concentration, because the gain amplification cannot further compensate for the absorption losses for lasing.

The optical properties include the scattering coefficient $\mu _s$, the reduced scattering coefficient $\mu _s'$ and the absorption coefficient $\mu _a$. The calculation formulas of $\mu _s$ and $\mu _s'$ of IL refer to [30,41] in detail. The Beer-Lambert’s law which can describe the attenuation of light by absorption was applied for the characterization of $\mu _a$ of black ink. The peak wavelength $\lambda _p$ of RLs was taken into the above calculation of the wavelength dependent optical properties. Considering the fluctuation of DRLs, five repeated measurements for each combination of the variables of ink concentration $C_{ink}$ and pump energy $E_p$ were carried out, leading to the mean value and coefficient of variation of $\lambda _p$ and the corresponding $\mu _s$, $\mu _s'$ and $\mu _a$. These values were employed for the whole study. It should be noted that the temperature is less likely to induce peak wavelength shift in the present study. According to the heat absorption equation of water, the absorption of the maximum pump energy of 40.86 mJ only induces a temperature rise of around 0.13$^\circ C$ on the 75 ml water solution. Therefore, the peak wavelength and the wavelength dependent optical properties were calculated only taking the varied ink concentration and pump energy into consideration.

Table 1 presents the characterization results of (ink) concentration dependent RL peak wavelength and the corresponding optical properties. The influence of pump energy was averaged out over the whole pump energy range. The low coefficients of variation indicate that the pump energy does not significantly alter the optical properties of the phantoms.

Table 1. Mean value and coefficient of variation (CV) of $\lambda _p$, $\mu _s$, $\mu _s'$ and $\mu _a$ averaged over all pump energies from 1.82 mJ to 40.86 mJ for each ink concentration.

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Although DRLs with a wide range of ink concentrations were applied for the measurements and analysis, only three ones with low (0%), medium (0.09%) and high (0.13%) ink concentrations are presented in results to representatively show the absorption influence on the RL spectral behaviors. Furthermore, only the DRL phantom without ink is presented as an example for the statistical analysis of Lévy regime and RSB effect.

2.3 Lasing threshold extraction

Five spectra were taken for the spectral study. More spectra are feasible but not necessary to be measured for the aim of averaging. Averaging was achieved by using the excitation laser with a long pulse duration of nanosecond instead of picosecond, i.e., temporal averaging. In addition, the spatial averaging was also obtained in this study by using the diffusive random laser. The Brownian motion of the scatterers in the colloidal system does the spatial averaging effect.

Three spectral parameters - peak intensity, full width at half maximum (FWHM) and peak wavelength - were extracted as RL threshold estimates. These three parameters were extracted from the Lorentzian fit of the averaged RL spectrum. The Lorentzian fit includes the spectral modes in addition to the continuous background for the fitting. In other words, the spectral parameters were extracted from the fitting spectra instead of the raw spectra.

To investigate the lasing threshold, the above three parameters were plotted as a function of $E_p$ for each ink concentration. Two linear lines were further utilized for the fitting of peak intensity in terms of $E_p$ as a standard method to get the lasing threshold of RLs. Same method was applied for the fitting of FWHM in terms of $E_p$. The pump energy where the two linear lines are intersected was defined as the standard lasing threshold $E_{th}$.

The above approaches for lasing threshold extraction were applied for the RL with one certain ink concentration. This might cause a problem because $\mu _a$ of ink influences the RL spectral features too. Therefore, a plot of the spectral parameters in a global sense is preferred. This was done with a Python function of "matplotlib.pyplot.contourf": a filled contour plot of the each of three RL spectral parameters as a function of both $E_p$ and $\mu _a$. The parameter of the extrapolation level was set as 100 to give a continuous view.

2.4 Implementation of Lévy and RSB statistics

Hundred repeated spectra were taken for the statistical study of Lévy regime and RSB effect. The phantom consists of 2$\times$10$^{-4}$ g/ml R6G, 5% IL and no ink. The Lévy statistic parameter $\alpha$ is obtained using the Python function of "scipy.stats.levy_stable" with the peak intensities of hundred RL spectra as the input parameter. The RSB parameter $q$ is calculated from the Eq. (1) with N=100.

3. Results

3.1 DRL spectra and lasing behaviors

In Fig. 2, the variations of RL spectral parameters including the peak intensity, FWHM and peak wavelength are plotted as a function of pump energies $E_p$. The results are obtained from three representative DRLs with low (0%), medium (0.09%) and high (0.13%) ink concentrations.

 

Fig. 2. (a)-(c) The change of peak intensity, FWHM and peak wavelength with pump energy for low, medium and high ink concentrations of 0% (a), 0.09% (b) and 0.13% (c), respectively. The first column demonstrates the DRL spectra (black curves) and the Lorentzian fits of the spectra (blue curves). The positions of peak wavelengths deduced from the Lorentzian fits are marked with blue dotted lines. The second column illustrates the dynamics of spectral features. The error bars represent the standard deviations of spectral features over five repeated measurements. The stationary points or regime of the peak shift are marked with circles. To more clearly visualize the slope change of the input-output curves in the second column, two interacted linear lines to extract the threshold value are added. The points of the saturated intensity in (a) are excluded in the fitting process.

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3.1.1 Threshold estimated from FWHM and peak intensity

In general, the peak intensity increases and the FWHM decreases with the increasing $E_p$ for each ink concentration. Furthermore, $E_{th}$ increases with the increasing $C_{ink}$ as shown from Fig. 2(a) to (c). By the mean of the standard threshold indicator with FWHM, $E_{th}$ of 4.1 and 12.26 mJ are obtained for DRLs with $C_{ink}$ of 0% and 0.09%, respectively. No accurate estimate of $E_{th}$ is extracted for the DRL with $C_{ink}$ of 0.13% because of the failure of the two-linear-lines fitting. In the case of peak intensity, a much larger and unstable $E_{th}$ is estimated for all DRLs in comparison to the estimation using FWHM. The threshold values evaluated from the peak intensity are not displayed here and the discrepancy is discussed in section 4.

3.1.2 Threshold estimated from peak wavelength

The peak wavelength does not shift monotonically with increasing $E_p$. There seems to be a stationary point or a stationary regime where the peak shift changes gradually from blueshift to redshift. Moreover, this stationary point seems to correspond to the standard lasing threshold point extracted from the peak intensity or FWHM. It suggests that the peak shift might be a potential threshold indicator in DRLs.

In addition, the rates of blueshift and redshift are different for different ink concentrations: both blueshift and redshift tend to be more progressive with increasing ink concentrations. For $C_{ink}$ of 0.13% (Fig. 2(c)), the wavelength redshift plateaus, potentially indicating a special threshold regime of DRLs. In this threshold regime, due to higher absorption losses, more pump energy is required to compensate the losses for lasing, making the gain-loss balance inefficient and yielding a progressive threshold dynamics. Moreover, no obvious redshift is observed in Fig. 2(c) even at the highest $E_p$, indicating that the stimulated emission for lasing is not dominant over the spontaneous emission yet (i.e., the above threshold regime is not reached). This might also explain the failure of the two-linear-lines fitting of FWHM as the losses are still not dominated by gain. From the spatial point of view, the higher ink concentration prevents the light propagating deeper into the diffusion volume where the unpumped dyes are located. As a result, the peak redshift caused by the reabsorption of the unpumped dyes is weakened too until no redshift is observed.

For DRLs without ink absorption (Fig. 2(a)), the peak wavelength shows a sublinear redshift with increasing $E_p$ above threshold. In comparison to the linear redshift for RLs with medium ink concentration (Fig. 2(b)), this saturated redshift indicates a saturated gain absorption for lasing so that a saturated RL emission intensity is expected as well. Although not presented here, it still should be pointed out that the saturated redshift becomes more obvious with increasing IL concentration (more scattering and stronger optical feedback) and even plateaus when the scattering strength is high enough. Either an exhaustive excitation of the fluorescent dye or an enhanced light leakage at sample boundaries occurs here leading to the saturation of the redshift.

The responses of the peak wavelength shift on pump energy for RLs with varied ink concentrations are illustrated in one plot in Fig. 3. It can be seen directly that the trend of peak shift from blueshift to redshift is valid for all RLs. Moreover, the RL with less absorption losses shows higher degree of blueshift/redshift. The stationary point of RL with less absorption losses occurs at longer wavelength. It proves again the peak wavelength shift is not artificial and furthermore the peak wavelength shift is highly dependent on the optical properties of the RLs, e.g., the absorption of the RL medium.

 

Fig. 3. Peak wavelength shift responds to the pump energy for RLs made of different ink concentrations.

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3.1.3 Agreement of threshold estimates

In general, all $E_{th}$ estimates increase with $\mu _a$ as shown in Fig. 4(a). This is because more absorption losses need to be compensated with higher pump energy. In Fig. 4(a), the values of $E_{th}$ extracted from peak shift match well with the $E_{th}$ extracted from the FWHM, while the $E_{th}$ deduced from the peak intensity shows a significant discrepancy when the additional ink absorption is present. Furthermore, the two threshold estimates of FWHM and peak shift exhibit a linear relationship with each other despite the variations at the lower $\mu _a$, as shown in Fig. 4(b). The agreement could be further improved when higher resolution of $E_p$ around the threshold is applied. The $E_{th}$ of RLs with $\mu _a$ of 12.51 $cm^{-1}$ ($C_{ink}$ of 0.13%) is lacking here because of the failure of the two-linear-lines fitting.

 

Fig. 4. (a) Threshold estimates using peak shift, FWHM and peak intensity. (b) Evaluation of the agreement of two threshold estimates from peak shift and FWHM. Each data point is obtained from the RL with one certain ink concentration. $C_{ink}$ is increased from the left corner to the right corner of the plot.

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3.2 DRL threshold regime

In the contour plot in Fig. 5, it can be seen that the threshold boundaries obtained from the FWHM (Fig. 5(a)) and peak intensity (Fig. 5(c)) do not match with each other. The non-lasing regime observed from the peak intensity is much larger than that from FWHM. This means that in terms of peak intensity, more $E_p$ is required for lasing or less $\mu _a$ is required to transit from lasing to non-lasing regime. This discrepancy of lasing threshold is discussed in section 4.

 

Fig. 5. The influence of $\mu _a$ and $E_p$ on the RL spectral properties of FWHM (a), peak wavelength (b) and peak intensity (c). The boundaries of lasing threshold are marked with I and II. The asterisks mark the $E_{th}$ extracted from the FWHM. The triangles mark the $E_{th}$ extracted from the peak intensity and the squares mark the $E_{th}$ extracted from the peak wavelength. The three sets of $E_{th}$ marked by the symbols correspond to the data in Fig. 4(a), respectively.

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In the contour plot of the peak wavelength in Fig. 5(b), in addition to the threshold boundary that previously obtained from the stationary point of the peak shift, there seems another threshold boundary above the lasing threshold. More importantly, the two threshold boundaries roughly correspond to those found from the FWHM and peak intensity, respectively. It is likely that a global threshold regime is revealed from the contour plot of the peak wavelength, and the threshold boundary I reveals the starting points of the lasing threshold regime. Besides, the size of the threshold regime becomes larger with increasing $\mu _a$.

Another interesting finding is the intensity saturation. The largest intensity is expected when the RL phantom is prepared without ink and pumped by the highest energy. However, a relatively weaker intensity is obtained, as shown in the lower right corner in Fig. 5(c). This observation is in accordance with the saturated redshift observed in Fig. 2(a).

3.3 Threshold estimation from Lévy statistics and RSB effect

The RL spectra replicas and spectral behaviors including the lasing modes behaviors are shown at first in Fig. 6. Estimated from the stationary point of the peak shift in Fig. 6(e), the $E_{th}$ is around 3.55 mJ, similar to the value obtained in Fig. 2(a). A faint variation is expected considering the variations of the newly prepared phantom. The fluctuations of RL spectra replicas are visualized in Fig. 6(a)-(d). Although the spikes on the top of the individual spectra are random, certain modes signatures are still preserved after averaging, for instance, the modes marked by the gray lines in Fig. 6(b)-(d).

 

Fig. 6. (a)-(d) Hundred RL spectra replicas and their averages for $E_p$ below (a), near (b), above (c) and well above (d) the lasing threshold. The black curves represent the averaged spectra. The gray lines mark the emerging peaks. (e) RL spectral behaviors as a function of pump energy $E_p$. The investigated phantom here contains no ink.

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The first dominating mode appears around $E_{th}$ as shown from Fig. 6(b). This dominating mode emerges at the center of the spectrum, where the maximum gain is located and compensates the losses for lasing. When keeping increasing $E_p$ above $E_{th}$, more modes get lasing and shown on the top of the spectra (Fig. 6 (d)). A multimode regime is reached. Due to the presence of fluorescence reabsorption and reemission, the maximum gain shifts to the longer wavelength side, leading to a redshift of the dominating lasing mode (see comparison of the dominating mode in Fig. 6(c) and (d)). What is not shown here is that for a stronger scattering phantom where the redshift plateaus with increasing $E_p$, a dominating mode does not shift and instead, increases the mode intensity with increasing $E_p$. This proves again that the lasing mode wavelengths here are mainly determined by the maximum gain.

The peak intensities of the above RL replicas were chosen to calculate $\alpha$ and the results are shown in Fig. 7(g). The onset of Lévy regime ($0 \,<\,\alpha \,<\, 2$) occurs at $E_p$ of 1.86 mJ, slightly preceding the laser threshold of 3.55 mJ estimated from the peak shift. It is likely that a stronger fluctuation due to the spontaneous emission happens at $E_p$ of 1.86 mJ and leads to a stronger Lévy signature. At $E_p$ of 3.55 mJ, the fluctuation is weakened as the stimulated emission starts dominating over the spontaneous emission such that the first lasing mode also shows up in Fig. 6(b). Similar discrepancy was reported, in which the authors took the individual peaks around the gain center for Lévy calculation and the overall peak intensity for typical threshold estimation [22]. A lower threshold value was obtained from the Lévy estimate as well. In Fig. 7(g), the Lévy threshold regime covers a range approximately from 2 mJ to 10 mJ. This regime basically agrees with the threshold regime revealed by the peak wavelength shift in Fig. 7(g) upper graph.

 

Fig. 7. (a)-(f) $P(q)$ distribution for $E_p$ below (a),(b), near (c), above (d) and well above (e),(f) $E_{th}$. (g) Lévy statistics of RL peak intensity and the RSB phase transition, represented by the deterministic parameters of $\alpha$ and $|q_{max}|$, respectively. The threshold regime is marked by the gray area. A colormap of the RL spectral features is plotted too for the comparison of the threshold regime.

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The RSB estimate used the whole spectral intensity for calculation and the results of $|q_{max}|$ are shown in Fig. 7(g). Interestingly in contrast to the literature that the onset of the threshold was denoted by the transition from $|q_{max}|\approx 0$ to $|q_{max}|\approx 1$, the threshold onset in current study is observed at the strongest RSB effect, i.e., at $E_p$ of 1.86 mJ where the correlation distribution wings are split the most, as shown in Fig. 7(b). Further increasing $E_p$ results in a decrease from $|q_{max}|\approx 1$ to $|q_{max}|\approx 0$, after which a Gaussian-like shape of the correlation distribution dominates (Fig. 7(e) and (f)). A regime is also revealed from the $|q_{max}|$ and comparable to the threshold regime from Lévy statistics and peak wavelength shift, indicating a true threshold regime from RSB estimate.

4. Discussion

The weakness of high lasing threshold of RL is well recognized. Although several approaches attempt to precisely characterize the threshold values [20,21,23,42], there are always discrepancies presented [22–24]. The difficulty lies on an unambiguous threshold definition, for example, eight distinct threshold definitions are summarized and discussed [43]. The equivalence of these definitions are met in a macroscopic laser, but not in micro-/nano-lasers where the phase transition from incoherent to fully coherent photons is gradual [43]. This exactly happens in RLs too and also is the source of the two remaining problems in this study: one is the discrepancy of the lasing thresholds from FWHM and peak intensity, the other one is the "contradictory" behavior of the RSB effect around the threshold in comparison to the literature.

The RL threshold was reached during the narrowing of FWHM, in particular when the first lasing mode emerged on the spectrum (Figs. 6(b) and (e)). However, the threshold extracted from the enhanced peak intensity took place at higher pump energy than that for the first lasing mode. A sketch of the above discrepancy is illustrated in Fig. 8(b). In this figure, the threshold points extracted from FWHM and peak intensity construct a threshold regime. This threshold regime defines a transition regime where the light emission is transited from totally spontaneous emission dominating to totally stimulated emission dominating when the pump energy is increased. Such threshold regime is barely to be observed in a general laser (Fig. 8(a)) where the laser system is a relatively less lossy system. The stimulated emission immediately dominates over the spontaneous emission when crossing the threshold.

 

Fig. 8. Sketch of the threshold behavior of (a) general lasers and (b) random laser. The threshold regime defines a transition regime where the light emission is transited from totally spontaneous emission dominating to totally stimulated emission dominating. In a general laser, the stimulated emission is immediately dominant over the spontaneous emission when the threshold pump energy $E_{th}$ is reached. A threshold regime is therefore so tiny that only a threshold point is extracted from both FWHM and intensity behaviors. In a random laser, the threshold regime is visualized, with the threshold point extracted from FWHM as the starting point and the threshold point extracted from intensity as the ending point. The presence of the threshold regime in random laser is because the random laser system is a multimode system and also is a relatively more lossy system. More losses are required to be compensated to get the lasing of multiple modes.

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The "delayed" threshold in random laser could be understood that more losses need to be compensated and more intensity is required for the lasing of other modes. In this context, the first lasing mode has a dominating influence on the FWHM narrowing, while the ensemble intensity of all modes gives rise to the pronounced increase of the RL peak intensity. The threshold difference from each lasing modes was observed in this study too (not shown) and was also numerically evidenced by other research: in a multimode RL, the threshold for the second lasing mode is higher compared to the first lasing mode [32]. In this circumstance, the FWHM behavior reveals the single mode lasing threshold, while the peak intensity reveals the multimode lasing threshold. More detailed investigation of the multimode interactions like self-saturation and cross-saturation is out of the scope of this study. Overall, they result in a much higher threshold extracted from the peak intensity. Such discrepancy was observed to be stronger in a RL phantom with higher ink concentrations as more absorption losses for each lasing mode need compensated additionally.

The discrepancy of RL threshold estimation using the RSB effect is common in literature [22–24]. In a weakly scattering optofluidic RL where the scatterers were fixed, the onset of RSB occurred even though $|q_{max}|\approx 0$ [22]. In the first RL experimental verification of RSB effect, no RSB signature was observed in a liquid RL phantom where scatterers moved among consecutive shots [23]. Later, other authors also carried out the RSB analysis on a fluid RL phantom, and pointed out that the lacking of RSB signature is attributed to the way lower pump energy. In this case, only weak fluctuation is impossible to probe a RSB effect [24]. To emphasize again, the RL threshold estimation using RSB effect requires identical experimental conditions such as quenched disorder system and an intrinsic phase transition (The replica nature of the pump laser was confirmed by the small fluctuation of the pump laser intensity). The former one is tricky as the quenched disorder could be also fulfilled in a liquid RL phantom if the motion of scatterers is frozen in the RL lifetime [24]. Otherwise, it is difficult to determine the appearance of RSB effect is caused by the intrinsic phase transition or by the fluctuation of the disorder system. To give an example, the onset of RSB effect appeared also in the Gaussian regime below the RL threshold [24]. Indeed, this is also the reason that the RSB effect was randomly observed below the lasing threshold in current study due to the possibly dominant fluctuation of the disorder over the laser fluctuation. Therefore, the onset of RSB not necessarily reflects the onset of the RL threshold.

Conversely, the onset of RL threshold must show RSB effect, probably even the strongest RSB effect where the distribution wings were separated the most (Fig. 7(b)). More remarkably, not only the onset but also the whole threshold regime showed RSB effect, despite the RSB wings were in a trend of broadening ($|q_{max}|$ decaying from 1 to 0 in Fig. 7(g)). Although the $|q_{max}|$ in most researches remained nearly constant around 1 above the threshold [22–24,27,44], still one research found the trend of decaying $|q_{max}|$ above the threshold in a solid-state RL [26]. The current study proves that such decaying RSB effect above the threshold is universal also in a liquid-state RL. The reason behind is the decreased fraction of incoherent emission which was found to have a higher non-zero correlation compared to the coherent part of the RL emission [44]. This point again agrees with the phase transition nature of RL: a fully coherent emission is gradually established in the threshold regime. In this context, the fluctuation contribution of the scatterers motion is negligible in comparison to the intrinsic phase transition (the quantum fluctuation statistics change when the emission is changed from the dominant spontaneous emission to stimulated emission, and the fluctuation effect of the stimulated emission dominates over the scatterer motion above the threshold for lasing). It is noted that another RSB regime showing increasing $|q_{max}|$ was also observed well above the threshold (appropriately 50 times larger than $E_{th}$) following the Gaussian-shape distribution of the fluctuation correlation. There might be another phase transition, but this regime is never explored in the RL field and is also out of the scope of the present study.

The current discussions are based on a diffusive random laser, which is made of colloidal scattering particles and fluorescent dyes. In principle, the lasing effect (i.e., the lasing threshold and its resultant spectral changes and statistical changes) of all random laser systems should be similar if not talking about the additional optical feedback mechanisms and nonlinear effects. Hence, the observation and discussion on the current diffusive bulk random laser system are also beneficial for other random laser systems.

5. Conclusion

The lasing behaviors of RLs are quite different from the macroscopic cavity lasers. Even though the RLs were generated from the bulky phantoms, the lasing behaviors of RLs are more like micro-/nano-lasers: the spontaneous (incoherent) emission is dominant by the stimulated (coherent) emission gradually, due to the existence of multimodes. This gradual change constructs a dynamical threshold regime which was visualized in this study by one simple spectral feature: the peak wavelength shift. The tunability was not only demonstrated, but also used to characterize the dynamic lasing threshold of the diffusive random laser. This theory may also apply to other random laser systems where the light diffusion and the effect of dye reabsorption and reemission are both present. This finding strengthens the understanding of the lasing nature of RLs, and might facilitate in potential manufacture of a coherence-tunable RL. Moreover, the peak wavelength could provide a measure of the coherence.

The threshold regime of RL revealed by the peak wavelength was also evidenced using the state-of-the-art approaches of Lévy statistics and RSB effect. The previous research of the RSB effect focused on the characterization of the lasing onset of RLs, while the present study revealed that the RSB effect is universal in the entire threshold regime of a liquid diffusive RL. This new finding reinforces the importance of RSB effect from the spin-glass theory, predicting not only the occurrence but also the crossover of the phase transitions. This observation and the physical reason behind provide a physical explanation of similar experimental observations which are never deeply understood, and might be useful in manipulating the system disorders and optical fluctuations, for instance, in the perspective of the multimode coupling.