## Abstract

The hybrid electronically addressable random (HEAR) laser is a novel type of random fiber laser that presents the remarkable property of selection of the fiber section with lasing emission. Here we present a joint analysis of the correlations between intensity fluctuations at distinct wavelengths and replica symmetry breaking (RSB) behavior of the HEAR laser. We introduce a modified Pearson coefficient that simultaneously comprises both the Parisi overlap parameter and standard Pearson correlation coefficient. Our results highlight the contrast between the correlations and presence or not of RSB phenomenon in the spontaneous emission behavior well below threshold, replica-symmetric ASE regime slightly below threshold, and RSB phase with random lasing emission above threshold. In particular, in the latter we find that the onset of RSB behavior is accompanied by a stochastic dynamics of the lasing modes, leading to competition for gain intertwined with correlation and anti-correlation between modes in this complex photonic phase.

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

## 1. Introduction

Random Fiber Lasers (RFLs) are the one-dimensional version of random lasers (RLs), which have in common the optical feedback mechanism required for laser operation. Whilst in regular lasers the feedback is generally achieved with a set of two mirrors, one being slightly transmissive at the lasing wavelength, in RLs and RFLs it results from the random scattering of photons in a disordered medium. This concept was first proposed by Letokhov in 1967 [1], but experimentally realized without ambiguity only in 1994 [2] using a colloid consisting of a laser dye Rh6G as the gain medium and TiO_{2} nanoparticles (250 nm) as scatterers, with the colloid in a cuvette pumped by the second harmonic of a nanosecond pulse duration Nd:YAG laser. Recent reviews on nanocomposite based RLs and more general reviews on two- and three-dimensional RLs can be found in [3–6].

In 2007 de Matos and co-authors demonstrated the first RFL using a similar gain, scattering, and pump source combination as in [2], but with the colloid inserted in a quasi-one-dimensional photonic crystal fiber, rather than in a three-dimensional cuvette [7]. Thereafter, the research field on RFLs has grown remarkably and much more efficient and technologically applicable RFL configurations have been developed [8–10]. Nowadays RFLs have found several relevant applications [8,10], with advances, e.g., in optical amplification [11] and effective use for sensing [12,13], as light sources for optical imaging [14,15], highly sensitive sensor for ultrasound detection [16], and high temperature sensing [17]. RFLs have also been exploited as platforms for studies of statistical physics complex phenomena, such as Lévy-like statistics [18–21] and photonic spin-glass phase with replica symmetry breaking (RSB) [22–25], as reviewed in [26], and turbulence-like behavior of emitted intensity [27]. Some quite interesting recent progress on RLs and RFLs also include studies on the statistics of power fluctuations, RSB regime, and RL spectroscopy in a diversity of photonic structures [28–32].

An important advance on RFL systems was the introduction of random fiber Bragg gratings in erbium doped fibers [33,34], followed by yet a new RFL configuration, now widely exploited, using Raman gain and Rayleigh scattering in conventional telecom fibers [35]. Brillouin gain can be also employed with Rayleigh scattering [36]. Another interesting development in RFLs concerns the use of hybrid schemes, whereby Brillouin-Raman [37], Brillouin-erbium [38], erbium-Raman [39], and a common cavity ytterbium-Raman [40] gain media have been exploited. Other gain media used along with Rayleigh scattering are semiconductor optical amplifiers (SOAs) [41,42]. Recently, a notable advance was achieved with the demonstration of a hybrid RFL using a SOA + erbium fiber dual gain combination which is electronically addressable, thus being able to spatially select the lasing position within the erbium portion of the random cavity [43]. Scattering in these HEAR (hybrid electronically addressable random) lasers arises from the random distribution of scatterers in the erbium fiber. The HEAR laser actually comprises of a weakly scattering system and its modes are extended.

In this work, we report on an experimental study, supported by theoretical analysis, of the intensity fluctuations in the HEAR laser presented in [43], which enabled us to simultaneously analyze and obtain, from a single set of measurements, the Parisi overlap parameter showing the phase transition from the photonic paramagnetic to the glassy phase with RSB, as well as the Pearson correlation coefficient related to the dynamics of intensity fluctuations in the HEAR laser. Our analysis comprises both the prelasing behavior below threshold and the random lasing emission regime above threshold. The use of Pearson coefficients to evaluate the fluctuation dynamics of RL systems was introduced in [44–47], but, to our knowledge, it has not been applied to RFLs so far. Also, we note that Refs. [44–46] did not explore the connection between the Pearson coefficient and the RSB properties, while Ref. [47] employed a covariant measure with normalization distinct from the standard one that would hamper our proposal of a single modified Pearson coefficient (see below), and also did not deepen the study of the emergence of the correlation and anti-correlation between intensity fluctuations, which are essential to understand the RSB glassy phase above threshold in RLs.

Although cavity-less, RFLs are not modeless [48–50] and the characterization of the correlations between modes in the random lasing regime can be suitably captured using the Pearson coefficient. Here we show as the main result that, by considering a modified Pearson equation, we can retrieve simultaneously both the Parisi overlap parameter as well as the standard Pearson coefficient, thus enabling a joint analysis of the role of the correlations between modes to the onset of the RSB phase above threshold in the HEAR laser. The modified Pearson coefficient proposed here can be applied equally to any RFL or RL configuration.

This article is organized as follows. In Section 2 we present details of the HEAR laser system studied in this work and the experimental setup. We also characterize the set of 1000 spectra obtained in the photonic regimes from well below to well above threshold. These spectra are considered in Section 3 for the calculation of the modified Pearson coefficient, with the joint analysis of the RSB and correlations results. At last, final remarks and conclusions are left to Section 4.

## 2. Experimental details and spectra characterization of the HEAR laser

The HEAR laser studied here was first demonstrated in [43]. It is a robust turnkey RFL configuration with the experimental setup depicted in Fig. 1(a). A semiconductor optical amplifier (SOA) is electrically pumped by 10-ns duration current pulses from a pulse generator externally triggered by a RF signal. Part of the optical pulse from the SOA is coupled via a long piece of passive fiber (230.5 m) to an erbium-doped fiber pumped at 1480 nm, which both amplifies the signal and backscatters light along its length. The choice of the fiber length was made to render the resonance frequency at a convenient value for the electronics available (∼500 kHz). In fact, it could be varied from several kHz to a few MHz [43]. It is also important to notice that the fiber end is angle cleaved, the pump light hardly reaches the end of the 90 m long EDF, and we experimentally found that the end reflection is insufficient to sustain laser action. Therefore, the feedback is indeed due to the Rayleigh scattering.

After passing again through coupler CP1, the amplified backscattered light recirculates in the SOA, which is electrically pumped again. Random lasing takes place when the amplification in the SOA and erbium-doped fiber (EDF) is sufficient to compensate for the weak ≈ −60 dB/m backscattering level of light in the EDF and additional cavity losses. By adjusting the current pulse frequency, it is possible to choose the section of the EDF that backscatters light in synchronism with the periodic gain of the SOA. In other words, the HEAR laser has an effective cavity length that can be adjusted electronically.

A polarization controller maximizes the gain of the hybrid cavity that incorporates two types of optical amplifiers. Coupler CP2 splits the cavity output for simultaneous spectral analysis in an optical spectrum analyzer (OSA, Agilent 861410B) and time domain characterization with a photodetector (PD, Thorlabs PDB460C, 200MHz) and oscilloscope. The length of the passive fiber is used to reduce the repetition rate of the HEAR laser to the 500-600 kHz range, which is convenient from the available equipment point-of-view. Figures 1(b) and 1(c) show, respectively, the linewidth reduction and output optical power as a function of the input pump power for the HEAR laser used in this work.

To analyze the intensity fluctuations, 1000 spectra shown in Fig. 2 were obtained for each relative pump power P/P_{th}, normalized by the threshold power, P_{th} = 120 mW. We investigated the regimes well below (Fig. 2(a)), below (Figs. 2(b) and 2(c)), slightly above (Fig. 2(d)), and well above (Figs. 2(e) and 2(f)) threshold. The spectrum in Fig. 2(a), for P/P_{th} = 0.1, is dominated by the characteristic broadband amplified spontaneous emission from the SOA, weakly modulated by the injection of the spontaneous emission from the EDF. In contrast, it can be seen in Figs. 2(b) and 2(c), respectively for P/P_{th} = 0.5 and 0.9, that the ASE from the EDF dominates, leading to the two well-known erbium bands with relatively short bandwidth, as compared with Fig. 2(a). Indeed, we note that Fig. 2(a) is plotted in the wavelength interval 1500-1600 nm, whereas in Figs. 2(b)–2(f) a narrower range (1530-1580 nm) is displayed. We also observe rather weak intensity fluctuations in the prelasing regime of Figs. 2(a)–2(c), which arise as a consequence of the Gaussian-distributed intensity with statistical properties governed by the central limit theorem (CLT) [18,19].

On the other hand, as the excitation power is increased above threshold, we notice in Figs. 2(d), 2(e) and 2(f), for P/P_{th} = 1.1, 1.5 and 2.0, respectively, a strong bandwidth narrowing and much larger intensity fluctuations. In contrast with the Gaussian statistical behavior below threshold (Figs. 2(a)–2(c)), this regime with random lasing emission has been characterized in an erbium-based RFL with random fiber Bragg gratings as displaying Lévy-like statistics with much larger variance in the distribution of emitted intensities [18,19].

The spectra shown in Fig. 2 will serve as the basis for the calculation of the modified Pearson coefficient and the analysis of dynamics correlations between intensity fluctuations and RSB behavior in the next section.

## 3. Results and discussion

To characterize the RSB phenomenon jointly with the correlations between modes, we introduce below a modified Pearson coefficient. Let ${I_{\gamma i}}$ denote the intensity emitted by the HEAR laser at the wavelength ${\lambda _i}$ in the spectrum $\gamma $. We also define ${I_i}$ as the intensity at the wavelength ${\lambda _i}$ averaged over the spectra and $\overline {{\Delta _{\gamma i}}} $ as the relative difference (fluctuation) with respect to this average, $\overline {{\Delta _{\gamma i}}} = {\Delta _{\gamma i}}/\sqrt {\mathop \sum \nolimits_K {{({{\Delta _{\gamma i}}} )}^2}} $, where ${\Delta _{\gamma i}} = {I_{\gamma i}} - {I_i}$ and capital Latin letters represent either the spectra (e.g., $K = \gamma $) or wavelengths (e.g., $K = i$). With these definitions, we propose the modified Pearson coefficient,

where above we use Einstein summation convention over repeated indexes. The normalization of $\overline {{\Delta _{\gamma i}}} $ by the factor $\sqrt {\mathop \sum \nolimits_K {{({{\Delta _{\gamma i}}} )}^2}} $ leads ${P_{MN}}$ to be restricted to the domain ${P_{MN}} \in [{ - 1,1} ]$.This modified Pearson coefficient comprises both the Parisi parameter ${q_{\gamma \beta }}$ of replica overlaps, which characterizes the photonic glassy phase with RSB, as well as the Pearson correlation ${C_{ij}}$ between intensity fluctuations. Indeed, on the one hand, by setting in Eq. (1) the spectrum indexes $M = \gamma $ and $N = \; \beta $ and the wavelength index $K = \; i$, we obtain the Parisi overlap parameter [22,23,51], expressed by

In the present context, each spectrum is considered a replica emitted by the HEAR laser, i.e., a copy of the its photonic signature generated under identical experimental conditions. The probability density function $P(q )$, analogue to the Parisi order parameter of spin glass theory in magnetism [51], describes the distribution of replica overlap values $q = {q_{\gamma \beta }}.$ It signals [51] a photonic replica-symmetric paramagnetic-like phase or a RSB glassy phase, respectively, if it presents a single maximum profile with *q*-values distributed around *q* = 0 (no RSB) or if a double peaked $P(q )$ emerges (RSB), with saturation occurring when the maxima $P(q )$ take place at the extreme values *q* = ± 1.

On the other hand, by considering in Eq. (1) the wavelength indexes $M = i$ and $N = \; j$ and the spectrum index $K = \gamma $, we write the standard Pearson correlation coefficient [44,47] between intensity fluctuations at wavelengths ${\lambda _i}$ and ${\lambda _j},$

We note that, differently from Eq. (2) in which the summations are over the wavelengths, in the Pearson coefficient ${C_{ij}}$ the sums are over the spectra $\gamma $ emitted at distinct times. In other words, Eq. (3) takes into account the dynamics evolution of the correlation between intensity fluctuations $.$ For instance, if two modes are present in the system with wavelengths ${\lambda _i}$ and ${\lambda _j}$ a null value of ${C_{ij}}$ implies that they behave statistically in an uncorrelated way over the time interval of the spectra series. Conversely, a positive (negative) ${C_{ij}}$ indicates that statistical fluctuations in the intensity of mode ${\lambda _i}$ are positively (negatively) correlated with those of mode ${\lambda _j}$, so that the spatially overlapped coupled modes share (compete for) gain along the time measurement series.

We start by identifying the presence or not of photonic RSB behavior in the HEAR laser. Figure 3 shows the distribution $P(q )$ of replica overlap values for the same excitation powers of Fig. 2. We first observe for P/P_{th} = 0.1 and 0.5 (Figs. 3(a) and 3(b), respectively) that the *q*-values are distributed around *q* = 0, consistently with the prelasing regime characterized by symmetric replicas and photonic paramagnetic-like phase with incoherent emission. We note in particular that the somewhat stronger fluctuations in Fig. 3(a) can be smoothed out towards a single-peaked $P(q )$ centered around *q* = 0 when a larger number of replicas (spectra) is considered. On the other hand, for P/P_{th} = 0.9 (Fig. 3(c)) the vicinity of the threshold marks the onset of RSB evidenced by a qualitative change in the distribution $P(q )$, which starts to present two maxima although not yet saturated at the extreme values *q* = ± 1. In fact, this saturation takes place only above threshold, as seen in Figs. 3(d), 3(e) and 3(f), respectively for P/P_{th} = 1.1, 1.5 and 2.0. In this case, a full RSB glassy state sets in, with the random lasing modes unable to oscillate in a synchronous way, although presenting nontrivial correlations that are inferred by the Parisi overlap parameter [51].

We now turn to the analysis of the correlations between intensity fluctuations on the light of the RSB results presented above. To understand how these two aspects are intrinsically intertwined, we remark that the photonic RSB glassy phase was first obtained in a Hamiltonian-like formulation of RL systems in both closed and open cavities (see the review [51] and references therein). Depending on the excitation energy, nonlinearity degree, and disorder strength of the active gain medium, the nonlinear interactions between modes with random amplitudes can lead to a rich photonic phase diagram, including a phase locking wave regime, a replica-symmetric mode locking laser phase with modes oscillating coherently with the same phase, and a RSB glassy regime with frustrated synchronous oscillation and nontrivial modes correlations [51]. In the present context of the HEAR laser, it is thus expected that the replica symmetric ASE and RSB random lasing regimes display not only different profiles of the Parisi overlap distribution $P(q )$ but also quite distinct signatures of correlations captured by the Pearson coefficient.

Figures 4 and 5 show heatmap plots of the HEAR laser with values of the Pearson coefficient representing cross-correlations between intensity fluctuations at different wavelengths ${\lambda _i}$ and ${\lambda _j}$, as well as self-correlations (diagonal lines with ${\lambda _i} = {\lambda _j}$). The color code essentially depicts in green absence of correlations between intensity fluctuations, whereas red and blue indicate correlated and anti-correlated fluctuations, respectively. We also remark that in these normalized plots the laser intensity at each wavelength is not apparent. Once again, we analyze in Figs. 4 and 5 the same excitation powers as in Figs. 2 and 3. We also display on the left column of Figs. 4 and 5 a few representative spectra for each excitation power (taken from shots 10, 100 and 1000), with a zoom in the wavelength range of interest around the maximum intensity of Fig. 2.

Figure 4 displays results for excitation powers below threshold. In Fig. 4(a) the Pearson coefficient was calculated for P/P_{th} = 0.1, with a relatively wide wavelength range covering the interval from 1500 nm to 1600 nm. No lasing modes are present in this regime well below threshold, with the mostly green heatmap plot signaling nearly uncorrelated intensity fluctuations when only spontaneous emission occurs. We also note the presence of some orange-to-red points with Pearson coefficient values up to 0.5 that can be possibly anticipating the onset of correlations observed at higher excitation powers. The thin diagonal red line represents self-correlations. This picture agrees with the replica-symmetric paramagnetic-like profile of $P(q )$ shown in Fig. 3(a).

As the excitation power is increased, the ASE regime is present for P/P_{th} = 0.5 and 0.9 (Figs. 4(b) and 4(c), respectively). In this case, on the right column of Figs. 4(b) and 4(c) we display the heatmaps for wavelengths in the much narrower range 1550-1555 nm around the maximum intensity of the band near 1552.5 nm. In this blown-up view, it is interesting to notice the onset of both red and blue regions in the heatmaps, indicating, respectively, the emergence of correlation and anti-correlation between different wavelengths of the emitted light. So, Figs. 4(b) and 4(c) denote that the dynamical evolutions of the intensity fluctuations in these regions are not statistically independent. Instead, in these cases they behave either correlatedly (red) or anti-correlatedly (blue), showing the beginning of a competition for gain, which is fully absent for P/P_{th} = 0.1 and is clearly evidenced in the random lasing regime above threshold. It may also imply that the roundtrip of photons back to the pumped region is defined before the threshold is reached. The presence of stronger correlations for P/P_{th} = 0.5 and 0.9, if compared with the spontaneous emission regime for P/P_{th} = 0.1, is also reflected in the wider distributions $P(q )$ observed in Figs. 3(b) and 3(c), which reach out to larger |*q*| values, although not yet displaying the bimodal profile with extreme side peaks of the RFL regime above threshold.

The RFL regime of the HEAR laser above threshold is considered in Fig. 5, which displays results for the relative excitation powers P/P_{th} = 1.1 (Fig. 5(a)), P/P_{th} = 1.5 (Fig. 5(b)), and P/P_{th} = 2.0 (Fig. 5(c)). An initial visual inspection already evidences relevant changes in the Pearson correlation profiles with respect to Fig. 4.

Although in Fig. 5 the lasing modes are bounded to the narrow bandwidth interval around 1552-1553 nm, we first note that the intensity fluctuations outside this range are no longer statistically independent, as indicated by the correspondent red regions in the heatmaps. Most interestingly, we now observe in the range 1552-1553 nm the presence of both correlation (red) and anti-correlation (blue) between lasing modes (see right column of Fig. 5) that share or compete for gain in the nonlinear regime above threshold. As shown in the selected spectra in the left column of Fig. 5, the sets of modes around 1552 nm and 1553 nm do not remain steady along the measurement time, but, instead, change stochastically from pulse to pulse with a complex dynamics that leads the gain distribution among the modes to be non-homogeneous and time dependent. For P/P_{th} = 1.1 in Fig. 5(a), with the modes around 1552 nm dominating the laser spectrum, the heatmap shows strong (self)correlation at the laser wavelength and strong anti-correlation between the laser peak and other wavelengths. This is a consequence of the competition for gain above the threshold, in which an increase in the intensity of the laser peak leads to a reduction in the intensity at other wavelengths.

Remarkably, for a higher excitation power, P/P_{th} = 1.5, the left column of Fig. 5(b) shows the emergence of another set of lasing modes around 1553 nm, which will compete for gain with the one around 1552 nm already observed in Fig. 5(a) for P/P_{th} = 1.1. This onset is also accompanied by a significant reduction of the emitted intensity at wavelengths outside the lasing range 1552-1553 nm. In this case, however, since the modes around 1552 nm in Fig. 5(b) are still prevalent over the others, as in Fig. 5(a), the overall qualitative picture of the heatmap in the interval around 1552-1553 nm (right column of Fig. 5(b)) is not too distinct from that of the right column of Fig. 5(a).

On the other hand, as the excitation power is increased further, Fig. 5(c) displays for P/P_{th} = 2.0 a more balanced competition for gain. As seen in the left column of Fig. 5(c), this competition is rather concentrated in the sets of lasing modes around 1552 nm and 1553 nm, and is also reflected in the dynamic alternation of the dominant intensity peak. Indeed, in the zoomed-in plot in the right column of Fig. 5(c) we notice two small blue squares, centered at the heatmap wavelength coordinates (1551.8 nm, 1552.7 nm) and (1552.7 nm, 1551.8 nm), which denote this anti-correlation between the two sets of competing modes around 1552 nm and 1553 nm. Due to the emergence of this balanced competition between modes that absorb and concentrate in a stochastic dynamical way essentially all the gain, an important qualitative change is noticed when the right columns of Figs. 5(b) and 5(c) are compared. In this case, apart from the strong correlation and anti-correlation between the competing lasing modes around 1552 nm and 1553 nm, the intensity fluctuations of these modes are nearly statistically independent (almost no correlation) from the intensity fluctuations at wavelengths outside the lasing range 1552-1553 nm, as indicated by the green regions in the heatmap of the right column of Fig. 5(c).

The above picture is consistent with the photonic RSB glassy phase of the HEAR laser above threshold. Indeed, at each pulse in the RSB regime a specific set of modes is activated in a nondeterministic way, leading to photonic frustration in the sense that the coherent oscillation of a given subset of coupled modes dominates and inhibits the coherent oscillation of the others. Accordingly, the emergence of RSB behavior coincides with the increase of the competition for gain evidenced in Fig. 5, which prevents the synchronous oscillation of modes while simultaneously yielding each pair of replicas to display either full correlation (*q* = 1) or anti-correlation (*q* = –1), in agreement with the results of Figs. 3(d)–3(f) for the Parisi overlap distribution $P(q )$.

## 4. Conclusions

In conclusion, we have presented in this work a joint analysis of the correlations between intensity fluctuations and RSB behavior of the HEAR laser through the introduction of a modified Pearson coefficient that simultaneously comprises both the Parisi overlap parameter and standard Pearson correlation coefficient.

The results presented above were obtained with a finite sweep time of the optical spectrum analyzer (∼1 sec) and are therefore an average of many single spectra. A very fast spectrometer capable of single shot spectral acquisition with adequate resolution would be a useful instrument, providing a better snapshot of the spectral content of the laser. It would allow monitoring the time evolution of the correlations for consecutive paths within the scattering media, a regime that is far from what was possible here.

The HEAR laser studied in this work is a quasi-one-dimensional RFL whose effective cavity length can be adjusted electronically, thus enabling to choose the section of the fiber that backscatters light in synchronism with the periodic gain of the SOA. In this system, we have studied and highlighted the contrast between the correlations between intensity fluctuations and the presence or not of RSB phenomenon in the spontaneous emission behavior well below threshold, ASE regime slightly below threshold, and RSB phase with random lasing emission above threshold. In particular, in the latter we have evidenced the role of the stochastic dynamics of the lasing modes to the competition for gain and frustration of modes phase synchrony in this complex photonic phase.

The electronic addressable lasing emission properties of HEAR lasers make these systems potential sources for important applications, such as in imaging and sensing. We hope our work contributes to advance the understanding of their emission regimes, as well as stimulates further theoretical and experimental research on these novel laser systems.

## Funding

Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Conselho Nacional de Desenvolvimento Científico e Tecnológico.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

## References

**1. **V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Zh. Eksp. Teor. Fiz. **53**, 1442 (1967); Sov. Phys. JETP 26, 835 (1968).

**2. **N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature **368**(6470), 436–438 (1994). [CrossRef]

**3. **A. S. L. Gomes, “Nanocomposite-based random lasers: a review on basics and applications,” in * Nanocomposites for Photonic and Electronic Applications*, edited by L. R. P. Kassab, S. J. L. Ribeiro, and R. Rangel-Rojo, eds., (Elsevier, Amsterdam, 2020), chap. 3.

**4. **F. Luan, B. Gu, A. S. L. Gomes, Y. Ken-Tye, S. Wen, and P. N. Prasad, “Lasing in nanocomposite random media,” Nano Today **10**(2), 168–192 (2015). [CrossRef]

**5. **D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. **4**(5), 359–367 (2008). [CrossRef]

**6. **H. Cao, “Review on latest developments in random lasers with coherent feedback,” J. Phys. A: Math. Gen. **38**(49), 10497–10535 (2005). [CrossRef]

**7. **C. J. S. de Matos, L. de S. Menezes, A. M. Brito-Silva, M. A. M. Gamez, A. S. L. Gomes, and C. B. de Araújo, “Random fiber laser,” Phys. Rev. Lett. **99**(15), 153903 (2007). [CrossRef]

**8. **S. K. Turitsyn, S. A. Babin, D. V. Churkin, I. D. Vatnik, M. Nikulin, and E. V. Podivilov, “Random distributed feedback fibre lasers,” Phys. Rep. **542**(2), 133–193 (2014). [CrossRef]

**9. **D. V. Churkin, S. Sugavanam, I. D. Vatnik, Z. Wang, E. V. Podivilov, S. A. Babin, Y. Rao, and S. K. Turitsyn, “Recent advances in fundamentals and applications of random fiber lasers,” Adv. Opt. Photon. **7**(3), 516 (2015). [CrossRef]

**10. **X. Du, H. Zhang, H. Xiao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “High-power random distributed feedback fiber laser: From science to application,” Ann. Phys. **528**(9-10), 649–662 (2016). [CrossRef]

**11. **P. Rosa, M. Tan, S. T. Le, I. D. Philips, J. D. Ania-Castañón, S. Sygletos, and P. Harper, “Unrepeatered DP-QPSK transmission over 352.8 km SMF using random DFB fibre laser amplification,” IEEE Photonics Technol. Lett. **27**(11), 1189–1192 (2015). [CrossRef]

**12. **Y. Xu, M. Zhang, P. Lu, S. Mihailov, and X. Bao, “Multi-parameter sensor based on random fiber lasers,” AIP Adv. **6**(9), 095009 (2016). [CrossRef]

**13. **H. Chen, S. Gao, M. Zhang, J. Zhang, L. Qiao, T. Wang, F. Gao, X. Hu, S. Li, and Y. Zhu, “Advances in random fiber lasers and their sensing application,” Sensors **20**(21), 6122 (2020). [CrossRef]

**14. **R. Ma, Y. J. Rao, W. L. Zhang, and B. Hu, “Multimode random fiber laser for speckle-free imaging,” IEEE J. Select. Topics Quantum Electron. **25**, 0900106 (2019). [CrossRef]

**15. **J. Y. Guo, W. L. Zhang, Y. J. Rao, H. H. Zhang, R. Ma, D. S. Lopes, I. C. X. Lins, and A. S. L. Gomes, “High contrast dental imaging using a random fiber laser in backscattering configuration,” OSA Continuum **3**(4), 759 (2020). [CrossRef]

**16. **Y. Xu, L. Zhang, S. Gao, P. Lu, S. Mihailov, and X. Bao, “Highly sensitive fiber random-grating-based random laser sensor for ultrasound detection,” Opt. Lett. **42**(7), 1353 (2017). [CrossRef]

**17. **J. Deng, D. V. Churkin, Z. Xu, and X. Shu, “Random fiber laser based on a partial-reflection random fiber grating for high temperature sensing,” Opt. Lett. **46**(5), 957 (2021). [CrossRef]

**18. **B. C. Lima, P. I. R. Pincheira, E. P. Raposo, L. de S. Menezes, C. B. de Araújo, A. S. L. Gomes, and R. Kashyap, “Extreme-value statistics of intensities in a cw-pumped random fiber laser,” Phys. Rev. A **96**(1), 013834 (2017). [CrossRef]

**19. **B. C. Lima, A. S. L. Gomes, P. I. R. Pincheira, A. L. Moura, M. Gagné, E. P. Raposo, C. B. de Araújo, and R. Kashyap, “Observation of Lévy statistics in one-dimensional erbium-based random fiber laser,” J. Opt. Soc. Am. B **34**(2), 293 (2017). [CrossRef]

**20. **E. P. Raposo and A. S. L. Gomes, “Analytical solution for the Lévy-like steady-state distribution of intensities in random lasers,” Phys. Rev. A **91**(4), 043827 (2015). [CrossRef]

**21. **R. Uppu and S. Mujumdar, “Lévy exponents as universal identifiers of threshold and criticality in random lasers,” Phys. Rev. A **90**(2), 025801 (2014). [CrossRef]

**22. **N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, L. Leuzzi, and C. Conti, “Experimental evidence of replica symmetry breaking in random lasers,” Nat. Commun. **6**(1), 6058 (2015). [CrossRef]

**23. **A. S. L. Gomes, B. C. Lima, P. I. R. Pincheira, A. L. Moura, M. Gagné, E. P. Raposo, C. B. de Araújo, and R. Kashyap, “Glassy behavior in a one-dimensional continuous-wave erbium-doped random fiber laser,” Phys. Rev. A **94**, 011801 (2016). [CrossRef]

**24. **P. I. R. Pincheira, A. F. Silva, S. I. Fewo, S. J. M. Carreño, A. L. Moura, E. P. Raposo, A. S. L. Gomes, and C. B. de Araújo, “Observation of photonic paramagnetic to spin-glass transition in specially-designed TiO_{2} particles-based dye-colloidal random laser,” Opt. Lett. **41**(15), 3459 (2016). [CrossRef]

**25. **D. Pierangeli, A. Tavani, F. Di Mei, A. J. Agranat, C. Conti, and E. DelRe, “Observation of replica symmetry breaking in disordered nonlinear wave propagation,” Nat. Commun. **8**(1), 1501 (2017). [CrossRef]

**26. **C. B. de Araújo, A. S. L. Gomes, and E. P. Raposo, “Lévy statistics and glassy behavior of light in random fiber lasers,” Appl. Sci. **7**(7), 644 (2017). [CrossRef]

**27. **I. R. R. González, B. C. Lima, P. I. R. Pincheira, A. A. Brum, A. M. S. Macêdo, G. L. Vasconcelos, L. de S. Menezes, E. P. Raposo, A. S. L. Gomes, and R. Kashyap, “Turbulence hierarchy in a random fibre laser,” Nat. Commun. **8**(1), 15731 (2017). [CrossRef]

**28. **P. S. Choubey, A. Sarkar, S. K. Varshney, and B. N. Shivakiran Bhaktha, “Random laser spectroscopy and replica symmetry breaking phase transitions in a solvent-rich polymer thin film waveguide,” J. Opt. Soc. Am. B **37**(8), 2505 (2020). [CrossRef]

**29. **A. Tehranchi and R. Kashyap, “Theoretical investigations of power fluctuations statistics in Brillouin erbium-doped fiber lasers,” Opt. Express **27**(26), 37508 (2019). [CrossRef]

**30. **Z. Zhou, L. Chen, and X. Bao, “High efficiency Brillouin random fiber laser with replica symmetry breaking enabled by random fiber grating,” Opt. Express **29**(5), 6532 (2021). [CrossRef]

**31. **S. Jagannathan, L. Ackerman, W. Chen, N. Yu, M. Cavillon, M. Tuggle, T. W. Hawkins, J. Ballato, and P. D. Dragic, “Random lasing from optical fibers with phase separated glass cores,” Opt. Express **28**(15), 22049 (2020). [CrossRef]

**32. **A. Tehranchi, V. L. Iezzi, and R. Kashyap, “Power fluctuations and random lasing in multiwavelength Brillouin erbium-doped fiber lasers,” J. Lightwave Technol. **37**(17), 4439–4444 (2019). [CrossRef]

**33. **N. Lizárraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Méndez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express **17**(2), 395 (2009). [CrossRef]

**34. **M. Gagné and R. Kashyap, “Demonstration of a 3 mW threshold Er-doped random fiber laser based on a unique fiber Bragg grating,” Opt. Express **17**(21), 19067 (2009). [CrossRef]

**35. **S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castañón, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nature Photon **4**(4), 231–235 (2010). [CrossRef]

**36. **M. Pang, X.Y. Bao, and L. Chen, “Observation of narrow linewidth spikes in the coherent Brillouin random fiber laser,” Opt. Lett. **38**(11), 1866 (2013). [CrossRef]

**37. **H. Ahmad, M. Z. Zulkifli, M. H. Jemangin, and S. W. Harun, “Distributed feedback multimode Brillouin–Raman random fiber laser in the S-band,” Laser Phys. Lett. **10**(5), 055102 (2013). [CrossRef]

**38. **C. Huang, X. Dong, S. Zhang, N. Zhang, and P. P. Shum, “Cascaded random fiber laser based on hybrid Brillouin-erbium fiber gains,” IEEE Photonics Technol. Lett. **26**(13), 1287–1290 (2014). [CrossRef]

**39. **S. Sugavanam, M. Z. Zulkifli, and D. V. Churkin, “Multi-wavelength erbium/Raman gain based random distributed feedback fiber laser,” Laser Phys. **26**(1), 015101 (2016). [CrossRef]

**40. **H. Wu, Z. Wang, Q. He, W. Sun, and Y. Rao, “Common-cavity ytterbium/Raman random distributed feedback fiber laser,” Laser Phys. Lett. **14**(6), 065101 (2017). [CrossRef]

**41. **Y. Xu, L. Zhang, L. Chen, and X. Bao, “Single-mode SOA-based 1kHz-linewidth dual-wavelength random fiber laser,” Opt. Express **25**(14), 15828 (2017). [CrossRef]

**42. **H. Shawki, H. Kotb, and D. Khalil, “Single-longitudinal-mode broadband tunable random laser,” Opt. Lett. **42**(16), 3247 (2017). [CrossRef]

**43. **W. Margulis, A. Das, J. P. von der Weid, and A. S. L. Gomes, “Hybrid electronically addressable random fiber laser,” Opt. Express **28**(16), 23388 (2020). [CrossRef]

**44. **M. Montinaro, V. Resta, A. Camposeo, M. Moffa, G. Morello, L. Persano, K. Kazlauskas, S. Jursenas, A. Tomkeviciene, J. V. Grazulevicius, and D. Pisignano, “Diverse regimes of mode intensity correlation in nanofiber random lasers through nanoparticle doping,” ACS Photonics **5**(3), 1026–1033 (2018). [CrossRef]

**45. **M. Leonetti, C. Conti, and C. Lopez, “The mode-locking transition of random lasers,” Nature Photon **5**(10), 615–617 (2011). [CrossRef]

**46. **L. F. Sciuti, L. A. Mercante, D. S. Correa, and L. De Boni, “Random laser in dye-doped electrospun nanofibers: Study of laser mode dynamics via temporal mapping of emission spectra using Pearson’s correlation,” J. Lumin. **224**, 117281 (2020). [CrossRef]

**47. **A. Sarkar, J. Andreasen, and B. B. N. S. Bhaktha, “Replica symmetry breaking in a weakly scattering optofluidic random laser,” Sci. Rep. **10**(1), 2628 (2020). [CrossRef]

**48. **J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, “Modes of random lasers,” Adv. Opt. Photon. **3**(1), 88 (2011). [CrossRef]

**49. **P. Tovar, G. Temporão, and J. P. von der Weid, “Longitudinal mode dynamics in SOA-based random feedback fiber lasers,” Opt. Express **27**(21), 31001 (2019). [CrossRef]

**50. **W. L. Zhang, R. Ma, C. H. Tang, Y. J. Rao, X. P. Zeng, Z. J. Yang, Z. N. Yang, Y. Gong, and Y. S. Wang, “All optical mode controllable Er-doped random fiber laser with distributed Bragg gratings,” Opt. Lett. **40**(13), 3181 (2015). [CrossRef]

**51. **F. Antenucci, A. Crisanti, M. Ibañez-Berganza, A. Marruzzo, and L. Leuzzi, “Statistical mechanics models for multimode lasers and random lasers,” Philos. Mag. **96**(7-9), 704–731 (2016). [CrossRef]