1. Introduction

Random lasers (RLs) with random feedback provided by the light scattering in a disordered medium replace the mirror-based reflection in conventional lasers [1–4]. As light propagates through the gain medium, numerous random scattering centers within this medium randomize photon paths, aiding in the retention of light energy. The first demonstration of one-dimensional optical fiber-based RL utilized a dye of TiO₂ colloidal within the hollow core of a photonic crystal fiber, forming a Random Fiber Laser (RFL) [5]. Thanks to versatile nonlinearity and disorder-involved dynamics, RFLs have been intensively exploited to provide a compelling analogy platform of photonic complex systems for research on diverse nonlinear phenomena, such as spin glass [6–8], turbulence-like behavior [9], and rogue waves [10,11].

The concept of replica symmetry breaking (RSB) initially addresses the issues of negative entropy and solution stability in spin-glass systems [12,13]. According to the spin glass theory, RSB, defined as the transition from the paramagnetic phase to the glass phase, indicates that the same system under identical conditions inherently exhibits different phase states. In RL-based complex systems, the interaction between disorder and fluctuation can be explicitly characterized by the statistical approach based on RSB [14]. The evidence of the RSB phenomenon in an RL was experimentally reported by employing a functionalized thiophene-based oligomer (T₅OC_x) in the amorphous solid state with planar geometry [15]. Subsequently, intensive discoveries of the RSB phenomenon in a variety of RL systems have been demonstrated, revealing the widespread presence of the RSB phenomenon in RL systems, such as dye lasers [7], liquid crystal lasers [16], semiconductors [17], fiber lasers [18–23], etc.

Attention has been paid not only to the experimental evidence of the existence of the RSB phenomenon in various RL systems but, more importantly, to the characteristics associated with the appearance or dynamics of the RSB. In a pioneer work, the occurrence of RSB in RL systems is predicted to be associated with the pump power [24], i.e., the pump power should be set at around the laser threshold. However, as the pump power of the RL is below or far above the threshold, RSB disappears. Moreover, the pump power of the RL would also play an important role in the evolution of the statistics of the order parameter q. The continuous increase of pump power above the laser threshold leads to the profile broadening of the probability distribution function (PDF) of the order parameter q [25,26]. Apart from the pump power, different feedback mechanisms have been demonstrated to enable the occurrence of RSB with distinct photonic spin glass behaviors in Brillouin random fiber laser (BRFL) [21], which shows that the presence or absence of RSB caused by different feedback mechanisms is related to the stability and couple of the lasing modes in RFLs. It revealed that mode coupling manipulation essentially provides a new approach for regulating RSB [26]. Recently, the tunability of RSB dynamics in a polymer random fiber laser by varying either the ambient temperature or the laser structure was experimentally demonstrated, which basically influenced the interaction and coupling of the excited random laser modes [27]. Furthermore, by varying the lengths of the gain fiber and the Rayleigh scattering fiber to control the density of random modes, a higher probability of observing transient dynamics of RSB can be achieved, thereby manipulating the observation of the RSB [28]. As the total length of the fiber increases, the random mode density increases while the average mode lifetime decreases, leading to the switching among different laser mode landscapes for the transient dynamics of the RSB. Nevertheless, the manipulation of RSB dynamics by adjusting the structure of the proposed random lasers, e.g., physically replacing gain and Rayleigh fiber length, is difficult to flexibly implemented. Approaches to generate and control RSB dynamics in a photonic complex system, particularly for its dynamical investigation, remain challenging.

In this work, we experimentally demonstrated the observation and harness of the RSB in a Brillouin random fiber laser photonic system with multi-Stokes-involved additive disorders. Incorporating the optical-heterodyne technique, a high-fidelity retrieval of the random mode evolution of the 1^st order Stokes light under an increased generation of Stokes light was achieved. By controlling the orders of excited Stokes light, the average mode lifetime of the 1^st order Stokes random laser modes turns out to be adjustable. Under various mode landscapes with different average mode lifetimes, distinctive statistical distributions of |q_max|_τ appeared when the scale of the sub-window was above (below) the average mode lifetime, indicating transient RSB states in a single random laser system without any laser structure changes. Ultimately, a roadmap of statistical q distributions under different EDFA powers and appropriately choosing the sub-window scales in terms of different average laser mode lifetimes for the RSB dynamics in such a photonic complex system was systematically carried out, thereby achieving an optically controlled photonic platform for comprehensively exploring the fundamental physics underlying the occurrence of the RSB phenomenon in disorder complex systems.

2. Experimental setup and principle

The setup of the proposed MWBRFL was depicted in Fig. 1, which comprised a main half-open random cavity incorporating a sub-fiber cavity. As the Brillouin gain medium, a 10.25 km single mode fiber (SMF) was connected to the main half-open random cavity through two optical circulators (CIRs 1 &2) and a 50/50 optical coupler (OC 3). Another spool of 6.23 km SMF was incorporated through CIR 2 to introduce distributed random feedback for the Stokes lasing resonance within the main half-open random cavity. Before the laser output, an isolator was placed at the end of the 6.23 km SMF to prevent any unwanted Fresnel reflections from occurring at the fiber end surface. The lasing Stokes lines were split by the OC 3 within the main half-open random cavity and then combined with the 90% input 1550 nm pump laser split by a 90/10 optical coupler (OC 1) through another 50/50 optical coupler (OC 2) within the sub-fiber loop. An EDFA was employed to boost the optical power before injecting it into the main half-open random cavity.

 

Fig. 1. Schematic of the MWBRFL and the tunable optical heterodyne system. OC – optical coupler, CIR – circulator, EOM – electro-optic modulator, SMF – single mode fiber, PD – photodetector, ISO – isolator, OSA – optical spectrum analyzer, ESA – electrical signal analyzer.

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To selectively obtain the 1^st order Stokes spectral components, an optical heterodyne detection scheme comprising a microwave source, an electro-optic modulator (EOM), and a photodetector (PD), was interfaced and beaten with a reference through an optical coupler (OC 5). The microwave source output was set at 10.80 GHz, and the pump light was modulated using an EOM to obtain upper and lower sidebands with a suppressed center carrier. With the longer-wavelength sideband as the reference light, the heterodyne frequency mixing with the 1^st order Stokes light for a beat frequency signal at the central frequency of 80 MHz was then obtained. Considering the Brillouin frequency shift of 10.88 GHz, the used PD with 350-MHz bandwidth was critical to individually detect the beat frequency signals only for the 1^st order Stokes since the beat signal between the 1^st order Stokes and reference lights ultimately located around 80 MHz while the beat signals among other orders Stokes and reference (typically much larger than 10.88 GHz) were far beyond the PD bandwidth. In this case, the frequency evolution of the 1^st order Stokes components can be effectively recorded with high fidelity.

In the proposed MWBRFL system, as the pump power exceeded the SBS threshold of the 10.25 km SMF, the 1^st order Stokes light was highly stimulated in the backward direction along the SMF gain media. 50% of the backward-propagating the 1^st order Stokes light was extracted from the main half-open random cavity via the OC 3 and launched into the EDFA in the sub-cavity. Combined with the 1^st order Stokes, the 2^nd Stokes injected back into the SMF gain media. In the main half-open random cavity, the remaining 50% of the 1^st order Stokes light was launched into the 6.23 km SMF, which hence provided random Rayleigh scattering feedback for its random lasing resonance. By increasing the power of the EDFA, higher order Stokes lights can be subsequently generated and involved in the interaction and gain competition with the 1^st order Stokes light, which eventually brought additional disorders to this photonic complex system for manipulating the mode landscapes of the 1^st order Stokes lasing emission. It should be pointed out that the power of the lower-order Stokes lights tended to be stabilized after the generation of subsequent higher-order Stokes components [29]. This provided a moderately small fluctuation of the photon lifetime in Stokes laser emission and benefits the investigation of the impact of multiple Stokes components on the transient dynamics of RSB in the 1^st order Stokes light.

Regarding the observation of transient dynamics of RSB, the continuous spectral components of 1^st order Stokes light were employed for time-series analysis, partitioned into equally-sized consecutive sub-windows. The total observation time T was divided into N_m groups, with each subgroup having a duration of, T_τ where τ = 1,2, …, N_m. Consequently, the overlap parameter between two replicas (α and β) within the sub-timewindow T_τ can be expressed as [28]:

Therefore, the time evolution of |q_max|_τ can be derived to further investigate the transient processes of RSB in the disordered photonic system of RFL. Besides, as mentioned in [28], for random laser systems above the threshold, two statistical states, i.e., the RSB and RS, coexisted fundamentally. When the average mode lifetime ${\bar{T}_d}$ in the random laser system was longer than the observation time T_τ, P(q) exhibited a unimodal distribution representative of the RS state. In the presence of multiple dominant random modes, if T_τ was longer than that of discrete random modes ${\bar{T}_d}$, P(q) showed a bimodal distribution representative of RSB. Therefore, by manipulating ${\bar{T}_d}$ of the 1^st order Stokes light, the transition from the RSB to the RS state can be achieved by properly selecting T_τ. With the excitation of high-order Stokes lights, gain competition within the EDFA and mutual coupling through the cascading SBS process occurred between the high-order and 1^st order Stokes lights in such proposed MWBRFL. These interactions resulted in a gain modification as well as an optical disturbance in the 1^st order Stokes random lasing oscillation, leading to a declining average random mode lifetime ${\bar{T}_d}$, which hence played a critical role in the observation of transient RSB. Ultimately, all-optical manipulation of transient RSB dynamics can be facilitated in such a photonic complex system.

3. Results and discussions

3.1 Optical power evolution

The spectrum of the MWBRFL output was collected by an OSA with a resolution of 0.017 nm. As shown in Fig. 2(a), by injecting 1550.078 nm pump light with the power of 0.6 mW, the wavelengths of the 1^st to 5^th order Stokes light subsequently generated in the proposed MWBRFL were 1550.160 nm, 1550.242 nm, 1550.324 nm, 1550.406 nm, and 1550.488 nm, respectively. It can be found that an identical wavelength interval of 0.082 nm between adjacent order of Stokes components, corresponded to the Brillouin frequency down-shift of 10.88 GHz in silica fibers around the laser wavelength of 1550 nm.

 

Fig. 2. (a) The output laser spectrum containing the 1^st to 5^th order Stokes lights. The capital Roman numerals represent the order of the generated highest-order Stokes light. (b) The power evolutions of the different generated Stokes lights. Here, the serial numbers ①, ②, ③, ④, ⑤ denoted the power of the EDFA of 6.2 mW, 12.2 mW,16.8 mW, 24.4 mW, 26.5 mW, respectively.

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To investigate the optical power evolution of generated Stokes lights, the optical power evolution of different orders Stokes with gradually increasing P_EDFA were shown in Fig. 2(b). Results showed that the 1^st order Stokes light appeared at P_EDFA of above 2.4 mW while higher orders Stokes components were subsequently activated by raising P_EDFA to 6.9 mW, 11.9 mW, 17.9 mW, and 25.5 mW, respectively. Meanwhile, the power efficiency of the 1^st order Stokes was reduced as the 2^nd order Stokes light was generated which inversely depleted the 1^st order Stokes via the SBS process in the main half-open random cavity, and its rejection in the sub-fiber loop also got boosted from the gain competition in the EDFA against the 1^st order Stokes. In this scenario, the 2^nd order Stokes actively interplayed with the 1^st order Stokes by laser mode coupling in cascading SBS process as well as gain competition in the EDFA. Similar situations could be found for the 3^rd to 5^th order Stokes. Consequently, the optical powers of all generated Stokes lights turned out to be claimed at a stable level, albeit with a moderated power fluctuation. It eventually introduced additional disorderliness in such a photonic complex system and opened new avenues for the manipulation of the RSB dynamics.

3.2 Mode density manipulation

In the multi-Stokes random fiber laser, the gain competition and power interplayed between the high-order Stokes light and the 1^st order Stokes light would randomly impose the perturbation and influence the probability of lasing mode switching in the 1^st order Stokes light, which eventually resulted in distinctive mode landscapes with the decreased average lifetime of the output laser mode. To validate it, the spectra of the beat signals of the 1^st order Stokes light with a spectral resolution of 10 kHz within 1000 ms as the highest-order of the generated Stokes lights increased from 1^st to 5^th were recorded and illustrated in Fig. 3. When the 1^st order was solely generated, the mode landscape with a stable single mode was found to be dominated over the observation time of up to 800 ms, as shown in Fig. 3(a). However, for the highest-order of 2^nd order, the excitation of the high-order Stokes light motivated the dominating mode interplay among 8 activated modes, as shown in Fig. 3(b). In Fig. 3(c)-3(e), as the highest-order of generated Stokes lights varied, the total number of the 1^st Stokes random modes were increasingly activated within the selective observation time scale of 1000 ms, leading to the mode landscape evolution under the different EDFA powers. Consequently, the more the number of Stokes components generated, the more random laser modes were excited for frequent mode landscape switching, which paved the way for the observation of transient dynamics of RSB. It was noticeable that by manipulating the number of the generated Stokes lights, the observation of RSB dynamics in terms of the 1^st order Stokes light can be all-optically controlled in such a single laser system without any physical reorganization of the laser configuration.

 

Fig. 3. Mode landscape observation of the 1^st order Stokes light in the proposed MWBRFL. The spectra of the beat signal of the 1^st order Stokes light within 1000 ms with the existing highest order Stokes light of (a) the 1^st, (b) the 2^nd, (c) the 3^rd, (d) the 4^th, (e) the 5^th order.

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Accordingly, the histogram of ${\bar{T}_d}$ and the mode hopping frequency (i.e., inverse proportion to the average mode duration) of the 1^st order Stokes light with respect to the number of generated Stokes lights were characterized, as shown in Fig. 4. ${\bar{T}_d}$ was calculated as the ratio of observation time T to the mode hopping frequency. As the number of Stokes lights varied from 1 to 5, ${\bar{T}_d}$ generally presented a declining tendency. Compared with ${\bar{T}_d}$ of 250.0 ms for only the 1^st order Stokes, ${\bar{T}_d}$ decreased to 1.2 ms as the number of generated Stokes lights raised to 5, ${\bar{T}_d}$ which was reduced by two orders of magnitude. It was evident that, with an increase in the number of the stimulated Stokes order, ${\bar{T}_d}$ exhibited a continuous decrease. When the quantities of Stokes light were 2, 3, and 4, the corresponding values of ${\bar{T}_d}$ were 125.0, 66.7, and 18.0 ms, respectively. Specifically, when the 5^th order Stokes light occurred, the mode hopping frequency of the 1^st order Stokes light reached an exceptionally high value of 833, which was consistent with the mode landscape in Fig. 3(e). Basically, the decrease of ${\bar{T}_d}$ reflected an increase in the mode hopping frequency, which led to more frequent mode hopping within the mode landscapes. Such mode hopping with adjustable switching frequency crucially laid the foundation for transient RSB observations as well as its dynamical manipulation.

 

Fig. 4. Histogram of the mode average duration ${\bar{T}_d}$ and mode hopping frequency of the 1^st order Stokes light with respect to the number of generated Stokes lights, corresponding to the five cases in Fig. 3.

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Furthermore, the impact of the random mode density on the RSB observation was investigated. The heat maps of q distributions with multi-mode of the 1^st order Stokes light radiation with different numbers of generated Stokes lights were illustrated in Fig. 5. During the observation time of 1000 ms, Parisi overlap parameters q were calculated based on multiple random laser modes. The random modes can partition replicas into N pure states, and the total number of q values between any two pure states reached N(N-1)/2, which led to the fractal structure of q distribution in Fig. 5. Compared to the occurrence of the 1^st order RSB with two pure states corresponding to two random activated modes, the complexity of a multi-mode random laser system with a large number of discrete q values indicated the emergence of dynamical RSB statistics in such photonic disorder system, even implying a fractal distribution structure of higher-order RSB, as seen in Fig. 5(a)–5(e). It was noteworthy that when the number of lasing random modes reached 55, there should be a representation of the q distribution covering the range of -1 to 1 in the order parameter matrix. However, the q distribution was primarily concentrated near 0, as depicted in Fig. 5(e). This phenomenon was mainly attributed to the nearly orthogonal nature of high-dimensional random vectors in the order parameter matrix [28].

 

Fig. 5. The q distribution with different modes density of the 1^st order Stokes light determined by different highest order Stokes from the 1^st to the 5^th order in the MWBRFL under P_EDFA of (a) 6.2 mW, (b) 12.2 mW, (c) 16.8 mW, (d) 24.4 mW, and (e) 26.5 mW.

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3.3 RSB transition observation

The transient RSB was subject to the duration of the observation and the average lifetimes of the random mode, which also led to the coexistence of RS and RSB phases in random laser systems [28]. It was found that, although RSB of the random laser statistics within a prolonged observation window was observed, RS could occasionally persist in sub-time windows. Such transient RSB dynamics were expected to exhibit a dependence on the magnitude relationship between the average laser mode duration ${\bar{T}_d}$ and the selected sub-window scales T_τ. To validate it, the long-term beat frequency signals of the 1^st order Stokes light under different P_EDFA were acquired with a time resolution of 1.0 ms and a whole 1000 ms observation time window T. Because the mode switching occurred within the whole observation time window that is much higher than current ${\bar{T}_d}$ (i.e., 250.0 ms at P_EDFA= 6.5 mW and 125.0 ms at P_EDFA= 12.5 mW), a bimodal distribution of q in correspondence to the RSB phase was depicted in Fig. 6(a) and 6(b). Regarding T_τ of 200 ms, both two selected whole observation windows T were consecutively each divided into five equal sub-windows. Distribution P(q) of Parisi overlap parameter q during the sub-windows were presented in Fig. 6(c) and 6(d), respectively. Since the selected scale of the sub-window, T_τ was smaller than ${\bar{T}_d}$, the overlap distribution P(q) in these five consecutive sub-windows of the whole observation time window in Fig. 6(a) were unimodal distributions, i.e., the RS state, as seen in Fig. 6(c). On the contrary, as T_τ exceeded ${\bar{T}_d}$, P(q) in all five sub-windows related to the whole observation time window in Fig. 6(b) showed the bimodal distribution, indicating the RSB state, as depicted in Fig. 6(d). In addition, according to the spectral calculation of the Parisi overlap parameter q for each sub-time window, |q_max|_τ was plotted in Fig. 6(e). Here, the defined parameter <|q_max|> represented the average value of |q_max|_τ for each window under sub-windows of different scales, reflecting the statistical distribution of |q_max|_τ as well as the RSB state in sub-windows for various sub-window scales. As the parameter <|q_max|> approached 1, RSB phases were predominantly presented in the sub-windows. Conversely, when <|q_max|> approached 0, RS phases were predominantly present in the sub-windows. As a result, an appropriate T_τ under different ${\bar{T}_d}$ can be flexibly selected for discovering the transition from the RSB to the RS phase. Note that, the occurrence of the RSB state in the random laser was no correlated with the intrinsic fluctuations of the excitation laser since it exhibits Gaussian distribution when such a cavity laser operates well beyond its threshold.

 

Fig. 6. Dynamics evolution observation of RSB transition. P(q) of the 1^st Stokes random lasing emission under ${\bar{T}_d}$ of (a) 250 ms (P_EDFA= 6.5 mW) and (b) 125 ms (P_EDFA= 12.5 mW). The insets show the beat frequency signals within a whole observation window T of 1000 ms. (c) P(q) within sub-time windows of the whole observation window in (a) with a sub-window scale T_τ of 200 ms. (d) P(q) within sub-time windows of the whole observation window in (b) with a sub-window scale T_τ of 200 ms. (e) |q_max| in the 5 subsequent sub-time windows of (c) and (d) with <|q_max|> of 0.990 and 0.006, respectively.

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3.4 All-optical manipulation of RSB dynamics

For such multi-Stokes random fiber laser, by increasing the EDFA power to alter the number of the generated Stokes light, the average mode lifetime of the 1^st order Stokes light ${\bar{T}_d}$ conversely reduced. On the other hand, given different values of ${\bar{T}_d}$, the choice of various observation times T_τ was critical to the realization of the transient dynamics of RSB. To clarify it, a roadmap of the statistical distribution of q subject to various observation time window T_τ and the EDFA power-dependent average mode lifetime ${\bar{T}_d}$ was carried out. With the increase of the EDFA power, as the highest order Stokes light varied from the 1^st to 5^th order, beat frequency signals of the 1^st order Stokes light during 1000 ms were consecutively collected. In Fig. 7, for each collected window, different scales of T_τ from 1 to 500 ms were chosen to investigate the statistics of |q_max|_τ within the evenly successive sub-windows. It is evident that when T_τ was generally greater than ${\bar{T}_d}$, <|q_max|> mostly approached the value of 1, indicating the appearance of the RSB state in such sub-windows. On the contrary, when T_τ was far smaller than ${\bar{T}_d}$, <|q_max|> turned out to drop down to 0, which signified that the RS state was dominating within such observation windows. It was worth meeting that as T_τ was slightly smaller than ${\bar{T}_d}$, <|q_max|> gradually transferred from being close to 1 to approaching 0 with the reduction of T_τ. In this specific region, the sharp variation of <|q_max|> indicated the dramatical RSB dynamics including the transition between the RSB to the RS. This phenomenon was intrinsically determined by the selective random laser mode landscapes during the specific observation window. Ultimately, the proposed multi-Stokes involved random laser system with tunable laser photon lifetime was proven to be a good candidate with abundant RSB dynamics, which can be optically manipulated by setting its internal optical amplifier powers-induced high order Stokes components generation under an appropriate observation T_τ.

 

Fig. 7. The heatmap of <|q_max|> with respect to different T_τ and ${\bar{T}_d}$. The red solid line represented the variation in the mode average duration of the 1^st order Stokes light ${\bar{T}_d}$ as the EDFA power increases, leading to an increase in the generation of the Stokes light.

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Note that, the relationship between the average mode lifetime of the 1^st order Stokes light and the observation window scale is essentially important to the observation of RSB dynamics. It is believed that such RSB dynamics with transient statistics would be also predicted in other random laser-based photonic complex systems based on distinctive gain mechanisms, such as Raman scattering or rare-earth doped fiber, providing high-resolution retrieval of random laser mode landscapes.

4. Conclusion

In summary, we experimentally demonstrated the manipulation of RSB transition in MWBRFL incorporating the optical heterodyne technique. By acquiring beat frequency signals of the 1^st order Stokes light shifted to a center frequency of tens of MHz, it had been demonstrated that varying the power of the EDFA allows for the excitation of more Stokes light, enabling the control of the average mode lifetime of the 1^st order Stokes light. As the number of stable Stokes lights generated increased from 1 to 5, the average lifetimes of the 1^st order Stokes lights exhibited a decreasing trend ranging from 250.0 ms to 1.2 ms. Furthermore, the increase in the number of generated Stokes lights also led to the incremental mode density of the 1^st order Stokes light. Owing to the inherent orthogonality of high-dimensional random mode vectors, the increase of the number of modes caused the q-value concentrated around on value of 0. In addition, by selecting different sub-window scales, if the chosen sub-window scale was smaller than the average mode-averaging lifetime, |q_max| of the sub-windows within the whole observation window whose |q_max| displayed RSB may exhibit RS, indicating a transition from RSB to RS in the overall observation window. These results offer a new paradigm for RSB dynamics universally occurring in diverse RL systems as an all-optically controlled photonic platform to discover the underlying physics of nonlinear phenomena, such as spin glass [30], Lévy distribution [31–33], extreme value events [34,35] and turbulence [36].