## Abstract

We present theoretical investigations including simulations and statistical analyses on the fluctuations of the temporal output power in a Brillouin erbium-doped fiber laser. The generation of even Stokes waves up to the 5^{th} order is considered by solving coupled-mode equations including SBS and Kerr nonlinearities. It is demonstrated that by increasing the EDFA pump power and generating a few orders of Stokes waves in such a laser, there are strong power fluctuations and rogue events are expected. Transition from Gaussian-like to levy-like regime is described as the power is increased from threshold resulting in the initial Stokes wave generation to well beyond threshold generating 5 effective even Stokes waves. Accordingly, phase portraits confirm increasing fluctuations as a function of the power. It is also shown that at the SBS lasing threshold, the output signals have the maximum correlation over replicas in round trips, nevertheless by enhancing the power, the correlation diminishes, which results in a full symmetry breaking and the system radiates in a chaotic manner and exhibits random laser behavior.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

During the past years, multi-wavelength Brillouin erbium-doped fiber lasers (MWBEFLs) have been extensively studied [1–6]. They work on the basis of the narrow gain bandwidth of stimulated Brillouin scattering (SBS) in a fiber [7–11] and the broadband gain of an erbium-doped fiber amplifier (EDFA) to generate cascaded SBS. They are promising because of their large tunability [12–14] and ability to generate many Brillouin shifted frequencies [15–16], as well as a narrow linewidth and low pump power threshold [17]. They have also potential applications in optical fiber sensing, amplification, lasing, slow light generation and microwave photonics [18–21].

Recently, it has been shown both theoretically and experimentally that the temporal evolutions of MWBEFLs signify the generation of chaotic pulses [22–23] comparable to the power fluctuations expected in random lasers including disorder and gain media without a cavity [24–25]. Also, generation of extreme and rogue events in random fiber lasers has been reported addressing turbulences in experiments [26–30]. Therefore, temporal and statistical properties of MWBEFLs are of fundamental and practical importance. One significant feature of random lasers that should be taken into account is the statistical behavior of the power fluctuations considering the probability density function (PDF). This behavior differs from the Gaussian-like statistics for weak fluctuations, transferring to a levy-like statistics for strong fluctuations and the PDF of output powers is then described by the *α*-stable distribution. Phase portraits are also used to investigate the fluctuations with enhancing the pump power [31–32]. A statistical analysis of the overlap of temporal replicas over round trips, comparable to what is used in replica symmetry case [33–39], is also useful to examine the behavior of such lasers.

In this work, the propagation of pump, Stokes and acoustic waves in an MWBEFL is simulated solving coupled-mode equations analytically considering the cascaded SBS and Kerr nonlinearities for the generation of up to 5 even Stokes waves and output signal. We then study the statistics of output power fluctuations. Two statistical regimes are recognized, namely, Gaussian-like around low pump power threshold and Levy-like at high pump power (beyond the threshold) presenting the first observation of Levy *α*-stable distribution in such a system. Moreover, we present phase portraits and analyze the correlation over replicas in an MWBEFL system, demonstrating a full symmetry breaking beyond the power threshold in accordance with our aforementioned findings.

## 2. MWBEFL set-up and theoretical results

The set-up of an MWBEFL system is depicted in Fig. 1. A Brillouin pump (BP) is made from a narrow-linewidth DFB laser amplified by an out-of-cavity EDFA 1 to feed the cascaded SBS via a 95:5 coupler. In each counter clockwise round trip in the upper ring, the BP wave and Stokes waves pass via the input coupler, amplified by the intra-cavity EDFA 2 for resonator gain and attenuated by the circulator 1, filter (to impede self-lasing cavity modes), fiber spool, output coupler and circulator 2 while achieving random phase shifts. The backward propagating waves passing counter clockwise in the lower ring, are attenuated by the filter, two circulators and the output coupler while obtaining random phase shifts as well. These trips provide the new propagating waves for the next round. The evolutions of the forward and backward propagating waves due to the cascaded SBS in a single-mode fiber are then simulated using the coupled-mode equations and considering boundary conditions given in [22]. In the system, even and odd orders of Stokes waves propagate in upper and lower loops and thus the BP and even-order Stokes waves forming a comb-like spectra with a channel spacing of twice the Brillouin frequency shift (2ν_{B} ∼ 21.7 GHz), are tapped via a 99:1 coupler as the output signal [23].

We examine the output power dynamics of the MWBEFL considering the generation of Stokes up to the 10^{th} order, resulting in 42 coupled-mode equations and numerically solve them by the method of characteristics [40–41]. The BP power is considered as a continuous wave at 1550.9 nm, amplified and launched into the system including a 2.5-km-long fiber (SMF-28) resulting in a round trip time of ∼12 µs. Moreover, to model the amplifier in each round trip, the equations describing the evolution of the EDFA pump, and signals (BP and Stokes lines) are used [42]. The parameters include a fiber loss coefficient of 0.2 dB/km, an effective mode area of 80 µm^{2}, a linear refractive index of ∼1.45, a nonlinear refractive index of ∼5×10^{−20} m^{2}/W, a Brillouin gain coefficient of ∼2.5×10^{−11} m/W, and a phonon lifetime of ∼10 ns. The temporal evolutions of optical powers of the output signals in the MWBEFL system are then simulated, for different EDFA pump powers to generate the 1^{st} even order Stokes waves (near and above power threshold) as well as all 5 even orders of Stokes waves (well above threshold).

Figures 2(a)–2(c) show power fluctuations versus time over a time window of 2000 µs (∼167 round trips) in the MWBEFL system for pump powers near threshold (20 mW), above threshold (30 mW) and much beyond it (2000mW), respectively. The frequency hopping of longitudinal modes within the Stokes bandwidths results in their random phase shifts and consequent chaotic interference which is deteriorated by increasing Stokes’ numbers [23]. Generating the first even Stokes wave, the weak fluctuations in Fig. 2(a) for a 20-mW pump power get stronger in Fig. 2(b) for a 30-mW power as a more powerful Stokes wave gets generated. Very strong power fluctuations in the temporal signal are observed when 5 even Stokes waves are generated in Fig. 2(c) for a 2000-mW pump power. The power fluctuations are larger for higher pump powers and the ratio of maximum power to average power is 2.20, 3.15 and 5.58 in Fig. 2(a), (b) and (c), respectively. Therefore, the occurrence of rogue events is more probable in the latter with 5 even Stokes waves, also observed in experimental measurements (See Fig. 2(a) in [22] for extreme events). It is worth noting that random MWBEFLs work in ways contrary to some conventional random lasers. In MWBEFLs at threshold, a weak Stokes wave get generated thus a perturbed pump as a weakly-fluctuating signal is available at the output. Above threshold, the pump and Stokes wave interact with random phases to make the output power thus it is highly-fluctuating. However, for other random lasers, the output power can be strongly or weakly-fluctuating at or above threshold, depending on their features [37,38].

## 3. Statistical analysis

To get a better insight, we deal with the statistical analysis of output power. This can be carried out by computing the PDF of temporal power data. To that end, histograms of normalized powers, *P*/*P*_{max} can be calculated for the pump powers of 20, 30 and 2000mW as illustrated in Figs. 2(a)–2(c), respectively. Basically, if the stochastic values presented by power show finite correlations and the second moment of its PDF is finite, the central limit theorem (CLT) guarantees that the related fluctuations are governed by the Gaussian dynamics. Alternatively, if its second moment is not finite, the generalized CLT states that the fluctuations are driven by the Levy statistics, described by the family of *α*-stable distributions [35], with the PDF of

*k*and is ${\Phi} = \tan \left( {\frac{{\pi \alpha }}{2}} \right)$ for $\alpha \ne 1$ and ${\Phi} ={-} \frac{2}{\pi }\log |k |$ for $\alpha = 1$ and $\alpha \in ({0,2} ]$, $\beta \in [{ - 1,1} ]$, $\gamma \in ({0,\infty } )$ and $\delta \in ({ - \infty ,\infty } )$ are the stability, skewness, scale and location parameter, respectively. $\alpha $ is an important parameter defining the type of statistics that characterizes the fluctuations of random variables. For

*α*= 2, the Gaussian statistics with weak fluctuations and the CLT result are recovered whilst fluctuations with relevant deviations from the Gaussian behavior are related to values in the range 0 <α < 2.

Figures 3(a)–3(c) depict the PDFs, *f* (*P*) obtained from the power data in Figs. 2(a)–2(c), as well as the respective fits using Eqs. (1) and (2). Fit values of the parameters are listed in Table 1. Figure 3(a) presents a Gaussian-like PDF and corresponding Gaussian fit for *α* = 2, just near the pump power threshold when the first Stokes wave begins to get generated leading to weak fluctuations. Figure 3(b) also shows a Gaussian-like PDF and corresponding fit for *α* = 1.8, when a stronger first Stokes wave is generated resulting in stronger fluctuations. The values of the location parameter *δ* for the Gaussian fits are *δ* = 0.48 and *δ* = 0.31 for *P* = 20 mW and *P* = 30 mW, as shown in Figs. 3(a) and 3(b), respectively. Moreover, above the threshold, a widening in the PDF associated with the larger *γ* is noticeable in Fig. 3(b), due to stronger fluctuations in power. Furthermore, a Levy-like PDF and corresponding Levy fit for α = 1.13, is observed in Fig. 3(c) when 5 even Stokes waves are generated. In Fig. 3(c) statistical analyses finally reveal that very strong power fluctuations result in a skewed power distribution. In all cases *β* = 1 represents the maximum skewness of the PDF, as the power has positive values. The insets in Figs. 3(a)–3(c) show the cumulative density functions (CDFs). The noticeable increase of the profile slopes of the CDFs also indicates the transition from Gaussian-like to Levy-like behavior.

To confirm the emergence of rogue events in a physical system two issues are generally considered. First, they are described using statistical behaviors with a skewed PDF of power. Second, based on the oceanic analogy the significant wave height (SWH) power, *P*_{SWH} corresponding to the mean height of the highest third of powers is calculated [43]. An extreme event is then detected when its power is more than twice the SWH. In other words, rouge waves are expected when *P*_{SWH}/*P*_{max} < 0.5. This power ratio is 0.57, 0.51 and 0.45, for the pump power of 20, 30 and 2000 mW, respectively. Thus, the presence of extreme events is more probable for the latter case.

Phase portrait is another means to detect the output power impairments [32]. To characterize the output power, it uses the joint PDF of power, *P _{N}* and its delayed version

*P*

_{N}_{+1}, where

*N*is the number of samples. This 2D PDF is sensitive to fluctuations. To generate the phase portrait, the 2D histogram of the signal and its delayed version is calculated and illustrated. For diagonal bars, it would give a less-fluctuated power distribution whilst for distributed bars with increased density, the temporal power fluctuations are well expected. Figures 4(a)–4(c) depict the 2D PDFs,

*f*(

*P*,

_{N}*P*

_{N}_{+1}) obtained from the power data in Figs. 2(a)–2(c). The confined signature in Figs. 4(a) and 4(b) manifests weak fluctuations however, the distributed signature in Fig. 4(c) shows strong fluctuations, in well agreement with previous findings.

In MWBEFLs, around threshold the system presents some coherence and correlation similar to random lasers above threshold [36–39] which can be analyzed in a way comparable to replica symmetry breaking (RSB). For this purpose, we have already calculated the output power in time ($\tau $) denoted here as $P(\tau )$. The overlap of the temporal evolution from shot to shot can be calculated by the correlation between powers of two shots *a* and *b* as

*a*is the replica index,

*M*is the number of temporal points and $\bar{P}(\tau )$ is the average over replicas for each temporal power component.

Figure 5 depicts histograms including the distribution, *P*(*q*) versus *q* calculated for *a*, *b* = 1,… 100 shots providing a total 100 × (100 − 1)/2 values of *q*. Three cases are illustrated in Figs. 5(a)–5(c) for enhancing EDFA pump power as shown in 2(a)-2(c). At pump power threshold the 1^{st} even Stokes wave and the output signal with minimum fluctuations is generated and thus *q* may have all possible values in the range (−1,1). Further, the distribution has two maxima at *q* = ± 1 and an emptied region around *q* = 0 revealing of one-step symmetry breaking. The one-step RSB shows suboptimal correlation as the pdf is observable for |q| <1 [37]. It is clear that at the power threshold, any two replicas have correlated power fluctuations. On the other hand, above the pump power threshold the system generates a more powerful 1^{st} even Stokes wave, the shots are less correlated as presented by a nearly zero-peaked distribution, as shown in Fig. 5(b) expressing that the correlation between power fluctuations in any two shots depends on the selected shots. By increasing the pump power much more, illustrated in Fig. 5(c) five even Stokes waves are generated so that all overlaps are centred around the zero value, denoting that the generated shots are uncorrelated, manifesting a full symmetry breaking. The full RSB shows that all shots fluctuate without significant correlation [37]. Therefore, when the lasing consists of a weak Stokes wave, it leads to weak pulses that follow a one-step symmetry breaking distribution while the presence of several Stokes waves results in the erratic intense pulses obeying a full symmetry breaking distribution with a high density around *q* = 0 as shown in Fig. 5(c).

## 4. Conclusion

It was demonstrated theoretically that there are great power fluctuations in time in an MWBEFL system to generate a few number of Stokes waves. Simulation and statistical analysis show that although the system has no intentional disorder, by increasing the power and number of Stokes waves, it shows random lasing behavior and becomes highly disordered. The PDF of output power differs from the Gaussian-like statistics at pump power threshold for weak fluctuations, transferring to the levy-like statistics at high pump powers causing strong fluctuations with the occurrence of rouge events. We have pointed out rogue events through our simulations, also seen in experimental measurements [22], and they are likely the first observations in such lasers, to the best of our knowledge. The phase portrait including 2D PDFs also confirm increasing fluctuations with enhancing the pump power. The correlation over replicas in an MWBEFL is also lost beyond the pump power threshold leading to full symmetry breaking, in accordance to our aforesaid findings and the system shows the signatures of a random laser.

## Funding

Natural Sciences and Engineering Research Council of Canada; Fonds de Recherche du Québec - Nature et Technologies.

## Disclosures

The authors declared that they have no conflicts of interest to this work.

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