The Raman random fiber laser (RRFL)^[1] has a simple structure and high tunability, and many researchers have studied its output characteristics in terms of wide-band tuning^[2], high power and high efficiency^[3,4], multi-wavelength operation^[5-7], mid-infrared wavelength laser generation^[8], etc. Due to the various merits of the RRFL, it has been recently explored in the fields of optical fiber sensing^[9,10], high-power laser facility^[11], and optical imaging^[12].

Particularly, as a significant issue, research on the temporal and spectral properties of RRFL output is crucial to reveal its physical qualities and has drawn a lot of attention. In terms of spectral characterizations, Sugavanam et al. measured the output spectra of the RRFL in real time and discovered the hidden stimulated Brillouin scattering (SBS) effect^[13]. Regarding the spectral intensity analysis, our group demonstrated that the statistical distribution of the RRFL has a transition from Gaussian to Lévy and finally back to Gaussian distribution with increasing pump power^[14], and other types of random laser excitation systems display similar variation process^[15,16]. In addition, ultra-narrow spectral patterns that spatially extend throughout the cavity are observed during the RRFL radiation process^[17].

In the study of time-domain statistical properties, since the spectral bandwidth is usually much larger than the electrical bandwidth of the measurement device, previous studies can be classified into two categories. The first effective method was to investigate the statistical properties by filtering the spectrum^[18-20], but it could not uncover the properties of the overall spectrum. The second approach involved exploring the integral statistical features directly, and researchers experimentally revealed that the time-domain intensity probability density function (PDF) of the RRFL correlates with the bandwidth of the oscilloscope (OSC) ^[21], which suggests that in bandwidth-limited situation the actual statistical characteristics are uncharted^[22]. However, previous works on the statistical features of the RRFL may not adequately disclose the physical nature with regard to the whole spectrum components since the detection bandwidth is limited.

When the RRFL spectral bandwidth is smaller than the measurement device bandwidth, the statistical features of the full bandwidth time-domain intensity of the RRFL are still unknown until now, but characterizing the time-domain properties of the RRFL under the full bandwidth condition is of practical importance in revealing the nature of the RRFL and offers precious guides for its application in many fields. For instance, intensity fluctuation statistics are crucial for studying the fundamental qualities of complicated phenomena such as the optical rogue wave^[23] and turbulence hierarchy^[24]. In random number generation^[25] and optical imaging domain^[26], temporal coherence and spatial coherence are important indicators for laser source selection, significantly affecting the random number production quality as well as the effectiveness of imaging. Selecting low coherence light sources in laser-driven inertial confinement fusion facilities could effectively suppress laser plasma instability^[11].

In this paper, the time-domain intensity statistical characteristics of the RRFL under the full bandwidth condition are studied thoroughly for the first time. This work also analyzes and explains both the modeling and experimental results in detail, revealing the RRFL’s distinctive properties. First, the output properties of the RRFL under full bandwidth condition are simulated utilizing the generalized nonlinear Schrödinger equations (NLSEs), showing that the intensity PDF of the RRFL depends on the pump power and the observing position, and it deviates inward from exponential distribution, indicating that different frequency components of the spectrum are correlated. Additionally, as the pump power increases, the PDF shrinks inward slightly, and the intensity fluctuations become smaller. Given the same pump power, the PDF of the RRFL at the front end of the fiber has more apparent fluctuations relative to that at the tail end. Afterward, based on a delicate laboratorial setup, the statistical properties of the RRFL are experimentally examined, and the results are consistent with the theoretical ones. This study could provide an important guideline for unveiling the physical characteristics of the RRFL and the suitability of the RRFL in diverse application scenarios.

In this section, the full bandwidth time-domain statistical properties of the RRFL are investigated theoretically, and the output spectral and temporal statistical properties of the RRFL at the fiber front end and tail end are both analyzed.

The simulation model is based on the generalized NLSEs. It takes the dispersion, loss, Kerr effect, stimulated Raman scattering (SRS), and Rayleigh scattering (RS) in the fiber into account, which can describe the output spectral and temporal dynamic properties of the RRFL effectively^[27] and can be expressed as follows: ∂up±∂z∓1vgs∂up±∂t±iβ2p2∂2up±∂t2±αp2up±=±iγp|up±|2up±∓gp(ω)2(|us±|2+⟨|us∓|2⟩)up±,(1)∂us±∂z±iβ2s2∂2us±∂t2±αs2us±∓ε(ω)2us∓=±iγs|us±|2us±±gs(ω)2(|up±|2+⟨|up∓|2⟩)us±,(2)where u is the complex envelope of the lightwave; p and s represent the pump and Stokes waves; + and − are the forward and backward propagating waves, respectively; vgs represents the inverse group velocities between the pump and Stokes waves; ω is the angular frequency of the lightwave; α, γ, β2, ε, and g are the linear fiber loss, Kerr coefficient, second-order dispersion, RS, and Raman gain, respectively.

In addition, the boundary conditions are described as follows: Pp+(0,ω,t)=Pin(ω)Tlp+Rlp(ω)Pp−(0,ω,t),(3)Pp−(L,ω,t)=Rrp(ω)Pp+(L,ω,t),(4)Ps+(0,ω,t)=Rls(ω)Ps−(0,ω,t),(5)Ps−(L,ω,t)=Rrs(ω)Ps+(L,ω,t),(6)where P is the lightwave power, Rl(ω) and Rr(ω) are the wavelength-dependent reflectivity at the left and right fiber ends, respectively, and Tl is the corresponding transmittance.

In contrast to the full open structure, the half-open configuration is insensitive to parasitic feedback, has a lower threshold, and includes a point feedback component that efficiently controls the bandwidth of the generated laser spectrum^[3,28]. So in order to study the full bandwidth time-domain statistical properties of the RRFL and facilitate the subsequent experimental validation, without loss of generality, a forward-pumped 1053 nm RRFL is exemplified, and the experimental structure is shown in Fig. 1, where the random distributed feedback is provided by an HI1060 Flex fiber. Its Raman gain spectrum is measured for the first time based on the on–off method^[29]; as shown in Fig. 2, it can be seen that the HI1060 Flex fiber has a larger Raman gain coefficient compared with single-mode fiber (SMF). Thus, a shorter feedback fiber can be applied in this work, and the total length is L=6km. Other parameter values in the simulation are summarized in Table 1. The simulation is iterated by the split-step Fourier method^[30]. During the simulation process, no averaging or filtering operations are performed on the time-domain intensity signal, but rather the output features are analyzed under the full bandwidth condition.

The simulation results are displayed in Fig. 3. Figure 3(a) shows the spectra of the forward RRFL at the fiber front end (z=0) and the tail end (z=L) at pump power of 0.69 W. The narrow spikes in the spectrum are similar to those previously observed in Refs. [13,17], and they may be associated with the interaction of processes such as SRS and RS during RRFL excitation. Figure 3(b) illustrates the intensity PDFs of the RRFL at the fiber tail end, which deviate inward from the exponential distribution, indicating there are correlations between different frequency elements of the spectrum^[19]. Besides, the intensity fluctuations become smaller with increasing pump power as the PDF shrinks inward, and the same tendency is observed in the fiber front end RRFL, suggesting that the laser becomes more stable. This trend may be because the higher the pump power is, the stronger nonlinear effects in the fiber and the more pronounced interaction effects between multiple frequency components.

Figure 3(c) exhibits the temporal intensity dynamics of the fiber tail end RRFL at pump power of 0.69 W. The rapid undulation of the signal intensity on the ns scale is evident, and there exist extreme events with intensities around 6 times the mean value. Such stochastic dynamic variations are similar to those shown in conventional resonant cavity based lasers such as the Raman fiber laser and ytterbium-doped fiber laser (YDFL)^[31,32]. The difference in the intensity PDF at different observation positions at the same pump power is shown in Fig. 3(d), and it reveals that the fiber front end PDF is closer to the exponential distribution compared to the tail end PDF; the distinction may be relevant to the dispersion and nonlinear effects in the transmission process^[33].

In this section, a compact experimental structure is designed to investigate the full bandwidth temporal statistical properties of the RRFL. The output spectral and temporal features of RRFL are analyzed first, and then the intensity PDF of the RRFL is measured and discussed in detail.

Aligning with the simulated structure, a forward-pumped configuration is conceived to explore the intensity statistical peculiarities of the RRFL experimentally, as shown in Fig. 1. A homemade 1011 nm YDFL^[34] acts as the pump source, and an optical isolator (ISO) located behind it prevents backward propagating light from affecting the pump output characteristics. Moreover, a high reflectivity fiber Bragg grating (FBG) with a central wavelength of 1052.75 nm and a bandwidth of 0.05 nm is utilized to provide narrowband point feedback so that we can investigate the statistical properties of the RRFL under the full bandwidth condition. Distributed RS and SRS in a 6 km HI1060 Flex fiber provide random feedback and Raman gain, respectively. It is worth noting that the cutoff wavelength of the HI1060 Flex fiber is about 930 nm, which can generate stable output and avoid transverse mode competition effects. To facilitate the subsequent analysis of the output characteristics of the RRFL, a 1011 nm/1053 nm wavelength division multiplexer (WDM) is fused to the fiber tail end to separate the pump and RRFL. In addition, the output characteristics of the forward RRFL at the fiber front end can be also examined conveniently through employing an optical coupler and WDM after the FBG.

The statistical characterization of the RRFL’s full bandwidth time-domain intensity has been experimentally explored with the aforementioned setup. First, the output power and spectral characteristics are recorded, and the results are shown in Fig. 4. The output 1053 nm RRFL power at Port A as a function of launched pump power is displayed in Fig. 4(a), and the excitation threshold of the RRFL is around 0.6 W. Increasing pump power leads to a nearly linear rise in output power, with a slope efficiency of ∼40%, and the efficiency can be further improved by optimizing the cavity length^[3]. Without compromising generality, three powers are selected for ensuing detailed study, i.e., 0.69 W, 0.77 W, 0.84 W. Figure 4(b) shows the spectra of the RRFL at Port A, which are measured by an optical spectrum analyzer (OSA) with 0.01 nm resolution. The spectra gradually broaden and become smooth with the pump power ramping up. Figure 4(c) describes the variation of both 10 dB and 3 dB bandwidths of the spectra, and 3 dB bandwidth of the spectrum steadily increases from 0.027 nm to 0.080 nm with the pump power rising from 0.69 W to 0.84 W. In addition, at pump power of 0.69 W, the spectra of the Port A and Port B are presented in Fig. 4(d), which indicates that the RRFL broadens due to nonlinear effects when transmitted in the fiber. Nonetheless, the absence of spikes on the spectra in the experiment similar to the simulation results is attributed to the restricted resolution and scanning speed of the OSA used in this work.

Then, in order to choose the suitable measuring device to acquire the time-domain signals of the RRFL under the full bandwidth condition, we analyze the radio frequency (RF) spectrum of the RRFL at Port A using a photodetector (PD) with a bandwidth of 40 GHz and a spectrum analyzer with a bandwidth of 43.5 GHz. As shown in Fig. 5(a), the bandwidth of the RRFL signal is roughly 15 GHz. After that, 1.25×107 data samples are recorded by using the same PD and an OSC with a bandwidth of 25 GHz at a sampling rate of 50 GSamples/s. Since the RRFL signal bandwidth is smaller than measured equipment bandwidth, statistical analysis is executed under the full bandwidth condition; thus, unlike previous work^[22], there is no frequency averaging effect, and actual temporal intensity information is obtained. Furthermore, by collecting several data sets for analysis, we precisely identify the nature of the RRFL output, preventing accidental data from occurring to undermine the dependability of the results.

The temporal signal of the RRFL at different observing positions is illustrated in Fig. 5(b), and the rapid and apparent intensity fluctuation on the ns scale can be seen. This may be due to the superposition of multiple optical transmission modes^[35], and such stochastic dynamics was also noted in previous research^[19,21]. The results also reveal that the RRFL exhibits quasi-continuous time-domain output features under the full bandwidth condition, which is different from the previous belief that the RRFL is a stable laser output^[1]. Besides, compared with the RRFL at Port A, the RRFL at Port B shows more obvious oscillations, and this may be related to the nonlinear effects in fiber during RRFL transmission.

Further, Fig. 5(c) depicts intensity PDFs of the RRFL at Port A at different pump powers, while the black dashed line shows the exponential distribution, which can be expressed as PDF(I(t)/⟨I(t)⟩)=exp(−I(t)/⟨I(t)⟩). It is well known that, for radiation consisting of statistically independent frequency components such as the amplified spontaneous emission (ASE) source, its intensity PDF follows the exponential distribution^[32,36]. However, the PDF of the RRFL deviates inward from exponential distribution, suggesting that the different frequency components of the spectrum are correlated^[19,32], and this result has been experimentally demonstrated in the spectral studies of the RRFL^[13]; this pertinence is illustrated here from a time-domain perspective.

It is worth noting that, in earlier researches exploring the statistical properties by filtering the spectrum, the intensity PDFs at different positions of the spectrum demonstrate exponential-like statistical distributions, both for the Raman fiber laser^[31] and for the RRFL^[36], which is different from our results. In addition, as the pump power increases, the intensity fluctuation becomes smaller, and the probability of extreme events decreases, a phenomenon that also occurs for the RRFL at Port B. The time-domain intensity PDF of the pump laser is also measured, which is independent of pump power, so the variation of RRFL’s PDF is not pump-induced, but its authentic features. More importantly, data samples are recorded employing an OSC with a bandwidth of 16 GHz for the analysis of the RRFL’s statistical properties. The results show identical ones to that in Fig. 5(c), indicating that there is no bandwidth-limited effect and the genuine temporal statistical properties are studied. The findings are significantly different from the previous results^[21], which demonstrated that the PDF shape gradually converges to a Gaussian one as the bandwidth of OSC decreases, implying that the bandwidth limitation severely affects the observation of the time-domain signal.

Figure 5(d) shows the intensity PDFs at Port A and Port B when the pump power is 0.69 W; similar to the results exhibited in Fig. 5(b), the PDF of the RRFL at Port B is more outwardly oriented than that of the fiber tail end, meaning a higher probability of extreme events. It can be seen that the experimental results are in good agreement with simulation ones, both revealing that the time-domain intensity PDF of the RRFL relies on the pump power and observing location, so flexibly choosing the appropriate pump power and acquisition position according to different application scenarios is feasible.

In this paper, for the first time to our best knowledge, the time-domain intensity statistical properties of the RRFL under full bandwidth condition are studied, both theoretically and experimentally. The findings indicate that, in contrast to the ASE source, there are correlations in different frequency components of the spectrum. The pump power has an impact on the intensity PDF of the RRFL, which deviates inward from an exponential distribution, and the intensity fluctuation becomes smaller with the increase of pump power. In addition, the PDF is also related to the location of the cavity, and the intensity fluctuation of the RRFL at the front end is more noticeable compared with that at the tail end. This suggests that dispersion and various nonlinear effects during the propagation process influence RRFL’s output characteristics, and we will look into this issue in depth in our subsequent research. The current work explicates the actual physical properties of the RRFL, supplying critical assistance for its significant applicability across an extensive range of realms.