The random fiber laser (RFL) utilizes Rayleigh scattering along the optical fiber to provide random distributed feedback^[1] with simple structure^[2], high efficiency^[3-6], wavelength agility^[7,8], and high-power output^[9,10]. The research on its physical properties has been very active, including nonlinear processes^[11], transient dynamics^[12], and optical rogue waves^[13]. These studies promote various applications of RFL involving optical fiber sensing^[14], optical imaging^[15], and vortex beams^[16].

The erbium-doped random fiber laser (ERFL) integrates appealing features of RFL and erbium-doped fiber (EDF) lasers. The former offers the traits mentioned above, while the latter introduces attractive physical mechanisms due to the erbium-doped gain, along with the advantage of having a low threshold^[17]. Consequently, the ERFL has not only obtained a range of applications^[18], but more importantly, it has also evolved into a promising bridge for the exploration of complex physics^[19]. For example, replica symmetry breaking is illustrated in ERFL, where a discernible transition from a photonic paramagnetic to a photonic spin-glass phase is verified^[20]. Complex mode hopping in ERFL is observed in experiments and has been explored in the frame of a phenomenological mathematical model^[21]. Additionally, the ERFL combines non-Hermitian random matrix theory that can be used to analyze the hindering temporal second-order intensity correlation^[22]. The aforementioned studies are all related to temporal features, forming the basis for a comprehensive investigation of ERFL.

In particular, the temporal statistical characteristic has been extensively studied in the field of RFL over the past decade. Initially, experiments can only be conducted under the limited bandwidth condition because the electrical bandwidth of the measuring device is typically smaller than the optical bandwidth of the RFL. In 2015, Gorbunov et al. measured statistical properties of RFL at different electrical and optical bandwidth ratios^[23], which can be used to approximately predict full-bandwidth features. For measuring the real intensities attributes, a filtering method is proposed to study part of spectral components. This group further analyzed the statistical features of Raman RFL based on the Lyot filter^[24]. Furthermore, Lévy spectral intensity statistics in a Raman RFL are investigated utilizing a fiber Fabry–Perot tunable filter^[25], which is a common behavior as that observed in ERFL^[26]. Based on the tunable filter, it is experimentally demonstrated that the statistical features of erbium/ytterbium co-doped RFL are highly dependent on the spectral location^[27]. However, the overall spectral characteristics remain unknown. Recently, our group explored the temporal statistical characteristics of a narrow-linewidth 1053 nm Raman RFL under full-bandwidth conditions. The experimental results are in good agreement with the simulation, revealing that its intensity probability density function (PDF) deviates from the exponential distribution and relies on the observing location^[28]. Constrained by the challenging generation of narrowband ERFL and the limited bandwidth of detection devices, the full-bandwidth time-domain statistical attributes of the ERFL require further exploration.

Previous research on the temporal statistical properties of RFL has mainly focused on feature description and revelation, while the current challenge in the application of RFL resides in tailoring these properties. As far as our knowledge extends, there has been limited research on this aspect within RFL systems. RFL with more random temporal fluctuations has been demonstrated to be a promising source for random bit generation featuring both high performance and simple configuration^[29] and is truly conducive to manipulating and further enhancing the efficiency of frequency doubling compared to amplified spontaneous emission (ASE)^[30]. In addition, such RFLs are great candidates for optical imaging, enabling near-perfect retrieval of ultrafast temporal objects^[31]. Conversely, in the field of optical fiber sensing^[14,32], a laser with more stable time-domain attributes has advantages in the transmission of sensing signals and in avoiding unnecessary nonlinear processes. Thus, it can reduce fading points and prevent the distortion of sensing results. Especially, in the inertial confinement fusion (ICF) field^[33], the RFL faces challenges related to temporal fluctuations as an alternative seed source. Larger temporal fluctuations can enhance the energy of backward stimulated Raman scattering and stimulated Brillouin scattering light, resulting in a significant reduction in energy acting on the target pellet. Therefore, possessing stable temporal characteristics is one of the keys to the successful application of RFL in ICF laser facilities. In general, different utilization scenarios have varying requirements for the temporal characteristics of RFL, ranging from more random to more stable. To better accommodate the demands of these applications, achieving tailoring time-domain features of RFL is of vital significance.

In this study, we explore the temporal statistical characteristics of ERFL under the full-bandwidth condition and achieve its tailoring from the perspective of fiber dispersion. First, a narrowband ERFL is specifically constructed, and its full-bandwidth intensity dynamics at different pump powers are measured. Statistical analysis based on the intensity PDF reveals an inward deviation from the exponential distribution. Second, the impact of fiber dispersion on the temporal properties of ERFL is investigated using fibers with different dispersion values and lengths. By extending the SMF length, i.e., increasing the accumulated dispersion of the system, the ERFL exhibits a more “random” state; by additionally introducing dispersion compensation fiber (DCF) for dispersion compensation, the ERFL can obtain a more “stable” state. This work reveals the intrinsic temporal dynamics of ERFL and offers inspiration for tailoring the temporal characteristics in RFL systems.

The structure of the ERFL is designed delicately to assure the full-bandwidth condition. The experimental setup of ERFL is shown schematically in Fig. 1. The 10-m-long EDF is pumped by a 1455 nm Raman fiber laser through a 1455/1550 nm wavelength division multiplexer (WDM1) to provide gain for the 1550 nm lasing. The inclusion of an isolator (ISO1) between the pump and WDM1 serves the purpose of avoiding back-reflections. The fiber Bragg grating (FBG) with 90% reflectivity, a center wavelength of 1550 nm, and a 3 dB bandwidth of 0.1 nm is connected to the 1550 nm port of WDM1. It acts as a point reflector to compromise a half-open cavity with the random distributed feedback in the 3-km-long single-mode fiber (SMF). The generated ERFL outputs at the end of SMF and is separated from the pump by an additional 1455/1550 nm WDM2. ISO2 placed between SMF and WDM2 plays the role of preventing backward light from affecting laser excitation. It is worth emphasizing that the choice of 3 km SMF ensures both slight nonlinear effects to prevent spectral broadening beyond the measuring bandwidth and sufficient random feedback to possess a low lasing threshold.

For spectrum measurements, an optical spectrum analyzer (OSA) with a 0.01 nm resolution is utilized while the time-domain signals are detected by a photodetector (PD) with a bandwidth of 40 GHz and an OSC with a bandwidth of 16 GHz. The power and spectral performances of ERFL are depicted in Fig. 2. With the optical signal-to-noise ratio (OSNR) as high as ∼50dB, as presented in Fig. 2(a), the effect of the pump to the results can be ignored. Figure 2(b) shows the measured output power of the 1550 nm random lasing as a function of the launched pump power. Five specific points highlighted in red are identified within the region of flat growth for subsequent study. The output spectra of selected points are presented in Fig. 2(c), while the inset depicts the spectral details in the red dashed box. As the pump power increases from 1.0 to 2.2 W, there is only slight broadening at the central lasing wavelength, which contains the majority of the power. This phenomenon resembles the spectra observed in the RFL based on active fiber^[34]. To provide a detailed evolution of the spectral linewidth versus the launched pump power, the 3 dB bandwidths are calculated and given in Fig. 2(d). The results reveal that the 3 dB bandwidth increases from ∼2.5 to ∼9.9GHz.

To measure the time-domain signals under the full-bandwidth condition, the bandwidths of the PD and OSC employed are much larger than the maximum 3 dB bandwidth (∼9.9GHz) of the random lasing plotted in Fig. 2(d). Under this circumstance, there is no effect of frequency average, allowing for the capture of real intensities and accurate statistics. For the purpose of analyzing the statistical attributes, 2×108 samples using the OSC at a 40 GSamples/s sampling rate are acquired. The intensity I(t) is normalized to its mean value ⟨I(t)⟩ according to the experimental data. As depicted in Fig. 3(a), the ERFL exhibits fluctuations on a sub-nanosecond timescale. The intensity PDFs of selected points are illustrated on a vertical logarithmic scale in Fig. 3(b). The black dashed line represents the exponential distribution, which is equal to the radiation consisting of statistically independent wavelength components with Gaussian statistics^[27], like the ASE source. The intensity PDF deviates from the exponential distribution inward, indicating some correlations among different wavelength components that exist in the ERFL. Moreover, the PDF remains almost the same as the pump power increases, suggesting that the ERFL’s operating status has not changed, which is a stable light source. In addition, the results maintain good consistency over the multiple measurements, indicating the accuracy of the statistical analyses.

The demand for the time-domain feature varies with the application scenario, highlighting the importance of tailoring this attribute. The characterization in the second section reveals that the ERFL serves as a stable seed source, which is beneficial for enhancing the reliability and accuracy of tailoring.

The analysis of the ERFL’s full-bandwidth PDF has displayed a deviation from the exponential distribution, which implies that correlations exist among different wavelength components. Previous studies^[35] mentioned that fiber dispersion could influence such correlations, theoretically allowing for the tailoring of the ERFL’s PDF distribution through the aspect of dispersion. Therefore, figuring out the impacts of fiber dispersion represents the initial step toward achieving tailoring temporal attributes.

Figure 4 illustrates the experimental configuration employed for investigating the effects of fiber dispersion. Without loss of generality, the ERFL operating at an output power of 210 mW with a pump power of 1.3 W is chosen as the seed source. The selection of SMF and DCF following the ERFL is determined by specific circumstances. In addition, subsequent studies into time-domain characteristics are all conducted under the full-bandwidth condition.

Especially, for accurate analyses of dispersion’s impacts on temporal statistical properties, we use the phase-shift method^[36] to measure dispersion values of the employed optical fibers, with results presented in Table 1. Two types of DCFs are utilized with different dispersion values, named DCF1 and DCF2.

At first, only SMF is connected to the ERFL, and the signals are collected after transmission at position b as presented in Fig. 4. The influence of transmission distance is considered with the SMF length set at 25, 50, and 75 km with results shown in Fig. 5, and the line colors in Fig. 5(a) are consistent with the meaning depicted in Fig. 5(b). Significant fluctuations on the sub-nanosecond time scale are observed with the increase in transmission distance, as displayed in Fig. 5(a). After traversing 25 km SMF, extreme events with intensity approximately 15 times higher than the average value have been recorded. Additionally, rare and intense events with peak powers up to 18 and 19 times the average power can be observed after traversing 50 and 75 km SMF, respectively. Similarly, the PDF is employed for analyzing statistical attributes, as illustrated in Fig. 5(b). When compared to the ERFL seed source represented by the blue curve, the distribution curves after transmission through three lengths of SMF all exhibit outward deviation. This variation suggests that the probability of extreme events increases after transmission, meaning a degradation in the time-domain stability. Meanwhile, it is worth noting that the degree of outward deviation of the distribution curve decreases with the same increased transmission distance, i.e., 25 km. According to this, the effect of transmission on the time-domain properties has an upper limit, which eventually converges to the exponential distribution like the ASE source.

The above evolution may be related to the accumulation of fiber dispersion. It can be noted that the SMF and DCF possess dispersion values with opposite signs as listed in Table 1. Therefore, we ingeniously utilize different combinations of these three optical fibers to explore the effects of dispersion on the temporal statistical features. Experiments are conducted based on the structure in Fig. 4 with SMF and DCF1, and signals are collected at position c. Figure 6 presents the results of the PDF alteration after dispersion compensation. There are two cases, the first one is insufficient compensation. A section of 2.2 km DCF1 is connected after the 25 km SMF for dispersion compensation, with the results depicted in Fig. 6(a). The orange and green lines’ relative positions reveal that the probability of temporal extreme events gets smaller after compensation by the DCF1. Although the green line has not returned to the position of the blue line representing the initial ERFL, it still shows the beneficial effect of dispersion compensation on stabilizing the time-domain characteristics. This is the case where dispersion compensation is insufficient because the absolute value of dispersion of the utilized 25 km SMF [25(km)×16.7(ps/nm/km)=417.5(ps/nm)] exceeds that of the employed 2.2 km DCF1 [2.2(km)×155(ps/nm/km)=341(ps/nm)] based on the data in Table 1. Next, we adopt a combination of optical fibers where the dispersion values almost cancel each other out. The second case is exact compensation. When the absolute values of dispersion of 21 km SMF [21(km)×16.7(ps/nm/km)=350.7(ps/nm)] and 2.2 km DCF1 are nearly equal, the PDF of the ERFL after transmitting (red line) returns to the initial position, as illustrated in Fig. 6(b). In a word, the results above first verify that the PDF variations observed in Fig. 5 are induced by fiber dispersion. Second, the effectiveness of dispersion compensation in stabilizing the time-domain characteristics are demonstrated. More specifically, the most stable time-domain output can be obtained when the system’s positive and negative dispersion values cancel each other out.

The aforementioned analyses indicate that dispersion compensation is effective to alleviate temporal fluctuation. As depicted in Fig. 1, the random feedback of the ERFL seed source is provided by SMF, and there are positive dispersion values naturally. We continue to utilize the configuration in Fig. 4 to improve the ERFL’s time-domain stability. Different from the preceding structure, there is only a section of 450 m DCF2 connecting to the ERFL. It is sufficient to compensate for the dispersion value of 3 km SMF. Figure 7 shows the PDF of the ERFL after DCF2 (pink line) has shifted inward compared to the initial state, meaning a reduction in the likelihood of signals with intensity significantly greater than the average value. The above results demonstrate that selecting appropriate optical fibers for dispersion compensation according to the system structure is an efficient and concise method for improving time-domain stability.

In addition, the spectra of the ERFL under the above various experimental conditions are presented in Fig. 8. For a clear comparison, we attenuate the power to the same level. The spectra remain almost constant at a resolution of 0.01 nm. Therefore, adjusting dispersion accumulation as proposed in this work could be an effective way to tailor the temporal characteristics without significantly changing the linewidth of the laser.

Here, we proceed to a more detailed discussion of the above results. Regarding the ERFL seed source as a unit, analysis is conducted at three positions marked as a, b, and c in Fig. 4. As shown in Fig. 3(b), the PDF deviates from the exponential distribution, which is equal to the radiation consisting of statistically independent wavelength components. It implies the existence of correlations among various wavelength components of the ERFL at position a.

The SMF is situated in the anomalous dispersion regime at the 1550 nm band with dispersion parameter D>0, meaning lightwaves with shorter wavelengths propagate more quickly than those with longer wavelengths. During transmission through the SMF, the anomalous dispersion leads to varying transmission speeds for lightwaves of different wavelengths. Upon reaching position b for subsequent analysis, the initially correlated wavelength components become “misaligned.” Moreover, the “misalignment” increases with the accumulation of dispersion until reaching a state of no correlation. Therefore, after transmission through 25, 50, or even longer 75 km SMF, the PDF approaches the exponential distribution as illustrated in Fig. 5(b), indicating the absence of correlations. In contrast to SMF, DCF is in the normal dispersion regime at the 1550 nm band with dispersion parameter D<0, where the lightwave of a shorter wavelength travels slower. When arriving at position c, the “misalignment” at position b due to anomalous dispersion in SMF is reduced by normal dispersion in DCF1, as depicted by the relative positions of the orange line and the green line in Fig. 6(a). When the values of normal dispersion and anomalous dispersion are equal, it means that the relative positions of the wavelength components would return to the original “aligned” state. This is exactly the case for the combination of 21 km SMF and 2.2 km DCF1 in Fig. 6(b), whose PDF goes back to the position of the initial ERFL. Similarly, there is also “misalignment” due to the presence of SMF with anomalous dispersion inside the ERFL seed source. This effect can be mitigated by connecting a section of DCF2 with normal dispersion outside the cavity, as shown in Fig. 7.

To summarize, PDF distributions that are either more inward or outward than those of the seed source can be achieved by accumulating or compensating for the system’s dispersion value, i.e., either a more random or more stable temporal output than the seed source can be tailored. In principle, the method of adjusting dispersion for tailoring temporal properties in this work could be adopted to other RFL systems including broadband ytterbium-doped RFL used in ICF^[33], Raman RFL utilized in fiber sensing^[37], and so on. This research provides guidance for applications requiring specific temporal attributes of RFL.

In this work, the full-bandwidth statistical properties of the narrowband ERFL are analyzed, and the tailoring is achieved from the aspect of fiber dispersion. In terms of characterization, the intensity PDF deviates from the exponential distribution inwards, indicating there are some correlations between different wavelength components. In terms of tailoring, dispersion accumulation breaks correlations among wavelength components, bringing the time-domain features closer to the ASE source. In contrast, dispersion compensation reduces fluctuations of the temporal signals, promoting a more convergent PDF distribution and allowing for a more stable time-domain output. This work reveals ERFL’s intrinsic temporal dynamics and paves a new way for tailoring the temporal statistical properties of RFL systems toward various demands in applications.