Conventional single-mode fiber lasers, constrained by the core diameter and waveguide characteristics of the fiber, only allow stable transmission of the fundamental transverse mode. In contrast, spatiotemporal mode-locked (STML) fiber lasers, as a new type of multimode laser system, have attracted significant attention due to their abundant spatial and temporal dimensional properties. In 2017, Wright et al. constructed the first STML fiber laser by adopting the nonlinear polarization rotation mode-locked technique^[1], achieving simultaneous locking of both transverse and longitudinal modes. This breakthrough not only provides a platform to investigate various STML solitons but also injects a powerful impetus for research on high-dimensional nonlinear phenomena. By manipulating parameters such as dispersion and nonlinearity within STML fiber lasers, researchers have observed various types of STML solitons, such as spatiotemporal self-similar solitons^[2], dissipative soliton resonances^[3], and spatiotemporal pulsation solitons^[4,5]. In addition to the solitons that exist as individual pulses mentioned above, there is another type of soliton existing in the form of a wave packet—noise-like pulses (NLPs)^[6]. Externally, NLPs appear as picosecond or nanosecond wave packets, but internally, they are composed of numerous sub-pulses with constantly varying amplitudes and durations^[7]. This unique time-domain structure endows NLPs with high energy, broad spectrum, and low coherence, which opens up novel technological paths for bio-imaging^[8], supercontinuum generation^[9], low-coherence spectral interferometry^[10,11], and interferometric diffusing wave spectroscopy^[12–15]. Recent research progress has been made in the study of STML NLPs. Xing et al. observed the switching dynamics of NLPs and Q-switching using the dispersive Fourier transform technique and verified that mode interactions play an important role in the generation of NLPs^[16]. Furthermore, STML NLPs also serve as an ideal carrier for triggering the beam self-cleaning effect. When high-intensity NLPs circulate within STML fiber lasers, the energy is coupled from higher-order modes to lower-order modes, causing the beam profile to evolve into a near Gaussian shape, thereby satisfying the requirements for both high energy and high beam quality^[17,18]. In addition, multimodal NLPs can also be generated in Mamyshev oscillators, pioneering new avenues for generating STML NLPs^[19].

On the other hand, dual-wavelength NLPs have shown great potential for spectral-domain optical coherence tomography^[20], owing to the combination of dual-channel spectral characteristics and the time-frequency characteristics of NLPs. At present, the research on dual-wavelength NLPs is mainly focused on single-mode fiber lasers, with studies reporting their generation^[21], switching between dual-wavelength and single-wavelength NLPs^[22], and their dependence on intracavity pump power^[23]. Various shapes of dual-wavelength NLPs have also been found, including rectangular, h-shaped, and trapezoidal shapes^[24–26]. Further wavelength-resolved investigations reveal that the shapes of NLPs at the two wavelengths can be either identical^[21] or different^[25], and that dual-wavelength NLPs are often superimposed by NLPs with different pulse widths at the two wavelengths. In contrast to single-mode fiber lasers, dual-wavelength NLPs in STML fiber lasers have been scarcely investigated. Zhang et al. observed synchronized dual-wavelength NLPs, which can switch to single-wavelength NLPs and exhibit splitting by adjusting the pump power^[27]. However, current research on STML dual-wavelength NLPs mainly focuses on power-related characteristics, while studies on wavelength-related characteristics and pulse shapes remain limited. Additionally, the simulation research on STML NLPs is still blank.

In this work, we report the generation of dual-wavelength STML noise-like square pulses (NLSPs) in a few-mode fiber laser based on the nonlinear amplifying loop mirror (NALM). The dual-wavelength NLSPs are synchronous and comprise two square pulses with distinct durations at each wavelength. To further analyze the experimental findings, we conduct numerical investigations into the mode-related and wavelength-related characteristics of the dual-wavelength NLPs. In terms of mode-related characteristics, it is found that different modes of NLP pulses exhibit distinct frequency components and pulse fine structures, and all of which vary in real time. Regarding wavelength-related characteristics, it is found that a wider spectrum, often associated with higher pulse energies, corresponds to a longer NLP envelope duration and a shorter sub-pulse duration, verifying the experimental results. In the simulation, it is found that the mode components of pulses at different wavelengths are distinct, resulting in differences in their beam profiles. These results enrich the comprehension of pulse characteristics in STML fiber lasers, and offer a new perspective for exploring spatiotemporal solitons.

The schematic of the experimental setup is depicted in Fig. 1. The laser utilizes a figure-eight structure based on the NALM mode-locked technique. A 2×2 optical coupler (OC) with a coupling ratio of 30:70 is used to combine the NALM and the unidirectional ring (UR). In the NALM ring, a 2-m-long erbium-doped fiber (EDF, Er1200-20/125, FORC, RAS), capable of supporting six transverse modes (LP01, LP11a, LP11b, LP21a, LP21b, and LP02), is pumped by a 980 nm continuous-wave laser diode via a 980/1550 nm wavelength division multiplexer (WDM; the fiber type of the input and output pigtails is MM-GSF-20/125-10 A). PC1 is used to regulate the saturable absorption effect of the NALM. Additionally, a 100-m-long few-mode fiber (FMF, YOFC, FM2011-A, 23/125 µm) is introduced to ensure that the clockwise and counterclockwise propagating lights within the loop accumulate a sufficiently nonlinear phase shift difference. The UR consists of a polarization-independent isolator (PI-ISO), PC2, and a 1×2 OC. The OC with a coupling ratio of 20:80 extracts 20% of the light for measuring. All passive fibers within the cavity are identical to the aforementioned FMFs, and the total cavity length is approximately 118.3 m. Arising from the differences in the core diameter and numerical aperture among the EDF, fiber pigtails of the WDM, and other passive FMFs, mismatch points exist at their splices, effectively counterbalancing intermode dispersion and introducing multimode interference filtering effects and wavelength tunability.

The characteristics of the STML pulse output from the cavity are measured using the corresponding instruments. A spectrum analyzer (Yokogawa, AQ6317C, resolution: 0.01 nm) is employed to measure the frequency components of the pulses. An oscilloscope (Teledyne Lecroy, 813Zi-B, 13 GHz) combined with a photodetector (Newport, Amplified Photoreceivers Models 1544-A, 12 GHz) is used to monitor the pulse-train. A radio frequency (RF) spectrum analyzer (Agilent, E4407B ESA-E SERIES, 26.5 GHz) is utilized to determine the pulse repetition rate. An autocorrelator (Femtochrome, FR-103WS) is employed to examine the fine structure of the pulses. Besides, a charge-coupled device (CCD) camera (Goldeye G-033SWIRTEC1) is used to capture the beam profile of the STML pulse.

Since the pulse requires sufficient gain to undergo soliton collapse and form the NLP, stable single-wavelength STML NLSP is achieved after increasing the pump power to 900 mW and carefully adjusting the PCs. The typical characteristics are presented in Fig. 2. Figure 2(a) shows the spectrum of the NLSP, exhibiting irregular modulations that arise from the interference between different transverse modes^[28,29]. The inset displays the corresponding beam profile. The pulse-train is shown in Fig. 2(b), revealing that the interval between adjacent pulses is approximately 572 ns. The inset indicates that the duration of the NLSP is 1.89 ns. Meanwhile, the RF spectrum in Fig. 2(c) demonstrates that the fundamental repetition rate of the pulse is 1.748 MHz, which corresponds to the adjacent pulse interval in Fig. 2(b). The inset of Fig. 2(c) shows the RF spectrum over a 1 GHz span, featuring a broad range of modulation periods corresponding to the pulse duration of 1.89 ns. The autocorrelation trace of the pulse is illustrated in Fig. 2(d), revealing a narrow spike riding on a wide base. The base corresponds to the duration of the NLSP envelope, while the spike indicates the average width of the sub-pulses within this envelope. Due to the NLSP’s duration exceeding the measurement range of the autocorrelator, we are unable to obtain the complete autocorrelation trace of the NLSP.

Given that two filtering effects coexist within the cavity, namely, the intrinsic filtering of the NALM and the multimode interference filtering caused by mismatched FMFs, this unique configuration enables the formation of multiple wavelengths. We achieve stable dual-wavelength NLSPs by carefully adjusting the PCs. Figure 3 presents the characteristics of the dual-wavelength NLSPs. The spectrum shown by the red curve in Fig. 3(a) comprises a narrower spectrum centered at 1550.1 nm and a broader spectrum centered at 1560.5 nm. The narrower spectral width at the shorter wavelength compared to the longer wavelength may result from the gain competition between the two wavelengths. Note that the spectrum contains many irregular modulations, which are the results of mode interference, and there is a continuous wave at 1568.8 nm. To verify the spatiotemporal mode locking of the two wavelengths, we spatially sample different beam locations of the output laser, as illustrated in the inset of Fig. 3(a), using a single-mode fiber. As illustrated in Fig. 3(a), the spectra corresponding to different sampling positions exhibit some differences, indicating that each sampling position contains different mode components. However, since the NLSPs are dominated by the fundamental mode, the spectral differences between the two sampling positions are not so obvious. The RF spectra are shown in Fig. 3(b). All sampling points exhibit the same repetition rate as the output pulse without sampling, demonstrating that the different transverse modes are transmitted synchronously. Figure 3(c) illustrates the double-scale autocorrelation trace of the NLSPs. Here, due to the low power after sampling, the autocorrelation traces of the pulses corresponding to the two sampling positions are not measured. Then, we resolve the individual pulses at these two wavelengths using a pulse shaper (Finisar, Waveshaper 4000 A), and the filtered spectra are shown in Fig. 3(d). Due to the limited filtering range of the Waveshaper, we are only able to filter the long-wavelength range up to 1568 nm. The corresponding RF spectra and pulse-trains, both before and after filtering, are presented in Figs. 3(e) and 3(f), respectively. The repetition rates of the pulses at both wavelengths are 1.748 MHz, corresponding to the pulse interval of 572 ns. This implies that the NLSPs at the two wavelengths are synchronous. The details of the pulse envelope are depicted in the inset of Fig. 3(f). The central wavelengths at 1550.1 and 1560.5 nm correspond to square pulse widths of 1.02 and 2.22 ns, respectively, implying that a narrower spectral bandwidth results in a shorter pulse duration, while a broader spectral bandwidth leads to a longer pulse duration. The dual-wavelength NLSPs are essentially a superposition of the two wavelength-resolved pulses. Consequently, the pulse duration of the dual-wavelength NLSPs is determined by the longer pulse duration of the two individual wavelength-resolved pulses, and it corresponds to the modulation period shown in the inset of Fig. 3(e). Note that the waveshaper employs a single-mode fiber, and we are unable to observe mode-dependent information for the pulses at the two filtered wavelengths.

To qualitatively understand the properties of synchronous dual-wavelength STML NLPs, we perform a numerical simulation based on the generalized multimode nonlinear Schrödinger equation^[30]. Given the complexity of accurately simulating the NALM ring, the simulation model approximates it as a saturable absorber (SA) whose behavior is dependent on the spatial and power characteristics of the spatiotemporal light field. The transmission of the SA is written as Aout(x,y,t)=Ain(x,y,t)1−a1+|Ain(x,y,t)|2/Isat,(1)where Ain(x,y,t) is the light field distribution before the SA, Aout(x,y,t) denotes the light field distribution after the SA, a is the modulation depth, and Isat is the saturation intensity. Here, a is set to 0.84, which corresponds to the modulation depth of the transmittance curve of the NALM ring. Six transverse modes in the cavity are considered. To provide a sufficiently large temporal range for the formation of NLPs, the time domain window is set to 2000 ps. The input pulse in the simulation is a single-wavelength Gaussian pulse with a total pulse energy of 0.001 nJ and a pulse width of 10 ps, which contains six transverse modes with equal energy. Given the presence of both NALM filtering and multimode interference filtering effects within the laser cavity to achieve multiwavelength operation, we introduce two Gaussian filters in the simulation to realize dual-wavelength NLPs. These filters have central wavelengths of 1548 and 1552 nm, with corresponding bandwidths of 4 and 6 nm, respectively. The synchronous dual-wavelength STML NLPs can be obtained by appropriately setting the cavity parameters. Specifically, the gain coefficient g0 is set to 5.5 dB/m, the gain saturation coefficient gsat is 1.5 nJ, and the saturable absorbed energy Isat is 2 W. Figure 4 illustrates the characteristics of the STML dual-wavelength NLPs in simulations. The formation of the spectrum of the dual-wavelength NLPs is displayed in Fig. 4(a). Although the spectrum evolves in real-time, it consistently maintains a dual-wavelength feature. The two wavelengths are centered at 1549 and 1551.5 nm. Notably, the spectral bandwidth of the 1549 nm wavelength is narrower compared to that of the 1551.5 nm wavelength. Figure 4(b) shows the temporal formation process of the dual-wavelength NLPs. Initially, under strong pumping conditions, the pulse forms rapidly and grows quickly. However, due to the limitations imposed by soliton area theory, the pulse reaches its variation limit and subsequently collapses. Following this collapse, small pulses emerge, acquiring energy from the collapsing pulse and rapidly growing to form new pulses, and so on. Throughout this process, the width of the pulse envelope gradually increases until the rate of new pulse formation is dynamically balanced with the rate of pulse collapse. It is worth noting that the pulse is transmitted as a whole envelope, but the fine structure within the envelope evolves dynamically in real-time, which results in the duration of the NLP envelope changing in real-time. During the evolution of the NLPs, some individual solitons are observed to separate from the NLP envelope [indicated by the white arrow in Fig. 4(b)]. This phenomenon occurs due to the intense motion and interaction among the solitons within the pulse^[31]. As a result, certain solitons are forced out of the NLP envelope and ultimately disappear, driven by competition with the dominant NLP structure. To better investigate the fine structure and mode-related properties of the STML NLPs, we examine the real-time characteristics of the mode-resolved pulse at the 300th round-trip. Figure 4(c) shows the spectra of the six transverse modes at this round-trip, revealing that each mode contains frequency components at both wavelengths, with these components exhibiting distinct variations across different modes. Figure 4(d) presents the fine structure of the real-time mode-resolved NLPs. It can be seen that the NLP envelope comprises numerous small pulses with varying intensities and pulse widths. This is attributed to the fact that the NLP is essentially a dynamic process involving the collapse and regeneration of these small pulses. Consequently, the fine structure of the NLP evolves differently with each round-trip. Comparing the fine structure of the mode-resolved pulses, we can discover that the characteristics of these pulses, including the intensity, duration, and the position of the envelope centered in the time domain, exhibit significant diversity among different modes. Notably, the centers of the pulse envelopes for different modes are not aligned in the time domain. It can be attributed to the large intermodal dispersion that arises from the long cavity length of hundreds of meters; despite the use of a gradual refractive index FMF, the significant walk-off of the mode-resolved pulses induced by this dispersion results in the distinct temporal positioning of their envelope centers.

In order to investigate the pulse characteristics at both wavelengths, we perform filtering operations on the pulses; the results are summarized in Fig. 5. Figures 5(a) and 5(d) depict the pulse evolution at 1549 and 1551.5 nm, respectively. It is evident that the pulse duration at 1551.5 nm is longer than that at 1549 nm. Here, the duration of the NLP is calculated by excluding the solitons that move away from the main pulse structure. To further examine the fine structure of the pulse, we analyze the details of the NLPs at two wavelengths at the 300th round-trip, as shown in Figs. 5(b) and 5(e). The insets of Figs. 5(b) and 5(e) show the zoomed-in sub-pulses at the two wavelengths. Compared to the pulse at 1549 nm, the pulse at 1551.5 nm exhibits a longer duration, and the widths of its internal sub-pulses are narrower. We speculate that the broader spectral width at 1551.5 nm implies a larger nonlinearity. According to the soliton area theory, the sub-pulses within the pulse are more prone to collapse and subsequently form additional sub-pulses, resulting in a longer pulse duration. Furthermore, we explored the mode-related information in two wavelength-resolved pulses. Figures 5(c) and 5(f) demonstrate the mode energy evolution of the wavelength-resolved pulses at 1549 and 1551.5 nm, respectively. In these figures, the solid lines of different colors represent the energy evolution of distinct modes, while the black dashed line indicates the total pulse energy. It is observed that the energy of the NLPs exhibits slight fluctuations even after stabilization. This behavior can be attributed to the inherent instability of NLPs compared to conventional solitons. Specifically, the continuous generation and annihilation of sub-pulses within the NLP envelope cause the total pulse energy to fluctuate around a certain value. Note that the higher total energy at 1551.5 nm enhances the nonlinearity of the pulse, leading to a broader spectrum and narrower pulse widths of the sub-pulses within the NLP. At both wavelengths, the fundamental mode is dominant; however, the energy ratios of the fundamental mode differ between the two wavelengths. At the 300th round-trip, the energy ratio of the fundamental mode at 1549 nm is 42.3%, whereas at 1551.5 nm, it is 55.4%. Despite the fluctuations in energy from one round-trip to another, overall, the energy ratio of the fundamental mode at 1551.5 nm is higher than that at 1549 nm. Consequently, the beam profile at 1551.5 nm is more biased towards the fundamental mode. In contrast, at 1549 nm, the beam profile is more inclined towards higher-order modes due to the superposition of higher-order transverse modes. Because of the significant challenges in achieving mode resolution, it is difficult to directly observe the mode-related information of dual-wavelength NLPs in our experiments to validate our simulations. Nevertheless, the simulation results offer valuable insights into the mechanisms underlying dual-wavelength synchronous NLPs in STML fiber lasers, serving as a useful guide for future experimental studies.

In conclusion, synchronous dual-wavelength STML NLSPs in a FMF laser are achieved. The filtering operation reveals that dual-wavelength NLSPs consist of two NLSPs with different durations corresponding to the two wavelengths. Furthermore, numerical simulation is performed to elucidate the mode-dependent and wavelength-dependent properties of these dual-wavelength NLPs. It is demonstrated that different frequency components and fine structures exist in NLPs of different modes, and they dynamically evolve over time. For wavelength-related characteristics, wavelength-resolved operations show that a wider spectrum is associated with a longer NLP envelope duration as well as a shorter sub-pulse duration, thereby validating the experimental results. Additionally, it is also found in the simulations that pulses of different wavelengths contain different mode components, resulting in differences in their beam profiles. These findings contribute to a deeper understanding of the intrinsic mechanism underlying STML NLPs and explore the potential applications of STML fiber lasers.