Rogue waves, characterized by unexpectedly large amplitudes, have been observed in various physical systems, including oceans, acoustics, and finance. In optics, their temporal dynamics have been studied extensively in single-mode fibers, but the spatial aspects remained largely unexplored due to limitations in measurement techniques and the single-mode nature of many fiber lasers. This paper addresses this gap by employing state-of-the-art compressed ultrafast photography (CUP) to observe the spatiotemporal dynamics of rogue waves in a multimode fiber laser cavity. Spatiotemporal mode-locking in multimode fiber lasers provides a unique environment for such studies, as it introduces spatial interactions into the mode-locking mechanism, creating a complex nonlinear system influenced by a universal, unstable attractor. Understanding rogue wave behavior in such systems is crucial for advancing our knowledge of nonlinear wave dynamics and has implications for applications ranging from optical communication to condensed matter physics.

The concept of rogue waves originates from anecdotal accounts of unusually large ocean waves. Their existence was confirmed instrumentally in 1995. The concept has since been extended beyond oceanography, encompassing rogue waves in acoustics, thermal, and even financial systems. In 2007, analogous temporally rare, high-intensity fluctuations were observed in optical supercontinuum generation. Subsequent research focused primarily on temporal rogue waves in single-mode fibers, examining the effects of noise on supercontinuum generation and soliton dynamics. Spatial studies were conducted using nonlinear systems like optical filamentation and linear systems with multimode fibers. Mode-locked fiber lasers, with their complex nonlinear transfer functions, offer a controlled environment to study rogue wave phenomena. Earlier studies investigated rogue waves in various laser configurations, including anomalous and normal dispersion cavities with different mode-locking regimes. Spatiotemporally mode-locked fiber lasers, based on graded-index multimode fibers (GRIN MMFs), have emerged as promising platforms due to their unique spatiotemporal dynamics. These lasers exhibit phenomena such as dissipative soliton pulses, bound-state solitons, harmonic mode-locking, and self-similar pulse propagation. The nonlinear beam cleaning effect, resulting from a universal unstable attractor in GRIN MMFs, has also been investigated; this attractor transforms input fields into high-quality beams. Prior work has demonstrated active control over the nonlinear mode-locking using wavefront shaping and studied real-time dissipative soliton dynamics using CUP.

The experimental setup consisted of a dispersion-managed spatiotemporally mode-locked fiber laser incorporating a Yb-doped multimode gain fiber, a GRIN multimode fiber, and a nonlinear polarization evolution (NPE) saturable absorber. The laser generated noise-like pulses with an average power of ~70 mW and a repetition rate of approximately 22 MHz under relaxed saturable absorber conditions. The CUP technique, capable of imaging at speeds up to 70 trillion frames per second, was employed to capture the real-time spatiotemporal dynamics of the laser output. Different DMD patterns (plane mask, slit mask, pseudorandom binary pattern) were used to acquire different types of data: direct streak camera images for temporal intensity analysis, sheared single pulse measurements for individual pulse shape analysis, and full spatiotemporal intrapulse dynamics. The temporal intensity fluctuations were analyzed to determine the probability distribution of intensities, comparing the experimental data to expected distributions of rogue wave events (e.g., long-tailed distributions). The spatial properties of the pulses were characterized using the CUP images, focusing on the beam profile and position of maximum intensity for both rogue and non-rogue events. Numerical simulations using the time-dependent beam propagation method were performed to understand the effect of pulse power on the spatial distribution of pulses propagating through the GRIN MMF section of the cavity. This simulation was based on a modified version of the complex Ginzburg-Landau equation that incorporates spatial coordinates and their related terms. The simulations used a symmetrized split-step Fourier method, leveraging GPU parallelization for efficient computation. The simulation parameters were chosen to reflect those of the experimental setup, and pulses with Gaussian temporal distributions were propagated through a modeled GRIN MMF.

The study revealed several key findings. First, the temporal intensity profile of the laser output exhibited long-tailed, non-Gaussian distributions characteristic of rogue waves, with pulses reaching intensities up to 13 times greater than the standard deviation. The intensity histogram followed an L-shaped distribution, further indicating the presence of rare, high-intensity events. Analysis of the streak camera images showed that rogue events were associated with pulses exhibiting tightly focused beam profiles centered around the beam axis, in contrast to the highly dispersed spatial distribution of non-rogue pulses. Quantitative analysis of the beam profiles, using 2D Gaussian fits, revealed a significant improvement in beam quality (FWHM ratio closer to 1) for pulses with rogue event intensities. The analysis of individual pulse shapes obtained using the slit mask showed a clear distinction between pulses below and above the rogue wave threshold (RWT). Pulses below the RWT exhibited irregular, jagged burst-like shapes, while rogue event pulses had well-structured shapes with minimal background. Spatiotemporal intrapulse dynamics, captured using the pseudorandom binary DMD pattern, confirmed the enhanced beam quality for rogue events. The numerical simulations showed that the power-dependent nonlinear beam cleaning effect within the GRIN MMF was responsible for the observed spatial shaping of rogue events. The high-intensity rogue events were found to be more effectively cleaned, focusing toward a more symmetrical and confined spatial distribution compared to lower intensity pulses. Statistical analysis indicated that both intensity and pulse energy changes are better fit by a Weibull distribution rather than a Rayleigh distribution, suggesting a different underlying mechanism compared to previous rogue wave observations in single-mode fibers.

The results demonstrate that rogue events in spatiotemporally mode-locked lasers differ significantly from those observed in single-mode lasers. Unlike the noise-like bursts reported in previous studies, rogue events in the multimode laser generate clean pulses with improved beam quality. The long-tailed, non-Gaussian intensity distributions and the enhanced spatial confinement of rogue events are consistent with the power-dependent nonlinear beam cleaning effect. The study reveals the influence of the GRIN MMF’s consistent attractor in shaping the spatiotemporal dynamics of rogue waves. The improved beam quality of high-intensity pulses is attributed to the power-dependent nature of this attractor. The findings suggest that rogue events in such lasers are inherently linked to this nonlinear spatial transformation. The Weibull distribution provides a better fit for the statistical properties of rogue events compared to the previously assumed Rayleigh distribution, which highlights the unique characteristics of the spatiotemporal mode-locking mechanism. These results highlight the importance of considering both spatial and temporal dynamics when investigating rogue waves in optical systems, particularly those involving multimode wave propagation.

This study presents, for the first time, real-time observations of optical rogue waves in a spatiotemporally mode-locked fiber laser. Using CUP, the researchers demonstrated that rogue events produce clean pulses with enhanced beam quality, a contrast to noise-like pulses seen in single-mode lasers. The findings underscore the impact of the power-dependent nonlinear beam cleaning effect and highlight the need to consider spatial dynamics in rogue wave research. The Weibull distribution accurately characterizes the observed statistics of rogue events, distinct from previous models. Future research could explore the influence of different cavity parameters and fiber designs on the generation and characteristics of rogue waves in spatiotemporally mode-locked lasers. Further investigations into the relationship between the complex Ginzburg-Landau equation and the observed dynamics could provide deeper insights into the underlying physical processes.

The study focuses on a specific laser configuration and fiber type. Generalizing these findings to other types of spatiotemporally mode-locked lasers or other multimode fiber configurations may require further investigation. The numerical simulations simplified certain aspects of the laser cavity, which might influence the accuracy of the results. The proprietary nature of the reconstruction algorithm used in CUP data processing prevents widespread validation and verification of results. Although the experimental data shows a good fit to a Weibull distribution, further analysis with larger datasets would strengthen the statistical significance.