Spatio-temporal-spectral imaging of non-repeatable dissipative soliton dynamics

Dissipative solitons (DSs), possessing universal particle-like properties, are localized waves found in various nonlinear systems across physics, chemistry, and biology. Their existence depends on a continuous energy exchange with the environment, balancing dispersion/diffraction with nonlinearity, and gain with loss. The formation of DSs is inherently multi-dimensional, involving space and time, leading to complex behavior. In ultrafast lasers, DSs are a widely used platform for generating energetic single-mode (SM) mode-locked femtosecond (fs) pulses. Recently, three-dimensional (3D) solitons have been discovered in multi-mode fibers (MMFs) and spatio-temporal mode-locking (STML) MMF lasers, where multiple transverse and longitudinal modes are simultaneously locked, creating complex structures from spatio-temporal-spectral (STS) interactions. The complexity contradicts the intuitive understanding that strong modal dispersion in MMFs would cause temporal walk-off among transverse modes, highlighting open questions about 3D solitons. Furthermore, noise-induced instabilities and dynamics complicate the behavior of 3D solitons when detuned from steady states. Observing these ultrafast dynamics is challenging, requiring picosecond (ps) temporal resolution, multiple degrees of freedom, and a long recording length, as transient DSs are unpredictable and non-repeatable. While single-shot technologies like temporal stretching and temporal lensing have enabled real-time measurements of non-repeatable 1D laser dynamics, revealing phenomena like rogue waves and soliton molecule internal motion, these methods lack the capability to fully observe multi-dimensional DSs. Mode-locked lasers have a low duty cycle (10⁻⁵), due to the ultrashort pulse duration compared to the round-trip time within the cavity. This temporal sparsity offers potential for applying compressed sensing techniques. This study introduces STS-CUP, operating at up to 2 trillion frames per second, to study non-repeatable dynamics of multimode DSs. Unlike previous methods, STS-CUP can capture both spectrally resolved mean modal profiles and long-term round-trip modal dynamics within the MM laser cavity.

The study builds upon previous research on dissipative solitons, their behavior in multimode fibers, and the challenges of observing their dynamics in real-time. The authors cite works on dissipative soliton theory, their application in mode-locked lasers, and the discovery of 3D solitons in multimode fibers and spatio-temporal mode-locked lasers. The literature review emphasizes the limitations of existing techniques for capturing the complex spatio-temporal-spectral dynamics of multimode dissipative solitons, particularly the lack of multi-dimensional, real-time observation capabilities. Previous single-shot techniques are mentioned, highlighting their success in 1D systems but their inadequacy for the multi-dimensional case. The authors also reference studies on compressed ultrafast spatio-temporal photography (CUST) and spatially and temporally resolved intensity and phase evaluation device (STRIPED FISH), contrasting their single-pulse focus with the capabilities of the presented STS-CUP system.

The research utilizes a multimode dissipative soliton (MM DS) laser system and a novel spatio-temporal-spectral compressed ultrafast photography (STS-CUP) system. The MM DS laser system is an STML fs MMF laser with a ring cavity using few-mode and multimode fibers, achieving mode-locking through nonlinear polarization rotation (NPR). The STS-CUP system consists of two channels: one for round-trip pulse (RTP) capture at up to 2 THz (500 fs between frames) and another for spectrally resolved single pulse (SRSP) capture, leveraging a free-space angular-chirp-enhanced delay (FACED) cavity to segment the spectrum and introduce wavelength-dependent time delays. The FACED cavity utilizes a diffraction grating and a pair of long mirrors to achieve spectral segmentation and time delays. The system is validated using a single-mode DS laser to generate and capture stable linear polarization modes (LP01, LP11, LP21), demonstrating the ability of STS-CUP to resolve spatial modes. Dynamic mode control is implemented using a broadband electro-optic modulator (EOM) to switch or rotate spatial modes of sequential round-trip pulses, further validating the system's capabilities. The main experiment involves imaging the non-repeatable STS dynamics of MM DSs in the STML MMF laser. This system uses a combination of few-mode and multimode graded-index fibers to support a large number of spatial modes and reduce modal dispersion. The MM nature of the STML DSs is verified using a standard CCD camera to obtain a time-averaged spatial profile. STS-CUP, however, captures the complex and stochastic modal dynamics in real-time. Spectrally resolved modal compositions of a single DS are also captured using SRSP mode at 2 trillion frames per second. The authors describe the FACED setup in detail, explaining how it segments the optical spectrum and introduces wavelength-dependent time delays. They also explain the EOM mode control method, where an EOM acts as an ultrafast wave plate to generate temporally varying spatial profiles. This allows the researchers to actively control the spatial modes of successive pulses, enabling more detailed investigation of the dynamics.

The study successfully demonstrates the real-time imaging of non-repeatable dissipative soliton dynamics using STS-CUP. The system's ability to capture both round-trip pulse dynamics and spectrally resolved single-pulse dynamics is validated. Using a single-mode DS laser and a few-mode fiber, the researchers successfully imaged and distinguished several distinct linear polarization modes (LP01, LP11, LP21) across consecutive round trips. They further demonstrated the ability to actively control and visualize dynamic spatial modes using an electro-optic modulator (EOM), switching between LP11 and LP21 modes or rotating an LP21 mode. In the multimode dissipative soliton laser, STS-CUP revealed the complex and stochastic nature of the spatio-temporal-spectral dynamics, showing temporal variations in spatial profiles, both randomly and periodically. The observed dynamics in the multimode system clearly deviate from time-averaged measurements obtained by standard CCD cameras, showcasing the importance of the real-time, multi-dimensional perspective offered by STS-CUP. The spectrally resolved imaging reveals variations in modal composition across the spectrum of a single pulse, indicating that the spectral components of an STML pulse are multimode and exhibit wavelength-dependent variations in modal composition. The ability to capture up to 60 consecutive round-trip events, even with a reduced streak rate, highlights the system's adaptability and capacity for handling complex, long-duration events. The observation of coexisting 3D solitons, ranging from one to three pulses depending on pump power, opens up possibilities for studying soliton molecule interactions. The achievement of spectral resolution potentially down to tens of picometers suggests future possibilities for even more detailed spectral analysis of the dynamics.

The findings address the research question by directly demonstrating the feasibility and effectiveness of real-time, multi-dimensional imaging of non-repeatable dissipative soliton dynamics. The results highlight the limitations of time-averaged measurements and underscore the need for a multi-dimensional, real-time approach. The ability to resolve both round-trip dynamics and spectrally resolved single-pulse behavior provides a significantly enhanced understanding of the complex interactions within multimode dissipative soliton systems. The observed stochastic and periodic variations in spatial profiles emphasize the inherent complexity and non-linear nature of these systems. The detailed spectral analysis further enhances this understanding by revealing wavelength-dependent modal composition, previously inaccessible with traditional measurement techniques. The success of this approach advances the field of nonlinear optics and provides a powerful tool for studying other complex lightwave phenomena, enabling further exploration of spatio-temporal mode-locking dynamics, soliton molecule interactions, and other complex nonlinear processes in multimode systems. The demonstrated capabilities are significant for applications in high-capacity communication, random bit generation, and the study of optical wave turbulence.

This study successfully demonstrated the application of spatio-temporal-spectral compressed ultrafast photography (STS-CUP) for real-time, multi-dimensional imaging of non-repeatable dissipative soliton dynamics. The findings highlight the limitations of traditional time-averaged measurements and showcase the advantages of real-time, multi-dimensional imaging. The successful observation of complex and stochastic behavior in multimode systems and the detailed spectral analysis of single pulses provide a significant advancement in the understanding of nonlinear dynamics. Future research directions could include exploring the limits of the technique for more complex soliton interactions, such as soliton collisions, as well as applying STS-CUP to investigate a wider range of nonlinear phenomena in different materials and systems.

The STS-CUP system has a maximum volumetric refresh rate of 50 Hz, limiting the observation of continuous dynamic evolutions over extended time periods. The compressed sensing principle is well-suited for periodic and sparse events, but its applicability to dense events like multiple soliton collisions might require adjustments in the system's parameters and trade-offs in resolution. While the system's spectral resolution is high, further improvements to the FACED device could potentially increase it even further. The study focuses on specific types of multimode dissipative soliton lasers and fibers; the generalizability of these findings to other systems needs further investigation.