Phase-tailored assembly and encoding of dissipative soliton molecules

The paper addresses how to programmatically control the internal dynamics of dissipative soliton molecules (DSMs) in mode-locked laser resonators to enable advanced optical information encoding. DSMs, formed by balanced dispersion/nonlinearity and gain/loss, can self-assemble into bound states with rich internal motions. Prior work has shown multisoliton interactions, supramolecular ordering via long-range forces, and control over temporal separation using pump modulation or dispersion management. However, precise and reversible manipulation of the molecular internal degrees of freedom—especially the relative phase and temporal separation between constituents—for deterministic switching among distinct bound states remains challenging. The authors propose programmable gain modulation to manipulate energy exchange between neighboring solitons, thereby tailoring their internal phase evolution. They aim to create four phase-defined regimes (singlet, negative-phase pair, stationary-phase pair, positive-phase pair) and demonstrate a quaternary encoding format, moving beyond pulse-counting binary schemes toward higher-capacity all-optical storage.

The introduction surveys developments in DSM physics and applications: (1) DSMs as particle-like entities with bound states and internal motions; (2) real-time tracking of transient multisoliton dynamics via time-stretch dispersive Fourier transform (TS-DFT) spectral interferometry, enabling observation of temporal separation and relative phase dynamics; (3) assembly of large soliton sequences using temporal optomechanical lattices to form supramolecular structures with controlled long-range interactions; (4) manipulation of intra-molecular dynamics through pump modulation and dispersion engineering (second-order group-velocity dispersion and dispersion losses) to switch among discrete binding states and implement quaternary encoding based on temporal separation. These works motivate finer phase-level control for multiple encoding formats and all-optical storage, highlighting the need for programmable, reversible tailoring of molecular phases within laser cavities.

An ultrafast fiber laser platform with nonlinear polarization rotation (NPR) mode-locking is constructed, comprising a programmable gain control module, a mode-locking module, and a real-time monitoring module. Key parameters: cavity repetition rate ~54.86 MHz, operation wavelength ~1565 nm. The gain (erbium-doped fiber, EDF) is modulated electronically using an arbitrary function generator (AFG 31000, Tektronix; 25 MHz modulation bandwidth), with a pump control transfer efficiency of ~52 mW/V. Intracavity polarization is adjusted via polarization controllers (PCs) and a polarizer to obtain stable DSM states. Real-time spectral interferometry (TS-DFT-based) monitors dynamic soliton assemblies, retrieving temporal separation τ and relative phase φ from spectral fringes. The authors define a phase-evolving velocity to quantify average phase evolution speed and visualize trajectories in a 3D interaction space involving τ, φ, and the phase-evolving velocity. Frame-by-frame manipulations are performed by stepwise pump increases from 212.7 mW to 255.6 mW in 1.3 mW steps; each frame comprises 1000 cavity roundtrips. The effects on total energy, τ, spectral modulation depth, and φ are recorded. For continuous switching, periodic electronic pump modulation with a 1.2 ms period is applied around a central pump level (e.g., 236.8 mW), yielding deterministic transitions among three dual-soliton regimes at representative pump levels (223.8 mW, 236.8 mW, 249.8 mW). Temporal separation and molecular phase are read out over entire harnessing cycles, and hysteresis during transitions is analyzed via first-order autocorrelation traces. Reversibility and fidelity across multiple harnessing periods are assessed using Pearson correlation coefficients between repeated switching cycles. For quaternary encoding, four phase-defined regimes are prepared at fixed pump powers: 171.7 mW (soliton singlet, SS; logical ‘0’), 213.3 mW (negative-phase, NP, soliton pair; ‘1’), 234.1 mW (stationary-phase, SP, soliton pair; ‘2’), and 254.9 mW (positive-phase, PP, soliton pair; ‘3’). Abrupt pump changes induce assembly (generation of a new singlet from femtosecond fluctuations) or dissociation. The assembly pathway includes raised relaxation oscillations, beating dynamics, transient bound state, and stable molecule; dissociation is faster. Successive real-time spectral interferograms document the switching and validate the encoding sequence.

• Deterministic formation and control of four phase-defined regimes: SS (‘none’ phase), NP (‘negative’ phase), SP (‘stationary’ phase), and PP (‘positive’ phase) by tailoring gain and thereby the energy exchange between adjacent solitons.
• In NP, the trailing pulse is weaker, leading to continuous negative phase accumulation; with increasing gain, intensity differences diminish. In SP, constituents have equal intensities and τ increases with gain. In PP, the trailing pulse becomes stronger, yielding positive phase evolution. Periodic energy exchange causes molecular vibration of τ, with each vibration period corresponding to a 2π change in φ.
• Frame-by-frame pump stepping (212.7→255.6 mW, 1.3 mW steps; 1000 roundtrips per frame) shows saturated total energy, increasing τ, and systematic variation of phase-evolving velocity; spectral modulation depth adjusts with pump.
• Continuous, reversible switching among NP, SP, and PP via periodic pump modulation (period 1.2 ms; transfer efficiency ~52 mW/V). Representative deterministic regimes at 223.8, 236.8, and 249.8 mW are demonstrated. The 3D interaction-space trajectories distinguish regimes by their phase-evolving velocities. Hysteresis is observed during regime transitions but can be reduced by faster electronic modulation.
• Phase-tailored quaternary encoding: mapping SS, NP, SP, PP to logical 0, 1, 2, 3 at pump powers 171.7, 213.3, 234.1, 254.9 mW, respectively. Real-time spectral streams validate continuous switching among all four states, including assembly/dissociation dynamics. The streams exhibit high reproducibility and robustness to timing jitter. Fidelity is quantified via Pearson correlation across repeated cycles (details in supplementary information).
• Dissociation of soliton pairs occurs faster than assembly, which follows a characteristic four-stage transient before stabilizing.

The findings show that controlled gain modulation can program the internal energy exchange within DSMs, enabling deterministic control of relative phase evolution and temporal separation. This directly addresses the challenge of precise, reversible manipulation of intra-molecular dynamics and extends encoding beyond binary pulse counting to phase-defined quaternary states. Real-time spectral interferometry provides an effective monitoring and feedback mechanism, and the observed 3D interaction-space trajectories serve as reliable identifiers of molecular regimes. The demonstrated continuous switching with high fidelity and resistance to timing jitter underscores the feasibility of robust, multi-level all-optical information storage and processing. Hysteresis during transitions highlights underlying gain and cavity dynamics but is manageable with optimized modulation, suggesting practical deployment is attainable with improved control electronics.

The work introduces and experimentally validates programmable phase tailoring of dissipative soliton molecules via electronically controlled gain modulation in a mode-locked fiber laser. Four distinct, phase-defined regimes (SS, NP, SP, PP) are harnessed to realize a quaternary encoding scheme, with deterministic and reversible switching confirmed by real-time spectral interferometry. The approach offers robust, high-fidelity, multi-level all-optical storage potential. Future research may pursue faster modulation to further mitigate hysteresis, closed-loop control using real-time phase feedback, integration with photonic platforms for compactness and stability, exploration of higher-order multilevel encodings, and quantitative evaluation of data rates and error performance under practical noise conditions.

• Hysteresis is observed during regime switching, indicating non-instantaneous and history-dependent dynamics; faster modulation reduces but may not eliminate it.
• Assembly is slower and more complex than dissociation, involving transient stages susceptible to external perturbations.
• Precise operation requires tight control of pump power and polarization, which may limit robustness across different cavities or environmental conditions.
• Quantitative metrics such as bit error rates, maximum achievable symbol rates, and absolute fidelity values are not explicitly reported in the main text.
• Generalizability to other laser platforms or wavelength regimes is not demonstrated within this work.