Soliton molecules in femtosecond fiber lasers: universal binding mechanism and direct electronic control

Sequences of ultrashort pulses form the basis of extremely precise laser applications ranging from femtosecond spectroscopy, to material microprocessing, to biomedical imaging. Dynamic patterns of temporal solitons—termed “soliton molecules”—inside mode-locked cavities provide yet unexplored means for generating reconﬁgurable arrangements of ultrashort pulses. Here, we demonstrate the external control of solitonic bound states in widespread erbium-doped ﬁber lasers via direct electronic modulation of the semiconductor pump source. This straightforward approach allows for switching between discrete soliton doublet states of picosecond separations, employing and relying on laser-intrinsic soliton interactions. We analyze the externally induced dynamics based on real-time switching data acquired by time-stretch dispersive Fourier transform spectroscopy and identify a universal bound-state formation mechanism different from broadly considered models. Owing to the ease of implementation and its intrinsic tunability, our control scheme is readily applicable to various laser platforms enabling, e.g., rapid multipulse measurements and tailored nonlinear light–matterinteractions.


INTRODUCTION
Advanced laser applications in science and technology rely on composing, modulating, and manipulating complex ultrashort pulse patterns in order to provide frequency-swept excitation, lownoise detection, and nonlinear light-matter interactions [1][2][3]. Intriguingly, multipulse operation and complex interactions are being reported for virtually all implementations of passive mode locking. Yet the intrinsic dynamics have been barely experimentally controlled nor practically exploited so far.
A broad range of mode-locking schemes has been demonstrated, featuring implementations via saturable absorber mode locking in bulk crystals, semiconductor absorber mirrors, nonlinear polarization rotation, nonlinear amplifying loop mirrors, and more [18,19]. As a result, Er:fiber lasers are established as a key model system for intracavity pulse dynamics in mode-locked sources [20,21]. In particular, pumping with semiconductor lasers enables fast and direct electronic modulation via miniaturized and lowcost pump sources for telecom amplifiers as employed previously in carrier-envelope stabilization of optical frequency combs [22].
Here, we demonstrate rapid external control of soliton bound states in a saturable-absorber mode-locked Er:fiber laser via direct electronic modulation of the pump power as sketched in Fig. 1(a).

A. External Perturbation
In the experiment, we initiate dual-pulse operation at the laser startup and implement a laser perturbation via current modulation of the semiconductor diode-laser driver. The electronic control signal is generated with an arbitrary waveform generator (AWG) and coupled to the laser driver circuit. The output signal of the 980 nm stabilized semiconductor laser allows for modulation bandwidths exceeding 10 MHz with a fall time of the pump power below 22 ns. The fiber oscillator is implemented in a ring resonator and incorporates passive mode locking via a semiconductor saturable absorber mirror (SAM) [23,24]. The typical mode of operation generates isolated solitary pulses of ∼500 fs duration at a repetition rate of 40 MHz.
We characterized the transfer dynamics of the intracavity pulse energy via sinusoidal cw-pump modulation, yielding modulation transfer up to a bandwidth of approximately 8 kHz. The transfer function corresponds to damped gain relaxation oscillations governed by the millisecond upper-state lifetime and high saturation energy. In the temporal domain, the intracavity energy displays a fall time of 29 µs upon transient reduction of the pump power via rectangular functions of varying durations as shown in Figs. 2(a) and 2(b), compatible with the gain relaxation oscillations around 10 kHz. We achieve a rapid perturbation of single-pulse operation by applying a sufficient external pump power reduction in order to initially induce and then to switch multipulse states. We map the transition between stable single-pulse and multipulse operation as a function of perturbation strengths, given by the cumulative pump energy reduction via drop depth P and drop duration τ . This transition map, displayed in Fig. 2(c), allows for the selection of control parameter values. As apparent in this representation, we achieve effective perturbation at a constant cumulative pump energy reduction given by the product P · τ and marked by the black line. The cumulative reduction corresponds to around 3% of the laser saturation energy.

B. Switching and Real-Time Spectroscopy
Starting at a stable bound state of two solitons, we rapidly switch between two dominant bound states (I and II) via injection of the control signal, i.e., via transient reduction of pump power. We track the resultant multisoliton dynamics via real-time detection with the TS-DFT spectral interferometry and a fast photodetector. The real-time oscilloscope features analog bandwidths of 16 GHz at 100 GSa sampling rate, and the group velocity dispersion (GVD) is set to 990 ps/nm. We trigger every real-time acquisition event via the corresponding external electrical modulation waveform.
The round-trip-resolved spectral interferograms and corresponding field autocorrelations (Fourier amplitudes of the spectra) synchronous to the external control signal of single switching events are shown in Figs. 3(a) and 3(b). Switching occurs from a stable bound state at 55.3 ps (state I) to a shorter binding separation of 14.5 ps (state II) via transient dissociation of the initial bound state. We assume a common central wavelength of 1567 nm for both pulses, in agreement with the extracted separations from scanning interferometric autocorrelations (see Supplement 1). The dissociation of the bound states requires sufficient reduction of intracavity energy, allowing for the transient induction of single-pulse operation. Subsequently, following the recovery of the pump energy, the single soliton experiences excess gain apparent as rapid spectral broadening and a power overshoot in the time-resolved spectra. At this stage, a new second soliton forms via  split-up from the first seeding pulse and, effectively, regenerates two closely spaced unbound pulses out of equilibrium. The corresponding autocorrelations in Fig. 3(b) resolve the trajectories of their intersoliton motion. The speed of relative motion is linked to the effective gain gradient experienced by the two succeeding pulses [25]. Prior to full recovery, the second soliton experiences a lower gain gradient as compared to the leading pulse, and thus the relative motion is fast. Upon full recovery, the second pulse saturates the gain comparably to the leading soliton, and the relative motion rapidly decreases. In case of no further interactions, multiple solitons typically stabilize in an equidistant harmonically mode-locked state.
As the soliton splitting and reformation of the double-pulse state arise from fluctuations, each individual event exhibits varying spectro-temporal dynamics that translate to different relative soliton motion in the early phase of the process.
Hence, soliton binding and stabilization occur either at the close separation τ II or later at the longer separation τ I , depending on the exact equilibration of the soliton intensities. This enables reverse switching between the short τ II and the long separation τ I as shown in the switching event of Fig. 3(c). Additionally, the regeneration of an initial state may be obtained as the alternative trajectory as shown in the example of Fig. 3(d). This dataset also displays a transient kink, revealing an instable binding separation at around 10 ps (extended view in the inset). Based on the employed strong perturbation via transient single-pulse operation, the reinitiation of the bound states is determined by subtle fluctuations. A tailored perturbation may enable the deterministic switching between bound states in analogy to results with Ti:sapphire solid-state lasers in a different regime [17].
We note that the relaxation to the stable separation in Fig. 3(d) was not entirely captured within this real-time dataset due to limited acquisition length, but the final state was given by the long separation of state I.

C. Universal Binding Mechanism
We now discuss the origin of soliton interactions and bound-state formation with general implications to a broad set of mode-locked laser sources. Attraction and repulsion of ultrashort temporal solitons in optical fiber may induce bound states of solitons or "soliton molecules," discovered as a result of direct interaction between closely spaced pulses [26] or upon long-range interaction, i.e., via dispersive waves [27]. Pulse evolution in mode-locked lasers-with additional gain, loss, and saturable absorption-is often described in the context of modified nonlinear Schrödinger equations or complex Ginzburg-Landau equations, and various bound states were found theoretically [28][29][30][31][32][33], yet the stability of these might vary significantly [34]. Experimentally, various resembling dynamics were observed in different systems, particularly reported within a large body of fiber laser studies (examples in Refs. [15,[35][36][37][38][39]). Despite highly reproducible experimental observations and remarkable insensitivity to the actual state of laser operation [36], the key experimental observable-the actual bound-state separation-is commonly not explained or quantitatively predicted by theoretical models. In contrast, the bound states mediated by reflected transverse acoustic waves in fibers present one rather exceptional mechanism in which theories are able to precisely account for experimental bound-state separations [16,[40][41][42][43]. Still, the interaction is subtle, and its importance often arises in long-range propagation or in custom-designed fibers [16,42].
Here, we identify a universal mechanism of temporal soliton bound-state formation akin to additive-pulse mode locking: A fraction of the leading soliton reflects at an optical interface of the cavity and counterpropagates to an adjacent interface. Following a second reflection, an "echo" is formed, which copropagates with the leading soliton and reaches the SAM with a time delay given by the twofold optical path length between both surfaces as sketched in Fig. 4 (green). Eventually, a second, initially unbound trailing soliton temporally superposes with the echo at the saturable absorber. As a result, the second soliton experiences slightly reduced nonlinear absorption, breaking the translational symmetry and stabilizing the soliton bound state at the distinguished separation. The mechanism can be considered as "long-range" binding without direct overlap of the pulses, and bound-state separations can range from few 100 fs up to the round-trip duration of several 10 ns. The mechanism is not limited to mode-locked lasers with saturable absorbers but applies to any ultrafast amplitude modulation.
In our experiments, the observed dominant bound-state separations are τ I = 55.3 ps and τ II = 14.5 ps. An analysis of intracavity feedback from the employed resonator components (detailed in Supplement 1) reveals that the shorter separation matches within the experimental accuracy to the round-trip propagation delay between the free-space path length between the SAM and the adjacent focusing lens surface of 15.9 ps or 2 · 2.38 mm/c with the vacuum speed of light c . The finite reflectivity of the antireflectivecoated lens surface is specified to below 0.5%. The round-trip duration and path length between the SAM and the opposing surface of the aspherical lens coincides with this separation of 54.7 ps or (2 · 2.38 mm + 2 · 1.6 · 3.64 mm)/c, with the group index of the lens material of 1.6 (see the sketch in Supplement 1). We performed sensitive linear autocorrelation measurements in single-pulse operation using an external scanning Michelson interferometer. The separations between subtle echo features match the separations obtained during bound-state operation. In addition, we detect a subtle delayed echo at around 10 ps that coincides with the characteristic transient kink or position of preferred restart of the second pulse in the real-time trajectories [see Figs. 3(b) and 3(d)]. This feature matches to backreflections inside the fiber-optical circulator between the collimator and the polarizer of ∼1.5 mm separation (see Supplement 1).
Whereas mode-locking dynamics in regular single-pulse operation are typically not impaired by spurious impedance mismatch due to the nonlinear "winner-takes-all" dynamics, here, the impact of such reflections onto multipulse interactions is of essential importance: a second soliton sensitively probes cavity perturbations from delayed temporal responses of intracavity components upon its relative motion. The adequate consideration of surface reflections appears yet to be largely missing in various discussions on soliton bound-state generation in a broad set of publications investigating soliton interactions in mode-locked lasers. Widely employed polarization optics and nonreciprocal elements, including fiber-optic isolators or circulators, typically include eight or more interfaces between different materials of millimeter (mm) lengths. Moreover, we note that our observations indicate no significant contributions from higher-order dispersion or nonlinear effects, dispersive wave interactions, nor acoustic feedback as primary sources of soliton binding. Consequently, previous lumped models that do not include the actual multicomponent laser structure are not capable of analyzing and predicting the bound-state separations frequently encountered in actual laser sources.
We emphasize that our proposed soliton coupling mechanism is based on linear reflections and thus features increased stability compared to potential interactions arising from higher-order effects [36]-whereas the latter sensitively depend on specific laser operation parameters, i.e., on pump power, dispersion, and peak intensities. Hence, our observations represent universal dynamics that are relevant for various mode-locked sources featuring multipulse interaction due to intracavity feedback.

D. Tunable Bound States
Finally, we directly apply the presented mechanism for the generation of variably tunable bound states. We insert an antireflective-coated thin plate (90 µm thickness, 0.1% reflectivity at 1550 nm) in the free-space section of the resonator prior to the focusing lens of the SAM. Using a mechanical delay stage, we are able to shift the plate along the propagation direction, effectively increasing the distance to the SAM. Upon initial perturbation or restart, we observe the formation of a novel bound state, the temporal separation of which is tunable via mechanically shifting the plate as presented in Fig. 4(b). We extracted the tunable separations from spectral interferograms as shown in Supplement 1. The slope is given by the free-space propagation delay for a roundtrip as 2/c . At this stage, we observe that the laser operation may be disturbed upon coarse mechanical intracavity motion, and the solitons may internally regroup or switch within the set of dominant bound-state separations.

CONCLUSIONS
In conclusion, we have experimentally demonstrated the rapid control of ultrashort soliton bound states via direct electronic modulation of the semiconductor pump-laser power, enabling the switching between two stable soliton-doublet states. Employing real-time spectroscopy, we map the switching trajectories and identify finite reflective feedback from intracavity components as the origin of bound-state separations, generating stable and highly reproducible temporal soliton separations. Moreover, we demonstrate the tunable generation of stable bound states via translation of an additional reflective intracavity plate. Given the ease of implementation and the flexible tunability of ultrashort separations, this versatile scheme appears highly applicable for future multipulse laser applications in high-speed measurements and nonlinear light-matter interactions.