Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions

Self-assembly of fundamental elements through weak, long-range interactions plays a central role in both supramolecular DNA assembly and bottom-up synthesis of nanostructures. Optical solitons, analogous in many ways to particles, arise from the balance between nonlinearity and dispersion and have been studied in numerous optical systems. Although both short- and long-range interactions between optical solitons have attracted extensive interest for decades, stable soliton supramolecules, with multiple aspects of complexity and flexibility, have thus far escaped experimental observation due to the absence of techniques for enhancing and controlling the long-range inter-soliton forces. Here we report that long-range soliton interactions originating from optoacoustic effects and dispersive-wave radiations can be precisely tailored in a fibre laser cavity, enabling self-assembly of large numbers of optical solitons into highly-ordered supramolecular structures. We demonstrate several features of such optical structures, highlighting their potential applications in optical information storage and ultrafast laser-field manipulation.


Results
Illustration of concept. The mode-locked fibre laser loop that we build to study soliton supramolecules is sketched in Fig. 1a (see details plotted in Supplementary Fig. 1 and described in Supplementary Note 1). A 2-m-long solid-core silica photonic crystal fibre (PCF) with a GHz-rate acoustic core resonance 37 is inserted into the laser cavity. Upon increasing the pump power of the Erdoped fibre amplifier (EDFA), a variety of stable supramolecular structures composed of a large population of optical solitons can be generated in the laser cavity, all the solitons involved in such structures being globally ordered by optomechanical interactions 5,33,34 . A long-lived acoustic wave in the PCF-core is driven coherently by the soliton sequence, and acts back on the pulses, linking them together by modulating their carrier frequencies, and forming a temporal optomechanical lattice with a period equal to a cycle of acoustic vibration 5 . This global lattice divides the laser cavity into many time-slots of identical length, each of which can accommodate multiple solitons (see Fig. 1a).
Long-range binding of multiple solitons in each optomechanical unit originates from the balance between attractive and repulsive inter-soliton forces. As the solitons ride on the acoustic wave whose wavefront (phase) velocity equals the group velocity of the soliton sequence 5 , the index modulation caused by the acoustic wave leads to a shift in the soliton carrier frequency as the soliton propagates (see Fig. 1b). The magnitude of this frequency shift is determined by the slope of the underlying index modulation 38 . Since the two solitons within one acoustic period are located at different positions (see Fig. 1b), their frequencies shift at rates that differ slightly from each other. The divergence in soliton frequencies, acting in concert with the group-velocity dispersion of the optical fibre, leads to an effective force of attraction. As illustrated in Fig. 1c, the first higher-frequency dispersive wave sideband 39,40 , shed from the first soliton, propagates faster than the soliton and eventually reaches the second soliton, perturbing it through cross-phase modulation 41 . Such dispersive wave perturbations effectively create a repulsive force between the two solitons 27,28 , balancing the force of attraction due to the optoacoustic effect (see theoretical modelling in Supplementary Note 2 with Supplementary Figs. 2 and 3). In principle, more than two solitons can be stably bound within one unit through the cascaded balance of long-range forces (see Fig. 1a, e).
The balance of these two long-range forces results in stable soliton spacing, effectively creating inter-solitonic springs that trap the solitons at each equilibrium position. As a result, noiseinduced fluctuations in soliton spacing (timing jitter) occur that are analogous to the thermal motion of a particle trapped in a harmonic potential (Fig. 1g). These can be modelled using a Langevin equation 42 (see Supplementary Note 3). When pulses are trapped in a harmonic potential, the timing jitter does not grow with time, even though the heat bath (noise source) continuously disturbs the system 42 Enabling techniques. Stable soliton supramolecules can be generated experimentally only by carefully designing the fibre laser cavity so that the evolution of long-range forces between hundreds of solitons is precisely controlled. Optoacoustic effects in conventional single-mode fibre are ultra-weak 28,37 , generally leading to harmonically mode-locked pulse trains with high noise levels 28,30,31 . In our laser cavity, in contrast, tight confinement of both optical and acoustic waves in the 1.95-μm-diameter PCF core leads to the enhancement of acoustic-wave-mediated intersoliton forces by more than two orders of magnitude 5,32 . The excellent robustness of the resulting temporal optomechanical lattice makes possible manipulation of the fine structure within each time-slot of the lattice 5 . In practice, tuning the acoustic-wave amplitude in the PCF core can be realised by detuning of the pulse repetition rate from the acoustic resonant frequency 32,33 , which itself can be adjusted by fabricating PCFs with different core diameters 32,37 .
On the other hand, dispersive-wave generation in soliton fibre lasers, leading to uncontrolled disturbance to the pulse train 6,26-28 , is widely regarded as undesirable, rather than as a source of exploitable inter-solitonic forces. The idea reported here is to stably balance long-range dispersive-wave and optoacoustic effects. In our laser cavity the strength of the dispersive waves shed by individual solitons, which determines the strength of repulsive inter-soliton forces (see Supplementary Note 2), must be tailored so as to counterbalance the attractive inter-soliton forces due to the strong optoacoustic effects. We reveal in the experiments that careful management of both cavity dispersion 44 and cavity loss are of great importance in determining the strengths and directions of the inter-soliton forces due to dispersive-wave radiation (see Methods).
Supramolecules with single solitons as building blocks. A typical soliton supramolecule with a chain of units, each containing 0, 1, 2, or 3 trapped solitons, is recorded using a fast detector and an oscilloscope. The resulting time-domain trace is shown in Fig. 2a, where the underlying grid is globally locked to the 1.887 GHz acoustic resonance in the PCF core (period 532 ps). The duration of individual solitons is measured to be 650 fs. This self-assembled solitonic structure, once formed, is robust, the pulse spacings in each unit being 80 ps between the first and second solitons and 70 ps between the second and third. Monitoring the soliton supramolecule over 1000 min (see Fig. 2b) reveals no measurable degradation in signal-to-noise ratio. This time interval corresponds to stable propagation over 12 billion kilometers (~84 astronomical units) in the freely-running fibre laser loop (see Supplementary Note 4 and Supplementary Figs. 6 for experimental details). The estimated pulse timing jitter is always below 5 ps, measured using the oscilloscope in persistencemode (see Fig. 2b). In order to monitor the carrier-wave phase of the solitons, we use a narrow-linewidth local oscillator in the form of a single-frequency fibre laser to heterodyne with the supramolecular soliton sequence 33 . The results reveal that the carrier-wave phases of assembled solitons are uncorrelated (see Supplementary Note 5 and Supplementary Figs. 7 for experimental details); in this respect, soliton supramolecules self-assembled through long-range forces differ from soliton molecules strongly bound by short-range forces 22 , when the carrier-waves are phase coherent.
In the experiments, we discover that the system tends to evolve into a supramolecular structure with roughly even distributions of optical solitons, although, since the structure directly emerges from noise by self-organization, the exact number of solitons differs randomly from time-slot to time-slot. Through careful adjusting both the laser pump power and intra-cavity polarization controllers, we can partially control the fine structure of the selfassembled soliton supramolecule, reproducibly generating structures in which every time-slot contained the same number of soliton units (single 5,33,34 , double, or triple) as shown in Fig. 2c, d. We can also generate a supramolecular soliton stream containing both triple and quadruple soliton units (see Fig. 2e). More Although so far we have not demonstrated independent control of individual solitons in the supramolecular structure, fast encoding of the supramolecular patterns should be possible by launching a timed sequence of writing pulses into the laser cavity or modulating the pump laser power 5,9,11 .
Elementary diversity. The elementary diversity of the supramolecular structures can be greatly increased by incorporating additional fundamental building blocks. For example, both single solitons and phase-locked soliton pairs [15][16][17] can be incorporated as building blocks in the supramolecular structure, as seen in the time-domain trace in Fig. 3a-c. This type of supramolecular structure is held together by both long-range and short-range inter-soliton interactions, in a manner reminiscent of atoms in biochemical supramolecules 36 , which self-assemble through a combination of short-range and long-range forces.
In the recorded time-domain sequence in Fig. 3a-c, the bandwidth (33 GHz) of the oscilloscope is not sufficient to resolve individual solitons in a pair, with the result that the soliton pairs appear as pulses with twice the amplitude of a single soliton. To overcome this limitation, we use a time-stretched dispersive Fourier transform (TS-DFT) to record a real-time interferogram of the soliton sequence 22 . In time-slots with higher amplitude pulses, strong fringes appear, confirming that they contain phase-locked soliton pairs with a short separation. Since many of the building blocks of this soliton supramolecule are soliton pairs with identical spacings and phase differences, second-harmonic autocorrelation can be used to directly measure the temporal spacing of the soliton pairs, which turned out to be Δt~4.5 ps (see Fig. 3d). In addition, using an optical spectral analyser we observe strong spectral interference with a period Δλ = 1.85 nm~λ 2 /(cΔt) (see Fig. 3e). Structural details of such supramolecules are shown in Supplementary Figs. 12 and 13 and described in Supplementary Note 8.
Experimentally we find that more types of elementary building blocks can exist in the assembly, which dramatically increases the complexity of the soliton supramolecule. For example, we observe that phase-drifting soliton pairs 45 (see Fig. 3f, g) and phasedlocked soliton-triplets (see Fig. 3h) can be included in supramolecules. (More details on these different soliton molecules in the supramolecular structures are given in Supplementary Figs. 14-16). Soliton molecules with different inner spacings and phase relations can coexist in the same soliton supramolecule, further increasing the structural complexity (see examples in Supplementary Figs. 17 and 18). Moreover, the spacing between different building blocks in the supramolecule can differ due to varied long-range forces. As shown in Fig. 3i, several characteristic internal spacings (75 ps, 42 ps and 50 ps) are observed in a soliton supramolecule composed of both single solitons and phase-locked soliton pairs, corresponding, respectively, to different interactions of pair-to-one, pair-to-pair, and one-toone interactions (see Supplementary Fig. 19 for more examples). The diversity of building blocks in the supramolecular structures greatly enriches the range of possible encoding strategies when such structures are used for carrying digital information. Instead of exclusively varying the soliton number in each time-slot, different soliton bound states (long-range or short-range) can be used in the encoding format. These bound states can easily be discriminated using fast detectors, and since the combined optical energy does not vary significantly from time-slot to time-slot, the optomechanical binding forces remain constant, leading to a stable optomechanical lattice.
Structural flexibility and reversibility. The weak nature of the long-range interactions renders the soliton supramolecules highly reconfigurable. For example, their inner structure can change in response to variations in the long-range, inter-soliton forces (see Fig. 4a). We find that the spacing between the long-range bound solitons in one unit of the optomechanical lattice can be continuously tuned over a large range while maintaining the overall supramolecular structure. By placing a tunable attenuator in the laser cavity (Fig. 1a) we are able to adjust the cavity gain and loss, permitting continuous tuning of the dispersive wave intensity. In particular, we are able to double the intensity (Fig. 4b), dramatically reinforcing the repulsive force between the solitons. As shown in Fig. 4c, this leads to an increase in the internal soliton spacing from 40 ps and 116 ps in a supramolecule with two solitons in every time-slot (see experimental details in Supplementary Note 9). Supramolecules with three solitons in every time-slot can also be tuned in the same way (See Supplementary  Fig. 20). We can also cycle the soliton spacing back and forth by adjusting the cavity length so as to vary the amplitude of the acoustic wave and thus the attractive force (see Supplementary  Fig. 21). The supramolecular pattern can switched to a new state by abruptly perturbing the system. Both addition and removal of individual solitons (see Fig. 5) are possible. For example (Fig. 5a), an abrupt increase of~15% in the EDFA pump power over~1 µs resulted in the generation of additional solitons (see experiment details described in Supplementary Note 10). The increased pump power leads to higher soliton intensities and thus lower group velocities, as seen in a sudden bending of the pulse trajectories (Fig. 5a), corresponding to variations in the round-trip time. We also find that, as a result of the increased background noise at higher pump power, noisy spikes sometimes turned into stable solitons after a transition period 46 (Fig. 5a). These newly created solitons are then incorporated into the supramolecule via the  Fig. 5b, corresponding to the region marked by the white arrow in Fig. 5a). In the experiments we can also remove solitons from a supramolecule (Fig. 5c) by decreasing the EDFA pump power by~10% over~1 µs, causing many double-soliton units to degrade into single-soliton units (see Fig. 5d for a zoom-in to a typical process of soliton fadeaway). Notably, the supramolecular structures that encounter abrupt perturbations in pump power, after experiencing a transient process of self-adjustment (see detailed experimental recordings in Supplementary Figs. 22-25), can quickly settle down to a stable structure, indicating a possible means of fine control (information encoding) of the supramolecular pattern.

Discussion
Although supramolecules are well-known biological and biochemical structures, they have not previously been observed in an optical setting. Weak long-range inter-soliton interactions make it possible to create solitonic supramolecules in fibre laser cavities passively mode-locked by optoacoustic effects. The temporal spacing between solitons in the supramolecule can be adjusted from tens to hundreds of picoseconds-easily resolvable with fast electronics. In contrast to conventional soliton molecules, which are localized structures composed of a small number of phasecorrelated solitons 13,[15][16][17] , supramolecules are large-scale structures, composed of a large number of solitons and tightly-bound soliton molecules, distributed over the entire fibre laser cavity with a built-in hierarchy. Biochemical and biological supramolecules 35,36 have many features in common with these newly-discovered optical counterparts, as well as important differences. On one hand, biological and biochemical supramolecular structures are typically complex three-dimensional structures that the optics cannot yet replicate. On the other hand, the underlying physics is similar. Elementary building blocks consisting of atoms and tightly bound molecules self-assemble to form supramolecules through interactions mediated by weak, long-range forces. The mode-locked fibre laser reported here, a fast one-dimensional optical platform with a single degree of freedom 13 , may be useful for emulating complex process in many-body biochemical and biological systems.
Within the taxonomy of nonlinear optics, soliton supramolecules are structurally protected solutions of dissipative nonlinear optical systems 22,47 . These systems include mode-locked fibre lasers 13,48 , optical bit-storage fibre loops 4-6 or optical fibre telecommunication systems with non-trivial Kerr nonlinearities 1,2,49 . By periodically  Fig. 4 Continuous tuning of the long-range forces between optical solitons. a Tailoring the pulse spacing within each unit of the supramolecular structure by varying the long-range, inter-soliton forces. b Showing how the spectral intensity in the first higher-frequency sideband decreases when the cavity loss is increased; the inset shows that the overall soliton bandwidth is almost independent of cavity loss. c The dependence of the soliton spacing on the side-band intensity. The experimental data-points, plotted as black squares (standard deviations are shown as error bars), agree well with the red theoretical curve (see Supplementary Note 2 and Note 9). d Four typical persistence-mode traces of the soliton supramolecule during the spacing tuning process, corresponding to the data-points (i), (ii), (iii), and (iv) in c.
introducing control elements 50 into supramolecular soliton systems with non-instantaneous nonlinearities, it may be possible to manipulate the supramolecules and form flexible, highly-ordered structures that are immune to perturbations. The soliton supramolecules reported here can be constructed from many different multi-soliton bound states, making it possible to encode information in non-binary formats using both long-range and short-range interactions. The ability to tailor long-range interactions between solitons in an optomechanical lattice may permit synthesis and control of highly-ordered, macroscopic optical structures, providing a promising platform for studying complex soliton molecules (e.g. their formation 22,51 , dissociation 13,52 , and vibrational modes 45,53 ) and optically simulating many-body systems with particle-like properties. The ability to synthesize highly ordered multi-soliton sequences may also be useful for improving laser micromachining 54 . The enriched dynamics of controllable long-range soliton interactions, and their analogies with the behaviour of chemical supramolecules (e.g. self-healing and self-replication), are also interesting research topics. Introducing spatiotemporal nonlinearities using multimode fibres 48 may in the future permit the formation of three-dimensional optical supramolecules, further enriching the physics and applications of the current system.

Methods
Mode-locked fibre laser. The experimental platform is an optoacoustically modelocked soliton fibre laser, with a pulse repetition rate that is locked to an acoustic resonance in the core of a photonic crystal fibre (PCF) inserted in the laser cavity. The cavity has net anomalous dispersion, ensuring that the fibre laser operates in the soliton regime. To achieve harmonic mode-locking and generate soliton supramolecules in the laser cavity, it is important to set a proper working point. First, the delay line in the cavity is adjusted so that a specific harmonic order of the cavity FSR falls within the optoacoustic gain spectrum of the PCF. Second, all the three fibre polarization controllers (FPCs) in the laser cavity are adjusted so that nonlinear polarization rotation (NPR) can induce an intensity-dependent cavity loss. Finally, the laser pump power and the intra-cavity attenuator are adjusted to enable self-assembly of a supramolecular structure. Once the desired soliton supramolecule is obtained, no further adjustment or any other stabilization technique is needed for long-term preservation of the supramolecular soliton pattern.
Diagnostic set-up. A time-domain trace of the laser output is acquired using a 30-GHz photodetector and a 33-GHz oscilloscope (OSC). The response time is 20 ps, limiting the minimum resolvable temporal features in all the plots recorded using the OSC. The timing jitter of the OSC in sampling is~2 ps, which sets the measurement error in obtaining the fine structure of the supramolecule. The duration of individual solitons is measured using a second-harmonic autocorrelator with a time resolution of 20 fs. The optical spectrum at the laser output is measured using an optical spectrum analyser with a resolution of 0.01 nm. We also performed measurements using time-stretched dispersive Fourier transformation (TS-DFT), using several-km-long SMF-28 fibre to characterize the soliton molecules in the supramolecules.
Tailoring of long-range interactions. The long-range interactions involved in soliton supramolecules can be tailored by tuning either the intensity of dispersive wave and thus the repulsive force between solitons, or the amplitude of the acoustic vibration in the PCF core, corresponding to the force of attraction between solitons. In order to tune the dispersive-wave intensity as shown in Fig. 4, we adjusted the intra-cavity tunable attenuator (TA), leading to higher cavity gain and therefore a stronger gain filtering effect, which significantly suppressed the Kelly-sideband intensity. During this dispersive-wave tuning process, the soliton spectral bandwidth remained almost unchanged due to strong gain saturation in the EDFA. See Supplementary Note 7 for more effects of dispersive-wave tailoring. Adding and removing solitons. Addition and removal of individual solitons is achieved by strongly perturbing the laser pump power. In the experiments, the output power of the pump laser diode is controlled using an electric pulse generator. The modulation responsivity of the pump laser diode is 106 mW/V, so that a 1 V variation in the driving electrical signal leads to a 106 mW variation in the laser output power. Before adding or removing solitons, it is important first to adjust the working point of the laser so that the supramolecular structure is stable.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability
The code we used in this paper is available from the corresponding author upon reasonable request.